Preface	6
	Contents	8
	1 Kinematics of Contact	19
	1.1 Reference Frame and the Initial Gap Function	20
	1.2 Establishment of a Contact Region	21
	1.2.1 Definition of Contact	22
	1.2.2 The Boundary Value Problem	22
	1.2.3 Signorini Problems	23
	1.2.4 Asymptotic Arguments	23
	1.2.5 The Discrete Problem	25
	1.3 Nonlinear Kinematics	26
	1.4 Almost Conformal Contact	27
	2 Three-Dimensional Frictionless Elastic Problems	30
	2.1 The Half-Space Approximation	30
	2.2 Normal Loading of the Half-Space	31
	2.2.1 The Point Force Solution	32
	2.2.2 Similarity, Equilibrium and Anisotropy	33
	2.2.3 The Composite Elastic Modulus	34
	2.3 Integral Equation Formulation	35
	2.3.1 Field-Point Integration	37
	2.3.2 Indentation by a Flat Elliptical Punch	37
	2.4 Galin's Theorem	40
	2.4.1 A Special Case	41
	2.5 Interior Stress Fields	42
	2.5.1 In-Plane Stress Components Near the Surface	42
	3 Hertzian Contact	45
	3.1 Transformation of Coordinates	45
	3.1.1 Cylinders and Spheres	47
	3.1.2 More General Cases	48
	3.2 Hertzian Pressure Distribution	49
	3.3 Strategy for Hertzian Contact Calculations	50
	3.3.1 Eccentricity of the Contact Area	50
	3.3.2 Dimensions of the Contact Area	51
	3.3.3 Highly Elliptical Contacts	54
	3.4 First Yield	55
	4 More General Problems for the Half-Space	58
	4.1 The Electrical--Mechanical Analogy	59
	4.1.1 Other Mathematical Analogies	61
	4.1.2 Boyer's Approximation	63
	4.1.3 Fabrikant's Approximation	64
	4.2 General Theorems for Frictionless Contact	66
	4.3 Superposition by Differentiation	70
	4.4 The Force--Displacement Relation	72
	4.4.1 Non-conformal Contact Problems	73
	5 Axisymmetric Contact Problems	77
	5.1 Green and Collins Solution	77
	5.1.1 The Flat Punch Solution	79
	5.2 Non-conformal Contact Problems	80
	5.3 Annular Contact Regions	82
	5.4 The Non-axisymmetric Cylindrical Punch	83
	5.5 The Method of Dimensionality Reduction (MDR)	84
	6 Two-Dimensional Frictionless Contact Problems	90
	6.1 The Line Force Solution	91
	6.2 Integral Equation Formulation	93
	6.2.1 Edge Conditions	94
	6.3 Incremental Solution of Non-conformal Contact Problems	98
	6.3.1 Symmetric Problems	98
	6.3.2 Bounded-Singular Problems	99
	6.4 Solution by Fourier Series	99
	6.4.1 Rigid-Body Rotation	100
	6.4.2 Galin's Theorem, Chebyshev Polynomials and Recurrence Relations	102
	6.5 Periodic Contact Problems	104
	6.5.1 Sinusoidal Contact Pressure	104
	6.5.2 Fourier Series Methods	105
	6.5.3 The Periodic Green's Function	106
	6.5.4 The Cotangent Transform	106
	6.5.5 Manners' Solution	107
	6.5.6 Westergaard's Problem	109
	6.6 The Smirnov--Sobolev Transform	110
	6.6.1 Inversion of the Transform	111
	6.6.2 Example: Uniform Loading Over the Circle	111
	6.6.3 Anisotropic Problems	112
	6.7 Displacements in Two-Dimensional Problems	113
	6.7.1 Kalker's Line Contact Theory	115
	7 Tangential Loading	121
	7.1 Kinematics	121
	7.1.1 Gross Slip and Microslip	122
	7.2 Green's Functions for Tangential Forces and Displacements	123
	7.2.1 Three-Dimensional [point] Loading	123
	7.2.2 Two-Dimensional [line] Loading	125
	7.2.3 Normal-Tangential Coupling	126
	7.3 Two-Dimensional Flat Rigid Punch with No Slip	127
	7.3.1 Uncoupled Problem	129
	7.3.2 Oscillatory Singularities	129
	7.4 Axisymmetric Flat Rigid Punch with No Slip	131
	7.5 The `Goodman' Approximation	133
	7.6 Uniform Tangential Displacement in a Prescribed Area	135
	7.6.1 Tangential Loading over a Circular Area	135
	7.6.2 Tangential Loading over an Elliptical Area	136
	7.6.3 Two Conjectures	138
	7.7 Non-conformal Contact Problems with No Slip	139
	7.7.1 Uncoupled Hertzian Contact with Tangential Loading	140
	7.7.2 The Coupled Axisymmetric Problem under Purely Normal Loading	141
	7.7.3 The Coupled Two-Dimensional Problem	142
	7.7.4 Relaxation Damping	144
	8 Friction Laws	149
	8.1 Amontons' Law	149
	8.1.1 Continuum Problems	150
	8.1.2 Two-Dimensional Problems	151
	8.1.3 Existence and Uniqueness Theorems	151
	8.2 The Klarbring Model	152
	8.2.1 General Loading Scenarios	154
	8.2.2 The Critical Coefficient of Friction	154
	8.2.3 Wedging	155
	8.3 Multinode Systems	156
	8.3.1 The Evolution and Rate Problems	157
	8.3.2 Algorithms for Two-Dimensional Problems with Time-Varying Forces	157
	8.3.3 History-Dependence and Memory	158
	8.3.4 Klarbring's P-Matrix Criterion	159
	8.4 Periodic Loading	160
	8.4.1 A Uniqueness Proof for Uncoupled Systems	161
	8.4.2 Shakedown	163
	8.4.3 Coupled Systems	163
	8.4.4 Asymptotic Approach to a Steady State	163
	8.5 A Simple Continuum Frictional System	164
	8.5.1 Unloading	167
	8.5.2 Periodic Loading	168
	8.5.3 Discrete Model of the Strip Problem	169
	8.5.4 The Inverse Problem	169
	8.6 More Complex Friction Laws	170
	8.6.1 Instabilities During Steady Sliding	171
	8.6.2 Velocity-Dependent Friction Coefficient	171
	8.6.3 Stick-Slip Vibrations	173
	8.6.4 Slip-Weakening Laws	174
	8.6.5 Rate-State Laws	175
	9 Frictional Problems Involving Half-Spaces	181
	9.1 Cattaneo's Problem	181
	9.2 The Ciavarella--Jäger Theorem	184
	9.2.1 Three-Dimensional Problems	186
	9.3 More General Loading Scenarios	187
	9.3.1 Constant Normal Force	187
	9.3.2 Variable Normal Force	188
	9.3.3 Memory and `Advancing Stick'	190
	9.4 The Effect of Bulk Stress	191
	9.4.1 Hertz Problem with Superposed Bulk Stress	191
	9.4.2 Combined Bulk Stress and Tangential Force	193
	9.5 Coupled Problems	196
	9.5.1 Indentation by a Two-Dimensional Flat Rigid Punch	196
	9.5.2 Normal Loading for More General Geometries	199
	9.5.3 Combined Normal and Tangential Loading	201
	9.5.4 Unloading	201
	9.5.5 Periodic Loading	202
	10 Asymptotic Methods	206
	10.1 Indentation by a Frictionless Rigid Punch	206
	10.1.1 Eigenfunction Series	208
	10.1.2 More General Frictionless Indentation Problems	209
	10.1.3 Non-conformal Problems	210
	10.1.4 Both Materials Deformable	211
	10.2 No-Slip Conditions	212
	10.3 Frictional Slip	213
	10.3.1 Slip-Separation Transition	214
	10.3.2 Slip--Stick Transition	215
	10.4 Indentation by an Elastic Wedge	216
	10.4.1 Right-Angle Wedge of the Same Material	217
	10.4.2 A Slipping Interface	218
	10.5 Local Fields	219
	10.5.1 The Flat and Rounded Indenter	220
	10.5.2 Fretting in Non-conformal Contact	222
	10.5.3 Edge Slip Zones with a Rigid Punch	223
	10.5.4 Slip Zones in Conformal Contact	225
	11 Receding Contact	232
	11.1 Characteristics of Receding Contact	233
	11.1.1 Examples of Receding Contact	234
	11.2 Frictional Problems	237
	11.2.1 Frictional Unloading	237
	11.3 Thermoelastic Problems	239
	11.4 Almost Conformal Contact Problems	240
	12 Adhesive Forces	244
	12.1 Adhesion Between Rigid Bodies	247
	12.2 The JKR Theory	248
	12.2.1 Axisymmetric Problems	249
	12.2.2 Indentation by a Sphere	250
	12.2.3 Energetic Considerations and Stability	252
	12.2.4 Hysteretic Energy Dissipation	254
	12.2.5 JKR Solution for More General Axisymmetric Bodies	254
	12.2.6 Guduru's Problem	256
	12.3 The Tabor Parameter	257
	12.3.1 An Adhesive Length Scale	259
	12.3.2 Limitations on the JKR Solution	260
	12.4 Solutions for Finite Tabor Parameter	261
	12.4.1 Jump-In at Large Tabor Parameter	262
	12.4.2 Simplified Force Laws	263
	12.4.3 Maugis' Solution	264
	12.4.4 The `double-Hertz' Approximation	267
	12.4.5 More General Axisymmetric Geometries	269
	12.5 Other Geometries	269
	12.5.1 Two-Dimensional Problems	269
	12.5.2 Elliptical Contact Area	270
	12.5.3 General Three-Dimensional Geometries	271
	13 Beams, Plates, Membranes and Shells	274
	13.1 Contact of Beams	274
	13.1.1 A Heavy Beam Lifted from the Ground	276
	13.1.2 Adhesive Forces	277
	13.1.3 Piston Ring in a Cylinder	278
	13.1.4 Two and Three-Dimensional Effects	281
	13.1.5 Matched Asymptotic Expansions	282
	13.2 Contact of Plates	285
	13.2.1 Displacement Due to a Concentrated Point Force	286
	13.2.2 Indentation by a Rigid Sphere	286
	13.3 Membrane Effects	288
	13.3.1 `Membrane Only' Solutions	289
	13.4 Contact of Shells	292
	13.5 Implications for Finite Element Solutions	296
	14 Layered Bodies	300
	14.1 Esll El: Plate on an Elastic Foundation	301
	14.1.1 Choice of Foundation Modulus	302
	14.1.2 Two-Dimensional Problems	302
	14.1.3 Three-Dimensional Problems	305
	14.2 Esgg El: Layer on a Rigid Foundation	306
	14.2.1 Frictionless Unbonded Layer	307
	14.2.2 Bonded Compressible Layer	309
	14.2.3 Bonded Incompressible Layer	309
	14.2.4 Flat Punch Problems	314
	14.2.5 Frictional Problems	315
	14.2.6 Effect of Adhesive Forces	315
	14.3 Winkler Layer on an Elastic Foundation	318
	14.3.1 Nonlinear Layers	319
	14.4 Fourier Transform Methods	320
	14.4.1 Elastic Layer Bonded to a Rigid Foundation	320
	14.4.2 Multilayered Bodies	324
	14.5 Functionally Graded Materials	324
	14.5.1 Exponential Variation of Modulus	325
	14.5.2 Power-Law Grading	326
	14.5.3 Linear Variation of Modulus	329
	15 Indentation Problems	333
	15.1 The Hardness Test	333
	15.2 Power-Law Material	334
	15.2.1 Graded Materials	336
	15.3 Other Constitutive Laws	337
	16 Contact of Rough Surfaces	339
	16.1 Bowden and Tabor's Theory of Friction	339
	16.1.1 The Ploughing Force	340
	16.1.2 Plastic Deformation at an Actual Contact	341
	16.1.3 The Effect of Surface Films	342
	16.2 Profilometry	343
	16.2.1 The Bearing Area Curve	344
	16.2.2 The Contact Problem	346
	16.3 Asperity Model Theories	347
	16.3.1 The Exponential Distribution	349
	16.3.2 The Gaussian Distribution	350
	16.3.3 The Plasticity Index	352
	16.4 Statistical Models of Surfaces	353
	16.4.1 Discrete Models	353
	16.4.2 Random Process Models	355
	16.4.3 Determining Asperity Parameters	361
	16.5 Fractal Surfaces	362
	16.5.1 Archard's Model	362
	16.5.2 Self-affine Fractals and the Fractal Dimension	362
	16.5.3 The Weierstrass Function	364
	16.5.4 Generating Realizations of Fractal Profiles and Surfaces	366
	16.6 Contact of Fractal Surfaces	369
	16.6.1 Majumdar and Bhushan's Theory	369
	16.6.2 Elastic Contact for a Fractal Surface	370
	16.6.3 The Weierstrass Profile	372
	16.6.4 Persson's Theory	374
	16.6.5 Implications for Coulomb's Law of Friction	378
	16.7 Adhesive Forces	379
	16.7.1 Asperity Model Predictions	380
	16.7.2 The Sinusoidal Profile	381
	16.7.3 Adhesion of Random Rough Surfaces	384
	16.8 Incremental Stiffness and Contact Resistance	385
	16.8.1 Asperity Model Predictions	386
	16.8.2 Clustering of Actual Contacts	387
	16.8.3 Bounds on Incremental Stiffness	388
	16.8.4 Persson's Theory of Incremental Stiffness	390
	16.8.5 Gaps and Fluid Leakage	391
	16.9 Finite-Size Effects	392
	16.9.1 Integral Equation Formulation	393
	16.9.2 Unit Cells and the Constriction Alleviation Factor	396
	16.9.3 Contact of Rough Spheres	397
	17 Thermoelastic Contact	405
	17.1 Thermoelastic Deformation	406
	17.1.1 Fourier Transform Solutions	406
	17.1.2 Steady-State Temperature	407
	17.1.3 Thermoelastic Distortion Due to a Point Heat Source	408
	17.1.4 Dundurs' Theorem	409
	17.1.5 Moving Heat Sources	410
	17.2 The Axisymmetric Thermoelastic Hertz Problem	411
	17.2.1 The Heat Conduction Problem	412
	17.2.2 Thermoelastic Distortion	413
	17.2.3 Solution of the Contact Problem	413
	17.3 Existence and Uniqueness	415
	17.3.1 A One-Dimensional Model	416
	17.3.2 Effect of a Thermal Interface Resistance	417
	17.3.3 Imperfect Thermal Contact	419
	17.3.4 The Hertz Problem Revisited	420
	17.3.5 Stability	420
	17.3.6 Contact of Dissimilar Materials	423
	17.3.7 Two-Dimensional Stability Problems	423
	17.4 Solidification Problems	425
	17.5 Frictional Heating	427
	17.5.1 The Rod Model	429
	17.5.2 Burton's Stability Analysis	430
	17.5.3 Out-of-Plane Sliding	431
	17.5.4 In-Plane Sliding	433
	17.5.5 Limiting Configurations	435
	17.5.6 Effect of Geometry	437
	17.5.7 Numerical Solutions	439
	18 Rolling and Sliding Contact	443
	18.1 Rigid-Body Kinematics	443
	18.1.1 Three-Dimensional Motions	445
	18.2 Johnson's Belt Drive Problem	448
	18.3 Tractive Rolling of Elastic Cylinders	451
	18.3.1 Dissimilar Materials	455
	18.3.2 Antiplane Loading	456
	18.3.3 Rolling of Misaligned Cylinders	456
	18.3.4 Three-Dimensional Rolling Contact Problems	457
	18.3.5 Kalker's Strip Theory	458
	18.3.6 The Incipient Sliding Solution	460
	18.3.7 Transient Problems	460
	18.3.8 Rail Corrugations	461
	18.4 Steady Sliding	462
	18.4.1 Two-Dimensional Problems	462
	18.4.2 Three-Dimensional Problems	464
	18.5 Wear	465
	18.5.1 Archard's Wear Law	465
	18.5.2 Long-Time Solution	466
	18.5.3 Transient Problems	467
	18.5.4 Galin's Eigenfunction Method	469
	18.5.5 Non-conformal Contact Problems	471
	18.6 Sliding of Rough Surfaces	472
	18.6.1 Flash Temperatures	473
	18.6.2 Bulk Temperatures	478
	18.6.3 Transient Asperity Interactions	479
	19 Elastodynamic Contact Problems	484
	19.1 Wave Speeds	485
	19.1.1 Rayleigh Waves	486
	19.2 Moving Contact Problems	487
	19.2.1 The Moving Line Force	487
	19.2.2 Integral Equation Formulation	488
	19.2.3 The Subsonic Problem	489
	19.2.4 The Speed Range cR	490
	19.2.5 The Solution of Slepyan and Brun	491
	19.2.6 The Transonic Solution c2	493
	19.2.7 The Superseismic Solution V>c1	494
	19.2.8 Three-Dimensional Problems	496
	19.3 Interaction of a Bulk Wave with an Interface	499
	19.3.1 SH-Waves Transmitted Across a Frictional Interface	499
	19.3.2 In-Plane Waves	505
	19.4 Interface Waves	507
	19.4.1 Slip Waves	508
	19.4.2 Slip Waves at a Sliding Interface	509
	19.4.3 Slip--Stick Waves	510
	19.5 Stability of Frictional Sliding	512
	19.6 Transient Elastodynamic Contact Problems	513
	19.6.1 Impulsive Line Force	513
	19.6.2 A Uniform Pressure Suddenly Applied	513
	19.6.3 Integral Equation Formulation of the Transient Contact Problem	514
	19.6.4 Normal Indentation by a Rigid Body	515
	19.6.5 Superseismic Indentation	516
	19.6.6 Self-Similar Indentation Problems	517
	19.6.7 Three-Dimensional Transient Problems	518
	20 Impact	522
	20.1 Hertz' Theory of Impact	523
	20.1.1 Duration of the Impact	524
	20.1.2 Homogeneous Sphere	526
	20.1.3 Range of Validity of the Theory	526
	20.1.4 The Superseismic Phase	527
	20.2 Impact of a Cylinder	528
	20.3 Oblique Impact	530
	20.3.1 The Equation of Motion	531
	20.3.2 The Tangential Contact Problem	532
	20.3.3 Complete Stick	532
	20.3.4 Gross Slip	535
	20.3.5 Partial Slip	535
	20.3.6 The Complete Trajectory	536
	20.3.7 Rebound Conditions	537
	20.4 One-Dimensional Bar Problems	538
	20.4.1 The Semi-infinite Bar	539
	20.4.2 The Infinite Bar	540
	20.4.3 Reflections	541
	20.4.4 The Impact Problem	542
	20.4.5 A Rigid Mass Impacting an Elastic Bar	542
	20.4.6 Frictional Problems	545
	20.4.7 Continuous Frictional Supports	547
	Appendix A Potential Function Solutions for Elasticity Problems	551
	A.1 Frictionless Problems	551
	A.2 Problems with Tangential Tractions	552
	A.3 Two-Dimensional Problems	554
	Appendix B Integrals over Elliptical Domains	555
	B.1 Mathematical Preliminaries	556
	B.1.1 The Singular Field n=0	557
	B.1.2 The Hertzian Field n=1	557
	B.2 Applications	558
	B.2.1 Normal Loading of an Isotropic Half-Space	558
	B.2.2 The Anisotropic Half-Space	559
	B.2.3 Tangential Loading of an Isotropic Half-Space	559
	B.3 Evaluation of Integrals	561
	Appendix C Cauchy Singular Integral Equations	562
	C.1 Integral Equations of the First Kind	562
	C.2 Integral Equations of the Second Kind	564
	Appendix D Dundurs' Bimaterial Constants	566
	References	568
	Index	588