MECH10_11_03_TB.qxd The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 1. Introduction Many engineering problems lead to the consideration of contact between surfaces. These include studies relat- ed to friction-induced vibration and noise, thermal and electrical contact resistance, and mechanical seals, bush- ing, fasteners, etc. Examination of contact characteristics encompasses equivalent contact stiffness and damping for vibration and noise, and true contact area for mechanical seals, and electrical and thermal resistance. Accounting for contact characteristics inherently necessitates charac- terization of surface topography and development of prob- abilistic models, which relate contact area, contact stiff- ness, contact load and separation of surfaces. These models are based on the presumption that a sur- face can, in effect, be represented by a distribution of asperities. As two surfaces are brought into contact, the macroscopic contact characteristic in question is a cumu- lative effect of localized interactions of the asperities. This approach has required the statistical formulation of a surface and statistical summation of microscopic contact effects to obtain probabilistic macroscopic expectation of ______________________________________ *Corresponding author E-mail: jdabdo@squ.edu.om the contact characteristic (contact area, load, and stiff- ness). Greenwood and Williamson (1966) pioneered "asper- ity-based model." The existing literature shows exten- sions of the Greenwood and Williamson (GW) model over the last three decades. The present probabilistic models of contact may be viewed with respect to the premise of elas- tic and plastic contact. The elastic models primarily rely on the Hertz theory of contact between two elastic bodies (Greenwood and Williamson, 1966; Greenwood and Tripp, 1967, 1970; Hisakado, 1974; Bush et al. 1975 and McCool, 1966). These models differ in their assumptions related to surface and asperity geometry and material properties. These extensions have included, for instance, the inclusion of the surface curvature effects Greenwood and Tripp, (1967), allowance for non-uniform curvature of asperity summits Hisakado, (1974) and presumption of average elliptic paraboloidal representation of asperity Bush et al. (1975). Other works presented advanced models for anisotropic materials McCool, (1986). Of par- ticular interest to the future goal of the present work is the surface model in which asperities are allowed to form con- tact on shoulders Greenwood and Tripp, (1967). The plastic models are based on the presumption that A Modified Approach in Modeling and Calculation of Contact Characteristics of Rough Surfaces J.A. Abdo* Mechanical and Industrial Engineering Department, College of Engineering, Sultan Qaboos University, P.O. Box 33, Al-Khod 123, Muscat, Sultanate of Oman Received 10 November 2003; accepted 30 July 2004 Abstract: A mathematical formulation for the contact of rough surfaces is presented. The derivation of the contact model is facilitated through the definition of plastic asperities that are assumed to be embedded at a critical depth with- in the actual surface asperities. The surface asperities are assumed to deform elastically whereas the plastic asperities experience only plastic deformation. The deformation of plastic asperities is made to obey the law of conservation of volume. It is believed that the proposed model is advantageous since (a) it provides a more accurate account of elastic- plastic behavior of surfaces in contact and (b) it is applicable to model formulations that involve asperity shoulder-to- shoulder contact. Comparison of numerical results for estimating true contact area and contact force using the proposed model and the earlier methods suggest that the proposed approach provides a more realistic prediction of elastic-plastic contact behavior. Keywords: Elastic-plastic, Contact, Friction, Rough surfaces, Contact model ¢¢üü∏∏îîàà°°ùŸŸGG~M ¤G πNG~àJ ¿G ¢VÎØj »àdGh áfôŸG ÒZ äCGƒàædG ∞jô©J ∫ÓN øe ” á«°VÉjôdG ä’OÉ©ŸG ¥É≤à°TG .áæ°ûÿG í£°S’G ¢SÉ«≤d »°VÉjQ êPƒ‰ Ω~≤J ábQƒdG Iòg : åëÑdG Gò¡d .ºé◊G ßØM ¿ƒfÉb ™Ñàj å«ëH ¿ôe ÒZ πµ°ûH »æëæJ áfôŸG ÒZ äCGƒàædG ɪæ«H ¿ôe πµ°ûH »æëæj ¿CG ¢VÎØj äCGƒàædG í£°S .»≤«≤◊G äCGƒàædG í£°S ‘ Ú©e É° jG åëÑdG Ω~≤j .»ÑfÉ÷G ¢Sɪà∏d ä’OÉ©e OÉéjG ‘ ∫ɪ©à°S’Gh ≥«Ñ£à∏d πHÉb (Ü) .¿ôŸG ÒZh ¿ôŸG í£°ùdG ¢SÉ“ ∑ƒ∏°S ÜÉ°ù◊ ÈcG ábO »£©j (CG) :ÚÑÑ°ùd á«∏° aG .É≤HÉ°S â“iôNG äÉ°SGQO èFÉàfh »°VÉjôdG êPƒªædG Gòg ΩG~îà°SG øe œÉædG ÚH ¢Sɪà∏d á«≤«≤◊G iƒ≤dGh áMÉ°ùŸG ôj~≤àd ájO~Y áfQÉ≤e ««MMÉÉààØØŸŸGG ääGGOOôôØØŸŸGG .¢Sɪà∏d »°VÉjQ êPƒ‰ ,áæ°ûÿG í£°S’G ,∑ɵàM’G ,¿ôŸG ÒZ- ¿ôŸG ¢SɪàdG : á áæ°ûÿG í£°SÓd ¢SɪàdG ¢üFÉ°üN ÜÉ°ù◊ ∫~©e »°VÉjQ êPƒ‰ 44 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 contact is dominated by plastic flow. Such models may be best suited for load ranges that warrant large degree of plastic flow. The earliest work on plastic contact model is attributed to Abbott and Firestone (1933). This model utilizes the geometrical intersection of a plane with a rough surface and presumes that contact flow pressure exists over the area of contact, obtained from geometrical intersection. Later works include two dimensional ran- dom process model Nayak, (1971), and investigations leading to the postulation that volume conservation dic- tates plastic flow (Pullen and Williamson, 1972; Nayak, 1973). The research work of Francis (1976, 1977), deal- ing with plastic deformation of contact, include empirical characterization using the results for spherical indentation. Liu et al. (2000) derived exponential functions from a fit- ting procedure applied to numerical results of the Gaussian height distribution offering analytical expres- sion for the Greenwood and Williamson (1966); elastic model, Change, et al. (1986), elastic-plastic model and Horng, (1998) elliptic elastic-plastic model. Whereas the elastic and plastic models are seen to be advantageous for extreme cases of loading, in a large number of engineering applications, contact loads may fall within ranges that do not warrant adequate representa- tion by either elastic or plastic model. This fact has led researchers to consider what is referred to as elastic-plas- tic models (Ishigaki et al. 1979, Chang et al. 1997). In the latter work, elastic-plastic model of contact is proposed based on the conservation of volume during plastic defor- mation. The work proposed by Chang et al. (1997) treat contact as elastic-plastic at macroscopic scale while at microscopic level it views an asperity to experience ini- tially elastic deformation followed by purely plastic defor- mation when a critical interference is reached. More recent works (Kucharski et al. 1994; Chang, 1986) have considered and modified contact models to treat elastic- plastic behavior of metallic coating. Finite element method solutions to elastic-plastic con- tact problems were also utilized. Liu et al. (2000) devel- oped a finite element method solution for an elastic-plas- tic contact problem. The finite element method presented by Kogu and Etsion (2002) suggested that the evolution of the elastic-plastic contact could be divided into three distinct stages. Kogu and Etsion (2003) presented an improved finite element-based model for the contact of rough surfaces. The contact parameters for a single asper- ity contact were predicted. The present work proposes a modified elastic-plastic model for surfaces in contact. The critical interference proposed by Chang et al. (1986) is utilized to define fic- titious asperities. In this manner two sets of asperities are defined: (1) those that are the actual surface asperities and (2) the fictitious asperities which can only deform plasti- cally. The concept of elastic and plastic asperities allows formulation of asperity deformation model which is also elastic-plastic at microscopic scale; in contrast to the model by Chang et al. (1986). Volume conservation is applied only to the plastic (fictitious) asperities. Comparisons of the contact load/separation values pre- dicted by the proposed model with that obtained through experiments as reported by Kucharski et al. (1994) demonstrate the effectiveness of the method in providing accurate prediction of contact characteristics. 2. The Mathematical Model Consider first the contact between a rough surface cov- ered with a number of asperities, that have spherical shape at their summits with average radius R and a nominally flat surface shown in Fig. 1. The summit heights, z, are z ys w d Statistical Representation (b) Rough surface Plane Surface Mean of asperity heights Mean of surface heights (a) wc Average asperity Fictitious asperity (c) Figure 1. Contact between a flat and a rough surface 45 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 assumed to have probability density function φ(z). The probability that an asperity develops contact with the flat plane is (Greenwood and Williamson, 1966): (1) where d is the separation based on asperity heights. For N asperities, the expected number of contacts will be: (2) where the total number of asperities, N, the density of asperities, η, and the nominal area, An, are related according to (3) For this type of contact, the interference w may be described as (Greenwood and Williamson, 1966): (4) In analyzing the contact, the laws describing the depend- ence of the local contact area, Ao, and the local contact load, Po, on w are employed. Hence, (5) The expected total contact area, A1, and the expected total contact load, P, are the statistical sums of the local contri- butions of each asperity. Therefore, (6) (7) The Hertzian contact area, Ao, contact load, Po, and the maximum contact pressure between one asperity, having interference w with a plane, Pm, are given as follows: (8) (9) (10) where, (11) In the equations above R is the average equivalent radius of curvature. In general, the maximum contact pressure is related to the hardness of the softer material, H, through (12) where K is the maximum contact pressure factor. Tabor (1977) showed that the plastic flow is reached when Pm= 0.6H. The critical interference at the inception of plastic deformation wc is defined by substituting Eq. (12) in Eqs. (10) and (14). (13) For w > wc, the contact is plastic. Using the critical interference and imposing the conservation of volume, Chang et al. (1997) derived the modified equations describing the contact area and load on an asperity: (14) and (15) It is appropriate here to emphasize the difference between the present work and that proposed by Chang et al. (1997). In the derivation of the equations, the authors assume that an asperity behavior is initially elas- tic. As the load is increased the elastic behavior continues to describe the deformation until a critical interference is reached. At this critical load and beyond, the asperity deforms as a purely plastic body. In the work of Chang et al. (1997) the elastic and plastic behavior do not occur simultaneously for an asperity. Hence, their formulation can only be characterized as an elastic-then-plastic model at microscopic scale. The method proposed in this paper is shown to repre- sent more accurately an elastic-plastic model of the con- tact, through the introduction of a fictitious surface that can only deform plastically. As shown in Fig. 1, the criti- cal interference, wc, is used to define a second surface. This second surface is obtained by displacement of every point on the surface by wc along the direction normal to the surface (see Fig. 2). As illustrated in Fig. 2, to obtain the mathematical description of the plastic asperity, the mapping of a point, A, on the surface to a point, B, on the plastic asperity must be considered. It is also noted that an asperity is described Fig. 2 in terms of a frame of ref- erence whose origin is at the asperity peak and ordinate points towards the mean plane Greenwood and Tripp, (1967). Therefore, ρy-frame is used to describe the origi- nal asperity, whereas, xy-frame is employed for the plas- tic asperity. ∫=> ∞ d dzzdz )()(Prob φ ∫= ∞ dc dzzNN )(φ nAN η= dzw −= ,)(00 wAA = )(00 wPP = dzzdzAAdA d nt )()()( 0 φη −∫= ∞ dzzdzPAdP d n )()()( 0 φη −∫= ∞ RwA π=0 2321 0 3 4 wERP = 21 0 0 2 2 3 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ == R wE A P Pm π 2 2 2 1 2 1 111 EEE νν − + − = KHPm = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −== w w Rw a A c2 4 2 0 π π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −== w w Rw a A c2 4 2 0 π π KH w w RwP c ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= 20 π The respective positions of points A and B are denoted by Ar and Br , as depicted in Fig. 2. The position of point B on the fictitious asperity is : 46 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 (16) (17) Equations (16) and (17) may be used to obtain the equa- tion describing the plastic asperity (see Appendix A for detail). To the first approximation, the equation of the plastic asperity is given as (Appendix A): (18) Since the plastic asperities only deform plastically, their introduction allows the reduction of Eqs. (14) and (15) to the well-known forms: (19) (20) Where, Rp represents the summit radius of curvature of the plastic asperity. Based on Eq. (18), this radius of curva- ture is (21) Clearly the formulation of the area of contact and con- tact force for two surfaces involves interactions of two sets of asperities: the original surface asperities which are assumed to deform elastically and plastic asperities that deform plastically. The next section presents this new mathematical model of contact. 3. Elastic-Plastic Contact Between a Rough Surface and a Plane The contact between one asperity on a rough surface and a plane is considered. Figure 3 illustrates two types of interactions. The first is the elastic contact between the plane and the surface asperity. If the interference, w, exceeds the critical interference wc, then the interaction also includes plastic contact. It is noted that the shaded volume representing the interference of the plastic asperi- ties and the plane contribute to the plastic portion of con- tact whereas the remaining volume of interference con- tributes to the elastic contact. Therefore if we let Q be the characteristic of contact (area or load) then it may be obtained by appropriately accounting for the aforemen- tioned interactions, that is: (22) where Qe1 corresponds to the elastic contact between the plane and surface asperity. However this includes, as shown in Fig. 3, a portion of the interference which is plastic. Therefore, the contribution due to elastic interfer- Figure 2. Plastic (fictitious) asperity shape ncAcB uwrjwr ++−= where, nu is the unit normal vector to the original asperity at point A. As usual , the unit vectors i and j are defined along ρ (or x) and y axes, respectively. Employing the notation of Greenwood and Tripp (1967), j R irA 2 2 1 ρρ += )(2 2 cwR x y − = wRA pπ20 = wKHRP pπ20 = )( cp wRR −= 221 pee QQQQ +−= Figure 3. Elastic-plastic interaction of an asperity with a plane 47 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 ence between the plane and the plastic asperity Qe2 must be subtracted to obtain the net elastic contribution. The contribution from plastic interaction due to the plastic interference of the plane and plastic asperity Qp2 is, then, added to the result. Using this approach the area of con- tact may be described as: (23) where, (24) (25) (26) (27) (28) (29) It is noteworthy to mention here that the plastic asper- ity peaks can be viewed as being farther away from the plane by wc. We have made the approximation as to the mean plane of plastic asperity being wc below the mean plane of the surface asperities. Therefore the limits of integration are shifted by wc as presented in Eqs. (25) and (26). Furthermore, in these equations the summit curva- ture corresponding to plastic asperities are used. Proceeding in a similar manner, the contact load may be written as: (30) where, (31) (32) (33) 4. Results: Evaluation of the Proposed Model The effectiveness of the proposed model is evaluated using the data and results given by Chang et al. (1997) and Kucharski et al. (1994). For convenience we shall refer to the former as CEB and the latter as KKPK models. The proposed model in this paper will be referred to as AFM model. In the ensuing discussion, results are presented in two subsections. In the first subsection the AFM model is evaluated and compared with the CEB model. In the sec- ond subsection AFM, CEB and KKPK models are tested against experimental results given by Kucharski et al. (1994). 4.1 Comparison of AFM and CEB models In the CEB model, as it is the case in our model, a dimensionless form of the probability density function is introduced: (34) To combine the material and surface topographic prop- erties in contact, the plasticity index, ψ, is introduced according to Greenwood and Williamson (1996): (35) The relation between separation h and σ of the sur- face microgeometry model and d and σs of the asperi- ty-based model is given as: (36) A form for plasticity index is obtained by substituting Eqs. (13) and (36) in (35), to give: (37) ])(5.0exp[)()2()( 2221* ss ss σ σ σ σ πφ −= − * 2 * 2 * 1 * )( epet AAAhA −+= ∫= ∞ − ** *** 1 )()( syh e dsswhA φπβ ∫= ∞ +− *** *** 2 )()( cwsyh pe dsswhA φπβ ∫= ∞ +− *** *** 2 )(2)( cwsyh pp dsswhA φπβ where φ(s) is the dimensionless form of the probability density function for summit height distribution . β, βp, w* and the dimensionless height of asperity , s, are defined as follows : σηβσηβ )(, cp wRR −== *** syhsw +−= σ z s = * 2 * 2 * 1 ** )( epe PPPhP −+= ∫⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∞ − ** 23* 21 ** 1 )(3 4 )( syh e dsswR hP φβ σ ∫⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∞ +− *** 23* 21 ** 2 )(3 4 )( cwsyh pe dsswR hP φβ σ ∫= ∞ +− *** *** 2 )(2)( cwsyh pp dsswE H KhP φβπ 21)( −= s cw σ ψ 22 4 22 10717.3 R s η σσ −× += 41 2 4 21 ) 10717.3 1()( 2 β σ π ψ −× −= RKH E The data employed by Chang et al. (1997) pertaining to steel on steel material is employed with the following parameters: MPaEE 521 1007.2 ×== , Brinell hardness MPaH 1960= , 29.021 == νν . The maximum contact pressure is taken to be based on 6.0=K , and β and σ/R are selected for different values of plasticity index from the experimental work of Nuri and Halling (1975). These values and the values of *cw and σ/σs, calculated from Eqs. (35) and (36), respectively, are shown in Table 1. Figure 4 illustrates the contact area versus separation, both given in dimensionless form, for the AFM and CEB models. While both models predict similar contact area for low plasticity index, the AFM model pr edicts higher values for materials of higher plasticity index. Figure 5 depicts the dimensionless 48 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 Figure 4. Contact area ration versus separation: Comparison between AFT and CEB models Figure 5. Dimensionless separation versus contact load: Comparison between AFT and CEB models Ψ ηη R σ σ/R σ/σs 0.5 0.0302 8.75x10-5 1.299192 0.8 0.374 2.00x10-4 1.001331 2.5 0.601 1.77x10-3 1.000515 Table 1. Plasticity index and surface topography C on ta ct A re a R at io , A t/A n Dimensionless Mean Separation D im en si on le ss M ea n Se pa ra tio n Dimensionless Load, P/AnE 49 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 (38) Therefore material with low plasticity index may be approximated as a purely elastic body. On the other hand, for high plasticity index (softer material), the contact is approximately totally plastic and the total expected dimensionless contact load for elastic-plastic contact of Eq. (30) is approximately: (39) Figure 6 depicts the contact area ratio versus dimen- sionless contact load for different values of, φ, as predict- ed by the AFM and CEB models. 4.2 Comparison of AFM, CEB and KKPK models It is the intent of this subsection to evaluate the pro- posed model (AFT) and present a comparative study of the model with those proposed by CEB and KKPK. In doing so we present the experimental results by Kucharski et al. (1994) as the basis of these comparisons. They pre- sented a finite element model of elastic-plastic contact (referred to as KKPK). They also performed measure- ment of contact load, area and approach. In their work, steel specimens are employed with Young's modulus E1 = 200,000 MPa, Poisson's ratio v1 = 0.3 and tensile yield strength Y = 400 MPa. The three dimensional profilom- etry of sand-blasted surface (E60), average over three samples, are given as follows Kucharski et al. (1994): As in Kucharski et al. (1994), 3Y/2 is employed instead of KH value in calculating the plastic contact con- tribution. Moreover, dimensionless approach is used instead of separation in presenting the results. In this case, approach is obtained by Kucharski et al. (1994): (40) where α is the approach, and zmax and h are the maxi- mum peak height and separation. Figure 7 illustrates the results. In this case dimension- less approach is with respect to the maximum summit height, Msh. Clearly, the proposed (AFM) model presents the most favorable agreement between the predicted con- tact load/approach values to that obtained by measure- ments. The accuracy of the AFM model is also attested to by the results shown in Fig. 8. The results show that the AFM model provides significantly more accurate predic- tion of contact load and contact area than CEB and KKPK models. 4.3 Elastic-Plastic Contact for Rough Surfaces In this section direct formulation of contact of rough- on-rough surfaces based on the work of Greenwood and Tripp (1967) is presented. The purpose here is to demonstrate the adaptability of the proposed contact model using plastic asperity concept and also a formal mathematical treatment of contact between two rough surfaces. Greenwood and Tripp (1967) intro- duced elastic as well as plastic models for two rough surfaces covered with paraboloidal asperities. Consideration of contact between two asperities results in the following description of interference (41) where z1 and z2 are the heights of asperities on surfaces one and two, respectively. r is the radial distance denot- ing the offset between the central lines of the two asperi- ties. For a summit radius of curvature R, the presumption of Hertzian contact leads to: (42) (43) The expected dimensionless contact load for elastic con- tact is (44) and that for plastic contact is (45) where, )( 15 216 )( **25 2** syhFR hP −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = σ πβ separation h* versus dimensionless load EA P n for different values of plasticity index , ϕ , as predicted by the AFM and CEB models. It is clear from the figure that the separation increases with the plasticity index for a given load. For low plasticity index (harder material), the contact is approximately totally elastic. As summarized in Table 1, lower plasticity index corresponds to larger critical interference, cw . This, in turn, increases e ffective separation for plast ic asperity as seen in Eqs. (32) and (33) . The result is diminished contributions of plastic asperities. Hence , in Eq. (30) 2eP and 2pP attain smaller value s, resulting in the following approximation : * 1 ** )( ePhP = * 2 ** )( pPhP = • Areal summit density 4.655=η mm-2 • Mean summit radius R = 30.0µm • 3D standard deviation of summit height distribution σ = 2.3µm • 3D maximum peak height zmax = 13.5 µm • 3D standard deviation of the surface height distribution σs = 2.95µm • 3D maximum summit height Msh = 9.9µm hza −= max )2/(221 rfzzw −+= w R A 20 π= 23 21 0 23 4 w R EP ⎟ ⎠ ⎞ ⎜ ⎝ ⎛= )(2)( **2 22** syhFE H hP −= βπ dssusuF u n n )()()( φ∫ −= ∞ 50 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 Figure 6. Contact area ratio versus contact load: Comparison between AFT amd CEB models 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.00E+00 5.00E-04 1.00E-03 1.50E-03 Dimensionless Load, P/AnE D im en si on le ss a pp ro ac h AFT CEB Experiment FEM Figure 7. Contact load versus approach: Comparison between AFT, CEB, experimental results of KKPK, and FEM model of KKPK Figure 8. Contact area versus dimensionless load: Comparison between AFT, CEB, experimental results of KKPK, and FEM model of KKPK C on ta ct A re a R at io , A t/A n C on ta ct A re a R at io , A t/A n D im en si on le ss A pp ro ac h Dimensionless Load, P/AnE Dimensionless Load, P/AnE Dimensionless Load, P/AnE 51 The Journal of Engineering Research Vol. 2, No. 1 (2005) 43-52 Hence, using the proposed method, the total dimension- less expected contact load for elastic-plastic behavior is: (46) where, (47) (48) (49) and σ denotes the standard deviation of sum of asperity heights on the two surfaces. 5. Conclusions A mathematical formulation of elastic-plastic contact has been presented in this paper. Using the definition of critical interference, the concept of elastic and plastic asperities have been developed in which a rough surface is represented by two surfaces. The first surface is the actual physical surface and is assumed to deform as a purely elastic body. The second surface is a fictitious one and it is derived from the physical surface and the critical interference. This surface is assumed to deform as a pure- ly plastic body. The development of the elastic and plas- tic surfaces has facilitated the mathematical formulation of elastic-plastic contact of rough surfaces. Comparison of the proposed model with the existing models for elas- tic-plastic contact have been performed. The measure- ment results of previously performed experiments on sand-blasted steel surface (Kucharski et al. 1994) have shown that the proposed model provides significant improvement over previous models in the prediction of contact load and area of contact. The proposed model (AFT) gives better prediction than CEB since in CEB for every asperity the elastic quantity due to plastic pressure Qe2 is not considered. Hence, the elastic and plastic behaviors do not occur simultaneously for an asperity. Therefore, their formulation can only be characterized as an elastic-then-plastic model at microscopic scale and elastic-plastic model at macroscopic scale. Appendix A Consider an asperity and assume that its shape is quad- ratic as proposed by Greenwood and Tripp (1971). The equation of the surface asperity is given by (A1) (A2) The unit tangential vector is obtained as (A3) Hence the unit normal vector is (A4) Then the description of the plastic asperity is obtained by (A5) or (A6) (A7) let then (A8) Therefore, the shape of the plastic (fictitious) asperity is given by: (A9) * 2 * 2 * 1 ** )( epet PPPhP −+= )( 15 216 )( **25 2** 1 se yhFR hP −⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = σ πβ )( 15 216 *** 25 2* 2 cspe wyhFR P +−⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = σ πβ )(2 ***2 22* 2 cspp wyhFE H P +−= βπ 2 2 1 ρ R y = j R ijyirA 2 2ρ ρρ +=+= )( 1 1 2 2 j R i R rd rd u A A t ρ ρ + + == )( 1 1 2 2 ji R R un +− + = ρ ρ jwuwrr cncAB −+= jw R w R i R R w r c c c B ) 1 2 () 1 1( 2 2 2 2 2 − + ++ + −= ρ ρ ρ ρ j R w R i R w r ccB ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −+⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −= 1 2 1 2ρ ρ )(2 2 cwR x y − = as shown in Fig. 2. The figure also illustrates fictitious plastic asperity whose shape is obtained by a displacement of cw along the normal to the quadratic curve. 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