ORIGINAL PAPER

Nonlocal Frictional Effects at Indentation of Elastic Materials

Denis Jelagin • Per-Lennart Larsson

Received: 28 November 2012 / Accepted: 11 June 2013 / Published online: 20 June 2013

� Springer Science+Business Media New York 2013

Abstract Indentation of elastic materials is investigated

numerically using the finite element method. Large defor-

mation theory is relied upon for accuracy. The study

focuses on nonlocal frictional effects on relevant indenta-

tion quantities in the microindentation regime. The inden-

tation quantities investigated include both local and global

ones. It is shown that nonlocal frictional effects are small

when global quantities are at issue, as is the case when

conventional (Coulomb) theory of friction is used, also

when these features are introduced at the ridges of a

Vickers indenter where stress gradients are substantial.

These effects are, however, shown to be of importance for

particular indenter geometries as far as local field variables

are concerned.

Keywords Indentation � Elastic materials � FEM �Nonlocal friction

1 Introduction

Indentation or hardness tests, associated with names such as

Brinell, Knoop, Vickers and Berkovich, have, for a long

time, been used to characterize conventional engineering

materials, such as metals and alloys. In recent years, such

tests have received increasing attention due to the develop-

ment of new experimental devices, such as the nanoindenter

[1], enabling the determination of the material properties

from very small samples at extremely small indentation

depths. Another reason for the renewed interest in indenta-

tion testing is the fact that, for many new engineering

materials such as ceramics, a standard uniaxial test often

fails to deliver reliable results, and accordingly, indentation

is the only alternative for material characterization. Fur-

thermore, indentation is a very convenient tool for deter-

mining the material properties of thin films or strings in

ready-to-use engineering devices. The indentation can be

used to determine the constitutive and fracture resistance

properties of the material by taking advantage of results

from earlier theoretical, numerical and experimental analy-

ses by, for example, Tabor [2], Johnson [3], Storakers et al.

[4], Larsson [5], Oliver and Pharr [6] and Wilshaw [7]. An

accurate description of the effect from the specimen’s

material and interface parameters on the stress field induced

in the specimen is an essential step in the analysis of the

indentation test data. Accordingly, in the present study, the

influence of the nonlocal frictional interactions on the local

and global indentation parameters is examined theoretically

and numerically.

Most commonly, indentation testing involves both

elastic and plastic deformation. This being in particular so

at sharp indentation where the absence of a characteristic

length in the problem indicates that plasticity enters the

problem immediately at contact (the exception being

indentation of materials described by nonlocal plasticity

where a characteristic length parameter is introduced from

the constitutive equation, cf. e.g. [8]). There are, however,

quite a few situations of substantial practical importance

where results for purely elastic deformations, pertinent to

different kinds of sharp indentation, are of immediate

significance. This concerns in particular indentation of

highly elastic polymer and rubber materials, where plastic

D. Jelagin (&)

Division of Highway and Railway Engineering, Royal Institute

of Technology, 10044 Stockholm, Sweden

e-mail: denis.jelagin@abe.kth.se

P.-L. Larsson

Department of Solid Mechanics, Royal Institute of Technology,

100 44 Stockholm, Sweden

123

Tribol Lett (2013) 51:397–407

DOI 10.1007/s11249-013-0172-4

deformation is negligible, but also when it comes to ana-

lytical modeling of unloading at indentation. The latter

issue is of interest when indentation is used in order to

determine the elastic properties of a material, based on the

load–displacement curve at initial unloading, cf. e.g. [6].

Regardless of whether or not both elastic and/or plastic

deformation is present at sharp indentation, the relation

between normal indentation load, P, and indentation depth,

h, can always be described as

P� h2 ð1Þ

The relation (1) is valid when the dimensions of the

contact area are small compared to the dimensions of the

indented material, and no length quantity is introduced

from, for example, the constitutive equation. This means

that (1) holds at, for example, Hookean elasticity and

classical Mises plasticity. For the case of cone indentation

of an elastic half-space, Sneddon [9] derived the explicit

relationship

P ¼ 2Eh2 tanða=2Þ= pð1� m2Þ� �

ð2Þ

where E and m are Young’s modulus and Poisson’s ratio,

respectively, of the half-space and a is the included angle

of the conical indenter. Numerous related formulas for

other types of indenters and constitutive equations have

been presented, cf. e.g. [3, 10, 11], but are not shown here

for brevity.

The case of spherical indentation differ from the sharp

one by the fact that here a characteristic length is intro-

duced from the curvature of the indenter. Accordingly,

plasticity will enter into the mechanical problem in a

gradual manner with increasing load, and the initial stage is

characterized by completely elastic deformation and is

known to be well described by the famous Hertzian [12]

relation:

P ¼ 4ER1=2h3=2= 3ð1� m2Þ� �

ð3Þ

where R is the radius of the indenting sphere and otherwise

the notation is introduced above in the context of Eq. (2).

It should be noticed that the Eqs. (2) and (3) are based

on the assumption of frictionless contact between the

indenter and the material. Investigations of frictional

effects at normal indentation are quite frequent in the lit-

erature, and this includes both elastic and plastic indenta-

tion (using both blunt and sharp indenters), and other

contact problems, cf. e.g. [13–19], at Coulomb friction.

The main conclusion from these efforts is that frictional

effects are small when global indentation variables, i.e.,

indentation force, depth and the contact radii, are con-

cerned. At the same time, it has been shown both numer-

ically [14, 20] and experimentally [20, 21] that the

presence of friction may have a profound influence on the

local field variables and in particular on the maximum

tensile stress location and magnitude. As a result, taking

friction into account changes both qualitatively and quan-

titatively the predictions concerning the initiation of so-

called Hertzian fracture in brittle specimens, cf. e.g. [21,

22]. This is a practically important phenomenon, and

considerable amount of work has been done to model and

predict the initiation and growth of such types of cracks, cf.

e.g., a review paper by Lawn [23]. One of the main reasons

behind the investigations of Hertzian fracture is a need to

improve the theoretical basis for the Hertzian fracture

test—a simple way to examine strength of brittle materials,

cf. e.g. [7].

It should, however, be emphasized that all the above-

mentioned investigations are pertinent to Coulomb fric-

tion, and this is not necessarily a good description of

frictional effects, in particular at small length scales in the

context of nano- or microindentation. Instead, in such a

situation, a nonlinear description of frictional contact can

be advantageous, and it is the present intention to inves-

tigate the behavior of mechanical indentation quantities

when using such an approach. Oden and Pires [24] pre-

sented an in-depth discussion about nonlocal friction

models where they pointed out that as the contact stresses

are transferred through the junctions formed by the

deformed surfaces asperities, the frictional mechanisms

have a nonlocal character. Furthermore, as commented

upon by Oden and Pires [24], there will always exist a

small tangential displacement between the contact points

in the ‘‘stick’’ region. These tangential displacements are

governed by the local elastoplastic deformations of the

asperities and can be accounted for, at least qualitatively,

by introducing nonlocal friction formulations where a

certain amount of elastic slip is allowed in stick regions. It

has to be emphasized, however, that the nonlinear friction

model discussed above is not able to fully capture the

micromechanics of rough frictional contact; in order to do

that one rather needs to rely on an explicit multi-asperity

contact models, cf. e.g., Olofsson and Hagman [25].

Furthermore, the influence of the surface roughness on the

global and local indentation quantities cannot be descri-

bed fully through a modified friction model alone as the

presence of the asperities on the interface also affects the

indentation parameters at frictionless contacts. In fact, the

effect the surface roughness has on the contact area,

indentation load and the surface pressure distribution

received a lot of attention in the literature, cf. e.g.,

Greenwood and Williamson [26], Johnson et al. [27] and

Carbone and Bottiglione [28]. At the same time, the

frictional model proposed in [24] has an advantage of

being a relatively simple way to account for the micro-slip

phenomena in the stick region as well as for its influence

on the local and global indentation parameters. To the

author’s knowledge, these nonlocal frictional effects have

398 Tribol Lett (2013) 51:397–407

123

not been considered previously in the context of blunt or

sharp indentation testing. Accordingly, this will be

attempted presently where, for clarity and convenience

(but not out of necessity), the constitutive behavior of the

indented material is restricted to (hypo) elasticity.

A qualitative discussion is, however, also provided

regarding the effects of the nonlinear friction law on the

indentation parameters at the presence of plasticity; and

computational results are reported for spherical indenta-

tion of rigid-ideal plastic materials in order to support the

arguments.

One of the motivations for the present work is the

recently published experimental investigations by Lim and

Chaudhri [29, 30]. These authors studied cone and Vickers

indentation of five highly elastic polymers and rubber

materials and reported deviations between theory [Eqs. (1)

and (2)] and experiments in case of Vickers indentation.

Lim and Chaudhri [30] suggested that frictional effects at

the ridges of the Vickers indenter might contribute to this

deviation. Presently, nonlocal frictional effects at elastic

indentation are investigated in a general manner, and the

findings of Lim and Chaudhri [30] are then evaluated in

this context.

In the analysis, straightforward finite element calcula-

tions will be adhered to, and in particular, the commercial

finite element package ABAQUS [31] is relied upon.

Nonlocal frictional effects are introduced according to the

formulation by Zhong [32] which differs from standard

Coulomb friction by the fact that friction is not evaluated

pointwise but instead over a small area. The investigation

includes elastic indentation using both blunt (spherical) and

sharp (cone and Vickers) indenters and is based on large

deformation theory for accuracy, in particular then at sharp

indentation. Both local and global indentation quantities

are of interest, and relevant results are presented for these

features.

2 Problem Formulation and Numerical Analysis

The present analysis concerns blunt and sharp indentation

of completely elastic materials. The indenter geometries to

be investigated include a spherical (blunt) indenter, a cone

indenter (in both these cases axisymmetry prevails) and a

Vickers pyramid indenter (resulting in a full 3D problem).

These indenters are shown in Fig. 1 in which also the

notation of the problem is presented. It should be noted that

P represents the normal indentation load in the negative x2-

direction (corresponding to the positive z-direction for the

axisymmetric cases).

In the analysis, quasi-static conditions are assumed

which implies equilibrium as

rij;j ¼ 0: ð4Þ

In (4), rij is the Cauchy stress, i.e., force per unit area in

the deformed system. Large deformations are accounted for

with the deformation rate tensor

Dij ¼ ð1=2Þðvi;j þ vj;iÞ ð5Þ

as the deformation measure used in the constitutive

description (to be detailed below). In (5), vi is the material

velocity. Note that in (4) and (5) derivation is carried with

θ

(a)

(b)

(c)

Fig. 1 Geometry and notation of the problem. a Spherical indenta-

tion. b Cone indentation. The included angle of the indenter is defined

as a = 180�- 2h. The problem is axisymmetric implying that the

cylindrical coordinate system and global indentation quantities

introduced in Fig. 1a are convenient to use. c Vickers indentation.

In this case h = 228 (a = 1368) according to standard

Tribol Lett (2013) 51:397–407 399

123

respect to xi being the coordinates representing the current

position of a material point.

As already discussed above, Eqs. (1)-(3) are valid at

indentation problems when the dimensions of the contact

area are small compared to the dimensions of the indented

material, and no length quantity is introduced from the

constitutive equation. This is the situation of interest in the

present case. The spherical contact solution by Hertz [12],

Eq. (3), is very well known and requires no further dis-

cussion. The solution by Sneddon [9], Eq. (2) concerning

cone indentation of an elastic half-space is perhaps less

known and deserves some further explanation. In the

analysis by Sneddon [9], it is assumed that the indenter is

perfectly sharp and rigid and that a small deformation

analysis is sufficient to describe the problem. Frictional

effects are, as mentioned above, not included in the anal-

ysis. In case of Vickers indentation, any corresponding

closed-form relation does not exist as the problem, being a

complete 3D-problem, then becomes too involved to be

analyzed by analytical methods. However, Giannakopoulos

et al. [33] used large-scale finite element calculations and

dimensional analysis and curve-fitted the relation

P ¼ 2:0746ð1� 0:1655m� 0:1737m2 � 0:1862m3ÞEh2=

ð1� m2Þ:ð6Þ

The basic features pertinent to Eq. (6) are linear

elasticity, linear kinematics, frictionless contact and a

perfectly sharp and rigid indenter. A corresponding solu-

tion for a Berkovich indenter (sharp pyramid indenter with

a triangular base) was subsequently presented by Larsson

et al. [34] but is left out for brevity (the results in [33] have

also been verified by FEM studies in [35]).

As been emphasized above, the relations (2), (3) and (6)

are all pertinent to an analysis based on linear kinematics.

At elastoplastic indentation, both for spherical and sharp

indenters, it has been shown that large deformation effects

can be substantial, cf. e.g. [36–38]. At purely elastic

deformations, these effects are reported to be less signifi-

cant as, for example, Giannakopoulos et al. [33] and

Larsson et al. [34] reported differences within a few per-

cent between large deformation and small deformation

results. Still though, there are differences between the two

sets of results, and aiming at high-accuracy solutions, it

was thought advisable to include nonlinear kinematics in

the present analysis.

Above, basically only the behavior of global indentation

properties has been discussed. This is so mainly because

such relations are more easily expressed in an analytical

manner. However, as mentioned previously, field variables

are of course also of considerable importance in the present

context and will be given detailed attention. In particular,

this concerns the influence of frictional effects on tensile

stresses close to the contact boundary.

As for the constitutive specification, remembering that

large deformations were accounted for a hypoelastic for-

mulation of Hooke’s law was relied upon yielding

Mrij ¼ E=ð1þ mÞð Þ dikdjl þ m= 1� 2mð Þð Þdijdkl

� �Dkl: ð7Þ

In (7), Drij is the objective Jaumann rate of the Cauchy

stress. Furthermore, Dij is the deformation rate tensor as

specified in (5), and dij is Kronecker’s delta. The material

constants E and m, Young’s modulus and Poisson’s ratio,

respectively, have also been presented previously above.

It remains then to formulate the frictional behavior in

the present analysis. In case of standard Coulomb friction,

this approach states that no relative motion between two

surfaces occurs if

seff\lp ð8Þ

applies. In (8), the effective shear stress is defined as

seff ¼ ðs12Þ2 þ ðs23Þ2� �1=2

ð9Þ

l is the coefficient of friction and p is the contact

pressure. If

seff ¼ lp ð10Þ

relative tangential slip between the two surfaces will be

present.

The Coulomb friction model is a so-called local friction

model, i.e., it applies pointwise over the contact region. In

the present analysis, the main interest is directed toward the

mechanical behavior when nonlocal frictional effects are

considered. A simple way to introduce such effects is to

approximate the stick conditions in Eq. (10) with a stiff

elastic behavior. The elastic stiffness is then chosen in such

a way that the relative motion from the position of zero

shear stress is bounded by a value cc. Such a nonlocal

friction model was suggested by Zhong [32] and can be

considered as a Coulomb model which is not applied

pointwise but weighted over a small area. Presently then,

the influence from cc on the mechanical behavior at elastic

contact will be investigated using the commercial FEM

program ABAQUS [27] where a corresponding nonlocal

friction model is implemented (even though the main focus

then is computational efficiency, see [39]).

In the last two decades, the number of finite element

analyses of sharp indentation testing has increased enor-

mously, cf. e.g. [5, 16, 18, 33–35, 40–46] (just to mention a

few), and can now be considered as a standard method for

numerical investigation of such problems. The analyses

include both 2D and 3D problems as well as different

forms of constitutive behavior (elastic, various elastoplastic

models and many others). Accordingly, in the presentation

400 Tribol Lett (2013) 51:397–407

123

of the present numerical approach below, only the essential

details of the analysis will be discussed.

In Fig. 2, the finite element mesh used in the analysis is

shown with axisymmetry taken into account for the conical

and spherical indenters in Fig. 2a. Eight-noded isopara-

metric quadrilateral elements were used. One may notice

the domain with especially dense mesh in the vicinity of

the contact region with the element size chosen to have at

least 50 elements in contact. The distance to the outer

boundary of the half-space was set to be at least 200a. At

the outer boundary of the half-space, both the horizontal

and vertical displacements were set to vanish. The load was

applied by prescribing a uniform vertical displacement to

the indenter’s upper boundary. A surface-based contact

formulation was used to model the interaction between two

elastic bodies. At normal contact behavior, a hard contact

formulation was used, where surfaces may not penetrate

each other. Two different formulations were used to cap-

ture frictional interactions on the interface. To model the

basic Coulomb friction, the Lagrange multiplier method

was employed which allows for sticking constraints to be

enforced exactly. In order to model nonlocal frictional

interactions, penalty method was used, and the maximum

allowable slip parameter was used to qualitatively capture

the effect of local asperity deformation. In case of a

Vickers indenter, 1/8 symmetry was enforced as shown in

Fig. 2b.

In order to validate the numerical procedure used

presently, the global and local field values found at fric-

tionless contact between a linear elastic half-space and a

rigid spherical indenter were compared with the analytical

solution due to Hertz [12]. The global values were found

to be accurate within 0.3 %, while the accuracy of the

local field values was found to be within 1 %. For the case

of a conical indenter, close agreement was found with

Sneddon’s solution when a very shallow cone was used

with the specimens Poisson’s ratio being 0.49. However,

in the case of a conical indenter geometry (half included

angle of 70.3�) as used below, it was found in a para-

metric study that the indentation load and contact radii

predicted numerically are approximately 10 % higher as

compared to the ones predicted with Sneddon’s solution.

This discrepancy is due to simplifying assumptions made

in Sneddon’s solution, as explained in detail in, e.g., Hay

et al. [41]. Hay et al. [41] also reported an approximately

10 % discrepancy between the numerical results and

Sneddon’s solution.

It remains now to discuss the parameters used in the

computational study. For a given profile geometry, there

are essentially four parameters governing the solution:

frictional law is described by the friction coefficient, l,

and the maximum allowable slip in stick region, cc; the

degree of elastic mismatch between the bodies in contact

is described by the Dundurs parameter, b, defined as

follows:

b ¼ 1� 2v1

G1

� 1� 2v2

G2

� �n2 1� v1

G1

þ 1� v1

G1

� �ð11Þ

where Gi; viði ¼ 1; 2Þ are elastic parameters, i.e., shear

modulus and Poisson’s ratio, of the specimen and the

indenter, respectively; finally, the normalized stress and

strain distributions in the specimen are influenced by its

Poisson’s ratio, #.

Jelagin and Larsson [46] presented a computational

study, where the influence of Coulomb friction on the

Fig. 2 Finite element meshes. a Axisymmetric indentation. b Vickers

indentation

Tribol Lett (2013) 51:397–407 401

123

Hertzian fracture initiation has been investigated for a

range of # and b values. In their study, the impact of

friction was found to be most profound at low specimens

Poisson’s ratio combined with the high b values as these

material combinations result in the highest tendency for

relative tangential slip between the bodies in contact.

Based on the results presented in [46], it was decided in the

present study to investigate the effect of nonlocal friction

for the case of a specimen with v ¼ 0:2 (corresponding to,

e.g., concrete or glass specimen) in contact with a rigid

indenter ðb ¼ 0:375Þ. The friction coefficient, l, has been

varied between 0 and 0.4. The magnitude of the maximum

allowable elastic slip in the stick region is, according to

Oden and Pires [24], related to the surface roughness and

the local mechanical properties of the materials in contact.

In order to cover a wide range of the possible cc values, in

the present study, the simulations have been performed

with cc=h ¼ 0; 0:1; 1. It should be clearly stated though that

pertinent experimental studies, devoted toward a consistent

physical understanding of the cc-parameter, cannot be

found in the literature. A certain quantitative guideline may

be obtained from Hagman and Olofsson [47] study where

the maximum amount of micro-slip (i.e., amount of tan-

gential displacement in the macro stick regime) has been

estimated numerically and measured experimentally for

steel and brass contact pairs at combined normal and tan-

gential loading. Based on the model with uniformly dis-

tributed ellipsoidal elastic asperities, Hagman and Olofsson

[47] predict the maximum elastic micro-slip in the stick

region to be in the range of approximately 0.2–0.5 lm.

Their experimental observations, however, indicate that the

model they use underestimates the amount of micro-slip,

which according to the authors may be due to the plastic

deformation of the asperities. As the micro-slip magnitudes

for pure normal loading contacts are not reported in the

literature, the present study is definitely directed toward a

qualitative, and not quantitative, understanding of the

influence from nonlocal frictional effects. cc=h ¼ 1 is

certainly a high value on a parameter related to surface

roughness, and it is therefore believed that the parameter

study conducted presently is covering the most significant

effects related to nonlocal friction. Having said this though,

it may be expected based on the results given in [47] that in

most situations of practical importance, cc will not exceed

several micrometers in magnitude and thus high values of

cc=h may be expected only at indentation done at micro-

and nano length scales.

3 Results and Discussion

Below, the numerical results will be presented and dis-

cussed. It deserves to mention once again, as also stated

above, that all results are pertinent to a value on Poisson’s

ratio being v ¼ 0:2 and a value on Dundurs parameter

being b ¼ 0:375. Furthermore, the friction coefficient, l,

has been varied between 0 and 0.4 and the allowable elastic

slip, cc, was given the values cc=h ¼ 0; 0:1; 1. It should be

emphasized in this context that cc=h ¼ 0 corresponds to

standard Coulomb friction with no nonlocal effects. The

results below are pertinent to the three different indenter

geometries shown in Fig. 1 (spherical indentation, cone

indentation and Vickers indentation).

It seems appropriate to start the presentation by showing

some results concerning the shear tractions on the speci-

men surface at frictional contact. This is depicted in Fig. 3

pertinent to a spherical indenter and with the frictional

coefficient l ¼ 0:2. For illustrative purposes, the contact

pressure distribution is also shown but multiplied with the

friction coefficient. Accordingly, the regions where shear

and scaled pressure distributions coincide are slip regions.

As it has been originally pointed out by Spence [13] for the

case of Coulomb friction, the relative radius of the stick

region is invariant of loading and contact geometry, pro-

vided that the profiles are smooth and convex. From the

results presented in Fig. 3, it is obvious that a nonlocal

friction formulation has a substantial influence on the shear

tractions, and at high values on cc, shear stresses almost

vanishes resembling a frictionless contact situation.

In Fig. 4 the corresponding results (at spherical contact)

for the radial surface displacements are shown for fric-

tionless contact and for finite friction ðl ¼ 0:2Þ with dif-

ferent values of the maximum allowable elastic slip. As it

may be seen in Fig. 4, the maximum radial surface dis-

placement at frictionless case is approximately 0:16h and is

Fig. 3 Normalized shear surface tractions at spherical indentation,

and frictional contact are shown for l ¼ 0:2 and cc=h ¼ 0; 0:1; 1. The

normalized pressure distribution is also shown multiplied with the

friction coefficient. The normalizing quantity pm is the average

contact pressure

402 Tribol Lett (2013) 51:397–407

123

approximately 50 % higher as compared to the case of

Coulomb friction. Basically, the same conclusion as in

Fig. 4 can be drawn as there is a very noticeable influence

from nonlocal frictional effects, and at high values on cc,

the frictionless contact situation is approached. It should be

emphasized though that, in particular when it comes to the

results in Fig. 4, nonlocal frictional effects are essentially

quantitative and not qualitative.

The radial stress distribution at the surface is shown in

Fig. 5 for the same frictional conditions and indenter

geometry as in Figs. 3, 4. In Fig. 5, the stress distributions

are presented in the region immediately outside of the

contact area, as this is where the maximum tensile stresses

are induced in specimen at elastic contact. As it has been

discussed in detail by many investigators, cf. e.g. [7, 21–

23], maximum surface radial stresses govern the Hertzian

fracture initiation in brittle materials. It is a well-known

fact that frictional effects reduce the maximum surface

tensile stress and move its location away from the contact

boundary. As shown in Fig. 5, the presence of Coulomb

friction on the interface reduces the maximum surface

radial stress more than twice as compared to the fric-

tionless case and moves its location from the contact

boundary to r ¼ 1:15a. From a practical point of view, it

is important to emphasize that as nonlocal frictional

effects reduces the shear tractions induced by indenter,

they result in a maximum radial stress value which is

significantly higher than at Coulomb friction, as shown in

Fig. 5. Furthermore, at nonlocal friction, the location of

maximum tensile stress moves closer to the contact

boundary. This result is indeed of substantial importance

when crack initiation and growth is at issue. Again, at

high values on cc, results are close to the frictionless

contact situation, and frictional effects have vanished.

Jelagin and Larsson [20] examined experimentally and

numerically the effect of friction on Hertzian fracture initi-

ation in glass. It was shown in their study that computational

predictions obtained with Coulomb friction taken into

account were in better agreement with experimental obser-

vations as compared to predictions based on frictionless

contact theory. It was, however, pointed out in [20] that even

with Coulomb friction taken into account, there is a certain

quantitative discrepancy between the experimental results

and computational findings, where essentially the influence

of friction on fracture loads was somewhat overestimated. In

[20], this discrepancy was attributed to the influence of the

random distribution of pre-existing defects in material sur-

face and was successfully accounted for based using Wei-

bull statistics. The nonlocal frictional effects may, however,

provide an alternative explanation for the experimental

observations, especially in situations where the contact

between rough surfaces is at issue.

Based on the results in Fig. 5, it seems important to

further emphasize some aspects concerning crack behavior

at spherical indentation in the context of nonlocal friction.

Therefore, in Fig. 6, the maximum tensile values for the

surface radial stresses are shown as function of the coef-

ficient of friction; l. This figure clearly indicates that

nonlocal effects are of significant importance and that these

effects intervene in a noticeable manner also at small

values on the coefficient of friction.

So far, only spherical indentation has been considered,

and it seems appropriate to now concentrate on a sharp

indentation. Accordingly, in Figs. 7, 8 and 9 results are

presented for the case of cone indentation with explicit

correspondence to Figs. 3, 4 and 6 in the spherical case.

Concerning general aspects of frictional effects at con-

tact, some features can be specifically commented upon.

Fig. 4 Normalized radial surface displacements are shown at spher-

ical indentation. Both frictionless and frictional contact, l ¼ 0:2 and

cc=h ¼ 0; 0:1; 1; are considered

Fig. 5 Normalized radial surface stresses are shown at spherical

indentation. Both frictionless and frictional contact, l ¼ 0:2 and

cc=h ¼ 0; 0:1; 1; are considered. The normalizing quantity pm is the

average contact pressure

Tribol Lett (2013) 51:397–407 403

123

For one thing, it can be seen in Fig. 7 that the slip region is

considerably smaller than that in case of spherical inden-

tation. Furthermore, the surface shear tractions are

reversed, i.e., they change sign, in a region close to the

symmetry axis. Indeed, as shown in Fig. 8, also the surface

radial displacements change sign in the center of the con-

tact area at sharp contact. It should be clearly emphasized

though that the effects discussed in this paragraph are not

due to nonlocal frictional effects but are a result of the

different mechanical behavior at sharp and blunt contact.

One may also observe in Fig. 8 that the maximum radial

surface displacement at frictionless cone indentation is

approximately 0:12h and is reduced by approximately

30 % at Coulomb friction.

As it may be seen from the results presented in Figs. 7

and 8, the effect of friction (both local and nonlocal for-

mulations) is somewhat smaller in the cone indentation

case as compared to the spherical one. This is due to the

fact that the relative radial displacements are somewhat

smaller in the cone case as shown in Fig. 8. As a result, as

depicted in Fig. 9, the effect from friction on the maximum

surface tensile stress is also somewhat less profound as

compared to the spherical case.

Above, only nonlocal frictional effects on local field

variables are considered. This is of course based on the

introductory discussion above as it has been shown in

Fig. 6 Maximum tensile values on the normalized radial surface

stresses are shown as function of the coefficient of friction, l; at

spherical indentation. The allowable elastic slip values cc=h ¼0; 0:1; 1; are considered. The normalizing quantity pm is the average

contact pressure

Fig. 7 Normalized shear surface tractions at cone indentation and

frictional contact are shown for l ¼ 0:2 and cc=h ¼ 0; 0:1; 1: The

normalized pressure distribution is also shown multiplied with the

friction coefficient. The normalizing quantity pm is the average

contact pressure

Fig. 8 Normalized radial surface displacements are shown at cone

indentation. Both frictionless and frictional contact, l ¼ 0:2 and

cc=h ¼ 0; 0:1; 1; are considered

Fig. 9 Maximum tensile values on the normalized radial surface

stresses are shown as function of the coefficient of friction, l; at cone

indentation. The allowable elastic slip values cc=h ¼ 0; 0:1; 1; are

considered. The normalizing quantity pm is the average contact

pressure

404 Tribol Lett (2013) 51:397–407

123

numerous studies and articles, cf. e.g. [13–19], that friction

does not have a noticeable effect on global contact

(indentation) properties. This result was presently con-

firmed, also when nonlocal frictional effects were

accounted for, as essentially no influence from friction at

all was found on the explicit values of the global inden-

tation quantities.

The conclusion just above also proved to be valid for

Vickers indentation. For brevity, no explicit results are

presented for this particular indenter geometry. In short

though, when it comes to, for example, the influence from

friction on local indentation quantities, Vickers indentation

results showed the same features as the corresponding

results for the axisymmetric indenters discussed in Figs. 3,

4, 5, 6, 7, 8 and 9.

It should be emphasized that the fact that there are no

influence from friction on global indentation properties is

particularly noticeable at elastic contact. At elastoplastic

deformation, friction can be of more interest, also when it

comes to global properties, cf. e.g., Carlsson et al. [17]. As

it is shown in [17] for the case of classical Coulomb fric-

tion, the presence of friction may alter the global inden-

tation parameters (i.e., mean pressure and the contact

radius) at rigid-ideally plastic deformations. In particular,

as reported in [17] for the limiting case of full adhesion, the

mean pressure is approximately 5 % higher and the contact

radii is approximately 5 % lower as compared to the ones

predicted in case of frictionless indentation. The influence

of Coulomb friction on surface radial and circumferential

stress components was found in [17] to be very substantial.

Namely, while at frictionless contact, both stress compo-

nents were compressive at rigid-ideally plastic indentation;

the presence of finite friction introduced a certain amount

of localized tension in both the radial and the circumfer-

ential direction.

Based on the results presented in Figs. 3, 4, 5, 6, 7, 8 and

9, the nonlocal frictional effects may be expected to result

in a response which lies between the ones predicted for the

Coulomb friction and the frictionless cases also in a rigid-

perfectly plastic situation. In order to confirm this, the

following computational study has been performed pres-

ently: the contact pair of spherical indenter and flat spec-

imen with parameters outlined above has been used to

model frictional indentation at rigid-perfectly plastic con-

ditions, the coefficient of friction, l was set to 0 and 0.4,

and at the presence of finite friction, the simulations have

been performed at cc=h ¼ 0; 1. Johnson [48] introduced a

parameter

K ¼ Ea

2ð1� v2ÞryRð12Þ

where ry is the flow stress at strain magnitude ey ¼0:4a=2R. Johnson [48] concluded that at K� 30, elastic

effects are negligible. In order to satisfy this condition, the

flow stress for the perfectly plastic material was set pres-

ently to ry ¼ 47MPa. Similarly to observations reported in

[17], it has been presently observed that the presence of

finite Coulomb friction results in smaller contact radius and

higher mean pressure as compared to the frictionless case.

Namely, the ratios between the contact area radii found at

the frictionless case and at finite friction areal¼0

al¼0:4¼ 1:02

andal¼0

al¼0:4¼ 1:01 for cc=h ¼ 0; 1 correspondingly. The

ratios for mean pressures were found to be pl¼0m

pl¼0:4m

¼ 0:97 and

pl¼0m

pl¼0:4m

¼ 0:99 again for cc=h ¼ 0; 1 correspondingly. In

Fig. 10, the surface radial stress distributions at rigid-per-

fectly plastic indentation are shown for the region imme-

diately outside the contact area; the results are shown for

the frictionless case as well for finite friction at cc=h ¼ 0; 1:

As it may be seen in Fig. 10, at plastic deformations, the

effect of friction on the surface radial stresses is opposite as

compared to the one reported above for purely elastic

indentation, this is due to the fact that the surface radial

displacement and thus the frictional tractions reverse their

direction. In particular, as one may notice in Fig. 10, while

at frictionless plastic indentation, the radial stresses are

compressive in the region around the contact area, the

presence of friction at cc=h ¼ 0 results in a small amount of

tensile stresses at the contact boundary. It may also be

observed in Fig. 10 that the presence of nonlocal frictional

effects results in the radial stress distribution which is

significantly lower as compared to the one obtained at

Coulomb friction on the interface and above the one found

at frictionless contact. It has to be pointed out that the

Fig. 10 Normalized radial surface stresses are shown at rigid-

perfectly plastic spherical indentation. Both frictionless and frictional

contact, l ¼ 0:4 and cc=h ¼ 0; 1; are considered. The normalizing

quantity pm is the average contact pressure

Tribol Lett (2013) 51:397–407 405

123

nonlocal friction effect on the surface radial stress illus-

trated in Fig. 10 is also qualitative, i.e., in contrast to the

Coulomb friction case, the radial stresses are compressive

everywhere at the specimen surface.

It may be concluded that the results presented above

indicate that the nonlocal frictional effects play a signifi-

cant role for the indentation response of materials in the

plastic regime. A thorough investigation of these effects

lies, however, beyond the framework of the present study

but will be undertaken in the future. One practical example

where a good understanding of frictional effects at com-

bined elastoplastic deformation is of significant importance

concerns the determination of residual stresses by inden-

tation testing, cf. e.g., Eriksson et al. [45]. Such a proce-

dure is most often carried out at the microindentation level

where nonlocal frictional effect can be very relevant.

4 Conclusions

Nonlocal frictional effects at elastic contact have been

studied. The main conclusions can be summarized as

follows:

• Frictional effects, local as well as nonlocal, have very

little influence on global indentation (contact) quantities.

• Nonlocal frictional effects on local field variables can

be substantial at blunt (spherical) contact

• Nonlocal frictional effects are of less importance at

sharp contact. Most often, frictionless contact results

are valid even at small values on the allowable elastic

slip parameter in a nonlocal frictional model.

• Nonlocal frictional effects will increase the maximum

tensile stresses (in comparison with corresponding

values given by an analysis based on Coulomb friction)

at the contact boundary.

It is suggested that future studies regarding this subject

should include plastic deformation in the analysis.

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