The Pennsylvania State University. The Graduate School. College of Engineering A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS

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1 The Pennsylvania State University The Graduate School College of Engineering A DETERMINISTIC-STATISTICAL MODEL FOR TRIBO-CONTACTS IN BOUNDARY LUBRICATION WITH LUBRICANT/SURFACE PHYSICOCHEMISTRY A Thesis in Mechanical Engineering by Huan Zhang 004 Huan Zhang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 005

2 The thesis of Huan Zhang was reviewed and approved* by the following: Liming Chang Professor of Mechanical Engineering Thesis Advisor Chair of Committee Marc Carpino Professor of Mechanical Engineering Seong H. Kim Assistant Professor of Chemical Engineering Richard C. Benson Professor of Mechanical Engineering Head of the Department of Mechanical and Nuclear Engineering *Signatures are on file in the Graduate School.

3 iii ABSTRACT The boundary-lubricated surface contact is truly an interdisciplinary process involving deformation, heat transfer, physicochemical interaction, and random-process probability. The objective of this thesis is to develop a surface contact model as a theoretical platform upon which to carry out the boundary lubrication research with a balanced consideration of all the four key aspects of the contact process. The modeling consists of three successive steps (1) elastoplastic finite element analysis of frictional asperity contacts, () modeling of contact systems with friction, and (3) modeling of a boundary lubrication process. Finite element analysis of frictional asperity contacts A finite element model is developed and systematic numerical analyses carried out to study the effects of friction on the deformation behavior of individual asperity contacts. The study reveals some insights into the modes of asperity deformation and asperity contact variables as functions of friction in the contact. The results provide guidance to analytical modeling of frictional asperity contacts and lay a foundation for subsequent work on system contact modeling. Modeling of contact systems with friction Analytical equations are developed relating asperity-contact variables to friction using contact-mechanics theories in conjunction with the finite element results. A system-level model is then derived from the statistical integration of the asperity-level equations. The model is a significant advancement of the Greenwood-Williamson types of system models by incorporating

4 iv contact friction. It also serves as the platform in the final step of model development for the boundary lubrication problem. Modeling of a boundary lubrication process On the basis of the above mechanical modeling, an asperity-based model is developed for the boundary-lubricated contact by incorporating other key aspects involved in the process. Four variables are used to describe an asperity contact under boundary lubrication conditions, including micro-contact area, friction force, load carrying capacity and flash temperature. In addition, three probability variables are used to define the interfacial state of an asperity junction that may be covered by various types of boundary films. Governing equations for the seven key asperity-level variables are derived based on first-principle considerations of asperity deformation, frictional heating, and formation/removal of boundary lubricating films. These coupled asperity-level equations, some of which are nonlinear, are solved iteratively and the solution is then statistically integrated to formulate the contact model for boundary lubrication systems. The results obtained from the model suggest that it may provide a framework for future investigation of the boundary lubrication process by integrating research advances in contact mechanics, tribochemistry, and other related fields.

5 v TABLE OF CONTENTS List of Figures...vii List of Tables...ix Nomenclature...x Acknowledgements...xii Chapter 1 Introduction Boundary Lubrication and Boundary-Lubricated Contact Important Aspects of Boundary-Lubricated Contact: Literature Review Mechanisms and Efficiency of Boundary Lubrication Contact Modeling: Unlubricated Surfaces Contact Modeling: Boundary-Lubricated Surfaces Flash Temperature Summary Research Objective, Approach and Outline...18 Chapter Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts....1 Introduction.... The Model Problem Results and Analysis Mode of Asperity Deformation Shape of the Plastic Zone Contact Size, Pressure and Load Capacity Summary...37 Chapter 3 A Mathematical Model of the Contact of Rough Surfaces with Friction Introduction Modeling Model Structure Asperity Contact Pressure Asperity Area of Contact Critical Normal Approaches System Variables Result Analysis...68

6 vi 3.4 Summary...76 Chapter 4 A Deterministic-Statistical Model of Boundary Lubrication Introduction Modeling Modeling Strategy Asperity Contact and Probability Variables System Variables Result Analysis Summary Chapter 5 Summary and Future Perspective The Deterministic-Statistical Model Perspective on Future Development...13 Bibliography...16

7 vii List of Figures Figure 1.1 Boundary lubricated contacts of two rough surfaces Figure.1 Half-cylinder contact model 39 Figure. Finite element mesh of the model problem 39 Figure.3 Figure.4 Effects of friction on the critical normal approaches: (a) linear scale; (b) logarithmic scale Plastic zones of the frictionless contact: (a) elastic-plastic transition; (b) onset of full plasticity Figure.5 Plastic zones of the contact with µ = 0.: (a) elastic-plastic transition; (b) onset of full plasticity Figure.6 Plastic zones of the contact with µ = 0.5: (a) elastic-plastic transition; (b) onset of full plasticity Figure.7 Plastic zones of the contact with µ = 1.0: (a) elastic-plastic transition; (b) onset of full plasticity Figure.8 Contact variables with δ = δ Figure.9 Shift and growth of the contact junction with δ = δ Figure.10 Contact variables with δ = 3δ Figure 3.1 Schematic of the equivalent contact system 79 Figure 3. Critical normal approaches and modes of asperity deformation 79 Figure 3.3 Figure 3.4 Figure 3.5 Slip-line field solution of a rigid-perfectly-plastic wedge under combined action of normal and tangential loading: (a) initial stage ( τ m < τ o ); (b) final stage ( τ m τ o ) Dimensionless first critical normal approach: D finite element results against 3D theoretical analysis Dimensionless second critical normal approach: finite element results and curve-fitting Figure 3.6 Surface mean separation as a function of load and friction coefficient 8

8 viii Figure 3.7 Asperity height distribution and mode of deformation of contacting asperities 83 Figure 3.8 Friction-induced load redistribution among asperities 83 Figure 3.9 Contribution of the friction-induced junction growth to the real area of contact 84 Figure 4.1 An individual boundary-lubricated asperity contact 115 Figure 4. Flowchart for the determination of the solution of an asperity contact 116 Figure 4.3 System-level friction coefficient as a function of load 117 Figure 4.4 Figure 4.5 Figure 4.6 Asperity shear stresses and asperity height: (a) ψ = 0.66; (b) ψ = 1.86; (c) asperity height distribution System-level contact and lubrication variables as functions of load: (a) degree of boundary protection; (b) surface separation; (c) real area of contact State of boundary lubrication in the operating parameter space: (a) system-level friction coefficient; (b) system boundary-lubrication protection

9 ix List of Tables Table 3.1 First critical normal approach as a function of the friction coefficient 85 Table 3. Percentage of elastically-deformed asperities in frictionless contact 85

10 x Nomenclature A l = area of asperity contact A n = nominal contact area A t = real area of contact E 1, E = elastic modulus E 1 1 ν 1 1 ν = equivalent elastic modulus, + E1 E F t = total friction force H = indentation hardness H a = lubricant/surface adsorption heat H r = bond destruction or chemical activation energy of the reacted film K c = substrate thermal conduct N A 3 = Avogadro constant ( mol -1 ) P m = average pressure of an asperity contact P mf = asperity contact pressure at the onset of plastic flow P my = asperity contact pressure at the inception of yielding R = asperity radius of curvature R c = molar gas constant ( J ( mol K) ) S a = probability of an asperity contact being covered by an adsorbed film S a = survivability of the adsorbed layer in an asperity contact S at = survivability of the adsorbed layer at the system level S n = probability of an asperity contact with no boundary protection S nt = probability of contact with no boundary protection at the system level S r = probability of an asperity contact being protected by a reacted film S r = survivability of the reacted film in an asperity contact S rt = survivability of the reacted film at the system level T b = bulk temperature T l = contact temperature of an the asperity junction T 1 = asperity flash temperature V = sliding velocity W t = total contact load a = radius of an asperity contact b π mk bt = adsorption coefficient k 1000 bt N Aϑ c = substrate specific heat

11 xi d = distance from the mean plane of asperity heights to the rigid flat f ( z) = distribution density function of the asperity height h = separation based on surface heights k A = friction-induced junction growth factor = δ µ k = upper bound of the junction growth factor at ( ) Al δ 3 k b = Boltzman constant ( J K ) m = lubricant/additive molecular weight t c = duration of an asperity contact t f = time to the break of the substrate/reacted film bonding z = asperity height z = distance between the mean of asperity heights and that of surface s heights α = constant in Tabor s equation β = ση R γ = activation or fluctuation volume of the reacted film δ = normal approach of asperity contact δ 1 = first critical normal approach δ = second critical normal approach η = area density of asperities κ = substrate thermal diffusivity µ = local friction coefficient l µ = system friction coefficient t υ,υ 1 = Poisson s ratio σ = standard deviation of surface heights σ a = standard deviation of asperity heights σ e = effective stress τ a = shear strength of the adsorbed layer τ m = average shear stress of an asperity contact τ n = shear strength of the substrate material τ r = shear strength of the reacted film ψ = plasticity index ϑ 34 = Planck constant ( J s )

12 xii Acknowledgements The completion of the thesis brings me to the end of my student life. I would like to take this opportunity to express my appreciation to all those who helped and supported me during my journey of learning. Without their guidance, help and patience, I would not be able to go this far. First and foremost, I am very grateful to my thesis advisor, Prof. Liming Chang, for introducing me to the exciting and challenging project, for his continuous guidance and encouragement from the day I met him more than five years ago. Since then he has inspired me in my research with his interest, dedication and enthusiasm for this study. At each stage of the research, I have benefited tremendously from his academic expertise, professional rigor, and solid grasp of the big picture. I especially appreciate the time and effort he put into reading and commenting many drafts of the thesis as it was taking shape. I want to also thank him for his knowledgeable advice and constructive criticism on every aspect of academic life which broadened my perspective, improved my research skills, and prepared me for future challenges. I would like to thank other members of my thesis committee, Professor Richard Benson, Professor Marc Carpino, and Dr. Seong Kim, for providing invaluable suggestions during the course of my research and generously sharing with me their deep understanding of this topic. I want to express my sincere thanks to Dr. Martin Webster and Dr. Andrew Jackson at ExxonMobil Technology Company for their consistent support and insightful comments.

13 xiii My special appreciation goes to Prof. Yongwu Zhao at Southern Yangtze University for his encouragement, advice, and fruitful discussions during his stay here at the Penn State University and when he is back in China. Many thanks are also due to my fellow students and research associates and all other friends at State College who have offered immediate and continuous support throughout the past five years. I wish to acknowledge ExxonMobil Technology Company for the financial support of the research project. I also would like to thank Prof. Stefan Thynell, Professorin-Charge of the Mechanical and Nuclear Engineering Graduate Programs, for his faith in my abilities and selecting me as a Graduate Teaching Fellow during the last semester of my Ph.D. This program has taught me many things which I cannot learn from any other experience. I am indebted to my parents, brother and sister for their enduring love and support, to my daughter for not spending as much time as I should, and to my dear wife, Jia, who have been with me through thick and thin and everything in between. Finally, I dedicate this thesis to my father, Shi-Chang Zhang, who lost his ability to speak two years ago.

14 Chapter 1 Introduction 1.1 Boundary Lubrication and Boundary-Lubricated Contact Boundary lubrication provides the basic protection to the bearing surfaces of machine components which operate at high load, low speed, or high temperature, such as o Gear/tooth, cam/tappet, and piston-ring/liner contacts; o Rolling element bearing at the pure sliding sites; o Journal bearings during the periods of start-up and shutdown. The effectiveness of boundary lubrication is critical to the service life of these components. In addition, boundary lubrication also plays an important role in the following devices or operations: o MEMS [1] and head/disk interface []; o CMP and the metal cutting and formation operations [3]; o Natural and artificial joints, such as those in the hip and in the knee, after periods of inactivity such as sleeping [4]. Therefore, knowledge of the surface contact behavior in boundary lubrication is essential to improve the performance of the above systems and procedures addressing the efficiency, safety, environment and other concerns. For example, such knowledge is invaluable in developing the strategies for controlling tribo-failure and minimizing wear,

15 and in designing the environmentally benign lubricants and additives. The objective of the current research is to enhance the understanding in the area by developing a theoretical model for the boundary-lubricated sliding contact of two rough surfaces. Figure 1.1 Boundary lubricated contacts of two rough surfaces The nominally flat bearing surfaces usually deviate from their prescribed geometry with microscopic irregularities. Under boundary lubrication conditions, two rubbing surfaces make frequent and random micro-contacts at their high spots or the asperities (as shown in Fig. 1.1). The load applied to the system is then mainly carried by the discrete asperity contacts and the total friction force is also the integration of local tangential resistance. During each asperity contact, a series of micro-scale processes of different nature proceed simultaneously and interact with each other in a number of ways. The direct mechanical response of two contacting asperities is their elastic or inelastic deformation which results in the asperity load support. This response is accompanied by a group of physical and chemical reactions among the substrate, additives, lubricants, and environment leading to the formation of low shear-modulus films in the contact junction. These films protect asperities from direct contact and effective lubrication is thus achieved. The protective boundary films may be ruptured and then the asperity contact takes place directly between the opposite metallic substrates. The local friction resistance may thus come from the shearing within the boundary films and/or that occurring at the

16 3 metallic surfaces. The shear stress, along with the sliding velocity, generates frictional heating in micro contact regions. As a result, high local temperatures of short duration, or so-called flash temperatures, may be aroused. The frictional heating process may facilitate the formation of the boundary lubricating films or deteriorate them by dissociation, desorption or oxidation. The state of these films or their integrity also depends on the levels of contact pressure and shear stress. This state, in turn, largely determines the shear stress and thus affects other micro-contact variables. In summary, the system-level tribological behavior under boundary lubrication conditions is collectively governed by multiple interactive asperity-level processes. On the other hand, the micro-contact processes may also be affected by the evolution of system features. For example, in the course of an asperity-to-asperity contact, the asperity temperature is composed of two components: the flash temperature and the bulk temperature. The latter is largely system specific, and governed by the overall heat generation and transfer. In addition, the geometrical characteristics of the rubbing surfaces may experience continuous progression resulting in dynamically changing conditions at each asperity contact. The above discussion indicates that the boundary lubrication processes exhibits diversity in their natures and scales. The corresponding contact modeling is therefore a truly interdisciplinary subject. The model should be developed based on the knowledge of the mechanisms of boundary films, the contact of rough surfaces, and the flash temperatures of asperity contacts. Significant advances have been made in these areas and the current understanding of each is summarized below from the modeling viewpoint to establish the theoretical framework and methodological focus for this thesis research.

17 4 1. Important Aspects of Boundary-Lubricated Contact: Literature Review 1..1 Mechanisms and Efficiency of Boundary Lubrication In boundary lubrication, two different types of protective films may be formed in an asperity junction to prevent the surface damage during sliding. A layer of organic compounds with polar end groups may be adsorbed on the surface. Meanwhile, an inorganic film may be produced by the chemical reaction between the substrate and the additives or lubricants. These boundary films usually reduce friction and increase the resistance of the system to surface failure such as seizure. For example, the formation of Fe Cl 3 films from chlorinate additive in PAO may raise the seizure load of a steel-steel system by a factor of 3-8 [5]. The system performance is thus largely controlled by the properties of the two types of boundary lubricating films, including their composition, structure, effectiveness, and shearing behavior. The generally accepted ideas about these important issues and the recent developments are briefly reviewed below for the adsorbed layer and the reacted film in sequence. A conceptual model has been proposed to explain the mechanism of boundary lubrication by the adsorption [6]. According to this model, the polar ends of organic lubricant or additive molecules are attached to the sliding surfaces with their hydrocarbon chains projected vertically upward. The molecular layers adsorbed on the opposite surfaces are only weakly interacted. The sliding of the two surfaces is then accomplished between the adsorbed layers resulting in a low interfacial friction. Therefore, the measured friction coefficient has often been used to characterize the relative lubrication

18 5 effectiveness of the adsorbed layers for various combinations of base lubricants, polar additives, and surfaces. It has been found that the effectiveness depends on the chain length of the hydrocarbon molecules [7-9], the molecular structure [10, 11], and the type of polar groups [1, 13]. The adsorbed layer is generally effective up to a critical interfacial temperature [14-16]. It is because high temperature corresponds to strong thermal desorption leading to a reduced fraction of surface that is covered by the adsorbed molecules. The fractional surfactant surface coverage θ or defect 1 θ has often been related to the interfacial temperature and the free energy of adsorption of the additive or lubricant to the surface. The simplest relationship for this purpose is the Langmuir adsorption isotherm [17], which assumes that the surface is energetically homogeneous and there is very small or zero net lateral interaction between adsorbate molecules. The applicability of the Langmuir isotherm in boundary lubrication studies has been verified experimentally for different additives and lubricants [14, 18, and 19]. In comparison, the Temkin isotherm may be more suitable in the case of heterogeneous surfaces and strong lateral interaction within the adsorbed layer [11, 13]. Another model is proposed to determine the fractional coverage based on the dwell-time of an adsorbed molecule at a particular surface site [0]. In addition to the interfacial temperature and adsorption energy, this model also accounts for the effect of sliding velocity. Assuming that the adsorbed layer is the only boundary lubricating film, direct metallic contact may occur as a result of the partial failure of this layer. The interfacial friction may then arise from both the shearing of the layer and the metallic contact. The

19 6 overall friction force can thus be related to the fractional surfactant surface coverage and the relation is given by [1], F A r [ θτ + ( θ ) τ ] = 1 (1.1) b m where A r is the real area of contact, τ b the shear strength of the boundary lubricating film, and τ m that of the substrate material. By assuming that the surfaces are fully covered by the adsorbate, the shear strength τ b may be determined on the basis of the measured frictional force and the knowledge of the real area of contact A r. However, this is difficult in real engineering situations due to the uncertainty involved in the estimation of A r and the possible desorption during the contact. In order to overcome this difficulty, a feasible approach is to deposit monolayers or multilayers of organic films on very smooth surfaces with simple contact geometry such as two crossed cylinders and a sphere against a plane. For these types of contact configuration, the area of contact could be calculated using the well-known Hertzian solution and the calculation may be verified experimentally, for example by multiple-beam interferometry. This approach was first used to study the shearing behavior of calcium stearate monolayers deposited on atomically smooth mica sheets [] and then extended to a variety of other organic films [3-6]. The results of these studies show that the film shear strength is dependent on the contact pressure and may be expressed in the following form [7]: n j τ b = τ 0 + µ j P (1.) j where τ 0 is the shear strength at zero pressure. In many cases of interest, τ 0 is small compared to other terms. The coefficients and exponents of the series in this expression

20 7 characterize the mechanical or rheological properties of the boundary lubricating films. In addition to the experimental studies, a theoretical model has been proposed relating the friction of two adsorbed layers on the opposite surfaces to the energy barrier between two adjacent equilibrium positions [8]. Without considering the dislocations and energy conservation, the predictions from this theory are much higher than the experimental results. Compared to the adsorbed layers, the reacted films in boundary lubrication systems are much more complex in terms of the formation, composition, structure, effectiveness, and mechanical properties. Typically, the reacted films are generated from the chemical reaction between the metal surface and the additive with one active element such as sulfur, phosphorus, chlorine, and boron [9, 30]. The corresponding formation process starts with the chemisorption of the additive on the metal surface. This is followed by the decomposition of the additive molecules leaving the active element chemically bonded to the surface. A thin film of metal salts is then formed and it may be mixed with oxides in the presence of moisture or in air atmosphere. Further growth of the film involves the diffusion of the active elements and metallic ions. Such a formation process is similar to that of the oxide layer on the surface. The growth of the film thickness may follow a linear law initially and a parabolic law afterwards, and may thus be described by the following equation [31]: 1 n Qn ρ f o h = Arn exp t n = 1 or (1.3) RT

21 8 where A n is the Arrhenius constant and Q n the activation energy of reaction. These two parameters are closely related to the type of metallic salt, which strongly depends on the availability of the active elements and the temperature at the interface. On the other hand, the reacted films may also be formed by a multifunctional additive containing two or more active elements. The most widely used multifunctional additives are the alkyl and aryl groups of zinc dithiophosphate (ZDTP), which usually form a boundary lubricating film of a multilayer structure. Starting from the substrate, this type of film composes of an inorganic layer of sulfates and oxides, a layer of short-chain polyphosphates and/or long-chain zinc polyphosphates, and a layer of organophosphates such as alkylphosphate. The transition between the two adjacent layers is gradual. The portion of each layer within the film depends not only on the properties of the lubricant, additive and substrate material, but also the severity of the sliding contact. More detailed information can be found in [30] and [3-34] on the structure and composition of the ZDTP films and the mechanism of action at the molecular level. In addition, the reacted films may include a multilayer of carboxylate formed from carboxylic acid additives [35, 36] and a thick layer of high-molecular weight organometallic compounds by the polymerization of additive-free oil minerals [37, 38]. The diversity of the reacted films formed in the boundary lubricated contact suggests that they may work by different mechanisms depending on their form, structure and properties. A very thin film of metal salts or oxides may act as a sacrificial layer of low shear strength. It is easily removed by the shear or cavitational forces along with the friction heating but is able to be reformed immediately to sustain continuous sliding. A prime example is the boundary film formed from the extreme pressure additives [39]. The

22 9 high-molecular polymeric film generated from base oil molecules may also work on the basis of repeated removal and repair [40]. In contrast, the metal salt-films derived from the antiwear additives are relatively thicker and usually much more tenacious. They are not easily removable during the sliding and the wear is thus controlled. As for the multilayer film resulting from ZDTP, each layer has different properties and functions [41]. The metal salts such as FeS has sufficiently high shear strength and serves as an adhesive layer as well as a seizure-resistant coating. The intermediate phosphate layer has high viscosity and its hardness is comparable to the mean contact pressure. It can flow plastically and may thus act as a protective layer against wear by eliminating the abrasive contribution of oxides. The outermost organic layer is mobile and has varying viscosity similar to the base oil ensuring that the shear plane is located within the boundary lubricating film. This layer also serves as a reservoir for the regeneration of polyphosphates. The reacted films described above may fail to provide effective protection to the surfaces when the films are removed during the contact. The failure process is strongly affected by the level of interfacial shear stress, frictional heating [9, 4], and contact pressure and plastic deformation [43, 44]. A number of models have been proposed to explain the film-failure in terms of the friction-induced temperature rise and/or the mechanical stresses. Accordingly, a group of criteria has been defined. The failure has often been attributed to the imbalance between the formation and the removal of the reacted films. Based on this hypothesis, a critical temperature condition has then been determined. In one of such studies [45], both the formation and removal rates have been measured and modeled as a function of interfacial temperature using the Arrhenius-type

23 10 expression in the form of Eq. (1.3). The failure occurs above a critical temperature when the removal rate is greater than the formation rate. For the system running at low speeds, the effects of frictional heating or interfacial temperature are negligible. The reacted films fail when the maximum interfacial stress exceeds the film or substrate shear strength and a stress criterion has thus been defined [46, 47]. The film failure has also been viewed as the result of the destruction of the chemical bonds between the active elements of additive molecules and the metal surface [48, 49]. From the energy transfer point of view, these mechanically stressed bonds can be broken by the combined action of the thermal energy from frictional heating and the distortion energy due to shearing. According to the thermal fluctuation theory of fracture [50], the typical lifetime of the bonds represents their resistance to the destruction and may thus be used to characterize the film-failure. The three types of models described above are deterministic, but the information about many of their input parameters is incomplete and the failure process itself also involves a certain degree of intrinsic uncertainty. Thus, a probabilistic approach is more appropriate to assess the likelihood of failure of the reacted films. This likelihood may be expressed as a probability similar to the fractional defect of the adsorbed layer. The probability may also be used to model the interfacial friction in combination with the knowledge of the film shearing properties. In addition to the formation, structure and effectiveness of the reacted films, their shearing behavior and other mechanical properties are also the key to understanding the mechanism of boundary lubrication. These aspects have thus been studied by many researchers for the reacted films formed during tribological testing using conventional tribometers and innovative scanning probe techniques. With a ball-on-flat configuration,

24 11 Tonck et al [51] measured the tangential stiffness by a microslip method for four types of tribo-films formed by pure paraffin, ZDTP, calcium sulphonate, and a friction modifier, respectively. The elastic shear moduli of these films were also determined and were found similar to those of high molecular weight polymers such as polystyrene. In addition, the results showed that the values of shear modulus would increase with the load except in the case of the friction modifier. More recently, nanoindentation has been widely used to measure the mechanical properties of the reacted films generated from a variety of lubricant additives [5-55]. It was observed that the film hardness and elastic modulus would increase with depth up to a few nanometers beneath the surface. Correspondingly, the resistive forces within the films might increase during the loading stage of the indentation to accommodate the increasing applied pressure. On the other hand, the lateral force microscopy has been used in combination with the atomic force microscopy to examine the frictional properties of the tribo-films formed in reciprocating Amsler tests [56, 57]. A linear relationship was revealed between the load and the friction force measured for micro regions of the tribo-films. This may be explained by the distribution of the hardness and modulus in depth observed in the nanoindentation tests. Therefore, the shearing behavior of the reacted films may also be described by Eq. (1.) in its linear form. Furthermore, the friction coefficient of the micro regions was found in good agreement with the macro results. The overall friction coefficient is thus indeed determined by the shearing of the reacted films covering the asperities. 1.. Contact Modeling: Unlubricated Surfaces For two nominally flat surfaces without lubrication, their contact takes place at distributed asperity junctions. The contact models predict the mechanical responses of

25 1 surfaces to the applied loading. These responses, including the size and spatial distribution of asperity contact spots, and the surface and subsurface stress fields around them, are dependent on the topography of surfaces and their material properties. Two major approaches have been used to model the contact of rough surfaces: stochastic and deterministic. The stochastic contact models can be further classified into two groups: statistical and fractal. These approaches or models are distinguished by the use of surface descriptions. The basic features of different approaches are briefly summarized below. A more comprehensive review including the discussion on their advantages and disadvantages can be found in ref. [58]. The statistical approach was first proposed by Greenwood and Williamson [59]. In this approach, the surface roughness is represented by asperities of simple geometrical shape and with predefined radii of curvature. The asperity heights are assumed to follow a statistical distribution. A rough surface is thus characterized by statistical parameters such as the standard deviation of surface heights and correlation length. A single asperityto-asperity contact is reduced to the deformation of two curved bodies in contact. Its solution may either be determined analytically using contact mechanics or expressed by the empirical formula from the finite element simulation. The surface contact is then modeled by relating the load and the real area of contact to their asperity-level counterparts by statistical integration. In many situations, the statistical parameters of surfaces have been found strongly dependent on the resolution of roughness-measuring instruments [60-6]. This phenomenon is due to the multiscale nature of the surface roughness which may be better

26 13 described by fractal geometry [63, 64]. The surface contact models are then developed based on the use of power spectrum and scaling laws characterized by scale-invariant quantities such as fractal dimension [65-69]. These models also take the system variables to be the integration of the asperity solution. However, each asperity is now represented by the size of the contact spot based on which its amplitude of deformation and radius of curvature are defined. The deterministic approach analyzes the computer generated surfaces or those represented by the digitized output of roughness measurement. The surface contact behavior may then be predicted numerically by the method of influence coefficients [70-77] and that based on the variational principle [78]. Compared to the statistical and fractal contact models, the numerical simulation uses the digital maps of rough surfaces and does not require any assumptions on asperity shape and distribution. In addition, this type of analysis may be able to naturally account for the interaction of deformation of adjacent contact spots. Significant advances have been made with the above approaches in the study of both frictionless and frictional dry contacts of rough surfaces. However, the models developed so far for the frictional contact appear to be largely oversimplified with some major assumptions. Two key phenomena, in the author s opinion, need to be addressed in modeling the frictional surface contact. One is that contacting asperities may deform elastically, elastoplastically or plastically. According to the results of frictionless indentation of a sphere on a plane, the normal load leading to initial yielding needs to increase more than 400 times to cause fully plastic flow [79]. The application of friction reduces the first critical normal load [80-8] and thus the elastic deformation regime. The

27 14 friction may also reduce the critical load related to plastic flow and the elastoplastic deformation regime. However, this transition regime may still be significant compared to the elastic regime. Hence, a high percentage of contacting asperities may be in the state of elastoplastic deformation for the contact of rough surfaces with or without friction. Moreover, a significant portion of asperities in contact may deform plastically in the frictional situation. For the frictionless contact, all the three possible deformation modes have been incorporated into several statistical models based on approximate analytical or finite element solutions of the elastoplastic asperity contact [83-85]. In contrast, there is no similar model for the frictional contact due to the lack of a systematic study of the elastoplastic behavior of contacting asperities with friction. The other key phenomenon is that the friction may significantly change the asperity pressure and contact area for those asperities in elastoplastic and, particularly, fully plastic deformation. Both experimental and theoretical studies have shown that, for a frictional plastic contact, the interfacial shear stress would lead to the growth of the asperity junction and reduction of the contact pressure [86-88]. Tabor [89] modeled these two trends using a flow equation derived for asperity junctions under the combined normal and tangential loading. The pressure and contact area of the plastic junctions have also been solved using slip-line field theory [90-95] and upper bound plasticity analysis [96]. For the surface contact, the effects of friction on the subsurface stresses have been modeled but the contact pressure and area are usually considered not to be altered by the friction. In summary, a mathematical model accounting for these two important issues should be formulated for the frictional contact of rough surfaces Contact Modeling: Boundary-Lubricated Surfaces

28 15 Under boundary lubrication conditions, the contact of two rough surfaces is also present in the form of distributed asperity contacts. In addition to the asperities, the boundary films covering them may be involved in the contact process. However, these films are very thin and thus it is reasonable to assume that the contact pressure and area are mainly determined by the asperity deformation. The contact response is mainly affected by the boundary films through their effects on the interfacial friction. Thus, the three approaches discussed in the last section may also be used to model the boundarylubricated surface contact if the shearing behavior of the boundary films is known. Many contact models have been developed for the boundary lubrication system using the statistical approach [97-104]. Besides the general contact response, these models predict the friction force as a function of load by summing up the local tangential resistance. The pressure and area of a single asperity contact are usually determined using the Hertzian elastic solution. In comparison, the finite element method has been used to analyze the mechanical responses of contacting asperities with nonlinear material properties [104]. For the determination of the friction force at the asperity junctions, there are several different formulations available. For example, Ogilvy [97] calculated the local friction force by assuming constant film shear strength and using the energy of adhesion, Blencoe and Williams [101] related the interfacial shear strength to the contact pressure according to empirical relations, and Ford [103] took account of the contribution from both interfacial adhesion and asperity deformation. In addition to the statistical models, direct numerical simulation has also been performed for the contact of rough surfaces to calculate the friction force resulting from adhesion and deformation [105]. This

29 16 deterministic model extends the method of influence coefficients to account for the effects of shear force on contact deformation. The study of the boundary-lubricated surface contact with the above models has provided some insights into the effects of the rheology of boundary layers, the substrate material properties, and the surface roughness on the system tribological behavior. However, there are significant rooms for advancements in many aspects and mathematical models with more insights may be developed. First, as mentioned in the last section, a large population of contacting asperities may be in either elastoplastic or fully plastic deformation. These two types of asperity contacts have not been properly considered. The important phenomena related to the two deformation modes, such as the pressure-shear stress coupling and the friction-induced junction growth, also need to be incorporated in to the model. Second, the adsorbed layer may be desorbed and the reacted film may be ruptured during the asperity contacts. Thus, the effectiveness of boundary lubrication at an asperity junction is characterized by intrinsic uncertainty. It would be of theoretical and practical significance to capture this uncertainty by modeling the kinetic behavior of the boundary lubricating films. Third, localized temperature rise or flash temperature may be caused by the intensive shear stress at asperity junctions. The increasing contact temperature in turn may significantly affect the kinetics of the boundary films and thus the interfacial shear stress. As reviewed in the next section, the flash temperature has been calculated or measured by a number of researchers. However, its interaction with the evolution of the boundary films has not been studied adequately in contact modeling Flash Temperature

30 17 The localized temperature rise due to frictional heating is an important characteristic of the dry and boundary- or mixed-lubricated sliding contact of rough surfaces. The rising temperature can be viewed as the thermal response of the contact and it may strongly affect the behavior of lubricating films, the properties of substrate materials, as well as most surface phenomena. Thus, the prediction of the interface temperature plays an important role in modeling the sliding contact behavior. The maximum or average temperature rise of single asperity contacts has been estimated based on the laws of energy conservation and heat conduction [ ]. Most of these analyses focused on the flash temperature of an individual square or circular contact. Gecim and Winer considered the cooling-off effect between two consecutive asperity contacts [11]. Bhushan proposed an approach to include the effects of frictional heating by neighboring asperity contacts [114]. The analysis of asperity flash temperatures has also been incorporated into different types of surface contact models to predict the interfacial temperature distribution [67, 68, and ]. For example, the fractal contact model developed by Wang and Komvopoulos [67, 68] included the analysis of the distribution of temperature rise at the interface. Based on a statistical contact model, Yevtushenko and Ivanyk [116] determined the temperature rise of contacting asperities and their thermal deformation for the sliding contact of rough surfaces under mixed lubrication conditions. In comparison, Qiu and Cheng [117] calculated the temperature rise at asperity contact spots which were the solution provided by a deterministic surface contact model [71].

31 Summary The above literature review shows that significant progress has been made in the understanding of different boundary lubrication mechanisms, the modeling of rough surfaces, and the calculation of flash temperature. Research has also been initiated to address the integral effects of these important aspects. For example, a failure criterion of boundary lubrication has been incorporated into a thermal contact model of rough surfaces [117]. However, only the elastic deformation and thermal desorption are considered. More recently, an asperity-contact model has been designed to calculate the tribological variables by simultaneously simulating the key processes involved but the solution obtained is not suitable to be integrated into a system model [119]. In summary, a comprehensive contact model needs to be developed to include the effects of multiple deformation modes of contacting asperities, the uncertainty of the boundary lubricating films, the flash temperature due to friction, and their interaction. 1.3 Research Objective, Approach and Outline This thesis aims to develop a surface contact model for the boundary lubrication system to gain more insights into its tribological behavior. For a given load, the model should be able to predict the asperity contact variables and their distribution, and the system friction coefficient and area of contact. The model should also factor in surface topography, material and lubricant properties, and other operating conditions in addition to the system load. In this research, the statistical approach is selected to relate the system contact variables to their asperity-level counterparts. The reason is that the statistical models are

32 19 able to identify the important trends in the effects of surface properties on the system contact behavior with relatively simple calculation. The key component of the research is thus the development of a deterministic model for a single asperity contact under boundary lubrication conditions. At the asperity level, the model needs to capture the characteristics of fundamental mechanical, physiochemical, and thermal processes involved in the boundary-lubricated contact. From the mechanical point of view, the model to be developed should cover the three possible deformation modes of contacting asperities under combined normal and tangential loading. For this purpose, the effects of friction on the pressure, area, and deformation mode of a single asperity contact are first explored using the finite element method since it is impossible to obtain the analytical solution directly. The finite element results are then combined with the contact mechanics theories to derive model equations for a frictional asperity contact involving the three possible deformation modes. These pure mechanical equations are used to describe the boundarylubricated asperity contact in conjunction with the expressions developed to calculate the flash temperature and to characterize the behavior of boundary films. The solution of all the asperity-level modeling equations is finally used to formulate the contact model for the boundary lubrication system by means of statistical integration. In summary, the thesis comprises three layers of modeling and analysis (1) elastoplastic finite element analysis of frictional asperity contacts, () modeling of contact systems with friction, and (3) modeling of a boundary lubrication process. Each layer of analysis is presented as a chapter in the main text and briefly described below.

33 0 Chapter : Finite element analysis of frictional asperity contacts A finite element model is developed and systematic numerical analyses carried out to study the effects of friction on the contact and deformation behavior of individual asperity contacts. The study reveals some insights into the modes of asperity deformation and asperity contact variables as function of friction in the contact. The results provide guidance to analytical modeling of frictional asperity contacts and lay a foundation for subsequent work on system modeling. Chapter 3: Modeling of contact systems with friction Analytical equations are developed relating asperity-contact variables to friction using the theory of contactmechanics in conjunction with the finite element results in chapter. By statistically integrating the asperity-level equations, a system-level model is developed and used to study the effects of the friction on the system contact behavior. It serves as the platform in the final step of model development for the boundary lubrication problem. Chapter 4: Modeling of a boundary lubrication process Based on the previous two layers of modeling, a deterministic-statistical model for the boundary-lubricated contact is developed by incorporating the essential aspects of boundary lubrication. Four variables are used to describe a single asperity contact, including micro-contact area, pressure, shear stress, and flash temperature. In addition, three probability variables are introduced to define the interfacial state of an asperity junction that may be covered by various boundary films. Governing equations for the seven key asperity-level variables are derived based on first-principle considerations of asperity deformation, frictional heating, and kinetics of boundary lubrication films. These asperity-scale equations are coupled and some of them are nonlinear. Their solution is thus obtained by an iterative

34 1 method and is statistically integrated to formulate the contact model for boundary lubrication systems. The model is then used to study the effects of surface roughness and operation parameters on the system tribological behavior. Each of the above three chapters is relatively self-contained though they are also well-connected. Finally, Chapter 5 concludes the thesis with a summary of the main contributions and some suggestions for future work.

35 Chapter Effects of Friction on the Contact and Deformation Behavior in Sliding Asperity Contacts.1 Introduction It is quite well recognized that the solid-to-solid contact between the surfaces of machine components is made at their surface asperities. These asperity contacts often play a significant role in the tribological performance of mechanical systems, especially under dry and boundary lubricated conditions. Greenwood and Williamson [56] established a framework for the statistical asperity-contact based models of two contacting surfaces. The concept was used in many areas of micro-tribology modeling, such as machine components in mixed lubrication [1], head-disk interface of computer disk-drive [13] and chemical-mechanical planarization of silicon wafer [14], to name just a few. The model of reference [56] does not include friction, which can significantly affect the behavior of the asperity contacts. A number of researchers have studied the effects of friction. For elastic contacts, the theory of elasticity is used to obtain closedform solutions. Poritsky and Schenectady [15] and Smith and Liu [16] calculated the subsurface stresses in frictional contacts under elastic plain-strain conditions. Hamilton and Goodman [17], Hamilton [18], and Sackfield and Hills [80] solved the threedimensional problem. The results show that the friction brings the point of the maximum shear stress closer to the surface and increases the compressive stress at the leading edge

36 3 and the tensile stress at the trailing edge of the contact. Johnson & Jefferis [81] studied the effects of friction on the plastic yielding in line contacts. Hills and Ashelby [8] and Sackfield and Hills [80] analyzed the problem for point contacts. The results show that the yielding would start at lower normal loads and the points of the initial yielding would move to the surface when the friction coefficient exceeds 0.3. For fully plastic contacts, the theory of plasticity may be used to obtain approximate solutions. McFarlane and Tabor [87, 88] studied the effects of friction in plastic contacts using the octahedral shear stress theory. The results show that, for a given normal load, the friction reduces the contact pressure and increases the contact area. Making use of the criterion of plastic flow for a two-dimensional body, Tabor [89] derived a flow equation for asperity junctions under the combined normal and tangential loading. With this equation, he explained the phenomenon of the junction growth and the high friction between clean metal surfaces that were observed in experiments. Johnson [9] and Collins [93] also solved the plastic frictional contact problems using the theory of slip-line field. In addition to the pressure reduction and junction growth, they concluded that the friction coefficient would reach a high value of about unity in the extreme. A large number of asperity contacts in a dry or boundary-lubricated system may be in elastic-plastic deformation. In this mode of deformation, analytical solutions are not readily available. The methods of finite elements are often used to study the effects of friction. Tian and Saka [19], Kral and Komvopoulos [130] and many others studied the contact of coated surfaces. Tangena and Wijnhoven [131] and Faulkner and Arnell [13] simulated the collision process of a pair of asperities. Nagaraj [133] and many others

37 4 analyzed contact problems with stick and slip. These numerical studies, however, largely focused on special problems. Fundamental issues have not been adequately addressed, such as the effects of friction on the mode of the asperity deformation, shape and size of the plastic zone in the micro-contact, and the asperity pressure, contact area and load capacity. In this chapter, a systematic finite element analysis is carried out to study sliding asperity contacts in elastic, elastic-plastic and fully plastic deformation. The analysis focuses on the above fundamental issues of the effects of friction to reveal some insights into the behavior of sliding asperity contacts. The modeling and results are presented in the next two sections.. The Model Problem The model of a deformable half-cylinder in sliding contact with a rigid flat is used in this chapter as illustrated in Fig..1. This two-dimensional plain-strain model should capture the essential effects of the friction on the contact and deformation behavior of an asperity contact while significantly simplifying the computational complexity. The material is assumed to be elastic-perfectly plastic with a Poisson s ratio of υ = 0. 3 and a ratio of Young s modulus to uni-axial yield stress of E / Y = 100. The choice of a high value of E / Y would result in a plastically deformed region in the contact that is much smaller than the cross-section area of the half-cylinder so that the results will be fairly independent of the latter and of the boundary conditions away from the contact. Furthermore, the results in the dimensionless form presented later in the chapter are essentially independent of the E / Y ratio so long as the region of plastic deformation is a

38 5 very small proportion of the bulk material, which is the case in actual asperity contacts. The normal loading to the contact is prescribed in terms of the approach of the rigid flat to the cylinder, δ, which is more meaningful than specifying a normal load for asperity contacts between two surfaces. The tangential loading, F, is given in terms of a shear stress distribution in the contact proportional to the pressure distribution: ( x ) µp( x) τ = (.1) where µ is a prescribed coefficient of friction and the pressure distribution is to be determined in the solution process. It should be pointed out that the contact between two bodies in gross sliding is of interest in this thesis study. In such a contact, the assumption of a uniform local friction coefficient defined by Eq. (.1) is theoretically feasible. The ratio of the local shear stress to the local pressure in a sliding contact can be extremely complex and often exhibits significant random behavior. A uniform µ as a parameter would represent a stochastic average that can be sensibly used to study the effects of friction on the contact. The solid modeling software I-DEAS is used to generate the finite element mesh of the model problem as shown in Fig... The mesh consists of 870 eight-node plane strain elements with a total number of,713 nodes. A substantial number of elements are allocated in the region around the contact. The commercial finite element code ABAQUS is used to simulate the sliding contact problem and small deformation is assumed in the finite element calculations. Zero-displacement boundary conditions are prescribed for the nodes at the bottom of the finite element model. The rigid-surface option is employed to mimic the rigid flat, which is constrained to move vertically. The normal loading to the

39 6 model asperity by means of a normal approach is realized by enforcing a vertical displacement to the flat. The adaptive automatic stepping scheme is implemented for loading. More detail descriptions of algorithms used to determine the contact nodes and contact conditions are given in the ABAQUS manual [134]. For a given combination of the normal approach and friction coefficient, the finite element calculations yield the pressure distribution and the width of the contact and the nodal von Mises stresses σ M. Then the average pressure and load capacity of the contact can be calculated. Furthermore, the first occurrence of a nodal stress of σ Y is used to determine the initial plastic yielding of the contact [135], and the stress contour of σ Y is used to determine the shape and size of the plastic zone. M = M The accuracy of the finite element model is evaluated. Mesarovic & Fleck [136] pointed out that the maximum relative error may be expressed as one-half of the ratio of the nodal spacing in the contact and the contact size. For the mesh given in Fig.. and under frictionless normal loading, about 1 surface nodes come into contact with the rigid flat when the initial yielding occurs in the model asperity. The error under this condition would then be under 10%. Indeed, the finite element results for an elastic frictionless contact compare favorably with the results from the Hertz theory, including the pressure distribution, contact width and location of the material point of initial yielding. Considering that a large portion of the analyses will be carried out for a greater number of surface nodes in the contact, the mesh arrangement of Fig.. should be fairly adequate. The adequacy of the finite element mesh is studied with additional evaluations. First, the results are essentially independent of the direction of sliding from either left or right. Second, the results are also essentially independent of the history of normal/tangential

40 7 loading (ie. changes of δ and µ ), which is sensible for small deformation of a nonwork-hardening asperity. Finally, the plastic zones for fully plastic contacts compare reasonably well with the slip-line analytical solutions by Johnson [9] and Collins [93]..3 Results and Analysis The contact pressure and sub-surface stresses are calculated for a range of the normal approach, δ, and friction coefficient, µ. The results are presented and analyzed to reveal the effects of friction on (1) the mode of asperity deformation, () the shape of micro-contact plastic zone and (3) the pressure, size and load capacity of the asperity contact..3.1 Mode of Asperity Deformation The state of the asperity deformation may be categorized into three regimes elastic, elastic-plastic and fully plastic. In an elastic contact, the von Mises stresses of all material points are less than the uni-axial yield strength of the material. In an elasticplastic contact, plastic yielding occurs at some material points, marking a transition from the elastic to fully plastic deformation. In a fully plastic contact, all material points around the contact enter plastic deformation and the ability of the asperity to take additional load is largely lost. For a frictionless contact, the transition from elastic-plastic to full plastic contact is often defined to be the point when all the nodal pressures in the contact largely reach the value of the material hardness, which is considered to be about equal to.8y [79]. For a frictional contact, this definition may not be used as the tangential loading can substantially bring down the pressure that can be developed. In this chapter, the elastic-plastic to full plastic transition is defined to be the condition under

41 8 which the von Mises stresses of all surface nodes in the contact region have reached the uni-axial yield stress of the material. It is noted from numerical results that, under the above condition, the contact pressure distribution is fairly uniform corresponding to full plasticity. Two critical values of the normal approach are defined to describe the modes of the asperity deformation. The first critical normal approach, δ 1, corresponds to the condition under which the initial yielding occurs in the contact, and the second one, δ, the condition under which the contact becomes fully plastic. The effects of the friction on the state of the asperity deformation may be studied by examining the values of the two critical normal approaches. Figure.3 shows the variations of δ 1 and δ as functions of the friction coefficient up to µ = 1.0; this µ value may be considered to be an upper bound based on Johnson [79]. The values of δ 1 and δ are plotted in the scale of δ 10, which is the first critical normal approach for the frictionless contact. For µ = 0, the normal approach causing the onset of fully plastic deformation of the contact is about forty times of δ 10. This large value of δ, which is of the same order of magnitude as those obtained for 3D circular contacts [84, 137], suggests a rather long transition from the elastic contact to the fully plastic contact. However, the elastic-plastic transition is rapidly reduced by the friction. The value of δ is only about 4δ 10 at µ = 0.3 and is further reduced to one half of δ 10 at µ = 1.0. The normal approach, or the contact force, causing the initial yielding of the contact is also reduced significantly by the friction. At µ = 0.3, for example, δ 1 is reduced to 0.7 of its zero-friction value of δ 10. This reduction accelerates at high friction values. At µ = 1.0, δ 1 is reduced to only about

42 9 0.14δ 10. The reduction of δ 1 with friction is more clearly seen in a log-scale shown in Fig..3 (b). It should be pointed out that the δ ~ µ curves in Fig..3 are numerical approximations dividing the regimes of asperity deformation. Numerical errors arise from the sizes of the finite element meshing and the stepping size of the normal approach δ in the solution process. The results of Fig..3 are obtained with a maximum stepping size δ = 0.01δ. The errors are sufficiently small and may not be further reduced given of 10 the assumptions and idealizations of the model problem. This is further supported by the fact that the δ 1 ~ µ curve in Fig..3 exhibits a similar trend as that for a circular contact derived analytically using the equations in references [79, 80]. The two curves of δ 1 and δ shown in Fig..3 describe the mode of the asperity deformation at a given friction coefficient and normal approach of the contact. The rapid reduction of δ with friction shown in Fig..3 (a) reveals a remarkable effect of the friction on the deformation in an asperity contact. With high friction, the contact may change from the state of elastic deformation to the state of fully plastic deformation with little elastic-plastic transition as the normal approach or the contact force increases. The large reductions of the two critical approaches with friction also signify significant reductions of the contact pressures at the points of transition of the mode of the asperity deformation. In a frictionless contact, the average contact pressure at the elastic-toelastic-plastic transition is 1.41 of the uni-axial yield stress, and it is about.60 at the elastic-plastic-to-plastic transition. With µ = 0.3, these two pressures are reduced to 1.3 and 1.79, respectively, and further reduced to 0.4 and 0.6 at µ = 1.0. The reductions in

43 30 the pressure are evidently due to the large shear stresses that are developed in the asperity contact. The finite element results may also be used to study the equation of the full plastic flow proposed by Tabor [89] that relates the pressure to the interfacial shear stress in the contact. This equation may be expressed as: p = + ατ H (.) where α is a constant, s the interfacial shear stress and H the indentation hardness of the material or the maximum pressure that can be developed in the contact. Taking H =. 6Y based on the finite element results with µ = 0, then a value for α in Eq. (.) can be determined for a given friction coefficient using the calculated pressure and surface shear stress at the normal approach of δ = δ. For the model problem with a friction coefficient up to µ = 1.0, the calculations of the nine data points along the δ ~ µ curve yield α values that are about 10 with low µ and 15 with high µ. These fairly uniform values of α lie in the range of values discussed in [89]..3. Shape of the Plastic Zone The behavior of the two critical normal approaches shown in Fig..3 is closely related to the effects of the friction on the shape and size of the plastic zone in the asperity contact. The problem of a frictionless contact is first studied. The location of the initial yielding is in the central region of the contact about 0.67 times the contact-halfwidth beneath the surface. Figure.4 shows the plastic zones for two values of the normal approach. One is at the halfway between δ 1 and δ and the other at δ,

44 31 corresponding to the mode of elastic-plastic deformation and the onset of full plastic flow, respectively. Under both loading conditions, the plastic zones are similar and are nearly of a circular shape. In the former, the subsurface initiated plastic deformation has grown substantially and has largely propagated to the contact surface except a thin layer that still remains elastic as shown in Fig..4 (a). In the latter, this thin surface layer has also become plastic while the plastic zone expands further with a diameter nearly three times as that of the former. The problems with friction are studied next. Figure.5 shows the results obtained with a friction coefficient of µ = 0.; the direction of the friction force is from the left to the right. The location of the initial yielding is shifted towards the leading edge of the contact at 0.53 times the contact-half-width beneath the surface and 0.65 to the right. With a normal approach corresponding to halfway into the elastic-plastic transition, the surface material at the trailing one half of the contact has become plastic while a surface layer at the leading one half is still elastic. This is in contrast to its frictionless counterpart of Fig..4 (a), where the plastic yielding at the surface starts in the central region of the contact. As the normal approach further increases, the plastic zone rapidly propagates towards the surface on the leading side. When full plasticity is reached in the contact, the plastic zone has expanded beyond the leading edge and is nearly of a rectangular shape of a depth that is 1.1 times the width, as shown in Fig..5 (b). Owing to the significant tangential loading in the contact, the value of the normal approach to bring about full plasticity is reduced to about 0.5 of that of the frictionless contact, and the width of the contact to about 0.7.

45 3 Figure.6 shows the results with a higher friction coefficient of µ = 0.5. With this high friction, the plastic yielding is initiated at the surface, one site at the leading edge and another immediately occurring thereafter at the trailing edge. The result of the two-site plastic yielding is consistent with an analytical approximation [79]. The two plastic sub-zones propagate and eventually unite as the normal approach increases. Halfway into the elastic-plastic transition, the plastic deformation is largely confined to near surface, and a small segment at the leading edge of the contact remains elastic. When full plasticity is reached, the plastic zone has not significantly propagated into the depth aside from a protruding-wing region that is developed towards the leading edge of the contact, as shown in Fig..6b. A protruding-wing shaped plastic zone of a lesser magnitude was obtained in the slip-line field solution reported in Collins [93] for a rigidperfectly plastic contact with high friction. The width of the contact in this case is only about 0.05 of that of its frictionless counterpart at the condition of full plasticity. Figure.7 shows the results with an even higher friction coefficient of µ = 1.0. Similar to the problem of µ = 0.5, the yielding initiates at the surface at both the leading and trailing edges of the contact. The two plastic sub-zones have not yet connected halfway into the elastic-plastic transition. Furthermore, at full plasticity, no protruding-wing shaped plastic zone of a significant magnitude is developed at the leading edge. The width of the contact is about 0.04 of the size for the frictionless problem when full plasticity is reached, and the plastic deformation is largely confined to a very thin surface layer in the contact region.

46 Contact Size, Pressure and Load Capacity It is of interest to study the effects of the friction on the contact variables including the junction size, pressure and load capacity of the asperity. For a meaningful study and results comparison, the normal approach is held constant while the friction coefficient is varied. Figure.8 shows the results obtained at a relatively low level of loading; the normal approach is set equal to the normal approach causing plastic yielding in a frictionless contact, δ 10. The results are plotted in the scale of their corresponding values with zero friction. With a relatively low friction coefficient of µ = 0.0 ~ 0.3, the effects are small on the three contact variables. At moderate friction of µ = 0.3 ~ 0.5, the contact pressure starts to decrease while the contact junction grows. At µ = 0.47, for example, the pressure is reduced to 0.84 of its frictionless value and the junction is increased to However, the load carried by the asperity is essentially unaffected due to the compensating effects of the pressure reduction and junction growth. At the higher level of the contact friction of µ = 0.5 ~ 1.0, the reduction in the pressure and the growth in the contact size becomes more intensified to about one half and two times their frictionless values at the extreme. The change in the load capacity is only modest with a maximum reduction of about 11% at µ = 1.0. The reduction of the pressure with friction in Fig..8 may be studied with Eq. (.). For a normal approach of δ = δ10, the contact is largely elastic when the friction coefficient is small. Therefore, it can accommodate some tangential traction without bringing about significant plastic deformation (ie. p + ατ is significantly less than H ). Consequently, the pressure is not affected by the friction. As the level of friction

47 34 increases, the amount of plastic deformation increases. At µ = 0.5, for example, δ 1 = 0. 36δ 10 and 1. 4δ 10 δ = as shown in Fig..3 (b) so that the contact is significantly plastic with the current normal approach of δ = δ10. As a result, the coupling between the normal and tangential loading in the asperity contact is more pronounced and the increase in the surface shear stress would be at the expense of the contact pressure. The contact eventually becomes fully plastic with a higher friction coefficient of µ > 0.6, and the tangential/normal coupling is even stronger and follows Eq. (.). The growth of the contact junction with friction may be studied by examining the shift of the junction in the direction of the friction force. Figure.9 shows the sizes of the contact junction at different levels of the friction coefficient along with the center locations of the junction. Up to a friction coefficient of µ = 0.38, the junction experiences little growth and its center location is virtually unchanged. This result may be attributed to the fact that the junction is largely elastic up to this level of the friction. The results however show a significant trend of the junction growth with the friction coefficient of µ = 0.38 ~ 0.47, yet a shift in the center of the contact junction is not visible. An examination of the critical normal approaches shown in Fig..3 suggests that, with δ = δ10, the degree of plastic deformation in the contact increases significantly in this range of the friction coefficient. Thus, the increase in the junction size is attributed to the contact becoming more plastic as, for a given normal approach (in a frictionless contact), the junction size is about twice as large for a plastic contact than for an elastic contact [79]. With an even higher friction level of µ = 0.47 ~ 0.6, the results in Fig..9 show that the junction growth becomes more pronounced accompanied by a significant

48 35 shift of the center of the junction, which is an indication of tangential plastic flow. In this range of the friction coefficient, the contact eventually reaches the state of full plasticity. The accelerated junction growth is attributed to two factors. One is the growth associated with the further increase of plastic deformation in the contact, and the other the tangential plastic flow induced by the friction force. For a friction coefficient beyond µ = 0.6, the trend of the junction growth and the shift of the center of the junction become somewhat moderated. In this range of the friction coefficient, the contact is now in the mode of full plasticity and the junction growth is primarily due to the friction-induced tangential plastic flow. Figure.10 shows the effects of the friction on the contact variables at a relatively high level of loading. The normal approach in this case is three times as large as that with which the results of Fig..8 are obtained. At this loading level, the pressure reduction and junction growth take place in the low range of the friction coefficient but the load capacity is virtually unchanged. In the median range of the friction, the pressure and the contact size become significantly more sensitive to the friction coefficient. At µ = 0.5, the pressure is reduced to 0.58 of its frictionless value while the junction size increased to The load capacity of the junction is still maintained at its frictionless level up to µ = 0.4 and then reduces for higher friction to a value of 0.93 at µ = 0.5. For higher friction coefficients, the pressure reduces further and so grows the junction. However, the results suggest that the junction growth in this case is not as pronounced as the pressure reduction in comparison with the results from the previous case of low loading. The results further show a limited junction growth at the high-end of the friction coefficient. As a result, the compensation of the junction growth to the pressure reduction becomes

49 36 less effective at this level of loading and the load capacity of the junction is significantly reduced by the effect of friction. At µ = 1.0, for example, the load capacity is reduced to 0.61 of its value for the frictionless contact. The limit in the junction growth shown in Fig..10 for relatively high contact loading is possibly due to the geometric effect of the asperity. A higher loading produces a larger contact size and a larger surface slope at the edges of the contact junction, particularly the leading edge because of the friction-induced tangential plastic flow. The tangential plastic flow and the surface slope are the two competing factors that determine the size and the growth of the contact junction. When the contact size is small, the slope is small and the junction growth is largely governed by the plastic flow leading to a large increase of the junction with friction. When the contact size is large, the surface slope at the leading edge is large and would ultimately limit further growth of the junction. It should be pointed out that a majority of the contacting asperities in the contact of rough surfaces might experience a level of loading that is significantly above that with which the contact-variable results in Fig..10 are obtained. For machine components such as bearings and engine cylinders, the radius of surface asperities may be taken as of the order of 10 µm [138] and the Young s modulus is around Pa. Then the normal approach causing plastic yielding of the contact in the absence of friction is of the order of magnitude of δ 10 = µm [79]. For relatively highly finished machine components, the surface RMS roughness is often significantly larger than 0.1 µm and thus the normal approaches of many contacting asperities can be significantly above 0.01 µm. In this situation, the loss of load capacity to the friction by these contacting asperities

50 37 could be more severe than that predicted in Fig..10. As a result, the average gap between the two surfaces would reduce so as to bring additional asperities into contact to support the applied load in the system..4 Summary This chapter conducts a finite element analysis of the effects of friction on the contact and deformation behavior in sliding asperity contacts. The analysis is carried out using two input variables. One is the normal approach of a rigid surface towards the asperity, and the other the coefficient of friction in the contact. Results are presented and analyzed to reveal the effects of friction on the mode of asperity deformation, the shape of micro-contact plastic zone, the contact pressure and size, and the asperity load capacity. The results lead to the following conclusions: 1) The friction in the contact can significantly reduce the normal approach that initiates the plastic yielding in the asperity and the normal approach that causes the asperity to become fully plastic. The reduction is more pronounced for the second critical normal approach so that, with a relatively high friction coefficient, the contact may change from the state of elastic deformation to the state of fully plastic deformation with little elastic-plastic transition as the normal approach or the contact force increases. ) The friction can significantly change the shape and reduce the size of the plastically deformed region in the asperity when the contact becomes fully plastic. The reduction is most pronounced at high friction coefficients, and the plastic deformation is largely confined to a thin surface layer in the contact.

51 38 3) The friction can have a large effect on the contact size, pressure and load capacity of the asperity. At low friction and a relatively small normal approach, these contact variables are not affected. With medium friction, the pressure is reduced and the contact size is increased; however the influence on the asperity load capacity is small due to a compensating effect between the pressure reduction and junction growth. With high friction, the pressure reduction continues but the junction growth is limited particularly for a large normal approach; the limit in the junction growth appears to be due to a geometric effect of the asperity. Consequently, the effect of the pressure-junction compensation becomes less effective and the asperity load capacity can be lost significantly. It should be emphasized that the finite element results presented in the dimensionless form given in this chapter are sufficiently general. Essentially the same results are obtained with different radii or material parameters of the model asperity as long as the region of plastic deformation in the contact is small so that the half-space assumption is fairly valid. Although the analyses are conducted using a line-contact model, the effects of friction in sliding asperity contacts of three-dimensional geometry should be basically the same and the same conclusions would have been reached. Therefore, the finite element results are used in the next chapter to guide the development of analytical modeling equations for frictional asperity contacts that lay a foundation for subsequent work on system contact modeling.

52 39 δ Rigid flat Sliding direction of the rigid flat Figure.1 Half-cylinder contact model Figure. Finite element mesh of the model problem

53 40 Critical normal approaches (a) δ 1 /δ 10 δ /δ δ1/ δ10 δ / δ 10 (b) Critical normal approaches Friction coefficient Figure.3 Effects of friction on the critical normal approaches: (a) linear scale; (b) logarithmic scale

54 41 Plastic deformation Elastic deformation Contact width Rigid flat (a) Asperity (b) Figure.4 Plastic zones of the frictionless contact (a) elastic-plastic transition; (b) onset of full plasticity (the top figure shows the zoom-in of the region in the dashed rectangle in (a))

55 4 Friction force Contact width (a) (b) Figure.5 Plastic zones of the contact with µ = 0. (a) elastic-plastic transition; (b) onset of full plasticity (the contact width in (b) is 0.7 of that of its frictionless counterpart in Fig.4)

56 43 (a) Contact width (b) Figure.6 Plastic zones of the contact with µ = 0.5: (a) elastic-plastic transition; (b) onset of full plasticity (the contact width in (b) is 0.05 of that of its frictionless counterpart in Fig.4)

57 44 (a) Contact width (b) Figure.7 Plastic zones of the contact with µ = 1.0: (a) elastic-plastic flow transition; (b) onset of full plasticity (the contact width in (b) is 0.04 of that of its frictionless counterpart in Fig.4)