i THREE-DIMENSIONAL MODELING OF ELASTO-PLASTIC SINUSOIDAL CONTACT UNDER TIME DEPENDENT DEFORMATION INCLUDING BOTH STRESS RELAXATION AND CREEP ANALYSIS by Amir Rostami A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the requirements for the Degree of Master of Science Auburn, Alabama August 3, 2013 Keywords: 3D Sinusoidal Contact, Elasto-plastic, Stress Relaxation, Creep, Finite Element Model Copyright 2013 by Amir Rostami Approved by Robert L. Jackson, Chair, Associate Professor of Mechanical Engineering Hareesh V. Tippur, McWane Professor of Mechanical Engineering Jeffery C. Suhling, Quina Distinguished Professor and Department Chair ii ABSTRACT Computational modeling of contact between rough surfaces has attracted a great deal of attention due to the developing technological needs of industry. Most of the early models of rough surface contacts assumed a cylindrical or spherical/ellipsoidal shape for the asperities on the surfaces. Due to high memory space and computational time requirements, researchers use simplified geometries to model the asperities or peaks on rough surfaces. Recent works tried to use a sinusoidal shape for asperities to improve the previous models. The sinusoidal geometry gives a better prediction of asperity interaction, especially for heavily loaded contacts. The effect of adjacent asperities is considered in sinusoidal contacts by using a symmetric boundary condition. Also, most of the multiscale contact models for rough surfaces use the Fourier series or Weierstrass profile to transform a rough surface to combination of sine and cosine functions. Therefore, it seems more reasonable to use a sinusoidal shape for asperities. In the current work, the transient effect of creep and stress relaxation in contact between sinusoidal surfaces is studied using FE simulations. A three-dimensional sinusoidal asperity is created, and is modeled in contact with a rigid flat surface. The material of the sinusoidal surface is modeled as elasto-plastic, bi-linear isotropic hardening solid. The Garofalo formula is used in the current work to model the transient behavior of creep and stress relaxation. Two load steps are used in commercial software ANSYS (version 13.0) to model the effect of creep and stress relaxation. The first load step is static deformation or the stress build-up stage that is used to pressurize the asperity by the rigid flat surface. The second load step is the transient process during which creep and stress relaxation occur. To verify the model, the results for the purely elastic and elasto-plastic cases (without the creep and stress relaxation effects) are compared to the previous works in the literature. Transient results under both constant displacement (stress relaxation) and constant force (creep) boundary conditions are presented and discussed. A parametric study is done to analyze the effect of the different material and geometrical properties and also the Garofalo constants on the iii transient results. In the end, empirical equations are developed for both contact area and contact pressure based on the FEM results. The empirical equations are dependent on the surface separation, aspect ratio, and the Garofalo formula constants. In the contact area and contact pressure results for stress relaxation, a critical interference or surface separation was found that the contact area and contact pressure showed different behaviors above and below this value. The aspect ratio rate, ( )ldtd /D , is introduced as a parameter that is independent from the height of the asperity during the stress relaxation process. This rate can be used in a multiscale contact model for rough surfaces to predict the real contact area as a function of time. iv ACKNOWLEDGEMENTS This thesis is based on the project that has been assigned and supported by the Siemens Corporate Technology under the title of ?The Development and Application of Multi-scale Friction Prediction Methods to Dynamic Actuator Systems?. I started working on this project from January 2012 along my M.Sc. studies in Auburn University under the supervision of Prof. Robert L. Jackson. I should thank Prof. Jackson who helped me a lot during this year and half, and I?ve benefited from his suggestions and ideas countlessly during the course of the project. He has been completely involved in every step of the project, and always supported me during the difficulties of the work. I should thank Mr. Andreas Goedecke and Mr. Randolf Mock in Siemens Corporate Technology. They have always supported and helped us with the data and info we needed during the project. I should also thank Prof. Hareesh V. Tippur and Prof. Jeffrey C. Suhling for serving on my thesis committee. In the same way, I should thank my lab mates in the Multi-scale Tribology Laboratory at Auburn University for their friendship and support and many shared laughs and hard work during this year and half. I should also mention the support of the most important people in my life, my parents, during my Master studies which of course they are far away from me, but, they have always been close to me in my heart. I will try my best to be a person that will make them proud. v TABLE OF CONTENTS ABSTRACT ?????????????????????????????.? ii ACKNOWLEDGEMENTS ???????????????????????... iv LIST OF FIGURES ???????????????????????????. vii LIST OF TABLES ???????????????????????????.. xi NOMENCLATURE ??????????????????????????... xii 1 INTRODUCTION ????????????????????????? 1 2 3 LITERATURE REVIEW ??????????????????????... 2.1 Introduction ?????????????????????????.. 2.2 Elastic Sinusoidal Contact of Single Asperity ?????????????. 2.3 Elasto-plastic Sinusoidal Contact of Single Asperity ??????????... 2.4 Average Surface Separation in Sinusoidal Contacts ??????????? 2.5 Creep and Stress Relaxation Effects in Single Asperity Contacts ?????? METHODOLOGY ????????????????????????.... 3.1 Introduction ?????????????????????????.. 3.2 Modeling and Simulation of the Static Deformation Step (Load Step 1) ??? 3.3 Verification of the Model Accuracy (Elastic Case) ???????????. 3.4 Verification of the Model Accuracy (Elasto-plastic Case) ????????... 3.5 Surface Separation Results for the Elastic Case ????????????.. 3.6 Surface Separation Results for the Elasto-plastic Case ?????????... 4 4 8 9 11 13 21 21 21 26 28 29 30 vi 3.7 Modeling and Simulation of the Stress Relaxation and Creep Effects (Load Step 2) ??????????????????????????????. 33 4 RESULTS ????????????????????????????... 4.1 Introduction ?????????????????????????.. 4.2 Stress Relaxation Results ????????????????????.... 4.3 Empirical Equations for the Stress Relaxation Case ??????????... 4.3.1 Empirical Fit for contact Area ??????????????????... 4.3.2 Empirical Fit for contact Pressure ?????????????????. 4.4 Base Height-dependency of the Stress Relaxation Results ????????.. 4.5 Creep Results ?????????????????????????. 4.6 Comparison between the Stress Relaxation and Creep Results ??????... 36 36 36 52 52 57 63 70 74 5 CONCLUSIONS ?????????????????????????? 78 BIBLIOGRAPHY ???????????????????????????... 80 APPENDICES ????????????????????????????? 85 A ?APDL? CODE USED FOR MODLING THE EFFECT OF STRESS RELAXATION ??????????????????????????.. 85 vii LIST OF FIGURES 1.1 Change in the stress and strain in a material due to (a) stress relaxation, and (b) creep ?.. 2 2.1 Contour plot of the three-dimensional sinusoidal surface geometry ????????.. 7 2.2 Rigid flat and the sinusoidal asperity (a) before contact and (b) during contact where surface separation, amplitude and the wavelength of the sinusoidal asperity are shown schematically ????????????????????????????? 12 2.3 A hemispherical asperity with radius, R , before and after loading, showing the contact radius, a , the displacement, ? , and the load, F ???????????????.. 15 2.4 Relaxation of force (solid line) and evolution of contact area (dashed line) with respect to normalized creep time in a (a) logarithmic and (b) conventional scale ???????... 17 3.1 Three-dimensional plot of the sinusoidal surface ???????????????.. 22 3.2 The steps to create the sinusoidal asperity: (a) creating the keypoints, (b) creating the lines, (c) creating the sinusoidal surface, and (d) adding volume to the sinusoidal surface.. 24 3.3 The element plot of the sinusoidal asperity and the rigid flat including the boundary conditions that are used for the geometry ??????????????????. 25 3.4 Comparison of the FEM elastic contact area results with JGH data and Jackson-Streator empirical equation ??????????????????????????? 27 3.5 Comparison of the FEM elasto-plastic contact area results with empirical equation provided by Krithivasan and Jackson ???????????????????... 28 3.6 Comparison of the FEM elastic results for average surface separation (shown by circles) with the JGH data (shown by crosses) and the new fit given by Eq. (38) (shown by solid line) ????????????????????????????????.. 29 3.7 The FEM elasto-plastic results for average surface separation for various yield strengths.. 30 3.8 The FEM elasto-plastic results for average surface separation for various aspect ratios ? 31 viii 3.9 The comparison of FEM elasto-plastic results for average surface separation (shown by small circles) with the new fit given by Eq. (39) (shown by solid line) ???????... 33 4.1 Plot of the three-dimensional sinusoidal asperity under constant displacement boundary condition ??????????????????????????????.. 37 4.2 The FEM results for the contact area and contact pressure versus time for reference parameters ?????????????????????????????? 38 4.3 The FEM results for contact pressure versus time for different surface separations (penetrations) ????????????????????????????.... 40 4.4 The FEM results for the contact area versus time for different surface separations (penetrations) over a short duration of time ?????????????????.. 41 4.5 The FEM results for the contact area versus time for different surface separations (penetrations) over a longer duration of time ????????????????? 42 4.6 The arrows show how the deformed material displaces to the void volume between the sinusoidal asperity and the rigid flat ????????????????????.. 43 4.7 The FEM results for the contact area and pressure versus time for different yield strength values ????????????????????????????.. 44 4.8 The FEM results for the contact area and pressure versus time for different Garofalo constant, 2 C , values ( GPaS y 1= )?????????????????...???? 45 4.9 The FEM results for the contact area and pressure versus time for different Garofalo constant, 2 C , values ( GPaS y 2= ) ?????????????????????. 46 4.10 The FEM results for the contact area and pressure versus time for different Garofalo constant, 2 C , values ( GPaS y 4= ) ?????????????????????. 47 4.11 The FEM results for the contact area and pressure versus time for different Garofalo constant, 1 C , values ??????????????????????????... 48 4.12 The FEM results for the contact area and pressure versus time for different elastic modulus, E , values ??????????????????????????... 49 4.13 The FEM results for the contact area and pressure versus time for different Poisson?s ratio, n , values ????????????????????????????.. 50 4.14 The FEM results for the contact area and pressure versus time for different aspect ratio, l/D ,values ?????????????????????????????... 51 ix 4.15 The FEM results and corresponding curve-fits for the normalized contact area for different surface separations, D/ o g ????????????????????.. 53 4.16 The FEM results and corresponding curve-fits for the normalized contact area for different aspect ratios, l/D ???????????????????????.. 54 4.17 The FEM results and corresponding curve-fits for the normalized contact area for different, 2 C , values ??????????????????????????.. 56 4.18 The FEM results and corresponding curve-fits for the normalized contact pressure for different aspect ratios, l/D ???????????????????????.. 58 4.19 The FEM results and corresponding curve-fits for the normalized contact pressure for different surface separations, D/ 0 g ????????????????????.. 59 4.20 The FEM results and corresponding curve-fits for the normalized contact pressure for different 2 C values ??????????????????????????... 61 4.21 The element plot of the sinusoidal asperity for a case with the (a) smaller height, and the case with the (b) doubled height ?????????????????????... 63 4.22 The von Mises stress plot for case with the (a) smaller height, and the case with the (b) doubled height ????????????????????????????.. 64 4.23 Contact area results for the case with the (a) smaller height, and the case with the (b) doubled height ????????????????????????????.. 65 4.24 Contact pressure results for case with the (a) smaller height, and the case with the (b) doubled height ????????????????????????????.. 65 4.25 The aspect ratio rate results for the case with the (a) smaller height, and the case with the (b) doubled height ??????????????????????????? 66 4.26 The normalized amplitude rate results for the different values of Garofalo constant, 2 C ... 67 4.27 The normalized amplitude rate results for the different values of aspect ratio, l/D ??... 68 4.28 The normalized amplitude rate results for the different values of surface separation, D/ o g ???????????????????????????????.... 69 4.29 Plot of the three-dimensional sinusoidal asperity under constant force boundary condition ??????????????????????????????.. 70 4.30 The FEM results for the contact area versus time for reference parameters, and contact pressure equal to GPap 8.0= ??????????????????????.. 71 x 4.31 The FEM results for the contact area versus time for different constant contact loads ?. 72 4.32 The FEM results for the contact area versus time for different Garofalo constant, 2 C values ???????????????????????????????... 73 4.33 The FEM results for the contact area versus time for different aspect ratio values, l/D ... 74 4.34 The von Mises stress (MPa) plot for the case under (a) constant force, and (b) constant displacement boundary conditions ????????????????????... 75 4.35 Contact area results for the case under (a) constant force (creep), and the case under (b) constant displacement (stress relaxation) boundary conditions ??????????. 76 4.36 Contact pressure results for the case under (a) constant force (creep), and the case under (b) constant displacement (stress relaxation) boundary conditions ????????? 77 xi LIST OF TABLES 2.1 Overview of parameter ranges used for the FE simulation [1] ??????????... 16 4.1 Reference properties ??????????????????????????. 38 4.2 Overview of the parameter ranges used for the FE simulations (constant displacement B.C.) ????????????????????????????????.. 39 4.3 Overview of the parameter ranges used for the FE simulations (constant force B.C.) ?... 72 xii NOMENCLATURE A area of contact B aspect ratio B? creep constant of the power law B ?? creep constant of the exponential law b creep constant of the exponential law C critical yield stress coefficient ( )41 ~ , -=iCC ii creep constants of the Garofalo, strain hardening, and modified time hardening model d material and geometry dependent exponent E elastic modulus E? reduced or effective elastic modulus f spatial frequency (reciprocal of wavelength) F contact force g average surface separation G normalized surface separation h height of the sinusoidal surface n creep constant of the Garofalo law H hardness * p average pressure for complete contact (elastic) xiii * ep p average pressure for complete contact (elasto-plastic) ave porp average pressure over entire asperity P normalized contact pressure y S yield strength t contact time t dimensionless contact time D amplitude of the sinusoidal surface l asperity wavelength n Poisson?s Ratio s stress e strain d interference between sinusoidal asperity and rigid surface ba, curve-fitting parameters for contact area dcba ???? ,,, curve-fitting parameters for contact pressure Subscripts 0 initial or at 0t = c critical value at onset of plastic deformation cr creep dependent parameter e elastic ep elasto-plastic JGH from model by Johnson et al. [2] 1 CHAPTER 1 INTRODUCTION The topic of contact between surfaces has been popular to researchers for many years. There exists many works on modeling the contact of surfaces starting around 1888 by Hertz [3] (originally developed to model optical contacts). All engineering surfaces are rough to some degree, therefore it is important to develop a model for the contact between rough surfaces. The main goal in modeling the contact between rough surfaces is to find a simple closed form solution for the real contact area. Most tribological effects such as friction, wear, adhesion, and electrical and thermal contact resistance are dependent on the real contact area between two contacting surfaces. Creep and stress relaxation are time dependent phenomenon which cause changes in the stress and strain in a material over time. For contacting surfaces, creep and stress relaxation cause changes in the contact area and contact pressure between surfaces as time passes. Stress relaxation refers to the stress relief of a material under constant strain condition (Fig. 1.1a), and creep describes how strain in a material changes under constant stress condition (Fig. 1.1b). Any material can experience creep if certain conditions are met. It could be metals at high temperatures, polymers at room temperatures, and any material under the effect of nuclear radiation. Although there is no recovered creep strain or reversible behavior under normal operating conditions, elastic deformations are still recovered. The goal of this thesis is to model the effect of creep and stress relaxation in contact between sinusoidal surfaces. 2 The change in contact area due to creep is considered as the reason why static friction changes over time, and why the dynamic friction is dependent on velocity [4-6]. The previous works on the creep effect in contact between surfaces used a cylindrical [4, 7-10] or spherical [1, 11, 12] geometry for the asperities. In this work, a sinusoidal geometry is developed and analyzed which is believed to be a more realistic geometry. The interaction between adjacent asperities, which is ignored in most previous works, is considered by assuming a sinusoidal asperity. Two kinds of boundary conditions are used in the current work to model the time dependent deformations: (i) constant displacement boundary condition (stress relaxation) and (ii) constant force boundary condition (creep). The hyperbolic sine function also known as the Garofalo formula is used to model the effect of the creep and stress relaxation [1]. The results for the contact area and contact pressure as they change over time are presented and discussed. Empirical equations are developed by fitting to the FEM results. These empirical equations can be put in a multiscale model to obtain the real contact area for a specific surface in contact as a function of time. This real contact area can Fig. 1.1 Change in the stress and strain in a material due to (a) stress relaxation, and (b) creep (a) (b) 3 be used to obtain a prediction for the time dependent static friction or the velocity dependent dynamic friction between surfaces. 4 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction Most of the previous models on the contact between rough surfaces assume a spherical shape [13-20] or ellipsoidal shape [21-23] for the geometry of the asperities on the surfaces. Archard [14] used a stacked model of spherical asperities and showed that although the relation between the contact area and load for a single asperity is nonlinear, by using a multi-scale model this relation becomes linear. More recent models consider a sinusoidal shape for the asperities because it seems to model the geometry of real surfaces better, especially for heavily loaded contacts [24-28]. It is shown for two-dimensional sinusoidal surfaces [26] and three-dimensional sinusoidal surfaces [27] that the average contact pressure increases past the conventional hardness, H , limit of y S3? obtained by assuming spherical geometries [29]. Several works have shown experimentally measured contact pressures much higher than three times the yield strength ( y S3? ) [30, 31]. Furthermore, the interaction between adjacent asperities is addressed by using a sinusoidal geometry with accurate boundary conditions which is overlooked in works based on spherical asperities. Also, most of the models consider the multi-scale nature of surface roughness by using a Fourier series or Weierstrass profile [25, 32-34], and since these series use superimposed harmonic waves, it is logical to use a sinusoidal shape instead of a spherical shape in modeling the asperities. The first models on the contact between rough surfaces using sinusoidal shaped asperities were mostly on the purely elastic contact. The elastic contact of two-dimensional sinusoidal surfaces was first solved by Westergaard [24]. Johnson et al. (JGH) [2] presented two asymptotic solutions 5 for the elastic contact of three-dimensional sinusoidal surfaces, but no analytical solution is available for the entire load range. Jackson and Streator [34] developed an empirical equation based on the JGH data [2] for the whole range of loading from early contact to complete contact. When two surfaces come into contact, it is primarily the peaks or asperities that carry the load and make the contact. Therefore, the small contact area caused by the asperities carries very high pressures, and plastic deformations are practically inevitable in most cases of contact between metallic rough surfaces. Gao et al. [26] considered plastic deformations in their sinusoidal contact model. They modeled a two-dimensional elastic-perfectly plastic sinusoidal contact using the finite element method (FEM). Krithivasan and Jackson [27] and Jackson et al. [28] considered both elastic and elasto-plastic contacts in three- dimensions in their work, and presented empirical equations for the contact area as a function of contact pressure for the whole range of contact. Their equations are used to verify the current model in the next sections. One of the important stages in contact between surfaces is when ?complete contact? happens. The definition of complete contact between rough surfaces depends on the specific case and geometry of the surfaces in contact. In contact between sinusoidal surfaces, it is defined as the state in which sinusoidal surfaces have completely flattened out and there is no gap in between the surfaces. The average pressure (including both contacting and non-contacting regions) that causes complete contact is an important parameter used to interpret the results, and is denoted by * p . Recent works extended the asperity contact models by including both normal and tangential loading [35], and loading and unloading [36-38]. However, the transient behavior of asperities under loading such as creep and stress relaxation has been neglected in these works. There have been a few works on the computational modeling of the creep [11, 12] and stress relaxation [1] effect in the contact between surfaces but their concentration has been mostly restricted to a single spherical asperity contact. Most of the earlier models on the effect of the creep in contact between asperities have assumed a rigid spherical punch indenting an elasto-plastic flat surface [39-43]. 6 Different effects in contact between rough surfaces such as the dwell-time dependent rise in static friction [5, 44], the velocity dependent dynamic friction [6-9] or friction lag and hysteresis [45] can be explained by the creep theory. Many experimental studies concerning the increase of friction with dwell time are published i.e. [46-50]. Malamut et al. [44] studied the effect of dwell time on the static friction coefficient due to creep for spherical contacts. The creep behavior depends on the temperature and stress level to which the material is exposed, and depends noticeably on the time duration of application of these conditions. The change in the real contact area between two solids due to time of stationary contact is the main motivation for modeling the creep and stress relaxation effects. The current analysis uses the same geometry used in Johnson et al. [2] and Krithivasan and Jackson [27] in order to compare the results in this paper to their works. The sinusoidal geometry is described by: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -= ? ?y ? ?x ?h 2 cos 2 cos1 (2.1) where h is the height of the sinusoidal surface, D is the amplitude of the sinusoidal surface, and l is the wavelength of the sinusoidal surface. The contour plot of the sinusoidal surface is shown in Fig. 2.1. 7 Quarter Used in FE Model Fig. 2.1 Contour plot of the three-dimensional sinusoidal surface geometry 8 2.2 Elastic Sinusoidal Contact of Single Asperity As mentioned before, Johnson et al. [2] developed asymptotic solutions for contact area of a perfectly elastic contact of three-dimensional sinusoidal shaped surfaces. In their work, p is defined as the average pressure on the surface (considering both contacting and non-contacting regions), and * p is defined as the average pressure that when applied to the surface causes complete contact. * p is given as: ?fE?2p * ?= (2.2) where D is the amplitude of the sinusoidal surface, f is the frequency or reciprocal of the asperity wavelength, l , and E ? is the equivalent elastic modulus which is given by: 2 2 2 1 2 1 111 EEE nn - + - = ? (2.3) 11 ,nE and 22 ,nE are the elastic modulus and Poisson?s ratio of the contacting surfaces. The flat surface is considered to be rigid in this work: ( )?? 2 E , and Eq. (2.3) reduces to: 2 1 1 n- =? E E (2.4) The Johnson et al. [2] solutions are applicable when *<< pp i.e. at the early stages of contact, and when p approaches * p *? pp i.e. near the complete contact. The equations are given as shown: *<< pp : 3/2 *2 1 8 3 )( ? ? ? ? ? ? ? ? = p p f A JGH p p (2.5) 9 * pp ? : ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? --= *2 2 1 2 3 1 1 )( p p f A JGH p (2.6) Jackson and Streator [34] developed an empirical equation based on the experimental and numerical data provided by Johnson et al. [2], linking Eqs. (2.5) and (2.6): for 8.0 * < p p : 04.1 * 2 51.1 * 1 )(1)( ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -= p p A p p AA JGHJGH (2.7) for 8.0 * ? p p : 2 )( JGH AA = (2.8) 2.3 Elasto-Plastic Sinusoidal Contact of Single Asperity Three-dimensional elasto-plastic contact between sinusoidal surfaces has been investigated by Krithivasan and Jackson [27] and Jackson et al. [28]. They showed that for the elasto-plastic contact, complete contact occurs much earlier (at the lower pressures) than elastic contact. Jackson et al. [28] presented an empirical equation for calculating the average pressure, * ep p , that causes complete contact for the elasto-plastic case. The equation is given below: ( ) 5/3 * * 7/4 11 ? ? ? ? ? ? ? ? +DD = c ep p p (2.9) where c D is the analytically derived critical interference, and is given by: f e E S y c n p 3/2 3 2 ? =D (2.10) 10 Jackson and Krithivasan?s [27] empirical equation for the contact area versus contact pressure for the elasto-plastic case is given by: ( ) ( ) 04.1 * 2 51.1 * 1 ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? -= ep JGH ep p p p A p p AA (2.11) where p A is given by d1 d 2 y d1 1 c p ? 4CS p3. 2 A 2A + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = (2.12) In the above equation, c A is the critical contact area for the sinusoidal contact based on spherical contacts [19], and is given by: 2 2 8 2 ? ? ? ? ? ? ? ? ? = E?f CS ? A y c (2.13) where the constant C is related to the Poisson?s ratio by: ( )?..C 7360exp2951= (2.14) The value of the constant d in Eq. (2.12) is given by: 2 1 d y S E dd ? ? ? ? ? ? ? ? D? = l (2.15) where, 8.3 1 =d and 11.0 2 =d are constants which are obtained empirically by curve-fitting to FEM results. 11 In the current work, Eq. (2.11) is used to confirm the first load step or stress build-up stage of the finite element model (before creep and stress relaxation initiate at 0=t ). 2.4 Average Surface Separation in Sinusoidal Contacts In many applications that require tight tolerances such as sealing and lubricated bearings, it is important to be able to predict the surface separation between contacting surfaces. The average separation also determines the volume of the space trapped between two surfaces which is a very important parameter in lubricated surfaces. Little work has been done to characterize the surface separation or gap between sinusoidal surfaces as a function of load. In Fig. 2.2, surface separation between a sinusoidal asperity and a rigid flat surface in two-dimensions is shown. The average surface separation before the contact is equal to the amplitude of the sinusoidal asperity, D=g , (Fig. 2.2a), and during the contact the average surface separation is less than the amplitude, D