Contact Mechanics and Friction (eBook)

Prof. Dr. Valentin L. Popovstudied physics and obtained his doctorate from the Moscow State Lomonosow University. He worked at the Institute of Strength Physics and Materials Science of the Russian Academy of Sciences. After a guest-professorship in the field of theoretical physics at the University of Paderborn, he has headed the department of System Dynamics and Friction Physics in the Institute of Mechanics at the Berlin University of Technology since 2002. His areas of interest, among others, include tribology, nanotribology, tribology at low temperatures, biotribology, the influence of friction through ultrasound, numerical simulation of frictional processes, research regarding earthquakes, as well as themes relating to materials sciences such as the mechanics of elastoplastic media with microstructure, strength of metals and alloys, and shape memory alloys. He is the joint editor of international journals and regularly organizes international conferences and workshops over diverse tribological themes.

Preface to the Second English Edition 6
From the Preface to the First English Edition 7
Preface to the First German Edition 8
Thanks 9
Table of Contents 10
1 Introduction 15
1.1 Contact and Friction Phenomena and their Applications 15
1.2 History of Contact Mechanics and the Physics of Friction 17
1.3 Structure of the Book 21
2 Qualitative Treatment of Contact Problems – Normal Contact without Adhesion 22
2.1 Material Properties 23
2.2 Simple Contact Problems 26
(1) Parallelepiped 26
(2) Thin Sheets 26
(3) Spherical Cap 27
(4) Contact between a thin elastic sheet on a rigid, cylindrical base with a rigid plane 28
2.3 Estimation Method for Contacts with a Three-Dimensional, Elastic Continuum 29
(1) Contact between a rigid, cylindrical indenter and an elastic half-space 29
(2) Contact between a rigid sphere and an elastic half-space 31
(3) Contact between a rigid cylinder and an elastic half-space 32
(4) Contact between a rigid cone and an elastic body 32
Problems 33
3 Qualitative Treatment of Adhesive Contacts 39
3.1 Physical Background 40
3.2 Calculation of the Adhesive Force between Curved Surfaces 44
3.3 Qualitative Estimation of the Adhesive Force between Elastic Bodies 45
3.4 Influence of Roughness on Adhesion 47
3.5 Adhesive Tape 48
3.6 Supplementary Information about van der Waals Forces and Surface Energies 49
Problems 50
4 Capillary Forces 55
4.1 Surface Tension and Contact Angles 55
4.2 Hysteresis of Contact Angles 59
4.3 Pressure and the Radius of Curvature 59
4.4 Capillary Bridges 60
4.5 Capillary Force between a Rigid Plane and a Rigid Sphere 61
4.6 Liquids on Rough Surfaces 62
4.7 Capillary Forces and Tribology 63
Problems 64
5 Rigorous Treatment of Contact Problems – Hertzian Contact 69
5.1 Deformation of an Elastic Half-Space being Acted upon by Surface Forces 70
5.2 Hertzian Contact Theory 73
5.3 Contact between Two Elastic Bodies with Curved Surfaces 74
5.4 Contact between a Rigid Cone-Shaped Indenter and an Elastic Half-Space 77
5.5 Internal Stresses in Hertzian Contacts 78
5.6 Method of Dimensionality Reduction (MDR) 81
Two preliminary basic steps of the MDR 81
Calculation steps of the MDR 82
Problems 84
6 Rigorous Treatment of Contact Problems – Adhesive Contact 94
6.1 JKR-Theory 95
6.2 Adhesive Contact of Rotationally Symmetrical Bodies 100
Problems 103
7 Contact between Rough Surfaces 107
7.1 Model from Greenwood and Williamson 108
7.2 Plastic Deformation of Asperities 113
7.3 Electrical Contacts 114
7.4 Thermal Contacts 118
7.5 Mechanical Stiffness of Contacts 119
7.6 Seals 119
7.7 Roughness and Adhesion 120
Problems 121
8 Tangential Contact Problems 127
8.1 Deformation of an Elastic Half-Space being Acted upon by Tangential Forces 128
8.2 Deformation of an Elastic Half-Space being Acted upon by a Tangential Stress Distribution 129
8.3 Tangential Contact Problems without Slip 131
8.4 Tangential Contact Problems Accounting for Slip 132
8.5 Absence of Slip for a Rigid Cylindrical Indenter 136
8.6 Tangential Contact of Axially Symmetrical Bodies 136
Problems 139
9 Rolling Contact 147
9.1 Qualitative Discussion of the Processes in a Rolling Contact 148
9.2 Stress Distribution in a Stationary Rolling Contact 150
A. Preparatory Step 150
B. Theory of Carter 151
Problems 156
10 Coulomb’s Law of Friction 161
10.1 Introduction 161
10.2 Static and Kinetic Friction 162
10.3 Angle of Friction 163
10.4 Dependence of the Coefficient of Friction on the Contact Time 164
10.5 Dependence of the Coefficient of Friction on the Normal Force 165
10.6 Dependence of the Coefficient of Friction on Sliding Speed 167
10.7 Dependence of the Coefficient of Friction on the Surface Roughness 167
10.8 Coulomb’s View on the Origin of the Law of Friction 168
10.9 Theory of Bowden and Tabor 170
10.10 Dependence of the Coefficient of Friction on Temperature 173
Problems 174
11 The Prandtl-Tomlinson Model for Dry Friction 183
11.1 Introduction 183
11.2 Basic Properties of the Prandtl-Tomlinson Model 185
11.3 Elastic Instability 189
11.4 Superlubricity 193
11.5 Nanomachines: Concepts for Micro and Nano-Actuators 194
Problems 198
12 Frictionally Induced Vibrations 203
12.1 Frictional Instabilities at Decreasing Dependence of the Frictional Force on the Velocity 204
12.2 Instability in a System with Distributed Elasticity 206
12.3 Critical Damping and Optimal Suppression of Squeal 209
12.4 Active Suppression of Squeal 211
12.5 Strength Aspects during Squeal 213
12.6 Dependence of the Stability Criteria on the Stiffness of the System 214
12.7 Sprag-Slip 219
Problems 221
13 Thermal Effects in Contacts 226
13.1 Introduction 227
13.2 Flash Temperatures in Micro-Contacts 227
13.3 Thermo-Mechanical Instability 229
Problems 230
14 Lubricated Systems 233
14.1 Flow between two parallel plates 234
14.2 Hydrodynamic Lubrication 235
14.3 “Viscous Adhesion” 239
14.4 Rheology of Lubricants 242
14.5 Boundary Layer Lubrication 244
14.6 Elastohydrodynamics 245
14.7 Solid Lubricants 249
Problems 250
15 Viscoelastic Properties of Elastomers 262
15.1 Introduction 262
15.2 Stress-Relaxation 263
15.3 Complex, Frequency-Dependent Shear Moduli 265
15.4 Properties of Complex Moduli 267
15.5 Energy Dissipation in a Viscoelastic Material 268
15.6 Measuring Complex Moduli 269
15.7 Rheological Models 270
15.8 A Simple Rheological Model for Rubber (“Standard Model”) 273
15.9 Influence of Temperature on Rheological Properties 275
15.10 Master Curves 276
15.11 Prony Series 277
15.12 Application of the Method of Dimensionality Reduction to Viscoelastic Media 281
Problems 282
16 Rubber Friction and Contact Mechanics of Rubber 289
16.1 Friction between an Elastomer and a Rigid Rough Surface 289
16.2 Rolling Resistance 295
16.3 Adhesive Contact with Elastomers 297
Problems 299
17 Wear 305
17.1 Introduction 305
17.2 Abrasive Wear 306
17.3 Adhesive Wear 309
17.4 Conditions for Low-Wear Friction 312
17.5 Wear as the Transportation of Material from the Friction Zone 313
17.6 Wear of Elastomers 314
Problems 317
18 Friction Under the Influence of Ultrasonic Vibrations 322
18.1 Influence of Ultrasonic Vibrations on Friction from a Macroscopic Point of View 323
I. Influence of vibration on static friction 323
II. The influence of vibration on kinetic friction 324
(1) Oscillation in the sliding direction 324
(2) Vibration perpendicular to sliding direction 326
18.2 Influence of Ultrasonic Vibrations on Friction from a Microscopic Point of View 328
18.3 Experimental Investigations of the Force of Static Friction as a Function of the Oscillation Amplitude 330
18.4 Experimental Investigations of Kinetic Friction as a Function of Oscillation Amplitude 332
Problems 334
19 Numerical Simulation Methods in Friction Physics 340
19.1 Many-Body Systems 341
19.2 Finite Element Method 342
19.3 Boundary Element Method 342
19.4 Boundary Element Method: Tangential Contact 344
19.5 Boundary Element Method: Adhesive Contact 345
19.6 Particle Methods 347
19.7 Method of Dimensionality Reduction 347
20 Earthquakes and Friction 348
20.1 Introduction 349
20.2 Quantification of Earthquakes 350
20.2.1 Gutenberg-Richter Law 351
20.3 Laws of Friction for Rocks 352
20.4 Stability during Sliding with Rate- and State-Dependent Friction 356
20.5 Nucleation of Earthquakes and Post-Sliding 359
20.6 Foreshocks and Aftershocks 362
20.7 Continuum Mechanics of Block Media and the Structure of Faults 363
20.8 Is it Possible to Predict Earthquakes? 367
Problems 368
Appendix 372
Appendix A - Normal Displacement under the Effect of Selected Pressure Distributions 372
a. Normal stress in a circle with the radius a according to the expression 372
b. Hertzian Stress Distribution 375
c. Uniform Stress Distribution in a Thin Circular Ring 376
Appendix B - Normal Contact of Axially Symmetrical Profiles 377
Appendix C - Adhesive Contact of Axially Symmetrical Profiles 380
Appendix D - Tangential Contact of Axially Symmetrical Profiles 381
Further Reading 384
Chapter 1 384
Chapter 2 384
Chapter 3 384
Chapter 4 385
Chapter 5 385
Chapter 6 385
Chapter 7 385
Chapter 8 386
Chapter 9 386
Chapter 10 386
Chapter 11 386
Chapter 12 387
Chapter 13 387
Chapter 14 387
Chapter 15 388
Chapter 16 388
Chapter 17 388
Chapter 18 388
Chapter 19 388
Chapter 20 389
Figure Reference 390
Index 392