Valentin L. Popov Contact Mechanics and Friction Physical Principles and Applications Second Edition 123 Contact Mechanics and Friction Valentin L. Popov Contact Mechanics and Friction Physical Principles and Applications Second Edition Valentin L. Popov TU Berlin Inst. Mechanik Berlin, Germany ISBN 978-3-662-53080-1 ISBN 978-3-662-53081-8 (eBook) DOI 10.1007/978-3-662-53081-8 Library of Congress Control Number: 2017933487 © Springer-Verlag GmbH Germany 2010, 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Germany The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany 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. v Preface to the Second English Edition The second English edition of “Contact Mechanics and Friction” was substantially revised compared to the 1st edition. This is especially related to the chapters on “contact mechanics” (normal contact with and without adhesion, tangential con- tact, and contact with elastomers), which were supplemented with many solutions for axially-symmetric contact problems. The chapter on “wear” now includes a discussion on fretting. The section of elasto-hydrodynamic lubrication was com- pletely rewritten and enriched by several problems. Furthermore, the chapter on numerical simulation methods in contact mechanics has been overhauled and now contains the basic ideas of the application of the boundary element method to ad- hesive contacts. I would like to thank M. Hess and R. Pohrt for their help in writing new sec- tions and problems. Also, thanks go to M. Hess and M. Popov for their corrections of this book, Ms. Wallendorf for drawing new figures, and Dr. J. Starcevic for ex- tensive support in the completion of the book. Berlin, September 2016 V.L. Popov vii From the Preface to the First English Edition The English edition of “Contact Mechanics and Friction” ....... 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 earthquake re- search. 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 ix Preface to the First 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. xi xii Preface to the First 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 ................................................. 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 xiii