DOCTORAL THES I S DOCTORAL THES I SLule University of TechnologyDepartment of Applied Physics and Mechanical Engineering Division of Machine Elements:ooo:+|issx: +o:-+|isrx: i+u-i+ -- oo + -- sr:ooo:+On the Effects of Surface Roughness in LubricationAndreas Almqvist2006:31On the Effects ofSurface Roughness in LubricationAndreas AlmqvistLule University of TechnologyDepartment of Applied Physics and Mechanical Engineering,Division of Machine Elements2006 : 31 | ISSN : 1402 1544 |ISRN:LTU-DT--06/31--SECover gure: An artistic visualization of an axial thrust pad bearing,see Fig. 1.1 for details.Title page gure: The pressure build-up on a single rough pad of the bearingpresented in Fig. 1.1.On the Effects ofSurface Roughness in LubricationCopyright c _ Andreas Almqvist (2006). This document is freely available athttp://epubl.ltu.se/1402-1544/2006/31or by contacting Andreas Almqvist,andreas.almqvist@ltu.seThe document may be freely distributed in its original form including the currentauthors name. None of the content may be changed or excluded without permis-sions from the author.ISSN: 1402-1544ISRN: LTU-DT--06/31--SEThis document was typeset in LATEX2.AbstractTribology is a multidisciplinary eld dened as the science and technology of in-teracting surfaces in relative motion, and embraces the study of friction, wear andlubrication. A typical tribological application is the rolling element bearing, seeFig. 1. Tribological contacts may also be found in other types of bearings, cam-mechanisms, gearboxes and hydraulic systems. Examples of bearings inside thehuman body are the operation of the human hip joint and the contact betweenteeth during chewing. To fully understand the operation of this type of applica-tion one has to understand the couplings between the lubricant uid dynamics,the structural dynamics of the bearing material, the thermodynamical aspects andthe resulting chemical reactions. This makes modeling tribological applications anextremely delicate task. Because of the multidisciplinary nature, such theoreti-Figure 1: A typical tribological applicationcal models lead to mathematical descriptions generally in the form of non-linearintegro-dierential systems of equations. Some of these systems of equations aresuciently well posed to allow numerical solutions to be carried out, resulting inaccurate predictions on performance.In this work, the inuence on performance of a surface microscopical nature, thesurface roughness, in contact interfaces between dierent types of machine elementcomponents is the subject of study. An example is the non-conformal lubricatedcontact between one of the rollers and the inner ring in the bearing depicted inFig. 1. The tribological contact controlling the operation of the human hip jointis also very similar to this. Another example of a non-conformal contact occursiwhen driving on rainy roads, where the hydrodynamic action of the water separatesthe tire. To enable investigations of these types of problems, dierent theoreticalmodels were studied; for the selected model, a numerical solution technique wasdeveloped within this project. This model is based on the Reynolds equationcoupled with the lm thickness equation. The numerical solution technique involvesa multilevel technique to facilitate the solution process. Results presented in thisthesis, utilizing this approach, study elementary surface features such as ridges andindentations passing each other inside the lubricated conjunction.The Reynolds equation is derived under the assumptions of thin uid lm andcreeping ow, and considers in its most general form shear thinning of the lubricant.This type of equation describes the hydrodynamic action of the lubricant owand may be used when the interfaces consist of either conformal or non-conformalconjunctions. Examples of applications having conformal interfaces are thrust- andjournal- bearings or the contact between the eye and a (optical) contact lens. SeeFig. 2 for a schematic illustration of a typical axial thrust pad bearing (the angleof pad inclination being highly exaggerated; in a realistic application this angleis generally only fractions of a degree). In such types of applications the loadFigure 2: A schematic illustration of an axial thrust pad bearingcarried by the interface is distributed over a fairly large area that under certaincircumstances helps to prevent mechanical deformation of the contacting surfaces.Such applications are said to operate in the hydrodynamic lubrication (HL) regime.Lubricant compressibility and cavitation are important aspects and have re-ceived some attention. However, the main objective when modeling HL has beento investigate and develop methods that enable the inuence of surface roughnessto be to be studied eciently.Homogenization is a rigorous mathematical concept that when applied to acertain problem may be regarded as an averaging technique as well as it providesinformation about the induced eects of local surface roughness. Homogeniza-tion inicts no restrictions on the surface roughness representation other than therepresentative part of the chosen surface roughness being assumed periodically dis-tributed and of course the assumptions of thin lm ow made through the Reynoldsequation. The homogenization process leads to a two sets of equations one for thelocal scale describing surface roughness, scale and one for the global scale describ-ing application geometry. The unequivocally determined coecients of the globalproblem, which may be regarded as ow factors, are obtained through the solutionof local problems. This makes homogenization an eminent approach to be usedinvestigating the inuence of surface roughness on hydrodynamic performance.In the present work, homogenization has been used to derive computation-ally feasible forms of problems originating from incompressible and compressibleReynolds type equations that describe stationary and unstationary ows in bothcartezian and cylindrical co-ordinates. This technique enables simulations of sur-face roughness induced eects when considering surface roughness descriptionsoriginating from measurements. Moreover, the application of homogenization facil-itates the interpretation of results. Numerical investigations following the homoge-nization process have been carried out to verify the applicability of homogenizationin hydrodynamic lubrication. Homogenization has also been shown here to enableecient analysis of rough hydrodynamically lubricated problems. Also of note,in connection to the scientic contribution within tribology, collaboration with agroup in applied mathematics has lead to the development of novel techniquesin that area. These ideas have also been successfully applied, with some resultspresented in this thesis.At start-ups, the contact in a rolling element bearing could be both starved anddrained from lubricant. In this case the hydrodynamic action becomes negligible interms of load carrying capacity. The load is carried exclusively by surface asperi-ties, the tribo lm, or both. This is hereby modeled as the unlubricated frictionlesscontact between rough surfaces, i.e. a contact mechanical approach. A variationalprinciple was used in which the real area of contact and the contact pressure distri-bution minimize the total complementary potential energy. The material model islinear elastic-perfectly plastic and the energy dissipation due to plastic deformationis accounted for. The numerics of this contact mechanical approach involve the fastFourier transformation (FFT) technique in order to facilitate the solution process.Investigation results of the contact mechanics of realistic surfaces are presentedin this thesis. In this investigation the variation in the real area of contact, theplasticity index and some surface roughness parameters due to applied load werestudied.AcknowledgmentGraduate studies are time-consuming, which means little time for other activities.This makes it hard when one is the father of three wonderful children. My greatestgratitude is therefore given to my wife Ulrika, who by being such fantastic motherand taking care of our children has made these studies possible. For all thoseinspiring moments with my children, I am also very grateful. Watching my childrenlearn and grow has really made everything easier for me. My parents, my wifesparents, our grandparents and other close relatives and friends also deserve mygratitude for all their support.I would like to thank my supervisor, Prof. Roland Larsson, for introducing meto the eld of tribology. For his enthusiasm, the inspiration he has given me andall his encouragement, I am especially grateful. He should also be acknowledgedfor always striving to progress in developing his talents as a supervisor.I would like to thank my associate supervisor, Dr. Inge Sderqvist, for hisguidance and support in the eld of Scientic Computing during the rst half ofmy Ph.D. studies. He was also a great source of inspiration and an excellent teacherduring my M.Sc studies, for which I am very grateful.Prof. Peter Wall is also gratefully acknowledged for introducing me to the con-cept of homogenization, the education in the eld and his invaluable co-operation.It should also be mentioned that Peter has been my associate supervisor duringthe second half of my Ph.D. studies.All co-authors and corresponding authors are, of course, also greatly acknowl-edged.Thanks to all my colleagues at the Division of Machine Elements for theirsupport, their co-operation and for introducing me to new concepts in tribology. Iam grateful to have had the opportunity to be a part of the division because of themany competent and friendly people working there, which led to the most pleasantand relaxed atmosphere.Special thanks goes to Tech. Lic. Fredrik Sahlin for all his co-operation, andhis support and help in scripting, especially in LATEX2 and Perl.My friend and former colleague, Dr. Torbjrn Almqvist, is gratefully acknowl-edged for his co-operation, enthusiasm and all our stimulating discussions.I will not forget all the help I have received, especially during the M.Sc. studies,from Dr. Reynold Nslund and for all stimulating and encouraging discussions.Finally, I would to acknowledge my sponsors during the Ph.D. studies; Fortum,Indexator, SKF Statoil, Volvo Car Corp. and the national research programmeHiMeC and the national graduate school in scientic computning, NGSSC, whichvwere both nanced by the Swedish Foundation for Strategic Research (SSF). Duringthe last two years I was also partly nanced by ProViking through Interface, whichis therefore acknowledged.ContentsI The Thesis 11 Introduction 31.1 Lubrication regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The boundary lubrication regime . . . . . . . . . . . . . . . . . . . . 41.3 The mixed lubrication regime . . . . . . . . . . . . . . . . . . . . . . 51.4 The full-lm lubrication regime . . . . . . . . . . . . . . . . . . . . . 51.4.1 Hydrodynamic lubrication . . . . . . . . . . . . . . . . . . . . 51.4.2 Elastohydrodynamic lubrication . . . . . . . . . . . . . . . . 61.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Modeling Full Film Lubrication 112.1 Surface roughness descriptions . . . . . . . . . . . . . . . . . . . . . 122.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.1 The Reynolds Equation . . . . . . . . . . . . . . . . . . . . . 142.2.2 The Film Thickness Equation . . . . . . . . . . . . . . . . . . 182.2.3 Lubricant compressibility . . . . . . . . . . . . . . . . . . . . 202.2.4 Lubricant Viscosity . . . . . . . . . . . . . . . . . . . . . . . . 212.2.5 The Force Balance Equation . . . . . . . . . . . . . . . . . . 212.3 Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 The present EHL model . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Dimensionless formulation . . . . . . . . . . . . . . . . . . . . 242.5 The present HL model . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Dimensionless formulation . . . . . . . . . . . . . . . . . . . . 293 The Contact Mechanical Problem 313.1 Statistical roughness models . . . . . . . . . . . . . . . . . . . . . . . 323.2 Deterministic roughness models . . . . . . . . . . . . . . . . . . . . . 323.3 Numerical solution techniques . . . . . . . . . . . . . . . . . . . . . . 333.4 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.5 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Dimensionless formulation . . . . . . . . . . . . . . . . . . . . . . . . 36vii4 Homogenization of Reynolds Equation 394.1 Multiple scales expansion - cartezian co-ordinates . . . . . . . . . . . 414.1.1 Stationary incompressible lubrication . . . . . . . . . . . . . . 424.1.2 Unstationary incompressible lubrication . . . . . . . . . . . . 464.1.3 Stationary compressible lubrication . . . . . . . . . . . . . . . 514.1.4 Unstationary compressible lubrication . . . . . . . . . . . . . 534.2 Multiple scales expansion - cylindrical co-ordinates . . . . . . . . . . 574.2.1 Stationary incompressible lubrication . . . . . . . . . . . . . . 584.2.2 Unstationary incompressible lubrication . . . . . . . . . . . . 624.2.3 Stationary compressible lubrication . . . . . . . . . . . . . . . 644.2.4 Unstationary compressible lubrication . . . . . . . . . . . . . 654.3 Two-scale convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3.1 Preliminaries and notation . . . . . . . . . . . . . . . . . . . 664.3.2 Homogenization of the Reynolds equation . . . . . . . . . . . 684.3.3 Properties of the homogenized matrix A . . . . . . . . . . . . 714.3.4 Corrector results . . . . . . . . . . . . . . . . . . . . . . . . . 724.3.5 Transversal and longitudinal roughness . . . . . . . . . . . . . 744.4 The technique of bounds - cartezian co-ordinates . . . . . . . . . . . 764.4.1 The governing Reynolds equation . . . . . . . . . . . . . . . . 764.4.2 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 774.4.3 Bounds of arithmetic-harmonic mean type . . . . . . . . . . . 804.4.4 Optimality of the A-H mean type bounds . . . . . . . . . . . 824.4.5 Bounds of Reuss-Voigt type . . . . . . . . . . . . . . . . . . . 834.5 The technique of bounds - cylindrical co-ordinates . . . . . . . . . . 854.5.1 The governing Reynolds equation . . . . . . . . . . . . . . . . 854.5.2 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . 864.5.3 Bounds of arithmetic-harmonic mean type . . . . . . . . . . . 864.5.4 Bounds of Reuss-Voigt type . . . . . . . . . . . . . . . . . . . 874.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 Surface Characterization 895.1 Fourier based ltering . . . . . . . . . . . . . . . . . . . . . . . . . . 896 Discretization and Numerical Solution Techniques 956.1 Numerics to be used in the eld of HL . . . . . . . . . . . . . . . . . 956.1.1 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . 966.1.2 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . 976.2 Numerics of the EHL line contact problem . . . . . . . . . . . . . . . 976.2.1 The Block-Jacobi method . . . . . . . . . . . . . . . . . . . . 986.2.2 A brief overview of the multilevel technique . . . . . . . . . . 1006.2.3 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . 1046.2.4 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . 1056.3 Numerics of the contact mechanical problem . . . . . . . . . . . . . . 1066.3.1 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . 1076.3.2 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . 1087 Verication of the EHL approach 1117.1 The CFD approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 CFD - Governing equations . . . . . . . . . . . . . . . . . . . . . . . 1127.3 The model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.3.1 Interpolation of solution data . . . . . . . . . . . . . . . . . . 1137.3.2 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 1147.4 The results of the comparison . . . . . . . . . . . . . . . . . . . . . . 1157.5 Discussion and concluding remarks . . . . . . . . . . . . . . . . . . . 1178 Simulations of Rough FL 1218.1 Hydrodynamic lubrication . . . . . . . . . . . . . . . . . . . . . . . . 1218.1.1 One rough surface . . . . . . . . . . . . . . . . . . . . . . . . 1228.1.2 Two rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . 1408.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 Elastohydrodynamic lubrication . . . . . . . . . . . . . . . . . . . . . 1518.2.1 The dierent overtaking situations . . . . . . . . . . . . . . . 1528.2.2 The Ridge-Ridge overtaking . . . . . . . . . . . . . . . . . . . 1538.2.3 The Ridge-Dent overtaking . . . . . . . . . . . . . . . . . . . 1548.2.4 The Dent-Dent overtaking . . . . . . . . . . . . . . . . . . . . 1548.2.5 The Dent-Ridge overtaking . . . . . . . . . . . . . . . . . . . 1548.2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1609 Simulation of Rough CM 1639.1 Varying the applied load . . . . . . . . . . . . . . . . . . . . . . . . . 1649.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16610 Concluding Remarks 16911 Future Work 173II Appended Papers 175A 177A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180A.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 181A.2.1 Boundary conditions and cavitation treatment . . . . . . . . 182A.3 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183A.3.1 The numerics for the CFD approach . . . . . . . . . . . . . . 183A.3.2 The numerics for the Reynolds approach . . . . . . . . . . . . 184A.3.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 185A.3.4 Interpolation of solution data . . . . . . . . . . . . . . . . . . 186A.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186A.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190B 195B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198B.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199B.2.2 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200B.2.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 201B.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 202B.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207C 211C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214C.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215C.3 Surface characterization . . . . . . . . . . . . . . . . . . . . . . . . . 216C.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217C.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220D 223D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226D.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 227D.3 Multiple Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228D.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230D.4.1 Discrete formulation . . . . . . . . . . . . . . . . . . . . . . . 230D.4.2 Convergence of pressure . . . . . . . . . . . . . . . . . . . . . 232D.4.3 Load carrying capacity . . . . . . . . . . . . . . . . . . . . . . 235D.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236E 239E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242E.2 The governing Reynolds type equations . . . . . . . . . . . . . . . . 242E.3 Homogenization (constant bulk modulus) . . . . . . . . . . . . . . . 244E.4 Homogenization in the incompressible case . . . . . . . . . . . . . . . 245E.5 Only one rough surface . . . . . . . . . . . . . . . . . . . . . . . . . . 248E.6 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249E.6.1 Incompressible case, L2 = L1 . . . . . . . . . . . . . . . . . . 249E.6.2 Incompressible case, L2 = 10L1. . . . . . . . . . . . . . . . . 250E.6.3 Constant bulk modulus, L2 = L1 . . . . . . . . . . . . . . . . 252E.6.4 Constant bulk modulus, L2 = 10L1 . . . . . . . . . . . . . . . 253E.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 255F 259F.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263F.2 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265F.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268F.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270F.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274G 275G.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279G.2 The governing Reynolds equation . . . . . . . . . . . . . . . . . . . . 280G.3 Homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281G.4 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283G.4.1 Bounds of arithmetic-harmonic mean type . . . . . . . . . . . 283G.4.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285G.4.3 Bounds of Reuss-Voigt type . . . . . . . . . . . . . . . . . . . 286G.5 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287G.6 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . 289G.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 294PrefaceThis thesis comprises the results from modeling and numerical simulations of bothlubricated and unlubricated contacts, that constitute tribological interfaces. Themain focus is on the inuence of surface topography on performance. The thesiswork was conducted at the Department of Applied Physics and Mechanical Engi-neering, the Division of Machine Elements, Lule University of Technology. Thescientic outcome is a number of publications that may be found in internationaljournals and in conference proceedings (with review procedure):[1] T. Almqvist, A. Almqvist, and R. Larsson. A comparison between compu-tational uid dynamic and Reynolds approaches for simulating transient ehlline contacts. Tribology International, 37:6169, 2004.[2] A. Almqvist and R. Larsson. The eect of two-sided roughness in rolling/slidingehl line contacts. In Proceedings of the 30th Leeds-Lyon Symposium on Tri-bology, Lyon, 2003.[3] A. Almqvist, F. Sahlin, and R. Larsson. An abbot curve based surfacecontact mechanics approach. In World Tribology Congress III, Washington,D.C., USA, Sep 2005.[4] F. Sahlin, A. Almqvist, S. B. Glavatskih, and R. Larsson. A cavitation al-gorithm for arbitrary lubricant compressibility. In World Tribology CongressIII, Washington, D.C., USA, Sep 2005.[5] A. Almqvist and J. Dasht. The homogenization process of the Reynoldsequation describing compressible ow. Tribology International, 39:9941002,2006.Some papers has been accepted for publication:[6] A. Almqvist, F. Sahlin, R. Larsson, and S. Glavatskih. On the dry elasto-plastic contact of nominally at surfaces. Accepted for publication in Tribol-ogy International, available online since 20 January 2006.[7] F. Sahlin, A. Almqvist, R. Larsson, and S. B. Glavatskih. A homogenizationmethod for developing rough surface ow factors in hydrodynamic lubrica-tion. Accepted for publication in Tribology International, 2005.[8] A. Almqvist, D. Lukkassen, A. Meidell and P. Wall. New concepts of homog-enization applied in rough surface hydrodynamic lubrication. Accepted forpublication in International Journal of Engineering Science, August 2006.xiiiSome has recently been submitted for publication:[9] A. Almqvist, R. Larsson, and P. Wall. The homogenization process of thetime dependent Reynolds equation describing compressible liquid ow. Re-search Report, No. 4, ISSN 1400-4003, Department of Mathematics, LuleUniversity of Technology, submitted for publication in Tribology International,2006.[10] A. Almqvist, E. K. Essel, L.-E. Persson, and P. Wall. Homogenization of theunstationary incompressible Reynolds equation. Submitted for publicationin Tribology International, May 2006.Of the 10 scientic contributions, 7 were specically chosen for this thesis, i.e.the papers [1, 2, 5, 6, 8, 9, 10].It should also be mentioned that the extremely well working co-operation be-tween the group in tribology and the group in applied mathematics at Lule Uni-versity of Technology, led to a rather extensive research report on the applicabilityof homogenization in hydrodynamic lubrication, i.e.[11] A. Almqvist,J. Dasht, S. Glavatskih, R. Larsson, P. Marklund, L.-E. Persson,F. Sahlin, and P. Wall Homogenization of the Reynolds equation. ResearchReport, No. 3, ISSN 1400-4003, Department of Mathematics, Lule Univer-sity of Technology, 2005.Sub-Division of Work inAppended PapersThis chapter describes the sub-division of work in the seven appended papers foundin Part II of this thesis.Paper A[1] T. Almqvist, A. Almqvist, and R. Larsson. A comparison between compu-tational uid dynamic and Reynolds approaches for simulating transient ehlline contacts. Tribology International, 37:6169, 2004.The development of the Reynolds equation based model and numerical solver wasperformed by A. Almqvist. T. Almqvist, who is the corresponding author, workedwith the modications of the CFD-software and performed the numerical simu-lations using it. A. Almqvist and T. Almqvist performed the analysis work andprepared the main part of the paper. R. Larsson is co-author.Paper B[2] A. Almqvist and R. Larsson. The eect of two-sided roughness in rolling/slidingehl line contacts. In Proceedings of the 30th Leeds-Lyon Symposium on Tri-bology, Lyon, 2003.The main part of the modeling and analysis was performed by A. Almqvist, thecorresponding author of this paper. R. Larsson is co-author and contributed withhis expertise.Paper C[6] A. Almqvist, F. Sahlin, R. Larsson, and S. Glavatskih. On the dry elasto-plastic contact of nominally at surfaces. Accepted for publication in Tribol-ogy International, available online since 20 January 2006.The modeling was performed by A. Almqvist, the corresponding author, and F.Sahlin. The numerical analysis was performed by A. Almqvist. R. Larsson and S.Glavatskih are co-authors.xvPaper D[5] A. Almqvist and J. Dasht. The homogenization process of the Reynoldsequation describing compressible ow. Tribology International, 39:9941002,2006.A. Almqvist and J. Dasht worked together with the modeling and the mathematicalanalysis; A. Almqvist, the corresponding author, developed the numerical techniqueand performed the numerical analysis.Paper E[10] A. Almqvist, E. K. Essel, L.-E. Persson, and P. Wall. Homogenization of theunstationary incompressible Reynolds equation. Submitted for publicationin Tribology International, May 2006.E. Essel is the corresponding author. The work was divided among the co-authorsas follows; the physical modeling is performed by A. Almqvist and P. Wall, mathe-matical analysis by P. Wall and E. Essel and the numerical analysis by A. Almqvist.L.-E. Persson is a co-author and contributed with his expertise.Paper F[9] A. Almqvist, R. Larsson, and P. Wall. The homogenization process of thetime dependent Reynolds equation describing compressible liquid ow. Re-search Report, No. 4, ISSN 1400-4003, Department of Mathematics, LuleUniversity of Technology, submitted for publication in Tribology International,2006.A. Almqvist, the corresponding author, worked with R. Larsson and in the modelingof the governing equation, helped P. Wall with the mathematical analysis, did thenumerical modeling and produced the numerical results. R. Larsson is co-author.Paper G[8] A. Almqvist, D. Lukkassen, A. Meidell and P. Wall. New concepts of homog-enization applied in rough surface hydrodynamic lubrication. Accepted forpublication in International Journal of Engineering Science, August 2006.P. Wall, D. Lukkassen and A. Meidell worked with the mathematical aspects con-nected to the arithmetic-harmonic type bounds. P. Wall and A. Almqvist derivedthe Reuss-Voigt bounds. A. Almqvist, the corresponding author, performed thenumerical modeling and analysis.Nomenclature Pressure-viscosity coecient Pa1 Dimensionless dynamic viscosity Dimensionless densityT Dimensionless step size in time st Step size in time sXi Dimensionless step size in Xi mxi Step size in spatial coordinate xi m Shear rate s1E Error measure

spatial Discretization error in space

time Discretization error in time

x Discretization error in space

y Discretization error in space Total potential complementary energyI Intergrid transfer operator Wavelength mL Functional operatoru Velocity eld, u = u(x, y, z) m s1x Spatial coordinates, x = (x, y, z) m Dynamic viscosity Pa s0 Dynamic viscosity at ambient pressure Pa s Poissons ratioxvii Integration domain 2D / 3D contact m / m2O(n) Mathematical order of the number n Solution variablei Topography of surface i mavg Mean surface height Density kg m30 Density at ambient pressure kg m3 Dimensionless stress function Total stress tensor PaUi Surface velocity, Ui = [u1i, u2i]Tm s1Um Mean surface velocity, Um = [u11 +u12, u21 +u22]T/2 m s1 Lubricant shear stress Pa0 Eyring stress Pa1 Lubricant shear stress at surface 1 Pam Lubricant midplane shear stress Paxixj Lubricant shear stress Paf Fourier transformation of f1 Dimensionless reduced wavelengthA Dent / Ridge Amplitude mb Hertzian half-width b = _(8wRx) / (E

) mC1 Constant C1 = 5.9 108C2 Constant C2 = 1.34d Elastic deformation mde Elastic deformation mdp Plastic deformation mE Modulus of elasticity PaE

Eective mod. of elast. 2/E

= (1 21)/E1 + (1 22)/E2 PaG Shear modulus of elasticity of the lubricant Pag Determinant of the metric tensorgs The gap between the undeformed surfaces mH Dimensionless lm thicknessh Film thickness mh0 Integration constant mHs Hardness of the softer material PaH00 Dimensionless integration constanthc Central lm thickness mhmin Minimum lm thickness mK Elastic deformation integral kernelL Moes dimensionless speed parameterli Amplitude variation of dent/ridgeL1 Discrete operator (Reynolds equation)L2 Discrete operator (Film thickness equation)Lxi Length parameter mM Moes dimensionless load parametermi Wavelength variation of dent/ridgeP Dimensionless pressurep Pressure PaP0 Constant in the viscosity expressionph Hertzian pressure ph = _(wE

) / (2Rx) Papmax Maximum pressure PaRa Average roughness mRi Radius of the surface i mRk KurtosisRq Root mean square (RMS) roughness mRx Reduced radius of curvature in x-dir. 1/Rx = 1/R11 + 1/R22 mRz Average maximum height mRsk SkewnessS Non-Newtionian slip factors Slide-to-roll ratio 2(u11u21)/(u11 +u21)T Dimensionless timet Time su Solution to the one-dimensional Poisson equationui Surface speed, ui = ui(x1, x2, x3) m s1us Sum of velocities us = u11 +u21 m s1W Applied load, 2D / 3D contact Pa m1/ Paxc Centre of dent / ridge xc (t) = xsu t mXi Dimensionless spatial coordinate mxi Spatial coordinate mxs Initial placement of dent / ridge mzvisc Pressure-viscosity indexxi Spatial frequency 1 / muij Speed of surface i in direction j m s1Part IThe Thesis1Chapter 1IntroductionMachines consist of machine elements and their safe and ecient operation relieson carefully designed interfaces between these elements. The functional design ofinterfaces covers geometry, materials, lubrication and surface topography, and anincorrect design may lead to both lowered eciency and shortened service life. Amisalignment due to the geometrical design could lead to large stress concentra-tions that in turn may lead to severe damage when mounting, a detrimental wearsituation, rapid fatigue during operation, etc. Large stress concentrations also im-plicitly imply a temperature rise because of the energy dissipation due to plasticdeformations. The choice of mating materials is also of great importance, e.g. elec-trolytic corrosion may drastically reduce service life. Contact fatigue due to lowductility would not only lower the service life but could lead to third body abrasiondue to spalling, which in turn could end up lowering the service life of other com-ponents. A lubricant serves several crucial objectives; when its main objective is tolower friction, the actions of additives are of concern. If the interface is subjectedto excessive wear, the lubricants ability to form a separating lm becomes evenmore crucial. In this case, the bulk properties of the lubricant have to be carefullychosen. At some scale, regardless of the surface nish, all real surfaces are roughand their topography inuences the contact condition.As implied above, these design parameters are mutually dependent, i.e. theyaect the way others inuence the operation of the system. For example, a changein geometry could require another choice of materials that may change the objec-tives of the lubricant and force the operation into another lubrication regime. Allfour design parameters are of great importance, though in this thesis work it is theeects induced by the surface topography that is of main interest.The inuence of surface roughness on performance has of course been investi-gated by many researchers in the eld, experimentally and numerically. However,because of the multidisciplinary nature of the eld and the complexity of the the-oretical models associated with tribological problems, the progress in the devel-opment of ecient, still user friendly software has not reached as far as in, e.g.,computational structural mechanics and computational uid dynamics. Moreover,the requirement on the density of the mesh to resolve not only the geometricalpart of the tribological contact but also the surface topography is dicult to meet.This thesis contributes to the eld of research through this connection. Namely,34 CHAPTER 1. INTRODUCTIONby development and implementation of rigorous theoretical models that allow foreective numerical treatment of some specically chosen rough tribological prob-lems. For example, within this thesis work, the outcome of an extremely successfulcollaboration has led to state-of-the-art results in applied mathematics [12] as wellas to a highly ecient numerical treatment of the Reynolds equation that incorpo-rates the eects induced by the surface roughness; for details, see Sections 4.4 and4.5.1.1 Lubrication regimesBecause of the mutual dependency of the design parameters as well as the change intheir objectives with dierent operating conditions, the surface topography inducedeects will vary.In tribology, lubrication is often used as target parameter, e.g. depending on theapplication and the operating conditions it is common to characterize the tribolog-ical contact by its lubrication regime. The lubricant regimes are often divided into:Boundary Lubrication (BL), Mixed Lubrication (ML) and Full Film Lubrication(FL).1.2 The boundary lubrication regimeIn the boundary lubrication (BL) regime, the lubricants hydrodynamic action isnegligible and the load is carried directly by surface asperities or by surface activeadditives (a so-called tribolm). Here, the surface topography is preferably chosento optimize the frictional behavior without increasing the rate of wear. To do this,one has to understand how the chemical processes are aected by the actual contactconditions, in terms of heat generation, pressure peaks, the real area of contact, etc.In this thesis this regime was modeled as the elasto-plastic, unlubricated contactbetween rough surfaces, i.e. as a contact mechanical problem. Within the eld,the well known elasto-plastic variational approach expressed as the minimizationof total complementary potential energy has been adopted and combined with thenumerical Fourier transform, known as fast Fourier transform (FFT), to ensure astable and eective numerical treatment of rough contact mechanics. This approachhelps to increase the understanding of how the surface roughness inuences theelastic deection, the plastic deformation (and plasticity index), the pressure buildup and the real area of contact. An in-depth understanding of this connection isrequired to rene the design of interfaces operating under these circumstances.As the hydrodynamical action of the lubricant increases, the contact mechanicalresponse becomes less severe in terms of pressure and real contact area, and atransition from the BL- to the ML- regime may therefore occur. The developedcontact mechanical tool is also applicable here and may be used to indicate thisinteresting transition, from a tribological angle of approach.1.3. THE MIXED LUBRICATION REGIME 51.3 The mixed lubrication regimeWhat characterizes the ML regime is that the load is carried by the lubricantshydrodynamical action, which may be inuenced by the elastic deection of thesurfaces, the tribolm, directly by surface asperities, or a combination thereof. Thismeans that the objectives of the surface topography are to support the hydrody-namic action of the lubricant, aid the elastic deection in rendering a smoothersurface, enable bonding of the surface active additives and optimize friction in thecontact spots without increasing wear.Direct modeling of mixed lubrication was not attempted during this Ph.D.project. However, as mentioned before, the contact mechanical approach maybe used to indicate a possible transition between the BL- and the ML-regimes.Similarly, modeling performed regarding full-lm lubrication has lead to numericalFL approaches that may be used to increase the understanding of the transitionfrom the FL- to the ML- regimes.1.4 The full-lm lubrication regimeWhen the hydrodynamic action of the lubricant fully separates the surfaces andthe load is carried totally by the lubricant lm, the contact enters the full lmlubrication (FL) regime. In the FL regime, traction may be reduced by carefullychosen topographies. Even though there is no direct contact, the topography mustalso prevent fatigue that may lead to excessive wear in the form of spalling in ahighly loaded contact.This regime is commonly sub-divided into hydrodynamic lubrication (HL) andelastohydrodynamic lubrication (EHL), since the performance is greatly aectedby the presence of elastic deections in the contact zone. In this thesis, the eectsinduced by the surface roughness in both these regimes have been studied.1.4.1 Hydrodynamic lubricationSlider bearings are typical examples of applications that, under certain conditionsoperate, in the hydrodynamic lubrication (HL) regime where the elastic deforma-tions of the bearing surfaces are suciently small to be neglected. For example, anaxial thrust pad bearing, as depicted in Fig. 1.1, consists of a conformal contactthat under certain circumstances may be assumed to operate in the hydrodynamiclubrication regime. Note that the angle of inclination of the pads, which is generallyonly a fraction of a degree, has been exaggerated in the gure. The problems thatarise when modeling these types of contacts are the large dierences in scales, i.e.That is, the global scale of the contact describing the geometry is several orders ofmagnitude larger than the local scale describing the surface topography/roughness.For this purpose a research group in applied mathematics with specializing inhomogenization was contacted and a cooperation was established. The outcomefrom this fruitful collaborative work was a variety of mathematically rigorous mod-els to be applied in the eld of hydrodynamic lubrication where the uid ow maybe governed by the Reynolds equation. This lead to highly eective numerical6 CHAPTER 1. INTRODUCTIONFigure 1.1: Schematics of an axial thrust pad bearingtools where the inuences of surface roughness are embedded in the derived ho-mogenized equations. More over, the equations are unambiguously determined andshows to naturally allow for parallelization. These tools enables studies of roughsurface hydrodynamically lubricated problems such as that arising in the bearingconguration visualized in Fig. 1.1. This means that the theoretical model concernsdierent types of the unstationary Reynolds equation in two dimensions.1.4.2 Elastohydrodynamic lubricationElastohydrodynamic lubrication (EHL) is the type of hydrodynamic lubricationwhere the elastic deformations of the contacting surfaces cannot be neglected. Thisis often the case in non-conformal (concentrated) contacts. For example, the con-tact between the roller and the raceway in a typical roller bearing, as shown inFig. 1.2, are most commonly designed to operate in the full-lm elastohydrody-namic lubrication regime. Note that the elastic deection is the key ingredientFigure 1.2: Schematics of a typical rolling element bearingin the lubrication process of this type of application. It is the highly compressed1.5. OBJECTIVES 7and extremely high viscous hydrodynamic action of the lubricant combined withthe elastic smoothening of the in-contact surface roughness that allows for a thinlubricating lm in the contacting interface.The actual contact zone for a rolling bearing is, in general, elliptic in shape.Depending on the design parameters previously mentioned and the actual runningconditions, the shape of the ellipse will change as shown in Fig. 1.3. In any case, thecontact region is small and the concentrated load implies a severe stress conditionthat will lead to a large elastic deection and possibly also plastic deformation.For a bearing in operation, it is the large elastic deection that causes fatigue,which in turn can lead to shortened service life due to, for example, spalling.When the contact is starved of lubricant, or when running conditions do not allowfor a hydrodynamic action that fully separates the surfaces, the risk for plasticdeformation increases.x1x2Roller widthFigure 1.3: Elliptic contacts _(x1/a)2+ (x2/b)2= 1_If b exceeds the minimum width of the raceway and the roller, the contact willbe then truncated and possibly lead to increased stresses in the material, at leastfor the unlubricated contact. In the case of a contact with b/a > 4, though withb still less than the minimum width of the raceway and the roller, the centerlinein the rolling direction can be approximated to a line contact, Evans et al. [13].This has motivated the choice of the theoretical unstationary, one-dimensional,elastohydrodynamical model adopted here. As with most tribological problems,this is numerically a very demanding problem that requires highly ecient solutionmethods. A numerical multilevel technique was adopted to remedy this problem.One of the drawbacks is the complexity of the tool, which unfortunately requiresthe skills of an engineer with a rather extensive theoretical background. Withthis numerical approach, the goal was to develop a coupled solver that solves theReynolds equation and the lm thickness equation simultaneously, and to developthe multilevel technique to also embrace the two-dimensional operators that arisedue to the coupling of the equations.1.5 ObjectivesThis work concerns the modeling of rough surface contact interfaces, with or with-out a separating lubricant in between. The main objective has been to investigate8 CHAPTER 1. INTRODUCTIONhow the surface topography inuences dierent contact conditions by means of nu-merical simulations. The long term goal is to develop engineering tools, i.e. toolswith low complexity, that enable ecient analysis and prediction of dierent con-tact conditions by considering the surface topography. There is also an eort increating a unied approach to be used for as a wide a variety of tribological prob-lems as possible. For this purpose the following specications of three dierentapproaches that may be used to increase the fundamental knowledge about surfaceroughness induced eects within tribology was postulated- To develop a robust yet still ecient rough surface solver to simulate elasto-hydrodynamic lubrication.- To develop a contact mechanical tool to study the contact between roughsurfaces.- To develop a tool to analyze hydrodynamically lubricated problems involvingrough surfaces.1.6 Outline of this thesisThe thesis found in Part I is self contained, with Part II comprising the workpresented in the papers A to G found in Part II. The content of the papers foundin Part II has in Part I been extended with more complete theoretical descriptionsand derivations. Some material not directly addressed in the papers has also beenincluded for completeness. Chapter 4, concerning homogenization of the Reynoldsequation, has been extended to also embrace the cylindrical form of the Reynoldsequation. This material has not yet been submitted to any journal, since thework was carried out simultaneously as this thesis was being nalized. However,this material will most likely be submitted if the author is granted the nancialsupport needed to continue his work in the eld.Chapter 2 of this thesis introduces the topic of full lm lubrication (FL). Thisis also the theoretical basis of Reynolds equation based approaches found in all thepapers presented in Part II. This chapter includes surface roughness modeling, i.e.the deterministic and the statistical way of dealing with surface roughness. Forclarity, this includes a derivation of the non-Newtonian Reynolds-Eyring equation(Conry et al. [14]) found in Section 2.2. In this section, the lm thickness equa-tion, pressure-density relations describing compressibility and pressure-viscosityrelations are given. Boundary conditions, dierent ways of modeling cavitationand the force-balance equation are also described here. Section 2.4 comprises theequations used in this thesis for modeling elastohydrodynamically lubricated prob-lems, while Section 2.5 comprises the equations used in this thesis for modelingpurely hydrodynamically lubricated problems.The theory that constitutes the basis for the dry elasto-plastic contact method,and thus Paper C, is found in Chapter 3 together with an introduction to the con-tact mechanics problem. Dierent models of the elastic contact are then discussedin connection to the present model. A description of how spectral analysis canbe used to determine the elastic deformation integral follows. Both the dimen-1.6. OUTLINE OF THIS THESIS 9sionless and discrete formulation and the numerical solution process for the dryelasto-plastic method are also provided.Chapter 4 deals theoretically with the application of homogenization techniqueson hydrodynamically lubricated problems governed by the Reynolds equation.Here, the formal method of multiple scales expansion is applied on the Reynoldsequation, which is also the theoretical basis of papers D, E and F. Two-scaleconvergence of this problem is also proven here. Moreover, the novel techniqueof nding bounds on the homogenized properties of the Reynolds problem is alsoaddressed. This forms the theoretical basis of Paper G.Simulating either rough FL or contact mechanics denitely requires some formof surface characterization as one of the preliminary processes. Chapter 5 addressesthis area, which is also a part of Paper C. Section 5.1 shows that the interpretationof the acquired data when using a deterministic model may be facilitated by makinguse of a Fourier based ltering technique. The technique is based on truncatedFourier series and is being applied to a sample topography.Chapter 6 describes discretization and numerical solution techniques for the dif-ferent types of models considered here. This includes a description of the discretepartial dierential equation that was implemented to solve the hydrodynamicallylubricated problem associated with the problems that have been homogenized (Sec-tion 6.1). This numerical solution technique was used in papers D, E, F and G.From papers A and B, the numerical Block-Jacobi method used to solve the cou-pled system, which arises when modeling EHL, consisting of the Reynolds equationand the lm thickness equation is found in Section 6.2. A multilevel technique usedto facilitate the numerical solution process of the one dimensional Poisson problemand a possible modication to facilitate the numerical solution of coupled systemsare then briey described. Section 6.3 describes the numerical contact mechanicalapproach, used in Paper C, which is a development of an existing numerical elasticmodel was modied to also account for plastic deformations, is described.In Chapter 7, which contains the results of Paper A, the Reynolds based ap-proach described in Section 2.4 is compared with a Computational Fluid Dynamics(CFD) based approach. This comparison consist of simulations of one stationaryand two dierent transient problems. The results are encouraging from severalviewpoints: verication of the codes, the possibilities to further develop the CFDapproach given by Almqvist and Larsson [15], and the justication of using aReynolds approach under the running conditions chosen.The topic of surface roughness induced eects that occur in the full lm lubri-cation regime is addressed in Chapter 8, which contains the results of Paper B.The results obtained using the theoretical models, described in Chapter 2, and thenumerical solution methods, described in 6, are presented.In Paper C the developed rough surface contact mechanics approach developedsimulates the contact mechanics of four dierent topographies. This is also thecontent of Chapter 9. The simulations are here restricted to two dimensions andproles taken from the measurements of four surfaces form the basis of the study.The content of the second part of this thesis, part II, is simply papers A, B, C,D, E, F and G, which form the scientic basis of the thesis.Chapter 2Modeling Full Film LubricationFactors such as load carrying capacity, traction, service life, etc., are determiningwhen designing any type of bearing. The performance of a thin lm lubricatedbearing is certainly aected by the operating conditions, the choice of lubricant,the thermal properties of the bearing surface material and the surface macro andmicro topography. As previously mentioned, the surface roughness is of maininterest here. The main reason to model full lm lubrication is of course that fullscale testing of machine elements, such as bearings, cam mechanism, gear boxes,etc., is very expensive and that modeling permits possible numerical simulationsthat reduce the cost. Scientically, modeling is of great importance, since it allowsthe gain of fundamental knowledge on the dierent acting mechanisms that controlthe operation of applications such as those previously mentioned. This due to thesimplicity in separating the eects caused by dierent parameters.In this chapter, the dierent adopted surface roughness/topography models areconsidered rst, followed by the previous work done in Section 2.1. The determin-istic model used in the EHL approach developed is here of concern. The need ofaveraging techniques, statistical models and the present HL approach based on theideas in homogenization are also discussed.The theoretical basis of the full lm lubrication models are described in Sec-tion 2.2. A derivation of a two-dimensional non-Newtonian Reynolds-Eyring partialdierential equation (PDE) in accordance with the work performed by Conry etal. [14] is given in Section 2.2.1, followed by the lm thickness equation in Sec-tion 2.2.2. This section also describes the elastic deection equation of the surfaces.Section 2.2.3 discusses the compressibility of the lubricant inside the conjunctionand describes two widely dierent ways of modeling density as a function of pressureare described. That is, one that may be used when modeling elastohydrodynamiclubrication (EHL) and one that may signicantly reduce the computational bur-den if used when modeling certain hydrodynamically lubricated (HL) applications,such as slider bearings. The lubricant Rheology is considered when deriving theReynolds-Eyring equation, though the viscosity of the lubricant and how it relatesto pressure is discussed separately in Section 2.2.4. Criteria on force-balance isdescribed in Section 2.2.5.The integro-dierential problem for the isothermal EHL line contact is consid-ered next. Basically, the present approach is an eort to consider the coupling1112 CHAPTER 2. MODELING FULL FILM LUBRICATIONbetween the governing equation for uid pressure (the Reynolds-Eyring equation)and the lm thickness equation that includes the elastic deection determined bythe uid pressure. The full EHL line contact problems consist of solving this cou-pled system, an equation for force balance and two semi-empirical equations for theuid density and the uid viscosity. The particular equations used for the presentEHL approach are presented in this section. Another objective set for the modelingof applications operating in the EHL regime is to adapt the multilevel techniques,such as the ones described by Venner and Lubrecht in [16], to facilitate the solutionprocess in solving rough EHL problems.Section 2.5 provides specic details on the governing equations used for thepresent HL approach discussed in Chapter 4. This includes, for example, infor-mation on lm thickness modeling, compressibility, the topic of force balance andsuitable non-dimensional forms.2.1 Surface roughness descriptionsThere are many ways to model surface topographies. Mathematically, the descrip-tions are either deterministic or statistic.In many theoretical studies of Full Film Hydrodynamic Lubrication (FL), usingdeterministic models, the one surface is considered smooth and one is consideredrough. This is a suitable approximation when modeling a rolling contact or acontact where the roughness of the moving surface is of minor importance. Forexample, in a pure rolling contact, the roughness of the two contacting surfacesmay be summed and the contact modeled as having an eective roughness and aperfectly smooth surface, an approach here referred to as one-sided roughness.When using a deterministic surface roughness description to study a lubricatedcontact that is subjected to sliding, the surface topographies of both surfaces areof signicance and a continuously changing eective surface roughness occurs. Theone-sided roughness approach is not valid in such situations and a two-sided rough-ness treatment is needed. Of course, the two-sided surface roughness treatmenthas been previously investigated, e.g. Evans et al., Tau et al., Chang, Venner andMorales-Espejel, Hooke [13, 17, 18, 19, 20].The highly detailed outcome of a deterministic approach in terms of local pres-sure, elastic deection, traction, etc., gives us a fundamental understanding of thedierent acting mechanisms controlling the performance of the lubricated contactand thus the performance of the machine element. In summary, whenever local in-formation about surface roughness induced eects is needed a deterministic modelis best suited. Both one-sided and two-sided roughness approaches have been ad-dressed in this thesis with the ndings presented in Chapter 8.However, a signicant amount of number of freedoms is introduced by the sur-face topography, i.e. the small scale surface features need to be resolved numeri-cally, requiring a very dense numerical grid. This implies signicant memory con-sumption and time consuming computations. One way to remedy this problem isto utilize statistical theory. The subject being considered in this thesis as a randomprocess is, of course, the surface roughness and it is the equation governing the uidow, here the Reynolds equation, that is being analyzed with the statistical tech-nique. The coecients determining the Reynolds equation are dependent on the2.1. SURFACE ROUGHNESS DESCRIPTIONS 13surface roughness through the lm thickness function; see Section 2.2. This meansthat these rapidly oscillating coecients in turn suggest some type of averaging.Applying a statistical method to analyze the Reynolds equation means losingsome local information to some extent and that the eects of surface topographywill be statistically inherited in the averaged determining equation. Christensen[21], utilizing statistical analysis, was able to derive averaged Reynolds equationswith the restriction that the roughness be either transversally- or longitudinally-oriented. Patir and Cheng [22], [23], [24] introduced the concept of ow-factorsthat were used as corrections of the corresponding smooth lm thickness so that theactual surface topography would be considered. Such an approach is not restrictedto the transversal and longitudinal cases and may be used with realistic threedimensional surface topographies. These ow-factors, which are the coecientsof the modied Reynolds type equation dened on a global (geometry) scale, areobtained by solving problems dened on a local (roughness) scale. This meansthat when the ow-factors for a specic surface roughness description have beenobtained, the numerical solution process of the modied Reynolds type equation isas fast as the corresponding smooth problem. This strongly justies of the use ofthis type of approach. The drawback of an approach based on the ndings of Patirand Cheng is the existing ambiguities in determining the ow-factors. For example,rigorous theoretical explanation on how many times the local problems should besolved to obtain the statistically correct result does not exist. Another ambiguityis how to truncate the solution of the local problem to reduce the inuence of theboundary conditions, Harp and Salant [25].Homogenization is a mathematical research area that includes studying PDEswith rapidly oscillating coecients. As mentioned above, the lm thickness func-tion that include the surface roughness is part of the coecients in the ReynoldsPDE. This suggests that the ideas in the area of homogenization are suitable andthat homogenization may be used to eciently average surface roughness inducedeects. Although being a well known technique mathematically, see e.g. Bensous-san et al., Oleinik et al., Braides and Defranceschi, Cioranescu and Donato, Pankov[26, 27, 28, 29, 30], homogenization has until recently not been used frequently inthe eld of tribology. For work considering the application of homogenization to hy-drodynamically lubricated problems, see e.g. Elrod, Bayada and Faure, Jai, Bayadaet al., Buscaglia and Jai, Buscaglia and Jai, Buscaglia et al., Buscaglia and Jai,Kane and Bou-Said, Kane and Bou-Said, Kane and Bou-Said, Wall, Sahlin et al.[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 7]. The rigorousness of this approach isvery appealing, with Chapter 8 summarizing the promising results in the aspect oflubrication. In comparison to the well-known method of Patir and Cheng, homoge-nization techniques also provide the possibility to study realistic three dimensionalsurface topographies. One major remark is that the homogenization process con-tains no ambiguities in determining the coecients of the modied Reynolds typeequation, in this case referred to as the homogenized Reynolds equation. Fur-ther, within this thesis, novel homogenization ideas are applied to the Reynoldsequation that lead to a new homogenization technique, making it possible for theengineer to perform systematic studies on the eects of realistic two- and three-dimensional surface roughness representations on the lubrication performance ofmachine elements, such as dierent types of slider bearings, Almqvist et al. [8].14 CHAPTER 2. MODELING FULL FILM LUBRICATION2.2 Governing equationsThis section explains the equations that govern the lubrication process of contactspossibly found in tribological applications such as rolling element bearings andslider bearings.2.2.1 The Reynolds EquationA modied Reynolds equation, based on on the Eyring theory of non-Newtonianow, is derived in one dimension. Johnson and Tevaarwerk [43] proposed thenonlinear constitutive equation for a lubricant under isothermal conditions givenby Eq. (2.2.1), = 1Gddt + 0 sinh_ 0_, (2.2.1)where G is the shear modulus of elasticity of the uid, is uid shear stress, is thedynamic viscosity, and 0 is the Eyring shear stress. This is a Maxwell rheologicalmodel where the total shear strain rate is the sum of an elastic term and a nonlinearviscous term based on the Eyrings theory of viscosity. The modied Reynoldsequation is derived from the Eyring equation, the nonlinear viscous portion ofEq. (2.2.1), and under the assumption of plain strain rate. Fig. 2.1 shows a uidx3

x31(x1, x2, 0)(x1 = C, x2, x3

= 0)u1(x1 = C, x2, x3

)x3x2x1Figure 2.1: A description of the contact regionelement in the thin lubricating lm between two solids. In this case the equationsof equilibrium in the are x1- and x2- direction take the forms:x1x3x3= px1, x2x3x3= px2, (2.2.2)where p is uid pressure. If the lm thickness is denoted by h(x), then accordingto Fig. 2.1, 0 x3

h(x) and x3

= x3x31. Assuming p = p (x), integration ofEq. (2.2.2) with respect to x3

yields:x1x3 = 11 +x3

px1, x2x3 = 21 +x3

px2, (2.2.3)2.2. GOVERNING EQUATIONS 15where i1 are the shear stress acting on surface 1. Substituting Eq. (2.2.3) into theconstitutive equation ( i = (0/) sinh(xix3/0)) yields: i = uix3= 0 sinh_i1 +x3

pxi0_. (2.2.4)Assuming that the velocity of the lower surface (x3

= 0) is U1 = [u11, u21]Tandthat the viscosity does not vary across the lm ,i.e., = (x1, x2), integration ofEq. (2.2.4) with respect to x3

gives the following expression for the velocity prolein the xi-direction:ui (x, x3

) = ui1 +_ x3

00 sinh_i1 +s pxi0_ds, (2.2.5)here x = (x1, x2) for simplicity, which after evaluation of the integral expressionbecomes:ui (x, x3

) = ui1 + 20 pxi_cosh_i1 +x3

pxi0_cosh_i10__. (2.2.6)Introduction of the mid plane shear stress:im = i1 + h2pxi(2.2.7)and a dimensionless function, dened as:i = h20pxi(2.2.8)makes it possible to rewrite Eq. (2.2.6) as:ui (x, x3

) =ui1 + 0h21i_cosh_im0i_1 2x3

h__ cosh_im0i__.(2.2.9)Application of the boundary condition ui (x, h) = ui2 and utilizing hyperbolicrelations gives:ui2 = ui1+0h2i_cosh_im0+ i_cosh_im0i__ =ui1 + 0hisinh_im0_sinh i.(2.2.10)Eq. (2.2.10) can be rearranged to allow for determination of the midplane shearstress as follows:sinh_im0_ = (ui2ui1)0hisinh i. (2.2.11)16 CHAPTER 2. MODELING FULL FILM LUBRICATIONThe mass ux per unit width, Mi (x), is dened as:Mi (x) =_ h0ui (x, x3

) dx3

. (2.2.12)After substitution of Eq. (2.2.9) into Eq. (2.2.12) and integration, the expressionfor mass ux per unit width becomes:Mi (x) = ui1h + 0h2i_ h2i_sinh_im0+ i_sinh_im0i__(2.2.13)hcosh_im0i__.Expanding and rearranging the terms in the brackets of Eq. (2.2.13) givesMi (x) = ui1h +0h22_sinh iisinh_im0_+(2.2.14)sinh ii coshi2icosh_im0__.Substitution of Eq. (2.2.11) into Eq. (2.2.14) yields:Mi (x) = (ui1 +ui2)2 h +0h22sinh ii cosh i2icosh_im0_= (ui1 +ui2)2 h +(2.2.15)0h22 isinh ii coshi3icosh_im0_= (ui1 +ui2)2 h +h312Si(x) pxi,whereSi(x) =_3 (sinh ii cosh i)3icosh_im0__The hyperbolic relation cosh2(x) sinh2(x) = 1 together with Eq. (2.2.11) maynow be used to yield the following expression for Si(x), referred to as the non-Newtonian slip factorSi (x) = 3 (i coshisinh i)3i1 +_(ui2ui1)0hisinh i_2. (2.2.16)2.2. GOVERNING EQUATIONS 17The equation of continuity yields2

i=1xi_ h(x)0ui (x, x3

) dx3

= 0, (2.2.17)which could also be expressed in terms of the mass ux per unit width as2

i=1xiMi (x) = 0. (2.2.18)Substituting Eq. (2.2.15) into Eq. (2.2.18) and using the expression of the non-Newtonian slip factor given by Eq. (2.2.16) nally yields the stationary two-dimensionalReynolds-Eyring equation ((h) Um) _h312Sp_ = 0, (2.2.19)where = [/x1, /x2]T,S =S1 00 S2andUm = [(u11 +u12) /2, (u21 +u22) /2]T.The representative lubricant stress 0, characterizes the transition from New-tonian to non-Newtonian uid behavior. When using the Eyring model an in-nitely large 0 characterizes a Newtonian uid, and by using LHospital rule itcan be shown that Si approaches unity. In this case Eq. (2.2.19) reduces to theconventional Reynolds equation, viz. ((h) Um) _h312p_ = 0, (2.2.20)governing Newtonian thin lm ow. The value of Si is always equal to or greaterthan one, which means that the eective viscosity /Si is always less or equal tothe bulk viscosity of the lubricant a.It is possible to re-state Eq. (2.2.19) (and, of course Eq. (2.2.20)) in unstationaryform, incorporating possible squeeze eects, according tot (h) = _h312Sp_ ((h) Um) . (2.2.21)Similarly the corresponding unstationary but non-Newtonian Reynolds equationyieldst (h) = _h312p_ ((h) Um) . (2.2.22)18 CHAPTER 2. MODELING FULL FILM LUBRICATIONh(x, t)x3x2x1Figure 2.2: A graphical illustration of the the lm thickness function h(x, t), whichis the gap between the upper red surface and the lower blue surface.2.2.2 The Film Thickness EquationA common way of modeling the lm thickness h(x, t) in lubrication ish(x, t) = h0 (x, t) +h2 (x, t) h1 (x, t) +de (x, t) , (2.2.23)see Fig. 2.2 for a graphical representation, where h0 (x, t) describes the lm thick-ness on the global scale including the geometry of the bearing, de (x, t) models theelastic deection and hi (x, t) describes the lm thickness on the local scale (thesurface topography/roughness). A simple slider bearing may be found in Fig. 2.3that illustrates the function h0(x, t) at a specic point in time. The functionsh0(x, t)1x3x1x2Figure 2.3: An illustration of the function h0(x, t) modeling a simple slider bearing.hi(x, t) may be originating from a surface topography measurement, see Fig. 2.4.Fig. 2.5 illustrates an undeformed rough slider bearing having the surface rough-ness descriptions (hi(x, t)) shown in Fig. 2.4 added to the geometrical descriptionof the simple bearing (h0(x, t)) found in Fig. 2.3.In general, the function h0(x, t) may be divided into two parts, i.e.,h0 (x, t) = h00 (t) +hg (x, t) , (2.2.24)where h00 models the global separation of the two bearing surfaces and hg modelsthe geometry. A time-dependent description of the geometrical part hg may be2.2. GOVERNING EQUATIONS 19h1(x, t)h2(x, t)Figure 2.4: The functions h1(x, t) (left) and h2(x, t) (right) exemplied.h0(x, t) +h2(x, t)h1(x, t)1x3x1x2Figure 2.5: An illustration of a rough slider bearing model.needed, e.g. when simulating a pivoted thrust pad bearing. This depends on themounting of the pad and if the angle of pad inclination needs to be determined bya force balance- a momentum balance- condition or both. Note that h00 will alsobe present, in the form of a constant, when simulating a stationary case.The elastic deection may be modeled using the Boussinesq approach. Thatis, the elastic deection at a specic point relates to the pressure at every pointwithin the domain throughde (x) =_ K (x s) p (s) ds +const. (2.2.25)where the integral kernel K is given byK (x s) = 4E

ln [x s[ , (2.2.26)andK (x1s1, x2s2) = 2E

1_(x1s1)2+ (x2s2)2, (2.2.27)for two- and three- dimensional problems respectively, and where the eective mod-ulus of elasticity E

is given by2E

= 1 21E1+ 1 22E2.20 CHAPTER 2. MODELING FULL FILM LUBRICATION2.2.3 Lubricant compressibilityIn this section some, for tribological applications, suitable compressibility ap-proaches, such as the Dowson-Higginson [44] and the model based on the lubricantbulk modulus, see e.g. Vijayaraghavan [45].The Dowson-Higginson, semi-empirical expression, for iso-thermal lubrication,yields (p) = aC1 +C2 (p pa)C1 + (p pa) , (2.2.28)where the constants C1 and C2 may be used to t this rational expression toexperimentally obtained data.Modeling the lubricant compressibility via a constant bulk modulus , for whichthe denition yields = p, (2.2.29)is another type of approach. It should be noted that, from a mathematical pointof view, this is an interesting transformation. Here the bulk modulus may betted to experimentally obtained data. This implies that the relation between thedensity and the pressure is of the form(p) = ae(ppa)/. (2.2.30)The exponential form restricts the usage to applications that are subjected to rea-sonably low hydrodynamic pressures, implying that this expression can not beused to simulate elastohydrodynamic lubrication. However, due to its mathemat-ical form it leads to the possibility to restate the Reynolds equation that governsNewtonian (S 1), iso-thermal (contact temperature does not change due to thehydrodynamic ow) and iso-viscous ( = const.) uid ow in a very simple form.Indeed, in dening the dimensionless density function w as w(x, t) = (p(x, t))/a,thenw = 1a

(p)p = 1e(ppa)/p = 1wp.In this case, the unstationary non-Newtonian Reynolds equation Eq. (2.2.22), as-suming that the velocity of the surfaces coincide with the x1-direction, is convertedto a linear equation expressed as t (wh) = _h3w_ x1(wh) , (2.2.31)where = 12a/, = 6aus/,and us = u11 +u12.Even though this linear PDE has restrictions, the particular form is of impor-tance. For example, when considering eects of internal cavitation as demonstratedin Section 2.3.2.3. CAVITATION 212.2.4 Lubricant ViscosityThere are several dierent ways to model viscosity. In some cases it is possible tomodel the viscosity as a constant, i.e. the iso-viscous case = const. When thecontact pressure is reasonably high (the limit for a specic uid may be foundthrough an experimental analysis) a pressure dependent viscosity model is needed.One such semi-empiric relation is given by the Barus equation(p) = a exp (p) , (2.2.32)There are also other semi-empiric expressions relating viscosity and pressure. Whenmodeling EHL in this thesis, the Roelands expression given by(p) = a exp_P0zvisc_1 +_1 + pP0_zvisc__, (2.2.33)has been used. The parameters , P0 and zvisc are mutually dependent, accordingtoP0zvisc= ln (a) + 9.67. (2.2.34)2.2.5 The Force Balance EquationTo enable simulation of applications, such as rolling element bearings or pivotedthrust pad bearings, some kind of force balance criterion and maybe also a mo-mentum balance criterion must be considered. This means that the dependentvariables of the modeled problem becomes h00(t) and p(x, t), i.e. a measurement ofthe response, from an applied load, in terms of lm thickness and pressure. As anexample, when modeling a rolling element bearing, Eq. (2.2.21) and a force balanceequation, i.e. Newtons second law in the x3 direction_p (x, t) dx = W (t) , (2.2.35)where W (t) is the applied load, needs to be satised simultaneously in order toretrieve the physically correct pressure distribution p (x, t). When modeling a piv-oted tilted pad bearing one needs also to incorporate momentum balance of thepad if the angle of pad inclination is being an unknown. It should be remarked thata force balance criterion as well as a momentum balance criterion may be neededstudying a stationary problem such as a pivoted thrust pad bearing.In other cases, it is possible to keep h00(t), constant during the numericalsimulation, and measure the response, due to disturbances induced for exampleby the surface roughness, in terms of a varying load carrying capacity (W(t)) andpressure (p(x, t)). This type of approach, of course, the advantage of being lesstime consuming when utilized in the numerical solution process.2.3 CavitationIn lubricated problems cavitation is an important phenomena. Cavitation occurswhen the lubricant is subjected to a rapid pressure drop. This might be induced22 CHAPTER 2. MODELING FULL FILM LUBRICATIONby geometry, of the specic application being modeled, related issues but somestudies also indicate that cavitation occurs on the surface roughness or at leastsurface texture scales, see e.g. Ryk et al, Etsion and Halperin or Brizmer et al.[46, 47, 48].One way to model cavitation using Reynolds type equations is by introduc-ing a so called switch function, see e.g. Elrod [49] and Vijayaraghavan and Keith[45, 50, 45, 51]. These approaches takes into account internal cavitation (cavita-tion that occurs inside the lubricant conjunction) and preserves continuity through-out the lubricated conjunction which may be of great importance when studyingcertain types of bearings, for example a journal bearing. In accordance with Vija-yaraghavan and Keith, Eq. (2.2.31) becomes t (wh) = _h3g(w)w_ x1(wh) , (2.3.1)where the switch function g(w) is dened asg(w) =_ 1, w 10, w < 1 (2.3.2)This type of approach is based on the theory that there are no pressure drivenow within the cavitated region, where the switch function becomes zero and thusterminates the pressure gradient in the Reynolds equation which then becomes t (wh) + x1(wh) = 0, (2.3.3)and the solution w is determined solely by the shape of the lm thickness functionh. This means that the point of reformation occurs when the h is small enough toproduce a w = 1.Since the governing Reynolds type equation for modeling of EHL applicationssuch as a rolling element bearing is by default a non-linear partial dierentialequation it becomes extremely computationally complex to incorporate this typeof switch function. Fortunately, when modeling EHL it has been observed thatinternal cavitation does not occur that frequently, at least for the surface rough-nesses that are possible to resolve numerically (10 millions degrees of freedom). Forapplications operating in the EHL regime it is common to incorporate a cavitationcriterion that becomes activated in the outlet region only. This yields a cavitationcriterionp pc, (2.3.4)where pc is the pressure at which cavitation rst occurs. This condition is beingused to ensure that all pressures smaller than pc obtained during the solutionprocess are removed. Remark that by imposing this condition on Eq. (2.2.21) itis not possible to handle internal cavitation as in the case when using Eq. (2.3.1).This is since mass continuity is ruined by removing all negative pressures.2.4 The present EHL modelIt is Eq. (2.2.21) combined with the equation for the lm thickness, an equationof force balance, and two semi-empiric equations for the density and the viscos-2.4. THE PRESENT EHL MODEL 23ity respectively, constitutes the basis of the EHD lubricated line contact problemstudied in this work.The Reynolds-Eyring equation that governs the line contact yieldst (h) = x_h312Spx_ x ((h) um) on I T, (2.4.1)with homogeneous Dirichlet conditions at boundaries, i.e. assuming that the pres-sure is zero at the boundaries. This assumption is justied by the fact that thesurrounding ambient pressure in for example a rolling element bearing is so smallcompared to the pressure developed inside the lubricated conjunction that it maybe neglected. um = (u1 +u2) /2, also note that here, ui represents the speed ofsurface i in the rolling direction (x). I is the spatial solution domain or interval,i.e. a x b and T is the solution domain in time. Note that a needs to bechosen so that no numerical truncation of the ow, i.e. numerical starvation, isobserved and that b has to be chosen so that the starting point of cavitation, dueto the rapidly divergent surfaces at the exit, lies within the solution domain.The non-Newtonian slip factor S is given byS (x) = 3 (cosh sinh )31 +_(u2 u1)0hsinh _2, (2.4.2)where = h20dpdx. (2.4.3)The equation that describes the lm thickness may be stated ash(x, t) = h00 (t) + x22Rx+de (x, t) +h2 (x, t) h1 (x, t) , (2.4.4)where Rx is the reduced radius of curvature in the x-direction given by 1/Rx =1/Rx1 +1/Rx2, de is the elastic deformation of the contacting solids and hi repre-sents the topography of surface i. Note that hg(x, t) = x2/ (2Rx) in this case.In Fig. 2.6 the EHL line contact region is illustrated together with the corre-sponding reduced geometry employed in the theoretical analysis. In one-dimension,the elastic deformation is given by the one-dimensional version of Eq. (2.2.25), viz.de (x, t) = 4E

_Iln [x s[ p (s, t) ds,where E

is the eective modulus of elasticity. As implicitly stated here, by eval-uating the above integral over I and not over R2, it is crucial to choose I so thatthe elastic deection at the boundaries is small enough to be neglected.For the present EHL approach, the Roelands expression given by Eq. (2.2.33)and Eq. (2.2.34) has been used, viz.(p) = a exp_P0zvisc_1 +_1 + pP0_zvisc__,P0zvisc= ln (a) + 9.67,24 CHAPTER 2. MODELING FULL FILM LUBRICATIONu1u2x3x1Rx1Rx2E1E2u1u2x3x1RxE

hg(x) x212RxFigure 2.6: EHL line contact geometry (left) and the corresponding reduced (right).where zvisc = 0.6 is the value that has been used in the numerical analysis, meaningthat if = 2.1108and a = 0.14 then P0 2.20108. The Dowson and Higginsonequation Eq. (2.2.28) (under the assumption pa = 0) is the semi-empiric expressionused to model the density, viz. (p) = aC1 +C2pC1 +p ,where it has been assumed that the ambient pressure equals zero. At all times, the1D correspondence of the force balance condition (2.2.35), i.e._Ip (x, t) dx = W (t) , (2.4.5)where I represents the spatial solution domain, determines the parameter h00 (t).Also, the cavitation condition (Eq. (2.3.4))p 0.is used to ensure that all negative pressures obtained during the solution processare removed. Thus it has been assumed that the cavitation pressure pc may beapproximated with pc = 0. In EHL with pressure build-up reaching to the levelsof GPa this is an approximation very well justied. As mentioned previously, byimposing this condition internal cavitation is not accounted for, as if Eq. (2.3.1) hadbeen used, since mass continuity is ruined by removing all negative pressures. Notethat during numerical simulation process performing the EHL results presented inSection 8.2 no internal cavitation is observed and mass continuity is thus preserved.2.4.1 Dimensionless formulationThe solution of the smooth Newtonian EHL (one-dimensional) line contact, asdescribed above, is dependent on the following seven input parametersw, E

, us, Rx, , a, P0 or zviscwhere us = (u1 +u2), ui represents the speed of surface i in the rolling directionand either P0 or zvisc is needed dependent on which of the that is being specied.2.4. THE PRESENT EHL MODEL 25For a smooth Reynolds-Eyring approach two extra input parameters are used, i.e.the relative speed;(u1u2) ,which could be replaced by an expression for the slide-to-roll ratios = 2 (u1u2) / (u1 +u2) ,and the Eyring shear stress0.For a rough Reynolds-Eyring approach the additional input parameters needed de-pend on how the surface topographies are modeled, e.g., for a single ridge/dentmodeled by an amplitude and a wavelength only there are 9 + 2 = 11 input para-meters.In order to restate the EHL line contact problem in dimensionless form thefollowing set of dimensionless parameters where introduced (e.g. Hamrock [52]),X = x/b, T = t/ (2b/us) ,H = h/_b2/Rx_, P = p/ph, = /0, = /0,(2.4.6)whereb =_8wRxE

, ph = 2wb =_ wE

2Rx(2.4.7)If this set of dimensionless parameters is used to restate the EHD lubricated linecontact problem in dimensionless form, it becomes clear that the smooth incom-pressible problem is, mathematically, a two-parameter problem, see Moes [53], i.e.,the solution