Contact Mechanics and Friction 1

fg Valentin L. Popov

Contact Mechanics and Friction

Physical Principles and Applications

123 Professor Dr. Valentin L. Popov Berlin University of Technology Institute of Mechanics Strasse des 17.Juni 135 10623 Berlin Germany [email protected]

ISBN 978-3-642-10802-0 e-ISBN 978-3-642-10803-7 DOI 10.1007/978-3-642-10803-7 Springer Heidelberg Dordrecht London New York

Library of Congress Control Number: 2010921669

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Dr. Valentin L. Popov studied 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 Ber- lin University of Technology since 2002. His areas of interest, among others, in- clude tribology, nanotribology, tribology at low temperatures, biotribology, the in- fluence 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 work- shops over diverse tribological themes.

Preface to the English Edition

The English edition of “Contact Mechanics and Friction” lying before you is, for the most part, the text of the 1st German edition (Springer Publishing, 2009). The book was expanded by the addition of a chapter on frictional problems in earth- quake research. Additionally, Chapter 15 was supplemented by a section on elasto-hydrodynamics. The problem sections of several chapters were enriched by the addition of new examples. This book would not have been possible without the active support of J. Gray, who translated it from the German edition. I would like to thank Prof. G. G. Ko- charyan and Prof. S. Sobolev for discussions and critical comments on the chapter over earthquake dynamics. Dr. R. Heise made significant contributions to the de- velopment and correction of new problems. I would like to convey my affection- ate thanks to Dr. J. Starcevic for her complete support during the composition of this book. I want to thank Ms. Ch. Koll for her patience in creating figures and Dr. R. Heise, M. Popov, M. Heß, S. Kürscher, and B. Grzemba for their help in proof- reading.

Berlin, November 2009 V.L. Popov

Preface to the German Edition

He who wishes to better understand the subject of Contact Mechanics and the Physics of Friction would quickly discover that there is almost no other field that is so interdisciplinary, exciting, and fascinating. It combines knowledge from fields such as the theories of elasticity and plasticity, viscoelasticity, materials sci- ence, fluid mechanics (including both Newtonian and non-Newtonian fluids), thermodynamics, electrodynamics, system dynamics, and many others. Contact Mechanics and the Physics of Friction have numerous applications ranging from measurement and system technologies on a nanoscale to the understanding of earthquakes and including the sheer overwhelming subject of industrial tribology. One who has studied and understands Contact Mechanics and the Physics of Fric- tion will have acquired a complete overview of the different methods that are used in the engineering sciences. One goal of this book is to collect and clearly present, in one work, the most important aspects of this subject and how they relate to each other. Included in these aspects is, first, the entirety of traditional Contact Mechanics including ad- hesion and capillarity, then the theory of friction on a macro scale, lubrication, the foundations of modern nanotribology, system dynamical aspects of machines with friction (friction induced vibrations), friction related to elastomers, and wear. The interplay between these aspects can be very complicated in particular cases. In practical problems, different aspects are always presented in new ways. There is no simple recipe to solve tribological problems. The only universal recipe is that one must first understand the system from a tribological point of view. A goal of this book is to convey this understanding. It is the solid belief of the author that the essential aspects of mechanical con- tacts and friction are often much easier than they appear. If one limits oneself to qualitative estimations, it is then possible to achieve an extensive qualitative un- derstanding of the countless facets of mechanical contacts and friction. Therefore, qualitative estimations are highly valued in this book. In analytical calculations, we limit ourselves to a few classical examples which we can then take as building blocks and apply them to understand and solve a wealth of problems with real applications. A large number of concrete tribological questions, especially if they deal with meticulous optimization of tribological systems, are not solvable in analytical form. This book also offers an overview of methods of Numerical Simulation for Contact Mechanics and Friction. One such method is then explained in detail, which permits a synthesis of several processes related to contact mechanics from different spatial ranges within a single model. Even though this book is primarily a textbook, it can also serve as a reference for the foundations of this field. Many special cases are presented alongside the theoretical fundamentals with this goal in mind. These cases are presented as ex- ercises in their respective chapters. The solutions are provided for every exercise along with a short explanation and results. x Preface to the German Edition

The basis of this textbook originates and is drafted from lectures that the author has conducted over Contact Mechanics and the Physics of Friction at the Berlin University of Technology, so that the material can be completed in its entirety in one or two semesters depending on the depth in which it is visited.

Thanks This book would not have been possible without the active support of my col- leagues. Several in the department of “System Dynamics and Frictional Physics,” from the Institute for Mechanics, have contributed to the development of the prac- tice exercises. For this, I thank Dr. M. Schargott, Dr. T. Geike, Mr. M. Hess, and Dr. J. Starcevic. I would like to express a heartfelt thanks to Dr. J. Starcevic for her complete support during the writing of this book as well as to Mr. M. Hess, who checked all of the equations and corrected the many errors. I thank Ms. Ch. Koll for her patience constructing figures as well as M. Popov and Dr. G. Putzar for their help with proofreading. I thank the Dean of Faculty V, Transportation and Machine Systems, for granting me a research semester, during which this book was completed.

Berlin, October 2008 V.L. Popov

Table of Contents

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...... 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 xii Table of Contents

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 by a 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 Table of Contents x iii

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 xiv Table of Contents

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 Table of Contents xv

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