Contact mechanics I: basics - mms2 - MINES ParisTech

Contact mechanics I: basics

Georges Cailletaud1 Stephanie Basseville1,2

Vladislav A. Yastrebov1

1Centre des Materiaux, MINES ParisTech, CNRS UMR 76332Laboratoire d’Ingenierie des Systemes de Versailles, UVSQ

WEMESURF short course on contact mechanics and tribology

Paris, France, 21-24 June 2010

Introduction Basic knowledges Elastic contact Inelastic contact Contact of composites

Table of contents

1 Introduction

2 Basic knowledges

3 Contact mechanics of elastic solids

4 Normal contact of inelastic solids

5 Contact of inhomogeneous bodies

G. Cailletaud, S. Basseville, V.A. Yastrebov — MINES ParisTech, UVSQ Contact mechanics I Paris, 21-24 June 2010 2/68


Plan

1 Introduction

2 Basic knowledges






Short historical sketch

Use and opposition to friction

Frictional heat - lighting of fire - more than [40 000 years ago].

Ancient Egypt -lubrication of surfaces with oil [5 000 years ago].




First studies on contact and friction

Leonardo da Vinci [1452-1519]first friction laws and manyother trobological topics;

Issak Newton [1687]Newton’s third law for bodiesinteraction;

Guillaume Amontons [1699]rediscovered firction laws;

Leonhard Euler [1707-1783]roughness theory of friction;

From Leonardo da Vinci’s notebook

Roughness theory of friction




First studies on contact and friction

Charles-Augustin de Coulomb [1789]friction independence on slidingvelocity and roughness; the influence ofthe time of repose.

Heinrich Hertz [1881-1882]the first study on contact ofdeformable solids;

Holm [1938],Ernst and Merchant [1940],Bowden and Tabon [1942]difference between apparent and realcontact areas, adhesion theory.

Photoelasticity analysis of Hertzcontact problem (shear stresses)

Apparent and real areas of contact



Practice VS theory

1900: Theory is several steps behind the practice

Theory Practice



Practice VS theory

1940: Theory is behind the practice

Theory Practice



Practice VS theory

1960: Theory catchs up with practice

Practice and Theory



Practice VS theory

1990: The trial-and-error testing becomming more andmore difficult. Theory leads practice.

Practice Theory



Plan

1 Introduction

2 Basic knowledges






Surface interaction properties

Surface properties:

Coefficient of friction

Adhesion

Wear parameters




Surface properties are not fundamental

Coefficient of friction /

Adhesion /

Wear parameters /






Adhesion /

Wear parameters /

Fundamental properties:

Volume:

Young’s modulus;Poisson’s ratio;shear modulus;yield stress;elastic energy;thermal properties.

Surface:

chemical reactivity;absorbtioncapabilities;surface energy;compatibility ofsurfaces;






Adhesion /

Wear parameters /

Fundamental properties:

Volume:


Surface:







Adhesion /

Wear parameters /

Fundamental propertiesare interdependent /

Volume:


Surface:







Adhesion /

Wear parameters /

More fundamental properties

solids are made of atoms;

atoms are linked by bonds;

many of the volume and surfaceproperties are the properties of thebonds.


Volume:


Surface:







Adhesion /

Wear parameters /

More fundamental properties

solids are made of atoms;

atoms are linked by bonds;

many of the volume and surfaceproperties are the properties of thebonds.


Volume:


Surface:




Material properties interdependence

Young’s modulus and yield strength interdependence [Rabinowicz, ]




Penetration hardness and yieldstress interdependence

[Rabinowicz, ]

Young’s modulus and melting temperatureinterdependence [Rabinowicz, ]




Thermal coefficient of expansionand Young’s modulus

interdependence [Rabinowicz, ]

Surface energy and hardness interdependence[Rabinowicz, ]



Real area of contact

Real area of contact depends on

normal load:real area of contact is proportional to thenormal load; coefficient of proportionality isinverse of the material hardness;

sliding distance:contact area might be 3(!) times as great asthe value before shear forces were first applied;

time: (for creeping materials)real area of contact increases with time;

surface energy:the higher the surface energy, the greater the

area of contact.

[Ref: Course of Julian Durand on surface roughness]









area of contact.


Ar ∼ F

Ar - real contact area, F -applied load









area of contact.


Ar =F

p

Ar - real contact area, F -applied load; p - hardness.









area of contact.


Ar =F

p









area of contact.


Ar =F

p









area of contact.


Ar =F

p



Engineering friction

First approximations: friction coefficient does not depend on

normal load

apparent area of contact

velocity

sliding surface roughness

time

Friction force direction is opposite to the sliding



Engineering friction

First approximations: friction coefficient does not depend on

normal load ,

apparent area of contact ,

velocity /

sliding surface roughness //,

time //,

Friction force direction is opposite to the sliding ,



Real friction :: normal load

First approximation:

friction coefficient doesnot depend on normalload.

Exceptions:at micro scale for small slidings (fig. 1);

for very large normal loads (metal forming)friction force is limited;

for very hard (diamond) or very soft (teflon)

materials:

generally T = cFα, α ∈ˆ

23; 1

˜;

thin hard coating and a softer substrate (fig.2).






Fig. 1. For very small sliding, theforce of friction is not proportionalto the normal force [Rabinowicz, ]




materials:


23; 1

˜;











materials:


23; 1

˜;











materials:


23; 1

˜;











materials:


23; 1

˜;


Fig. 2. In case of hard surface layer on a softer substrate, at moderate loads friction isdetermined by the hard surface, higher load brakes the coating and softer material begins

to define the frictional properties [Rabinowicz, ]



Real friction :: normal force

Friction coefficient versus tangential movement; experiments from[Courtney-Pratt and Eisner, 1957]



Real friction :: friction direction


friction force direction isopposite to the sliding.

Exceptions:the direction of the friction force remains within[178; 182] degrees to sliding direction (fig. 1);

the difference is higher for oriented surfaceroughnesses.








Fig. 1. Change of the direction of friction force with sliding[Rabinowicz, ]








Fig. 1. Change of the direction of friction force with sliding[Rabinowicz, ]



Real friction :: apparent area and roughness


Friction coefficient does not dependon the apparent area of contact.

Exceptions:very smooth and very clean surfaces.


Friction coefficient does not dependon sliding surface roughness.

Exceptions:very smooth or very rough surfaces(fig. 1).



















Fig. 1. Friction roughness influences the coefficient of friction [Rabinowicz, ]



Real friction :: time and velocity


Friction coefficient does not dependon time.

Exceptions:creeping materials.


Friction coefficient does not dependon sliding velocity.

Exceptions:if material behaves differently atdifferent loading rate, then thefriction depends on the slidingvelocity;










Static coefficient of friction evolution with time










Static coefficient of friction evolution with time Kinetic friction decreases with increasing slidingvelosity



Real friction :: velocity







Friction coefficient slightly decreses withincreasing velocity of sliding, titanium on

titanium [Rabinowicz, ]







Friction coefficient slightly decreses withincreasing velocity of sliding, titanium on

titanium [Rabinowicz, ]




Friction coefficient dependence on velocity ofsliding for lubricated surfaces [Rabinowicz, ]




Friction coefficient increases and decreases withincreasing velocity of sliding, hard on soft (steel

on lead, steel on indium) [Rabinowicz, ]




Friction coefficient dependence on velocity ofsliding for lubricated surfaces [Rabinowicz, ]



Three scales of contact study

Nanoscale:Study of molecular junctions, van

des Waals forces and Casimir effect.

Microscale:Roughness and microstructure study

Macroscale:Stress-strain state of contacting

solids



Plan

1 Introduction

2 Basic knowledges






Macroscopic contact

Signorini contact law (1933)r

u

n

n

Fn ≤ 0

un ≤ 0

Fnun = 0

Compliance contact lawr

u

n

n

[Kragelsky, 1982], [Oden-Martins, 1985]−Fn = Cn(un)

mn+

[Song, Yovanovich, 1987]

−Fn = C1ec2u

2n



Hertz theory (1882)

Geometry of smooth, non-conforming surface in contact

P

z

0

1S

δuz

2

P

δ 2

δ2

2δx−y plane

2 a

δ1

uz

1

S2

Expression of the profile of each surface

z1 =1

2R′1

x21 +

1

2R′′1

y21

z2 = −„

1

2R′2

x21 +

1

2R′′2

y22

«where R′

i and R′′i are the principal radii of curvature

of the surface i .

Separation between the two surfaces

h = z1 − z2 = Ax2 + By2

Displacement

uz1+ uz2

+ h = δ1 + δ2

[Johnson, 1996]



Hertz theory (1882)

Assumptions in the Hertz theory:

The surface are continuous and non-conforming, a << R

The strains are small, a << R

Each solid can be considered as an elastic half-space, a << R1,2, a << l

The surfaces are frictionless, q − x = qy = 0

Applications

1 Solids of revolution

2 Two-dimensional contact of cylindrical bodies

Note1

E∗=

1− ν1

E1+

1− ν2

E2



Hertz theory : Solids of revolution

Simple case of solids of revolution

Principal radii of curvatureR′

i = R′′i = Ri , i = 1, 2

Boundary conditions for the displacement

uz1+ uz2

= δ − (1/2R)r2

Pressure distributionp = p0{1− (r/a)2}1/2

Consequences

Pressure

p0 =

6PE∗2

P3R2

!1/3

Radius of the contact circle

a =

„3PR

4E∗2

«1/3

Displacement

δ =

9P2

16RE∗2

!1/3



Hertz theory : Solids of revolution

Distributions of stresses

σrp0

(x = 0, z) =σθp0

(x = 0, z)

= −(1 + ν){1− (z/a)tan−1(a/z)}

+ 12 (1 + z2/a2)−1

σzp0

(x = 0, z) = −(1 + z2/a2)−1

Maximum shear stress τ1 = 12 |σr − σθ|

(τ1)max = 0.31p0 at the deph of 0.48a (for ν = 0.3)



2D contact of cylindrical bodies

�O

x

z

a ap0

p(x)

q(x)

�M(x, y)

σxx

σzz

τxz

1E∗ =

1−ν1E1

+1−ν2

E2

1R = 1

R1+ 1

R2

p(x) = p0(1− (x/a)2)−1/2

a =q

4PRπE∗

p0 =q

PE∗πR



2D contact of cylindrical bodies

Example : cylinder/plateDistributions of normal pressure (Hertz) and tangential stress

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;�<=;?>@�A @CBD�E D5FG?HJI GK�L�MONPRQTSVUW X Y Z

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σxx (x = 0, z) = − p0a {(a

2 + 2z2)(a2 + z2)−1/2 − 2z}

σzz (x = 0, z) = −p0a(a2 + z2)−1/2

τmax (x = 0, z) = p0a{z − z2(a2 − z2)−1/2}

σxz = σxy = σyz = 0



Macroscopic friction 1/2

Tresca 8>>>>><>>>>>:

|Ft | ≤ g

If |Ft | < g , then Vslide = 0

If |Ft | = g , ∃λ > 0 such Vslide = −λFt

Coulomb 8>>>>><>>>>>:

|Ft | ≤ µ|Fn|

If |Ft | < µ|Fn|, then Vslide = 0 stick

If |Ft | = µ|Fn|, ∃λ > 0 such Vslide = −λFt slip



Macroscopic friction 2/2

Regularized Coulomb [Oden, Pires, 1983], [Raous, 1999]

|Ft | = −µφi (Vslide) |Fn|φ1 = Vslide√

V 2slide+ε2

φ2 = tanh(

Vslide

ε

)

Variable friction

8>>>>><>>>>>:

|Ft | ≤ Ct(ut)mt

If |Ft | < Ct(ut)mt , then Vslide = 0

If |Ft | = Ct(ut)mt , ∃λ > 0 such Vslide = −λRt



Transition toward the slip

Definition of slidingRelative peripheral velocity of the surfaces at their point of contactSliding of non-conforming elastic bodies

S

Fixed slider

V

P

Q

Qx

z

S2

1

2a

Sliding contact

QuestionThe tangential traction due to the friction at thecontact surface influences the size and shape of thecontact area or the distribution of normal pressure ?

Evaluation of the elastic stresses and displacementsBasic premise of the Hertz theory

Relationship between the tangential traction and thenormal pressure

|q(x, y)|p(x, y)

=|Q|P

= µ Coulomb’s law

ApplicationCylinder sliding perpendicular to its axis

[Johnson, 1996, Goryacheva, 1998]



Cylinder sliding perpendicular to its axis

Distributions of normal pressure (Hertz) and tangential traction

p(x) = 2Pπa

q1− ( x

a )2

q(x) = ±µp(x) = ±µ 2Pπa

q1− ( x

a )2

(1)

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Stress componentsσxx (x, z = 0) = −p0

nq1 − ( x

a)2 + 2µ x

a

o

σzz (x, z = 0) = −p0

q1 − ( x

a)2

σyy (x, z = 0) = ν(σxx + σzz )

τxz (x, z = 0) = −µp0

q1 − ( x

a)2

τxy (x, z = 0) = τyz (x, z = 0) = 0



Partial slip

Relation between slip zone (c) and contact zone (a)

−a +a

q1(x) = µ p0

(

1−x

2

a2

)1/2

+ =

−a +a−c +c

q2(x) = µc

ap0

(

1−x

2

c2

)1/2

q(x) = q1(x) − q2(x)

−a +a−c +c

glissement

zone collee

c

a=

s1−

Q

µP

If x < c : stick condition. The local contact shear stress is

τxz = µp0

r1− (

x

a)2 −

c

aµp0

r1− (

x

c)2

If c < x < a: slip condition. The local contact shear stress is

τxz = µp0

r1− (

x

a)2



Friction instability : “stick-slip” 1/4

Definiton of the stick-slip :

Intermittent relative motion between the contact surfaces, alternation of slip and stick.

time

F

F

s

d

time

F

time

F

Phenomenon occurs at various scales:

Macroscopic : discontinuities in the gravity center displacement of contact body and loads.

Microscopic : location of the phenomenon at the interface




The stick-slip is a coupling result :

The dynamic response of the friction systemstiffness, damping, inertia

Friction dynamic at the interfaceDifference between static (µs ) and dynamic (µd ) friction coefficientµs and µd dependence on the sliding velocity and time

A simple stick-slip model

��

��

m

kF

X

v

Friction law

F

v

F

F

s

d

Fig : Plot of frictional force vs. time.




During sliding, the problem is:

mx − Fd = −kx x(0) =Fs

kx(0) = v

and the solution is

x(t) =1

k{(Fs − Fd )cos(ωt) + Fd} +

v

ωsin(ωt), ω =

rk

m

or the velocity v is negligible compared to dx/dt:

x(t) ≈1

k{(Fs − Fd )cos(ωt) + Fd}

Characteristic time of sliding

Tinertia = 2π

rm

k

The force F oscillates between Fs and 2Fd − Fs .

F

t

F

d2F − Fs

Tinertia

s




Fig : Regions of stable and stick-slip motion.

The red curve in the parameters plane at the other parameters being fixed, demarcates the regionsof stable and unstable motion.



Plan

1 Introduction

2 Basic knowledges






Examples

(a) Vickers indenta-tion test, palladiumglasses

(b) Contact zone underVickers indenter, zirconiumglasses

(c) Sracth resistance of soda-Lime Silica Glasses



Hill’s theory: Elastic-plastic indentation

Cavity model of anelastic-plastic indentation cone

daPlastic

Elastic

dh

dcc

da

a

da

aβ

rcore

du(r)

AssumptionsWithin the core:Hydrostatic component of stress p

Outside the core:Radial symmetry for stresses and displacement

At the interface ( between core and plastic zone)Hydrostatic stress (in the core)= radial component ofstress (in the external zone)

The radial displacement on r=a during an increment dhmust accommodate the volume of material.



Characteristic

In the plastic zone: a ≤ r ≤ cσrY = −2ln(c/r)− 2/3

σθY = −2ln(c/r) + 1/3

In the elastic zone: r ≥ cσrY = − 2

3

`cr

´3σθY = 1

3

`cr

´3where Y denotes the value of the yield strees of material in simple shear and simple compression.

Core pressure

pY = −[ σr

Y ]r=a = 23 + 2ln(c/a)

Radial displacement

du(r)dc = T

E {3(1− ν)(c2/r2)− 2(1− 2ν)(r/c)}

Conservation volume

2πa2du(a) = πa2dh = πa2tan(β)da

9>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>;

Pressure in the core,for an incompressible material

p

Y=

2

3

1 + ln

„Etan(β)

3Y

«ff



Unloading indentation: elastic strain energy

Example: spherical indenter

R R’

aa

R ρ

a aBefore loading Under loading After unloading

Residual depth δ − δ′: Estimation of the energy dissipated ∆W in one cycle of the load∆W = α

RPdδ where α is the hysteresis-loss factor. (α = 0, 4% for hard bearing steel)

W = 25

„9E∗

2P5

16R

«4a3“

1R −

1ρ

”= 3P

E∗

a =“

3P4E′

”1/3

δ = a2

R =“

9P2

16RE′3

”1/3

9>>>>>>>>>>>=>>>>>>>>>>>;

δ′ = 9πPpm

16E′2with pm = 0.38Y in fully plastic state

PPY

= 0, 38(δ′/δY )2



Sharp indentation

Schematic illustration of a typical P-hreponse of an elasto-plastic material

Characterisation of P-h reponseDuring the loading,

P = Ch2 Kick’s law

During the unloading,

dPudh

˛hm

initial slope

hr Residual indentation depthafter complete unloading

Three independent quantities

[?]



Plastic behavior

The power law elasto-plastic stress-strainbehavior

Model

σ =

8><>:Eε for σ ≤ σy

Rεn for σ ≥ σy

with

E Young’s modolusR a strength coefficientn the strain hardening exponentσy the initial yield stress

Assumption :The theory of plasticity with the von Mises effect stress.

Parameters for an elasto-plastic behaviorE, ν, σy , n



Dimensional analysis

ObjectivePrediction of the P − h reponse from elasto-plastic properties

Application of the universal dimensionless functions : the Π theorem

Material parameter set

(E , ν, σy , n) or (E , ν, σr , n) or (E , ν, σy , σr )

Load P

P = P(h, E∗, σy , n) or P = P(h, E∗, σr , n) or P = P(E , ν, σy , σr )

with

E∗ =

1− ν2

E+

1− ν2i

Ei

!−1

Unload

Pu = Pu(h, hm, E , ν, Ei , νi , σr , n) or Pu = Pu(h, hm, E∗, σr , n)



Determination of hm

Application of the dimensional analysis during the load

Load Applying the Π theorem in dimensional analysis

P = P(h, E∗, σy , n) C = Ph2 = σrΠ1

“E∗σr

, n”

P = P(h, E∗, σr , n) C = Ph2 = σyΠ

A1

“E∗σy

, σrσy

”P = P(E , ν, σy , σr ) C = P

h2 = σrΠB1

“E∗σr

,σyσr

”

with Π are dimensionless functions.

And then hm !



Determination of hr

Application of the dimensional analysis during the unload

dPu

dh=

dPu

dh(h, hm, E∗, σr , n)

thusdPu

dh= E∗hΠ0

2

„hm

h,

σr

E∗, n

«Consequently,

dPu

dh

˛h=hm

= E∗hmΠ02

„1,

σr

E∗, n

«= E∗hmΠ2

„σr

E∗, n

«Or

Pu = Pu(h, hm, E∗, σr , n) = E∗Πu

„hm

h,

σr

E∗, n

«Finaly,

Pu = 0 implies 0 = Πu

„hm

hr,

σr

E∗, n

«whether

hr

hm= Π3

„σr

E∗, n

«and then hr !Πi ,i =1,2,3 can be used to relate the indentaion reponse to mechanical properties.



FE Vickers indentation test

Maximum penetration hmaxs

3.11 µm

Parameters

Elastic E=81600MPa, ν = 0, 36

Elasto-plastic model E=81600MPa, ν = 0, 36,σy = 1610MPa

Drucker-Prager model E=81600MPa, ν = 0, 36,σt

y = 1600MPa, σcy = 1800MPa

[Laniel, 2004]



Computation results

Penetration depth

von Misesmax Residual stress



Computation results

Elasto-plastic contact reponse



Computation results

Elasto-plastic contact reponse



Spherical indentation on a single crystal

HypothesisSphere radius : R=100µmCopper and zinc single crystals : crystal plasticitySilicon substrate : isotropic elasticMaximum penetration hmax

s : 3.5 µ m

[Casal and Forest, 2009]



Contact reponse

(d) Elastic anisotropic contact reponse (e) Elasto-plastic anisotropic contact reponse



von Mises stress fields

Figure: (a) f.c.c and (b) h.c.p crystals. Penetration depth: hs = 1.25µm



Plastic zone morphology

(a) f.c.c copper crystals (b) h.c.p zinc crystals

Figure: Penetration depth: hs = 3.5µm



Plan

1 Introduction

2 Basic knowledges






Bounds for the global coefficient of friction

��9Lubricant µ1 = 0.2

PPPi Coating µ2 = 0.8

µ(x) =q(x)

p(x)µ =

∫µ(x)p(x)dS∫

p(x)dS

Uniform stress ≡∑

µi fi ≤ µ ≤∑

µici fi∑ci fi

≡ Uniform strain

with ci = Ei

(1−ν2i )

(1−νi )2

(1−2νi )

[Dick and Cailletaud, 2006]



Bounds for the global coefficient of friction

Cont A ν E (GPa) C (GPa) µ Cont B ν E (GPa) C (GPa) µ

Comp 1 0.32 8 11.45 0.1 Comp 1 0.15 55 58.08 0.1Comp 2 0.15 55 58.08 0.5 Comp 2 0.32 08 11.45 0.5

00.050.1

0.150.2

0.250.3

0.350.4

0.450.5

0 20 40 60 80 100

µ

Component 2 (%)

P1=P2εy1=εy2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 20 40 60 80 100µ

Component 2 (%)

P1=P2εy1=εy2

Case A: µ1 = 0.1, E1 = 11.45GPa Case B: µ1 = 0.5, E1 = 58GPaµ2 = 0.5, E2 = 58GPa µ2 = 0.1, E2 = 11.45GPa



FE computations VS analytic estimation

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

CO

F

Component 2 (%)

analyticanalytic

P1-10P2-10

-8

-7

-6

-5

-4

-3

-2

-1

0

-0.01 -0.005 0 0.005 0.01

norm

aliz

ed c

onta

ct p

ress

ure

x (mm)

CComp.1CComp.2

P1-10: pComp.1, pComp.2P2-10: pComp.1, pComp.2

comp.1comp.2 comp.2



Different CSL geometries 1/2

Bulk material Component 1 Component 2

E (GPa) 119 8 55ν 0.29 0.32 0.15

R0 ( MPa) - 200 500



Different CSL geometries 2/2

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

CO

F

Component 2 (%)

analyticanalyticS20_elR20_elL20_elLxx_el

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

norm

aliz

ed c

onta

ct p

ress

ure

x (mm)

CComp.1CComp.2

S20_el: pComp.1, pComp.2R20_el: pComp.1, pComp.2L20_el: pComp.1, pComp.2Lxx_el: pComp.1, pComp.2

comp.2 comp.1 comp.2

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

-0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025

norm

aliz

ed c

onta

ct p

ress

ure

x (mm)

CComp.1CComp.2

S20_el: pComp.1, pComp.2R20_el: pComp.1, pComp.2L20_el: pComp.1, pComp.2Lxx_el: pComp.1, pComp.2

comp.2 comp.2comp.1



Influence of the number of dimples

10% comp 2 70% comp 2

ESD type 2a (µm) lESD (µm) nb.ESD lESD (µm) nb.ESDR10 360 11.1 32.4 33.3 10.8R20 360 22.2 16.2 66.6 5.4R40 360 44.4 8.1 133.3 2.7

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

CO

F

Component 2 (%)

R10_elR20_elR40_elS20_elS40_el



Influence of plastic deformations

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1

CO

F

Component 2 (%)

analyticanalyticR40_elR40_plS40_elS40_pl

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

-0.04 -0.02 0 0.02 0.04

norm

aliz

ed c

onta

ct p

ress

ure

x (mm)

CComp.1CComp.2

R40_el: pComp.1, pComp.2R40_pl: pComp.1, pComp.2

comp.1comp.2 comp.2

00.0038

0.00760.011

0.0150.019

0.0230.027

0.0310.035

0.0380.042

0.0460.05



Conclusion

Estimation of the upper and lower bound.

The friction coefficient depend on

the CSL geometrythe dissimilarity of the CSL component materialsthe compliance of substrate and counter body



Summary

Contact and friction

complicated phenomena;depend on many material properties;not yet well elaborated.

Analytical solutions

hertzian contact;nonlinear material;friction;stick-slip instabilities.

Numerical analysis

examples of indendation tests;analysis of heterogeneous friction.


Thank you for your attention!

Georges Cailletaud <[email protected]>Stephanie Basseville <[email protected]>Vladislav A. Yastrebov <[email protected]>


Casal, O. and Forest, S. (2009).

Finite element crystal plasticity analysis of spherical indentation.

Computational Materials Science, 45:774–782.

Dick, T. and Cailletaud, G. (2006).

Analytic and FE based estimations of the coefficient of friction of compositesurfaces.

Wear, 260:1305–1316.

Goryacheva, I. (1998).

Contact mechanics in tribology.

London.

Johnson, K. (1996).

Contact mechanics.

Cambridge.

Laniel, R. (2004).

Simulation des procı¿½dı¿½s d’indentation et de rayage par ı¿½lı¿½ments finis etı¿½lı¿½ments disctincts.

Rabinowicz, E.

Friction and Wear.


Documents

Contact mechanics I: basics - mms2 - MINES ParisTech