Deterministic repeated contact of rough surfaces

Available online at www.sciencedirect.com Wear 264 (2008) 349 358 Deterministic repeated contact of rough surfaces J. Jamari, D.J. Schipper Laboratory for Surface Technology and Tribology, Faculty of Engineering Technology, University of Twente, Drienerloolaan 5, Postbus 217, 7500 AE Enschede, The Netherlands Received 14 July 2006; received in revised form 16 February 2007; accepted 22 March 2007 Available online 15 May 2007 Abstract A theoretical and experimental study to analyze the contact behavior of the repeated stationary deterministic contact of rough surfaces is presented in this paper. Analysis focuses on the contact of rough surfaces on asperity level without bulk deformation. Contact area and deformation of isotropic and anisotropic contacting surfaces were simulated for each loading cycle and were plotted along with the experimental results. Very good agreement is found between the theoretical prediction and the measurement. It was found that the contact behavior becomes elastic soon after the first loading has been applied for the same normal load. 2007 Elsevier B.V. All rights reserved. Keywords: Contact mechanics; Elastic plastic contact; Asperity; Loading unloading 1. Introduction For many engineering applications knowledge about contact between surfaces is of fundamental importance to understand friction, wear, lubrication, friction-induced vibrations and noise, thermal and electrical contact resistance, etc. Therefore, the interest in studying the contact between rough surfaces is very high. This can be seen in the reviewing papers of Bhushan [1], Liu et al. [2], Barber and Ciavarella [3] and Adams and Nosonovsky [4]. Several approaches have been proposed to study the behavior of the contacting surfaces: statistical [5 11], fractal [12] and direct or numerical simulation [13 16]. The statistical contact model is pioneered by Greenwood and Williamson [5] where a nominal flat surface is assumed to be composed by hemi-spherical asperities of the same radius and the height of the asperities is represented by a well-defined statistical distribution function (i.e., Gaussian). In the fractal theory, the scale-dependency of the rough surface is diminished; however, the method can be applied only for fractal related surfaces. For numerical contact modeling of surfaces, most of the proposed models consider pure elastic deformation; numerical elastic plastic contact models are rarely reported. Another Corresponding author. Tel.: +31 53 4892463; fax: +31 53 4894784. E-mail address: j.jamari@ctw.utwente.nl (J. Jamari). important consideration to numerical contact modeling is the computational cost. Karpenko and Akay [17] have introduced a fast numerical method to calculate the friction force between two rough surfaces for the elastic contact situation. Contact force distribution is computed, using the local contact geometry, until the sum of the local contact forces equals the normal load. This iterating procedure is similar with the procedure as used in the present paper. The method of Karpenko and Akay [17] has been applied to the elastic plastic contact situation for the contact of rough wavy surfaces [18]. However, in their elastic plastic contact analysis, a contact point deforms elasto-plastically once its contact pressure reaches the hardness of the softer material while it has been widely accepted [1,2] that the elastic plastic condition starts when the mean contact pressure reach about 0.4 times the hardness of the softer material. There are many models available to analyze the contact of rough surfaces in the elastic, elastic plastic and fully plastic contact regimes, however, the experimental investigations are rarely published. A theoretically and experimentally study of the deterministic contact between rough surfaces in the fully plastic contact regime has been proposed by Jamari et al. [19]. Results showed that the proposed model correlate very well with the experimental investigation. Most mechanical contact pairs carry their service load not just once but for a large number of (repeated) cycles. The application ranges from micro/nano-systems, such as MEMS microswitches [20], to normal systems, such as rail-wheel con- 0043-1648/$ see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2007.03.024

350 J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 tact [21]. Several studies have been performed to study the elastic plastic contact behavior for the loading and unloading case. By Finite Element Analysis calculation, Vu-Quoc et al. [22] studied the relation between force and displacement of the loading unloading of contacting spherical bodies and Li et al. [23] analyzed the loading unloading of the contact between a rigid flat with an elastic plastic sphere or a rigid sphere with an elastic plastic half-space. Recently, Jones [24] modeled the loading and unloading of a rough surface based on the statistical work of Greenwood and Williamson [5]. This paper presents an analysis of the loading and unloading of a stationary deterministic contact between rough surfaces. Rough surfaces are modeled based on the measured real surfaces. Asperity or roughness deformation without any bulk deformation is considered. The model introduces a new method to determine the initial asperity geometry and the remaining or deformed asperity geometry after unloading. In order to verify the proposed model, an experimental investigation has been performed. 2. Model 2.1. Model condition In the proposed deterministic contact of rough surfaces, the deformation of the contacting surfaces is studied only on asperity level without any surface bulk deformation, therefore, a criterion as was proposed in [25,26] is used to make sure that the contact system is operating in the asperity deformation regime only. A deformation criterion C m is defined, according to [25], as: C m = p 0 H [ 1 exp ( 2 3α 3/5 )] < 0.6 (1) where H is the hardness of the softer material. p 0 and α are the maximum Hertzian contact pressure and the roughness parameter, respectively, defined as [27]: ( 16PE 2 ) 1/3 p 0 = π 3 R 2 (2) ( 16RE 2 ) 1/3 α = σ 9P 2 (3) where P is the load, R the radius of the curved body and σ is the standard deviation of the height distribution. The effective elastic modulus E is defined as: 1 E = 1 v2 1 + 1 v2 2 (4) E 1 E 2 where v 1, v 2, and E 1, E 2, are respectively the Poisson s ratio and the elasticity modulus of surfaces 1 and 2. 2.2. Asperity determination Schematic representation of the stationary repeated contact between rough surfaces is presented in Fig. 1(a). Here, the contact loading and unloading takes place at the same position. Fig. 1. Repeated contact of rough surfaces. There are several ways to analyze the asperity-based deterministic contact of rough surfaces. Greenwood [28], for instance, introduced a way to calculate the asperity properties using the summit method. In this method, a summit is defined as a local surface height higher than its neighboring points whether based on five or nine points. Recently, Jamari et al. [19] proposed a new method to determine the contacting asperity properties based on volume conservation. Contact areas (Fig. 1(b)) are determined based on the measured rough surface loaded by a certain load P or separation given between the means of surface heights. These areas are converted into elliptical cross-sectional areas (Fig. 1(c)) by conserving the total area of each asperity of Fig. 1(b). Other parameters such as asperity radii and asperity heights are determined based on the volume conservation method, see Figs. 2 and 3. Details of the method can be found in Appendix A. Fig. 2 shows the discrepancy of the asperity properties between the nine-points summit based asperity and the asperity volume conservation method. An example of surface protrusion and its cut-off height is demonstrated in Fig. 2(a). In Fig. 2(b) (d), at the left-side are the properties obtained by the volume conservation method and at the right-side are the properties by using the nine-point summit method. Fig. 2(b) (d) presenting the asperities property for the indentation by a flat of ω 1, ω 2 and ω 3, respectively. As can be seen from Fig. 2(b) (d), the number of asperity in contact (micro-contacts) changes as the indentation or separation changes. For the summit method, this number always increases as the indentation increases. For the volume conservation method, this number is changing depending on the surface geometry and the location of the asperities in contact depending to the indentation. For d = ω 3, for instance, there are 40 asperities in contact according to the summit method whilst for the volume conservation method there is only 1 large asperity which represents a real surface in contact much better than the summit method. It can be concluded that the summit

J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 351 Fig. 2. 3D surface asperities and its properties. (a) Model asperity and its centre profile (b) (d) location micro-contact points (not size) as a function of indentation 1 to 3, respectively. Left volume conservation method and right the nine-point summit method.

352 J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 Fig. 3. Asperity height determination (a) surface, (b) original asperity represented by paraboloid and (c) height z of paraboloid. method represents the micro-contacts well only for the very small interference whilst the volume conservation method the micro-contacts are represented very well for all the interferences or cut-off heights. Therefore, in the present model the volume conservation method is used. The curvatures of the asperity based on the volume conservation method, see Appendix A, are defined as: a new asperity geometry, and as a result shifts the operating contact regime to the elastic contact situation. In the proposed model also an elastic perfectly plastic material property without any strain hardening effect is considered. Therefore, the elastic plastic elliptical asperity contact model of Jamari and Schipper [29] is employed. κ x = 4π V A 2 L y L x (5) κ y = κ x L 2 x L 2 y (6) where V is the volume and A is the contact area. V and A are determined from the micro-geometry measurement, therefore, the elliptic paraboloid representing the actual asperity has the same volume and contact area as the as measured asperity. After determining the asperity curvatures, by Eqs. (5) and (6), another important parameter in the present deterministic contact model is the asperity height. Fig. 3 shows the method how the asperity height is determined. The asperity height, z, is determined based on the calculated radius in x direction R x and the diameter of the cross-sectional elliptical contact area in x direction L x as: z = d + (L x/2) 2 (7) 2R x where d is the contact separation measured from the mean contact of the surface height of the contacting surfaces. 2.3. Elastic plastic elliptical asperity contact model The elastic plastic to fully plastic contact condition is considered in the present analysis, however, for subsequence loading the asperity changes due to plastic deformation, resulting into Fig. 4. Flow diagram for the contact calculation of repeated stationary contacts.

J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 353 Fig. 5. Matching procedure: surface before test (a), surface after test (b) and their difference (c). 2.4. Calculation procedure Principally, the procedure to simulate the repeated stationary contact situation is similar to the calculation of single contact as for instance was presented in [19]. The only difference is that for the repeated stationary contacts the new surfaces, as a result of the previous loading condition, are used as the input for the next loading cycle. Fig. 6. Setup of the experiment. Fig. 4 shows schematically the iterative procedure for analyzing the repeated stationary contact of rough surfaces using the volume conservation method. The 3D surface height data of the contacting surfaces, z 1 (x, y) and z 2 (x, y) are taken from a roughness measurement and are used as inputs. The hardness properties of the material H of the surfaces are obtained from indentation tests. A normal load is applied to bring these surfaces into contact. In the present study, analysis of the contact is based on the contact interference; therefore, the load is calculated based on the separation between the surfaces. For this purpose, a separation is given as an initial guess. At this separation, there will be a number of asperities in contact (microcontacts). Subsequently, all the contact input parameters such as curvatures, height and location of each asperity in contact are determined based on the volume conservation method as was discussed early. Given all the contact input parameters, the contact load, the contact area, etc. can be calculated readily from the analysis of a single asperity as was discussed in Section 2.3 and formulated in [29]. Summation of all the micro-contact loads (P i ) gives the macro-contact load (input load), therefore, an iterative procedure is applied by changing the separation until the difference between summation of all the micro-contacts and the applied load reach a certain criterion, ε. When the criterion ε (=0.01) is satisfied the iteration loop is terminated and the contact parameters such as load, contact area, plastic deformation, etc. for each asperity are taken from the last separation. Here, the surface topography z 1 (x, y) and z 2 (x, y) are the output from the contact model simulation. These outputs are used as the input for the next loading cycle. The condition where there

354 J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 Fig. 7. Isotropic aluminium surface before contact is applied (a) and location (not size) of the corresponding asperities (b). is no plastic deformation anymore, i.e., the difference between the subsequent input and output values of the surface topography satisfies a certain running-in surface criterion ε r (=0.01), is referred as the run-in condition. 3. Experiment 3.1. Specimens Experiments are performed by loading a hard and smooth curved surface onto a deformable nominally flat rough surface. A hard Silicon Carbide ceramic SiC ball (H =28GPa, E = 430 GPa and v = 0.17) with a diameter of 6.35 mm was used as hard smooth curved indenter. To comply with the assumption of perfectly smooth surface as was used in the present analysis, the r.m.s. roughness of the ceramic ball of 0.01 m was chosen. An elastic perfectly plastic aluminium (H = 0.24 GPa, E = 75.2 GPa and v = 0.34), as was used for the single asperity contact experiment in [30], was used for the rough flat surface specimen. Isotropic and anisotropic flat surfaces were used. The r.m.s. roughness is respectively, 1.3 and 1.4 m for the isotropic and the anisotropic flat surfaces. 3.2. Matching and stitching tool The plastic deformation of the flat surface by a spherical body after being loaded against each other, for instance, is distinctive but the plastic deformation of the asperities (without bulk deformation) cannot be observed with the naked eye. Sloetjes et al. [31] have developed a new robust technique to determine the changes of the surface topography locally by matching the measurement images before and after an indentation test. Procedure of the final matching process is described in Fig. 5. Fig. 5(a) shows the 3D measurement result of the surface before the test. This surface is brought into a contact with a hard spherical indenter. This surface is measured again after the indentation test and is presented in Fig. 5(b). In this case, to determine the change of the surface cannot be done by simply subtracting the surface before and after the test since the surface after the test has translated and rotated relatively to the surface taken before the test, therefore, the matching process is needed. In the matching process the mutual translations and rotations are determined by aligning or repositioning both surfaces. Once these mutual translations and rotations are found the surfaces are matched and the difference image is determined readily by subtracting the matched surfaces before and after test which results in Fig. 5(c). Fig. 8. Contact area of the isotropic aluminium surface after the first load cycle: (a) model and (b) experiment.

J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 355 3.3. Experiment details Fig. 6 shows the experimental setup used in the present experiments. The setup can bear a maximum load of about 300 N. The spherical and the flat specimens were cleaned with acetone and dried in air prior to any test. An optical interference microscope was used for measuring the three-dimensional surface roughness of the surface before and after a test. As is shown in Fig. 6, anx Y table which is controlled by stepper motors is used to positioning the flat specimen from the loading position (position A) to the surface measuring position (position A ) and the other way around. The flat surface was measured at the statically optical interference microscope (position A ) before the loading test. After finishing the surface measurement in position A, the flat surface was moved to the loading position (position A). In this loading position the statically mounted spherical specimen was moved down by a loading screw and subsequently loaded by a dead weight load system. The contact region was lubricated in order to reduce possible effects of friction on the contact condition. The load was applied to the contact system for 30 s and then removed by lifting up the loading arm manually. Next, the flat surface is brought to the surface measuring position (position A ) by using the X Y table. Before measuring the surface after loading with the optical interference microscope, again the flat specimen was cleaned and dried using the same procedure as was described earlier. Subsequently, the surface is acquisitioned by the interference microscope for surface topography data. Until this step, the first loading cycle is finished. To continue the test for the next loading cycle the same procedure is applied. When position A is reached, the load is gently reapplied. All the surface topography images from every loading cycle are matched with the initial surface topography image. This was done separately by a personal computer. 4. Results and discussions Fig. 7 to Fig. 9 show the deformation results for the isotropic surface. Elastic perfectly plastic aluminium as was used in [30] Fig. 9. Profile of the matched and stitched isotropic aluminium surface: (a) x-profile at y = 120 m and (b) y-profile at x = 129 m; n = number of load cycles. has a hardness constant for full plasticity c h of 0.71 and the inception of fully plastic deformation ω 2 equals 80 times the inception of the deformation where the first yield occurs ω 1 for circular contact, or c A = 160. These constants are used in the Fig. 10. Anisotropic aluminium surface before load is applied (a) and location (not size) of the corresponding asperities (b).

356 J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 Fig. 11. Contact area of the anisotropic aluminium surface after the first load cycle: (a) model and (b) experiment. contact simulations. The explanation of these constants can be found in [30]. Fig. 7(a) presents the initial isotropic surface before brought in contact with a hard smooth ball. A load of 0.2 N was applied to the contact. The dashed line represents macroscopic contact area. The position of the asperities in contact due to the applied load is presented in Fig. 7(b). The location (not size) of the corresponding asperities is represented by dot markers in this figure. 0.2 N load together with the r.m.s. roughness value R q of 1.3 m and its material properties yields a C m value for the criterion of Eq. (1) of 0.16 which means that deformation of the surface on asperity level is expected. As can be seen from Fig. 9, there is no bulk deformation, all the plastic deformation occurs at asperity level so that the deformation criterion works accurately. The contact area prediction by the model for the cycle n =1 is depicted in Fig. 8(a). Impression of the size of the geometry of the asperities may be illustrated from this figure. The experimental results of the contact area from the contact system are presented in Fig. 8(b). As can be seen, the model predicts the contact area very well. More insight in the model prediction and the measured results may be seen in the plot of the asperities in x and y direction as presented in Fig. 9. The model predicts the change of the surface topography accurately. Experimental results show that there is almost no difference between the profile for cycle 1, 2, 3, 10 and 20. In the first loading cycle the asperity deforms elastic, elastic plastic or fully plastic depending on its geometry, location and height of the contact indentation. The contact area for each asperity is developed to support the load. When the load is removed, plastic deformation will modify the surface geometry due to elastic plastic and fully plastic deforming asperities which normally increase the degree of conformity to the counter-surface (indenter). If the same load is reapplied at exactly the same position (stationary), the same contact area as for the first loading will be developed. In this second loading step the asperities deform elastically as a result of the residual stresses and geometrical changes induced by the first loading, therefore there is no change of the surface topography or of the contact area. Residual stresses will increase the yielding stress and the change of the geometry will reduce the level of applied stresses. In the present model the change of the surface geometry is the only factor. Since the same loading condition is applied for cycle 1, 2, 3, 10 and 20, therefore, the same phenomena are observed. Fig. 12. Profile of the matched and stitched anisotropic aluminium surface: (a) x-profile at y = 118 m and (b) y-profile at x = 126 m; n = number of load cycles.

J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 357 Results of the anisotropic aluminium surface are presented in Figs. 10 12. The initial anisotropic aluminium surface and its asperity location are shown in Fig. 10. The same load as was applied to the isotropic aluminium surface is used for the anisotropic surface. Fig. 11 presents the result of the contact area of the model prediction and of the experimental investigation for the first cycle. From this figure it can be seen that the surface is dominantly represented by a single elliptic asperity. With an r.m.s. roughness R q value of 1.4 m, it yields a C m value of 0.15 so that asperity deformation is expected. This is confirmed by the experimental results of Fig. 12 where the plot of the x and y profile are drawn. Results for cycle 1, 2, 3, 10 and 20 are similar to the isotropic aluminium surface, i.e., there are almost no differences in the contact area and the surface topography. Small deviations between the model prediction and the experimental results are observed for the y profile in Fig. 12(b). This may be caused by the assumption of the rigid indenter where, in fact, there is some elasticity of the ceramic ball indenter. 5. Conclusions Theoretical and experimental studies have been performed to analyze the deterministic contact behavior of the repeated stationary contact of rough surfaces. Bulk deformation of the surface is excluded in the analysis and concentrated to the contact deformation on asperity level. Contact area and deformation of the isotropic and anisotropic contacting surfaces were simulated for every loading cycle and were plotted along with the experimental findings. Results show that the theoretical model predicts the measurement results very well. In the repeated stationary contact of rough surfaces, it was found that the contact behavior becomes elastic soon after the first loading has been applied for the same normal load. This is because of the degree of conformity of the contacting surfaces is reached, so that the developed contact area supports the applied load elastically. Acknowledgements The financial support of SKF ERC B.V. Nieuwegein, The Netherlands and The Dutch Technology Foundation (STW) are gratefully acknowledged. Appendix A. Volume conservation method for determining asperity geometry In this appendix, a method to model the micro-contacts of real rough surfaces which asperities are represented by elliptical paraboloids will be described. The size of the asperities is based on a volume conservation method. An elliptical paraboloid is defined as a paraboloid having an elliptical cross-section in the xy-plane and paraboloids in the xzand the yz-plane respectively, see Fig. 2. In mathematical form, this elliptical paraboloid is expressed as: x 2 a 2 + y2 b 2 z c = 0 (A.1) where x, y, and z are the coordinate system and a, b, c are constants. The volume displaced, V, due to contact of this elliptical paraboloid is the same as the volume above a certain cut-off height (volume conservation), hence: xhigh yhigh zhigh V = dx dy dz (A.2) x low y low z low The limits of integral in Eq. (A.2) are determined as follows. For the z-coordinate the upper integration limit is the cut-off height ω of the micro-contact as: z low = ω (A.3) and the lower limit is determined rearranging Eq. (A.1) into: z high = c a 2 b 2 (x2 b 2 + y 2 a 2 ) (A.4) For the y-coordinate the following equations are valid at the edge of the contact: c a 2 b 2 (x2 b 2 + y 2 a 2 ) = ω (A.5) Solving y in Eq. (A.5) gives: y low = b ac c(a 2 ω cx 2 ) (A.6) y high = b ac c(a 2 ω cx 2 ) (A.7) Now, the integration limits for x are left to consider. These can be determined by substituting y low = y high = 0 into Eqs. (A.6) and (A.7) which results: ω x low = a (A.8) c ω x high = a (A.9) c By substituting Eqs. (A.3) (A.9) into Eq. (A.2) prior to integration and simplifying yields: V = π ba 2 c ω2 (A.10) The length of the micro-contact area in x-direction L x is calculated by subtracting Eq. (A.9) by Eq. (A.8) as: ω L x = 2a (A.11) c and the length of the micro-contact area in y-direction L y is calculated by subtracting Eq. (A.7) by Eq. (A.6) at x = 0 as: L y = 2 b ca ac 2 ω (A.12) Substituting Eqs. (A.11) and (A.12) into Eq. (A.10) results a new expression for the volume as: V = 1 8 πl xl y ω (A.13) The curvature is defined as the second derivative of the elliptical paraboloid, thus the curvature κ x in x-direction and the

358 J. Jamari, D.J. Schipper / Wear 264 (2008) 349 358 curvature κ y in y-direction are found by applying the second derivative to Eq. (A.1) as: κ x = 2 a 2 c κ y = 2 b 2 c (A.14) (A.15) A combination is made by rearranging Eqs. (A.10) (A.15): κx κ y = 2 c ab = 8ω = 4π V L x L y A 2 (A.16) κ x = L2 y κ y L 2 (A.17) x Substituting Eq. (A.17) into Eq. (A.16) and rearranging gives the final expressions for the elliptic paraboloid that will be used for fitting the real micro-contact region as: κ x = 4π V A 2 L y L x κ y = κ x L 2 x L 2 y References (A.18) (A.19) [1] B. Bhushan, Contact mechanics of rough surfaces in tribology: multiple asperity contacts, Tribol. Lett. 4 (1998) 1 35. [2] G. Liu, Q.J. Wang, C. Lin, A survey of current models for simulating the contact between rough surfaces, Tribol. Trans. 42 (1999) 581 591. [3] J.R. Barber, M. Ciavarella, Contact mechanics, Int. J. Solids Struct. 37 (2000) 29 43. [4] G.G. Adams, M. Nosonovsky, Contact modelling forces, Tribol. Inter. 33 (2000) 431 442. [5] J.A. Greenwood, J.B.P. Williamson, Contact of nominally flat surfaces, Proc. R. Soc. Lond. A295 (1966) 300 319. [6] W.R. Chang, I. Etsion, D.B. Bogy, An elastic plastic model for the contact of rough surfaces, ASME J. Tribol. 109 (1987) 257 263. [7] J.H. Horng, An elliptic elastic plastic asperity microcontact model for rough surface, ASME J. Tribol. 120 (1998) 82 88. [8] Y. Zhao, D.M. Maietta, L. Chang, An asperity microcontact model incorporating the transition from elastic deformation to fully plastic flow, ASME J. Tribol. 122 (2000) 86 93. [9] Y.R. Jeng, P.Y. Wang, An elliptical microcontact model considering elastic, elastoplastic, and plastic deformation, ASME J. Tribol. 125 (2003) 232 240. [10] L. Kogut, I. Etsion, A finite element based elastic plastic model for the contact of rough surfaces, Tribol. Trans. 46 (2003) 383 390. [11] R.L. Jackson, I. Green, A statistical model of elasto-plastic asperity contact between rough surfaces, Tribol. Inter. 39 (2006) 906 914. [12] A. Majumdar, B. Bhushan, Fractal model of elastic plastic contact between rough surfaces, ASME J. Tribol. 113 (1991) 1 11. [13] M.N. Webster, R. Sayles, A numerical model for the elastic frictionless contact of real rough surfaces, ASME J. Tribol. 108 (1986) 314 320. [14] X. Liang, Z. Linqing, A numerical model for the elastic contact of threedimensional real rough surfaces, Wear 148 (1991) 91 100. [15] N. Ren, S.C. Lee, A contact simulation of three-dimensional rough surfaces using moving grid method, ASME J. Tribol. 115 (1993) 597 601. [16] L. Chang, Y. Gao, A simple numerical method for contact analysis of rough surfaces, ASME J. Tribol. 121 (1999) 425 432. [17] Y.A. Karpenko, A. Akay, A numerical model of friction between rough surfaces, Tribol. Inter. 34 (2001) 531 545. [18] Y.A. Karpenko, A. Akay, A numerical method for analysis of extended rough wavy surfaces in contact, ASME J. Tribol. 124 (2002) 668 679. [19] J. Jamari, M.B. de Rooij, D.J. Schipper, Plastic deterministic contact of rough surfaces, ASME J. Tribol., accepted. [20] S. Majumder, N.E. McGruer, G.G. Adams, A.P. Zavracky, P.M. Morrison, J. Krim, Study of contacts in an electrostatically actuated microswitch, Sensor Actuators A 93 (2001) 19 26. [21] A. Kapoor, F.J. Franklin, S.K. Wong, M. Ishida, Surface roughness and plastic flow in rail wheel contact, Wear 253 (2002) 257 264. [22] L. Vu-Quoc, X. Zhang, L. Lesberg, A normal force-displacement model for the contacting spheres accounting for plastic deformation: force-driven formulation, ASME J. Appl. Mech. 67 (2000) 363 371. [23] L.-Y. Li, C.-Y. Wu, C. Thornton, A theoretical model for the contact of elastoplastic bodies, Proc. Instn. Mech. Engrs. 216C (2002) 421 431. [24] R.E. Jones, Models for contact loading and unloading of a rough surface, Int. J. Eng. Sci. 42 (2004) 1931 1947. [25] J. Jamari, D.J. Schipper, Deformation due to contact between a rough surface and a smooth ball, Wear 262 (2007) 138 145. [26] J. Jamari, Running-in of Rolling Contacts, Ph.D. Thesis, University of Twente, The Netherlands. [27] J.A. Greenwood, K.L. Johnson, E. Matsubara, A Surface roughness parameter in hertz contact, Wear 100 (1984) 47 57. [28] J.A. Greenwood, A unified theory of surface roughness, Proc. Roy. Soc. Lond. A393 (1984) 133 157. [29] J. Jamari, D.J. Schipper, An elastic plastic contact model of ellipsoid bodies, Tribol. Lett. 21 (2006) 262 271. [30] J. Jamari, D.J. Schipper, Experimental investigation of fully plastic contact of a sphere against a hard flat, ASME J. Tribol. 128 (2006) (2006) 230 235. [31] J.W. Sloetjes, D.J. Schipper, P.M. Lugt, J.H. Tripp, The determination of changes in surface topography using image processing techniques, in: Proceeding of the International Tribology Conference, Nagasaki, 2000, pp. 241 246.