View
2
Download
0
Category
Preview:
Citation preview
The Pennsylvania State University
The Graduate School
College of Engineering
A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS
IN BOUNDARY LUBRICATION
WITH LUBRICANTSURFACE PHYSICOCHEMISTRY
A Thesis in
Mechanical Engineering
by
Huan Zhang
copy 2004 Huan Zhang
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
May 2005
The thesis of Huan Zhang was reviewed and approved by the following
Liming Chang Professor of Mechanical Engineering Thesis Advisor Chair of Committee
Marc Carpino Professor of Mechanical Engineering
Seong H Kim Assistant Professor of Chemical Engineering Richard C Benson Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering
Signatures are on file in the Graduate School
iii
ABSTRACT
The boundary-lubricated surface contact is truly an interdisciplinary process
involving deformation heat transfer physicochemical interaction and random-process
probability The objective of this thesis is to develop a surface contact model as a
theoretical platform upon which to carry out the boundary lubrication research with a
balanced consideration of all the four key aspects of the contact process The modeling
consists of three successive steps ndash (1) elastoplastic finite element analysis of frictional
asperity contacts (2) modeling of contact systems with friction and (3) modeling of a
boundary lubrication process
Finite element analysis of frictional asperity contacts ndash A finite element model is
developed and systematic numerical analyses carried out to study the effects of friction
on the deformation behavior of individual asperity contacts The study reveals some
insights into the modes of asperity deformation and asperity contact variables as
functions of friction in the contact The results provide guidance to analytical modeling of
frictional asperity contacts and lay a foundation for subsequent work on system contact
modeling
Modeling of contact systems with friction ndash Analytical equations are developed
relating asperity-contact variables to friction using contact-mechanics theories in
conjunction with the finite element results A system-level model is then derived from the
statistical integration of the asperity-level equations The model is a significant
advancement of the Greenwood-Williamson types of system models by incorporating
iv
contact friction It also serves as the platform in the final step of model development for
the boundary lubrication problem
Modeling of a boundary lubrication process ndash On the basis of the above
mechanical modeling an asperity-based model is developed for the boundary-lubricated
contact by incorporating other key aspects involved in the process Four variables are
used to describe an asperity contact under boundary lubrication conditions including
micro-contact area friction force load carrying capacity and flash temperature In
addition three probability variables are used to define the interfacial state of an asperity
junction that may be covered by various types of boundary films Governing equations
for the seven key asperity-level variables are derived based on first-principle
considerations of asperity deformation frictional heating and formationremoval of
boundary lubricating films These coupled asperity-level equations some of which are
nonlinear are solved iteratively and the solution is then statistically integrated to
formulate the contact model for boundary lubrication systems
The results obtained from the model suggest that it may provide a framework for
future investigation of the boundary lubrication process by integrating research advances
in contact mechanics tribochemistry and other related fields
v
TABLE OF CONTENTS
List of Figures vii
List of Tables ix
Nomenclaturex
Acknowledgementsxii
Chapter 1 Introduction 1
11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4
121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18
13 Research Objective Approach and Outline 18
Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22
21 Introduction 22 22 The Model Problem24 23 Results and Analysis27
231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33
24 Summary37
Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48
31 Introduction 48 32 Modeling51
321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65
33 Result Analysis68
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
The thesis of Huan Zhang was reviewed and approved by the following
Liming Chang Professor of Mechanical Engineering Thesis Advisor Chair of Committee
Marc Carpino Professor of Mechanical Engineering
Seong H Kim Assistant Professor of Chemical Engineering Richard C Benson Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering
Signatures are on file in the Graduate School
iii
ABSTRACT
The boundary-lubricated surface contact is truly an interdisciplinary process
involving deformation heat transfer physicochemical interaction and random-process
probability The objective of this thesis is to develop a surface contact model as a
theoretical platform upon which to carry out the boundary lubrication research with a
balanced consideration of all the four key aspects of the contact process The modeling
consists of three successive steps ndash (1) elastoplastic finite element analysis of frictional
asperity contacts (2) modeling of contact systems with friction and (3) modeling of a
boundary lubrication process
Finite element analysis of frictional asperity contacts ndash A finite element model is
developed and systematic numerical analyses carried out to study the effects of friction
on the deformation behavior of individual asperity contacts The study reveals some
insights into the modes of asperity deformation and asperity contact variables as
functions of friction in the contact The results provide guidance to analytical modeling of
frictional asperity contacts and lay a foundation for subsequent work on system contact
modeling
Modeling of contact systems with friction ndash Analytical equations are developed
relating asperity-contact variables to friction using contact-mechanics theories in
conjunction with the finite element results A system-level model is then derived from the
statistical integration of the asperity-level equations The model is a significant
advancement of the Greenwood-Williamson types of system models by incorporating
iv
contact friction It also serves as the platform in the final step of model development for
the boundary lubrication problem
Modeling of a boundary lubrication process ndash On the basis of the above
mechanical modeling an asperity-based model is developed for the boundary-lubricated
contact by incorporating other key aspects involved in the process Four variables are
used to describe an asperity contact under boundary lubrication conditions including
micro-contact area friction force load carrying capacity and flash temperature In
addition three probability variables are used to define the interfacial state of an asperity
junction that may be covered by various types of boundary films Governing equations
for the seven key asperity-level variables are derived based on first-principle
considerations of asperity deformation frictional heating and formationremoval of
boundary lubricating films These coupled asperity-level equations some of which are
nonlinear are solved iteratively and the solution is then statistically integrated to
formulate the contact model for boundary lubrication systems
The results obtained from the model suggest that it may provide a framework for
future investigation of the boundary lubrication process by integrating research advances
in contact mechanics tribochemistry and other related fields
v
TABLE OF CONTENTS
List of Figures vii
List of Tables ix
Nomenclaturex
Acknowledgementsxii
Chapter 1 Introduction 1
11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4
121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18
13 Research Objective Approach and Outline 18
Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22
21 Introduction 22 22 The Model Problem24 23 Results and Analysis27
231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33
24 Summary37
Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48
31 Introduction 48 32 Modeling51
321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65
33 Result Analysis68
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
iii
ABSTRACT
The boundary-lubricated surface contact is truly an interdisciplinary process
involving deformation heat transfer physicochemical interaction and random-process
probability The objective of this thesis is to develop a surface contact model as a
theoretical platform upon which to carry out the boundary lubrication research with a
balanced consideration of all the four key aspects of the contact process The modeling
consists of three successive steps ndash (1) elastoplastic finite element analysis of frictional
asperity contacts (2) modeling of contact systems with friction and (3) modeling of a
boundary lubrication process
Finite element analysis of frictional asperity contacts ndash A finite element model is
developed and systematic numerical analyses carried out to study the effects of friction
on the deformation behavior of individual asperity contacts The study reveals some
insights into the modes of asperity deformation and asperity contact variables as
functions of friction in the contact The results provide guidance to analytical modeling of
frictional asperity contacts and lay a foundation for subsequent work on system contact
modeling
Modeling of contact systems with friction ndash Analytical equations are developed
relating asperity-contact variables to friction using contact-mechanics theories in
conjunction with the finite element results A system-level model is then derived from the
statistical integration of the asperity-level equations The model is a significant
advancement of the Greenwood-Williamson types of system models by incorporating
iv
contact friction It also serves as the platform in the final step of model development for
the boundary lubrication problem
Modeling of a boundary lubrication process ndash On the basis of the above
mechanical modeling an asperity-based model is developed for the boundary-lubricated
contact by incorporating other key aspects involved in the process Four variables are
used to describe an asperity contact under boundary lubrication conditions including
micro-contact area friction force load carrying capacity and flash temperature In
addition three probability variables are used to define the interfacial state of an asperity
junction that may be covered by various types of boundary films Governing equations
for the seven key asperity-level variables are derived based on first-principle
considerations of asperity deformation frictional heating and formationremoval of
boundary lubricating films These coupled asperity-level equations some of which are
nonlinear are solved iteratively and the solution is then statistically integrated to
formulate the contact model for boundary lubrication systems
The results obtained from the model suggest that it may provide a framework for
future investigation of the boundary lubrication process by integrating research advances
in contact mechanics tribochemistry and other related fields
v
TABLE OF CONTENTS
List of Figures vii
List of Tables ix
Nomenclaturex
Acknowledgementsxii
Chapter 1 Introduction 1
11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4
121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18
13 Research Objective Approach and Outline 18
Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22
21 Introduction 22 22 The Model Problem24 23 Results and Analysis27
231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33
24 Summary37
Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48
31 Introduction 48 32 Modeling51
321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65
33 Result Analysis68
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
iv
contact friction It also serves as the platform in the final step of model development for
the boundary lubrication problem
Modeling of a boundary lubrication process ndash On the basis of the above
mechanical modeling an asperity-based model is developed for the boundary-lubricated
contact by incorporating other key aspects involved in the process Four variables are
used to describe an asperity contact under boundary lubrication conditions including
micro-contact area friction force load carrying capacity and flash temperature In
addition three probability variables are used to define the interfacial state of an asperity
junction that may be covered by various types of boundary films Governing equations
for the seven key asperity-level variables are derived based on first-principle
considerations of asperity deformation frictional heating and formationremoval of
boundary lubricating films These coupled asperity-level equations some of which are
nonlinear are solved iteratively and the solution is then statistically integrated to
formulate the contact model for boundary lubrication systems
The results obtained from the model suggest that it may provide a framework for
future investigation of the boundary lubrication process by integrating research advances
in contact mechanics tribochemistry and other related fields
v
TABLE OF CONTENTS
List of Figures vii
List of Tables ix
Nomenclaturex
Acknowledgementsxii
Chapter 1 Introduction 1
11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4
121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18
13 Research Objective Approach and Outline 18
Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22
21 Introduction 22 22 The Model Problem24 23 Results and Analysis27
231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33
24 Summary37
Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48
31 Introduction 48 32 Modeling51
321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65
33 Result Analysis68
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
v
TABLE OF CONTENTS
List of Figures vii
List of Tables ix
Nomenclaturex
Acknowledgementsxii
Chapter 1 Introduction 1
11 Boundary Lubrication and Boundary-Lubricated Contact 1 12 Important Aspects of Boundary-Lubricated Contact Literature Review 4
121 Mechanisms and Efficiency of Boundary Lubrication4 122 Contact Modeling Unlubricated Surfaces 11 123 Contact Modeling Boundary-Lubricated Surfaces14 124 Flash Temperature 16 125 Summary18
13 Research Objective Approach and Outline 18
Chapter 2 Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts22
21 Introduction 22 22 The Model Problem24 23 Results and Analysis27
231 Mode of Asperity Deformation 27 232 Shape of the Plastic Zone 30 233 Contact Size Pressure and Load Capacity 33
24 Summary37
Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction 48
31 Introduction 48 32 Modeling51
321 Model Structure 51 322 Asperity Contact Pressure 53 323 Asperity Area of Contact55 324 Critical Normal Approaches60 325 System Variables 65
33 Result Analysis68
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
vi
34 Summary76
Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication86
41 Introduction 86 42 Modeling88
421 Modeling Strategy 88 422 Asperity Contact and Probability Variables 90 423 System Variables 100
43 Result Analysis104 44 Summary113
Chapter 5 Summary and Future Perspective121
51 The Deterministic-Statistical Model121 52 Perspective on Future Development123
Bibliography 126
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
vii
List of Figures
Figure 11 Boundary lubricated contacts of two rough surfaces 2 Figure 21 Half-cylinder contact model 39 Figure 22 Finite element mesh of the model problem 39 Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale 40
Figure 24 Plastic zones of the frictionless contact
(a) elastic-plastic transition (b) onset of full plasticity 41
Figure 25 Plastic zones of the contact with micro = 02
(a) elastic-plastic transition (b) onset of full plasticity 42
Figure 26 Plastic zones of the contact with micro = 05
(a) elastic-plastic transition (b) onset of full plasticity 43
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic transition (b) onset of full plasticity 44
Figure 28 Contact variables with 10δδ = 45 Figure 29 Shift and growth of the contact junction with 10δδ = 46 Figure 210 Contact variables with 103δδ = 47 Figure 31 Schematic of the equivalent contact system 79 Figure 32 Critical normal approaches and modes of asperity deformation 79 Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under
combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
80
Figure 34 Dimensionless first critical normal approach 2D finite element
results against 3D theoretical analysis 81
Figure 35 Dimensionless second critical normal approach finite element results
and curve-fitting 81
Figure 36 Surface mean separation as a function of load and friction coefficient 82
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
viii
Figure 37 Asperity height distribution and mode of deformation of contacting
asperities 83
Figure 38 Friction-induced load redistribution among asperities 83 Figure 39 Contribution of the friction-induced junction growth to the real area
of contact 84
Figure 41 An individual boundary-lubricated asperity contact 115 Figure 42 Flowchart for the determination of the solution of an asperity contact 116 Figure 43 System-level friction coefficient as a function of load 117 Figure 44 Asperity shear stresses and asperity height
(a) ψ = 066 (b) ψ = 186 (c) asperity height distribution 118
Figure 45 System-level contact and lubrication variables as functions of load
(a) degree of boundary protection (b) surface separation (c) real area of contact
119
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection
120
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
ix
List of Tables
Table 31 First critical normal approach as a function of the friction coefficient 85 Table 32 Percentage of elastically-deformed asperities in frictionless contact 85
x
Nomenclature
lA = area of asperity contact
nA = nominal contact area
tA = real area of contact
1E 2E = elastic modulus
lowastE = equivalent elastic modulus 1
2
22
1
21 11
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus+
minusEEνν
tF = total friction force H = indentation hardness
aH∆ = lubricantsurface adsorption heat
rH∆ = bond destruction or chemical activation energy of the reacted film cK = substrate thermal conduct
AN = Avogadro constant ( 231002213676 times mol-1) mP = average pressure of an asperity contact
mFP = asperity contact pressure at the onset of plastic flow
mYP = asperity contact pressure at the inception of yielding R = asperity radius of curvature
cR = molar gas constant (831451 ( )KmolJ sdot )
aS = probability of an asperity contact being covered by an adsorbed film
aS prime = survivability of the adsorbed layer in an asperity contact
atS prime = survivability of the adsorbed layer at the system level
nS = probability of an asperity contact with no boundary protection
ntS = probability of contact with no boundary protection at the system level
rS = probability of an asperity contact being protected by a reacted film rS prime = survivability of the reacted film in an asperity contact rtS prime = survivability of the reacted film at the system level
bT = bulk temperature
lT = contact temperature of an the asperity junction
1T∆ = asperity flash temperature V = sliding velocity
tW = total contact load a = radius of an asperity contact
0b = adsorption coefficient
123
210002
minus
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛sdotsdot
ϑπ
A
bb N
TmkTk
c = substrate specific heat
xi
d = distance from the mean plane of asperity heights to the rigid flat ( )zf = distribution density function of the asperity height
h = separation based on surface heights Ak = friction-induced junction growth factor Alk = upper bound of the junction growth factor at ( )microδδ 2=
bk = Boltzman constant ( KJ10380661 23minustimes ) m = lubricantadditive molecular weight
ct = duration of an asperity contact
ft = time to the break of the substratereacted film bonding z = asperity height
sz = distance between the mean of asperity heights and that of surface heights
α = constant in Taborrsquos equation β = Rση γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact
1δ = first critical normal approach 2δ = second critical normal approach
η = area density of asperities κ = substrate thermal diffusivity
lmicro = local friction coefficient
tmicro = system friction coefficient
21 υυ = Poissonrsquos ratio σ = standard deviation of surface heights
aσ = standard deviation of asperity heights
eσ = effective stress
aτ = shear strength of the adsorbed layer
mτ = average shear stress of an asperity contact
nτ = shear strength of the substrate material
rτ = shear strength of the reacted film ψ = plasticity index ϑ = Planck constant ( sJ10626086 34 sdottimes minus )
xii
Acknowledgements
The completion of the thesis brings me to the end of my student life I would like
to take this opportunity to express my appreciation to all those who helped and supported
me during my journey of learning Without their guidance help and patience I would not
be able to go this far
First and foremost I am very grateful to my thesis advisor Prof Liming Chang
for introducing me to the exciting and challenging project for his continuous guidance
and encouragement from the day I met him more than five years ago Since then he has
inspired me in my research with his interest dedication and enthusiasm for this study At
each stage of the research I have benefited tremendously from his academic expertise
professional rigor and solid grasp of the big picture I especially appreciate the time and
effort he put into reading and commenting many drafts of the thesis as it was taking
shape I want to also thank him for his knowledgeable advice and constructive criticism
on every aspect of academic life which broadened my perspective improved my research
skills and prepared me for future challenges
I would like to thank other members of my thesis committee Professor Richard
Benson Professor Marc Carpino and Dr Seong Kim for providing invaluable
suggestions during the course of my research and generously sharing with me their deep
understanding of this topic I want to express my sincere thanks to Dr Martin Webster
and Dr Andrew Jackson at ExxonMobil Technology Company for their consistent
support and insightful comments
xiii
My special appreciation goes to Prof Yongwu Zhao at Southern Yangtze
University for his encouragement advice and fruitful discussions during his stay here at
the Penn State University and when he is back in China Many thanks are also due to my
fellow students and research associates and all other friends at State College who have
offered immediate and continuous support throughout the past five years
I wish to acknowledge ExxonMobil Technology Company for the financial
support of the research project I also would like to thank Prof Stefan Thynell Professor-
in-Charge of the Mechanical and Nuclear Engineering Graduate Programs for his faith in
my abilities and selecting me as a Graduate Teaching Fellow during the last semester of
my PhD This program has taught me many things which I cannot learn from any other
experience
I am indebted to my parents brother and sister for their enduring love and
support to my daughter for not spending as much time as I should and to my dear wife
Jia ldquowho have been with me through thick and thin and everything in betweenrdquo Finally
I dedicate this thesis to my father Shi-Chang Zhang who lost his ability to speak two
years ago
Chapter 1
Introduction
11 Boundary Lubrication and Boundary-Lubricated Contact
Boundary lubrication provides the basic protection to the bearing surfaces of
machine components which operate at high load low speed or high temperature such as
o Geartooth camtappet and piston-ringliner contacts
o Rolling element bearing at the pure sliding sites
o Journal bearings during the periods of start-up and shutdown
The effectiveness of boundary lubrication is critical to the service life of these
components In addition boundary lubrication also plays an important role in the
following devices or operations
o MEMS [1] and headdisk interface [2]
o CMP and the metal cutting and formation operations [3]
o Natural and artificial joints such as those in the hip and in the knee after periods
of inactivity such as sleeping [4]
Therefore knowledge of the surface contact behavior in boundary lubrication is essential
to improve the performance of the above systems and procedures addressing the
efficiency safety environment and other concerns For example such knowledge is
invaluable in developing the strategies for controlling tribo-failure and minimizing wear
2
and in designing the environmentally benign lubricants and additives The objective of
the current research is to enhance the understanding in the area by developing a
theoretical model for the boundary-lubricated sliding contact of two rough surfaces
Figure 11 Boundary lubricated contacts of two rough surfaces
The nominally flat bearing surfaces usually deviate from their prescribed
geometry with microscopic irregularities Under boundary lubrication conditions two
rubbing surfaces make frequent and random micro-contacts at their high spots or the
asperities (as shown in Fig 11) The load applied to the system is then mainly carried by
the discrete asperity contacts and the total friction force is also the integration of local
tangential resistance During each asperity contact a series of micro-scale processes of
different nature proceed simultaneously and interact with each other in a number of ways
The direct mechanical response of two contacting asperities is their elastic or inelastic
deformation which results in the asperity load support This response is accompanied by a
group of physical and chemical reactions among the substrate additives lubricants and
environment leading to the formation of low shear-modulus films in the contact junction
These films protect asperities from direct contact and effective lubrication is thus
achieved The protective boundary films may be ruptured and then the asperity contact
takes place directly between the opposite metallic substrates The local friction resistance
may thus come from the shearing within the boundary films andor that occurring at the
3
metallic surfaces The shear stress along with the sliding velocity generates frictional
heating in micro contact regions As a result high local temperatures of short duration or
so-called flash temperatures may be aroused The frictional heating process may
facilitate the formation of the boundary lubricating films or deteriorate them by
dissociation desorption or oxidation The state of these films or their integrity also
depends on the levels of contact pressure and shear stress This state in turn largely
determines the shear stress and thus affects other micro-contact variables In summary
the system-level tribological behavior under boundary lubrication conditions is
collectively governed by multiple interactive asperity-level processes
On the other hand the micro-contact processes may also be affected by the
evolution of system features For example in the course of an asperity-to-asperity contact
the asperity temperature is composed of two components the flash temperature and the
bulk temperature The latter is largely system specific and governed by the overall heat
generation and transfer In addition the geometrical characteristics of the rubbing
surfaces may experience continuous progression resulting in dynamically changing
conditions at each asperity contact
The above discussion indicates that the boundary lubrication processes exhibits
diversity in their natures and scales The corresponding contact modeling is therefore a
truly interdisciplinary subject The model should be developed based on the knowledge
of the mechanisms of boundary films the contact of rough surfaces and the flash
temperatures of asperity contacts Significant advances have been made in these areas
and the current understanding of each is summarized below from the modeling viewpoint
to establish the theoretical framework and methodological focus for this thesis research
4
12 Important Aspects of Boundary-Lubricated Contact Literature
Review
121 Mechanisms and Efficiency of Boundary Lubrication
In boundary lubrication two different types of protective films may be formed in
an asperity junction to prevent the surface damage during sliding A layer of organic
compounds with polar end groups may be adsorbed on the surface Meanwhile an
inorganic film may be produced by the chemical reaction between the substrate and the
additives or lubricants These boundary films usually reduce friction and increase the
resistance of the system to surface failure such as seizure For example the formation of
Fe2Cl3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel
system by a factor of 3-8 [5] The system performance is thus largely controlled by the
properties of the two types of boundary lubricating films including their composition
structure effectiveness and shearing behavior The generally accepted ideas about these
important issues and the recent developments are briefly reviewed below for the adsorbed
layer and the reacted film in sequence
A conceptual model has been proposed to explain the mechanism of boundary
lubrication by the adsorption [6] According to this model the polar ends of organic
lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon
chains projected vertically upward The molecular layers adsorbed on the opposite
surfaces are only weakly interacted The sliding of the two surfaces is then accomplished
between the adsorbed layers resulting in a low interfacial friction Therefore the
measured friction coefficient has often been used to characterize the relative lubrication
5
effectiveness of the adsorbed layers for various combinations of base lubricants polar
additives and surfaces It has been found that the effectiveness depends on the chain
length of the hydrocarbon molecules [7-9] the molecular structure [10 11] and the type
of polar groups [12 13]
The adsorbed layer is generally effective up to a critical interfacial temperature
[14-16] It is because high temperature corresponds to strong thermal desorption leading
to a reduced fraction of surface that is covered by the adsorbed molecules The fractional
surfactant surface coverage θ or defect θminus1 has often been related to the interfacial
temperature and the free energy of adsorption of the additive or lubricant to the surface
The simplest relationship for this purpose is the Langmuir adsorption isotherm [17]
which assumes that the surface is energetically homogeneous and there is very small or
zero net lateral interaction between adsorbate molecules The applicability of the
Langmuir isotherm in boundary lubrication studies has been verified experimentally for
different additives and lubricants [14 18 and 19] In comparison the Temkin isotherm
may be more suitable in the case of heterogeneous surfaces and strong lateral interaction
within the adsorbed layer [11 13] Another model is proposed to determine the fractional
coverage based on the dwell-time of an adsorbed molecule at a particular surface site [20]
In addition to the interfacial temperature and adsorption energy this model also accounts
for the effect of sliding velocity
Assuming that the adsorbed layer is the only boundary lubricating film direct
metallic contact may occur as a result of the partial failure of this layer The interfacial
friction may then arise from both the shearing of the layer and the metallic contact The
6
overall friction force can thus be related to the fractional surfactant surface coverage and
the relation is given by [21]
( )[ ]mbrAF τθθτ minus+= 1 (11)
where rA is the real area of contact bτ the shear strength of the boundary lubricating
film and mτ that of the substrate material By assuming that the surfaces are fully
covered by the adsorbate the shear strength bτ may be determined on the basis of the
measured frictional force and the knowledge of the real area of contact rA However this
is difficult in real engineering situations due to the uncertainty involved in the estimation
of rA and the possible desorption during the contact In order to overcome this difficulty
a feasible approach is to deposit monolayers or multilayers of organic films on very
smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere
against a plane For these types of contact configuration the area of contact could be
calculated using the well-known Hertzian solution and the calculation may be verified
experimentally for example by multiple-beam interferometry This approach was first
used to study the shearing behavior of calcium stearate monolayers deposited on
atomically smooth mica sheets [22] and then extended to a variety of other organic films
[23-26] The results of these studies show that the film shear strength is dependent on the
contact pressure and may be expressed in the following form [27]
sum+=j
njb
jPmicroττ 0 (12)
where 0τ is the shear strength at zero pressure In many cases of interest 0τ is small
compared to other terms The coefficients and exponents of the series in this expression
7
characterize the mechanical or rheological properties of the boundary lubricating films In
addition to the experimental studies a theoretical model has been proposed relating the
friction of two adsorbed layers on the opposite surfaces to the energy barrier between two
adjacent equilibrium positions [28] Without considering the dislocations and energy
conservation the predictions from this theory are much higher than the experimental
results
Compared to the adsorbed layers the reacted films in boundary lubrication
systems are much more complex in terms of the formation composition structure
effectiveness and mechanical properties Typically the reacted films are generated from
the chemical reaction between the metal surface and the additive with one active element
such as sulfur phosphorus chlorine and boron [29 30] The corresponding formation
process starts with the chemisorption of the additive on the metal surface This is
followed by the decomposition of the additive molecules leaving the active element
chemically bonded to the surface A thin film of metal salts is then formed and it may be
mixed with oxides in the presence of moisture or in air atmosphere Further growth of the
film involves the diffusion of the active elements and metallic ions Such a formation
process is similar to that of the oxide layer on the surface The growth of the film
thickness may follow a linear law initially and a parabolic law afterwards and may thus
be described by the following equation [31]
n
nrno t
RTQ
Ahf1
exp ⎥⎦
⎤⎢⎣
⎡∆sdot⎟
⎠⎞
⎜⎝⎛minus=∆ρ n = 1 or 2 (13)
8
where An is the Arrhenius constant and Qn the activation energy of reaction These two
parameters are closely related to the type of metallic salt which strongly depends on the
availability of the active elements and the temperature at the interface On the other hand
the reacted films may also be formed by a multifunctional additive containing two or
more active elements The most widely used multifunctional additives are the alkyl and
aryl groups of zinc dithiophosphate (ZDTP) which usually form a boundary lubricating
film of a multilayer structure Starting from the substrate this type of film composes of
an inorganic layer of sulfates and oxides a layer of short-chain polyphosphates andor
long-chain zinc polyphosphates and a layer of organophosphates such as alkyl-
phosphate The transition between the two adjacent layers is gradual The portion of each
layer within the film depends not only on the properties of the lubricant additive and
substrate material but also the severity of the sliding contact More detailed information
can be found in [30] and [32-34] on the structure and composition of the ZDTP films and
the mechanism of action at the molecular level In addition the reacted films may include
a multilayer of carboxylate formed from carboxylic acid additives [35 36] and a thick
layer of high-molecular weight organometallic compounds by the polymerization of
additive-free oil minerals [37 38]
The diversity of the reacted films formed in the boundary lubricated contact
suggests that they may work by different mechanisms depending on their form structure
and properties A very thin film of metal salts or oxides may act as a sacrificial layer of
low shear strength It is easily removed by the shear or cavitational forces along with the
friction heating but is able to be reformed immediately to sustain continuous sliding A
prime example is the boundary film formed from the extreme pressure additives [39] The
9
high-molecular polymeric film generated from base oil molecules may also work on the
basis of repeated removal and repair [40] In contrast the metal salt-films derived from
the antiwear additives are relatively thicker and usually much more tenacious They are
not easily removable during the sliding and the wear is thus controlled As for the
multilayer film resulting from ZDTP each layer has different properties and functions
[41] The metal salts such as FeS has sufficiently high shear strength and serves as an
adhesive layer as well as a seizure-resistant coating The intermediate phosphate layer has
high viscosity and its hardness is comparable to the mean contact pressure It can flow
plastically and may thus act as a protective layer against wear by eliminating the abrasive
contribution of oxides The outermost organic layer is mobile and has varying viscosity
similar to the base oil ensuring that the shear plane is located within the boundary
lubricating film This layer also serves as a reservoir for the regeneration of
polyphosphates
The reacted films described above may fail to provide effective protection to the
surfaces when the films are removed during the contact The failure process is strongly
affected by the level of interfacial shear stress frictional heating [29 42] and contact
pressure and plastic deformation [43 44] A number of models have been proposed to
explain the film-failure in terms of the friction-induced temperature rise andor the
mechanical stresses Accordingly a group of criteria has been defined The failure has
often been attributed to the imbalance between the formation and the removal of the
reacted films Based on this hypothesis a critical temperature condition has then been
determined In one of such studies [45] both the formation and removal rates have been
measured and modeled as a function of interfacial temperature using the Arrhenius-type
10
expression in the form of Eq (13) The failure occurs above a critical temperature when
the removal rate is greater than the formation rate For the system running at low speeds
the effects of frictional heating or interfacial temperature are negligible The reacted films
fail when the maximum interfacial stress exceeds the film or substrate shear strength and
a stress criterion has thus been defined [46 47] The film failure has also been viewed as
the result of the destruction of the chemical bonds between the active elements of
additive molecules and the metal surface [48 49] From the energy transfer point of view
these mechanically stressed bonds can be broken by the combined action of the thermal
energy from frictional heating and the distortion energy due to shearing According to the
thermal fluctuation theory of fracture [50] the typical lifetime of the bonds represents
their resistance to the destruction and may thus be used to characterize the film-failure
The three types of models described above are deterministic but the information about
many of their input parameters is incomplete and the failure process itself also involves a
certain degree of intrinsic uncertainty Thus a probabilistic approach is more appropriate
to assess the likelihood of failure of the reacted films This likelihood may be expressed
as a probability similar to the fractional defect of the adsorbed layer The probability may
also be used to model the interfacial friction in combination with the knowledge of the
film shearing properties
In addition to the formation structure and effectiveness of the reacted films their
shearing behavior and other mechanical properties are also the key to understanding the
mechanism of boundary lubrication These aspects have thus been studied by many
researchers for the reacted films formed during tribological testing using conventional
tribometers and innovative scanning probe techniques With a ball-on-flat configuration
11
Tonck et al [51] measured the tangential stiffness by a microslip method for four types of
tribo-films formed by pure paraffin ZDTP calcium sulphonate and a friction modifier
respectively The elastic shear moduli of these films were also determined and were
found similar to those of high molecular weight polymers such as polystyrene In
addition the results showed that the values of shear modulus would increase with the
load except in the case of the friction modifier More recently nanoindentation has been
widely used to measure the mechanical properties of the reacted films generated from a
variety of lubricant additives [52-55] It was observed that the film hardness and elastic
modulus would increase with depth up to a few nanometers beneath the surface
Correspondingly the resistive forces within the films might increase during the loading
stage of the indentation to accommodate the increasing applied pressure On the other
hand the lateral force microscopy has been used in combination with the atomic force
microscopy to examine the frictional properties of the tribo-films formed in reciprocating
Amsler tests [56 57] A linear relationship was revealed between the load and the friction
force measured for micro regions of the tribo-films This may be explained by the
distribution of the hardness and modulus in depth observed in the nanoindentation tests
Therefore the shearing behavior of the reacted films may also be described by Eq (12)
in its linear form Furthermore the friction coefficient of the micro regions was found in
good agreement with the macro results The overall friction coefficient is thus indeed
determined by the shearing of the reacted films covering the asperities
122 Contact Modeling Unlubricated Surfaces
For two nominally flat surfaces without lubrication their contact takes place at
distributed asperity junctions The contact models predict the mechanical responses of
12
surfaces to the applied loading These responses including the size and spatial
distribution of asperity contact spots and the surface and subsurface stress fields around
them are dependent on the topography of surfaces and their material properties
Two major approaches have been used to model the contact of rough surfaces
stochastic and deterministic The stochastic contact models can be further classified into
two groups statistical and fractal These approaches or models are distinguished by the
use of surface descriptions The basic features of different approaches are briefly
summarized below A more comprehensive review including the discussion on their
advantages and disadvantages can be found in ref [58]
The statistical approach was first proposed by Greenwood and Williamson [59]
In this approach the surface roughness is represented by asperities of simple geometrical
shape and with predefined radii of curvature The asperity heights are assumed to follow
a statistical distribution A rough surface is thus characterized by statistical parameters
such as the standard deviation of surface heights and correlation length A single asperity-
to-asperity contact is reduced to the deformation of two curved bodies in contact Its
solution may either be determined analytically using contact mechanics or expressed by
the empirical formula from the finite element simulation The surface contact is then
modeled by relating the load and the real area of contact to their asperity-level
counterparts by statistical integration
In many situations the statistical parameters of surfaces have been found strongly
dependent on the resolution of roughness-measuring instruments [60-62] This
phenomenon is due to the multiscale nature of the surface roughness which may be better
13
described by fractal geometry [63 64] The surface contact models are then developed
based on the use of power spectrum and scaling laws characterized by scale-invariant
quantities such as fractal dimension [65-69] These models also take the system variables
to be the integration of the asperity solution However each asperity is now represented
by the size of the contact spot based on which its amplitude of deformation and radius of
curvature are defined
The deterministic approach analyzes the computer generated surfaces or those
represented by the digitized output of roughness measurement The surface contact
behavior may then be predicted numerically by the method of influence coefficients [70-
77] and that based on the variational principle [78] Compared to the statistical and fractal
contact models the numerical simulation uses the digital maps of rough surfaces and
does not require any assumptions on asperity shape and distribution In addition this type
of analysis may be able to naturally account for the interaction of deformation of adjacent
contact spots
Significant advances have been made with the above approaches in the study of
both frictionless and frictional dry contacts of rough surfaces However the models
developed so far for the frictional contact appear to be largely oversimplified with some
major assumptions Two key phenomena in the authorrsquos opinion need to be addressed in
modeling the frictional surface contact One is that contacting asperities may deform
elastically elastoplastically or plastically According to the results of frictionless
indentation of a sphere on a plane the normal load leading to initial yielding needs to
increase more than 400 times to cause fully plastic flow [79] The application of friction
reduces the first critical normal load [80-82] and thus the elastic deformation regime The
14
friction may also reduce the critical load related to plastic flow and the elastoplastic
deformation regime However this transition regime may still be significant compared to
the elastic regime Hence a high percentage of contacting asperities may be in the state
of elastoplastic deformation for the contact of rough surfaces with or without friction
Moreover a significant portion of asperities in contact may deform plastically in the
frictional situation For the frictionless contact all the three possible deformation modes
have been incorporated into several statistical models based on approximate analytical or
finite element solutions of the elastoplastic asperity contact [83-85] In contrast there is
no similar model for the frictional contact due to the lack of a systematic study of the
elastoplastic behavior of contacting asperities with friction The other key phenomenon is
that the friction may significantly change the asperity pressure and contact area for those
asperities in elastoplastic and particularly fully plastic deformation Both experimental
and theoretical studies have shown that for a frictional plastic contact the interfacial
shear stress would lead to the growth of the asperity junction and reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] For the surface contact the effects of
friction on the subsurface stresses have been modeled but the contact pressure and area
are usually considered not to be altered by the friction In summary a mathematical
model accounting for these two important issues should be formulated for the frictional
contact of rough surfaces
123 Contact Modeling Boundary-Lubricated Surfaces
15
Under boundary lubrication conditions the contact of two rough surfaces is also
present in the form of distributed asperity contacts In addition to the asperities the
boundary films covering them may be involved in the contact process However these
films are very thin and thus it is reasonable to assume that the contact pressure and area
are mainly determined by the asperity deformation The contact response is mainly
affected by the boundary films through their effects on the interfacial friction Thus the
three approaches discussed in the last section may also be used to model the boundary-
lubricated surface contact if the shearing behavior of the boundary films is known
Many contact models have been developed for the boundary lubrication system
using the statistical approach [97-104] Besides the general contact response these
models predict the friction force as a function of load by summing up the local tangential
resistance The pressure and area of a single asperity contact are usually determined using
the Hertzian elastic solution In comparison the finite element method has been used to
analyze the mechanical responses of contacting asperities with nonlinear material
properties [104] For the determination of the friction force at the asperity junctions there
are several different formulations available For example Ogilvy [97] calculated the local
friction force by assuming constant film shear strength and using the energy of adhesion
Blencoe and Williams [101] related the interfacial shear strength to the contact pressure
according to empirical relations and Ford [103] took account of the contribution from
both interfacial adhesion and asperity deformation In addition to the statistical models
direct numerical simulation has also been performed for the contact of rough surfaces to
calculate the friction force resulting from adhesion and deformation [105] This
16
deterministic model extends the method of influence coefficients to account for the
effects of shear force on contact deformation
The study of the boundary-lubricated surface contact with the above models has
provided some insights into the effects of the rheology of boundary layers the substrate
material properties and the surface roughness on the system tribological behavior
However there are significant rooms for advancements in many aspects and
mathematical models with more insights may be developed First as mentioned in the
last section a large population of contacting asperities may be in either elastoplastic or
fully plastic deformation These two types of asperity contacts have not been properly
considered The important phenomena related to the two deformation modes such as the
pressure-shear stress coupling and the friction-induced junction growth also need to be
incorporated in to the model Second the adsorbed layer may be desorbed and the reacted
film may be ruptured during the asperity contacts Thus the effectiveness of boundary
lubrication at an asperity junction is characterized by intrinsic uncertainty It would be of
theoretical and practical significance to capture this uncertainty by modeling the kinetic
behavior of the boundary lubricating films Third localized temperature rise or flash
temperature may be caused by the intensive shear stress at asperity junctions The
increasing contact temperature in turn may significantly affect the kinetics of the
boundary films and thus the interfacial shear stress As reviewed in the next section the
flash temperature has been calculated or measured by a number of researchers However
its interaction with the evolution of the boundary films has not been studied adequately in
contact modeling
124 Flash Temperature
17
The localized temperature rise due to frictional heating is an important
characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough
surfaces The rising temperature can be viewed as the thermal response of the contact and
it may strongly affect the behavior of lubricating films the properties of substrate
materials as well as most surface phenomena Thus the prediction of the interface
temperature plays an important role in modeling the sliding contact behavior
The maximum or average temperature rise of single asperity contacts has been
estimated based on the laws of energy conservation and heat conduction [106-115] Most
of these analyses focused on the flash temperature of an individual square or circular
contact Gecim and Winer considered the cooling-off effect between two consecutive
asperity contacts [112] Bhushan proposed an approach to include the effects of frictional
heating by neighboring asperity contacts [114] The analysis of asperity flash
temperatures has also been incorporated into different types of surface contact models to
predict the interfacial temperature distribution [67 68 and 116-118] For example the
fractal contact model developed by Wang and Komvopoulos [67 68] included the
analysis of the distribution of temperature rise at the interface Based on a statistical
contact model Yevtushenko and Ivanyk [116] determined the temperature rise of
contacting asperities and their thermal deformation for the sliding contact of rough
surfaces under mixed lubrication conditions In comparison Qiu and Cheng [117]
calculated the temperature rise at asperity contact spots which were the solution provided
by a deterministic surface contact model [71]
18
125 Summary
The above literature review shows that significant progress has been made in the
understanding of different boundary lubrication mechanisms the modeling of rough
surfaces and the calculation of flash temperature Research has also been initiated to
address the integral effects of these important aspects For example a failure criterion of
boundary lubrication has been incorporated into a thermal contact model of rough
surfaces [117] However only the elastic deformation and thermal desorption are
considered More recently an asperity-contact model has been designed to calculate the
tribological variables by simultaneously simulating the key processes involved but the
solution obtained is not suitable to be integrated into a system model [119] In summary
a comprehensive contact model needs to be developed to include the effects of multiple
deformation modes of contacting asperities the uncertainty of the boundary lubricating
films the flash temperature due to friction and their interaction
13 Research Objective Approach and Outline
This thesis aims to develop a surface contact model for the boundary lubrication
system to gain more insights into its tribological behavior For a given load the model
should be able to predict the asperity contact variables and their distribution and the
system friction coefficient and area of contact The model should also factor in surface
topography material and lubricant properties and other operating conditions in addition
to the system load
In this research the statistical approach is selected to relate the system contact
variables to their asperity-level counterparts The reason is that the statistical models are
19
able to identify the important trends in the effects of surface properties on the system
contact behavior with relatively simple calculation The key component of the research is
thus the development of a deterministic model for a single asperity contact under
boundary lubrication conditions
At the asperity level the model needs to capture the characteristics of
fundamental mechanical physiochemical and thermal processes involved in the
boundary-lubricated contact From the mechanical point of view the model to be
developed should cover the three possible deformation modes of contacting asperities
under combined normal and tangential loading For this purpose the effects of friction on
the pressure area and deformation mode of a single asperity contact are first explored
using the finite element method since it is impossible to obtain the analytical solution
directly The finite element results are then combined with the contact mechanics theories
to derive model equations for a frictional asperity contact involving the three possible
deformation modes These pure mechanical equations are used to describe the boundary-
lubricated asperity contact in conjunction with the expressions developed to calculate the
flash temperature and to characterize the behavior of boundary films The solution of all
the asperity-level modeling equations is finally used to formulate the contact model for
the boundary lubrication system by means of statistical integration
In summary the thesis comprises three layers of modeling and analysis ndash (1)
elastoplastic finite element analysis of frictional asperity contacts (2) modeling of
contact systems with friction and (3) modeling of a boundary lubrication process Each
layer of analysis is presented as a chapter in the main text and briefly described below
20
Chapter 2 Finite element analysis of frictional asperity contacts ndash A finite
element model is developed and systematic numerical analyses carried out to study the
effects of friction on the contact and deformation behavior of individual asperity contacts
The study reveals some insights into the modes of asperity deformation and asperity
contact variables as function of friction in the contact The results provide guidance to
analytical modeling of frictional asperity contacts and lay a foundation for subsequent
work on system modeling
Chapter 3 Modeling of contact systems with friction ndash Analytical equations are
developed relating asperity-contact variables to friction using the theory of contact-
mechanics in conjunction with the finite element results in chapter 2 By statistically
integrating the asperity-level equations a system-level model is developed and used to
study the effects of the friction on the system contact behavior It serves as the platform
in the final step of model development for the boundary lubrication problem
Chapter 4 Modeling of a boundary lubrication process ndash Based on the previous
two layers of modeling a deterministic-statistical model for the boundary-lubricated
contact is developed by incorporating the essential aspects of boundary lubrication Four
variables are used to describe a single asperity contact including micro-contact area
pressure shear stress and flash temperature In addition three probability variables are
introduced to define the interfacial state of an asperity junction that may be covered by
various boundary films Governing equations for the seven key asperity-level variables
are derived based on first-principle considerations of asperity deformation frictional
heating and kinetics of boundary lubrication films These asperity-scale equations are
coupled and some of them are nonlinear Their solution is thus obtained by an iterative
21
method and is statistically integrated to formulate the contact model for boundary
lubrication systems The model is then used to study the effects of surface roughness and
operation parameters on the system tribological behavior
Each of the above three chapters is relatively self-contained though they are also
well-connected Finally Chapter 5 concludes the thesis with a summary of the main
contributions and some suggestions for future work
22
Chapter 2
Effects of Friction on the Contact and Deformation Behavior
in Sliding Asperity Contacts
21 Introduction
It is quite well recognized that the solid-to-solid contact between the surfaces of
machine components is made at their surface asperities These asperity contacts often
play a significant role in the tribological performance of mechanical systems especially
under dry and boundary lubricated conditions Greenwood and Williamson [56]
established a framework for the statistical asperity-contact based models of two
contacting surfaces The concept was used in many areas of micro-tribology modeling
such as machine components in mixed lubrication [122] head-disk interface of computer
disk-drive [123] and chemical-mechanical planarization of silicon wafer [124] to name
just a few
The model of reference [56] does not include friction which can significantly
affect the behavior of the asperity contacts A number of researchers have studied the
effects of friction For elastic contacts the theory of elasticity is used to obtain closed-
form solutions Poritsky and Schenectady [125] and Smith and Liu [126] calculated the
subsurface stresses in frictional contacts under elastic plain-strain conditions Hamilton
and Goodman [127] Hamilton [128] and Sackfield and Hills [80] solved the three-
dimensional problem The results show that the friction brings the point of the maximum
shear stress closer to the surface and increases the compressive stress at the leading edge
23
and the tensile stress at the trailing edge of the contact Johnson amp Jefferis [81] studied
the effects of friction on the plastic yielding in line contacts Hills and Ashelby [82] and
Sackfield and Hills [80] analyzed the problem for point contacts The results show that
the yielding would start at lower normal loads and the points of the initial yielding would
move to the surface when the friction coefficient exceeds 03
For fully plastic contacts the theory of plasticity may be used to obtain
approximate solutions McFarlane and Tabor [87 88] studied the effects of friction in
plastic contacts using the octahedral shear stress theory The results show that for a given
normal load the friction reduces the contact pressure and increases the contact area
Making use of the criterion of plastic flow for a two-dimensional body Tabor [89]
derived a flow equation for asperity junctions under the combined normal and tangential
loading With this equation he explained the phenomenon of the junction growth and the
high friction between clean metal surfaces that were observed in experiments Johnson
[92] and Collins [93] also solved the plastic frictional contact problems using the theory
of slip-line field In addition to the pressure reduction and junction growth they
concluded that the friction coefficient would reach a high value of about unity in the
extreme
A large number of asperity contacts in a dry or boundary-lubricated system may
be in elastic-plastic deformation In this mode of deformation analytical solutions are not
readily available The methods of finite elements are often used to study the effects of
friction Tian and Saka [129] Kral and Komvopoulos [130] and many others studied the
contact of coated surfaces Tangena and Wijnhoven [131] and Faulkner and Arnell [132]
simulated the collision process of a pair of asperities Nagaraj [133] and many others
24
analyzed contact problems with stick and slip These numerical studies however largely
focused on special problems Fundamental issues have not been adequately addressed
such as the effects of friction on the mode of the asperity deformation shape and size of
the plastic zone in the micro-contact and the asperity pressure contact area and load
capacity
In this chapter a systematic finite element analysis is carried out to study sliding
asperity contacts in elastic elastic-plastic and fully plastic deformation The analysis
focuses on the above fundamental issues of the effects of friction to reveal some insights
into the behavior of sliding asperity contacts The modeling and results are presented in
the next two sections
22 The Model Problem
The model of a deformable half-cylinder in sliding contact with a rigid flat is used
in this chapter as illustrated in Fig 21 This two-dimensional plain-strain model should
capture the essential effects of the friction on the contact and deformation behavior of an
asperity contact while significantly simplifying the computational complexity The
material is assumed to be elastic-perfectly plastic with a Poissonrsquos ratio of 30=υ and a
ratio of Youngrsquos modulus to uni-axial yield stress of 1200 =YE The choice of a high
value of YE would result in a plastically deformed region in the contact that is much
smaller than the cross-section area of the half-cylinder so that the results will be fairly
independent of the latter and of the boundary conditions away from the contact
Furthermore the results in the dimensionless form presented later in the chapter are
essentially independent of the YE ratio so long as the region of plastic deformation is a
25
very small proportion of the bulk material which is the case in actual asperity contacts
The normal loading to the contact is prescribed in terms of the approach of the rigid flat
to the cylinder δ which is more meaningful than specifying a normal load for asperity
contacts between two surfaces The tangential loading F is given in terms of a shear
stress distribution in the contact proportional to the pressure distribution
( ) ( )xpx microτ = (21)
where micro is a prescribed coefficient of friction and the pressure distribution is to be
determined in the solution process It should be pointed out that the contact between two
bodies in gross sliding is of interest in this thesis study In such a contact the assumption
of a uniform local friction coefficient defined by Eq (21) is theoretically feasible The
ratio of the local shear stress to the local pressure in a sliding contact can be extremely
complex and often exhibits significant random behavior A uniform micro as a parameter
would represent a stochastic average that can be sensibly used to study the effects of
friction on the contact
The solid modeling software I-DEAS is used to generate the finite element mesh
of the model problem as shown in Fig 22 The mesh consists of 870 eight-node plane
strain elements with a total number of 2713 nodes A substantial number of elements are
allocated in the region around the contact The commercial finite element code ABAQUS
is used to simulate the sliding contact problem and small deformation is assumed in the
finite element calculations Zero-displacement boundary conditions are prescribed for the
nodes at the bottom of the finite element model The rigid-surface option is employed to
mimic the rigid flat which is constrained to move vertically The normal loading to the
26
model asperity by means of a normal approach is realized by enforcing a vertical
displacement to the flat The adaptive automatic stepping scheme is implemented for
loading More detail descriptions of algorithms used to determine the contact nodes and
contact conditions are given in the ABAQUS manual [134] For a given combination of
the normal approach and friction coefficient the finite element calculations yield the
pressure distribution and the width of the contact and the nodal von Mises stresses Mσ
Then the average pressure and load capacity of the contact can be calculated
Furthermore the first occurrence of a nodal stress of YM =σ is used to determine the
initial plastic yielding of the contact [135] and the stress contour of YM geσ is used to
determine the shape and size of the plastic zone
The accuracy of the finite element model is evaluated Mesarovic amp Fleck [136]
pointed out that the maximum relative error may be expressed as one-half of the ratio of
the nodal spacing in the contact and the contact size For the mesh given in Fig 22 and
under frictionless normal loading about 12 surface nodes come into contact with the rigid
flat when the initial yielding occurs in the model asperity The error under this condition
would then be under 10 Indeed the finite element results for an elastic frictionless
contact compare favorably with the results from the Hertz theory including the pressure
distribution contact width and location of the material point of initial yielding
Considering that a large portion of the analyses will be carried out for a greater number of
surface nodes in the contact the mesh arrangement of Fig 22 should be fairly adequate
The adequacy of the finite element mesh is studied with additional evaluations First the
results are essentially independent of the direction of sliding from either left or right
Second the results are also essentially independent of the history of normaltangential
27
loading (ie changes of δ and micro ) which is sensible for small deformation of a non-
work-hardening asperity Finally the plastic zones for fully plastic contacts compare
reasonably well with the slip-line analytical solutions by Johnson [92] and Collins [93]
23 Results and Analysis
The contact pressure and sub-surface stresses are calculated for a range of the
normal approach δ and friction coefficient micro The results are presented and analyzed
to reveal the effects of friction on (1) the mode of asperity deformation (2) the shape of
micro-contact plastic zone and (3) the pressure size and load capacity of the asperity
contact
231 Mode of Asperity Deformation
The state of the asperity deformation may be categorized into three regimes ndash
elastic elastic-plastic and fully plastic In an elastic contact the von Mises stresses of all
material points are less than the uni-axial yield strength of the material In an elastic-
plastic contact plastic yielding occurs at some material points marking a transition from
the elastic to fully plastic deformation In a fully plastic contact all material points
around the contact enter plastic deformation and the ability of the asperity to take
additional load is largely lost For a frictionless contact the transition from elastic-plastic
to full plastic contact is often defined to be the point when all the nodal pressures in the
contact largely reach the value of the material hardness which is considered to be about
equal to 28Y [79] For a frictional contact this definition may not be used as the
tangential loading can substantially bring down the pressure that can be developed In this
chapter the elastic-plastic to full plastic transition is defined to be the condition under
28
which the von Mises stresses of all surface nodes in the contact region have reached the
uni-axial yield stress of the material It is noted from numerical results that under the
above condition the contact pressure distribution is fairly uniform corresponding to full
plasticity
Two critical values of the normal approach are defined to describe the modes of
the asperity deformation The first critical normal approach 1δ corresponds to the
condition under which the initial yielding occurs in the contact and the second one 2δ
the condition under which the contact becomes fully plastic The effects of the friction on
the state of the asperity deformation may be studied by examining the values of the two
critical normal approaches Figure 23 shows the variations of 1δ and 2δ as functions of
the friction coefficient up to micro = 10 this micro value may be considered to be an upper
bound based on Johnson [79] The values of 1δ and 2δ are plotted in the scale of 10δ
which is the first critical normal approach for the frictionless contact For micro = 0 the
normal approach causing the onset of fully plastic deformation of the contact is about
forty times of 10δ This large value of 2δ which is of the same order of magnitude as
those obtained for 3D circular contacts [84 137] suggests a rather long transition from
the elastic contact to the fully plastic contact However the elastic-plastic transition is
rapidly reduced by the friction The value of δ2 is only about 104δ at micro = 03 and is
further reduced to one half of 10δ at micro = 10 The normal approach or the contact force
causing the initial yielding of the contact is also reduced significantly by the friction At
micro = 03 for example 1δ is reduced to 07 of its zero-friction value of 10δ This
reduction accelerates at high friction values At micro = 10 1δ is reduced to only about
29
014 10δ The reduction of 1δ with friction is more clearly seen in a log-scale shown in
Fig 23 (b) It should be pointed out that the microδ ~ curves in Fig 23 are numerical
approximations dividing the regimes of asperity deformation Numerical errors arise from
the sizes of the finite element meshing and the stepping size of the normal approach δ∆
in the solution process The results of Fig 23 are obtained with a maximum stepping size
of 10010 δδ =∆ The errors are sufficiently small and may not be further reduced given
the assumptions and idealizations of the model problem This is further supported by the
fact that the microδ ~1 curve in Fig 23 exhibits a similar trend as that for a circular contact
derived analytically using the equations in references [79 80]
The two curves of 1δ and 2δ shown in Fig 23 describe the mode of the asperity
deformation at a given friction coefficient and normal approach of the contact The rapid
reduction of 2δ with friction shown in Fig 23 (a) reveals a remarkable effect of the
friction on the deformation in an asperity contact With high friction the contact may
change from the state of elastic deformation to the state of fully plastic deformation with
little elastic-plastic transition as the normal approach or the contact force increases The
large reductions of the two critical approaches with friction also signify significant
reductions of the contact pressures at the points of transition of the mode of the asperity
deformation In a frictionless contact the average contact pressure at the elastic-to-
elastic-plastic transition is 141 of the uni-axial yield stress and it is about 260 at the
elastic-plastic-to-plastic transition With micro = 03 these two pressures are reduced to 123
and 179 respectively and further reduced to 042 and 062 at micro = 10 The reductions in
30
the pressure are evidently due to the large shear stresses that are developed in the asperity
contact
The finite element results may also be used to study the equation of the full plastic
flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the
contact This equation may be expressed as
222 Hp =+ατ (22)
where α is a constant s the interfacial shear stress and H the indentation hardness of the
material or the maximum pressure that can be developed in the contact Taking
YH 62= based on the finite element results with micro = 0 then a value for α in Eq (22)
can be determined for a given friction coefficient using the calculated pressure and
surface shear stress at the normal approach of 2δδ = For the model problem with a
friction coefficient up to micro = 10 the calculations of the nine data points along the
microδ ~2 curve yield α values that are about 10 with low micro and 15 with high micro These
fairly uniform values of α lie in the range of values discussed in [89]
232 Shape of the Plastic Zone
The behavior of the two critical normal approaches shown in Fig 23 is closely
related to the effects of the friction on the shape and size of the plastic zone in the
asperity contact The problem of a frictionless contact is first studied The location of the
initial yielding is in the central region of the contact about 067 times the contact-half-
width beneath the surface Figure 24 shows the plastic zones for two values of the
normal approach One is at the halfway between 1δ and 2δ and the other at 2δ
31
corresponding to the mode of elastic-plastic deformation and the onset of full plastic
flow respectively Under both loading conditions the plastic zones are similar and are
nearly of a circular shape In the former the subsurface initiated plastic deformation has
grown substantially and has largely propagated to the contact surface except a thin layer
that still remains elastic as shown in Fig 24 (a) In the latter this thin surface layer has
also become plastic while the plastic zone expands further with a diameter nearly three
times as that of the former
The problems with friction are studied next Figure 25 shows the results obtained
with a friction coefficient of micro = 02 the direction of the friction force is from the left to
the right The location of the initial yielding is shifted towards the leading edge of the
contact at 053 times the contact-half-width beneath the surface and 065 to the right
With a normal approach corresponding to halfway into the elastic-plastic transition the
surface material at the trailing one half of the contact has become plastic while a surface
layer at the leading one half is still elastic This is in contrast to its frictionless counterpart
of Fig 24 (a) where the plastic yielding at the surface starts in the central region of the
contact As the normal approach further increases the plastic zone rapidly propagates
towards the surface on the leading side When full plasticity is reached in the contact the
plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of
a depth that is 11 times the width as shown in Fig 25 (b) Owing to the significant
tangential loading in the contact the value of the normal approach to bring about full
plasticity is reduced to about 025 of that of the frictionless contact and the width of the
contact to about 027
32
Figure 26 shows the results with a higher friction coefficient of micro = 05 With
this high friction the plastic yielding is initiated at the surface one site at the leading
edge and another immediately occurring thereafter at the trailing edge The result of the
two-site plastic yielding is consistent with an analytical approximation [79] The two
plastic sub-zones propagate and eventually unite as the normal approach increases
Halfway into the elastic-plastic transition the plastic deformation is largely confined to
near surface and a small segment at the leading edge of the contact remains elastic
When full plasticity is reached the plastic zone has not significantly propagated into the
depth aside from a protruding-wing region that is developed towards the leading edge of
the contact as shown in Fig 26b A protruding-wing shaped plastic zone of a lesser
magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigid-
perfectly plastic contact with high friction The width of the contact in this case is only
about 005 of that of its frictionless counterpart at the condition of full plasticity Figure
27 shows the results with an even higher friction coefficient of micro = 10 Similar to the
problem of micro = 05 the yielding initiates at the surface at both the leading and trailing
edges of the contact The two plastic sub-zones have not yet connected halfway into the
elastic-plastic transition Furthermore at full plasticity no protruding-wing shaped plastic
zone of a significant magnitude is developed at the leading edge The width of the contact
is about 004 of the size for the frictionless problem when full plasticity is reached and
the plastic deformation is largely confined to a very thin surface layer in the contact
region
33
233 Contact Size Pressure and Load Capacity
It is of interest to study the effects of the friction on the contact variables
including the junction size pressure and load capacity of the asperity For a meaningful
study and results comparison the normal approach is held constant while the friction
coefficient is varied Figure 28 shows the results obtained at a relatively low level of
loading the normal approach is set equal to the normal approach causing plastic yielding
in a frictionless contact 10δ The results are plotted in the scale of their corresponding
values with zero friction With a relatively low friction coefficient of micro = 00 ~ 03 the
effects are small on the three contact variables At moderate friction of micro = 03 ~ 05 the
contact pressure starts to decrease while the contact junction grows At micro = 047 for
example the pressure is reduced to 084 of its frictionless value and the junction is
increased to 119 However the load carried by the asperity is essentially unaffected due
to the compensating effects of the pressure reduction and junction growth At the higher
level of the contact friction of micro = 05 ~ 10 the reduction in the pressure and the growth
in the contact size becomes more intensified to about one half and two times their
frictionless values at the extreme The change in the load capacity is only modest with a
maximum reduction of about 11 at micro = 10
The reduction of the pressure with friction in Fig 28 may be studied with Eq
(22) For a normal approach of 10δδ = the contact is largely elastic when the friction
coefficient is small Therefore it can accommodate some tangential traction without
bringing about significant plastic deformation (ie 22 ατ+p is significantly less than
2H ) Consequently the pressure is not affected by the friction As the level of friction
34
increases the amount of plastic deformation increases At micro = 05 for example
101 360 δδ = and 102 421 δδ = as shown in Fig 23 (b) so that the contact is significantly
plastic with the current normal approach of 10δδ = As a result the coupling between the
normal and tangential loading in the asperity contact is more pronounced and the increase
in the surface shear stress would be at the expense of the contact pressure The contact
eventually becomes fully plastic with a higher friction coefficient of micro gt 06 and the
tangentialnormal coupling is even stronger and follows Eq (22)
The growth of the contact junction with friction may be studied by examining the
shift of the junction in the direction of the friction force Figure 29 shows the sizes of the
contact junction at different levels of the friction coefficient along with the center
locations of the junction Up to a friction coefficient of micro = 038 the junction
experiences little growth and its center location is virtually unchanged This result may be
attributed to the fact that the junction is largely elastic up to this level of the friction The
results however show a significant trend of the junction growth with the friction
coefficient of micro = 038 ~ 047 yet a shift in the center of the contact junction is not
visible An examination of the critical normal approaches shown in Fig 23 suggests that
with 10δδ = the degree of plastic deformation in the contact increases significantly in
this range of the friction coefficient Thus the increase in the junction size is attributed to
the contact becoming more plastic as for a given normal approach (in a frictionless
contact) the junction size is about twice as large for a plastic contact than for an elastic
contact [79] With an even higher friction level of micro = 047 ~ 062 the results in Fig 29
show that the junction growth becomes more pronounced accompanied by a significant
35
shift of the center of the junction which is an indication of tangential plastic flow In this
range of the friction coefficient the contact eventually reaches the state of full plasticity
The accelerated junction growth is attributed to two factors One is the growth associated
with the further increase of plastic deformation in the contact and the other the tangential
plastic flow induced by the friction force For a friction coefficient beyond micro = 062 the
trend of the junction growth and the shift of the center of the junction become somewhat
moderated In this range of the friction coefficient the contact is now in the mode of full
plasticity and the junction growth is primarily due to the friction-induced tangential
plastic flow
Figure 210 shows the effects of the friction on the contact variables at a relatively
high level of loading The normal approach in this case is three times as large as that with
which the results of Fig 28 are obtained At this loading level the pressure reduction
and junction growth take place in the low range of the friction coefficient but the load
capacity is virtually unchanged In the median range of the friction the pressure and the
contact size become significantly more sensitive to the friction coefficient At micro = 05
the pressure is reduced to 058 of its frictionless value while the junction size increased to
154 The load capacity of the junction is still maintained at its frictionless level up to micro
= 04 and then reduces for higher friction to a value of 093 at micro = 05 For higher
friction coefficients the pressure reduces further and so grows the junction However the
results suggest that the junction growth in this case is not as pronounced as the pressure
reduction in comparison with the results from the previous case of low loading The
results further show a limited junction growth at the high-end of the friction coefficient
As a result the compensation of the junction growth to the pressure reduction becomes
36
less effective at this level of loading and the load capacity of the junction is significantly
reduced by the effect of friction At micro = 10 for example the load capacity is reduced to
061 of its value for the frictionless contact
The limit in the junction growth shown in Fig 210 for relatively high contact
loading is possibly due to the geometric effect of the asperity A higher loading produces
a larger contact size and a larger surface slope at the edges of the contact junction
particularly the leading edge because of the friction-induced tangential plastic flow The
tangential plastic flow and the surface slope are the two competing factors that determine
the size and the growth of the contact junction When the contact size is small the slope
is small and the junction growth is largely governed by the plastic flow leading to a large
increase of the junction with friction When the contact size is large the surface slope at
the leading edge is large and would ultimately limit further growth of the junction
It should be pointed out that a majority of the contacting asperities in the contact
of rough surfaces might experience a level of loading that is significantly above that with
which the contact-variable results in Fig 210 are obtained For machine components
such as bearings and engine cylinders the radius of surface asperities may be taken as of
the order of 10 microm [138] and the Youngrsquos modulus is around 205times1011 Pa Then the
normal approach causing plastic yielding of the contact in the absence of friction is of the
order of magnitude of 01010 =δ microm [79] For relatively highly finished machine
components the surface RMS roughness is often significantly larger than 01 microm and
thus the normal approaches of many contacting asperities can be significantly above 001
microm In this situation the loss of load capacity to the friction by these contacting asperities
37
could be more severe than that predicted in Fig 210 As a result the average gap
between the two surfaces would reduce so as to bring additional asperities into contact to
support the applied load in the system
24 Summary
This chapter conducts a finite element analysis of the effects of friction on the
contact and deformation behavior in sliding asperity contacts The analysis is carried out
using two input variables One is the normal approach of a rigid surface towards the
asperity and the other the coefficient of friction in the contact Results are presented and
analyzed to reveal the effects of friction on the mode of asperity deformation the shape
of micro-contact plastic zone the contact pressure and size and the asperity load
capacity The results lead to the following conclusions
1) The friction in the contact can significantly reduce the normal approach that
initiates the plastic yielding in the asperity and the normal approach that causes
the asperity to become fully plastic The reduction is more pronounced for the
second critical normal approach so that with a relatively high friction coefficient
the contact may change from the state of elastic deformation to the state of fully
plastic deformation with little elastic-plastic transition as the normal approach or
the contact force increases
2) The friction can significantly change the shape and reduce the size of the
plastically deformed region in the asperity when the contact becomes fully plastic
The reduction is most pronounced at high friction coefficients and the plastic
deformation is largely confined to a thin surface layer in the contact
38
3) The friction can have a large effect on the contact size pressure and load capacity
of the asperity At low friction and a relatively small normal approach these
contact variables are not affected With medium friction the pressure is reduced
and the contact size is increased however the influence on the asperity load
capacity is small due to a compensating effect between the pressure reduction and
junction growth With high friction the pressure reduction continues but the
junction growth is limited particularly for a large normal approach the limit in the
junction growth appears to be due to a geometric effect of the asperity
Consequently the effect of the pressure-junction compensation becomes less
effective and the asperity load capacity can be lost significantly
It should be emphasized that the finite element results presented in the
dimensionless form given in this chapter are sufficiently general Essentially the same
results are obtained with different radii or material parameters of the model asperity as
long as the region of plastic deformation in the contact is small so that the half-space
assumption is fairly valid Although the analyses are conducted using a line-contact
model the effects of friction in sliding asperity contacts of three-dimensional geometry
should be basically the same and the same conclusions would have been reached
Therefore the finite element results are used in the next chapter to guide the development
of analytical modeling equations for frictional asperity contacts that lay a foundation for
subsequent work on system contact modeling
39
Rigid flat
δ
Figure 21 Half-cylinder contact model
Sliding direction of the rigid flat
Figure 22 Finite element mesh of the model problem
40
Figure 23 Effects of friction on the critical normal approaches
(a) linear scale (b) logarithmic scale
35
0 02 04 06 08 1 0
5
10
15
20
25
30
35
40 δ1δ10
δ2δ10 (a)
0 02 04 06 08 1 10 -1
10 0
10 1
10 2
δ1 δ10 δ2 δ10
Crit
ical
nor
mal
app
roac
hes
(b)
Crit
ical
nor
mal
app
roac
hes
Friction coefficient
41
Figure 24 Plastic zones of the frictionless contact (a) elastic-plastic transition (b) onset of full plasticity
(the top figure shows the zoom-in of the region in the dashed rectangle in (a))
(a)
(b)
Contact width
Elastic deformation Plastic deformation
Rigid flat
Asperity
42
Figure 25 Plastic zones of the contact with micro = 02 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 027 of that of its frictionless counterpart in Fig 24)
(a)
(b)
Contact width
Friction force
43
(a)
Figure 26 Plastic zones of the contact with micro = 05 (a) elastic-plastic transition (b) onset of full plasticity
(the contact width in (b) is 005 of that of its frictionless counterpart in Fig 24)
Contact width
(b)
44
Figure 27 Plastic zones of the contact with micro = 10
(a) elastic-plastic flow transition (b) onset of full plasticity (the contact width in (b) is 004 of that of its frictionless counterpart in Fig 24)
(b)
Contact width (a)
45
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 28 Contact variables with 10δδ =
46
-3 -2 -1 0 1 2 3 0
05
1
15
micro=10
micro =07
micro =038
Contact center Friction force
Contact size
Fric
tion
coef
ficie
nt
Figure 29 Shift and growth of the contact junction with 10δδ =
47
0 02 04 06 08 10
05
1
15
2
25 PressureContact size Load capacity
Friction coefficient
Con
tact
var
iabl
es
Figure 210 Contact variables with 103δδ =
48
Chapter 3
A Mathematical Model of the Contact of Rough Surfaces with
Friction
31 Introduction
The contact between two nominally flat but rough surfaces is of great importance
in the study of the tribological behavior of mechanical systems Since the true contacts
are made at randomly distributed surface peaks or asperities asperity-based models have
often been used to study surface contact phenomena
A typical asperity contact-based model incorporates individual asperity contact
solutions into statistical descriptions of surfaces Greenwood and Williamson initiated
this approach in 1966 [59] In the GW model the rough surface was taken to consist of
hemispherically tipped asperities with an identical radius The asperity heights were
assumed to follow an isotropic Gaussian distribution The contact between two rough
surfaces was further converted to a contact between an equivalent rough surface and a
rigid flat plane By applying the Hertzian elastic contact solution to the distributed
asperities the GW model related the real area of contact and system contact load to the
mean separation of the surfaces Handzel-Powierza et al [139] verified this model
experimentally within the range of elastic deformation and for quasi-isotropic surfaces
However they also found that the theoretical prediction by the GW model would become
invalid when a significant portion of contacting asperities no longer deform elastically
The GW model has been extended mainly in two ways One is to treat other asperity
49
contact geometries including random radii of asperity curvatures [140] elliptic
paraboloidal asperities [141] and anisotropic surfaces [142 143] The other is to consider
asperity inelastic deformation such as an elastic-plastic model based on the volume
conservation of plastically deformed asperities [144] and a model incorporating the
transition from elastic deformation to fully plastic flow [84]
The aforementioned models assume frictionless contacts However any sliding
contact of surfaces involves friction which can be significant For a surface contact with
friction an asperity-based model may also be developed from the variables of frictional
asperity contacts A number of researchers have studied frictional contact of surfaces
using such a scheme For elastic contacts the asperity pressure and area are slightly
affected by the friction [79] and the two variables may be determined using the Hertz
theory Using this relation in combination with the expressions for adhesive forces
Francis [99] and Ogilvy [97] modeled the system contact variables and the friction
coefficient as functions of the separation of the mean surfaces Ogilvy [97] also modeled
a plastic contact system by assuming that all contacting asperities deform plastically and
that the asperity pressure and contact area are not affected by the friction Chang et al
[145] devised an elastic-plastic frictional surface model in which some asperities deform
elastically and others in full plastic flow It is assumed that the area of asperity contact is
determined from the Hertz solution and that only elastically deformed asperities
contribute to the friction force
The above researchers have made some fundamental contributions to the study of
frictional effects in the contact of rough surfaces However they have not considered two
key phenomena in frictional contacts One is that a contacting asperity may deform
50
elastically elastoplastically or plastically and the friction can largely change the mode of
the asperity deformation Johnson [79] showed that in a frictionless asperity contact the
contact force causing fully plastic flow could be 400 as large as the contact force leading
to the initial yielding According to the finite element study in the last chapter the
difference between the two contact forces is reduced by friction but is still significant
Thus a high percentage of the asperity contacts of rough surfaces may be in the state of
elastoplastic deformation The other key phenomenon is that the friction may
significantly change the asperity pressure and contact area for those asperities in
elastoplastic and particularly fully plastic deformation Both experimental and
theoretical studies have shown that for a frictional plastic contact the interfacial shear
stress can cause large growth of the asperity junction and large reduction of the contact
pressure [86-88] Tabor [89] modeled these two trends using a flow equation derived for
asperity junctions under the combined normal and tangential loading The pressure and
contact area of the plastic junctions have also been solved using slip-line field theory [90-
95] and upper bound plasticity analysis [96] To the authorrsquos knowledge a mathematical
model including these two key phenomena has not been formulated for the frictional
contact of rough surfaces
In Chapter 2 a finite element model has been used to study the effects of friction
on the asperity contact in all the three modes of deformation This chapter uses the finite
element results in conjunction with the theory of contact mechanics to model frictional
asperity contacts in the regimes of elastic elastoplastic and fully plastic deformation
including the junction growth and the coupling between contact pressure and shear stress
The asperity-scale equations are then used to build a mathematical model for the
51
frictional contact between two nominally flat surfaces The modeling is described next
and results presented
32 Modeling
321 Model Structure
In this chapter the framework established by Greenwood and Williamson [59] is
used to model the sliding contact between two rough surfaces As illustrated in Fig 31
the concept of equivalent rough surface is used The material properties of the equivalent
surface are taken to be a combination of those of the two surfaces in contact
Consider a single contact point of the surface shown in Fig 31 The normal
loading to the contact is prescribed in terms of the approach of the rigid flat to the
asperity
dz minus=δ (31)
where z is the height of the asperity and d the distance from the mean plane of asperity
heights to the rigid flat The friction force F is measured in terms of the average
interfacial shear stress in the asperity contact that is assumed to be proportional to the
average contact pressure
mm Pmicroτ = (32)
where micro is the coefficient of friction taken to be an input parameter in this chapter It
should be pointed out that the frictional sliding contact between two surfaces is studied
52
In such a contact the assumption of a uniform friction coefficient for all asperities is
theoretically feasible to study the effects of the frictional loading
The asperity pressure and area of contact depend on both the normal approach and
the friction coefficient Or
( )microδ mm PP = (33)
( )microδ ll AA = (34)
For a given surface separation d and friction coefficient micro the real area of contact and
the contact load of the system are calculated by statistically integrating the above two
asperity contact variables
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus= microηmicro (35)
( ) ( ) ( )dzzfdzWAdWd lnt intinfin
minus= microηmicro (36)
where ( )zf is the probability distribution of asperity heights and ( )microdzWl minus the
asperity contact force which is equal to the product of asperity contact pressure and area
A key component of the modeling is to develop expressions for the asperity
contact variables in terms of normal approach and friction coefficient With a given
friction coefficient a contacting asperity experiences three deformation stages as the
normal approach increases elastic elastic-plastic and fully plastic The transition of the
deformation mode is characterized by two critical normal approaches ( )microδ1 and ( )microδ 2
The finite element results in Chapter 2 have shown that both ( )microδ1 and ( )microδ 2 largely
53
decreases with micro as illustrated in Fig 32 The asperity contact pressure and area are
first formulated as functions of δ and micro in each of the three deformation regimes Then
the dependence of the two critical normal approaches on the friction coefficient is
modeled Finally the equations used to determine the system variables from the asperity
contact solutions are presented
322 Asperity Contact Pressure
Consider a contacting asperity in elastic deformation It is defined by the normal
approach δ below ( )microδ1 Under such a condition the tangential loading generally has
small effects on the contact pressure and area [79] Therefore the two variables are
assumed to be only dependent on the normal approach The asperity contact pressure is
then given by [79]
( )21
34 ⎟
⎠⎞
⎜⎝⎛=
REPm
δπ
microδ δ le ( )microδ1 (37)
When δ is increased beyond )(2 microδ plastic flow occurs For a frictionless
contact the asperity contact pressure at 02 )(
==
micromicroδδ or 20δ reaches its maximum
possible value or the indentation hardness of the material H Thus the frictionless
asperity contact pressure for 20δδ ge can be written as
( ) HP m ==0
micro
microδ 20δδ ge (38)
54
For a frictional contact the asperity pressure in fully plastic deformation depends on how
much interfacial shear stress is developed in the contact The pressure and shear stress
may be related by the Tabor equation [89]
222 HP mm =+ατ ( )microδδ 2ge (39)
Combining this equation with mm Pmicroτ = yields a general expression for the asperity
pressure in a fully plastic contact
( )( ) 2121
αmicro
microδ+
=HPm ( )microδδ 2ge (310)
With the asperity pressure determined for both ( )microδδ 1le and ( )microδδ 2ge a
pressure expression can be obtained for a contact in elastoplastic deformation For a
frictionless elastoplastic contact Francis [146] characterized the pressure as a logarithmic
function of the normal approach Based on that Zhao et al [84] derived an expression of
pressure in terms of the first and second critical approaches 10δ and 20δ
( ) ( )1020
10
lnlnlnln
δδδδ
δminusminus
minus+= mYmFmYm PPPP 2010 δδδ ltlt (311)
where mYP is the asperity contact pressure at the inception of yielding or at 10δδ = and
mFP is the pressure at 20δδ = and is equal to H It is assumed that the logarithmic
relation also holds when friction is present Equation (311) may then be generalized to
calculate the contact pressure of a frictional asperity contact in the elastoplastic regime
For a given normal approach and friction coefficient the pressure expression is given by
55
( ) ( ) ( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicromicromicroδ
12
1
lnlnlnlnminus
minusminus+= mYmFmYm PPPP
( ) ( )microδδmicroδ 21 ltlt (312)
In this equation ( )micromYP is the pressure at ( )microδδ 1= calculated using Eq (37) and
( )micromFP is the pressure for ( )microδδ 2ge determined by Eq (310)
323 Asperity Area of Contact
The asperity contact area is determined first for a frictionless contact When the
normal approach is smaller than 10δ the area of contact is given by the Hertz theory [79]
( ) δπmicroδmicro
RAl ==0
10δδ le (313)
With a normal approach equal to or greater than 20δ the asperity is in fully plastic flow
Its area of contact may be determined by the Abbott and Firestone model [147] and is
given by
( ) δπmicroδmicro
RAl 20=
= 20δδ ge (314)
For the asperity with a normal approach between 10δ and 20δ Zhao et al [84] and Jeng
and Wang [148] modeled the area of contact using a polynomial function which smoothly
joins Eqs (313) and (314) The resulting area expression is given by
( ) δπδδmicroδmicro
RAl )231( 320
primeprimeminusprimeprime+==
2010 δδδ lele (315)
where ( ) ( )102010 δδδδδ minusminus=primeprime
56
Next the area of a frictional asperity contact is modeled According to previous
experimental and theoretical studies [87-89] the tangential loading would cause the
growth of the asperity junction The amount of junction growth depends on the interfacial
shear stress and the mode of deformation Thus the asperity contact area may be
expressed as the frictionless area ( )0
=micro
microδlA multiplied by a junction growth factor that
is a function of both the normal approach and the friction coefficient ( )microδ Ak
( ) ( ) )0( δmicroδmicroδ lAl AkA = (316)
A model for )( microδAk is developed below to calculate the asperity contact area from the
above equation For elastic deformation the area of contact is assumed to be unaffected
by the tangential force Furthermore there is no growth at 0=micro Therefore
( ) 01 equivmicroδAk ( )microδδ 1le or 0=micro (317)
Next for fully plastic deformation defined by ( )microδδ 2ge the asperity contact pressure
and shear stress remains constant for a given friction coefficient Therefore it is
reasonable to assume that ( )microδ Ak also reaches an upper bound ( )microAlk at ( )microδδ 2=
Or
( ) ( )micromicroδ AlA kk equiv ( )microδδ 2ge (318)
Within the range between ( )microδδ 1= and ( )microδδ 2= the shear stress increases with the
normal approach and is approximated by a logarithmic function of δ according to Eq
(312) Thus a similar approximation scheme may be used to model ( )microδ Ak in the same
range to give
57
( ) ( )[ ] ( )( ) ( )microδmicroδ
microδδmicromicroδ
12
1
lnlnlnln11minus
minusminus+= AlA kk ( ) ( )microδδmicroδ 21 ltlt (319)
The upper-bound junction growth function ( )microAlk defined in Eq (318) needs to
be modeled to complete the modeling of the asperity contact area This function may be
determined by first transforming it into a function of the interfacial shear stress ( )mAlk τprime
For an asperity in fully plastic deformation Eq (310) in conjunction with Eq (32)
yields a relation between the shear stress and the friction coefficient
( )( ) 2121
αmicro
micromicroδτ+
=H
m ( )microδδ 2ge (320)
Now consider an asperity subjected to both normal and tangential loading and is in fully
plastic flow Under such a condition the characteristics of the junction growth may be
captured by the slip-line field solution of a rigid-perfectly-plastic wedge As shown by
Johnson [92] schematically illustrated in Fig 33 the tangential force causes the plastic
zone to be shifted in the direction of the force and a volume of material to be
agglomerated at the leading shoulder of the wedge A similar shifting and agglomerating
process is also revealed by the finite element results in the last chapter This process is
intensified as the shear stress increases and is likely to be the cause of the friction-
induced junction growth Both the slip-line field solution and the finite element results
show that the shift of the plastic-zone and the agglomeration of the material level off as
the interfacial shear stress approaches to the shear strength of the substrate oτ At this
point the upper-bound function ( )mAlk τprime or )(microAlk reaches its maximum value 0Alk
which is estimated next
58
Figure 33 (b) shows a schematic of the slip-line field solution of a rigid-perfectly-
plastic wedge with om ττ asymp With such a high interfacial shear stress the plastic
deformation is largely confined to the thin surface layer [92] The finite element results in
Chapter 2 also exhibit similar features Consequently volume conservation requires that
the material agglomerated at the leading edge occupies a volume equal to that of the apex
segment of the wedge that would have penetrated into the flat surface The slip-line
solution further suggests that the shape of the agglomerated material is similar to that of
the penetrated segment of the wedge Thus the amount of the junction growth l∆ may be
approximated by
( )w
ibl
αsin=∆ (321)
where ib is the semi-width of the frictionless contact at the given normal approach of the
wedge The size of contact with friction is then given by
( ) iw
bl 2sin2
11 ⎥⎦
⎤⎢⎣
⎡+=
α (322)
The maximum junction-growth factor 0Alk is the ratio of l to ib2 and so
( )wAlk
αsin2110 += (323)
A cylindrical asperity may be approximated as a wedge with a semi-angle Wα
approaching o90 Equation (323) then yields 510 =Alk for this case A value of
410 =Alk is chosen in this study to model the junction growth of spherical asperities
59
The choice is based on the above order-of-magnitude analysis in conjunction with the
consideration that the asperity load-capacity decreases with friction
For an asperity contact in fully plastic deformation the upper-bound junction
growth function ( )mAlk τprime or )(microAlk increases from unity to 0Alk as the interfacial shear
stress mτ increases from zero to oτ This increase may be divided into two stages based
on the analysis of the junction growth by Kayaba and Kato [149] and the finite element
results in the last chapter In the first stage the junction growth is very mild before the
shear stress reaches a value of om ττ 90~80= In the second stage of om ττ rarr it
largely accelerates to reach the maximum value of 0Alk Therefore the following
piecewise linear function is used to model ( )mAlk τprime
( )( )
( )⎪⎪⎩
⎪⎪⎨
⎧
geminusminus
sdotminus+
ltlesdotminus+=prime
cmc
cmAlcAlAlc
cmc
mAlc
mAl
kkk
kk
ττττττ
ττττ
τ
00
011 (324)
In this study 11=Alck and oc ττ 850= are used to describe the mild junction growth in
the first stage Finally transforming ( )mAlk τprime in Eq (324) back into the original upper-
bound junction growth function )(microAlk using Eq (320) yields
( )( )
( )( ) ( )
( )( )⎪⎪
⎩
⎪⎪
⎨
⎧
ge+minus
+minusminus+
ltle+
minus+
=
c
c
cAlcAlAlc
c
c
Alc
Al Hkkk
Hk
kmicromicro
αmicroττ
αmicroτmicro
micromicroαmicroτ
micro
micro
2120
212
0
212
1
1
01
11
(325)
where cmicro from Eq (320) is related to cτ by
60
212)(
minus
⎥⎦
⎤⎢⎣
⎡minus= α
τmicro
cc
H (326)
The value of cmicro is around 03 with oc ττ 850= implying that significant junction growth
can take place at a modest friction coefficient Equations (316) (319) and (325) form a
complete set to model the junction growth of the asperity contact area
The frictional asperity contact pressure and area have been expressed above in
terms of δ and micro within different ranges of normal approach separated by ( )microδ1 and
( )microδ 2 The two critical normal approaches are determined in the next section using
contact-mechanics theories in conjunction with finite element results
324 Critical Normal Approaches
The first and second critical normal approaches divide the asperity deformation
into three modes elastic elastoplastic and fully plastic Referring to Fig 32 both of
them decrease as the friction coefficient increases Their dependence on the friction
coefficient is modeled below Consider the first critical normal approach ( )microδ1 It
corresponds to the initial yielding of a contacting asperity The yield of material is
assumed to be governed by von Misesrsquo shear strain-energy criterion [135]
3
2
2YJ = (327)
where 2J is the second stress tensor invariant and Y the yield strength of the material
This invariant is defined in terms of the stress components by
61
( ) ( ) ( )[ ] 222222
2 6 zxyzxyxxzzzzyyyyxxJ τττ
σσσσσσ+++
minus+minus+minus= (328)
For a frictionless contact the von Mises criterion may be simplified to a linear relation
between the contact pressure and the yield strength [144]
YkP YmY = (329)
A typical value of Yk is 1067 Substituting Eq (37) into Eq (329) an expression for
( ) 1001 δmicroδmicro
==
is obtained and is given by
REYkY
2
2
10 43
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
πδ (330)
When friction exists the von Mises yielding criterion should be applied to the
resultant stresses caused by both normal and tangential loading In the case of elastic
deformation Hamilton [128] assumed that the actions of these two types of loading are
largely independent of each other Under this assumption the principle of superposition
is applicable and the resultant stress filed is given by
Tij
Nijij σσσ += (331)
where Nijσ and T
ijσ are the stress fields induced in the asperity by the normal and the
tangential loading respectively For a spherical asperity Hamilton [128] derived the
expressions of Nijσ and T
ijσ which may be written in the following functional form
( ) mijLij PZYX microσσ primeprimeprime= (332)
62
where ijLσ is a dimensionless function of the friction coefficient and the position within
the asperity The position is defined by the coordinates normalized by the radius of the
asperity contact a axX prime=prime ayY primeprime=prime and azZ prime=prime As a result the second stress
tensor invariant can also be expressed in a similar functional form
( ) 222 mL PZYXJJ microprimeprimeprime= (333)
where LJ 2 is also a dimensionless function of position and friction coefficient With the
pressure mP given by Eq (37) 2J is shown to be a linear function of the normal
approach
( )R
EZYXJJ Lδ
πmicro
2
22 34 ⎟⎟
⎠
⎞⎜⎜⎝
⎛primeprimeprime= (334)
For a given friction coefficient the initial yielding takes place at the position
( mX prime mY prime mZ prime ) where the function LJ 2 reaches its maximum ( )micromax2LJ Combining Eqs
(327) and (334) yields the condition of initial yielding of a frictional asperity contact
( ) ( )3
34 21
2
max2 YR
EJ L =⎟⎟⎠
⎞⎜⎜⎝
⎛ microδπ
micro (335)
From this equation the first critical normal approach is determined and is given by
( ) ( ) REY
J L
2
max2
1 43
⎟⎠⎞
⎜⎝⎛=π
micromicroδ (336)
The value of ( )microδ1 may be normalized by 10δ and the ratio of ( ) 101 δmicroδ is given by
63
( ) ( )( )micromicroδ
max2
max21
0
L
L
JJ
=prime (337)
Due to the complexity of the original stress expressions only numerical results are
available for ( )micromax2LJ and thus ( )microδ1 Table 31 presents the calculated values of the
normalized first critical normal approach ( )microδ1prime for a range of friction coefficient
Similar results are obtained for a cylindrical asperity by the finite element method in
Chapter 2 as illustrated in Figure 34
The second critical normal approach ( )microδ 2 defines the onset of fully plastic
deformation of the contacting asperity For a frictionless contact Johnson [79] proposed a
criterion for the onset based on a group of experimental and numerical results The
criterion is given by
402 asymplowast
YRaE (338)
where 2a is the radius of the contact area This radius is related to the frictionless second
critical normal approach 20δ by Eq (314) to give
( ) 21202 2 δRa = (339)
Substituting Eq (339) into Eq (338) an expression for 20δ is then obtained and is given
by
REY 2
20 800 ⎟⎠⎞
⎜⎝⎛asympδ (340)
64
With the availability of 20δ the second critical approach ( )microδ 2 can now be
determined The determination is based on the results that the theoretically determined
)(1 microδ is closely matched by the finite element results for a cylindrical asperity It is
sensible to assume that the normalized second critical approach ( ) 2022 δmicroδδ =prime is also
similar to that obtained from the finite element results An approximate expression can
then be determined for ( )microδ 2prime by curve-fitting the finite element results of the 2D model
in the last chapter to give
( ) 028083184374)(log 22 +minus=prime micromicromicroδ (341)
Equation (341) is obtained by a least-square regression of the data points using a
quadratic equation relating 2logδ and micro as shown in Fig 35 It should be mentioned
that Eq (341) is derived for the friction coefficient up to 10 as the finite element
calculation has only been performed in this range For the friction coefficient larger than
10 the ratio of ( )microδ 2 to ( )microδ1 is taken to be constant Or
( )( )
( )( )
11
2
1
2
=
=micro
microδmicroδ
microδmicroδ 01gemicro (342)
Since both 1δ and 2δ are substantially reduced at such a high friction coefficient this
approximation should not cause any significant error Using Eqs (340) to (342) along
with Eq (336) ( )microδ 2 is determined for any given friction coefficient
In summary the asperity contact pressure is expressed in terms of the normal
approach and the friction coefficient by Eqs (37) (310) and (312) depending on the
value of δ It is presented below for convenience
65
( )
( )
( ) ( ) ( )[ ] ( )( ) ( ) ( ) ( )
( )( )⎪
⎪⎪
⎩
⎪⎪⎪
⎨
⎧
ge+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
microδδαmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
2212
2212
1
1
21
1
lnlnlnln
34
H
PPP
RE
P mYmFmYm
(343)
The area of asperity contact is the product of the frictionless contact area 0|)( =micromicroδlA
and the junction growth function )( microδAk The expressions of the two functions are also
repeated below
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (344)
and
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln11
01
Al
AlA
k
kk (345)
where )(microAlk is given by Eq (325)
325 System Variables
The asperity contact equations developed in previous sections are now used to
model the frictional sliding-contact between two nominally flat rough surfaces The real
area of contact and contact load of the system are related to the corresponding asperity-
level variables by Eqs (35) and (36) The two system variables are functions of the
66
surface separation and friction coefficient They are also dependent on both material and
topographical properties of the surfaces The material characteristics are described by
Youngs modulus Brinell hardness and Poissons ratio Since the solution of an asperity
contact is expressed in terms of its height the probability distribution of asperity heights
is then used in Eqs (35) and (36) to calculate the two system variables Accordingly the
parameters based on the asperity heights are used to describe the surface However the
surface is usually characterized by the parameters related to the surface heights
Therefore all the variables in Eqs (35) and (36) need to be expressed in terms of the
second set of surface parameters such as the standard deviation of surface heights σ The
relation between these two sets of surface parameters was provided by Nayak [150]
The two surface contact variables may be normalized by the system parameters
The real area of contact is normalized by the nominal contact area nA and the contact
load by the product of nA and lowastE The following steps are taken to complete the
normalization The asperity pressure is normalized by the equivalent Youngrsquos modulus
lowastE and the area of asperity contact by the product of σ and R Meanwhile all the other
variables of length scale in Eqs (35) and (36) are normalized by σ The resulting
dimensionless system contact variables are given by
( ) ( ) ( )
dzzfdzAdAd lt intinfin
minus= microβmicro (346)
( ) ( ) ( ) ( )
dzzfdzPdzAdWd mlt intinfin
minusminus= micromicroβmicro (347)
67
where RAA ll σ = Epp mm = Rησβ = )()( zfzf σ= σ dd = and
σ zz = As shown in Fig 31 of the equivalent contact system d is equal to szh minus
and so )( ss zhzhd minus=minus= σ Here h is the gap between the mean plane of the rough
surface and the rigid flat and sz the difference between the mean plane of surface heights
and that of asperity heights If the asperity heights follow a Gaussian distribution their
probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(348)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (349)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (346) and (347) However three of them β aσσ
and sz are all dependent on another parameter sα which measures the spectrum
bandwidth of the surface roughness [150] Their expressions in terms of sα are given by
[138]
πα
σηβ sR3
481
== (350)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (351)
68
( ) 21
4
ssz
πα=lowast (352)
The surface roughness is therefore characterized by two independent parameters sα and
Rσ
33 Result Analysis
The model developed above is uedd to investigate the frictional contact behavior
of two nominally flat surfaces Using numerical integration the surface separation and
real area of contact are obtained and presented over a range of loading conditions and a
set of surfaces characterized by plasticity indices The statistical features of individual
asperity contacts are also examined to provide insights into the effects of friction on the
system contact behavior
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa and Poissons ratio
3021 ==υυ The constant α in the Taborrsquos equation or Eq (39) may be estimated by
considering an extreme situation Under high vacuum with pressures of 101021 minustimesminus torr
a very high friction coefficient of the order of 10 or higher is observed for clean metal
surfaces [89 151] In this case the shear stress approaches the substrate shear strength 0τ
and the shear flow is observed As a result the real area of contact increases substantially
and the pressure much reduced In the extreme the Taborrsquos equation yields
( )20τα H= (353)
69
Since YH 3asymp and 0213 τasympY for many metal materials in the spherical indentation [79]
the value of α is selected to be 27 according to the above equation The surface
asperities are assumed to have a Gaussian distribution As mentioned in the modeling
section the surface geometry is thus described by two parameters Rσ and sα Based
on experimental data given in [152] the value of Rσ is chosen to be in the range of
41001 minustimes to 31002 minustimes approximating smooth to rough surfaces A number of studies of
surface contacts [84 138] show that the other parameter sα takes a value ranging from
15 to 10 It is also known that this parameter would tend to be a constant for a given type
of finishing operation [138] Without loss of generality sα = 5 is used in the calculation
According to Eqs (350) ndash (352) the corresponding values of β aσσ and sz are
00455 1104 and 1009 respectively
The combined effect of surface roughness and material properties may be
measured by the plasticity index defined by [59]
( ) 2110δσψ a= (354)
According to Eq (330) 10δ is proportional to ( )2lowastEY Thus the plasticity index
measures the relative degree of surface roughness to material strength For a frictionless
contact it is also directly related to the likelihood that plastic deformation takes place
The contact is purely elastic if ψ is substantially less than one and a significant number
of asperity contacts are plastic when ψ is around unity The results of the system contact
variables are presented next for surfaces with a number of ψ values
70
Figure 36 examines the effects of friction on the relation between the separation
and load The results are obtained for the contact at three different values of the plasticity
index =ψ 066 093 and 186 For the steel surfaces studied in this chapter the three
values of the plasticity index correspond to low medium and high degrees of surface
roughness of Rσ = 10 20 and 41008 minustimes respectively The separation-load curve is
not affected by friction when the friction coefficient is sufficiently small particularly for
a low plasticity index With a high plasticity index however the effects of friction on the
surface separation become significant Relatively large reductions of the surface
separation are predicted particularly under high contact load The results of Fig 36 may
be analyzed by examining the asperity-scale contact behavior and its statistical
characteristics
Referring to Fig 31 the asperities with heights larger than the separation d are
in contact Among them those with heights ranging from d to 10δ+d deform elastically
when there is no friction Figure 37 shows the distribution curve of the asperity heights
normalized by aσ The area below the curve to the right of ad σ gives the percentage of
the asperities that are in contact With 00=micro the elastically deformed asperities fall in
the interval between ad σ and ( ) ad σδ10+ The area under the distribution curve
within this interval corresponds to the population of the asperities in frictionless elastic
contact Thus the percentage of all the contacting asperities in elastic deformation eφ is
given by
71
( )( )int
intinfin
+
=
10
d
d
de
dzzf
dzzfδ
φ
(355)
Table 32 presents the values of eφ for different plasticity indices and a number of
loading conditions defined by the surface separations
In the case of =ψ 066 the ratio of aσδ10 is about 23 Table 32 shows that
without friction the majority of contacting asperities would deform elastically When
friction is present an effective plasticity index may be similarly defined following Eq
(354)
( ) ( )[ ] 211 microδσmicroψ ae = (356)
In addition to surface roughness and material properties this effective plasticity index is a
function of friction coefficient The friction leads to a decrease of )(1 microδ and thus an
increase of the effective plasticity index As a result some of the asperities originally in
the elastic regime now deform at least partially plastically For a friction coefficient
smaller than 30=micro the asperities experiencing the deformation transition are in the
early stage of elastic-plastic regime Their contact pressure might decrease slightly but
compensated by the friction-induced junction growth so that the load capacities of these
asperities are not reduced For a higher friction coefficient a certain percentage of
asperities go deep into the elastoplastic regime or even fully plastic The increase in the
contact area can no longer compensate the reduction of the contact pressure As a result
these asperities lose a significant part of their load capacity To support the given load
72
the separation of the surfaces is reduced to bring more asperities into contact and to have
the asperities of smaller heights carry a larger portion of the load
For the surface with a higher plasticity index of =ψ 093 the ratio of aσδ10 is
about 11 Referring to Table 32 a substantial population of contacting asperities
undergoes inelastic deformation at 00=micro although the majority still deform elastically
With friction the deformation becomes more severe and more asperities become
elastoplastic or fully-plastic At 20=micro the value of ( )microδ1 is above 1090 δ According
to Eq (356) the effective plasticity index only increases about 5 This implies that
there is only a small portion of asperities in severe elastoplastic deformation for the
friction coefficient within the range of 00 to 02 Withmicro greater than 02 a significant
reduction of the surface separation develops and the reduction becomes more pronounced
with a higher friction coefficient In the case of 70=micro for example the reduction
reaches a value about σ130 at a load of 4103 minuslowast times=nt AEW For the surface with an
even higher plasticity index of =ψ 186 the ratio of aσδ10 is below 03 Results in
Table 32 suggest that the elastically deformed asperities only make a small contribution
to the overall load capacity in the case of 00=micro Therefore the percentage of asperities
with a decreased load capacity is significant even at a relatively low friction level Fig
36 (c) shows that a large reduction of the surface separation is generated with a modest
friction coefficient of 30=micro
The friction-induced reduction of the surface separation can be examined by
considering the load-redistribution among asperities of different heights Let the load
taken by an asperity of height z be ( )microzWl Then the load carried by the asperities of
73
heights between z and dzz + is given by ( ) ( )dzzfzWl micro An asperity-load density
function may be defined to characterize the load distribution among asperities of different
heights and is given by
( ) ( ) ( )zfWzW
zft
lW
micromicro
= (357)
where tW is the system load Figure 38 shows the distribution function )( microzfW along
the asperity height with =ψ 186 4104 minuslowast times=nt AEW and a number of friction
coefficients As the friction coefficient is increased the distribution curve shifts towards
the asperities of smaller heights and its peak value decreases This shift is accompanied
by the reduction of the surface separation that brings additional asperities into contact A
close examination of the distribution curves however reveals that the load carried by
these additional asperities is a small portion of the total load This portion of the load is
geometrically equal to the area below the curve to the left of point od It is 03 with
30=micro and 45 with 70=micro Thus the friction largely causes the applied load to
redistribute among the asperities that have already been in contact The shift of the
distribution curves in the manner shown in Fig 38 implies that the asperities of larger
heights give up some load which is redistributed among asperities of smaller heights
The load-redistribution is closely associated with the change of the modes of deformation
of the asperities which provides a measure of the contact severity In the case of 00=micro
about 30 of the total load is carried by the asperities in elastic contact and the
remaining by the asperities in elastoplastic deformation At 50=micro the contacting
asperities deforming elastically carry only 03 of the system load the asperities in
74
elastoplastic deformation contribute 407 and the remaining 59 is by the fully plastic
asperities As the friction coefficient is further increased to 70=micro these three
percentages change to 01 100 and 899 respectively and the contact severity is
much increased
In addition to reducing the surface separation and changing the asperity load
distribution the friction increases the total real area of contact This increase consists of
two parts One part is due to the reduction of surface separation As a result a larger
population of asperities is brought into contact and the asperities originally in contact are
subjected to higher normal approaches The other part is due to the friction-induced
junction growth of the asperities in elastoplastic and fully plastic contacts This part is
more critical as the contribution from the junction growth to the total real area of contact
reflects the degree of tangential flow and thus provides a measure of the friction-induced
contact instability The friction-induced junction growth may be characterized at the
system level by
( ) ( )( )micro
microφ
0
dAdAdA
t
ttAj
minus= (358)
where ( )microdAt is the real area of contact and ( )0δtA is its frictionless counterpart
Figure 39 shows Ajφ as a function of the contact load at different friction levels
and for the three plasticity indices The results indicate that the junction growth mainly
depends on the friction and the plasticity index and is not very sensitive to the applied
load At a low plasticity index of =ψ 066 as shown in Fig 39 (a) the junction growth
due to friction contributes very little to the total contact area for the friction coefficient up
75
to 50=micro Under a contact load of 4102 minuslowast times=nt AEW for example the ratio of the real
area of contact tA to the nominal contact area nA is about 466 in the frictionless case
At 50=micro the ratio nt AA increases to 51 and the value of Ajφ is about 30 This
can be explained by the fact that the frictionless second critical normal approach 20δ is
very large compared to the standard deviation aσ For =ψ 066 the value of aσδ 20 is
larger than 200 according to Eqs (330) and (340) If there is no friction most of the
contacting asperities are in elastic deformation as shown in Table 32 The additional
tangential loading reduces both the first and second critical normal approaches and a
certain population of asperities deform inelastically Then the junction growth occurs at
these asperities The higher the friction coefficient the larger the population of asperities
in inelastic deformation and so is the contribution made by the junction growth
However even with 50=micro most of the elastically-deformed asperities are still in the
early stage of the transition from ( )microδδ 1= to ( )microδδ 2= For example the normalized
density function given by Eq (349) has a value below 4102 minustimes at an asperity height of
az σ = 4 which is about half of the value of ( ) aσmicroδmicro 502 =
As a result the friction only
causes very small junction growth suggesting that the contact system with a low plasticity
index remains fairly stable up to a relatively large friction coefficient With an even
larger friction coefficient the values of )(1 microδ and )(2 microδ are further reduced and the
junction growth may eventually become significant At a friction coefficient of 70=micro
for example the value of nt AA becomes 57 and that of Ajφ is increased to about
10 Since this amount of junction growth is concentrated on asperities of large heights
the local instability developed at these asperities may induce some adverse tribological
76
behavior at the system level In the case of =ψ 093 the value of aσδ 20 is much
reduced Table 32 shows that the frictionless contact already involves a significant
population of asperities in elastoplastic or fully plastic deformation The number of these
asperities is further increased by friction Thus a larger portion of the real area of contact
comes from the junction growth as shown in Fig 39 (b) This portion is over 16 for the
contact with 4102 minuslowast times=nt AEW and 70=micro The tangential plastic flow is significantly
more severe than the case of =ψ 066 With an even higher plasticity index the friction-
induced junction growth could be much more pronounced At ψ = 186 as shown in Fig
39 (c) the value of Ajφ is over 11 under a load of 4102 minuslowast times=nt AEW and with a
friction coefficient of micro = 04 and Ajφ reaches 25 with micro = 07 This high level of
friction-induced junction growth and tangential plastic flow would likely be a source of
tribo-instability that can lead to scuffing failure of the system
34 Summary
This paper develops an asperity-based model for the frictional sliding-contact of
rough surfaces Model equations for asperity contact variables are first derived using
theories of contact mechanics in conjunction with finite element results The equations
include the effects of friction on the modes of deformation of the asperity and asperity
pressure and area of contact The asperity-scale equations are then used to formulate a
contact model of the surfaces by means of statistical integration The model is used to
study the effects of the friction on the system contact behavior The results lead to the
following conclusions
77
1) For a contact system with a friction coefficient lower than 10=micro the friction
has little impact on the contact behavior even for a relatively rough and soft
surface with a plasticity index around =ψ 20
2) For a contact system of a given plasticity index the friction beyond a certain level
can significantly reduce the surface separation and increase the real contact of
area The reduction of the surface separation is closely associated with the load-
redistribution among asperities of different heights which increases system
contact severity
3) The percentage contribution to the real area of contact of the surfaces by the
friction-induced junction growth increases with the friction coefficient and the
plasticity index Since this increase is closely associated with the degree of
tangential flow of the surface materials it may provide a measure of friction-
induced contact instability of the tribo-system
The contact model presented in this chapter assumes a uniform friction
coefficient In reality the friction coefficient in an asperity junction may vary
significantly depending on the local contact conditions particularly in boundary
lubrication It can reach a very high value in severe situations such as metal-to-metal
contact due to the damage of boundary lubrication films The junction growth or local
instability may lead to system-level instability even though the overall friction
coefficient is not too high Therefore the surface contact model for boundary lubrication
systems should be able to take account of the variation and distribution of friction
78
coefficients among all contacting asperities A model of this ability is developed in the
next chapter based on the above modeling of contact systems with friction
79
Figure 31 Schematic of the equivalent contact system
Figure 32 Critical normal approaches and modes of asperity deformation
0 02 04 06 08 1 10
-1
10 0
10 1
10 2
Fully plastic
Elastic deformation
Elastic-plastic ( ) 102 δmicroδ
( ) 101 δmicroδ
micro
10δδ
δ
Mean plane of surface heights Mean plane of asperity heights
h sz
dz
Equivalent rough surface Rigid flat
80
Figure 33 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading (a) initial stage ( om ττ lt ) (b) final stage ( om ττ asymp )
(redrawn from ref [92])
αw αw
P
F
Plastically deformed region
(b) 2bi
αw αw
P
Q
Plastically deformed region
(a)
∆l
81
Figure 34 Dimensionless first critical normal approach 2D finite element results against 3D theoretical analysis
Figure 35 Dimensionless second critical normal approach finite element results and curve-fitting
0 02 04 06 08 101
05
1
Finite element resultsTheoretical rsults
micro
0 02 04 06 08 110-2
10-1
100Finite element resultsCurve-fitting results
micro
δ2δ20
δ1δ10
82
0 2 4 6x 10-4
05
1
15
2
0 2 4 6 8x 10-4
05
1
15
2
0 02 04 06 08 1
x 10-3
05
1
15
2
Figure 36 Surface mean separation as a function of load and friction coefficient
micro = 00 ~ 03 micro = 07 nt AEW lowast
(a) ψ = 066
nt AEW lowast
(b) ψ = 093
nt AEW lowast
micro = 00 ~ 02
micro = 04
micro = 07
micro = 03
micro = 0 ~ 01
σh
(c) ψ = 186
micro = 07
micro = 05
σh
σh
83
Figure 37 Asperity height distribution and mode of deformation of contacting asperities
Figure 38 Friction-induced load redistribution among asperities ( 861=ψ and 4104 minuslowast times=nt AEW )
-4 -2 00
01
02
03
04
05
(d+δ10)σa
I II III
f(zσa)
2 4 dσa
zσa
-1 0 1 2 3 4 5 6 70
02
04
06
08
Wf
az σ
30=micro
00=micro
70=micro
od
84
0 2 4 6x 10-4
0
005
01
015
02
025
0 2 4 6x 10-4
0
005
01
015
02
025
0 02 04 06 08 1x 10-3
0
005
01
015
02
025
Figure 39 Contribution of the friction-induced junction growth to the real area of contact
Ajφ
nt AEW lowast
nt AEW lowast
nt AEW lowast
Ajφ
Ajφ
micro = 04 micro = 05
micro = 07
micro = 04
micro = 07
micro = 02
micro = 04
micro = 07
(a) ψ = 066
(b) ψ = 093
(c) ψ = 186
micro = 03
85
Table 31 First critical normal approach as a function of the friction coefficient ( 30=υ ) micro 0 01 02 03 04 05 075 10 15 ( )microδ1prime 1 0985 0932 0820 0593 0420 0215 0130 0062
Table 32 Percentage of elastically-deformed asperities in frictionless contact
lowasth
ψ 05 075 10 15 20
066 947 965 978 991 997093 622 687 745 836 898186 151 184 220 294 367
86
Chapter 4
A Deterministic-Statistical Model of Boundary Lubrication
41 Introduction
Mathematical modeling is an important element to study the tribological behavior
of boundary-lubricated systems In boundary lubrication the surface asperities carry a
large portion of the applied load and the friction force is the sum of individual asperity-
level tangential resistance Therefore a sensible approach to model a boundary
lubrication system is to incorporate individual asperity contact solutions into statistical
descriptions of surfaces Such an approach was first proposed by Greenwood and
Williamson [59] for the frictionless contact of surfaces
Following the framework of the GW model [59] many asperity contact-based
models have been developed for the boundary lubrication system [97 101 104 105 120
and 121] In these models the system-level load and tangential force and the real area of
contact are solved by integrating the corresponding asperity-level variables For each
contacting asperity the contact pressure and area are usually determined using the
Hertzian elastic solution In comparison there are several different formulations for the
determination of the friction force at the asperity junctions For example Ogilvy [97]
calculated the local friction force by assuming constant shear strength of the interfacial
film and using the energy of adhesion Blencoe and Williams [101] related the interfacial
shear strength to the contact pressure according to empirical relations and Komvopoulos
87
[120] took account of the local resistance from both the asperity deformation and the
interfacial adhesive shearing
For the boundary lubrication systems the asperity contact-based models
developed so far have provided some insights into the effects of the rheology of boundary
layers the substrate material properties and the surface roughness on the system
tribological behavior However significant room exists for advancement in many aspects
and mathematical models with more insight can be developed First a large population of
the contacting asperities may be in either elastoplastic or fully plastic deformation
Important phenomena related to the two deformation modes such as the pressure-shear
stress coupling and the friction-induced junction growth have not been adequately
studied Second the contacting asperities under boundary lubrication are protected by
physically adsorbed or chemically reacted interfacial films The shear strength of these
films is dependent on the contact pressure and the dependence has been incorporated into
some surface contact models [101] On the other hand the adsorbed layer may be
desorbed [14] and the reacted film may be ruptured [153] during the asperity contacts
Thus the effectiveness of boundary lubrication at an asperity junction is characterized by
intrinsic uncertainty It would be of theoretical and practical significance to capture this
uncertainty by modeling the kinetic behavior of the boundary lubricating films in
conjunction with probability theory Third the intensive shear stresses at the asperity
junctions can generate high flash temperatures which in turn affect the integrity of the
boundary films and thus the interfacial shear stresses and asperity pressure Although the
flash temperature has been calculated or measured by a number of researchers [106-115]
its interdependence with the state of the boundary films has not been studied In
88
summary the mode of micro-contact deformation the kinetics of the adsorbed layers and
the reacted films and the temperature rising due to friction are all important aspects in
boundary lubrication Although extensive work has been conducted on each of these
aspects respectively research addressing their integral effects is limited Recently a
micro-contact model [119] has been designed to fill this gap It calculates the tribological
variables during a collision of two asperities by simultaneously simulating the key
processes involved However the approach is not suitable for an asperity-based contact
model of surfaces
A mathematical model is presented in this chapter for the contact of rough
surfaces in boundary lubrication The surface contact is viewed as distributed asperity
contacts in a random process Seven asperity event-average variables are defined to
characterize an individual asperity contact in boundary lubrication The governing
equations for the seven variables are derived from first-principle considerations of the
asperity deformation frictional heating and the state of boundary films These equations
are solved simultaneously and the asperity-level solution is further integrated to calculate
the tribological variables at the system level The modeling process is described next
followed by results and discussion
42 Modeling
421 Modeling Strategy
This chapter develops an asperity-contact based model for the boundary-
lubricated sliding contact between two surfaces which is illustrated by Fig 11 Similar to
the system contact model developed in Chapter 3 as shown in Fig 31 the concept of a
89
single equivalent rough surface is used The contact between two rough surfaces is
converted to a contact between an equivalent rough surface and a rigid flat plane Each
contact point of the equivalent surface corresponds to a sliding contact between two
asperities on the original surfaces
The modeling starts by considering an individual boundary-lubricated asperity
contact illustrated in Fig 41 During the course of the contact several processes proceed
simultaneously and interact with each other in a number of ways The asperity deforms
under the combined action of tangential and normal loading The temperature in the
micro-contact rises as a result of the frictional heating The stresses and temperature
affect the state of the boundary film in the asperity junction which in turn affects the
mechanical and thermal behavior of the micro-contact Four micro contact variables are
used to characterize the asperity-level event involving these processes They are the
asperity contact pressure and area mP and 1A shear stress mτ and flash temperature
1T∆ In addition the interfacial condition of an asperity junction may be in one of three
states or their combination The asperity may be covered by the lubricantadditive
molecules adsorbed on the surface protected by surface oxides or other reacted films or
in direct contact without boundary protections Because of the intrinsic uncertainty
involved in a boundary-lubricated asperity contact it may not be possible to determine
the state of micro-boundary lubrication in absolute terms Accordingly three probability
variables introduced in [119] are used to describe this state The first variable aS is the
probability of the asperity junction covered by an adsorbed film the second variable rS
the probability of the junction protected by a reacted film and the third nS the
90
probability of contact with no boundary protection These probability variables take
values of less or equal to one and they sum to unity
1=++ nra SSS (41)
The three probability variables may be interpreted using the fuzzy set theory [154]
Taking each of the three possible contact states as a fuzzy set the corresponding
probability variable may then represent the membership degree of the interfacial film as a
whole into this set
At a given moment the random asperity contacts developed in the contact of two
surfaces are in general at different stages of asperity collision A typical asperity contact
event may be meaningfully described using the time-averages of the four micro contact
variables and the three probability variables over the duration of the contact For
simplicity the same symbols are used to represent the corresponding asperity event-
average variables The next section derives the governing equations for the seven event-
average variables based on first-principle considerations of asperity deformation
frictional heating and asperity interfacial condition Since these processes are interrelated
the governing equations are coupled and an iterative procedure is then used to solve them
for the seven event variables of an individual asperity contact Finally the system-level
tribological and probability variables are determined by statistically integrating the
asperity-level results in the random process
422 Asperity Contact and Probability Variables
Consider the junction formed during an asperity-to-asperity contact which is
represented by a single asperity contact of the equivalent surface shown in Fig 31 The
91
area of the junction and the contact pressure may be expressed in terms of the asperity
normal approach δ and the local friction coefficient lmicro Such expressions have been
derived in the last chapter for the contacting asperity in any of the three modes of
deformation elastic elastoplastic or fully plastic The pressure expression is given by
[ ]
( )⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪
⎨
⎧
minusge
+
ltltminus
minusminus+
le⎟⎠⎞
⎜⎝⎛
=
lowast
ndeformatioplasticFullyH
ndeformatioticElastoplasPPP
ndeformatioElasticRE
P
l
l
ll
ll
llmYlmFlmY
l
lm
)(
1
)()()(ln)(ln
)(lnln)()()(
)(3
4
)(
2212
21
12
1
121
microδδ
αmicro
microδδmicroδmicroδmicroδ
microδδmicromicromicro
microδδδπ
microδ
(42)
where lmicro is equal to mm Pτ and )(1 lmicroδ and )(2 lmicroδ are the two critical normal
approaches categorizing the asperity deformation into the three deformation modes The
expressions for )(1 lmicroδ and )(2 lmicroδ are also derived in Chapter 3 and other symbols in
Eq (42) are defined in the nomenclature The area of the asperity contact is given by
( ) )0()( δmicroδmicroδ llAll AkA = (43)
where )0(δlA is the frictionless asperity contact area and )( lAk microδ is a junction growth
function due to friction Of the two functions )0(δlA is derived in ref [84] and is given
by
( ) ( )⎪⎩
⎪⎨
⎧
geltltprimeminusprime+
le=
=
20
201032
10
0
2231
δδδπδδδδπδδ
δδδπmicroδ
micro
RR
RAl (44)
92
where [ ] [ ])0()0()0( 121 δδδδδ minusminus=prime The junction growth function )( lAk microδ is
formulated in the last chapter and is given by
( )( )
( )[ ] ( )( ) ( ) ( ) ( )
( ) ( )⎪⎪⎩
⎪⎪⎨
⎧
ge
ltltminus
minusminus+
le
=
llAl
llll
llAl
l
lA
k
kk
microδδmicro
microδδmicroδmicroδmicroδ
microδδmicro
microδδ
microδ
2
2212
1
1
lnlnlnln
11
01
(45)
where )( lAlk micro is the upper bound of the junction growth at )(2 lmicroδδ = discussed in
detail in Chapter 3
At a given δ the asperity contact pressure and area may be calculated from the
above three equations if the local friction coefficient lmicro is known For the current
problem mml Pτmicro = is a variable to be determined instead of an input parameter as in
the last chapter The asperity shear stress mτ which is needed to determine lmicro may be
considered as the interfacial shear strength in the sliding junction This shear strength
generally varies with the state of micro-boundary lubrication which is characterized by
the three interfacial probability variables defined earlier It may be estimated as the
weighted average of the shear strengths of the three possible interfacial states with aS
rS and nS being the weighting factors
nnrraam SSS ττττ ++= (46)
where aτ rτ and nτ are the interfacial shear strengths of the adsorbed layer the reacted
film and with no boundary protection respectively Among them nτ may be taken as
the shear strength of the substrate material The shear strengths of the boundary layers
93
aτ and rτ are in general dependent on the asperity pressure Empirical shear strength-
pressure relations have been obtained for different lubricantsurface pairs by experimental
studies These relations can be written as a polynomial of the form [27]
)(
0)(
ij
nji
jP ⎥⎦
⎤⎢⎣
⎡+= summicroττ i = a or r (47)
where 0τ is the shear strength at zero pressure In many cases of interest its value is
small compared to other terms The coefficients and exponents of the series in this
equation are parameters characterizing the rheological properties of the boundary
lubricant layers Various specific forms of Eq (47) have been used to study the effects of
boundary-film properties on the system tribological behavior [100 101] In this study the
linear form is used as a first-order approximation
The three probability variables in Eq (46) need to be modeled to determine the
interfacial shear stress mτ The modeling makes use of two additional probability
variables One is the survivability of the adsorbed film in the course of an asperity contact
aS prime and the other the survivability of the reacted film rS prime Each of them takes a value of
unity if the integrity of the corresponding film is intact On the other hand aS prime goes to
zero when the adsorbed layer is largely desorbed and so does rS prime if the reacted film is
mostly damaged The values of aS prime and rS prime are determined by modeling the thermal
desorption of the adsorbed layer and the damage of the reacted film
The survivability of the adsorbed layer aS prime is modeled first In an asperity
junction the adsorbed layer is unlikely to be continuous due to thermal desorption [14]
94
and substrate plastic deformation [26] It is sensible to equal the survivability of the
adsorbed layer to its fractional surface coverage which has been used to characterize the
effectiveness of boundary lubrication via the adsorbed layer [29] Therefore an
appropriate adsorption model may be selected to determine aS prime based on the fundamental
aspects of the structure of adsorbed molecules and the interactions among them Of the
adsorption models available the Langmuirrsquos isotherm [17] assumes that the surface is
energetically uniform and no lateral interactions are involved between adsorbed
molecules It has the advantage of giving a simple equation for the adsorption process
and being used to directly analyze the experimental results [18] Therefore the
Langmuirrsquos isotherm is chosen in this study as a first-order approximation It is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛primeminus
prime=⎟⎟
⎠
⎞⎜⎜⎝
⎛ ∆
a
a
lc
am S
STR
HPb
1exp0 (48)
For a given contact pressure and temperature aS prime is solved from the above equation by a
numerical method
Next consider the survivability of the reacted film rS prime during an asperity contact
The film may be ruptured resulting from the destruction of the chemical bond between
the film and the substrate Thus rS prime may be related to the lifetime of the substratefilm
bonding ft The bonding can be broken up by adsorbing the thermal energy from
frictional heating andor the distortion energy due to shearing According to the thermal
fluctuation theory of fracture [50] ft may be determined using the Zhurkovrsquos equation
[155]
95
⎟⎟⎠
⎞⎜⎜⎝
⎛ minus∆=
lc
erf TR
Htt
γσexp0 (49)
where 0t is the period of a single elemental thermal fluctuation with a magnitude of 10-13
sec rH∆ the bond destruction or chemical activation energy of the reacted film γ its
activation or fluctuation volume in which active failure occurs and eσ the effective
stress and lT the junction temperature representing the mechanical and thermal loading
on the film Since the rupture of the reacted film is more likely developed along the
interface the effective stress eσ in Eq (49) may be directly related to the interfacial
shear stress mτ In addition the film rupture usually starts from a micro defect in the
asperity junction and the micro defect may be viewed as a micro crack The development
of the micro crack is then controlled by the shear stress within a small element at the edge
of the crack Due to the existence of the micro crack eσ or the maximum shear stress at
the interface may be expressed as
mse C τσ = (410)
where sC is a factor reflecting the intensification of the shear stress within a small
element at the edge of a micro crack This factor is of the order of ddl λ where dλ is
the size of the small element at the crack edge and of the order of interatomic spacing or
100 Aring and dl the length of the micro crack usually of the order of 101nm Thus the value
of sC is of the order of 10 With ft determined by Eq (49) the survivability rS prime may
now be estimated by comparing ft with the duration of the contact which is given by
96
Vatc 2= Dividing ct into a number of very short periods of time t∆ the probability
that the reacted film will fail within t∆ is given by
fr ttS ∆=primeminus1 (411)
and the corresponding survivability of the film is equal to
fr ttS ∆minus=prime 1 (412)
Assuming that the total number of dt is n ( ttc ∆= ) the survivability of the film through
the asperity contact is then given by
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=⎟
⎟⎠
⎞⎜⎜⎝
⎛ ∆minus=prime
infinrarrinfinrarr
f
c
n
f
c
n
n
fnr
tt
ntt
ttS
exp
1lim1lim (413)
The survivability in this form may also be deduced from the exponential failure-time
distribution model [156]
The two survivability variables aS prime and rS prime are now used to determine the three
contact probability variables According to the analysis by surface enhanced Raman
spectroscopy [157] and the electrochemical study [158] the adsorption of lubricant
molecules usually occurs on the top of the reacted film Thus there is no effective
protection for the substrate surface if the reacted film is damaged and the probability of
contact without boundary protection is given by
rn SS primeminus= 1 (414)
97
By Eq (41) rS prime can then be expressed as the sum of aS and rS
rra SSS prime=+ (415)
The probability of contact covered by an adsorbed layer may then be written as
ara SSS primeprime= (416)
Combining Eq (415) and (416) the probability of contact protected by the reacted film
is given by
( )arr SSS primeminusprime= 1 (417)
Six of the seven asperity event-average variables have been modeled above The
last one the contact temperature lT in the asperity junction needs to be determined In
general lT comprises two components
lbl TTT ∆+= (418)
where bT is the bulk temperature and lT∆ is the flash temperature caused by the
frictional heating in the asperity contact In this study the bulk temperature is taken to be
an operating parameter while the flash temperature is determined based on a model
developed by Tian and Kennedy [115] They derived the formulation of lT∆ for the
elastic and plastic contacts respectively In the case of an elastic contact or ( )lmicroδδ 1le
the pressure distribution at the asperity junction is parabolic and so is that of the shear
stress The flash temperature is thus calculated with a parabolic circular heat source and
is given by
98
2211 874087408260
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 1le (419)
where 11 2 κVaPe = and 22 2 κVaPe = are the Peclet numbers of the asperity pair For a
plastic contact or ( )lmicroδδ 2ge the pressure and thus the shear stress are almost uniformly
distributed over the asperity junction The expression for lT∆ is then derived with a
uniform circular heat source and is given by
2211 658065806880
ecec
ml PKPK
VaT
+++=∆
τ ( )lmicroδδ 2ge (420)
Additional derivation is needed for the elastoplastic contact with a normal approach of
( ) ( )ll microδδmicroδ 21 ltlt In this deformation regime the frictional heating can be viewed as
the combination of a parabolic heat source and a uniform one It is sensible to assume the
corresponding flash temperature takes a form similar to Eqs (419) and (420) Therefore
a generalized expression of the flash temperature for the whole range of normal approach
is given by
( ) ( )( ) ( ) 2211 eTceTc
mTl PGKPGK
VaDT
+++=∆
δδτδ
δ (421)
In this equation ( ) 8260=δTD and ( ) 8740=δTG for ( )lmicroδδ 1le and are denoted as
TeD and TeG respectively Similarly ( ) 6880=δTD and ( ) 6580=δTG for ( )lmicroδδ 2ge
and are called TpD and TpG respectively For an elastoplastic contact TD and TG may
be approximated by linear interpolation and are given by
99
( ) ( )( ) ( ) ( )TeTp
ll
lTeT DDDD minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (422)
and
( ) ( )( ) ( ) ( )TeTp
ll
lTeT GGGG minus
minusminus
+=microδmicroδ
microδδδ
12
1 ( ) ( )ll microδδmicroδ 21 ltlt (423)
The above modeling process provides a complete set of equations for the contact
and probability variables that characterize a single asperity contact under boundary
lubrication Equations (42) (43) and (46) define the asperity contact pressure mP area
lA and shear stress mτ Equations (414) (416) and (417) calculate the three contact
probability variables Equation (421) provides an expression for the flash temperature
lT∆ Supplementary equations are also developed to determine other variables involved
in the seven key equations such as the two survivability variables aS prime and rS prime Each one
of the modeling equations is coupled with some others and some of them are highly
nonlinear Thus these equations can only be solved iteratively for given material and
lubricant properties asperity geometry asperity normal approach and sliding velocity
Starting from initial estimates of the three interfacial probability variables an iteration
procedure is outlined below
1) Solve Eqs (42) ndash (47) for the frictional asperity contact pressure area and shear
stress for given normal approach and contact probability variables
2) Calculate the flash temperature lT∆ from the frictional asperity contact solution
using Eq (421)
100
3) Estimate the survivability of the adsorbed layer aS prime using Eq (48)
4) Estimate the survivability of the reacted film rS prime using Eq (413)
5) Determine the three contact probability variables using Eqs (414) (416) and
(417)
6) Calculate the shear stress mτ using Eq (46)
7) Check the convergence by comparing the current shear stress result with its
previous value If the accuracy requirement is satisfied stop the iteration
Otherwise go back to step 1)
This procedure is also illustrated by the flowchart in Fig 42 At the end of the iteration
the seven asperity event-average variables and other supplementary variables are
determined They are the solution of an individual asperity contact
423 System Variables
The tribological variables of the boundary lubrication system are determined next
Given a surface separation Fig 31 shows that there are many numbers of asperity
contacts of different normal approaches The variables in each of these contacts may be
determined using the procedure described in the preceding section The following
statistical integrals are then used to model the asperity-contact random process to
determine the load friction force and the real area of contact at the system level
( ) ( ) ( ) ( )dzzfdzAdzPAdW ld mnt minusminus= intinfin
η (424)
101
( ) ( ) ( ) ( )dzzfdzAdzAdFd lmnt intinfin
minusminus= τη (425)
( ) ( ) ( )dzzfdzAAdAd lnt intinfin
minus=η (426)
where z is the height of the asperity ( )zf its probability distribution d the distance
from the mean plane of asperity heights to the rigid flat and dz minus the approach of the
rigid flat to the asperity or δ With the system load tW and friction force tF determined
the system-level friction coefficient may be calculated by
ttt WF=micro (427)
In addition the asperity-level probability variables may be integrated to generate a group
of system-level probability variables to measure the overall effectiveness of boundary
lubrication For example the system-level probability of contact with no boundary
protection and the system-level survivability of the reacted film and that of the adsorbed
layer are given by
( ) ( )
( )intint
infin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (428)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (429)
( ) ( )
( )intint
infin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (430)
102
Similarly the mean flash temperature among the contacting asperities may be calculated
by
( ) ( )
( )intint
infin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (431)
The three system-level contact variables tW tF and tA may be normalized by
system parameters Their dimensionless expressions are given by
( ) ( ) ( ) ( )
dzzfdzAdzPdWd lmt intinfin
minusminus= β (432)
( ) ( ) ( ) ( )
dzzfdzAdzdFd lmt intinfin
minusminus= τβ (433)
( ) ( ) ( )
dzzfdzAdAd tt intinfin
minus= microβmicro (434)
where ntt AEWW = ntt AEFF = EPP mm = Emm ττ = RAA ll σ =
ntt AAA = Rησβ = σ dd = )()( zfzf σ= and σ zz = As shown in Fig 31
of the equivalent contact system d is equal to szh minus and so )( ss zhzhd minus=minus= σ
The system-level probability variables and the mean flash temperature may also be
expressed in a similar dimensionless manner as follows
( ) ( )( )int
intinfin
infinminus
=
d
d n
ntdzzf
dzzfdzSS (435)
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d r
rtdzzf
dzzfdzSS (436)
103
( ) ( )( )int
intinfin
infinminusprime
=prime
d
d a
atdzzf
dzzfdzSS (437)
( ) ( )( )int
intinfin
infinminus∆
=∆
d
d l
ldzzf
dzzfdzTT (438)
Finally assume that the asperity heights have a Gaussian distribution of standard
deviation aσ Their probability distribution function is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
2
50exp2
1
aa
zzfσσπ
(439)
And the dimensionless distribution function )( zf is given by
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛minus⎟⎟
⎠
⎞⎜⎜⎝
⎛= lowastlowastlowast 2
2
50exp21 zzf
aa σσ
σσ
π (440)
Four surface parameters including β aσσ sz and Rσ are needed to determine the
system contact solution from Eqs (432) ndash (438) As discussed in Chapter 3 three of
them β aσσ and sz are related to the parameter measuring the spectrum bandwidth
of the surface roughness or sα Their expressions in terms of sα are given by [138]
πα
σηβ sR3
481
== (441)
21896801
minus
⎟⎟⎠
⎞⎜⎜⎝
⎛minus=
sa α
σσ (442)
104
( ) 21
4
ssz
πα=lowast (443)
It should also be noticed that the asperity flash temperature is related to the
absolute value of the contact size according to Eq (421) Thus the asperity radius R
needs to be given Based on the surface descriptions in refs [122 138] the area density
of the asperities η is specified and then R determined from Eq (441) in conjunction
with the Rσ parameter Therefore the surface roughness is characterized by three
independent parameters sα Rσ and η
43 Result Analysis
The model is used to study the sliding contact behavior between two rough
surfaces in boundary lubrication The results are obtained and presented for a set of
surfaces characterized by their plasticity indices and a range of system load and sliding
velocity
The contact of steel-on-steel surfaces is considered with Youngs modulus
1121 10072 times== EE Pa Brinell hardness 910961 times=H Pa Poissons ratio 3021 ==υυ
and tensile strength 3HY = The constant α in Eq (42) was estimated to be around
27 in the last chapter The substrate thermal properties are defined by the thermal
conductivity =cK 40wmK density 7800=ρ kgm3 and specific heat =c 500JmK
Two parameters are used to describe the surface adsorption of the lubricant molecules
They are the adsorption heat aH∆ and the average molecular weight m of the adsorbate
The value of aH∆ is taken to be 40kJmol corresponding to relatively strong
105
physisorption of the lubricantadditive to the surface [159] The value of m is assumed to
be 600amu representative of the combination of general lubricants and additives [160]
Two other parameters the bond destruction energy rH∆ and the activation volume γ
are used to characterize the reacted film on the surface The value of rH∆ is chosen to be
120kJmol and that of γ 36 times 10-5 m3mol These two values are selected based on the
experimental results of polymers [155] considering that the reacted film can be viewed
as high-molecular-weight organo-metallic polymers [161 162] The proportional
constant relating the interfacial shear strength to the asperity pressure in Eq (47) is
chosen to be 050=amicro for the adsorbed layer and 150=rmicro for the reacted film which
are reasonable values [163] The surface asperities are assumed to have a Gaussian
distribution As mentioned in the modeling section the surface geometry of this
distribution is described by three parameters Rσ sα and η Based on experimental
data given in [152] the value of Rσ is chosen to be in the range of 41001 minustimes to
31002 minustimes representing smooth to rough surfaces The value of sα is chosen to be 50 as
discussed in Chapter 3 According to Eqs (441) ndash (443) the corresponding values of β
aσσ and sz are 00455 1104 and 1009 respectively The area density of surface
asperities is usually in the range of -2mm2000 to -2mm4000 [122 138] In this study
-2mm3000=η is used Finally the boundary lubrication system is assumed to nominally
operate at a sliding velocity of =V 10ms and a bulk temperature of =bT 50˚C
The effect of contact force on the system friction is studied first A higher load
dependence of the friction would suggest a higher degree of tribo-instability of the
boundary lubrication system Figure 43 shows the results for surfaces of different
106
degrees of roughness represented by a series of plasticity indices ψ = 066 093 186
and 255 The plasticity index is defined by [59]
( ) 2110δσψ a= (444)
where 10δ is the first critical normal approach of a frictionless asperity contact with
which plastic yielding takes place In this study the values of the plasticity index chosen
above correspond to low to high degrees of surface roughness of Rσ = 01 02 08 and
31051 minustimes respectively For the relatively smooth surface with a low plasticity index the
results show that the friction coefficient at the system level is low and is almost
independent of the load At ψ = 066 for example the value of tmicro varies very slightly
around 0055 This value is close to the assumed ratio of the shear strength of the
adsorbed layer to the contact pressure It suggests that the surface is well protected by an
adsorbed layer of lubricantadditive molecules and the corresponding system-level
survivability of the adsorbed layer atS prime calculated by Eq (437) is nearly 100 A further
examination shows that most of the contacting asperities deform elastically The
correlation between the system tribological behavior and its asperity level origin will be
discussed in detail later In the case of ψ = 093 the mode of deformation of the
contacting asperities are basically elastic or early elastoplastic and similar results of the
system friction coefficient are obtained On the other hand the system friction coefficient
increases with the load for systems of plasticity index significantly higher than unity At
ψ = 186 the value of tmicro nearly doubles from 0056 to 0101 as the load increases from
5 10557 minustimes=tW to 4 10658 minustimes=tW Within the same load range the probability of
107
overall surface protection rtS prime decreases from nearly unity to 967 The probability of
unprotected contact at the system level ntS emerges and it is about 33 at the high end
of the load This probability is small but mainly contributed by the few asperities of large
heights which are in fully plastic deformation This group of asperities would carry a
significant portion of load if they are well protected by the boundary films However the
protection becomes damaged in these junctions and the shear stress approaches the shear
strength of the substrate As a result these asperities lose their load carrying capacity
causing the significant increase in the system friction coefficient With an even higher
plasticity index of ψ = 255 the friction coefficient at the system level increases
dramatically from 1520=tmicro to 5630=tmicro within a load range narrower than that for
the case of ψ = 186 Even under a relatively low load of 5 10557 minustimes=tW the system
friction coefficient is above rmicro = 015 which is the assumed shear strength-contact
pressure ratio of the reacted film At this load a close examination reveals that the
boundary lubrication fails in a significant number of asperity junctions The
corresponding value of the probability of surface protection is about 994=primertS The
probability decreases to about 70 for a higher load of 4 10984 minustimes=tW Many more
asperities lose their load capacity as the boundary films in these junctions are deteriorated
leading to the drastic increase of the friction which suggests a possibility of tribo-
instability
It should be pointed out that each of the above four groups of results is obtained
for a constant plasticity index In reality the continuous operation may change the
roughness of the bearing surfaces and the properties of the near-surface material leading
108
to an increasing or decreasing plasticity index A reduction of the plasticity index
corresponds to a healthy run-in process while an increase indicates some tribo-instability
For a given system the current model may be used to determine whether a run-in process
is needed by studying the friction behavior around the intended operating point If the
friction coefficient is sensitive to the operating parameters such as load or sliding velocity
the system should go through a run-in period at mild conditions to reduce its plasticity
index On the other hand the run-in may not be needed if the friction coefficient is
insensitive to the operating conditions as a result of the combined effects of boundary
lubricant material and surface finish
The behavior of the system friction with the load is rooted in the scattering
tribological behavior of distributed asperity contacts Figure 44 presents the shear stress
in an asperity junction as a function of asperity height the probability distribution
function of the asperity heights is also shown in the figure for reference The analysis is
performed for two systems of low and high plasticity indices ψ = 066 and ψ = 186 For
each system the results are presented at three values of the surface separation =σh 05
10 and 20 which are used to represent different levels of loading In the system with ψ
= 066 almost all the contacting asperities deform elastically for the three given values of
σh The asperity pressures are not very high and the areas of contact are relatively
small In these asperity junctions both the adsorbed layer and the reacted film are largely
intact The interfacial shear stress increases continuously with the asperity height and the
asperity-level friction coefficients are slightly higher than amicro = 005 At the given
nominal sliding velocity of =V 10ms only low flash temperatures are generated The
low pressure friction and flash temperature of the asperity contacts suggest that there is
109
no significant coupling among the deformation the frictional heating and the condition
of the boundary films The contacting asperities can thus be viewed as very stable At the
system level the resulting friction coefficient also has a value close to amicro = 005 and it is
almost independent of the load as shown in Fig 43 Next the tribological behavior of the
asperity contacts is examined for the relatively rough system of ψ = 186 When the
asperity height is below some critical value Figure 44 (b) shows that the shear stress in
the asperity junction also increases continuously with the height similar to the case of ψ =
066 The asperities in this group may be considered as stable For the asperities with a
height above a critical value the shear stress jumps to a value close to the shear strength
of the substrate A close examination of the results reveals that these asperities are in
fully plastic deformation as a result of the strong coupling among the physical and
chemical processes involved The frictional heating accelerates the thermal desorption of
the adsorbed layer and the rupture of the reacted film The damage of these films in turn
increases the interfacial shear stress as well as the frictional heating Consequently the
boundary films in these asperity junctions fail to provide effective protection The shear
stress then approaches the substrate shear strength and the asperity contact pressure is
largely reduced leading to a high asperity-level friction coefficient This group of
asperities may thus be considered as unstable The size of the group is measured by the
area ua shown in Fig 44 (c) which increases as the surface separation decreases The
above two groups of results show that the emergence of unstable contacting asperities
and their population are related to the value of the plasticity index and the load The
system tribological behavior is thus also affected by these two parameters In practice the
possible variation of the plasticity index during the operation may significantly change
110
the number of the unstable asperities For example a successful run-in process reduces
the plasticity index and pushes to the right the critical position of the shear stress-asperity
height relation shown in Fig 44 (b) The number of unstable asperities is reduced to a
low level so that they do not induce a tribo-instability to the system
It is interesting to examine how the condition of boundary lubrication may affect
the surface separation and the real area of contact of the system from the results of a
frictionless contact For illustration purposes the sliding velocity between the two
contacting surfaces is used to alter the condition of the boundary lubrication which may
be defined by the probability variable rtS prime of the overall boundary-film protection
Figure 45 present the rtS prime results as a function of the applied load for two sliding
velocities of =V 10ms and 40ms the separation gap of the surfaces and the real area
of contact are also presented under these conditions as well as for frictionless contacts At
a light load such as 3 10080 minustimes=tW the sliding velocity up to 40 ms has a negligible
effect on the boundary film and the value of rtS prime decreases only slightly from 999 to
987 as the sliding velocity increases from =V 10ms to =V 40ms Consequently
the calculated surface gap and the real area of contact are essentially the same as those
calculated assuming frictionless contact For heavier loads the sliding velocity may
increasingly deteriorate the boundary-film protection by thermal desorption of the
lubricant molecules adsorbed on the surface and by mechanical rupture of the reacted
surface film As a result the asperity load capacity may be reduced leading to a
significant decrease of the surface separation and significant increase of the real area of
contact Results in Fig 45 show that with a load of 3 1060 minustimes=tW the boundary-film
111
protection is 198=primertS with =V 10ms and decreases to 387=primertS when the
sliding velocity increases to =V 40ms For =V 10ms the gap between the two
surfaces is about the same as that for frictionless contact but it is reduced by about 27
when the system slides at =V 40ms Similar results are shown for the calculated real
area of contact With =V 40ms the area increases more than 50 from that for the
frictionless contact It should be pointed out that this increase is largely due to tangential
plastic flow of the asperity contacts that lose the boundary-film protection and it may
play a key role in the system tribo-instability An analysis of the contributions of the
tangential plastic flow to the real area of contact is presented in Chapter 3
The model may also be used to study the tribological behavior of the boundary
lubrication system in key parameter spaces The load and the sliding velocity are chosen
to define a key space since it is of particular interest to determine the limits of the two
operating parameters as guidelines for the design of tribological components [164 165]
Figure 46 presents the contours of the system friction coefficient tmicro and surface
protection probability rtS prime in this operating space The results show that the value of tmicro
increases with the two operating parameters and that of rtS prime decreases In addition a
given level of friction coefficient usually corresponds to a specific level of boundary
protection and is also related to a certain degree of plastic deformation
Considering 20=tmicro for example the corresponding value of the surface protection
probability is around 90=primertS and about 30 of the real area of contact is due to the
asperities in fully plastic deformation Based on experimental observations the surface
and subsurface plastic flow may precede scuffing a catastrophic system failure [43 165]
112
The scuffing may be more attributed to the tangential flow of the plastically deformed
asperities which may be measured by the contribution of the junction growth to the real
area of contact Corresponding to 20=tmicro this contribution is about 6 Thus the two
contour patterns shown in Fig 46 may be used to evaluate the tribo-severity of the
boundary lubrication system Accordingly the load-velocity plane may be divided into
two different regions In the high load-high velocity region the contours crowd together
and exhibit high gradients between adjacent levels The system may have a high
possibility of instability Left to this region this possibility decreases as the friction
coefficient and surface protection probability become insensitive to the two operating
parameters The transition regime between the above two regions may define the limits of
safe operation This transition regime has been related to the critical temperature for a
system in which the tendency to failure is controlled by the competitive formation and
removal of oxides [45] For a more general system considered in the current study the
transition regime may correspond to a critical level of plastic deformation or junction
growth which needs to be determined experimentally
It should also be mentioned that the above results are obtained for given bulk
temperature and surface plasticity index In reality the bulk temperature may be elevated
under high load andor high velocity since the system cooling in these severe situations is
not as effective as in the mild operations As a result the operating conditions may have
more dramatic effects on the system behavior in the high load-high velocity regime For
example the system friction coefficient may become even higher and its contours may be
more crowded compared to the results presented in Fig 47 (a) Separately the plasticity
index of the bearing surfaces may either increase or decrease during the operation The
113
pattern of the two types of contours and the region of high tribo-severity may thus change
accordingly Although limited by the lack of reliable data about the above two factors
more insight may be gained into their effects on the lubrication performance and the
effects of other factors through a systematic parametric study with the current model
Insights may also be gained by further developing the model considering the thermal
balance and the progression of surface topography
44 Summary
An asperity-based model is developed for the sliding contact of two rough
surfaces in boundary lubrication Four variables are used to describe an individual
asperity contact including micro-contact area pressure interfacial shear stress and flash
temperature Furthermore three probability variables are used to define the interfacial
state of the asperity junction The asperity-level modeling equations are derived from the
theories of contact mechanics flash temperature kinetics of boundary films and random-
process probability These equations are then used to formulate a contact model of the
surfaces by means of statistical integration Results from the model may be summarized
in the following
1) For relatively smooth and hard surfaces the boundary lubrication is effective at
both the asperity and system levels over a relatively wide range of load and
sliding velocity The resulting system friction coefficient is low and insensitive to
load and speed
2) For relatively rough and soft surfaces a significant group of contacting asperities
may lose boundary-film protection and experience a high level of local friction
114
At a given sliding velocity the number of these unstable asperities increases with
the load leading to a significant increase in the system friction coefficient
3) For a given system a friction coefficient sensitive to the operating parameters
suggests that the system should go through a run-in period to reduce the surface
plasticity index and thus the number of unstable asperity contacts On the other
hand the run-in may not be needed if this sensitivity is absent
4) The condition of boundary lubrication may strongly affect the system contact
behavior Under a given load an increase in the sliding velocity may deteriorate
the boundary-film protection leading to a significant decrease of the surface
separation and a significant increase of the real area of contact
5) The space of operating parameters may be divided into two regions according to
the tribo-severity evaluated from the contour pattern of the system friction
coefficient or the surface protection probability in this space The transition
between these two regions may be related to a critical degree of asperity plastic
deformation or junction growth
A more systematic parametric study can be conducted with the current model to
gain more insights into the effects of material and lubricant properties in boundary
lubrication The structure of the model is flexible enough for further development and
improvement by incorporating research advances in contact mechanics tribochemistry
and other related fields
115
Figure 41 An individual boundary-lubricated asperity contact
116
|error| lt ε
End
Initial guess of local contact probabilities
Start
Solve Pm Al and microl from Eqs (42) ndash (45)
Calculate ∆Tl with Eq (421)
Calculate Sa with Eq (48)
Calculate Sr with Eq (413)
Calculate Sa Sr and Sn with Eqs (414) (416) and (417)
Calculate τm with Eq (46)
error = τm ndash τm
Calculate τm with Eq (46)
τm = τm
Figure 42 Flowchart for the determination of the solution of an asperity collision
117
ψ = 066
ψ = 093
ψ = 186
ψ = 255
0 02 04 06 08 1
x 10-3
0
02
04
06
08
Figure 43 System-level friction coefficient as a function of load
( =V 10ms and =bT 50˚C)
tmicro
nt AEW lowast
118
hσ = 05
hσ = 10
hσ = 20 0
005
01
015
02
-1 0 2 4 60
01
02
03
04
05
Figure 44 Asperity shear stresses and asperity height distribution (a) ψ = 066 (b) ψ = 186 (c) asperity height distribution
( =V 10ms and =bT 50˚C)
z
nm ττ
nm ττ
0
02
04
06
08
1
-1 0 1 2 3 4 5 60
01
02
03
04
05
zσ
(b)
(a)
nm ττ
f(zσ)
Asperity height
Shea
r stre
ss
Shea
r stre
ss
Dis
tribu
tion
dens
ity
(c) au
119
0 02 04 06 08 1x 10-3
08
082
084
086
088
09
092
094
096
098
1
0 02 04 06 08 1x 10-3
05
1
15
2
0 02 04 06 08 1x 10-3
0
002
004
006
008
01
012
Figure 45 System-level contact and lubrication variables as functions of load (a) degree of boundary protection (b) surface separation (c) real area of contact
(ψ = 186 and =bT 50˚C)
σh
No-sliding
=V 10ms
=V 40ms
nt AEW lowast
nt AA
No-sliding =V 10ms
=V 40ms
(b)
(c)
nt AEW lowast
rtS prime
=V 10ms
=V 40ms
(a)
nt AEW lowast
120
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
01
01
01
01
02
02
02
03
03
03
04
04
05
06
0 2 4 6 8 10
1
2
3
4
5
6
7
8
9x 10-4
099
099
095
095
095
09
09
09
085
085
08
08
075
07
Figure 46 State of boundary lubrication in the operating parameter space
(a) system-level friction coefficient (b) system boundary-lubrication protection (ψ = 186 and =bT 50˚C)
(b) rtS prime
(a) tmicro
nt AEW lowast
V (ms)
V (ms)
nt AEW lowast
121
Chapter 5
Summary and Future Perspective
This thesis research develops an interdisciplinary surface contact model for
boundary lubrication systems based on a balanced consideration of key processes of
different natures involved in the contact The major efforts and conclusions of the
research are summarized below along with visions of future trends
51 The Deterministic-Statistical Model
The modeling process consists of three successive phases which are outlined as
follows
1) Finite Element Analysis of a Single Frictional Asperity Contact
A systematic finite element analysis is first carried out to study the effects of
friction on the deformation behavior of a single asperity contact The results show that
the friction in contact can significantly affect the mode of asperity deformation With a
relatively high friction coefficient the contact may change from the state of elastic
deformation to the state of fully plastic deformation with little elastic-plastic transition as
the contact force increases The friction can also significantly change the shape and size
of plastically deformed zone At high friction coefficients the plastic deformation is
largely confined to a thin surface layer in the contact In addition the friction causes the
reduction of pressure and the growth of asperity junction in the case of elastoplastic or
fully-plastic contact These results are presented in the dimensionless form and the
conclusions drawn from them are sufficiently general The insights gained in the analysis
122
are used in the second part as a foundation for the analytical modeling of frictional
asperity and surface contacts
2) A Elastic-Plastic Contact Model of Rough Surfaces with Friction
A statistical asperity-based model is developed for the frictional contact between
two nominally flat surfaces using the finite element results in the first part and the theory
of contact mechanics This model significantly advances the Greenwood-Williamson
types of system contact models by adding the dimension of friction as well as
incorporating the three possible modes of asperity deformation The model is able to
capture the essential effects of friction on the surface contact behavior These effects are
reflected by the reduction of surface separation and the increasing real area of contact
The model is also able to determine the contribution from the friction-induced junction
growth to the real area of contact The level of this contribution may be a measure of the
system tribo-instability Moreover the model provides a basis for further refinement and
development Although assuming a uniform friction coefficient at the interface it lays a
foundation for the study of boundary lubrication in which the friction may vary
dramatically among contacting asperities
3) A Deterministic-Statistical Model of the Boundary-Lubricated Surface Contact
The third part of the modeling process is the core of this thesis It models the
boundary-lubricated surface contact by incorporating the physicochemical and thermal
aspects of the problem into the mechanical contact model developed in the second part
In this interdisciplinary model an individual asperity contact under boundary lubrication
conditions is viewed as an event A group of deterministic and probabilistic variables are
123
defined or selected to characterize such a contact process or event The governing
equations for these variables are derived based on a balanced consideration of asperity
deformation frictional heating and the kinetics of boundary films These asperity-level
equations are solved iteratively and the solution is then integrated to formulate the
contact model for the boundary lubrication system This model is capable of relating the
system tribological behavior defined by the friction coefficient the real area of contact
and the effectiveness of boundary films to surface roughness operation conditions and
material and lubricant properties It is thus able to evaluate the safety of operation and the
tribo-stability through parametric study or sensitivity analysis regarding the range of
different factors Furthermore the modeling equations of asperity variables and their
solution as well as the statistical integration can be viewed as interrelated modules The
model is thus an open-ended framework allowing each module to be updated by
incorporating research advances in related fields Some possible directions of future
development are discussed in the next section
52 Perspective on Future Development
The final model developed in this thesis provides a tool to study the tribological
behavior of the boundary lubrication system in a greater depth of understanding than any
previous model One of the immediate applications of the model is a systematic
parametric study or sensitivity analysis on the effects of various important factors
involved in the boundary-lubricated contact An example is the analysis carried out in
Chapter 4 on the contour of the system friction coefficient and that of the degree of
boundary protection in the operation space defined by the load and sliding velocity
These contour patterns may reveal insights into the tribo-instability of the system and the
124
safety of operation More insights may be gained into these two issues by conducting
similar parametric study with the model on different groups of factors In this way the
coupling effects and relative importance of each group of factors can be easily identified
The insights provided by the parametric study may help define the guidelines for
controlling the tribo-severity
The model also provides a framework which may be refined or extended in many
different ways This framework is developed with a flexible structure consisting of a few
interrelated modules The model may thus be improved at the asperity level andor the
system level by updating individual modules and refining their interaction For example
the current model assumes that the asperity contacts are independent of each other and
they are not affected by previous ones Thus one way to improve the asperity-level
modeling is to consider the mechanical and thermal interaction among neighboring
asperity contacts The other way is to consider the cumulative effects of consecutive
contacts on the asperity flash temperature and the effectiveness of boundary lubrication
In addition the competition between the formation and the rupture or removal of the
boundary films may be considered to refine the model For this purpose it is important to
include in the model the up-to-date and balanced information about the properties and
behavior of these films At the system level the surface plasticity index and the bulk
temperature are currently taken to be fixed parameters In reality they may either
increase or decrease during the contact process depending on the operation conditions
material properties and other factors Their evolution may significantly affect the
dominant deformation mode of contacting asperities and the state of boundary
125
lubrication Therefore a possible extension is to capture the trends of evolution by
modeling the global thermal balance and the progression of surface topography
The further development of the model may be related to its structure which is
characterized by the way to describe the surface topography The current model combines
the statistical surface descriptions with the ability to take account of interactive micro-
mechanical physicochemical and thermal processes involved in the contact This ability
is the core of the model and it may also be combined with the fractal or deterministic
types of surface descriptions to develop the corresponding surface contact models
Moreover a contact model of a totally new structure may be developed by viewing the
interfacial contact region as a network whose nodes are the asperity junctions From the
network point of view the system failure damage such as scuffing may be taken to be the
catastrophic collapse starting from a small number of nodes As summarized by Johnson
[166] many social artificial and natural networks crash in such a way These complex
systems have also been found to be similar in their structures and inter-node linkages
following some universal organizational principles The contact model of network
structure may open a new window to the boundary lubrication system and then lead to a
more insightful understanding of its failure mode and tribo-severity
126
Bibliography
1 Bhushan B 2001 ldquoTribology on the Macroscale to Nanoscale of Microelectro-mechanical System Materials a Reviewrdquo Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 215 (J1) 1-18
2 Marchon B 2002 ldquoThe Physics of Boundary Lubrication at the HeadDisk
Interfacerdquo Boundary and Mixed Lubrication Science and Application Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 217-225
3 Podgornik B Jacobson S and Hogmark S 2003 ldquoDLC Coating of Boundary
Lubricated Components ndash Advantages of Coating One of the Contact Surfaces Rather than Both or Nonerdquo Tribology International 36 (11) 843-849
4 BNJ Persson 1998 Sliding Friction Physical Principles and Applications
Springer-Verlag Berlin 5 Kotvis P V Lara J Surerus K and Tysoe W T 1996 ldquoThe Nature of the
Lubricating Films Formed by Carbon Tetrachloride under Conditions of Extreme Pressurerdquo Wear 201 (1-2) 10-14
6 Hardy W B and Doubleday I 1922 ldquoBoundary Lubrication ndash The Paraffin
Seriesrdquo Proc R Soc London Ser A 100 (707) 550-574 7 Bowden F P and Tabor D 1950 Friction and Lubrication of Solids Part I
Clarendon Press Oxford UK 8 Zisman W A 1959 ldquoDurability and Wettability Properties of Monomolecular Films
of Solidsrdquo Friction and Wear (ed R Davies) Elsevier Amsterdam the Netherlands pp 110-148
9 Jahanmir S 1985 ldquoChain Length Effects in Boundary Lubricationrdquo Wear 102 (4)
331-349 10 Studt P 1981 ldquoThe Influence of the Structure of Isomeric Octadecanols on their
Adsorption from Solution on Iron and their Lubricating Propertiesrdquo Wear 70 (3) 329-334
11 Jahanmir S and Beltzer M 1986 ldquoAn Adsorption Model for Friction in Boundary Lubricationrdquo ASLE Transactions 29 (3) 423-430
12 Godfrey D 1965 ldquoLubrication mechanism of tricresyl phosphate on steelrdquo ASLE
Transactions 8 (1) 1-11
127
13 Jahanmir S and Beltzer M 1986 ldquoEffect of Additive Molecular Structure on Friction Coefficient and Adsorptionrdquo ASME Journal of Tribology 108 (1) 109-116
14 Frewing J J 1944 ldquoThe Heat of Adsorption of Long-Chain Compounds and Their
Effect on Boundary Lubricationrdquo Proc R Soc London Ser A 182 (990) 270-285 15 Askwith T C Cameron A and Crouch R F 1966 ldquoChain Length of Additives in
Relation to Lubricants in Thin Film and Boundary Lubricationrdquo Proc R Soc London Ser A 291 (1427) 500-519
16 Rowe C N 1966 ldquoSome Aspects of the Heat of Adsorption in the Function of a
Boundary Lubricantrdquo ASLE Transactions 9 100-111 17 Langmuir I 1918 ldquoThe Adsorption of Gases on Plane Surfaces of Glass Mica and
Platinumrdquo Journal of American Chemistry Society 40 1361-1402 18 Grew W J S and Cameron A 1972 ldquoThermodynamics of Boundary Lubrication
and Scuffingrdquo Proc R Soc London Ser A 327 (1568) 47-57 19 Biresaw G Adhvaryu A Erhan S Z and Carriere C J 2002 ldquoFriction and
Adsorption Properties of Normal and High-Oleic Soybean Oilsrdquo Journal of the American Oil Chemistsrsquo Society 79 (1) 53-58
20 Kingsbury E P 1958 ldquoSome Aspects of the Thermal Desorption of a Boundary
Lubricantrdquo Journal of Applied Physics 29 (6) 888-891 21 Bowden F P Gregory J N and Tabor D 1945 ldquoLubrication of Metal Surfaces
by Fatty Acidsrdquo Nature (London) 156 (3952) 97-101 22 Bailey A I and Courtney-Pratt J S 1955 ldquoThe Area of Real Contact and the
Shear Strength of Monomolecular Layers of a Boundary Lubricantrdquo Proc R Soc London Ser A 227 (1171) 500-515
23 Israelachvili J N 1973 ldquoThin Film Studies Using Multiple-Beam Interferometryrdquo
Journal of Colloid and Interface Science 44 (2) 259-272 24 Israelachvili J N and Tabor D 1973 ldquoThe Shear Properties of Molecular Filmsrdquo
Wear 24 (3) 386-390 25 Briscoe B J and Evans D C B 1982 ldquoThe Shear Properties of Langmuir-
Blodgett Layersrdquo Proc R Soc London Ser A 380 (1779) 389-407 26 Timsit R S and Pelow C V 1992 ldquoShear Strength and Tribological Properties of
Stearic Acid Film ndash Part I on Glass and Aluminum Coated Glassrdquo ASME Journal of Tribology 114 (1) 150-158
128
27 Williams J A 2002 ldquoAdvances in the Modeling of Boundary Lubricationrdquo Boundary and Mixed Lubrication Proceedings of the 28th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 37-48
28 Sutcliffe M J Taylor S R and Cameron A 1978 ldquoMolecular asperity theory of
boundary frictionrdquo Wear 51 (1) 181-192 29 Sethuramiah A 2003 Lubricated Wear Science and Technology (Tribology Series
42) Elsevier Amsterdam the Netherlands 30 Pawlak Z 2003 Tribochemistry of Lubricating Oils (Tribology Series 45) Elsevier
Amsterdam the Netherlands 31 Quinn T F J 1983a ldquoReview of Oxidational Wear ndash Part I Recent Developments
and Future Trends in Oxidational Wear Researchrdquo Tribology International 16 (5) 257-271
32 Gellman A J and Spencer N D 2002 ldquoSurface Chemistry in Tribologyrdquo
Proceedings of the Institution of Mechanical Engineers Part J Journal of Engineering Tribology 216 (J6) 443-461
33 Georges J-M 1997 ldquoSome Surface Science Aspects of Tribologyrdquo New Directions
in Tribology (ed I M Hutchings) Mechanical Engineering Pub Bury St Edmunds UK pp 67-82
34 Barnes A M Bartle K D and Thibon V R A 2001 ldquoA Review of Zinc
Dialkyldithiophosphates (ZDDPS) Characterisation and Role in the Lubricating Oilrdquo Tribology International 34 (6) 389-395
35 Ratoi M Anghel V Bovington C H and Spikes H A 2000 ldquoMechanisms of
oiliness additivesrdquo Tribology International 33 (3-4) 241-247 36 Randles S J Roberts A J and Cain R B 1991 ldquoEnvironmentally Considerate
Lubricants for the Automotive and Engineering Industriesrdquo Chemicals for the Automotive Industry (ed J A G Drake) the Royal Society of Chemistry Special Publication no 93 pp 165-178
37 Cavdar B and Ludema K C 1991 ldquoDynamics of Dual Film Formation in
Boundary Lubrication of Steels ndash Part I Functional Nature and Mechanical Propertiesrdquo Wear 148 (2) 305-327
38 Hsu S M 1997 ldquoBoundary Lubrication Current Understandingrdquo Tribology Letters
3 (1) 1-11 39 Batchelor A W and Stachowiak G W 1986 ldquoSome Kinetic Aspects of Extreme
Pressure Lubricationrdquo Wear 108 (2) 185ndash199
129
40 Hsu S M 2003 ldquoMolecular Basis of Lubricationrdquo Tribology International (article
in press) 41 Bec S Tonck A Georges J-M Coy R C Bell J C and Roper G W 1999
ldquoRelationship between Mechanical Properties and Structures of Zinc Dithiophosphate Anti-Wear Filmsrdquo Proc R Soc London Ser A 455 (1992) 4181-4203
42 Sethuramiah A Okabe H and Sakurai T 1973 ldquoCritical Temperatures in EP
Lubricationrdquo Wear 26 (2) 187ndash206 43 Ludema KC 1984 ldquoA Review of Scuffing and Running-in of Lubricated Surfaces
with Asperities and Oxides in Perspectiverdquo Wear 100 (1-3) 315ndash331 44 Batchlor AW Stachowiak G W and Cameron A 1986 ldquoThe Relationship
between Oxide Films and the Wear of Steelsrdquo Wear 113 (2) 203-223 45 Cutiongco E C and Chung Y W 1994 ldquoPrediction of Scuffing Failure Based on
Competitive Kinetics of Oxide Formation and Removal - Application to Lubricated Sliding of AISI-52100 Steel on Steelrdquo Tribology Transactions 37 (3) 622-628
46 Wang L Y Yin Z F Zhang J Chen C-I and Hsu S 2000 ldquoStrength
measurement of thin lubricating filmsrdquo Wear 237 (2) 155-162 47 Zhang C Cheng H S and Wang Q J 2004 ldquoScuffing behavior of piston-pinbore
bearing in mixed lubrication - Part II Scuffingrdquo Tribology Transactions 47 (1) 149-156
48 Hsu SM and Klaus EE 1979 ldquoSome chemical effects in boundary lubrication Part I Base oilndashmetal interactionrdquo ASME Transactions 22 (2) 135-145
49 Hsu S M and Zhang X H 1996 ldquoLubrication Traditional to Nano-lubricating
Filmsrdquo Micro-Nanotribology and Its Applications Proceedings of the NATO Advanced Study Institutes (ed B Bhushan) Kluwer Academic Boston MA pp 399-411
50 Cherepanov G P 1997 Methods of Fracture Mechanics Solid Matter Physics
Kluwer Academic Publishers Dordrecht the Netherlands 51 Tonck A Kapsa P Sabot 1986 ldquoMechanical-Behavior of Tribochemical Films
under a Cyclic Tangential Load in a Ball-Flat Contactrdquo ASME Journal of Tribology 108 (1) 117-122
52 Warren O L Graham J F Norton PR Houston J E and Milchaske TA
1998 ldquoNanomechanical Properties of Films Derived from Zincdialkyldithio-phosphaterdquo Tribology Letters 4 (2) 189-198
130
53 Graham J F McCague C and Norton P R 1999 ldquoTopography and Nano-
mechanical Properties of Tribochemical Films Derived from Zinc Dalkyl and Diaryl Dithiophosphatesrdquo Tribology Letters 6 (3-4) 149-157
54 Ye J P Kano M and Yasuda Y 2002 ldquoEvaluation of Local Mechanical
Properties in Depth in MoDTCZDDP and ZDDP Tribochemical Reacted Films Using Nanoindentationrdquo Tribology Letters 13 (1) 41-47
55 Aktary M McDermott M T and McAlpine G A 2002 ldquoMorphology and
nanomechanical properties of ZDDP antiwear films as a function of tribological contact timerdquo Tribology Letters 12 (3) 155-162
56 Pidduck A J and Smith G C 1997 ldquoScanning Probe Microscopy of Automotive
Anti-Wear Filmsrdquo Wear 212 (2) 254-264 57 Miklozic K T Graham J and Spikes H 2001 ldquoChemical and Physical Analysis
of Reaction Films Formed by Molybdenum Dialkyl-dithiocarbamate Friction Modifier Additive Using Raman and Atomic Force Microscopyrdquo Tribology Letters 11 (2) 71-81
58 Bhushan B 1998 ldquoContact Mechanics of Rough surfaces in Tribology Multiple
Asperity Contactrdquo Tribology Letters 4 (1) 1-35 59 Greenwood J A and Williamson J B P 1966 ldquoContact of Nominally Flat
Surfacesrdquo Proc R Soc London Ser A 295 (1442) 300-319 60 Sayles R S and Thomas T R 1979 ldquoMeasurements of the Statistical Micro-
geometry of Engineering Surfacesrdquo ASME Journal of Lubrication Technology 101(4) 409-417
61 Bhushan B Wyant J C and Meiling J 1988 ldquoA New Three-Dimensional Non-
Contact Digital Optical Profilerrdquo Wear 122 (3) 301-312 62 Greenwood J A 1992 ldquoProblems with Surface Roughnessrdquo Fundamentals of
Friction Microscopic and Microscopic Processes (ed I L Singer et al) Kluwer Academic Boston MA pp 57-76
63 Majumdar A and Bhushan B 1990 ldquoRole of Fractal Geometry in Roughness
Characterization and Contact Mechanics of Rough Surfacesrdquo ASME Journal of Tribology 112 (2) 205ndash216
64 Ganti S and Bhushan B 1996 ldquoGeneralized Fractal Analysis and Its Applications
to Engineering Surfacesrdquo Wear 180 (1) 17ndash34
131
65 Majumdar A and Bhushan B 1991 ldquoFractal Model of ElasticndashPlastic Contact between Rough Surfacesrdquo ASME Journal of Tribology 113 (1) 1ndash11
66 Bhushan B and Majumdar A 1992 ldquoElasticndashPlastic Contact Model of Bi-Fractal
Surfacesrdquo Wear 153 (1) 53ndash64 67 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part I ndash Elastic Contact and Heat Transferrdquo ASME Journal of Tribology 116 (4) 812-822
68 Wang S and Komvopoulos K 1994 ldquoA Fractal Theory of the Interfacial
Temperature Distribution in the Slow Sliding Regime Part II ndash Multiple Domains Elastoplastic Contact and Applicationrdquo ASME Journal of Tribology 116 (4) 824-832
69 Yan W and Komvopoulos K 1998 ldquoContact Analysis of Elastic-Plastic Fractal
Surfacesrdquo Journal of Applied Physics 84 (7) 3617-3624 70 MN Webster and RS Sayles 1986 ldquoA Numerical Model for the Elastic Frictionless
Contact of Real Rough Surfacesrdquo ASME Journal of Tribology 108 (3) 314ndash320 71 Ren N and Lee S C 1993 ldquoContact Simulation of Three-Dimensional Rough
Surfaces Using Moving Grid Methodrdquo ASME Journal of Tribology 116 (4) 597ndash601 72 S Bjoumlrklund and S Andersson 1994 ldquoA Numerical Method for Real Elastic
Contacts Subjected to Normal and Tangential Loadingrdquo Wear 179 (1-2) 117ndash122 73 Mayeur C Sainsot P and Flamand L 1995 ldquoNumerical Elastoplastic Model for
Rough Contactrdquo ASME Journal of Tribology 117 (3) 422-429 74 Lee SC and Ren N 1996 ldquoBehavior of Elastic-Plastic Rough Surface Contacts as
Affected by Surface Topography Load and Material Hardnessrdquo Tribology Transactions 39 (1) 67ndash74
75 Yu M M H and Bushan B 1996 ldquoContact Analysis of Three-Dimensional Rough
Surfaces under Frictionless and Frictional contactrdquo Wear 200 (1-2) 265ndash280 76 Kalker J J Dekking F M Vollebregt E A H 1997 ldquoSimulation of Rough
Elastic Contactsrdquo ASME Journal of Mechanics 64 (2) 361ndash368 77 Sui PC 1997 ldquoAn Efficient Computation Model for Calculating Surface Contact
Pressures using Measured Surface Roughnessrdquo Tribology Transactions 40 (2) 243-250
78 Tian X and Bhushan B 1996 ldquoA Numerical Three-Dimensional Model for the
Contact of Rough Surfaces by Variational Principlerdquo ASME Journal of Tribology 118 (1) 33ndash42
132
79 Johnson K L (1985) Contact Mechanics Cambridge University Press Cambridge 80 Sackfield A and Hills D 1983 ldquoSome Useful Results in the Tangentially Loaded
Hertzian Contact Problemrdquo Journal of Strain Analysis 18 (2) 107-110 81 Johnson K L and Jefferis J A 1963 ldquoPlastic Flow and Residual Stresses in
Rolling and Sliding Contactrdquo Symposium on Fatigue Rolling Contact the Institution of Mechanical Engineers pp 54 -65
82 Hills D A and Ashelby D W 1982 ldquoThe Influence of Residual Stresses on
Contact Load Bearing Capacityrdquo Wear 75 (2) 221-240 83 Chang W R 1997 ldquoAn Elastic-Plastic Contact Model for a Rough Surface with an
Ion-Plated Soft Metallic Coatingrdquo Wear 212 (2) 229-237 84 Zhao Y Maietta D and Chang L 2000 ldquoAn Asperity Micro-Contact Model
Incorporating the Transition from Elastic Deformation to Fully Plastic Flowrdquo ASME Journal of Tribology 122 (1) 86-93
85 Kogut L and Etsion I 2003 ldquoA finite element based elastic-plastic model for the
contact of rough surfacesrdquo Tribology Transactions 46 (3) 383-390 86 Parker R C and Hatch D 1950 ldquoThe Static Friction Coefficient and the Area of
Contactrdquo Proc Phys Soc Sec B 63 (3) 185-197 87 McFarlane J F and Tabor D 1950 ldquoAdhesion of Solids and the Effect of Surface
Filmsrdquo Proc R Soc London Ser A 202 (1069) 224-243 88 McFarlane J F and Tabor D 1950 ldquoRelation between Friction and Adhesionrdquo
Proc R Soc London Ser A 202 (1069) 244-253 89 Tabor D 1959 ldquoJunction Growth in Metallic Friction the Role of Combined
Stresses and Surface Contaminationrdquo Proc R Soc London Ser A 251 (1266) 378-393
90 Green A P 1954 ldquoPlastic Yielding of Metal Junctions due to Combined Shear and
Pressurerdquo Journal of Mechanics and Physics of Solids 2 (8) 197-211 91 Green A P 1955 ldquoFriction between Unlubricated Metals a Theoretical Analysis of
the Junction Modelrdquo Proc R Soc London Ser A 228 (1173) 191-204 92 Johnson K L 1968 ldquoDeformation of a Plastic Wedge by a Rigid Flat Die under the
Action of a Tangential Forcerdquo Journal of the Mechanics and Physics of Solids 16 (6) 395-402
133
93 Collins I F 1980 ldquoGeometrically Self-Similar Deformations of a Plastic Wedge under Combined Shear and Compression Loading by a Rough Flat Dierdquo International Journal of Mechanical Sciences 22 (12) 735-742
94 Challen J M and Oxley P L B 1979 ldquoDifferent Regimes of Friction and Wear
Using Asperity Deformation Modelsrdquo Wear 53 (2) 229-243 95 Lisowski Z and Stolarski T 1981 ldquoAn Analysis of Contact between a Pair of
Surface Asperities during Slidingrdquo ASME Journal of Applied Mechanics 48 (3) 493-499
96 Edwards C M and Halling J (1968) ldquoAn Analysis of the Interaction of Surface
Asperities and Its Relevance to the Value of the Coefficient of Frictionrdquo Journal of Mechanical Engineering Science 10 (2) 101-121
97 Ogilvy J A 1991 ldquoNumerical Simulation of Friction between Contacting Rough
Surfacesrdquo Journal of Physics D Applied Physics 24 (11) 2098-2109 98 Ogilvy J A 1993 ldquoPredicting the friction and durability of MoS2 Coatings using a
Numerical Contact Modelrdquo Wear 160 (1) 171-180 99 Francis H A 1977 ldquoApplication of Spherical Indentation Mechanics to Reversible
and Irreversible Contact between Rough Surfacesrdquo Wear 45 (2) 221-269 100 Williams J A and Xie Y 1996 ldquoFriction of Sliding Surfaces Carrying
Adsorbed Lubricant Layersrdquo the Third Body Concept Interpretation of Tribological Phenomena Proceedings of the 22nd Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 651-664
101 Blencoe K A and Williams J A 1997 ldquoFriction of Sliding Surfaces Carrying
Boundary filmsrdquo Wear 203-204 722-729 102 Bressan J D Genin G M and Williams J A 1999 ldquoThe Influence of
Pressure Boundary Film Shear Strength and Elasticity on the Friction Between a Hard Asperity and a Deforming Softer Surfacerdquo Lubrication at the Frontier Proceedings of the 25th Leeds-Lyon Symposium on Tribology (ed D Dowson et al) Elsevier Amsterdam the Netherlands pp 79-90
103 Ford I J 1993 ldquoRoughness effect on friction for multi-asperity contact between
surfacesrdquo Journal of Physics D Applied Physics 26 (12) 2219ndash2225 104 Tworzydlo WW Cecot W Oden JT and Yew CH 1998 ldquoComputational
Micro- and Macroscopic Models of Contact and Friction Formulation Approach and Applicationsrdquo Wear 220 (2) 113ndash140
134
105 Karpenko Y A and Akay A 2001 ldquoA numerical model of friction between rough surfacesrdquo Tribology International 34 (8) 531-545
106 Blok H 1937 ldquoTheoretical Study of Temperature Rise at Surface of Actual
Contact under Oiliness Lubrication Condition General Discussion on Lubricationrdquo General Discussion of Lubrication Proceedings of the Institution of Mechanical Engineers 2 222-235
107 Jaeger J C 1942 ldquoMoving Sources of Heat and the Temperature at Sliding
Contactsrdquo Proc R Soc New South Wales 76 203-224 108 Archard J F 1958-1959 ldquoThe Temperature of Rubbing Surfacesrdquo Wear 2 (6)
438-455 109 Ling F F and Pu S L 1964 ldquoProbable Interface Temperatures of Solids in
Sliding Contactrdquo Wear 7 (1) 23-34 110 Francis H A 1971 ldquoInterfacial Temperature Distribution within a Sliding
Hertzian Contactrdquo ASLE Transactions 14 (1) 41-54 111 Barber J R 1970 ldquoThe Conduction of Heat from Sliding Solidsrdquo International
Journal of Heat and Mass Transfer 13 (5) 857-869 112 Gecim B and Winer W O 1985 ldquoTransient Temperatures in the Vicinity of an
Asperity Contactrdquo ASME Journal of Tribology 107 (3) 333ndash342 113 Kuhlmann-Wilsdorf D ldquoSample Calculations of Flash Temperatures at a Silver-
Graphite Electric Contact Sliding on Copperrdquo Wear 107 (1) 71-90 114 Bhushan B 1987 ldquoMagnetic Head-Media Interface Temperatures Part 1 ndash
Analysisrdquo ASME Journal of Tribology 109 (2) 243ndash251 115 Tian X and Kennedy F E 1994 ldquoMaximum and Average Flash Temperatures
in Sliding Contactsrdquo ASME Journal of Tribology 116 (1) 167-174 116 Yevtushenko A A and Ivanyk E G 1995 ldquoStochastic Contact Model of
Rough Frictional Heating Surfaces in Mixed Friction Conditionsrdquo Wear 188 (1-2) 49-55
117 Qiu L and Cheng H S 1998 ldquoTemperature Rise Simulation of Three-
Dimensional Rough Surfaces in Mixed Lubricated Contactrdquo ASME Journal of Tribology 120 (2) 310-318
118 Vick B and Furey M J 2001 ldquoA Basic Theoretical Study of the Temperature
Rise in Sliding Contact with Multiple Contactsrdquo Tribology International 34 (12) 823-829
135
119 Zhang H Chang L Webster M N and Jackson A 2003 A Micro-Contact
Model for Boundary Lubrication with LubricantSurface Physicochemistry ASME Journal of Tribology 125 (1) 8-15
120 Komvopoulos K 1991 ldquoSliding Friction Mechanisms of Boundary Lubricated
Layered Surfaces Part IIndashndashTheoretical Analysisrdquo STLE Tribology Transactions 34 (2) 281ndash291
121 MT Bengisu and A Akay 1997 ldquoRelation of Dry-Friction to Surface
Roughnessrdquo ASME Journal of Tribology 119 (1)18ndash25 122 Johnson K L Greenwood J A and Poon S Y 1972 ldquoA Simple Theory of
Asperity Contact in Elastohydrodynamic Lubricationrdquo Wear 19 (1) 91-108 123 Gui J and Marchon B 1995 ldquoA Stiction Model for a Head-Disk Interface of a
Rigid-Disk Driverdquo Journal of Applied Physics 78 (6) 4206-4217 124 Zhao Y and Chang L 2002 ldquoA Micro-Contact and Wear Model for Chemical-
Mechanical Polishing of Silicon Wafersrdquo Wear 252 (3-4) 220-226 125 Poritsky H and Schenectady N Y 1950 ldquoStresses and Deflection of Cylindrical
Bodies in Contact with Application to Contact of Gears and of Locomotive Wheelsrdquo ASME Journal of Applied Mechanics 17 191-201
126 Smith J O and Liu C K 1953 ldquoStresses Due to Tangential and Normal Loads
on an Elastic Solidrdquo ASME Journal of Applied Mechanics 20 157-166 127 Hamilton G M and Goodman L E 1966 ldquoThe Stress Field Created by a
Circular Sliding Contactrdquo ASME Journal of Applied Mechanics 33 371-376 128 Hamilton G M 1983 ldquoExplicit Equations for the Stresses beneath a Sliding
Spherical Contactrdquo Proceedings of the Institution of Mechanical Engineers Part C Mechanical Engineering Science 197 53-59
129 Tian H and Saka N 1991 ldquoFinite-Element Analysis of an Elastic-Plastic 2-
Layer Half-Space Sliding Contactrdquo Wear 148 (2) 261-285 130 Kral E R and Komvopoulos K 1996 ldquoThree-Dimensional Finite Element
Analysis of Surface Deformation and Stresses in an Elastic-Plastic Layered Medium Subjected to Indentation and Sliding Contact Loadingrdquo ASME Journal of Applied Mechanics 63 (2) 365-375
131 Tangena A G and Wijnhoven P J M 1985 ldquoFinite Element Calculations on
the Influence of Surface Roughness on Frictionrdquo Wear 103 (4) 345-354
136
132 Faulkner A and Arnell R D (2000) ldquoThe Development of a Finite Element Model to Simulate the Sliding Interaction Between Two Three-Dimensional Elastoplastic Hemispherical Asperitiesrdquo Wear 114 (1-2) 114-122
133 Nagaraj H S 1984 ldquoElastoplastic Contact of Bodies with Friction under Normal
and Tangential Loadingrdquo ASME Journal of Tribology 106 (4) 519 ndash 526 134 ABAQUS 2000 V62 Userrsquos Manual Pawtucket RI Hibbitt Karlsson amp
Sorensen Inc 135 Irving H S and Francis A C 1992 Elastic and Inelastic Stress Analysis
Prentice Hall Englewood Cliffs NJ 136 Mesarovic S D J and Fleck N A 1999 ldquoSpherical Indentation of Elastic-
Plastic Solidsrdquo Proc R Soc London Ser A 455 (1987) 2707-2728 137 Kogut L and Etsion I 2002 ldquoElastic-Plastic Contact Analysis of a Sphere and
a Rigid Flatrdquo ASME Journal of Applied Mechanics 69 (5) 657-662 138 McCool J I 1986 ldquoComparison of Models for the Contact of Rough Surfacesrdquo
Wear 107 (1) 37-60 139 Handzel-Powierza Z Klimczak T and Polijaniuk A 1992 ldquoOn the
Experimental Verification of the Greenwood-Williamson Model for the Contact of Rough Surfacesrdquo Wear 154 (1) 115-124
140 Whitehouse D J and Archard J F 1970 ldquoThe Properties of Random Surfaces
of Significance in their Contactrdquo Proc R Soc London Ser A 316 (1524) 97-121 141 Bush A W Gibson R D and Thomas T R 1975 ldquoThe Elastic Contact of a
Rough Surfacerdquo Wear 35 (1) 15-20 142 Bush A W Gibson R D and Keogh G P 1979 ldquoStrongly Anisotropic
Rough Surfacesrdquo ASME Journal of Lubrication Technology 101 (1) 15-20 143 McCool J I and Gassel S S 1981 ldquoThe Contact of Two Rough Surfaces
having Anisotropic Roughness Geometryrdquo Proceedings of the ASLE Energy Sources Technology Conference ASLE Special Publication Sp-7 pp 29-38
144 Chang W R Etsion I and Bogy DP 1987 ldquoAn Elastic-Plastic Model for the
Contact of Rough Surfacesrdquo ASME Journal of Tribology 109 (2) 257-263 145 Chang W R Etsion I And Bogy D B 1988 ldquoStatic Friction Coefficient
Model for Metallic Rough Surfacesrdquo ASME Journal of Tribology 110 (1) 57-63
137
146 Francis H A 1976 ldquoPhenomenological Analysis of Plastic Spherical Indentationrdquo ASME Journal of Engineering Materials and Technology 76 (2) 272-281
147 Abbott EJ and Firestone FA 1933 ldquoSpecifying Surface Quality ndash A Method
Based on Accurate Measurement and Comparisonrdquo Mechanical Engineering 55 (9) 569-572
148 Jeng Y R and Wang P Y 2003 ldquoAn Elliptical Microcontact Model
Considering Elastic Elastoplastic and Plastic Deformationrdquo ASME Journal of Tribology 125 (2) 232-240
149 Kayaba T and Kato K 1978 ldquoTheoretical Analysis of Junction Growthrdquo
Technology Report Tohoku University 43 (1) 1-10 150 Nayak P R 1971 ldquoRandom Process Model of Rough Surfacerdquo ASME Journal
of Lubrication Technology 93(3) 398-407 151 McFadden C F and Gellman A J 1998 ldquoMetallic friction the effect of
molecular adsorbatesrdquo Surface Science 409 (2) 171-182 152 Nuri K A and Halling J 1975 ldquoThe Normal Approach between Rough Flat
Surfaces in Contactrdquo Wear 32 (1) 81-93 153 Shpenkov G P 1995 Friction Surface Phenomena (Tribology Series 29)
Elsevier Amsterdam the Netherlands 154 Zimmermann H J 2001 Fuzzy Set Theory and Its Application (fourth edition)
Kluwer Academic Publishers Boston MA 155 Zhurkov S N 1965 ldquoKinetic Concept of the Strength of Solidsrdquo International
Journal of Fracture Mechanics 1 (4) 311-323 156 Johnson R A 2000 Probability and Statistics for Engineers (sixth edition)
Prentice-Hall Upper Saddle River NJ 157 Hu Z S Hsu S M and Wang P S 1992 ldquoTribochemical and
Thermochemical Reactions of Stearic-Acid on Copper Surfaces Studied by Infrared Microspectroscopyrdquo Tribology Transactions 35 (1) 189-193
158 Su Y Y 1997 ldquoElectrochemical study of the interaction between fatty acid and
oxidized copperrdquo Tribology International 30 (6) 423-428 159 Tompkins L S 1978 Chemisorption of Gases on Metals Academic Press
London
138
160 Denis J Briant J and Hipeaux J-C 2000 Lubricant Properties Analysis amp Testing Editions Technip Paris
161 Belin M Martin J M Amnsot J L Dexpert H and Lagarde P 1984
ldquoMixed Lubrication with a Complex Ester as a Friction Modifierrdquo ASLE Transactions 27 (4) 398-404
162 Gates R S Jewett K L and Hsu S M 1989 ldquoA Study on the Nature
of Boundary Lubricating Film Analytical Method Developmentrdquo Tribology Transactions 32 (4) 423-430
163 Ashby M F and Jones D R H 1980 Engineering Materials a Introduction
to Their Properties and Applications Pergamon Press Oxford 164 Yang Z and Chung Y 1997 ldquoSurface Science Perspective of Tribological
Failurerdquo Tribology Letters 3 (1) 19-26 165 Sheiretov T Yoon H and Cusano C 1998 ldquoScuffing under Dry Sliding
Conditions ndash Part I Experimental Studiesrdquo Tribology Transactions 41 (4) 435ndash446 166 Johnson G 2000 ldquoFirst Cells Then Species Now the Webrdquo The New York
Times Company httpwwwracemattersorgcomplexsystemshtm
VITA
Huan Zhang received his BS and MS in Engineering Mechanics from Jiaotong
University Xirsquoan China in 1990 and 1993 respectively He then worked as a lecturer in
the School of Power and Energy Technology in Jiaotong University Xirsquoan
In August 1999 the author came to the Pennsylvania State University for the
PhD program in Mechanical Engineering He has been a Graduate Research Assistant in
the Tribology Group since then He also worked as a Graduate Teaching Fellow for one
semester
Huan Zhang is a student member of STLE (the Society of Tribologist and
Lubrication Engineers)
Recommended