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DOCTORA L T H E S I S

DOCTORA L T H E S I S

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Machine Elements

2006:31|: 102-15|: - -- 06 ⁄31 -- 

2006:31

On the Effects of Surface

Roughness in Lubrication

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2006:31

On the Effects of

Surface Roughness in Lubrication

Andreas Almqvist

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering, Division of Machine Elements

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Title page figure: The pressure build-up on a single rough pad of the bearing presented in Fig. 1.1.

On the Effects of

Surface Roughness in Lubrication

Copyright c Andreas Almqvist (2006). This document is freely available at http://epubl.ltu.se/1402-1544/2006/31

or by contacting Andreas Almqvist,

andreas.almqvist@ltu.se

The document may be freely distributed in its original form including the current author’s name. None of the content may be changed or excluded without permis-sions from the author.

ISSN: 1402-1544

ISRN: LTU-DT--06/31--SE

This document was typeset in LATEX 2ε

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Abstract

Tribology is a multidisciplinary field defined as the science and technology of in-teracting surfaces in relative motion, and embraces the study of friction, wear and lubrication. A typical tribological application is the rolling element bearing, see Fig. 1. Tribological contacts may also be found in other types of bearings, cam-mechanisms, gearboxes and hydraulic systems. Examples of bearings inside the human body are the operation of the human hip joint and the contact between teeth during chewing. To fully understand the operation of this type of applica-tion one has to understand the couplings between the lubricant fluid dynamics, the structural dynamics of the bearing material, the thermodynamical aspects and the resulting chemical reactions. This makes modeling tribological applications an extremely delicate task. Because of the multidisciplinary nature, such

theoreti-Figure 1: A typical tribological application

cal models lead to mathematical descriptions generally in the form of non-linear integro-differential systems of equations. Some of these systems of equations are sufficiently well posed to allow numerical solutions to be carried out, resulting in accurate predictions on performance.

In this work, the influence on performance of a surface microscopical nature, the

surface roughness, in contact interfaces between different types of machine element

components is the subject of study. An example is the non-conformal lubricated contact between one of the rollers and the inner ring in the bearing depicted in Fig. 1. The tribological contact controlling the operation of the human hip joint is also very similar to this. Another example of a non-conformal contact occurs

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models were studied; for the selected model, a numerical solution technique was developed within this project. This model is based on the Reynolds equation coupled with the film thickness equation. The numerical solution technique involves a multilevel technique to facilitate the solution process. Results presented in this thesis, utilizing this approach, study elementary surface features such as ridges and indentations passing each other inside the lubricated conjunction.

The Reynolds equation is derived under the assumptions of thin fluid film and creeping flow, and considers in its most general form shear thinning of the lubricant. This type of equation describes the hydrodynamic action of the lubricant flow and may be used when the interfaces consist of either conformal or non-conformal conjunctions. Examples of applications having conformal interfaces are thrust- and journal- bearings or the contact between the eye and a (optical) contact lens. See Fig. 2 for a schematic illustration of a typical axial thrust pad bearing (the angle of pad inclination being highly exaggerated; in a realistic application this angle is generally only fractions of a degree). In such types of applications the load

Figure 2: A schematic illustration of an axial thrust pad bearing

carried by the interface is distributed over a fairly large area that under certain circumstances 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 been to investigate and develop methods that enable the influence of surface roughness to be to be studied efficiently.

Homogenization is a rigorous mathematical concept that when applied to a certain problem may be regarded as an averaging technique as well as it provides information about the induced effects of local surface roughness. Homogeniza-tion inflicts no restricHomogeniza-tions on the surface roughness representaHomogeniza-tion other than the representative part of the chosen surface roughness being assumed periodically dis-tributed and of course the assumptions of thin film flow made through the Reynolds equation. The homogenization process leads to a two sets of equations one for the local scale describing surface roughness, scale and one for the global scale

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describ-ing application geometry. The unequivocally determined coefficients of the global problem, which may be regarded as flow factors, are obtained through the solution of local problems. This makes homogenization an eminent approach to be used investigating the influence 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 compressible Reynolds type equations that describe stationary and unstationary flows in both cartezian and cylindrical co-ordinates. This technique enables simulations of sur-face roughness induced effects when considering sursur-face roughness descriptions originating 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 homogehomoge-nization in hydrodynamic lubrication. Homogenization has also been shown here to enable efficient analysis of rough hydrodynamically lubricated problems. Also of note, in connection to the scientific contribution within tribology, collaboration with a group in applied mathematics has lead to the development of novel techniques in that area. These ideas have also been successfully applied, with some results presented in this thesis.

At start-ups, the contact in a rolling element bearing could be both starved and drained from lubricant. In this case the hydrodynamic action becomes negligible in terms of load carrying capacity. The load is carried exclusively by surface asperi-ties, the tribo film, or both. This is hereby modeled as the unlubricated frictionless contact between rough surfaces, i.e. a contact mechanical approach. A variational principle was used in which the real area of contact and the contact pressure distri-bution minimize the total complementary potential energy. The material model is linear elastic-perfectly plastic and the energy dissipation due to plastic deformation is accounted for. The numerics of this contact mechanical approach involve the fast Fourier transformation (FFT) technique in order to facilitate the solution process. Investigation results of the contact mechanics of realistic surfaces are presented in this thesis. In this investigation the variation in the real area of contact, the plasticity index and some surface roughness parameters due to applied load were studied.

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Acknowledgment

Graduate 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 greatest gratitude is therefore given to my wife Ulrika, who by being such fantastic mother and taking care of our children has made these studies possible. For all those inspiring moments with my children, I am also very grateful. Watching my children learn and grow has really made everything easier for me. My parents, my wife’s parents, our grandparents and other close relatives and friends also deserve my gratitude for all their support.

I would like to thank my supervisor, Prof. Roland Larsson, for introducing me to the field of tribology. For his enthusiasm, the inspiration he has given me and all his encouragement, I am especially grateful. He should also be acknowledged for always striving to progress in developing his talents as a supervisor.

I would like to thank my associate supervisor, Dr. Inge Söderqvist, for his guidance and support in the field of Scientific Computing during the first half of my Ph.D. studies. He was also a great source of inspiration and an excellent teacher during 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 field and his invaluable co-operation. It should also be mentioned that Peter has been my associate supervisor during the 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 their support, their co-operation and for introducing me to new concepts in tribology. I am grateful to have had the opportunity to be a part of the division because of the many competent and friendly people working there, which led to the most pleasant and relaxed atmosphere.

Special thanks goes to Tech. Lic. Fredrik Sahlin for all his co-operation, and his support and help in scripting, especially in LATEX 2ε and Perl.

My friend and former colleague, Dr. Torbjörn 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 Näslund 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 programme HiMeC and the national graduate school in scientific computning, NGSSC, which

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Contents

I

The Thesis

1

1 Introduction 3

1.1 Lubrication regimes . . . 4

1.2 The boundary lubrication regime . . . 4

1.3 The mixed lubrication regime . . . 5

1.4 The full-film lubrication regime . . . 5

1.4.1 Hydrodynamic lubrication . . . 5

1.4.2 Elastohydrodynamic lubrication . . . 6

1.5 Objectives . . . 7

1.6 Outline of this thesis . . . 8

2 Modeling Full Film Lubrication 11 2.1 Surface roughness descriptions . . . 12

2.2 Governing equations . . . 14

2.2.1 The Reynolds Equation . . . 14

2.2.2 The Film Thickness Equation . . . 18

2.2.3 Lubricant compressibility . . . 20

2.2.4 Lubricant Viscosity . . . 21

2.2.5 The Force Balance Equation . . . 21

2.3 Cavitation . . . 21

2.4 The present EHL model . . . 22

2.4.1 Dimensionless formulation . . . 24

2.5 The present HL model . . . 27

2.5.1 Dimensionless formulation . . . 29

3 The Contact Mechanical Problem 31 3.1 Statistical roughness models . . . 32

3.2 Deterministic roughness models . . . 32

3.3 Numerical solution techniques . . . 33

3.4 Governing equations . . . 33

3.5 Spectral analysis . . . 35

3.6 Dimensionless formulation . . . 36 vii

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4.1.1 Stationary incompressible lubrication . . . 42

4.1.2 Unstationary incompressible lubrication . . . 46

4.1.3 Stationary compressible lubrication . . . 51

4.1.4 Unstationary compressible lubrication . . . 53

4.2 Multiple scales expansion - cylindrical co-ordinates . . . 57

4.2.1 Stationary incompressible lubrication . . . 58

4.2.2 Unstationary incompressible lubrication . . . 62

4.2.3 Stationary compressible lubrication . . . 64

4.2.4 Unstationary compressible lubrication . . . 65

4.3 Two-scale convergence . . . 66

4.3.1 Preliminaries and notation . . . 66

4.3.2 Homogenization of the Reynolds equation . . . 68

4.3.3 Properties of the homogenized matrix A . . . 71

4.3.4 Corrector results . . . 72

4.3.5 Transversal and longitudinal roughness . . . 74

4.4 The technique of bounds - cartezian co-ordinates . . . 76

4.4.1 The governing Reynolds equation . . . 76

4.4.2 Homogenization . . . 77

4.4.3 Bounds of arithmetic-harmonic mean type . . . 80

4.4.4 Optimality of the A-H mean type bounds . . . 82

4.4.5 Bounds of Reuss-Voigt type . . . 83

4.5 The technique of bounds - cylindrical co-ordinates . . . 85

4.5.1 The governing Reynolds equation . . . 85

4.5.2 Homogenization . . . 86

4.5.3 Bounds of arithmetic-harmonic mean type . . . 86

4.5.4 Bounds of Reuss-Voigt type . . . 87

4.6 Conclusions . . . 88

5 Surface Characterization 89 5.1 Fourier based filtering . . . 89

6 Discretization and Numerical Solution Techniques 95 6.1 Numerics to be used in the field of HL . . . 95

6.1.1 Discrete formulation . . . 96

6.1.2 Solution methods . . . 97

6.2 Numerics of the EHL line contact problem . . . 97

6.2.1 The Block-Jacobi method . . . 98

6.2.2 A brief overview of the multilevel technique . . . 100

6.2.3 Discrete formulation . . . 104

6.2.4 Solution method . . . 105

6.3 Numerics of the contact mechanical problem . . . 106

6.3.1 Discrete formulation . . . 107

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7 Verification of the EHL approach 111

7.1 The CFD approach . . . 111

7.2 CFD - Governing equations . . . 112

7.3 The model problem . . . 112

7.3.1 Interpolation of solution data . . . 113

7.3.2 Error estimation . . . 114

7.4 The results of the comparison . . . 115

7.5 Discussion and concluding remarks . . . 117

8 Simulations of Rough FL 121 8.1 Hydrodynamic lubrication . . . 121

8.1.1 One rough surface . . . 122

8.1.2 Two rough surfaces . . . 140

8.1.3 Conclusions . . . 151

8.2 Elastohydrodynamic lubrication . . . 151

8.2.1 The different overtaking situations . . . 152

8.2.2 The Ridge-Ridge overtaking . . . 153

8.2.3 The Ridge-Dent overtaking . . . 154

8.2.4 The Dent-Dent overtaking . . . 154

8.2.5 The Dent-Ridge overtaking . . . 154

8.2.6 Conclusions . . . 160

9 Simulation of Rough CM 163 9.1 Varying the applied load . . . 164

9.2 Conclusions . . . 166 10 Concluding Remarks 169 11 Future Work 173

II

Appended Papers

175

A 177 A.1 Introduction . . . 180

A.2 Governing equations . . . 181

A.2.1 Boundary conditions and cavitation treatment . . . 182

A.3 Numerics . . . 183

A.3.1 The numerics for the CFD approach . . . 183

A.3.2 The numerics for the Reynolds approach . . . 184

A.3.3 Error estimation . . . 185

A.3.4 Interpolation of solution data . . . 186

A.4 Results . . . 186

A.5 Discussion . . . 187

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B.2 Theory . . . 199

B.2.1 Equations . . . 199

B.2.2 Numerics . . . 200

B.2.3 Error estimation . . . 201

B.3 Results and discussion . . . 202

B.4 Conclusions . . . 207 C 211 C.1 Introduction . . . 214 C.2 Theory . . . 215 C.3 Surface characterization . . . 216 C.4 Results . . . 217 C.5 Conclusions . . . 220 D 223 D.1 Introduction . . . 226 D.2 Governing Equations . . . 227 D.3 Multiple Scales . . . 228 D.4 Numerical results . . . 230 D.4.1 Discrete formulation . . . 230 D.4.2 Convergence of pressure . . . 232

D.4.3 Load carrying capacity . . . 235

D.5 Conclusions . . . 236

E 239 E.1 Introduction . . . 242

E.2 The governing Reynolds type equations . . . 242

E.3 Homogenization (constant bulk modulus) . . . 244

E.4 Homogenization in the incompressible case . . . 245

E.5 Only one rough surface . . . 248

E.6 Numerical results . . . 249

E.6.1 Incompressible case, L2= L1 . . . 249

E.6.2 Incompressible case, L2= 10L1. . . . 250

E.6.3 Constant bulk modulus, L2= L1 . . . 252

E.6.4 Constant bulk modulus, L2= 10L1. . . 253

E.7 Concluding remarks . . . 255

F 259 F.1 Introduction . . . 263 F.2 Homogenization . . . 265 F.3 Discretization . . . 268 F.4 Numerical Results . . . 270 F.5 Conclusions . . . 274

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G 275

G.1 Introduction . . . 279

G.2 The governing Reynolds equation . . . 280

G.3 Homogenization . . . 281

G.4 Bounds . . . 283

G.4.1 Bounds of arithmetic-harmonic mean type . . . 283

G.4.2 Optimality . . . 285

G.4.3 Bounds of Reuss-Voigt type . . . 286

G.5 Discretization . . . 287

G.6 Numerical results and discussion . . . 289

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Preface

This thesis comprises the results from modeling and numerical simulations of both lubricated and unlubricated contacts, that constitute tribological interfaces. The main focus is on the influence of surface topography on performance. The thesis work was conducted at the Department of Applied Physics and Mechanical Engi-neering, the Division of Machine Elements, Luleå University of Technology. The scientific outcome is a number of publications that may be found in international journals and in conference proceedings (with review procedure):

[1] T. Almqvist, A. Almqvist, and R. Larsson. A comparison between compu-tational fluid dynamic and Reynolds approaches for simulating transient ehl line contacts. Tribology International, 37:61–69, 2004.

[2] A. Almqvist and R. Larsson. The effect of two-sided roughness in rolling/sliding ehl 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 surface contact 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 Congress

III, Washington, D.C., USA, Sep 2005.

[5] A. Almqvist and J. Dasht. The homogenization process of the Reynolds equation describing compressible flow. Tribology International, 39:994Ű1002, 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 flat 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 homogenization method for developing rough surface flow 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 for publication in International Journal of Engineering Science, August 2006.

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[9] A. Almqvist, R. Larsson, and P. Wall. The homogenization process of the time dependent Reynolds equation describing compressible liquid flow.

Re-search Report, No. 4, ISSN 1400-4003, Department of Mathematics, Luleå University of Technology, submitted for publication in Tribology International,

2006.

[10] A. Almqvist, E. K. Essel, L.-E. Persson, and P. Wall. Homogenization of the unstationary incompressible Reynolds equation. Submitted for publication in Tribology International, May 2006.

Of the 10 scientific contributions, 7 were specifically 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 Luleå Uni-versity of Technology, led to a rather extensive research report on the applicability of 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. Research

Report, No. 3, ISSN 1400-4003, Department of Mathematics, Luleå Univer-sity of Technology, 2005.

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Sub-Division of Work in

Appended Papers

This chapter describes the sub-division of work in the seven appended papers found in Part II of this thesis.

Paper A

[1] T. Almqvist, A. Almqvist, and R. Larsson. A comparison between compu-tational fluid dynamic and Reynolds approaches for simulating transient ehl line contacts. Tribology International, 37:61–69, 2004.

The development of the Reynolds equation based model and numerical solver was performed by A. Almqvist. T. Almqvist, who is the corresponding author, worked with the modifications of the CFD-software and performed the numerical simu-lations using it. A. Almqvist and T. Almqvist performed the analysis work and prepared the main part of the paper. R. Larsson is co-author.

Paper B

[2] A. Almqvist and R. Larsson. The effect of two-sided roughness in rolling/sliding ehl 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, the corresponding author of this paper. R. Larsson is co-author and contributed with his expertise.

Paper C

[6] A. Almqvist, F. Sahlin, R. Larsson, and S. Glavatskih. On the dry elasto-plastic contact of nominally flat 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.

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[5] A. Almqvist and J. Dasht. The homogenization process of the Reynolds equation describing compressible flow. Tribology International, 39:994Ű1002, 2006.

A. Almqvist and J. Dasht worked together with the modeling and the mathematical analysis; A. Almqvist, the corresponding author, developed the numerical technique and performed the numerical analysis.

Paper E

[10] A. Almqvist, E. K. Essel, L.-E. Persson, and P. Wall. Homogenization of the unstationary incompressible Reynolds equation. Submitted for publication in Tribology International, May 2006.

E. Essel is the corresponding author. The work was divided among the co-authors as 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 the time dependent Reynolds equation describing compressible liquid flow.

Re-search Report, No. 4, ISSN 1400-4003, Department of Mathematics, Luleå University of Technology, submitted for publication in Tribology International,

2006.

A. Almqvist, the corresponding author, worked with R. Larsson and in the modeling of the governing equation, helped P. Wall with the mathematical analysis, did the numerical 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 for publication 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 derived the Reuss-Voigt bounds. A. Almqvist, the corresponding author, performed the numerical modeling and analysis.

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Nomenclature

α Pressure-viscosity coefficient Pa−1 ¯

μ Dimensionless dynamic viscosity ¯

ρ Dimensionless density

ΔT Dimensionless step size in time s Δt Step size in time s ΔXi Dimensionless step size in Xi m

Δxi Step size in spatial coordinate xi m

˙γ Shear rate s−1

E 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 energy I Intergrid transfer operator

Λ Wavelength m

L Functional operator

u Velocity field,u = u (x, y, z) m s−1

x Spatial coordinates,x = (x, y, z) m

μ Dynamic viscosity Pa s

μ0 Dynamic viscosity at ambient pressure Pa s

ν Poissonťs ratio

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O (n) Mathematical order of the number n

Φ Solution variable

ψi Topography of surface i m

ψavg Mean surface height

ρ Density kg m−3

ρ0 Density at ambient pressure kg m−3

Σ Dimensionless stress function

σ Total stress tensor Pa

Ui Surface velocity,Ui = [u1i, u2i] T

m s−1

Um Mean surface velocity,Um= [u11+ u12, u21+ u22]T/2 m s−1

τ Lubricant shear stress Pa

τ0 Eyring stress Pa τ1 Lubricant shear stress at surface 1 Pa

τm Lubricant midplane shear stress Pa

τxixj Lubricant shear stress Pa



f Fourier transformation of f 

1 Dimensionless reduced wavelength

A Dent / Ridge Amplitude m

b Hertzian half-width b =(8wRx) / (πE) m

C1 Constant C1= 5.9 108 C2 Constant C2= 1.34 d Elastic deformation m de Elastic deformation m dp Plastic deformation m E Modulus of elasticity Pa

E Effective mod. of elast. 2/E= (1− ν21)/E1+ (1− ν22)/E2 Pa

G Shear modulus of elasticity of the lubricant Pa

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gs The gap between the undeformed surfaces m

H Dimensionless film thickness

h Film thickness m

h0 Integration constant m

Hs Hardness of the softer material Pa

H00 Dimensionless integration constant

hc Central film thickness m

hmin Minimum film thickness m

K Elastic deformation integral kernel

L Moes dimensionless speed parameter

li Amplitude variation of dent/ridge

L1 Discrete operator (Reynolds equation) L2 Discrete operator (Film thickness equation)

Lxi Length parameter m M Moes dimensionless load parameter

mi Wavelength variation of dent/ridge

P Dimensionless pressure

p Pressure Pa

P0 Constant in the viscosity expression ph Hertzian pressure ph=



(wE) / (2πRx) Pa

pmax Maximum pressure Pa

Ra Average roughness μm

Ri Radius of the surface i m

Rk Kurtosis

Rq Root mean square (RMS) roughness μm

Rx Reduced radius of curvature in x-dir. 1/Rx= 1/R11+ 1/R22 m

Rz Average maximum height μm

Rsk Skewness

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T Dimensionless time

t Time s

u Solution to the one-dimensional Poisson equation

ui Surface speed, ui= ui(x1, x2, x3) m s−1 us Sum of velocities us= u11+ u21 m s−1

W Applied load, 2D / 3D contact Pa m−1 / Pa

xc Centre of dent / ridge xc(t) = xs− u t m

Xi Dimensionless spatial coordinate m

xi Spatial coordinate m

xs Initial placement of dent / ridge m

zvisc Pressure-viscosity index

ωxi Spatial frequency 1 / m

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Part I

The Thesis

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Chapter 1

Introduction

Machines consist of machine elements and their safe and efficient operation relies on carefully designed interfaces between these elements. The functional design of interfaces covers geometry, materials, lubrication and surface topography, and an incorrect design may lead to both lowered efficiency and shortened service life. A misalignment due to the geometrical design could lead to large stress concentra-tions that in turn may lead to severe damage when mounting, a detrimental wear situation, rapid fatigue during operation, etc. Large stress concentrations also im-plicitly imply a temperature rise because of the energy dissipation due to plastic deformations. The choice of mating materials is also of great importance, e.g. elec-trolytic corrosion may drastically reduce service life. Contact fatigue due to low ductility would not only lower the service life but could lead to third body abrasion due 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 to lower friction, the actions of additives are of concern. If the interface is subjected to excessive wear, the lubricant’s ability to form a separating film becomes even more crucial. In this case, the bulk properties of the lubricant have to be carefully chosen. At some scale, regardless of the surface finish, all real surfaces are rough and their topography influences the contact condition.

As implied above, these design parameters are mutually dependent, i.e. they affect the way others influence the operation of the system. For example, a change in geometry could require another choice of materials that may change the objec-tives of the lubricant and force the operation into another lubrication regime. All four design parameters are of great importance, though in this thesis work it is the effects induced by the surface topography that is of main interest.

The influence of surface roughness on performance has of course been investi-gated by many researchers in the field, experimentally and numerically. However, because of the multidisciplinary nature of the field and the complexity of the the-oretical models associated with tribological problems, the progress in the devel-opment of efficient, still user friendly software has not reached as far as in, e.g., computational structural mechanics and computational fluid dynamics. Moreover, the requirement on the density of the mesh to resolve not only the geometrical part of the tribological contact but also the surface topography is difficult to meet. This thesis contributes to the field of research through this connection. Namely,

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by development and implementation of rigorous theoretical models that allow for effective numerical treatment of some specifically chosen rough tribological prob-lems. For example, within this thesis work, the outcome of an extremely successful collaboration has led to state-of-the-art results in applied mathematics [12] as well as to a highly efficient numerical treatment of the Reynolds equation that incorpo-rates the effects induced by the surface roughness; for details, see Sections 4.4 and 4.5.

1.1

Lubrication regimes

Because of the mutual dependency of the design parameters as well as the change in their objectives with different operating conditions, the surface topography induced effects will vary.

In tribology, lubrication is often used as target parameter, e.g. depending on the application 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 regime

In the boundary lubrication (BL) regime, the lubricant’s hydrodynamic action is negligible and the load is carried directly by surface asperities or by surface active additives (a so-called tribofilm). Here, the surface topography is preferably chosen to optimize the frictional behavior without increasing the rate of wear. To do this, one has to understand how the chemical processes are affected by the actual contact conditions, 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 contact between rough surfaces, i.e. as a contact mechanical problem. Within the field, the well known elasto-plastic variational approach expressed as the minimization of total complementary potential energy has been adopted and combined with the numerical Fourier transform, known as fast Fourier transform (FFT), to ensure a stable and effective numerical treatment of rough contact mechanics. This approach helps to increase the understanding of how the surface roughness influences the elastic deflection, the plastic deformation (and plasticity index), the pressure build up and the real area of contact. An in-depth understanding of this connection is required to refine the design of interfaces operating under these circumstances.

As the hydrodynamical action of the lubricant increases, the contact mechanical response becomes less severe in terms of pressure and real contact area, and a transition from the BL- to the ML- regime may therefore occur. The developed contact mechanical tool is also applicable here and may be used to indicate this interesting transition, from a tribological angle of approach.

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1.3. THE MIXED LUBRICATION REGIME 5

1.3

The mixed lubrication regime

What characterizes the ML regime is that the load is carried by the lubricant’s hydrodynamical action, which may be influenced by the elastic deflection of the surfaces, the tribofilm, directly by surface asperities, or a combination thereof. This means that the objectives of the surface topography are to support the hydrody-namic action of the lubricant, aid the elastic deflection in rendering a smoother surface, enable bonding of the surface active additives and optimize friction in the contact 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 may be used to indicate a possible transition between the BL- and the ML-regimes. Similarly, modeling performed regarding full-film lubrication has lead to numerical FL approaches that may be used to increase the understanding of the transition from the FL- to the ML- regimes.

1.4

The full-film lubrication regime

When the hydrodynamic action of the lubricant fully separates the surfaces and the load is carried totally by the lubricant film, the contact enters the full film lubrication (FL) regime. In the FL regime, traction may be reduced by carefully chosen topographies. Even though there is no direct contact, the topography must also prevent fatigue that may lead to excessive wear in the form of spalling in a highly loaded contact.

This regime is commonly sub-divided into hydrodynamic lubrication (HL) and elastohydrodynamic lubrication (EHL), since the performance is greatly affected by the presence of elastic deflections in the contact zone. In this thesis, the effects induced by the surface roughness in both these regimes have been studied.

1.4.1

Hydrodynamic lubrication

Slider bearings are typical examples of applications that, under certain conditions operate, in the hydrodynamic lubrication (HL) regime where the elastic deforma-tions of the bearing surfaces are sufficiently small to be neglected. For example, an axial thrust pad bearing, as depicted in Fig. 1.1, consists of a conformal contact that under certain circumstances may be assumed to operate in the hydrodynamic lubrication regime. Note that the angle of inclination of the pads, which is generally only a fraction of a degree, has been exaggerated in the figure. The problems that arise when modeling these types of contacts are the large differences in scales, i.e. That is, the global scale of the contact describing the geometry is several orders of magnitude larger than the local scale describing the surface topography/roughness. For this purpose a research group in applied mathematics with specializing in homogenization was contacted and a cooperation was established. The outcome from this fruitful collaborative work was a variety of mathematically rigorous mod-els to be applied in the field of hydrodynamic lubrication where the fluid flow may be governed by the Reynolds equation. This lead to highly effective numerical

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Figure 1.1: Schematics of an axial thrust pad bearing

tools where the influences of surface roughness are embedded in the derived ho-mogenized equations. More over, the equations are unambiguously determined and shows to naturally allow for parallelization. These tools enables studies of rough surface hydrodynamically lubricated problems such as that arising in the bearing configuration visualized in Fig. 1.1. This means that the theoretical model concerns different types of the unstationary Reynolds equation in two dimensions.

1.4.2

Elastohydrodynamic lubrication

Elastohydrodynamic lubrication (EHL) is the type of hydrodynamic lubrication where the elastic deformations of the contacting surfaces cannot be neglected. This is 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 in Fig. 1.2, are most commonly designed to operate in the full-film elastohydrody-namic lubrication regime. Note that the elastic deflection is the key ingredient

Figure 1.2: Schematics of a typical rolling element bearing

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1.5. OBJECTIVES 7 and extremely high viscous hydrodynamic action of the lubricant combined with the elastic smoothening of the in-contact surface roughness that allows for a thin lubricating film 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 running conditions, the shape of the ellipse will change as shown in Fig. 1.3. In any case, the contact region is small and the concentrated load implies a severe stress condition that will lead to a large elastic deflection and possibly also plastic deformation. For a bearing in operation, it is the large elastic deflection 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 allow for a hydrodynamic action that fully separates the surfaces, the risk for plastic deformation increases.

x1

x2

Roller width

Figure 1.3: Elliptic contacts(x1/aξ)2+ (x2/bξ)2= 1



If bξ exceeds the minimum width of the raceway and the roller, the contact will

be then truncated and possibly lead to increased stresses in the material, at least for the unlubricated contact. In the case of a contact with bξ/aξ > 4, though with

still less than the minimum width of the raceway and the roller, the centerline

in 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 efficient solution methods. A numerical multilevel technique was adopted to remedy this problem. One of the drawbacks is the complexity of the tool, which unfortunately requires the skills of an engineer with a rather extensive theoretical background. With this numerical approach, the goal was to develop a coupled solver that solves the Reynolds equation and the film thickness equation simultaneously, and to develop the multilevel technique to also embrace the two-dimensional operators that arise due to the coupling of the equations.

1.5

Objectives

This work concerns the modeling of rough surface contact interfaces, with or with-out a separating lubricant in between. The main objective has been to investigate

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how the surface topography influences different contact conditions by means of nu-merical simulations. The long term goal is to develop engineering tools, i.e. tools with low complexity, that enable efficient analysis and prediction of different con-tact conditions by considering the surface topography. There is also an effort in creating a unified approach to be used for as a wide a variety of tribological prob-lems as possible. For this purpose the following specifications of three different approaches that may be used to increase the fundamental knowledge about surface roughness induced effects within tribology was postulated

- To develop a robust yet still efficient rough surface solver to simulate elasto-hydrodynamic lubrication.

- To develop a contact mechanical tool to study the contact between rough surfaces.

- To develop a tool to analyze hydrodynamically lubricated problems involving rough surfaces.

1.6

Outline of this thesis

The thesis found in Part I is self contained, with Part II comprising the work presented in the papers A to G found in Part II. The content of the papers found in Part II has in Part I been extended with more complete theoretical descriptions and derivations. Some material not directly addressed in the papers has also been included for completeness. Chapter 4, concerning homogenization of the Reynolds equation, has been extended to also embrace the cylindrical form of the Reynolds equation. This material has not yet been submitted to any journal, since the work was carried out simultaneously as this thesis was being finalized. However, this material will most likely be submitted if the author is granted the financial support needed to continue his work in the field.

Chapter 2 of this thesis introduces the topic of full film lubrication (FL). This is also the theoretical basis of Reynolds equation based approaches found in all the papers presented in Part II. This chapter includes surface roughness modeling, i.e. the deterministic and the statistical way of dealing with surface roughness. For clarity, this includes a derivation of the non-Newtonian Reynolds-Eyring equation (Conry et al. [14]) found in Section 2.2. In this section, the film thickness equa-tion, pressure-density relations describing compressibility and pressure-viscosity relations are given. Boundary conditions, different ways of modeling cavitation and the force-balance equation are also described here. Section 2.4 comprises the equations used in this thesis for modeling elastohydrodynamically lubricated prob-lems, while Section 2.5 comprises the equations used in this thesis for modeling purely 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. Different models of the elastic concon-tact are then discussed in connection to the present model. A description of how spectral analysis can be used to determine the elastic deformation integral follows. Both the

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dimen-1.6. OUTLINE OF THIS THESIS 9 sionless and discrete formulation and the numerical solution process for the dry elasto-plastic method are also provided.

Chapter 4 deals theoretically with the application of homogenization techniques on hydrodynamically lubricated problems governed by the Reynolds equation. Here, the formal method of multiple scales expansion is applied on the Reynolds equation, which is also the theoretical basis of papers D, E and F. Two-scale convergence of this problem is also proven here. Moreover, the novel technique of finding bounds on the homogenized properties of the Reynolds problem is also addressed. This forms the theoretical basis of Paper G.

Simulating either rough FL or contact mechanics definitely requires some form of surface characterization as one of the preliminary processes. Chapter 5 addresses this area, which is also a part of Paper C. Section 5.1 shows that the interpretation of the acquired data when using a deterministic model may be facilitated by making use of a Fourier based filtering technique. The technique is based on truncated Fourier 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 discrete partial differential equation that was implemented to solve the hydrodynamically lubricated problem associated with the problems that have been homogenized (Sec-tion 6.1). This numerical solu(Sec-tion 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 equation and the film thickness equation is found in Section 6.2. A multilevel technique used to facilitate the numerical solution process of the one dimensional Poisson problem and a possible modification to facilitate the numerical solution of coupled systems are then briefly described. Section 6.3 describes the numerical contact mechanical approach, used in Paper C, which is a development of an existing numerical elastic model was modified 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 stationary and two different transient problems. The results are encouraging from several viewpoints: verification of the codes, the possibilities to further develop the CFD approach given by Almqvist and Larsson [15], and the justification of using a Reynolds approach under the running conditions chosen.

The topic of surface roughness induced effects that occur in the full film 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 the numerical solution methods, described in 6, are presented.

In Paper C the developed rough surface contact mechanics approach developed simulates the contact mechanics of four different topographies. This is also the content of Chapter 9. The simulations are here restricted to two dimensions and profiles 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 scientific basis of the thesis.

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Chapter 2

Modeling Full Film Lubrication

Factors such as load carrying capacity, traction, service life, etc., are determining when designing any type of bearing. The performance of a thin film lubricated bearing is certainly affected by the operating conditions, the choice of lubricant, the thermal properties of the bearing surface material and the surface macro and micro topography. As previously mentioned, the surface roughness is of main interest here. The main reason to model full film lubrication is of course that full scale testing of machine elements, such as bearings, cam mechanism, gear boxes, etc., is very expensive and that modeling permits possible numerical simulations that reduce the cost. Scientifically, modeling is of great importance, since it allows the gain of fundamental knowledge on the different acting mechanisms that control the operation of applications such as those previously mentioned. This due to the simplicity in separating the effects caused by different parameters.

In this chapter, the different adopted surface roughness/topography models are considered first, 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 of averaging techniques, statistical models and the present HL approach based on the ideas in homogenization are also discussed.

The theoretical basis of the full film lubrication models are described in Sec-tion 2.2. A derivaSec-tion of a two-dimensional non-Newtonian Reynolds-Eyring partial differential equation (PDE) in accordance with the work performed by Conry et al. [14] is given in Section 2.2.1, followed by the film thickness equation in Sec-tion 2.2.2. This secSec-tion also describes the elastic deflecSec-tion equaSec-tion of the surfaces. Section 2.2.3 discusses the compressibility of the lubricant inside the conjunction and describes two widely different ways of modeling density as a function of pressure are described. That is, one that may be used when modeling elastohydrodynamic lubrication (EHL) and one that may significantly 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 the Reynolds-Eyring equation, though the viscosity of the lubricant and how it relates to pressure is discussed separately in Section 2.2.4. Criteria on force-balance is described in Section 2.2.5.

The integro-differential problem for the isothermal EHL line contact is consid-ered next. Basically, the present approach is an effort to consider the coupling

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between the governing equation for fluid pressure (the Reynolds-Eyring equation) and the film thickness equation that includes the elastic deflection determined by the fluid 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 the fluid density and the fluid viscosity. The particular equations used for the present EHL approach are presented in this section. Another objective set for the modeling of 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 solution process in solving rough EHL problems.

Section 2.5 provides specific details on the governing equations used for the present HL approach discussed in Chapter 4. This includes, for example, infor-mation on film thickness modeling, compressibility, the topic of force balance and suitable non-dimensional forms.

2.1

Surface roughness descriptions

There 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), using deterministic models, the one surface is considered smooth and one is considered rough. This is a suitable approximation when modeling a rolling contact or a contact where the roughness of the moving surface is of minor importance. For example, in a pure rolling contact, the roughness of the two contacting surfaces may be summed and the contact modeled as having an effective roughness and a perfectly smooth surface, an approach here referred to as ‘one-sided roughness’.

When using a deterministic surface roughness description to study a lubricated contact that is subjected to sliding, the surface topographies of both surfaces are of significance and a continuously changing effective surface roughness occurs. The one-sided roughness approach is not valid in such situations and a ‘two-sided’ rough-ness treatment is needed. Of course, the two-sided surface roughrough-ness treatment has been previously investigated, e.g. Evans et al., Tau et al., Chang, Venner and Morales-Espejel, Hooke [13, 17, 18, 19, 20].

The highly detailed outcome of a deterministic approach in terms of local pres-sure, elastic deflection, traction, etc., gives us a fundamental understanding of the different acting mechanisms controlling the performance of the lubricated contact and thus the performance of the machine element. In summary, whenever local in-formation about surface roughness induced effects is needed a deterministic model is best suited. Both one-sided and two-sided roughness approaches have been ad-dressed in this thesis with the findings presented in Chapter 8.

However, a significant amount of number of freedoms is introduced by the sur-face topography, i.e. the small scale sursur-face features need to be resolved numeri-cally, requiring a very dense numerical grid. This implies significant memory con-sumption and time consuming computations. One way to remedy this problem is to utilize statistical theory. The subject being considered in this thesis as a random process is, of course, the surface roughness and it is the equation governing the fluid flow, here the Reynolds equation, that is being analyzed with the statistical tech-nique. The coefficients determining the Reynolds equation are dependent on the

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2.1. SURFACE ROUGHNESS DESCRIPTIONS 13 surface roughness through the film thickness function; see Section 2.2. This means that these rapidly oscillating coefficients in turn suggest some type of averaging.

Applying a statistical method to analyze the Reynolds equation means losing some local information to some extent and that the effects of surface topography will be statistically inherited in the averaged determining equation. Christensen [21], utilizing statistical analysis, was able to derive averaged Reynolds equations with the restriction that the roughness be either transversally- or longitudinally-oriented. Patir and Cheng [22], [23], [24] introduced the concept of flow-factors that were used as corrections of the corresponding smooth film thickness so that the actual surface topography would be considered. Such an approach is not restricted to the transversal and longitudinal cases and may be used with realistic three dimensional surface topographies. These flow-factors, which are the coefficients of the modified Reynolds type equation defined on a global (geometry) scale, are obtained by solving problems defined on a local (roughness) scale. This means that when the flow-factors for a specific surface roughness description have been obtained, the numerical solution process of the modified Reynolds type equation is as fast as the corresponding smooth problem. This strongly justifies of the use of this type of approach. The drawback of an approach based on the findings of Patir and Cheng is the existing ambiguities in determining the flow-factors. For example, rigorous theoretical explanation on how many times the local problems should be solved to obtain the statistically correct result does not exist. Another ambiguity is how to truncate the solution of the local problem to reduce the influence of the boundary conditions, Harp and Salant [25].

Homogenization is a mathematical research area that includes studying PDE’s with rapidly oscillating coefficients. As mentioned above, the film thickness func-tion that include the surface roughness is part of the coefficients in the Reynolds PDE. This suggests that the ideas in the area of homogenization are suitable and that homogenization may be used to efficiently average surface roughness induced effects. 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 in the field of tribology. For work considering the application of homogenization to hy-drodynamically lubricated problems, see e.g. Elrod, Bayada and Faure, Jai, Bayada et 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 is very appealing, with Chapter 8 summarizing the promising results in the aspect of lubrication. In comparison to the well-known method of Patir and Cheng, homoge-nization techniques also provide the possibility to study realistic three dimensional surface topographies. One major remark is that the homogenization process con-tains no ambiguities in determining the coefficients of the modified Reynolds type equation, in this case referred to as the homogenized Reynolds equation. Fur-ther, within this thesis, novel homogenization ideas are applied to the Reynolds equation that lead to a new homogenization technique, making it possible for the engineer to perform systematic studies on the effects of realistic two- and three-dimensional surface roughness representations on the lubrication performance of machine elements, such as different types of slider bearings, Almqvist et al. [8].

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2.2

Governing equations

This section explains the equations that govern the lubrication process of contacts possibly found in tribological applications such as rolling element bearings and slider bearings.

2.2.1

The Reynolds Equation

A modified Reynolds equation, based on on the Eyring theory of non-Newtonian flow, is derived in one dimension. Johnson and Tevaarwerk [43] proposed the nonlinear constitutive equation for a lubricant under isothermal conditions given by Eq. (2.2.1), ˙γ = 1 G dt + τ0 μ sinh  τ τ0  , (2.2.1) where G is the shear modulus of elasticity of the fluid, τ is fluid shear stress, μ is the dynamic viscosity, and τ0 is the Eyring shear stress. This is a Maxwell rheological

model where the total shear strain rate is the sum of an elastic term and a nonlinear viscous term based on the Eyring’s theory of viscosity. The modified Reynolds equation is derived from the Eyring equation, the nonlinear viscous portion of Eq. (2.2.1), and under the assumption of plain strain rate. Fig. 2.1 shows a fluid

x3 x31 (x1, x2, 0) (x1= C, x2, x3= 0) u1(x1= C, x2, x3) x3 x2 x1

Figure 2.1: A description of the contact region

element in the thin lubricating film between two solids. In this case the equations of equilibrium in the are x1- and x2- direction take the forms:

∂τx1x3 ∂x3 = ∂p ∂x1 , ∂τx2x3 ∂x3 = ∂p ∂x2 , (2.2.2) where p is fluid pressure. If the film thickness is denoted by h (x), then according to Fig. 2.1, 0≤ x3≤ h (x) and x3= x3− x31. Assuming p = p (x), integration of Eq. (2.2.2) with respect to x3 yields:

τx1x3 = τ11+ x3 ∂p

∂x1, τx2x3= τ21+ x3  ∂p

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2.2. GOVERNING EQUATIONS 15 where τi1are the shear stress acting on surface 1. Substituting Eq. (2.2.3) into the

constitutive equation ( ˙γi= (τ0/μ) sinh (τxix30)) yields:

˙ γi= ∂ui ∂x3 = τ0 μ sinh τi1+ x3∂x∂pi τ0 . (2.2.4) Assuming that the velocity of the lower surface (x3 = 0) is U1 = [u11, u21]T and that the viscosity does not vary across the film ,i.e., μ = μ (x1, x2), integration of

Eq. (2.2.4) with respect to x3gives the following expression for the velocity profile in the xi-direction: ui(x, x3) = ui1+ x3 0 τ0 μ sinh τi1+ s∂x∂pi τ0 ds, (2.2.5) here x = (x1, x2) for simplicity, which after evaluation of the integral expression

becomes: ui(x, x3) = ui1+ τ02 μ∂x∂p i cosh τi1+ x3∂x∂p i τ0 − cosh  τi1 τ0  . (2.2.6) Introduction of the mid plane shear stress:

τim= τi1+ h 2 ∂p ∂xi (2.2.7) and a dimensionless function, defined as:

Σi= h 0 ∂p ∂xi (2.2.8) makes it possible to rewrite Eq. (2.2.6) as:

ui(x, x3) = ui1+ τ0h 1 Σi  cosh  τim τ0 − Σi  1−2x3 h  − cosh  τim τ0 − Σi  . (2.2.9)

Application of the boundary condition ui(x, h) = ui2 and utilizing hyperbolic

relations gives: ui2= ui1+ τ0h 2μΣi  cosh  τim τ0 + Σi  − cosh  τim τ0 − Σi  = ui1+ τ0h μΣi sinh  τim τ0  sinh Σi. (2.2.10)

Eq. (2.2.10) can be rearranged to allow for determination of the midplane shear stress as follows: sinh  τim τ0  =μ (ui2− ui1) τ0h Σi sinh Σi . (2.2.11)

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The mass flux per unit width, Mi(x), is defined as:

Mi(x) =

h 0

ρui(x, x3) dx3. (2.2.12)

After substitution of Eq. (2.2.9) into Eq. (2.2.12) and integration, the expression for mass flux per unit width becomes:

Mi(x) = ρui1h + ρ τ0h 2μΣi hi  sinh  τim τ0 + Σi  − sinh  τim τ0 − Σi  (2.2.13) h cosh  τim τ0 − Σi  .

Expanding and rearranging the terms in the brackets of Eq. (2.2.13) gives

Mi(x) = ρui1h + ρ τ0h2 sinh Σi Σi sinh  τim τ0  + (2.2.14) sinh Σi− Σicosh Σi Σ2i cosh  τim τ0  .

Substitution of Eq. (2.2.11) into Eq. (2.2.14) yields:

Mi(x) = (ui1+ ui2) 2 ρh + ρτ0h2 sinh Σi− Σicosh Σi Σ2i cosh  τim τ0  = (ui1+ ui2) 2 ρh + (2.2.15) ρτ0h2 Σi sinh Σi− Σicosh Σi Σ3i cosh  τim τ0  = (ui1+ ui2) 2 ρh + ρh3 12μSi(x) ∂p ∂xi , where Si(x) = 3 (sinh Σi− Σicosh Σi) Σ3i cosh  τim τ0 

The hyperbolic relation cosh2(x)− sinh2(x) = 1 together with Eq. (2.2.11) may now be used to yield the following expression for Si(x), referred to as the

non-Newtonian slip factor

Si(x) = 3 (Σicosh Σi− sinh Σi) Σ3i  1 +  μ (ui2− ui1) τ0h Σi sinh Σi 2 . (2.2.16)

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2.2. GOVERNING EQUATIONS 17 The equation of continuity yields

2  i=1 ∂xi h(x) 0 ρui(x, x3) dx3 = 0, (2.2.17)

which could also be expressed in terms of the mass flux per unit width as

2  i=1 ∂xi Mi(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) finally yields the stationary two-dimensional Reynolds-Eyring equation ∇ · ((ρh) Um)− ∇ ·  ρh3 12μS∇p  = 0, (2.2.19) where∇ = [∂/∂x1, ∂/∂x2]T, S = ⎡ ⎣ S1 0 0 S2 ⎤ ⎦ and Um= [(u11+ u12) /2, (u21+ u22) /2]T.

The representative lubricant stress τ0, characterizes the transition from

New-tonian to non-NewNew-tonian fluid behavior. When using the Eyring model an infi-nitely large τ0 characterizes a Newtonian fluid, and by using L’Hospital rule it

can be shown that Si approaches unity. In this case Eq. (2.2.19) reduces to the

conventional Reynolds equation, viz.

∇ · ((ρh) Um)− ∇ ·  ρh3 12μ∇p  = 0, (2.2.20) governing Newtonian thin film flow. The value of Si is always equal to or greater

than one, which means that the effective viscosity μ/Si is always less or equal to

the bulk viscosity of the lubricant μa.

It is possible to re-state Eq. (2.2.19) (and, of course Eq. (2.2.20)) in unstationary form, incorporating possible squeeze effects, according to

∂t(ρh) =∇ ·  ρh3 12μS∇p  − ∇ · ((ρh) Um) . (2.2.21)

Similarly the corresponding unstationary but non-Newtonian Reynolds equation yields ∂t(ρh) =∇ ·  ρh3 12μ∇p  − ∇ · ((ρh) Um) . (2.2.22)

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h(x, t)

x

3

x

2

x

1

Figure 2.2: A graphical illustration of the the film thickness function h(x, t), which is the gap between the upper red surface and the lower blue surface.

2.2.2

The Film Thickness Equation

A common way of modeling the film thickness h (x, t) in lubrication is

h (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 film

thick-ness on the global scale including the geometry of the bearing, de(x, t) models the

elastic deflection and hi(x, t) describes the film thickness on the local scale (the

surface topography/roughness). A simple slider bearing may be found in Fig. 2.3 that illustrates the function h0(x, t) at a specific point in time. The functions

h0(x, t)

ω1 x3

x1 x2

Figure 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 description

of 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 h00models the global separation of the two bearing surfaces and hg models

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2.2. GOVERNING EQUATIONS 19

h1(x, t)

h2(x, t)

Figure 2.4: The functions h1(x, t) (left) and h2(x, t) (right) exemplified.

h0(x, t) + h2(x, t)

h1(x, t)

ω1 x3

x1 x2

Figure 2.5: An illustration of a rough slider bearing model.

needed, e.g. when simulating a pivoted thrust pad bearing. This depends on the mounting of the pad and if the angle of pad inclination needs to be determined by a force balance- a momentum balance- condition or both. Note that h00 will also be present, in the form of a constant, when simulating a stationary case.

The elastic deflection may be modeled using the Boussinesq approach. That is, the elastic deflection at a specific point relates to the pressure at every point within the domain through

de(x) =

−∞

K (x− s) p (s) ds + const. (2.2.25) where the integral kernel K is given by

K (x− s) = − 4 πE ln|x − s| , (2.2.26) and K (x1− s1, x2− s2) = 2 πE 1  (x1− s1)2+ (x2− s2)2 , (2.2.27) for two- and three- dimensional problems respectively, and where the effective mod-ulus of elasticity E is given by

2 E = 1− ν12 E1 + 1− ν22 E2 .

(44)

2.2.3

Lubricant compressibility

In this section some, for tribological applications, suitable compressibility ap-proaches, such as the Dowson-Higginson [44] and the model based on the lubricant bulk modulus, see e.g. Vijayaraghavan [45].

The Dowson-Higginson, semi-empirical expression, for iso-thermal lubrication, yields

ρ (p) = ρa

C1+ C2(p− pa)

C1+ (p− pa)

, (2.2.28) where the constants C1 and C2 may be used to fit this rational expression to experimentally obtained data.

Modeling the lubricant compressibility via a constant bulk modulus β, for which the definition yields

β = ρ∂p

∂ρ, (2.2.29)

is another type of approach. It should be noted that, from a mathematical point of view, this is an interesting transformation. Here the bulk modulus β may be fitted to experimentally obtained data. This implies that the relation between the density and the pressure is of the form

ρ(p) = ρae(p−pa)/β. (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 be used to simulate elastohydrodynamic lubrication. However, due to its mathemat-ical form it leads to the possibility to restate the Reynolds equation that governs Newtonian (S ≡ 1), iso-thermal (contact temperature does not change due to the hydrodynamic flow) and iso-viscous (μ = const.) fluid flow in a very simple form. Indeed, in defining the dimensionless density function w as w(x, t) = ρ(p(x, t))/ρa,

then

∇w = ρ−1

a ρ(p)∇p = β−1e(p−pa)/β∇p = β−1w∇p.

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 converted to a linear equation expressed as

Γ ∂t(wh) =∇ ·  h3∇w− Λ ∂x1(wh) , (2.2.31) where Γ = 12μa/β, Λ = 6μaus/β, and us= u11+ u12.

Even though this linear PDE has restrictions, the particular form is of impor-tance. For example, when considering effects of internal cavitation as demonstrated in Section 2.3.

References

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