akay a numerical model of friction between rough surfaces

8/11/2019 Akay a Numerical Model of Friction Between Rough Surfaces

1/15

Tribology International 34 (2001) 531545www.elsevier.com/locate/triboint

A numerical model of friction between rough surfacesYu. A. Karpenko, Adnan Akay

*

Carnegie Mellon University, Mechanical Engineering Department Pittsburgh, PA 15213, USA

Received 2 March 1999; received in revised form 28 February 2000; accepted 9 May 2001

Abstract

This paper describes a computational method to calculate the friction force between two rough surfaces. In the model used,

friction results from forces developed during elastic deformation and shear resistance of adhesive junctions at the contact areas.Contacts occur between asperities and have arbitrary orientations with respect to the surfaces. The size and slope of each contactarea depend on external loads, mechanical properties and topographies of surfaces. Contact force distribution is computed byiterating the relationship between contact parameters, external loads, and surface topographies until the sum of normal componentsof contact forces equals the normal load. The corresponding sum of tangential components of contact forces constitutes the frictionforce. To calculate elastic deformation in three dimensions, we use the method of inuence coefcients and its adaptation to shearforces to account for sliding friction. Analysis presented in Appendix A gives approximate limits within which inuence coefcientsdeveloped for at elastic half-space can apply to rough surfaces. Use of the method of residual correction and a successive gridrenement helped rectify the periodicity error introduced by the FFT technique that was used to solve for asperity pressures. Theproposed method, when applied to the classical problem of a sphere on a half-space as a benchmark, showed good agreement withprevious results. Calculations show how friction changes with surface roughness and also demonstrate the methods efciency. 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Sliding contact; Surface roughness; Coefcient of friction

1. Introduction

Elastic deformation and adhesion are among the manydifferent processes that contribute to resistance to rela-tive motion during sliding contact [1,2]. The resistiveforces on rough surfaces develop at the true contact areasbetween asperities. As surfaces slide against each other,different pairs of asperities come into contact. Location,size, and orientation of the contact areas, and the contactforces that develop at each, depend on asperity distri-butions, mechanical properties of the surfaces, externalloads, and relative motion.

Progress of friction models between rough surfacesclosely correlates with advances in experimental investi-gations of contact forces viz., [27] and theoreticalmodeling of surface roughness viz., [2,711]. Experi-ments show that shear stress at points of true contact (orlocal friction) depends on the contact pressure and in

* Corresponding author. E-mail address: [email protected] (A. Akay).

0301-679X/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0301 -679X (01)0 0044-5

most cases such a dependence is linear [24,7]. Interfa-cial shear stresses may result from a combination of shear resistance of adhesion [2,7] and contaminants inthe interface [3,4]. However, studies also show that forrough surfaces adhesion usually has negligible effect onthe contact area and pressure [5,6].

One of the pioneering attempts at using multi-asperitymodels to relate friction to elastic deformation and shearstrength of adhesion dates back to the early works of Krag-elsky [2]. He started with a surface model in which theasperities were considered as an assembly of rods of differ-ent length with their lower ends xed in a rigid base.Applying a statistical approach, he expressed friction coef-cient in a general form as a function of deformationbehavior of surface material, height distribution of therods, and the dependence of local friction on the contactpressure. For ease of analysis, Kragelsky assumed elasticrods that had linear height distribution, and that the localfriction varied linearly with the contact pressure [2,7]. Hisanalytical results showed an increase in coefcient of fric-tion as surface roughness decreases, which agreed wellwith the corresponding experimental studies [1,2].


2/15

532 Yu. A. Karpenko, A. Akay / Tribology International 34 (2001) 531545

Modeling of surface roughness progressed with thestudy of contact of rough surfaces by Greenwood andWilliamson (G W) [8]. They assumed that surfaces arecomposed of hemispherically tipped asperities with auniform radius of curvature. In accordance with theexperimental observations, Greenwood and Williamson

approximated the distribution of asperity heights abouta mean plane as Gaussian [8]. For mathematical con-venience, contact between two rough surfaces was simu-lated as a contact between an equivalent rough surfaceand a rigid at plane. The equivalent rough surface isobtained by summing up the heights of the two real sur-faces and by an effective modulus that represents theirelastic moduli [8,12]. Whitehouse and Archard [9]extended the G W model [8] by allowing for the randomradii of curvature of the asperity tips. Nayak [10] intro-duced the techniques of random process theory into theanalysis of Gaussian rough surfaces. He related surfacestatistics such as the distribution of summit heights, thedensity of summits, the mean surface gradient, and themean curvature of summits to the power spectral densityof a pro le of the surface. Random process theory wasalso used by Bush et al. [11], who approximated thesummits of surface asperities of random heights and cur-vatures by paraboloids with the same principal curva-tures.

Francis generalized the asperity models describedabove to model sliding friction between rough surfaces[13]. By using Nayak s approach [10], he simulatedGaussian engineering surfaces with asperity shapes thatare paraboloidal only at their vertices but otherwise have

random heights and curvatures. To calculate elasticdeformation at each asperity contact, Francis used theHertz solution for elastic circular paraboloids [12]. Tomodel the sliding resistance at true contacts, he utilizedempirical relations based on data in [3,4]. By statisticallysumming over all asperity contacts, Francis expressedthe total contact area, load, and the friction coef cientas functions of the separation of the mean surfaces.

Ogilvy removed the need for the simplifying assump-tions about the geometry of asperities by presenting anumerical model to predict the friction force betweenrough surfaces [14]. The Ogilvy model considers contactbetween two numerically-generated Gaussian surfacesand also uses the concept of equivalent surface rough-ness [12]. Ogilvy calculated elastic deformation of asperities using the Hertz theory and accounted for theirplastic deformation with a simple plasticity theory[1,14]. Under conditions of steady sliding [12], Ogilvycalculated friction force due to adhesion by using twomethods that she called microscopic and macro-scopic. When using the microscopic method, the totalfriction force becomes the sum of the adhesive forcesrequired to break every junction [15]. In the macroscopicapproach, the total friction force is related to the totaltrue contact area under the assumption of constant shear

strength of adhesive junctions, but without consideringthe details of each asperity contact. Ogilvy observed aqualitative agreement between the numerical predictionsof her model and corresponding experimental results[14].

The concept of equivalent roughness used in previous

studies presumes that the true contact areas always haveparallel orientation to the mean planes. To account forcontacts at asperity slopes that may have arbitrary orien-tations [16,17], Ford developed a multi-asperity contactmodel [18]. He combined a previous model by Tabor[17, ch. 10] for a sliding contact point over triangularasperities with the statistical approach by Greenwoodand Williamson [8] for frictionless contact betweennominally- at surfaces. By extending Ogilvy s method[14], Ford removed the restriction of asperity peaks tohemispherical shapes in the G W model [8,18]. Fordshowed that, for small asperity slopes, single asperityanalysis [17] overestimates the in uence of asperityslopes on friction compared with multiple-asperity con-tact analysis.

Other studies that address the problem of frictionbetween rough surfaces include an extension of Tabor smodel [17] by Seal [19] to the case where sliding is notalong the principle slopes. Seal showed that experimen-tally observed anisotropy in friction may be explainedby sideway excursions during sliding. Suh and Sin [20]found asperity deformations to be signi cant for staticcoef cient of friction but not in the case of sliding fric-tion as compared with friction due to adhesion. In aninvestigation of the in uence of roughness on rollingcontact fatigue, Bailey and Sayles [21] considered con-tact between two-dimensional rough surfaces. Theyshowed that presence of asperities within contact regionsbrings the maximum orthogonal and principal shearstresses to occur at or very close to the surface [21]. Huet al. [22] extended the model in [21] to three-dimen-sional rough surfaces and showed that during steady slid-ing the amplitude and location of the maximum shearstress in the subsurface region continuously change.Tworzydlo et al. [23] developed a constitutive model of contact interface by combining nite element analysis of surface asperities, empirical relations for local friction[3,4], and statistical homogenization techniques [10,13].To validate their model, they measured sliding frictionforce between two rough surfaces made of steel andcompared the friction coef cient obtained experimen-tally with its value predicted under assumptions similarto those in Ogilvy s macroscopic model of friction [14].Based on an energy balance analysis at the interface of two-dimensional rough surfaces, Bengisu and Akay [24]related the interface forces to the relative normal andtangential velocities of contacting surfaces as a functionof asperity deformation and adhesive shear forces. Inanother study [25], the same authors demonstrated howcontacts at asperity slopes give rise to force components


3/15

533Yu. A. Karpenko, A. Akay / Tribology International 34 (2001) 531 545

in a direction normal to sliding and, thus, couple slidingand normal motions of the surfaces.

This paper describes a computational method to calcu-late friction force between three-dimensional rough sur-faces under conditions of steady sliding [12]. Themethod assumes that friction results from elastic defor-

mation of asperities and shearing of adhesive junctions.The model used here has two features that distinguish itfrom most of the previous ones: local friction dependson contact pressure [2 4,7], and contacts take place atthe slopes of asperities and may have arbitrary orien-tations, similar to that developed by Bengisu and Akay[24,25].

The interdependence of contact forces and (size andslope of) contact areas makes it necessary to computecontact forces and areas iteratively. The methoddescribed below represents the relationship between adistribution of contact forces and the resulting displace-ment eld by using the method of in uence coef cients[12,26], originally developed for normal forces, adaptedhere to also include locally-tangential forces. The presentmethod uses the empirical binomial law of friction[2,7] to represent shear strength of adhesion at each con-tact instead of Coulomb s law [21,22].

Use of a successive grid re nement technique [27]increases the computational ef ciency of iterations inreaching a balance between normal components of con-tact and external forces. Details of the computationalmethod are outlined later in the paper, followed by anexample that demonstrates computation of friction para-meters.

2. Description of the method

The contact model described below for sliding frictionbetween two rough surfaces restricts itself to shallowasperities such that the rms value of their slopes is lessthan 0.1 rad [1,2]. In such cases, the interlocking effectthat leads to shearing (fracture) of the asperities does notdevelop [2,20].

3. The contact model

The contact model described in this section considerstwo rough surfaces sliding [12] along a straight line par-allel to the mean planes of the surfaces. The directionopposite that of sliding de nes the tangential directionand the normal to the mean planes de nes the normaldirection of the coordinate system. The surfaces comeinto contact under the action of an external normal load,P . Fig. 1(a,b) portray an exaggerated cross-section of surfaces, designated as 1 and 2, just prior to and follow-ing contact. In both cases, the dotted lines represent themean planes of the surfaces. Function h( x, y) denotes the

Fig. 1. Diagram of the rough surfaces: (a) before contact; (b) afterdeformation. Positions of the mean planes of half-spaces, (1) and (2),before and after deformation are shown with dotted lines.

separation between the two surfaces prior to deformation[Fig. 1(a)]. As shown in Fig. 1(b), d represents the com-bined deformation of the surfaces at the rst contactpoint 1 . The plane parallel to the mean planes of the sur-faces containing the point of rst contact, O, establishesthe plane of reference. The global coordinate axes foreach surface have a common origin at O , with X 1Y 1 and X 2Y 2 planes that coincide with the plane of reference.The axes Z 1 and Z 2 , point away from the surfaces intothe bodies [Fig. 1(a)].

The contact model used here only considers the contri-butions of forces developed at each asperity contactresulting from their elastic deformation and shear resist-ance due to adhesion. Under an applied normal load P,surface asperities come into contact and deform as

1 As the maximum value of the combined deformation at each con-tact between the surfaces d represents the approach between the sur-faces.


4/15

534 Yu. A. Karpenko, A. Akay / Tribology International 34 (2001) 531 545

depicted in Fig. 2(a), where dotted lines represent theundeformed shapes of the asperities and solid lines rep-resent their deformed shapes. The model allows for truecontacts between surfaces to take place at the slopes of their asperities, which may have arbitrary orientations(Fig. 1). Under such conditions, local normals at contact

points do not necessarily coincide with the normal to themean planes of the surfaces [24,25]; see Fig. 2(a). Thecorresponding spatial distribution of contact pressure, f ( x, y), that results from asperity deformation is rep-resented schematically in Fig. 2(b).

The resistance of an adhesive junction to shearingtakes place in the local tangential direction and, in effect,constitutes the local friction [24]. Following Kragelskyet al. [7], the shear stress due to local friction, r , isexpressed with a binomial expression

r t b f , (1)

where t and b are empirical constants. For many metalsand polymers b=0.02 0.25; for metal metal sliding

Fig. 2. Schematic description of asperities in contact (a) and the cor-responding contact force distributions (b). Dotted lines represent unde-formed shapes of the asperities. Solid lines represent their deformedslopes. Positions of the matching surface points before the contact andafter the deformation are denoted with 1 and 2, and 1* and 2*, respect-ively, where d h( x, y) describes their contact approach.

pairs t =2.530 MPa and for metal polymer pairst =0.20.5 MPa [7].

The aggregate of the normal components of defor-mation and local friction forces at all asperities through-out the contact region yields the normal force on thesurfaces [24,25]; Fig. 2(b):

N A

( f cosq x r sinq x)cos q yd xd y, (2)

where A is the true contact area. The spatial derivatives,q x, and q y, of the deformed surfaces at contact pointsrepresent the contact slopes with respect to x and y direc-tions in the global coordinate system. Under the assump-tion of quasi-static equilibrium, N must balance the loadP . The corresponding sum of the projections of contactforces in the sliding direction constitutes the frictionforce, F :

F A

( f sinq x r cosq x)cos q yd xd y, (3)

and the friction coef cient, m, is obtained from theirratio:

mF P

. (4)

Thus, the local friction and deformation forces each con-tribute to both vertical and tangential interface forces, N and F .

4. Computational method

The computational method described here utilizes thecomplex relationship that exists among external loads,material properties and topography of surfaces, and con-tact parameters. The contact parameters consist of con-tact forces, contact areas, and deformation at each con-tact. Their magnitudes, orientation, and distributiondepend on external loads, and on the topography, or dis-tribution, of asperities on the surfaces as describedbelow.

Contact deformations on the surfaces collectivelymanifest themselves as the approach, or relative nominaldeformation, between the surfaces. Thus, for any valueof approach between the surfaces under external loads,there exists a corresponding distribution of contactforces, contact areas, and asperity deformations. Sinceat quasi-static equilibrium, local contact forces betweensurfaces balance the external loads on a friction pair, therelationship between external loads and contact para-meters can be expressed in terms of approach d [12].The solution procedure described later iterates thisrelationship by adjusting the approach between the twosurfaces until equilibrium is established between theexternal normal load P and the normal contact force N


5/15


P N (d ) 0. (5)

Following conventional contact analyses [12,21,26],we assume also that true contact area is contained withinthe kinematic interference area and that the relative tan-gential displacements are negligibly small. Then, therelationship between approach and local asperity dis-placements may be described geometrically as the kine-matic interference of the surfaces, schematically shownin Fig. 2. As illustrated in Fig. 2(a), the local normaldisplacements u z1( x, y) and u z2( x, y) relate to the approachd through the function h( x, y):

u z1 u z2 [d (6a)

h( x, y)]cos q xcosq y within contact area

u z1 u z2 [d (6b)

h( x, y)]cos q xcosq y outside contact area

Because displacement at any point on an elastic sur-face depends on forces everywhere on the surface, calcu-lation of local displacements requires consideration of both the normal and tangential forces at all contactsthroughout the surface. However, contact forces them-selves depend on displacement at the points of contactand, therefore, are not known a priori. Use of in uencecoef cients, which represent the relationship betweendisplacement at a point and the forces that cause it[12,26], makes it possible to calculate normal displace-ments as a linear superposition of displacements due to

forces at all contact points.Two types of in uence coef cients are necessary to

calculate friction. In addition to the conventional in u-ence coef cients that represent normal displacementsa ( x, y) at any surface point due to a unit normal pressureacting over a grid element [26], in uence coef cientsrepresenting normal displacements c( x, y) due to a unittangential traction, uniform over a grid element, are alsorequired to model the effects of local friction [12]. Twocomplete sets of such in uence coef cients for all gridelements, one set each for normal and shear tractions,constitute the corresponding exibility matrices used forcomputations as described below.

Equations describing contact forces and the total eldof local normal displacements are expressed in matrixform as:

[a ][f ] [c][r ] [u ] (7)

where [ u ] represents normal displacements due to alllocal normal and shear loads; [ a ] and [c] are the exi-bility matrices due to locally normal and tangential trac-tions of unit intensity over each grid element, respect-ively. Matrices [ f ] and [ r ] represent the local contactpressure and local friction, where [ f ] is assumed to bepositive inside the true contact area and to vanish outside

[12]. The local friction depends, in part, on the contactpressure, [ f ], and use of Eq. (1) makes it possible torepresent their relationship as:

[r ] t b[f ]. (8)

The computational method described here uses Eq.(7), along with Eq. (8), to obtain force distributions [ f ]and [ r ] for a given displacement eld [u ]. Thus, the sol-ution of Eq. (7) requires an initial estimate of [ u ], whichis obtained here from the kinematic interference of thesurface topographies.

Solving Eq. (7) rst requires determining the exi-bility matrices [ a ] and [c]. Matrices of in uence coef-cients essentially de ne a set of Green s functionsamong the contact points. Analytical expressions forinuence functions that describe force displacementrelationships exist only for an elastic half-space with aat surface [12]. Although commonly used for roughsurfaces as well [21,22,26], applicability of classicalinuence functions to oblique contacts has not beenreported. The analysis described in Appendix A, basedon analytical solutions of two related problems, givesapproximate limits within which in uence functions maybe used for oblique contacts. The analysis in AppendixA combines the response in a half-space to a uniformpressure on a narrow strip of its at boundary [12] withthe response of an in nite wedge under arbitrarily dis-tributed surface tractions [28]. As shown in AppendixA, inuence functions for an elastic half-space are appli-cable to rough surfaces within limits. For the cases con-

sidered in this study where the rms value of asperityslopes varies between 0.125 and 2.5 , the analysis inAppendix A shows that the maximum error associatedwith the application of the conventional in uence coef-cients to rough surfaces is less than 2%.

5. Computational procedure

The computational procedure followed here consistsof an iterative search for a value of d that provides aglobal force balance between the external normal loadand the sum of the normal components of contact forces[12,26]; Eq. (5). The process begins, as the rst step,with the construction of matrices [ a ] and [c] in Eq. (7)with the use of a moving grid method described in [29].Iteration consists of solving Eq. (7) for the distributionof local forces [ f ] and [ r ] for a given displacement eld[u ], initially estimated from kinematic interference of surfaces for an assumed value of approach d ; Eq. (6a).The exibility matrix [ a ] in Eq. (7) is inverted using anFFT-based technique [30].

For the computations, discretized rectangular tractionelements represent the nominal contact area on each sur-face with ( M M ) number of matching points located at


6/15


the center of each element viz., [12,22,26]. Each match-ing point has assigned to it a surface height relative toa selected datum. All traction elements and matchingpoints are measured with respect to the global referenceframe, which also acts as a reference frame for theinuence coef cients in the exibility matrices. A grid

size with M M traction elements results in a exibilitymatrix of size M 2 M 2 .

The solution process consists of three nested iterationsdescribed by a ow chart in Fig. 3. The rst iterationdetermines, from among those identi ed by kinematicinterference, the actual contact areas that have positivecontact pressure. Because kinematically-based inter-ference between surfaces does not account for their elas-tic deformation, the rst iteration reconciles deformedsurface geometry and contact forces. The second iter-ation seeks a balance between external normal load andthe sum of the normal contact forces between the sur-faces. The third iteration increases the resolution of con-tact areas by progressively decreasing the tractionelement size.

The solution process starts by bringing the surfacestogether for an assumed value of approach where contactis detected through kinematic interference. Upon detec-tion of contact, matrix [ u ] is constructed from the kinem-atic interference relations in Eq. (6a) for that particularvalue of approach d . Eq. (7) is solved iteratively for [ f ]and [ r ] corresponding to the kinematically obtained con-tact displacements in [ u ] by using the exibility matrix[c] and the inverse of matrix [ a ]. At each iteration, thosecontact areas for which contact pressure is not positive

are identi ed and excluded from the calculations [12,30].Following convergence, the contact parameters still con-tain an error referred in literature as periodicity error[31,32]. This error results from the periodic extensionof contact domain introduced implicitly into the contactanalysis by Fourier transformation [31,32]. Assumingthat such an error is equivalent to the residual of thesolution vector [ f ], the method of residual correction isapplied to Eq. (7) to modify the vector [ f ] so that itsresidual vanishes [27]. The rst iteration thus yields dis-tributions of contact areas, local friction and pressuresthat correspond to the assumed value of approach d .

The second iteration adjusts the assumed value of d until a balance is reached between the external normalload and sum of the normal components of all contactforces [12,26]. When balance is reached, (i) the corre-sponding d represents the approach that would resultfrom the applied load, and (ii) the sum of the corre-sponding tangential forces represents the friction force.

The third iteration improves the resolution of contactarea distribution. In this iteration, however, to increasecomputational ef ciency, results of the previous stepwith coarser grids, are interpolated using cubic poly-nomials for use with the new, higher-resolutionelements. The solution process starts again with Eq. (7)

and continues until the desired convergence is achieved.Within this iteration, the computational effort to reachconvergence is greatly reduced because the periodicityerror introduced by FFT is already recti ed.

For the computations reported below, the initial spatialresolution begins with a grid size of 128 128 for a

1.61 1.61 mm nominal contact area, followed by doub-ling it to 256 256, and later quadrupled to 512 512grids. The corresponding element sizes change from acoarse 12.612.6 m element to a medium 6.36.3m element and nally to a ne 3.153.15 m elementsize. The results reported below are obtained using thene mesh. The tolerances used for convergence are:| [f ]/[f ] 10 5 and |( P N )/ P | 104.

The iterative computational method described aboveis applied in the next section to steady sliding of a sphereon an elastic-half-space to compare with previouslyreported analytical and numerical results [12,33].

6. Contact of a sphere with a at surface

To validate the numerical method described above, weapply it to the case of a sphere sliding steadily on a half space. The choice of a sphere and half-space providescomparisons with previously reported analytical studiesof frictionless contact [12,33] and numerical and experi-mental studies of roughness effects [2,7,14,18]. Choiceof materials was motivated by the availability of empiri-cal values for the coef cients t and b in the binomialexpression used for local friction, or resistance to shearof adhesive junctions [7].

6.1. Surface roughness

In the examples reported below, asperity height distri-bution, f ( z), and auto-correlation function, R zz, charac-terize the statistical distribution of the numerically-generated roughness for each surface. Both surfaces haveGaussian height distributions and an isotropic bell-shaped auto-correlation function (ACF) with a corre-lation radius of 104 m that de nes where the value of ACF reduces to 10% of its value at the origin [34]. Theseproperties represent freshly ground surfaces [34].

For the computations, the asperity height distributionsfor the ball and the at surface are normalized to havea unit rms value (or standard deviation) s =1. This nor-malization makes it convenient to obtain actual heightdistributions for any surface with a different roughnessamplitude by simply multiplying the normalized valueswith the desired value s . The ACF is expressed in dis-crete form by

R zz(k ,m) s 2e2.3[ k / l x)

2 + m / l y)2 ] (9)

where k =0,1,2, % ,l x,m=0,1,2, % ,l y, and l x and l y, represent


7/15


Fig. 3. Flow chart of the computer program that calculates iteratively sliding contact parameters between two rough surfaces.

the non-dimensional correlation lengths in x and y direc-tions [35]. For the isotropic surfaces considered here,l x=l y=l=33.

In keeping with the established practice in the litera-ture, roughness effects on friction and contact pressureare examined as a function of equivalent roughness, s e,of the two surfaces [12]. For the results reported below,a range of equivalent roughnesses is considered by com-

bining one surface roughness for the sphere with a seriesof surface roughnesses for the half-space. The roughnessof sphere surface has standard deviations of s (s )=0.084m for its asperity heights and s (s )q x=s (s )q y=0.125 for itsslope distributions. Standard deviations for the at sur-face roughness vary between s ( fs)=0.084 and 1.680 mfor its asperity heights, and between s ( fs)q x =s ( fs)q y =0.125 and2.50 for its slope distributions. Computations are car-


8/15


Table 1Numerical values of the parameters used in calculations. The data forthe binomial law of friction ( t =15 MPa and b=0.08) are taken for asteel-copper pair [7]

Surface R(m) E (GPa) v

Ball 0.14 200 0.30Falt 118 0.30

ried out for normal loads P=14, 42, and 70 N. Theempirical values for the coef cients in the binomialexpression in Eq. (1) [7] and other relevant parametersused in the computations are given in Table 1.

Numerically generated rough surfaces (Fig. 4) used inthe following computations have a spatial resolution of =3.15 m along both x and y directions. However,

because ACF and height distribution together do notuniquely describe a surface topography, different sets of random inputs can lead to different surface represen-tations while having the same statistical characteristics R zz and s [14,35]. As pointed out in earlier studies[14,18], such differences in surface representations leadto a variation in the values of the computed contact para-meters. Thus, unless otherwise noted, the results reportedbelow represent the average value of those that corre-spond to a set of 75 surface representations obtainedusing uncorrelated random inputs for given values of s eand R zz.

7. Model validation

Validation of the model described above consists of a comparison of numerical results with selected cases inthe literature. The cases described below include simul-ation of steady sliding of a smooth sphere against a at

Fig. 4. Normalized surface roughness for the half-space (a) and theball (b).

surface [12], frictionless contact between a smoothsphere and a rough surface [12,33], and contact betweentwo rough surfaces [2,7].

It is known that friction slightly shifts the pressuredistribution and contact area between smooth surfacesof a hard sphere and a more elastic half-space toward

the trailing edge viz., [12]. Computations using Cou-lomb s law of friction show a semi-elliptical distributionfor the tangential contact stress similar in shape to thenormal stress distribution, but reduced in amplitude inproportion to the friction coef cient. On the other hand,use of constant shear strength at the interface producesa uniform tangential stress across the contact area, inde-pendent of the normal pressure [14]. The binomial lawof friction, Eq. (1), produces a tangential contact stressdistribution as the sum of a constant shear stress t dis-tributed uniformly over the contact area and a Coulomb-like shear term, proportional to the contact pressure.Thus, near the edges of the contact area where contactpressure tends to zero, the Coulomb-like term also van-ishes, leaving the adhesive shear stress t to dominate thevalue of local friction (Fig. 5).

Frictionless contact refers to contact deformation inthe absence of local friction. Greenwood and Tripp [33]examined the in uence of roughness on contact pressurebetween a sphere and a half-space. They reported thatthe maximum value of the effective contact pressure isless than that of the Hertzian stress for the correspondingsmooth surfaces. They further showed that, betweenrough surfaces, contact and contact pressure can existoutside the Hertz contact region calculated for smooth

surfaces. By setting the constants t and b in Eq. (8) to

Fig. 5. Contact parameters between a sphere sliding against a half-space, both with smooth surfaces: (a) distribution of normal (a) andtangential stresses (b) in the sliding direction; (b) the correspondingdistributions in the direction normal to the sliding direction. Dottedlines represent the distributions of the Hertzian contact stress and thetangential stress with Coulomb law of friction, where P=70 N.


9/15


zero and allowing contacts to take place only at the tipsof asperities, the present method similarly simulates theeffects of roughness on contact pressure distribution.Results presented in Fig. 6(a c) show good agreementwith those reported by Greenwood and Tripp [12,33].An example of the distribution of pressure peaks

between the rough surfaces (that exceed the maximumHertz stress) illustrated in Fig. 6(d) also shows thatpressure peaks may exist outside the Hertz region indi-cated by a0 , radius of the Hertz contact area.

Computation of pressure distribution between tworough surfaces with and without local friction delineateswhat in uence local friction, or shear strength of adhesion, may have on contact parameters. Results showthat the local friction has negligible in uence on contactdeformation, which agrees with previously reportedexperimental results [2,7]. Fig. 7 portrays a cross-sectionof sphere and at surface just prior to [Fig. 7(a)] andduring contact [Fig. 7(b)], which illustrates the deformedasperities and deformed mean plane of the at surface.Deformed surface pro le (dotted line) calculated byomitting shear adhesion maps nearly identically onto theprole obtained by including shear adhesion effects.

The benchmark cases considered provide a validationof the numerical method proposed here. In addition,results show that the combined deformation of asperities

Fig. 6. Pressure distribution between a ball and a half-space with rough surfaces normalized with respect to the maximum Hertzian pressure f 0between corresponding smooth surfaces, f 0=163.9 MPa: (a) (c) effective pressure distributions when s e=0.068; 0.511 and 2.69 m, correspondingly;(d) contact domain when the equivalent surface roughness s e=0.511 m. Solid line the present model; broken line Hertz pressure, and dasheddotted line the model by Greenwood and Tripp [12,33]. The radius of the Hertz contact a0=0.452 mm; the parameter a is a ratio between thesurface roughness s e and the bulk compression given by the Hertz theory, d 0 [12].

Fig. 7. Cross-section of the ball and half-space prior to contact (a)and in contact (b), where P=70 N and equivalent surface roughness

s e=0.511 m. The dashed line indicates the mean plane of the half-space; the dotted line the contact deformation in the absence of fric-tion.


10/15


and of the mean plane of the at surface has a compara-ble magnitude to the average height of asperities. Conse-quently, the deformed contact areas may not have thesame statistical characteristics they had prior to theirdeformation and, thus, they may no longer have a Gaus-sian distribution as discussed later.

8. Effects of roughness

Results show that surface roughness reduces frictionin a complex manner. Friction coef cient m, as shownin Fig. 8(a), decreases exponentially with increasing sur-face roughness. Its rate of change asymptoticallyapproaches a constant value beyond s e 0.5 m, similarto results reported in [2,14]. As shown in Fig. 8(b), thetotal contact area changes with s e in the same manner as m. Thus under constant load, the mean contact pressure, f

m= N / A, increases with roughness. Results in Fig. 8(c)

show that, under constant load, friction coef cient andcontact area both have nearly identical rates of changewith respect to s e.

An explanation for the m s e dependence may be

Fig. 8. Predicted variation of sliding contact parameters between a sphere and a half-space with their equivalent roughness s e and plasticity indexy [8]: (a) variation of friction coef cient; (b) of contact area; (c) of their rates of change, where solid line- d m /ds e /|d m /ds e |s e=0, and dashed dottedline d A /ds e /|d A /ds e |s e=0.

given by extending the shear adhesion term in Eq. (1)to rough surfaces. Similar to the application by Singer[36] to smooth contact of ball bearings and at surfaces,an analogous expression that approximates m for roughsurfaces may be obtained as follows. Substituting for r in Eqs. (2) and (3) and restricting in contact to the

asperity tips ( q 0), friction coef cient can be written as:

m A(t + b f )d xd y

A

f d xd y

t A N

bt

f m b (10)

At high contact pressures such that, t / f m b, b dominatesthe value of m. In accordance with simple plasticitytheory [1,14], the upper limit for the mean contact press-ure may be represented by the hardness, H , of the softermaterial. Hence, the smallest value of m (for smallasperity slopes) may be approximated as:

mt

H b (11)


11/15


For the case considered here ( H Cu =800 MPa [7]), thelowest value m reaches is estimated as 0.1. This resultapproximates well the asymptotic value m reaches inFig. 8.

Plasticity index y and its dependence on surfaceroughness [8] provides an explanation for the critical

value of s e=0.5 m about which friction becomes lesssensitive to further increase in roughness. For Gaussiansurfaces, y , may be approximated as [14]:

y 2.2 E H

s el

, (12)

where E is the effective elastic modulus [12]. As shownin Fig. 8, y reaches the threshold value of unity at abouts e 0.5 m beyond which contact ceases to be purelyelastic [8] and falls outside the conditions of the modelused here. In such a case, Eq. (11) may be used to esti-mate the friction coef cient.

The variations in numerically generated surfacedescriptions, due to the differences in initial conditionsdescribed earlier, introduce uctuations to m throughuctuations in the contact area A. In the results presentedin (Fig. 8), contact area A uctuates by 7.5% about itsmean value with a corresponding variation in m of approximately 1.0%. The difference between the vari-ations of contact area and friction results from the pres-ence of a constant shear adhesion term b in Eq. (10).Computations with b=0 in Eq. (8) shows variation of mto be the same as that for A, 7.5% the same order of magnitude as in [18] in which local friction was takento be independent of contact pressure.

Results also show the existence of a binormal compo-nent Q of the contact forces (normal to the direction of motion in the tangential plane). As illustrated in Fig. 9,

Fig. 9. Binormal component of the interface force Q (normal to thedirection of motion in the tangential plane) between rough surfaces of a sphere sliding against a half-space, where P=70 N.

Q has a very small amplitude in comparison with thatof the normal load P. Binormal force results from thedeformation of asperities. Although it has a very smallmagnitude, in cases where high level of accuracy isrequired, binormal force may become important [19].

Compared with shear adhesion, forces resulting from

elastic deformation of asperities have negligible directcontribution to the total friction force, as shown in Fig.10(a). However, their indirect contribution to frictionthrough adhesive forces makes elastic deformation anecessary part of the model and the computations.

In the cases considered here, both the total frictionforce and its component due to elastic deformation varywith the direction of relative motion, as observed pre-viously [18]. As shown in Fig. 10, the difference, whilevery small in absolute terms, depends on surface rough-ness. The source of anisotropy of friction force, and itscomponents, relates to the distribution of surface rough-ness within the contact areas. Although the asperityheights and slopes on undeformed surfaces both haveGaussian distributions, corresponding distributionswithin the much smaller set of true contact areas (that arenow deformed) may no longer remain Gaussian. Resultsshow that the distribution of the asperity heights andtheir slopes at contact areas differ from their originalGaussian distributions, as illustrated in Fig. 11. In parti-cular, the distribution of deformed asperity slopes exhib-its an overall negative skewness indicating a non-Gaus-sian feature, see Fig. 11(d).

The ef ciency of the proposed approach comparesfavorably with those reported in the literature for static

contact between a half-space and a sphere with a roughsurface. Table 2 summarizes CPU times spent by using

Fig. 10. Relative change in friction force due to the change in thedirection of sliding of a sphere against a half-space, where P=70 N.The upper gure shows resistance to sliding due to elastic deformationforce normalized with respect to friction force. The lower gure showsthe relative difference between the total friction forces.


12/15


Fig. 11. Roughness distribution of undeformed and deformed at surfaces, where P=70 N: (a) height distribution of undeformed surface in theregion of the nominal contact area, s ( fs )=1.680 m; (b) slope distribution of undeformed surface, s ( fs)q =2.5; (c) height distribution of deformedsurface at the real contact area, and (d) slope distribution of asperities in contact.

Table 2Comparison of computing time between variational approach [37] and matrix inversion approach based on the developed computation technique.The speci cations of the hardware: Silicon Graphics Indigo R-4000/100 MHz workstation (60.5 SPECfp92) [37] and PC with Pentium processor90 MHz (70.5 SPECfp92). Cases 1 4 in our calculations correspond to the following set of grids: 100 100, 128 128, 256 256 and 512 512

Case Matrix Inversion by FFT Variational principle [37]Contact points Computing time Contact points Computing timeInitial Final Initial Final

1 9801 1821 0.07 h 9717 1711 1 h2 14697 4371 0.42 h 3 59171 17383 7.21 h 65310 16517 4 days

4 23715 68951 2.1 days

the present method and those by a variational approachas reported in [37] to solve for the true contact area andpressure, in both cases without local friction. Using themethod described above, computations for contact area A, friction coef cient m, and the maximum contact press-ure f max converge up to two decimal places with the useof ne mesh. Table 3 gives representative examplesthat show convergence of calculation results.

9. Summary

The computational method described in this paper cal-culates friction force by summing the forces that resistthe relative motion at the true contact areas between tworough surfaces. The model for the resistive forcesincludes those due to local friction and elastic defor-mation. The size and location of true contact areas


13/15


Table 3Numerical values of the tribological parameters calculated via successive grid re nement

s half-space (m) 0 0.504 1.008 1.680 sphere (m) 0 0.084 0.084 0.084Parameters of the Mesh, M M model

128 256 512 128 256 512 128 256 512 128 256 512

A/10 1 (mm 2) 6.54 6.44 6.43 1.35 1.26 1.22 0.77 0.71 0.70 0.50 0.45 0.44 /10 1 2.20 2.19 2.18 1.09 1.07 1.06 1.01 0.97 0.95 0.94 0.91 0.89f max /10 1 (GPa) 1.63 1.64 1.64 17.9 18.5 19.1 30.8 31.4 32.1 50.2 52.1 52.8

depend on the topography as well as on the contactforces that develop at each area. Arbitrary orientation of contacts at asperity slopes further complicates this inter-dependence between contact areas and forces. Theresulting complex relationship is solved with an iterativetechnique. The solution process includes adapting themethod of in uence coef cients to shear forces.

Contacts at asperity slopes give rise to force compo-nents in both the sliding direction and normal to it, thus,coupling sliding and normal motions. Coupling of fric-tion and normal forces has an important role in dynamicresponse of friction-excited systems. Use of pressure-dependent local friction and consideration of defor-mation of contacts at asperity slopes, set this model apartfrom the previous similar models. The numerical resultspresented above show dependence of friction on surfaceroughness and direction of motion. The reason for direc-tional dependence is shown to be the non-Gaussian dis-tribution of roughness within the contact region. It was

shown that a surface with an undeformed Gaussianroughness distribution may have a non-Gaussian rough-ness distribution within the much smaller, non-contigu-ous true contact areas.

The present method handles the complexities associa-ted with roughness of surfaces and oblique contactsef ciently using a nested iterative approach. The compu-tational ef ciency of the method outlined above is a fav-orable alternative to existing ones.

Acknowledgements

The authors wish to acknowledge helpful commentsof the reviewers.

Appendix A

The purpose of the following analysis is to determineapproximately the error associated in computing contactstresses on rough surfaces with the use of conventionalinuence functions developed for at half-spaces. Corre-sponding analytical expressions that describe displace-ment distribution due to a unit force on a rough surface

Fig. 12. Effects of surface waviness on surface displacement due touniform pressure acting over the narrow strip on the boundary.

do not yet exist and their development is beyond thescope of this work. Similarly, direct numerical evalu-ation of such in uence coef cients using theory of elas-ticity is too computationally intensive to be of practicaluse. The analysis presented below gives estimated limitsof applicability by an approximate analysis.

As a rst approximation, a one-dimensional sinusoidalprole describes surface roughness as illustrated in Fig.12. To further simplify the analysis, a wedge tip on theat half-space approximates each crest of the wavy sur-face (Fig. 13). The analysis begins by determining theeffects of a periodic pro le on surface displacement dueto a uniform pressure acting over the narrow strip (a x a ,a l ) on the at boundary of the elastic half-space [12]. Simulating an asperity as a wedge tip cangive its stress strain eld following the work of Tranter[28]. The analysis described here yields in uence func-tions for a wedge tip by calculating displacements nor-mal to the mean plane of the surface ( y-direction) inresponse to a unit amplitude pressure applied on its facesnear the tip (Fig. 13). Comparison of in uence functionsfor a wedge tip with those for a at surface [12] estab-lishes approximately the error that results from using the


14/15


Fig. 13. Effects of simulated asperity on in uence functions shownas relative differences in comparison to the conventional case of a

at surface.

conventional in uence functions in calculating displace-ment on a rough surface.

The effects of surface roughness on the displacementeld of a half-space are found by superposing a roughsurface pro le on the free at surface in equations thatdescribe the stress eld in a half-space. The equationsthat represent the 2-D stress eld produced by a uniformpressure having an intensity p per unit length along the z-axis and acting over the strip ( a x a ) are given by,

s x p

2p {2( q

1q

2) (sin2 q

1sin2 q

2)} (A1a)

s y p2p

{2( q1 q2) (sin2 q1 sin2 q2)} (A1b)

t yx p2p (cos2

q1 cos2 q2) (A1c)

where tan q1,2 = y /( x a ), x , and 0 y , [12]. Asinusoidal surface pro le describes the roughness of the surface

y 2hasin 2p x l (A2)

where the amplitude ha represents an asperity height andis much smaller than the wavelength ha l . Substitutionof (Eq. A2) into (Eq. A1a) gives a rst-order approxi-mation of the stress eld, s x, s y and t xy on the sinus-oidal surface.

A stress-free boundary on such a periodically wavysurface requires pseudo surface forces X and Y to com-pensate for the ctitious boundary stresses which theperiodic pro le of the surface induces,

X =(s xcosq+t xysinq)Y =(s ysinq+t xycosq)

, (A3)

where q is the slope of the boundary.Surface displacements due to the pseudo forces are

calculated as a function of surface waviness and com-pared in Fig. 12 with those due to a strip loading of the at half-space. Not surprisingly, the effect of surfacewaviness is smaller for shallower waviness amplitudes.

Of particular interest in these results is the con nementof displacement to within the tip of the asperity whichsuggests that it may be permissible to use individualasperity models and neglect coupling between theirstrain elds, as shown below.

In the analysis given above, each crest of the sinus-oidal surface pro le represents an asperity. In turn, thetip of a wedge placed on a at half-space approximatesan asperity as illustrated in Fig. 13. An initial stress dis-tribution at the base of a wedge tip is estimated fromthe expressions for stress distribution for a full (in nite)wedge. The stresses in a wedge s q , s r and t r q , given interms of polar coordinates, are [28]:

p r 2a

(s q s r )p cosg cosqp 2g +sin2 g

(A4)

0

P (x)sin{ xlog( a / r )}dx,

p r 2a

(s q s r ) p cosg cosqp 2g sin2 g

0[P (x) xQ(x)]

sin{ xlog( a / r )}1 x2

dx (A5)

0[Q(x) xP (x)]cos{ xlog( a / r )}

1 x2 dx,

p r a

t r q0

T (x)cos{ xlog( a / r )}d x, (A6)

where P(x), Q(x) and T (x) are given by

P (x)

cos( g +q)cosh( p /2g +q)x+cos( g q)cosh( p /2g q)x

xsin2 g +sinh( p 2g )x , (A7)

Q(x)

sin( g +q)sinh( p /2g +q)

x+sin( g q)sinh( p /2g q)xxsin2 g +sinh( p 2g )x

, (A8)

T (x)

cos( g +q)sinh( p /2g +q)xcos( g q)sinh( p /2g q)x

xsin2 g +sinh( p 2g )x , (A9)

An iterative approach using (Eqs (A4) (A9)) yields thecombined stress strain eld of a wedge tip located on ahalf-space by matching boundary conditions at the baseof the wedge tip and the half-space where they areattached. However, for a rst-order approximation, weonly match the load at the interface. Total displacement


15/15


on the wedge tip ( l /2 x l /2 and 0 y hasp ) resultsfrom a combination of local wedge displacement anddisplacement of the half space resulting from the loadon the wedge tip.

The inuence coef cients calculated as describedabove are compared with the in uence coef cients given

by the classical approach. The results, presented in Fig.13, give the difference between displacement on asmooth elastic half-space loaded over the strip a x a by unit pressure and one on a rough surface at anangle to the surface mean plane. This difference causedby a wedge tip is con ned to within the immediate vicin-ity of the load suggesting that the changes in in uencecoef cients due to asperity slopes are primarily local andmost signi cant within 3a x 3a .

Although the approximate analysis given above showsthat conventional in uence functions can be used forasperities with small slopes, this technique can beapplied directly to cases where asperities have largerslopes. In such cases, only a few of the terms in theexibility matrices need to be modi ed by using thetechnique developed here. For the range of surface top-ography considered in the numerical examples in thispaper, the maximal asperity slope did not exceed 6.5 .Hence, the use of classical in uence coef cients is per-missible for calculation of rough surfaces as describedabove.

References

[1] Bowden FP, Tabor D. London: Oxford University Press, 1954.[2] Kragelsky IV. Friction and wear. Washington: Butterworths,1965.

[3] Boyd J, Robertson BP. The friction properties of various lubri-cants at high pressures. Trans ASME 1945;67:51 9.

[4] Briscoe BJ, Scruton B, Willis FR. The shear strength of thinlubricant lms. Proc Roy Soc A 1973;333:99 114.

[5] Gane N, Pfaelzer PF, Tabor D. Adhesion between clean surfacesat light loads. Proc Roy Soc A 1974;340:495 517.

[6] Fuller KNG, Tabor D. The effect of surface roughness on theadhesion of elastic solids. Proc Roy Soc A 1975;345:327 42.

[7] Kragelsky IV, Dobychin MN, Kombalov VS. Friction and wearcalculation methods. New York: Pergamon Press, 1982.

[8] Greenwood JA, Williamson JBP. Contact of nominally at sur-faces. Proc Roy Soc A 1966;295:300 19.

[9] Whitehouse DJ, Archard JF. The properties of random surfaces of signi cance in their contact. Proc Roy Soc A 1970;316:97 121.[10] Nayak PR. Random process model of rough surfaces. ASME J

Lubr Technol 1971;93:398 407.[11] Bush AW, Gibson RE, Thomas TR. The elastic contact of a rough

surface. Wear 1975;35:87 111.[12] Johnson KL. Contact mechanics. Cambridge: Cambridge Univer-

sity Press, 1987.[13] Francis HA. Application of spherical indentation mechanics to

reversible and irreversible contact between rough surfaces. Wear1977;45:221 69.

[14] Ogilvy JA. Numerical simulations of friction between contactingrough surfaces. J Phys D, Appl Phys 1991;24:2098 109.

[15] Chang WR, Etsion I, Bogy DB. Adhesion model for metallicrough surfaces. ASME J Trib 1988;110:50 8.

[16] Ernst H, Merchant ME. Surface friction of clean metals a basic

factor in the metal cutting process. In: Proceedings Special Sum-mer Conference on Friction and Surface Finish, MassachusettsInst. Technology, Cambridge, MA, 1940;76 101.

[17] Tabor D. In: Field JE, editor. The properties of diamond. Oxford:Academic Press, 1979.

[18] Ford IJ. Roughness effect on friction for multi-asperity contactbetween surfaces. J Phys D, Appl Phys 1993;26:2219 25.

[19] Seal M. The friction of diamond. Phil Mag A 1981;43(3):587 94.[20] Suh NP, Sin HC. On genesis of friction. Wear 1981;69:91 114.[21] Bailey DM, Sayles RS. Effect of roughness and sliding friction

on contact stresses. ASME J Trib 1991;113:729 34.[22] Hu Y-Z, Barber GC, Zhu D. Numerical analysis for the elastic

contact of real rough surfaces. STLE Trans 1999;42(3):443 51.[23] Tworzydlo WW, Cecot W, Oden JT, Yew CH. Computational

micro- and macroscopic models of contact and friction: formu-

lation. Approach and applications. Wear 1998;220:113 40.[24] Bengisu MT, Akay A. Relation of dry-friction to surface rough-ness. ASME J Trib 1997;119:18 25.

[25] Bengisu MT, Akay A. Stick slip oscillations: dynamics of fric-tion and surface roughness. J Acoust Soc Am1999;105(1):194 205.

[26] Webster MN, Sayles RS. A numerical model for the elastic fric-tionless contact of real rough surfaces. ASME J Bib1986;108:314 20.

[27] Akai TJ. Applied numerical methods for engineers. New York:John Wiley and Sons Inc, 1994.

[28] Tranter CJ. The use of the mellin transform in nding the stressdistribution in an in nite wedge. Quart J Mech Appl Math1948;1:125 30.

[29] Ren N, Lee SC. Contact simulation of three-dimensional rough

surfaces using moving grid method. ASME J Trib1993;115:597 601.[30] Jiang X, Hua DY, Cheng HS, Ai X, Lee SIC. A mixed elastohyd-

rodynamic lubrication model with asperity contact. ASME J Trib1999;121:481 91.

[31] Stanley HM, Kato T. An FFT-based method for rough surfacecontact. ASME J Trib 1997;119:481 5.

[32] Ai X, Sawamiphakdi K. Solving elastic contact between roughsurfaces as an unconstrained strain energy minimization by usingCGM and FFT techniques. ASME J Trib 1999;121:481 91.

[33] Greenwood JA, Tripp JH. The elastic contact of rough spheres.ASME Trans, Series E, J Appl Mech 1967;34:153 9.

[34] Peklenik J. New developments in surface characterization andmeasurement by means of random process analysis. Proc InstMech Eng 1967;182(3):108 26.

[35] Hu YZ, Tonder K. Simulation of 3-D random rough surface by2-D digital lter and Fourier analysis. Int J Mach Tools Manufact1992;32(1/2):83 90.

[36] Singer IL. Solid lubrication processes. In: Singer IL, Pollock HM,editors. Fundamentals of friction: macroscopic and microscopicprocesses. Dordrecht: Kluwer, 1992:237 61.

[37] Tian X, Bhushan B. A numerical three-dimensional model forthe contact of rough surfaces by variational principle. ASME JTrib 1996;118:33 42.

akay a numerical model of friction between rough surfaces

Documents