Contact Mechanics and Friction: Physical Principles and Applications

1 Introduction ........................................................................................................ 1
1.1 Contact and Friction Phenomena and their Applications ............................. 1
1.2 History of Contact Mechanics and the Physics of Friction .......................... 3
1.3 Structure of the Book ................................................................................... 7
2 Qualitative Treatment of Contact Problems – Normal Contact without
Adhesion ................................................................................................................ 9
2.1 Material Properties ..................................................................................... 10
2.2 Simple Contact Problems........................................................................... 13
2.3 Estimation Method for Contacts with a Three-Dimensional, Elastic
Continuum ....................................................................................................... 16
Problems .......................................................................................................... 20
3 Qualitative Treatment of Adhesive Contacts ................................................. 27
3.1 Physical Background ................................................................................. 28
3.2 Calculation of the Adhesive Force between Curved Surfaces ................... 32
3.3 Qualitative Estimation of the Adhesive Force between Elastic Bodies ..... 33
3.4 Influence of Roughness on Adhesion ........................................................ 35
3.5 Adhesive Tape ........................................................................................... 36
3.6 Supplementary Information about van der Waals Forces and Surface
Energies ........................................................................................................... 37
Problems .......................................................................................................... 38
4 Capillary Forces ............................................................................................... 43
4.1 Surface Tension and Contact Angles ......................................................... 43
4.2 Hysteresis of Contact Angles ..................................................................... 47
4.3 Pressure and the Radius of Curvature ........................................................ 47
4.4 Capillary Bridges ....................................................................................... 48
4.5 Capillary Force between a Rigid Plane and a Rigid Sphere ...................... 49
4.6 Liquids on Rough Surfaces ........................................................................ 50
4.7 Capillary Forces and Tribology ................................................................. 51
Problems .......................................................................................................... 52
5 Rigorous Treatment of Contact Problems – Hertzian Contact .................... 57
5.1 Deformation of an Elastic Half-Space being Acted upon by Surface Forces.58
5.2 Hertzian Contact Theory ............................................................................ 61
5.3 Contact between Two Elastic Bodies with Curved Surfaces ..................... 62
5.4 Contact between a Rigid Cone-Shaped Indenter and an Elastic Half-Space..65
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5.5 Internal Stresses in Hertzian Contacts ....................................................... 66
5.6 Method of Dimensionality Reduction (MDR) ........................................... 69
Problems .......................................................................................................... 72
6 Rigorous Treatment of Contact Problems – Adhesive Contact .................... 83
6.1 JKR-Theory ............................................................................................... 84
6.2 Adhesive Contact of Rotationally Symmetrical Bodies ............................. 89
Problems .......................................................................................................... 92
7 Contact between Rough Surfaces .................................................................... 97
7.1 Model from Greenwood and Williamson .................................................. 98
7.2 Plastic Deformation of Asperities ............................................................ 103
7.3 Electrical Contacts ................................................................................... 104
7.4 Thermal Contacts ..................................................................................... 108
7.5 Mechanical Stiffness of Contacts ............................................................. 109
7.6 Seals ......................................................................................................... 109
7.7 Roughness and Adhesion ......................................................................... 110
Problems ........................................................................................................ 111
8 Tangential Contact Problems ........................................................................ 117
8.1 Deformation of an Elastic Half-Space being Acted upon
Forces ...................................................................................... 118
8.2 Deformation of an Elastic Half-Space being Acted upon
Stress Distribution ................................................................ 119
8.3 Tangential Contact Problems without Slip .............................................. 121
8.4 Tangential Contact Problems Accounting for Slip .................................. 122
8.5 Absence of Slip for a Rigid Cylindrical Indenter ..................................... 126
8.6 Tangential Contact of Axially Symmetrical Bodies ................................ 126
Problems ........................................................................................................ 129
9 Rolling Contact ............................................................................................... 137
9.1 Qualitative Discussion of the Processes in a Rolling Contact.................. 138
9.2 Stress Distribution in a Stationary Rolling Contact ................................. 140
Problems ........................................................................................................ 146
10 Coulomb’s Law of Friction ......................................................................... 151
10.1 Introduction ............................................................................................ 151
10.2 Static and Kinetic Friction ..................................................................... 152
10.3 Angle of Friction .................................................................................... 153
10.4 Dependence of the Coefficient of Friction on the Contact Time ........... 154
10.5 Dependence of the Coefficient of Friction on the Normal Force ........... 156
10.6 Dependence of the Coefficient of Friction on Sliding Speed ................. 157
10.7 Dependence of the Coefficient of Friction on the Surface Roughness .. 157
10.8 Coulomb’s View on the Origin of the Law of Friction .......................... 158
10.9 Theory of Bowden and Tabor ................................................................ 160
by Tangential
by a Tangential
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10.10 Dependence of the Coefficient of Friction on Temperature ................ 163
Problems ........................................................................................................ 164
11 The Prandtl-Tomlinson Model for Dry Friction ....................................... 173
11.1 Introduction ........................................................................................... 173
11.2 Basic Properties of the Prandtl-Tomlinson Model ................................. 175
11.3 Elastic Instability ................................................................................... 179
11.4 Superlubricity ........................................................................................ 183
11.5 Nanomachines: Concepts for Micro and Nano-Actuators ..................... 184
Problems ........................................................................................................ 188
12 Frictionally Induced Vibrations .................................................................. 193
12.1 Frictional Instabilities at Decreasing Dependence
on the Velocity .......................................................... 194
12.2 Instability in a System with Distributed Elasticity ................................. 196
12.3 Critical Damping and Optimal Suppression of Squeal .......................... 199
12.4 Active Suppression of Squeal ................................................................ 201
12.5 Strength Aspects during Squeal ............................................................. 203
12.6 Dependence of the Stability Criteria on the Stiffness of the System ..... 204
12.7 Sprag-Slip .............................................................................................. 209
Problems ........................................................................................................ 211
13 Thermal Effects in Contacts ....................................................................... 217
13.1 Introduction ........................................................................................... 218
13.2 Flash Temperatures in Micro-Contacts .................................................. 218
13.3 Thermo-Mechanical Instability .............................................................. 220
Problems ........................................................................................................ 221
14 Lubricated Systems ...................................................................................... 225
14.1 Flow between two parallel plates ........................................................... 226
14.2 Hydrodynamic Lubrication .................................................................... 227
14.3 “Viscous Adhesion” ............................................................................... 231
14.4 Rheology of Lubricants ......................................................................... 234
14.5 Boundary Layer Lubrication .................................................................. 236
14.6 Elastohydrodynamics ............................................................................. 237
14.7 Solid Lubricants ..................................................................................... 241
Problems ........................................................................................................ 242
15 Viscoelastic Properties of Elastomers ......................................................... 255
15.1 Introduction ........................................................................................... 255
15.2 Stress-Relaxation ................................................................................... 256
15.3 Complex, Frequency-Dependent Shear Moduli ..................................... 258
15.4 Properties of Complex Moduli............................................................... 260
15.5 Energy Dissipation in a Viscoelastic Material ....................................... 261
15.6 Measuring Complex Moduli .................................................................. 262
of the Frictional Force
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15.7 Rheological Models ............................................................................... 263
15.8 A Simple Rheological Model for Rubber (“Standard Model”) .............. 266
15.9 Influence of Temperature on Rheological Properties ............................ 268
15.10 Master Curves ...................................................................................... 269
15.11 Prony Series ......................................................................................... 270
15.12 Application of the Method of Dimensionality Reduction
Media .................................................................................... 274
Problems ........................................................................................................ 275
16 Rubber Friction and Contact Mechanics of Rubber ................................. 283
16.1 Friction between an Elastomer and a Rigid Rough Surface ................... 283
16.2 Rolling Resistance ................................................................................. 289
16.3 Adhesive Contact with Elastomers ........................................................ 291
Problems ........................................................................................................ 293
17 Wear .............................................................................................................. 299
17.1 Introduction ............................................................................................ 299
17.2 Abrasive Wear ....................................................................................... 300
17.3 Adhesive Wear ....................................................................................... 303
17.4 Conditions for Low-Wear Friction ........................................................ 306
17.5 Wear as the Transportation of Material from the Friction Zone ............ 307
17.6 Wear of Elastomers ................................................................................ 308
Problems ........................................................................................................ 311
18 Friction Under the Influence of Ultrasonic Vibrations ............................. 317
18.1 Influence of Ultrasonic Vibrations on Friction from
of View ........................................................................ 318
18.2 Influence of Ultrasonic Vibrations on Friction from
of View ......................................................................... 323
18.3 Experimental Investigations of the Force of Static Friction
of the Oscillation Amplitude .................................................... 325
18.4 Experimental Investigations of Kinetic Friction as a Function
Problems ........................................................................................................ 329
19 Numerical Simulation Methods in Friction Physics .................................. 335
19.1 Many-Body Systems .............................................................................. 336
19.2 Finite Element Method .......................................................................... 337
19.3 Boundary Element Method .................................................................... 337
19.4 Boundary Element Method: Tangential Contact .................................... 339
19.5 Boundary Element Method: Adhesive Contact ...................................... 340
19.6 Particle Methods .................................................................................... 342
19.7 Method of Dimensionality Reduction .................................................... 342
to Viscoelastic
a Macroscopic Point
a Microscopic Point
as a Function
of Oscillation Amplitude ............................................................................... 327
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20.1 Introduction ........................................................................................... 344
20.2 Quantification of Earthquakes ............................................................... 345
20.2.1 Gutenberg-Richter Law .................................................................. 346
20.3 Laws of Friction for Rocks .................................................................... 348
20.4 Stability during Sliding with Rate- and State-Dependent Friction ........ 351
20.5 Nucleation of Earthquakes and Post-Sliding ......................................... 354
20.6 Foreshocks and Aftershocks .................................................................. 358
20.7 Continuum Mechanics of Block Media and the Structure of Faults ...... 358
20.8 Is it Possible to Predict Earthquakes? .................................................... 362
Problems ........................................................................................................ 363
Appendix ............................................................................................................ 367
Appendix A - Normal Displacement under the Effect of Selected
Distributions ................................................................................... 367
Appendix B - Normal Contact of Axially Symmetrical Profiles ................... 372
Appendix C - Adhesive Contact of Axially Symmetrical Profiles ................ 375
Appendix D - Tangential Contact of Axially Symmetrical Profiles .............. 376
Further Reading ................................................................................................ 379
Figure Reference ............................................................................................... 385
Index ................................................................................................................... 387