Available online at www.sciencedirect.com

Mechanics of Materials 40 (2008) 206–219

www.elsevier.com/locate/mechmat

Effects of mechanical properties and surface frictionon elasto-plastic sliding contact

S.C. Bellemare a,b, M. Dao a, S. Suresh a,*

a Department of Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue,

Cambridge, MA 02139, USAb Simpson, Gumpertz & Heger, 41 Seyon Street, Suite 1-500, Waltham, MA 02453, USA

Received 25 April 2007; received in revised form 31 July 2007

Abstract

Indentation hardness has been used extensively for material characterization and many recent computational studieshave established quantitative relationships between elasto-plastic mechanical properties and the response in instrumentedindentation. In contrast, very few studies have systematically quantified the effect of the plastic deformation characteristicson the frictional sliding response of metals and alloys. Building upon dimensional analysis and finite element computa-tions, a parametric study was carried out to extend our previous work to different contact friction conditions. For a widerange of elasto-plastic and contact friction parameters, we established closed form universal functions, for various contactconditions, that relate elasto-plastic properties (Young’s modulus, yield strength, and power law hardening exponent) tosteady state frictional sliding response (scratch hardness, pile-up height and overall sliding frictional coefficient). Distribu-tion of the plastic strain beneath the indenter was studied to rationalize the deformation modes versus elasto-plastic prop-erties and pile-up. In parallel, experiments were conducted for the effect of plastic flow characteristics on the frictionalsliding (or scratch) response under different surface friction conditions. Pure copper and a brass alloy were heat-treatedto vary yield strength and strain hardening exponent and the contact friction coefficient was varied by applying a liquidlubricant on the surface. Frictional sliding experiments were conducted using a nanoindentation testing system, wheregrain size and alloy composition were found to influence the response. Although variations in the frictional slidingresponse versus yield strength, strain hardening and friction were invariably coupled, the combined computational andexperimental approach enabled us to isolate the relative contributions of each parameter. The results clearly demonstratedthat an increase in the strain hardening exponent can significantly decrease the pile-up height, with known and furtherpotential implications for the evaluation of tribological damage.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Frictional sliding; Scratch test; Elasto-plastic properties; Contact mechanics; Friction

0167-6636/$ - see front matter � 2007 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2007.07.006

* Corresponding author. Tel.: +1 617 253 3320.E-mail address: ssuresh@mit.edu (S. Suresh).

1. Introduction

Material hardness is a mechanical property refer-ring to the normal contact force that a material cansupport per projected unit area of contact. Indenta-

.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 207

tion hardness tests have been used extensively formaterial characterization and also as a basis for pre-dicting the tribological response (Hutchings, 1992;Fischer-Cripps, 2000; Gouldstone et al., 2007).Indentation tests were traditionally based on anestimate of the residual area of contact or the rem-nant penetration depth, but developments and com-mercialization of depth-sensing instrumentedindentation systems have enabled continuous mea-surement of the force and displacement during load-ing and unloading. Following these advances, manystudies have examined the contact mechanics ofinstrumented indentation. Dimensional analysisand finite element methods (FEM) were employedto quantify relationships between the measuredforce–displacement (P–h) response and elasto-plas-tic properties (Dao et al., 2001; Mata et al., 2002;Matsuda, 2002; Tunvisut et al., 2002; Bucailleet al., 2003; Chollacoop et al., 2003; Cao and Lu,2004; Cheng and Cheng, 2004; Oliver and Pharr,2004; Ogasawara et al., 2005; Wang et al., 2005).Other studies also investigated experimentally theindentation response of various materials usinginstrumented systems (Schwaiger et al., 2003;VanLandingham, 2003; Schuh and Nieh, 2004).

As compared to normal indentation, few studieshave systematically investigated the mechanics forfrictional sliding (Bucaille et al., 2001; Bucaille andFelder, 2002; Subhash and Zhang, 2002; Younand Kang, 2004; Fang et al., 2005). In the steadystate regime of frictional sliding, the normal forceis maintained constant and the tangential displace-ment induces material flow and the formation ofridges or pile-ups on each side of the scratch scar.Under appropriate contact conditions, our relatedearlier computational study predicted a strong con-nection between the frictional sliding response andmaterial elasto-plastic properties (Bellemare et al.,2007). In fact, the effects of initial yield strengthand plastic strain hardening exponent were quanti-fied and isolated for their contribution to scratchhardness and pile-up height. This quantitativeapproach to study frictional sliding was proposedas an experimental tool for material characteriza-tion, but also as a simple predictor for the tribolog-ical response of materials. Recently, a similar set ofstudies were carried out to determine and comparethe hardness and friction response for a range ofstrain hardening characteristics (Wredenberg andLarsson, 2007). Their results showed a representa-tive plastic strain of 35% for an indenter apex angleof 68� (Wredenberg and Larsson, 2007), versus

33.6% for an apex angle of 70.3� in our earlier study(Bellemare et al., 2007). Although these values aresimilar, our previous work indicated that the repre-sentative plastic strain is considerably smaller forhigh strain hardening materials. Several other previ-ous experimental studies have used frictional slidingexperiments (Zhang et al., 1994, 1995; Liang et al.,1995; Deuis et al., 1996; Wilson et al., 2000; Bolducet al., 2003), but the underlying interpretation pro-vided only partial information about the relativecontributions to the overall frictional slidingresponse in terms of the material and contactparameters.

Contact friction significantly influences the slid-ing contact response. For a sliding contact wheresignificant plastic deformation develops underneaththe indented surface, we envision two componentsto the total or overall friction coefficient, which isdefined as ratio of tangential force over normalforce. The first component is from the local interac-tion between the indenter and material surfaces innormal contact, i.e. surface friction. This surfacefriction can be accounted for using Amontons’slaw with an appropriate friction coefficient and itcan be directly affected by lubrication. The secondcomponent to the overall friction is from the workfor plastically deforming the surface and it shouldbe influenced by the contact geometry and elasto-plastic properties, although a recent study suggestedindependence on properties for relatively soft mate-rials (Wredenberg and Larsson, 2007).

For normal indentation, FEM computationssuggested a significant influence of the friction coef-ficient on the pile-up behavior (Mesarovic andFleck, 1999; Carlsson et al., 2000; Mata and Alcala,2004). Experimentally, lubricants decrease theindentation hardness (Atkinson and Shi, 1989; Shiand Atkinson, 1990) and increase the hardness infrictional sliding (Brookes and Green, 1979; Broo-kes, 1981). From the viewpoint of the contact geom-etry evolution, the frictional force pushes thematerial downward during normal indentationwhile it pushes the material upward and to the frontand side in frictional sliding. This fundamental dif-ference explains the reverse effect of friction in fric-tional sliding versus normal indentation, but a moredetailed analysis is needed for a quantitative predic-tion of the effect of friction in sliding contact.

In the present study, we used dimensional analy-sis and large scale finite element computations, toextend our previous theoretical framework (Belle-mare et al., 2007) to include the influence of various

208 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

contact friction conditions. We established a set ofclosed form universal functions to relate elasto-plas-tic properties, i.e. Young’s modulus, yield strengthand power law hardening exponent, to steady statefrictional sliding response, i.e. scratch hardness,pile-up height and overall sliding frictional coeffi-cient, with respect to various contact friction condi-tions. In parallel, we conducted a comprehensive setof sliding contact experiments on a model materialsystem to investigate the effect of plastic flow char-acteristics and surface friction parameters on thefrictional sliding response, or scratch response.The results are compared with predictions fromour theoretical/computational results. Based on acomparison of experiments with computationalresults, the effect of plastic strain hardening on thedeformation field is also discussed.

401

501

601

ss (

MP

a)

700

600

500

400401

501

601

700

600

500

400

2. Experimental and computational methods

2.1. Material system

Copper was selected as a model material systembecause large variations in the strain hardeningexponent can be introduced by controlling the grainsize and composition. Commercially pure copper(99.9%) and a single phase copper–30 wt% zincalloy were obtained from Noranda Inc. (Pointe-Claire, Canada) in the form of cold worked sheetsthat were 0.6 mm thick. Standard dog-bone speci-mens were machined out of the sheets. Aftermachining, the specimens were divided into threegroups that were then heat-treated for recrystalliza-tion at temperatures of 450, 600 or 700 �C for 3 h.The microstructure of each alloy was then charac-terized in detail, including a quantification of theaverage grain size �d as listed in Table 1. With pureCu and the brass alloy, a total of six material condi-tions were investigated. In the absence of other sig-nificant changes to the microstructure, the

Table 1Material conditions tested with average grain size, initial yieldstrength and strain hardening exponent based on tensile testresults

Material T (�C) �d (lm) ry (MPa) n

Cu 450 20 ± 8 145 0.13Cu 600 150 ± 30 44 0.27Cu 700 380 ± 50 28 0.29Cu–Zn 450 27 ± 6 45 0.35Cu–Zn 600 76 ± 10 15.5 0.41Cu–Zn 700 180 ± 20 7 0.45

conditions with the different heat-treatments pro-vided the opportunity to study the specific effect ofthe grain size on the frictional sliding response.

Tensile tests were carried out on all materials toquantify the plastic deformation response. Prior totesting, the specimens were marked with ink at spec-ified interval distances to independently measure theplastic strain at maximum tensile strength. Thestress–strain curves were corrected for machine/specimen compliance and consistency was obtainedbetween the critical engineering strain, i.e. strain attensile strength, and the permanent elongation ofthe specimens measured using the marking tech-nique. True stress versus true strain power law hard-ening was used to fit the experimental data:

r ¼ ry 1þ Ery

ep

� �n

; ð1Þ

where ep is the equivalent plastic strain, ry is the ini-tial yield strength, E is the Young’s modulus of thematerial and n is the strain hardening exponent. Theresults from this fit are summarized in Table 1 andthree example curves are shown in Fig. 1. In the fit-ting procedure, more weight was given to the laterpart of the experimental curve where the plasticstrain is most significant.

2.2. The frictional sliding experiments

All specimens were mechanically polished to asurface roughness of less than ±5 nm and tested

0.0 0.2 0.3 0.4 0.5013

101

201

301

Fitted Cu-Zn 450oCCu 450oCCu-Zn 700oC

Tru

e S

tre

True Strain

300

200

100

00.1 0.

013

101

201

301300

200

100

0

Fig. 1. True stress versus true strain curves for three differentmaterials and the associated fitted function using power lawstrain hardening.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 209

on a commercial nanoindentation test system(NanotestTM, Micro Materials Ltd., Wrexham, Uni-ted Kingdom). The indenter was a conical diamondwith an apex angle h of 70.3� and a tip radius of2 lm. For the conditions of penetration depth inves-tigated, the size of the scratches was sufficientlylarge to consider the indenter as perfectly conical.The experiments were carried out under constantnormal load, P, at a velocity of 10 lm/s and overa total distance of 1500 lm, which was sufficientto attain steady state conditions after approximately300 lm and continue to generate a region of validsteady state profile. After the experiment, a seriesof at least 30 cross-sectional residual profiles wereobtained over the steady state regime by using aTencor P10 profilometer (KLA-Tencor, San Jose,California). The profilometer was equipped with aconical diamond probe which had an apex angleof 45� and a tip radius of 2 lm. The steady stateregime was also observed with a Leo VP438 scan-ning electron microscope (Leo Electron MicroscopyInc., Thornwood, New York).

Schematic drawings of frictional sliding areshown in Fig. 2, where the frictional sliding process,a cross-sectional view of symmetry plane during thesteady state stage, and a cross-sectional view of theresidual scratch profile are presented. Graphicalrepresentations for the pile-up height hp, the resid-ual penetration depth hr and the contact radius ar

are defined. The contact radius ar can be used

hh

x y

z

Indenter

Initial surface level

Stage 1: Normal indentation

Stage 2: Transient regime of frictional sliding

Stage 3: Steady statfrictional sliding

x y

z

Indenter

Initial surface level

Stage 1: Normal indentation

Stage 2: Transient regime of frictional sliding

Stage 3: Steady statfrictional sliding

a

Fig. 2. (a) A schematic drawing of the frictional sliding process, (b) a cr(c) a cross-sectional view of the residual scratch profile. All solid linedeformed.

directly to calculate the overall resistance to pene-tration using the traditional definition of hardness(Tabor, 1951; Johnson, 1985; Williams, 1996;Fischer-Cripps, 2000; Gouldstone et al., 2007)

H S ¼2Ppa2

r

; ð2Þ

where P is the applied normal load. Assuming theabsence of significant size effects, the main advan-tage of the conical geometry is the size-indepen-dence due to self similarity. With this assumption,the simple ratio of hp/hr provides an indication onthe tendency of the material to form a pile-up.

After hardness and pile-up profiles, friction is thethird important parameter to the frictional slidingresponse. Because friction occurs at two differentlevels, we will separate the overall friction coefficientas

ltot ¼F t

P¼ la þ lw; ð3Þ

where Ft is the total tangential force, la is the coef-ficient of friction for the normal contact and lw isthe friction contribution besides la. The parameterla is governed by Amontons’s law of friction whichspecifies the ratio between the normal pressure andthe local tangential traction. The value of la willbe varied experimentally using an isostearic acid(Century 1105, Arizona Chemical, Jacksonville,Florida) as a liquid lubricant. The lubricant had a

P

Fthp

r h

x

y

θ

P

Fthp

r h

x

y b

z hphr

ary

z hphr

ary c

e

Indenter motion

e

oss-sectional of the symmetry plane in the steady state regime ands represent the top free surface of the material being plastically

210 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

viscosity of 70 cps at 25 �C and it contained mainlyC18 branched chains (59%) and C18 cyclic chains(11%).

2.3. Dimensional analysis and computational model

setup

For steady state frictional sliding of elasto-plasticmaterials, the contact conditions can now be ana-lyzed and predicted in detail. To simplify the elasticcontributions from the material and the indenter,we used the reduced modulus (Johnson, 1985)

E� ¼ ð1� m2ÞE

þ ð1� m2i Þ

Ei

� ��1

; ð4Þ

where Ei and mi are the Young’s modulus and Pois-son’s ratio of the indenter, respectively. For the dia-mond indenter used in the experiments,Ei = 1100 GPa and mi = 0.07 are given (Mat-Web:www.matweb.com, 2006). The elastic con-stants specified for polycrystalline Cu were takento be: E = 110 GPa and m = 0.35 (Mat-Web:www.matweb.com, 2006). These propertieswere assumed to be isotropic for the conditionstested.

For a fixed cone apex angle of h = 70.3� and afixed friction coefficient of la = 0.15, three newdimensionless functions have been defined usingdimensional analysis and evaluated numericallythrough a comprehensive parametric study (Belle-mare et al., 2007). Under the assumptions for fixedh and la, these functions predict the frictional slid-ing response based on elasto-plastic properties andthe following closed-form functions:

Pa n;ry

E�� �

¼ H S

ry

� �¼ a1ðnÞ ry

E�� �a2ðnÞ

�; ð5Þ

Pb n;ry

E�� �

¼ hp

hr

¼ Pb;RPðnÞ�

1þ ry

X bðnÞE�� �pbðnÞ

" #and

ð6Þ

Pc n;ry

E�� �

¼ F t

P

� �¼ ltot

¼ Pc;RP

1þ ry

X cðnÞE�� �pcðnÞ

" #; ð7Þ

where the subscript ‘RP’ indicate the value ofthe function at the limit of rigid-plastic properties,the variable Ft is for the overall lateral forceand the variable ltot is for the overall friction coef-ficient. Simple numerical expressions were provided(Bellemare et al., 2007) for the other numerical

terms for the sub-functions of n(a1(n),a2(n),Pb,RP(n), Xb(n), pb(n), Xc(n) and pc(n)) and the con-stant Pc,RP. All functions are smooth and with amonotonic variation, except Xb(n) which has aminimum at n = 0.4. With these functions and theirunderlining assumptions, one can specify the elasto-plastic properties of a material and predict the fric-tional sliding response in terms of normalizedscratch hardness Hs/ry(Pa), ratio of pile-up height(Pb) and overall friction coefficient ltot(Pc).

In their most general form, these dimensionlessfunctions are expressed as:

Pa ¼ f n;ry

E�; la; h

� �; ð8Þ

Pb ¼ f n;ry

E�; la; h

� �; and ð9Þ

Pc ¼ f n;ry

E�; la; h

� �: ð10Þ

For the current study, we specifically investigate theeffect of the friction coefficient la, which will add adimension to the dimensionless functions Pa, Pb

and Pc presented in Eqs. (5)–(7). On the other hand,the parameter h remains fixed at 70.3� in this study.Full three-dimensional models were used becausethe stress and strain fields generated by frictionalsliding cannot be approximated using two-dimen-sional or axisymmetric FEM. The complete meshdomain contained 170,000 reduced integration 8-noded elements. The finite element computationswere performed using the general purpose FEMsoftware package ABAQUS (Simulia, Providence,Rhode Island, USA). The solution method was ex-plicit and based on Eulerian boundaries where themesh remains stationary. Additional details on themeshing procedure and model validation can befound elsewhere (Bellemare et al., 2007). The ap-proach was well tested for mesh refinement and con-vergence, and for the independence of the solutionmethod adapted.

Finite element solutions were obtained for fixedvalues of the friction coefficient la = 0, 0.08, 0.2 or0.3. For the materials with a plastic strain hardeningexponent n 6 0.2, la was limited to a maximumvalue of 0.2 (for n < 0.35) or 0.3 (for n > 0.35)because higher la could generate physical andnumerical instability due to excessive pile-up. Inthe complete parametric study a minimum of 6 casesof ry/E* were explored for any combination of n

and la covered, adding a total of 90 cases to the pre-vious study (Bellemare et al., 2007) that focused onla = 0.15. The same procedure as in the previous

10-4 10-3 10-2 10-1

2

3

10

100

n = 0.02

n = 0.1

n = 0.2

n = 0.35

μa values

00.080.150.2 or 0.3

HS / σ

y

y / Eσ *

n = 0.5

Fig. 3. Effect of the friction coefficient la on the normalizedhardness versus normalized yield strength relationship.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 211

study was used to extract hp, hr and ar from thenodal position of the residual profile. The FEMresults from both studies were all incorporated intothe new dimensionless functions so as to develop acomprehensive understanding of the effects of E*,ry, n and la on frictional sliding.

3. Results and discussion

3.1. Quantitative descriptions of contact sliding with

friction

Extending our previous study (Bellemare et al.,2007), we now consider the effect of varying frictioncoefficient la on the sliding contact response.Computationally, three parameters can be variedindependently: the friction coefficient la, the normal-ized initial yield strength ry/E*, and the strain hard-ening exponent n. We quantified the effect of each ofthese parameters within the framework of the gen-eral dimensionless functions Pa, Pb and Pc (Eqs.(8)–(10)). Although we could have used new func-tions to represent the new data, we found that thegeneral dependency on n and ry/E* was very similarto that in our earlier analysis (Eqs. (4)–(7)). There-fore, the influence of the friction coefficient wasincorporated by adding penalty terms in the formerequations (Bellemare et al., 2007). The new sub-functions and their coefficients were determined byminimization of the residuals. These universaldimensionless functions represent fits to the FEMresults, and the constructed fitting functions will becompared with and verified against experimentalmeasurements for a range of cases in Section 3.2.

For the normalized scratch hardness HS/ry, thebest fit was obtained by using the function

Pa ¼HS

ry

� �

¼ a1ðnÞ þ nCa1ðlaÞ½ � ry

E�� �½a2ðnÞþCa2ðlaÞ�

ð11Þ

with

Ca1ðlaÞ ¼ 0:12� 0:64=½1þ e30ðla�0:1Þ�;Ca2ðlaÞ ¼ 0:006� 0:0278=½1þ e25ðla�0:1Þ�

and the sub-functions (a1(n) and a2(n)) as evaluatedpreviously (Bellemare et al., 2007). For hardness,the contribution from the friction coefficient la is in-cluded with the terms that previously depended onlyon the strain hardening exponent n. The sub-func-tions Ca1 and Ca2 are exponential growth functions

and their values vary monotonically with la. Theywere selected to minimize the error when fitting allthe data throughout the range of numerically simu-lated material properties.

Fig. 3 presents the function Pa and the associatedFEM data points on log–log plots of the normalizedhardness HS/ry versus the normalized yield strengthry/E*. The data for the five different values of thestrain hardening exponent n are well separated.For each value of n the effect of the surface frictioncoefficient la is more limited, but the followingtrends are seen:

• For n ffi 0.35, the effect of friction is negligible;• For n > 0.35, the hardness increases with increas-

ing friction coefficient la;• For n < 0.35, the hardness decreases with increas-

ing friction coefficient la.

For n < 0.35, the decrease in hardness could be dueto an increase in the amount of material beingpushed to the side of indenter, increasing the areaof contact through a higher pile-up. Although thechanges appear limited on this log–log plot, theeffect on hardness can reach 5–15% depending onthe conditions.

The second function studied is Pb for the nor-malized pile-up height hp/hr. The behavior of thisfunction should be asymptotic on both sides withlittle influence of yield strength for rigid-plasticmaterials and with a residual height of zero in the

212 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

elastic limit. The logistic function selected in previ-ous study respects both of these limits and it allowsfor a simple expression of the pile-up height:

Pb¼hp

hr

¼Pb;RPðnÞCb;RPðlaÞ 1þ ry

X bðnÞCXbðlaÞE�� �pbðnÞ

" #,

ð12Þ

with

Cb;RPðlaÞ ¼ 0:909þ 0:627la;

CXbðlaÞ ¼ 0:651þ 1:21la þ 7:61l2a;

and the sub-functions of n (Pb,RP(n), Xb (n) andpb(n)) as evaluated previously (Bellemare et al.,2007). Fig. 4 presents the function Pb evaluated atthe different values of n and la for which we haveFEM results. The sets of curves for the specified val-ues of n illustrate in more detail the combined influ-ence of n and ry/E* on hp/hr. Within each set ofthese curves, the normalized pile-up height hp/hr al-ways increases when the friction coefficient la in-creases. This increase in height can be associatedwith an increase in the interaction forces that pushthe material to the front and sides of the indenter.

The absolute value of the offset in hp/hr caused byvariations in la is significant for the strain harden-ing exponent range: 0.02 6n 6 0.35. For the largestvalue of n = 0.5, where the transition between pile-up and sink-in is approached, the effect of frictionis limited. The sink-in phenomenon is a different

10-4 10-3 10-2 10-1

0.0

0.2

0.4

0.6

0.8

1.0

n = 0.5

n = 0.35

n = 0.2

μa values

00.080.150.2 or 0.3

h p / h r

σy / E*

n = 0.02

Fig. 4. Effect of the frictional coefficient la on the normalizedpile-up height versus normalized yield strength relationship.

behavior than the pile-up behavior reported in thispaper because the motion of the plastically deform-ing material is all downward ahead of the tip. Thissink-in was observed during simulation for n = 0.5and ry/E* > 0.005, but the results are not presentedhere. With sink-in, there is no real pile-up and it isdifficult to ascertain the area of contact from theresidual profile. Including sink-in behavior in theequations would require a more detailed analysis.At the same time, it would probably find limitedpractical applications because very few hardenedductile materials have properties beyond the rangecovered in this study.

To illustrate the effect of the strain hardeningexponent on the pile-up, a semi-quantitativedescription was developed for the evolution of thedeformation zone with plastic properties. Data forthe equivalent plastic strain were extracted fromthe elements located in the unloaded region and ata distance ar/2 from the symmetry plane and theyare reported in Fig. 5 as a function of the distancebeneath the scratch surface. Fig. 5a presents a seriesof equivalent plastic strain contour plots for a rela-tively soft material where ry/E* = 0.001. As thestrain hardening exponent decreases, the equivalentplastic strain near the surface increases significantly.For the strain distribution beneath the surface, theplastic strain deceases less rapidly for the materialswith a higher hardening exponent. The plastic strainis more distributed with increasing n, which is con-sistent with the decrease in the pile-up height asthe flow of material extends further beneath mate-rial/indenter interface.

The initial yield strength also influences theplastic strain distribution. Fig. 5b presents a simi-lar series of results for a harder material wherery/E* = 0.01. Although the general shape of thesecurves remains the same as those for the softermaterial, there is a general and significant decreasein the magnitude of the plastic strain. When usedtogether, Fig. 5a and b provide a description ofthe evolution of equivalent plastic strain for differ-ent strain hardening exponent and initial yieldstrength values. The hardening exponent signifi-cantly affects the distribution, while the initial yieldstrength clearly influences the average magnitude ofthe equivalent plastic strain.

The third and last universal function Pc is for theoverall friction coefficient ltot or the ratio of the lat-eral force of interaction Ft over the normal force P

between the indenter and the surface. The effect offriction was incorporated to yield

a

0.00.0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

n values 0.1 0.2 0.35

Equ

ival

ent p

last

ic s

trai

n

Distance beneath the scar normalized by hr

b

0.00.0 0.5 1.0 1.5 2.0 2.50

1

2

3

4

n values0.10.2

0.35

Equ

ival

ent p

last

ic s

trai

n

Distance beneath the scar normalized by hr

Fig. 5. Magnitude of the equivalent plastic strain beneath the indenter for (a) ry/E* = 0.001 and (b) ry/E* = 0.01.

10-4 10-3 10-2 10-1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

μa = 0

μa = 0.08

μa = 0.15

μa = 0.2

n values 0.02 0.1 0.2 0.35 0.5

Fric

tion

coef

ficie

nt (

μ tot=

μ a+μ w

)

σy / E*

μa = 0.3

Fig. 6. Effect of the frictional coefficient la on the overall frictioncoefficient versus normalized yield strength relationship.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 213

Pc¼F t

P

� �¼ ltot¼laþlw¼la

þðPc;RPCc;RPðlaÞ�laÞ 1þ ry

X cðnÞCX cðlaÞE�� �pcðnÞ

" #,

ð13Þ

with

Cc;RPðlaÞ ¼ 0:586þ 2:6la þ 0:877l2a;

CX cðlaÞ ¼ 0:8þ 1:33la þ 0:235l2a

and the sub-functions of n (Pc,RP, Xc(n) and pc(n))as evaluated previously (Bellemare et al., 2007).Without losing generality, ltot is defined to be thesum of two contributions from the quantities la

and lw (see Eq. (3)), where lw encompasses all con-tributions to the lateral force that cannot be ac-counted for by Amontons’s law of friction with acoefficient la evaluated from the nominally elasticcontact between two bodies. Therefore, lw accountsfor all increases in Ft that are required to deformthe material plastically and to move it under andto the side of the indenter, leaving the path for theadvancing indenter. Consequently, lw is the onlyterm in the equation that depends on mechanicalproperties.

Fig. 6 graphically represents the function Pc

where the sets of adjacent curves are now for a givenvalue of the friction coefficient la and the differentseries are for the five different values of la. Thecurves were plotted using Eq. (9) while the FEMdata points are also included to illustrate the fitting

accuracy. Although the effect of material propertieson the geometrical friction coefficient is smaller thanthe pile-up height, the variations over the rangestudied are still significant. The large range of mate-rial parameters used in the current study is probablythe origin for the discrepancy between the effect offriction that we found and other published resultswhere the effect of material parameters on geometri-cal friction was found limited over a narrowerrange of elasto-plastic properties (Wredenberg and

214 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

Larsson, 2007). It is evident that, for a given set ofelasto-plastic properties, the lw term consistentlyincreases with la. Another feature that persists forthe different values of la is that friction is indepen-dent of yield strength and strain hardening exponentat the limit of a rigid-plastic material.

3.2. Experimental results and correlations with

computational results

An experimental study was undertaken to mea-sure and control the friction coefficient. The valueof la was measured through a repeated frictionalsliding test in which a spherical tip with a radiusof 100 lm was used for repeatedly sliding over thesame area 12 times. The normal load P was fixedat 1 N and the material was a high strength speci-men of pure copper with an indentation hardnessof 1.5 GPa. Under those conditions, the ratchetingor deepening of the impression progressivelyreduced and became negligible after approximately8 passes. Fig. 7 shows the raw friction signalobtained during such an experiment. The frictioncoefficient is found to decrease from an initial valueof approximately 0.22 for the first pass to a steadystate value of approximately 0.14. This techniqueof measuring la may result in a small overestima-

00 2 4 6 8 100.00

0.05

0.10

0.15

0.20

0.25

Pass # 1

2 3 5 12

12(lub)

Fric

tion

coef

ficie

ntμ to

t

Sliding distance normalized by R

Fig. 7. Experimental determination of the friction coefficient la.From top to bottom the curves represent the coefficient of frictionltot for an increasing number of passes over the same area until asteady state is reached. The dotted lime at the bottom is for thesteady state with lubrication.

tion with a potential for a limited amount of plastic-ity in steady state, but it offers a reasonable estimateof la in an efficient manner.

To vary experimentally la, we used the isostearicacid as a liquid lubricant. The lubricant was addedto the surface and the tip after carrying out the loadand displacement calibrations. With the relativelysmall velocity of the tip, the conditions were bound-ary lubrication. To verify the influence on la, theexperiment with a spherical tip presented in Fig. 7was followed by another experiment under the sameconditions but with lubricant. For each pass, thefriction coefficient was significantly lower with alubricant and Fig. 7 presents the results for thesteady state regime. The use of lubrication causedthe steady state friction coefficient to decrease from0.14 to 0.11. This 25% decrease could have severalbeneficial effects including a reduction in chip for-mation on hard materials with limited plastic strainhardening exponent. The variation also allows for amore comprehensive comparison between computa-tional predictions and experimental results.

After measuring experimentally the surface fric-tion coefficient, we performed frictional slidingexperiments with the conical tip. The tests were onpure Cu and Cu–Zn, at a normal load P = 2 N,and for both the unlubricated and the lubricatedcases. For each material condition, the experimentwas repeated five times and profilometry was carriedout on each profile. From these profiles, the scratchparameters were calculated and compared with pre-dictions using the dimensionless functions, Eqs.(11)–(13). For each of the three parameters underthe unlubricated condition, Table 2(panel a) indi-cates a maximum difference between the experimentand the predictions of at most 7%, 13% and 5% forthe hardness, pile-up and friction, respectively. Inaddition, the differences between experiments andpredictions indicate the absence of a definite trend,suggesting that the overall dependence of slidingbehavior on elasto-plastic properties is correctlypredicted over the range of properties studied. Sim-ilarly, Table 2(panel b) shows the maximum differ-ence of 11% for lubricated case for hardness andpile-up ratio between experiments and predictions.

The effect of different friction coefficients on nor-malized pile-up height is summarized in Fig. 8. Thematerials shown in Fig. 8 are classified by theirvalue of the strain hardening exponent n. The valuesof the pile-up height is consistently lower with alubricant, but the effect of lubrication is foundto progressively decrease with increasing n and

Table 2Experimental results compared with the predictions made using the dimensionless functions: (a) unlubricated friction (la = 0.14) and (b)lubricated friction (la = 0.11)

Material (treatment T in �C) Properties HS (GPa) hp/hr ltot

(a) ry (MPa) n Exp. FEM D (%) Exp. FEM D (%) Exp. FEM D (%)

Cu(450) 145 .13 0.66 0.71 �7 .7 0.69 3 .42 .406 3Cu(600) 44 .27 0.62 0.66 �6 .57 0.52 13 .43 .409 5Cu(700) 28 .29 0.60 0.55 �1 .51 0.50 5 .42 .410 2Cu–Zn(450) 45 .35 1.13 1.16 �3 .44 0.40 12 .41 .407 1Cu–Zn(600) 15.5 .41 0.95 0.90 5 .3 0.33 �8 .40 .410 �2Cu–Zn(700) 7 .45 0.81 0.78 4 .28 0.28 1 .41 .411 0

Material (T in �C) Properties HS (GPa) hp/hr

(b) ry (MPa) n Exp. FEM D (%) Exp. FEM D (%)

Cu (450) 145 .13 0.71 0.73 �2 0.63 0.68 �7Cu (600) 44 .27 0.70 0.67 4 0.47 0.50 �7Cu (700) 28 .29 0.62 0.56 11 0.45 0.49 �7Cu–Zn (450) 45 .35 1.19 1.17 2 0.41 0.39 4Cu–Zn (600) 15.5 .41 0.92 0.91 1 0.29 0.33 �11Cu–Zn (700) 7 .45 0.78 0.78 0 0.28 0.28 1

The materials are listed in order of increasing n.

0.1 0.2 0.3 0.4 0.50.2

0.3

0.4

0.5

0.6

0.7

0.8

(7)(16)

(45)

(28)

(44)

Not lubricated Lubricated

Nor

mal

ized

pile

-up

heig

hth p/

h r

Strain hardening exponent n

(145)

Fig. 8. Influence of lubrication on the experimentally measuredvalues of the pile-up height. The numbers in parenthesis are thevalues of the initial yield strength in units of MPa for each case.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 215

eventually vanish for n = 0.5. For the scratch hard-ness, the results presented in Table 2(panel b) areconsistent with the predictions from the simulationfor n below 0.35 where the hardness increased withdecreasing la. Above n = 0.35, a decrease in hard-ness was observed for all three Cu–Zn alloys to anextent slightly larger than that predicted com-putationally.

The overall coefficient of friction ltot can be mea-sured readily using an instrumented nanoindenter

wherein the frictional sliding experiment is per-formed. For all six ductile materials investigated,Table 2(panel a) shows a maximum difference of lessthan 7% in the value of ltot. A reasonable agreementwas obtained between the experiments and the pre-dictions. However, at least for the range of condi-tions studied, it would be difficult to use only thefriction information to differentiate between thematerials. A similar observation was also madefrom a previous study on nickel where the effect ofgrain-size refinement and large variations in yieldstrength did not significantly change the overall fric-tion coefficient (Bellemare et al., 2007). Althoughthe friction coefficient ltot can be readily monitoredduring an experiment, the two scratch parametersthat are most sensitive to elasto-plastic propertiesare definitely the hardness and the normalizedpile-up height. In other words, for the conditionsexplored in this work, relatively little variation inoverall friction coefficient was found.

For illustration purposes, we now extract repre-sentative experimental pile-up profiles from the dif-ferent conditions listed in Table 2 and discuss themwith respect to the dimensionless functions. Aslisted in Table 2, with the same heat-treatment con-dition, the pile-up height is always lower in Cu–Znthan in pure Cu. From the dimensionless functionin Eq. (12), a decrease in yield strength, as seenwith Cu–Zn, is predicted to always increase thepile-up height. Therefore, any significant decrease

-10 -5 0 5 10-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

pure Cu Cu-Zn

Sur

face

hei

ght n

orm

aliz

ed b

y h r

Distance to symmetry plane normalized by hr

-10 -5 0 5 10-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Pure Cu Cu-Zn

Sur

face

hei

ght n

orm

aliz

ed b

y h r

Distance to symmetry plane normalized by hr

a

b

Fig. 9. Cross-section profiles for the experiments on the materialsrecrystallized at (a) the lowest temperature of 450 �C and (b) thehighest temperature of 700 �C. There are five data sets for each ofthe two materials.

216 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

in pile-up height observed with Cu–Zn is necessarilydue to the increase in the strain hardening exponentbecause the initial yield strength is lower for Cu–Znand a lower yield strength would normally increasethe pile-up. This experimental comparison suggestsa dominant effect of the strain hardening exponent.It is also in agreement with Eq. (12) that predicts alimited sensitivity of pile-up height on the initialyield strength ry for the range of elasto-plasticproperties covered in this study. Therefore, in thefollowing discussion, we primarily focus on the var-iation of n to interpret the results from theseexperiments.

A strong correlation can be identified betweenthe strain hardening exponent n and the normalizedpile-up height. Fig. 9a presents five individual resid-ual profiles for each of the two materials recrystal-lized at the lowest temperature of 450 �C, acondition for which n is 0.13 for pure Cu and 0.35for Cu–Zn. On this figure, the direction of motionof the indenter is normal to the plane of the image.Although the profiles of the two materials are simi-lar in general shape, the average value of hp/hr

decreases from 0.70 to 0.44 as n increases from0.13 to 0.35. Similarly, Fig. 9b presents profiles forrecrystallization at the highest temperature wheren is 0.29 for pure Cu and 0.45 for Cu–Zn. Withthe decrease in n between pure Cu and Cu–Zn, theaverage hp/hr decreases by nearly one half, from0.51 to 0.28. With an experimental scatter in heightof the order of ±0.05 and excellent reproducibilitybetween the different scratches, the differencesbetween pure Cu and Cu–Zn are significant and wellbeyond the level of fluctuations and consistent withthe finite element predictions. For these two specificexamples, the initial yield strength of the materialwas lower for the high hardening case, which wouldhave increased the pile-up height based on Eq. (12).Therefore, the decrease in hp/hr can only be due tothe increase in n. In fact, the decrease and variationbetween the conditions shown in Fig. 9 would havebeen even more significant without the difference ininitial yield strength between the materials. Thus,the effects of n alone would be higher than the differ-ences shown in Fig. 9.

Fig. 10 presents secondary electron imagesobtained in a scanning electron microscope(SEM). For each of the four material conditions,the images present the steady state regime and thefinal termination of the experiment for an indentermoving downward. In the steady state regime, thepile-up is more regular for the samples recrystallized

at 450 �C than for those recrystallized at 700 �C. Atleast in Cu–Zn, the preferential orientation of thedeformation bands illustrates an effect of individualgrains. For pure Cu recrystallized at 700 �C, thereare also changes in the orientation of the bands onthe free surface that are consistent with the intrinsiceffect of grains. Since the grains are larger for thehighest recrystallization temperature of 700 �C, theintrinsic effect of grains with different orientationscould well explain the variability in scar width andsurface features.

Fig. 10. Top surface image over the steady regime and the termination of a scratch for: (a) pure Cu recrystallized at 450 �C, (b) Cu–Znalso at 450 �C, (c) pure Cu at 700 �C and (d) Cu–Zn also at 700 �C. The indenter was traveling from top to bottom.

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 217

Intrinsic effects from the microstructure causedlocal fluctuations in the scratch pattern, includingscar width and pile-up height. However, it shouldbe noted that the average values remained consis-tent with the FEM predictions which are based ona continuum formulation. Due to the lateral dis-placement, frictional sliding offers the advantageover normal indentation to probe a larger volumeof material and can generate averaged values ofelasto-plastic properties from a single experiment.

4. Conclusions

The frictional sliding contact of elasto-plasticmaterials was studied experimentally and computa-tionally. The following conclusions can be drawn:

1. High plastic strain hardening significantlydecreases the normalized pile-up height for thematerial left on each side of the scratch scar. Amore refined grain microstructure reduces the

218 S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219

variability in the frictional sliding process, and italso decreases the normalized pile-up height.

2. High plastic strain hardening or yield strengthdecreases the magnitude of the equivalent plasticstrain underneath the indenter. Strain hardeningalso distributes the strain to a greater distancebeneath the surface of contact.

3. Dimensionless functions developed in our paral-lel study (Bellemare et al., 2007) on instrumentedfrictional sliding were modified to include theeffect of the friction coefficient. For materialsfor which the strain hardening exponent is below0.35, scratch hardness increases if frictiondecreases.

4. The most significant effect of friction is toincrease the normalized pile-up height. The effectdecreases for very large strain hardening expo-nents. No previous studies had reported the spe-cific effect of friction on the frictional slidingresponse of a large variety of materials.

5. An isostearic acid used as a boundary lubricantcan decrease by 25% the friction coefficientbetween the surface and the diamond tip. Theexperimental effect of lubrication on the hardnessand pile-up was consistent with our finite elementpredictions.

The frictional sliding experiment can be well con-trolled and designed to consistently yield results inagreement with the computational predictions. Itcould become an alternative or a complement to anormal indentation test.

Acknowledgements

The authors would like to acknowledge the finan-cial support of the Defense University Research Ini-tiative on Nano Technology (DURINT) which isfunded at MIT by ONR under grant N00014-01-1-0808, as well as a research grant provided by Sch-lumberger Limited. Post-Graduate Fellowship fromthe Natural Science and Engineering ResearchCouncil of Canada is also acknowledged. Finally,special thanks to Professor Ivan Dickson fromEcole Polytechnique de Montreal for coordinatingthe supply of the materials used for this study.

References

Atkinson, M., Shi, H., 1989. Friction effect in low load hardnesstesting of iron. Materials Science and Technology 5 (6), 613–614.

Bellemare, S., Dao, M., Suresh, S., 2007. The frictional slidingresponse of elasto-plastic materials in contact with a conicalindenter. International Journal of Solids and Structures 44(6), 1970–1989.

Bolduc, M., Terreault, B., Reguer, A., Shaffer, E., St-Jacques,R.G., 2003. Optimum tribological improvement of aluminumusing oxygen plasma source ion implantation. Journal ofMaterials Research 18 (8), 1761–1764.

Brookes, C.A., 1981. Scratch and indentation hardness ofcrystals. Philosophical Magazine A 43 (3), 529–543.

Brookes, C.A., Green, P., 1979. Anisotropy in the scratchhardness of cubic crystals. Proceedings of the Royal Societyof London Series A – Mathematical and Physical Sciences 368(1732), 37–57.

Bucaille, J.L., Felder, E., 2002. Finite-element analysis ofdeformation during indentation and scratch tests on elastic-perfectly plastic materials. Philosophical Magazine A –Physics of Condensed Matter Structure Defects and Mechan-ical Properties 82 (10), 2003–2012.

Bucaille, J.L., Felder, E., Hochstetter, G., 2001. Mechanicalanalysis of the scratch test on elastic perfectly plastic materialswith the three-dimensional finite element modeling. Wear 249(5–6), 422–432.

Bucaille, J.L., Stauss, S., Felder, E., Michler, J., 2003. Determi-nation of plastic properties of metals by instrumentedindentation using different sharp indenters. Acta Materialia51 (6), 1663–1678.

Cao, Y.P., Lu, J., 2004. Depth-sensing instrumented indentationwith dual sharp indenters: stability analysis and correspond-ing regularization schemes. Acta Materialia 52 (5), 1143–1153.

Carlsson, S., Biwa, S., Larsson, P.-L., 2000. On frictional effectsat inelastic contact between spherical bodies. InternationalJournal of Mechanical Sciences 42 (1), 107–128.

Cheng, Y.T., Cheng, C.M., 2004. Scaling, dimensional analysis,and indentation measurements. Materials Science & Engi-neering R-Reports 44 (4–5), 91–149.

Chollacoop, N., Dao, M., Suresh, S., 2003. Depth-sensinginstrumented indentation with dual sharp indenters. ActaMaterialia 51 (13), 3713–3729.

Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A.,Suresh, S., 2001. Computational modeling of the forward andreverse problems in instrumented sharp indentation. ActaMaterialia 49 (19), 3899–3918.

Deuis, R.L., Subramanian, C., Yellup, J.M., 1996. Abrasive wearof aluminium composites – a review. Wear 201 (1–2), 132–144.

Fang, L., Cen, Q., Sun, K., Liu, W., Zhang, X., Huang, Z., 2005.FEM computation of groove ridge and Monte Carlo simu-lation in two-body abrasive wear. Wear 258 (1–4 SPEC ISS),265–274.

Fischer-Cripps, A.C., 2000. Introduction to Contact Mechanics.Springer-Verlag, New York.

Gouldstone, A., Chollacoop, N., Dao, M., Li, J., Minor, A.M.,Shen, Y.-L., 2007. Indentation across size scales and disci-plines: recent developments in experimentation and modeling.Acta Materialia 55 (12), 4015–4039.

Hutchings, I.M., 1992. Tribology: Friction and Wear of Engi-neering Materials. CRC Press, Kent.

Johnson, K.L., 1985. Contact Mechanics. Cambridge UniversityPress, London.

Liang, Y.N., Ma, Z.Y., Li, S.Z., Li, S., Bi, J., 1995. Effect ofparticle size on wear behaviour of sic particulate-reinforced

S.C. Bellemare et al. / Mechanics of Materials 40 (2008) 206–219 219

aluminum alloy composites. Journal of Materials ScienceLetters 14 (2), 114–116.

Mata, M., Alcala, J., 2004. The role of friction on sharpindentation. Journal of the Mechanics and Physics of Solids52 (1), 145–165.

Mata, M., Anglada, M., Alcala, J., 2002. A hardness equation forsharp indentation of elastic-power-law strain-hardening mate-rials. Philosophical Magazine A: Physics of CondensedMatter, Structure, Defects and Mechanical Properties 82 (10SPEC), 1831–1839.

Matsuda, K., 2002. Prediction of stress–strain curves of elastic–plastic materials based on the vickers indentation. Philosoph-ical Magazine A: Physics of Condensed Matter, Structure,Defects and Mechanical Properties 82 (10 SPEC), 1941–1951.

MatWeb: www.matweb.com, 2006. Material Property Data.Automation Creations, Inc.

Mesarovic, S.D., Fleck, N.A., 1999. Spherical indentation ofelastic–plastic solids. Proceedings of the Royal Society ofLondon Series A – Mathematical Physical and EngineeringSciences 455 (1987), 2707–2728.

Ogasawara, N., Chiba, N., Chen, X., 2005. Representative strainof indentation analysis. Journal of Materials Research 20 (8),2225–2234.

Oliver, W.C., Pharr, G.M., 2004. Measurement of hardness andelastic modulus by instrumented indentation: advances inunderstanding and refinements to methodology. Journal ofMaterials Research 19 (1), 3–20.

Schuh, C.A., Nieh, T.G., 2004. A survey of instrumentedindentation studies on metallic glasses. Journal of MaterialsResearch 19 (1), 46–57.

Schwaiger, R., Moser, B., Dao, M., Chollacoop, N., Suresh, S.,2003. Some critical experiments on the strain-rate sensitivityof nanocrystalline nickel. Acta Materialia 51 (17), 5159–5172.

Shi, H., Atkinson, M., 1990. A friction effect in low-load hardnesstesting of copper and aluminum. Journal of Materials Science25 (4), 2111–2114.

Subhash, G., Zhang, W., 2002. Investigation of the overallfriction coefficient in single-pass scratch test. Wear 252 (1–2),123–134.

Tabor, D., 1951. The Hardness of Metals. Clarendon press,Oxford.

Tunvisut, K., Busso, E.P., O’Dowd, N.P., Brantner, H.P., 2002.Determination of the mechanical properties of metallic thinfilms and substrates from indentation tests. PhilosophicalMagazine A: Physics of Condensed Matter, Structure,Defects and Mechanical Properties 82 (10 SPEC),2013–2029.

VanLandingham, M.R., 2003. Review of instrumented indenta-tion. Journal of Research of the National Institute ofStandards and Technology 108 (4), 249–265.

Wang, L., Ganor, M., Rokhlin, S.I., 2005. Inverse scalingfunctions in nanoindentation with sharp indenters: determi-nation of material properties. Journal of Materials Research20 (4), 987–1001.

Williams, J.A., 1996. Analytical models of scratch hardness.Tribology International 29 (8), 675–694.

Wilson, S., Hawthorne, H.M., Yang, Q., Troczynski, T., 2000.Sliding and abrasive wear of composite sol–gel aluminacoated al alloys. Surface and Coatings Technology,389–396.

Wredenberg, F., Larsson, P.-L., 2007. On the numerics andcorrelation of scratch testing. Journal of Mechanics ofMaterials and Structures 2 (3), 573–594.

Youn, S.W., Kang, C.G., 2004. A study of nanoscratch exper-iments of the silicon and borosilicate in air. Materials Scienceand Engineering A 384 (1–2), 275–283.

Zhang, Z., Zhang, L., Mai, Y.-W., 1994. Modelling friction andwear of scratching ceramic particle-reinforced metal compos-ites. Wear 176 (2), 231–237.

Zhang, Z.F., Zhang, L.C., Mai, Y.-W., 1995. Particle effects onfriction and wear of aluminium matrix composites. Journal ofMaterials Science 30 (23), 5999–6004.