Contact Mechanics and Friction

Table of Contents
1 Introduction
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 25
3.1 Physical Background 26
3.2 Calculation of the Adhesive Force between Curved Surfaces 30
3.3 Qualitative Estimation of the Adhesive Force between Elastic Bodies 31
3.4 Influence of Roughness on Adhesion 33
3.5 Adhesive Tape 34
3.6 Supplementary Information about van der Waals Forces and Surface  Energies 35
Problems 36

4 Capillary Forces 41
4.1 Surface Tension and Contact Angles 41
4.2 Hysteresis of Contact Angles 45
4.3 Pressure and the Radius of Curvature 45
4.4 Capillary Bridges 46
4.5 Capillary Force between a Rigid Plane and a Rigid Sphere 47
4.6 Liquids on Rough Surfaces 48
4.7 Capillary Forces and Tribology 49
Problems 50
5 Rigorous Treatment of Contact Problems – Hertzian Contact 55
5.1 Deformation of an Elastic Half-Space being Acted upon by Surface   Forces 56
5.2 Hertzian Contact Theory 59
5.3 Contact between Two Elastic Bodies with Curved Surfaces 60
5.4 Contact between a Rigid Cone-Shaped Indenter and an Elastic  Half-Space 63
5.5 Internal Stresses in Hertzian Contacts 64
Problems 67

6 Rigorous Treatment of Contact Problems – Adhesive Contact 71
6.1 JKR-Theory 72
Problems 77

7 Contact between Rough Surfaces 81
7.1 Model from Greenwood and Williamson 82
7.2 Plastic Deformation of Asperities 88
7.3 Electrical Contacts 89
7.4 Thermal Contacts 92
7.5 Mechanical Stiffness of Contacts 93
7.6 Seals 93
7.7 Roughness and Adhesion 94
Problems 95

8 Tangential Contact Problems 105
8.1 Deformation of an Elastic Half-Space being Acted upon by  Tangential Forces  106
8.2 Deformation of an Elastic Half-Space being Acted upon  Tangential Stress Distribution 107
8.3 Tangential Contact Problems without Slip 109
8.4 Tangential Contact Problems Accounting for Slip 110
8.5 Absence of Slip for a Rigid Cylindrical Indenter 114
Problems 114

9 Rolling Contact 119
9.1 Qualitative Discussion of the Processes in a Rolling Contact 120
9.2 Stress Distribution in a Stationary Rolling Contact 122
Problems 128

10 Coulomb’s Law of Friction  133
10.1 Introduction 133
10.2 Static and Kinetic Friction 134
10.3 Angle of Friction 135
10.4 Dependence of the Coefficient of Friction on the Contact Time 136
10.5 Dependence of the Coefficient of Friction on the Normal Force 137
10.6 Dependence of the Coefficient of Friction on Sliding Speed 139
10.7 Dependence of the Coefficient of Friction on the Surface Roughness 139
10.8 Coulomb’s View on the Origin of the Law of Friction 140
10.9 Theory of Bowden and Tabor 142
10.10 Dependence of the Coefficient of Friction on Temperature 145
Problems 146

11 The Prandtl-Tomlinson Model for Dry Friction 155
11.1 Introduction 155
11.2 Basic Properties of the Prandtl-Tomlinson Model 157
11.3 Elastic Instability 161
11.4 Superlubricity 165
11.5 Nanomachines: Concepts for Micro and Nano-Actuators 166
Problems 170

12 Frictionally Induced Vibrations 175
12.1 Frictional Instabilities at Decreasing Dependence of the Frictional  Force on the Velocity 176
12.2 Instability in a System with Distributed Elasticity 178
12.3 Critical Damping and Optimal Suppression of Squeal 181
12.4 Active Suppression of Squeal 183
12.5 Strength Aspects during Squeal 185
12.6 Dependence of the Stability Criteria on the Stiffness of the System 186
12.7 Sprag-Slip 191
Problems 193

13 Thermal Effects in Contacts  199
13.1 Introduction 200
13.2 Flash Temperatures in Micro-Contacts 200
13.3 Thermo-Mechanical Instability 202
Problems 203

14 Lubricated Systems 207
14.1 Flow between two parallel plates 208
14.2 Hydrodynamic Lubrication 209
14.3 “Viscous Adhesion”  213
14.4 Rheology of Lubricants 216
14.5 Boundary Layer Lubrication 218
14.6 Elastohydrodynamics 219
14.7 Solid Lubricants 222
Problems 223

15 Viscoelastic Properties of Elastomers 231
15.1 Introduction 231
15.2 Stress-Relaxation 232
15.3 Complex, Frequency-Dependent Shear Moduli 234
15.4 Properties of Complex Moduli 236
15.5 Energy Dissipation in a Viscoelastic Material 237
15.6 Measuring Complex Moduli 238
15.7 Rheological Models 239
15.8 A Simple Rheological Model for Rubber (“Standard Model”)  242
15.9 Influence of Temperature on Rheological Properties 244
15.10 Master Curves 245
15.11 Prony Series 246
Problems 250

16 Rubber Friction and Contact Mechanics of Rubber 255
16.1 Friction between an Elastomer and a Rigid Rough Surface 255
16.2 Rolling Resistance 261
16.3 Adhesive Contact with Elastomers 263
Problems 265

17 Wear 271
17.1 Introduction 271
17.2 Abrasive Wear 272
17.3 Adhesive Wear 275
17.4 Conditions for Low-Wear Friction 278
17.5 Wear as the Transportation of Material from the Friction Zone 279
17.6 Wear of Elastomers 280
Problems 283

18 Friction Under the Influence of Ultrasonic Vibrations 285
18.1 Influence of Ultrasonic Vibrations on Friction from a Macroscopic  Point of View 286
18.2 Influence of Ultrasonic Vibrations on Friction from a Microscopic  Point of View 291
18.3 Experimental Investigations of the Force of Static Friction as a  Function of the Oscillation Amplitude 293
18.4 Experimental Investigations of Kinetic Friction as a Function of  Oscillation Amplitude 295
Problems 297

19 Numerical Simulation Methods in Friction Physics 301
19.1 Simulation Methods for Contact and Frictional Problems:  An Overview 302
19.1.1 Many-Body Systems 302
19.1.2 Finite Element Methods 303
19.1.3 Boundary Element Method 304
19.1.4 Particle Methods 305
19.2 Reduction of Contact Problems from Three Dimensions to One Dimension 306
19.3 Contact in a Macroscopic Tribological System 307
19.4 Reduction Method for a Multi-Contact Problem 311
19.5 Dimension Reduction and Viscoelastic Properties 315
19.6 Representation of Stress in the Reduction Model 316
19.7 The Calculation Procedure in the Framework of the Reduction  Method 317
19.8 Adhesion, Lubrication, Cavitation, and Plastic Deformations in the  Framework of the Reduction Method 318
Problems 318

20 Earthquakes and Friction 323
20.1 Introduction 324
20.2 Quantification of Earthquakes 325
20.2.1 Gutenberg-Richter Law 326
20.3 Laws of Friction for Rocks 327
20.4 Stability during Sliding with Rate- and State-Dependent Friction 331
20.5 Nucleation of Earthquakes and Post-Sliding 334
20.6 Foreshocks and Aftershocks 337
20.7 Continuum Mechanics of Block Media and the Structure of Faults 338
20.8 Is it Possible to Predict Earthquakes 342
Problems 343
Appendix 347
Further Reading 351
Figure Reference 357
Index 359


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