Contact Mechanics (eBook)

James Richard Barber graduated in Mechanical Sciences from the University of Cambridge in 1963. He then joined British Rail, who later sponsored his research at Cambridge between 1965 and 1968 on the subject of thermal effects in braking systems. In 1969 he became a Lecturer and later Reader in Solid Mechanics at the University of Newcastle upon Tyne, U.K.  He moved to the University of Michigan, Department of Mechanical Engineering in 1981. His current research interests are in solid mechanics with particular reference to thermoelasticity, contact mechanics and tribology. He is a Chartered Engineer in the U.K., a Fellow of the Institution of Mechanical Engineers and has engaged extensively in consulting work in the field of stress analysis for engineering design. Dr. Barber is author of two books and numerous articles in the fields of Elasticity, Thermoelasticity, Contact Mechanics, Tribology, Heat Conduction and Elastodynamics and he is a member of the editorial boards of the International Journal of Mechanical Sciences and the Journal of Thermal Stresses. 

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< V<
19.2.5 The Solution of Slepyan and Brun 491
19.2.6 The Transonic Solution c2< V<
19.2.7 The Superseismic Solution V> c1
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