Mechanical Engineering Series

Frederick F. Ling Series Editor

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1. Angeles, Fundamentals of Robotic Mcchanical Systcms: Theory, Methods, and Algorithms

P. Basu, C. Kefa, and L. 1estin, Boilers and Burners: Design and Theory

1.M. Berthelot, ComllOsite Materials: Mechanical Behavior and Structural Analysis

U. Busch-Vishniac, Electmmechanical Sensors and Actuators

1. Chakrabarty, Applied Plasticity

G. Chryssolouris, Laser Machining: Thcory and Practice

V.N. Constantinescu, Laminar Viscous Flow

G.A. Costello, Thcory of Wirc Ropc, 2nd cd.

K Czolczynski, Rotordynamics of Gas-Luhricated Journal Bcaring Systcms

M.S. Darlow, Balancing of High-Speed Machinc.'y

1.F. Doyle, Nonlincar Analysis of Thin-Wallcd Structur{'s: Statics, Dynamics, and Stability

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P.A Engel, Stnlctural Anal~'sis of Printcd Circuit Board Systcms

AC. Fischer-Cripps, Introduction to Contact Mcchanics

AC. Fischer-Cripps, Nanoindcntation

1. Garcia de 1al6n and E. Bayo, Kincmatic and Dynamic Simulation of Multibody Systems: The Rcal-Time Challcnge

W.K Gawronski, Dynamics and Contml of Structures: A Modal Approach

KC. Gupta, Mechanics and Control of Robots

1. Ida and 1.P.A Bastos, Electromagnctics and Calculations of Fields

M. Kaviany, Pl'inciplcs of Convective Heat Transfer, 2nd cd,

M. Kaviany, Principles of Heat Transfcr in Pomus Mcdia, 2nd cd,

E.N. Kuznetsov, Underconstnlincd Structural Systems

(contin1led after Index)

Frederick F. Ling W. Michael Lai Don A. Lucca

Fundamentals of Surface Mechanics With Applications

Second Edition

With 218 Figures

, Springer

Frederick F. Ling Manufacturing Systems Center The University of Texas at Austin Austin. TX 78712. USA

Don A. Lucca School of Mechanical and

Aerospace Engineering Oklahoma State University Stillwater, OK 74078, USA

Series Editor Frederick F. Ling

W. Michael Lai Department of Mechanical Engineering School of Engineering Columbia University New York, NY 10027, USA

Ernest F. GIoyna Regents Chair in Engineering Department of Mechanical Engineering The University of Texas at Austin Austin. TX 78712-1063. USA

and William Howard Hart Professor Emeritus Department of Mechanical Engineering,

Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy. NY 12180-3590. USA

Library of Congress Cataloging-in-Publication Data Ling, Frederick F. (Frederick Fongsun), 1927-

Fundamentals of surface mechanics: with applications I Frederick F. Ling, W. Michael Lai, Don A. Lucca.

p. cm.-(Mechanical engineering series) Includes bibliographical references. ISBN 978-1-4684-9562-1 ISBN 978-0-387-21776-5 (eBook) DOI 10.1007/978-0-387-21776-5 I. Surfaces (Technology) 2. Continuum mechanics. 3. Tribology. I. Lai, W. Michael,

1930- II. Lucca, Don A. III. Mechanical engineering series (Berlin, Germany) TA418.7 .L53 2002 620'.44--{1c21 2001060200

ISBN 978-1-4684-9562-1 Printed on acid-free paper.

First edition, Surface Mechanics, John Wiley & Sons, New York, NY © 1973

© 2002 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 2nd edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

9 8 765 432 I SPIN 10863824

Camera-ready pages prepared from the authors' LaTeX2e Scientific Word files.

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Series Preface

Mechanical engineering, an engineering discipline borne of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solu­tions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for informa­tion in contemporary areas of mechanical engineering.

The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of consult­ing editors on the advisory board, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the next page of this volume. The areas of concentration are: applied mechanics; biomechanics; computational mechanics; dynamic systems and control; energetics; mechanics of materials; processing; thermal science; and tribology.

Austin, Texas Frederick F. Ling

Mechanical Engineering Series

Frederick F. Ling Series Editor

Advisory Board

Applied Mechanics

Biomechanics

Computational Mechanics

Dynamical Systems and Control

Energetics

Mechanics of Materials

Processing

Production Systems

Thermal Science

Trihology

F.A. Leckie University of California, Santa Barbara

V.c. Mow Columbia University

H.T. Yang University of California, Santa Barbara

K.M. Marshek University of Texas, Austin

J.R. Welty University of Oregon, Eugene

I. Finnie University of California, Berkeley

K.K. Wang Cornell University

G.-A. Klutke Texas A&M University

A.E. Bergles Rensselaer Polytechnic Institute

W.O. Winer Georgia Institute of Technology

Preface to the Second Edition

This book evolved from the monograph, Surface Mechanics, which was pub­lished in 1973 by Wiley Interscience. Coauthor, Professor W. Michael Lai, has given a graduate course at Rensselaer Polytechnic Institute and subse­quently at Columbia University under the title, Surface Mechanics; more recently, coauthor, Professor Don A. Lucca of Oklahoma State University, has also given a graduate course under the same title. As the original mono­graph evolved into the current book, the coauthors have chosen to name the text, Fundamentals of Surface Mechanics with Applications.

We note that many of the original objectives are preserved in this text, in particular, rigorously derived generic, surface quantities pertaining to so­lutions for governing differential equations within continuum theories, for example, surface temperature and surface deformation. The availability of rigorous generic solutions at the surface of a single body allows the formu­lation of two-body or intersurface problems through the use of a singular integral equation. From the technological viewpoint, within the framework of surface mechanics there are a host of intersurface problems where sur­faces are in relative motion. This means, in anticipation of the formulation of intersurface problems, that the aforementioned generic solutions would be needed in a Eulerian frame of reference in general. Such is the case with heat conduction problems. In deformation problems, unless the rela­tive speed of motion mentioned above is substantial when compared with the speed of sound of the solid body under discussion, the use of a Eulerian frame of reference is unnecessary. As such, treatment of classical contact problems of elasticity remains valid here, for example. The exception would

viii Preface to the Second Edition

be the case of contact problems, which would involve relative motion, in thermo elasticity, for example.

In terms of applications, the intervening quarter of a century since the appearance of Surface Mechanics has brought a vast number of interesting technologically motivated problems and solutions, especially computer so­lutions. Of course, it is not our intention nor is it feasible to include all that has been published during the said period. We are referencing some of this work, while including other research where appropriate; we call attention to a survey of literature by Ling and Kennedy1in 1990 in order to have a measure of the magnitude of the literature.

There are eight chapters and an appendix in this book. Chapter I, In­troduction, is a survey of field equations of classical continuum mechanics and various constitutive equations. Chapter 2 is on surface temperatures in moving bodies. Chapter 3 is on stress and deformation fields in half­space and layered elastic media. Chapter 4 is on thermal stress and asso­ciated deformation fields in half-space and layered media. Chapter 5 is on viscoelasticity. Chapter 6 is on perfect plasticity. Chapter 7 is on rough surfaces. Chapter 8 is on applications. The Appendix addresses singular integral equations and their inverses or methods for finding inverses. We note that, for specific model problems, Chapters 2 through 6 deal with mathematically smooth surfaces. Inasmuch as plastic behavior is primarily nonlinear and computation methods are required to solve plasticity prob­lems in general, we have chosen the subfield of perfect plasticity, which is treated in Chapter 6. That is to say, we used substantially slip-line theories there. Also, for our purpose, plasticity is used to model elements of rough surfaces under various loading; rough surfaces, of course, is the subject of Chapter 7. Moreover, the material in Chapters 2 through 4 were developed into the present form for a one-semester graduate course.

Support of the National Science Foundation, Division of Design, Manu­facture, and Industrial Innovation through Grants DMI-9713605 (FFL) and DMI-9713747 (DAL), and the Alexander von Humboldt Stiftung (DAL) is greatly appreciated. We would like to express our sincere thanks to Diane E. Compton and Roberta DeAngelis for their skillful preparation of the manuscript. Finally, we thank Matthew J. Klopfstein and Rudy Ghisleni for their careful proofreading.

Austin, Texas New York, New York Stillwater, Oklahoma May 2002

Frederick F. Ling W. Michael Lai

Don A. Lucca

1 "Contact and Surface Mechanics," F.F. Ling and F.E. Kennedy, Jr., Achievements in Tribology, L.B. Sibley and F.E. Kennedy, editors., American Society of Mechanical Engineers, TRIB-Vol. 1, 129-149, 1990.

Preface to Surface Mechanics

Surface phenomena have in recent years been receiving increasing atten­tion in the study of materials science and engineering, notably the field of activity that has acquired a new designation-tribology. Surface behavior of all types is becoming of interest, and the awareness of the fact that bulk behavior of materials is often critically affected by surface conditions is increasing. This field of endeavor, so designated, covers several traditional academic disciplines. The name "materials" not only underscores the in­terdependence of various disciplines for fruitful pursuits but also brings to the fore the interrelatedness of several professional fields.

This book was written to satisfy a twofold purpose. The first is to set down concrete examples dealing with one facet of the body of interrelated knowledge, that of surface mechanics, which seems appropriate in filling a void within the larger context of surface physics. The second is to pro­vide a collection of basic tools relevant to quantitative studies of problems involving surfaces.

Classical continuum mechanics are used throughout, that is, the field theory which assumes that the material body is indefinitely divisible while retaining its defining properties or continuum physics. In choosing the phe­nomenological method for describing materials, its preference over electron, molecular, or atomic theories of material is not inferred. In fact, newer con­tinuum theories to be described briefly include first approximation of mi­croscopic phenomena. Moreover, while the bulk of the text is analytical and theoretical, much of the theories are buttressed by decades of experience accrued through experimental mechanics.

x Preface to Surface Mechanics

By limiting the scope to surface mechanics, the second purpose is better illuminated, that is, to offer a set of critical ingredients for solving problems encountered in various professions while not committing space to profes­sional problems. Not the least important is that quantitative tools are often catalytic to new discoveries in surface chemistry and surface physics.

Surface mechanics pertains to surfaces, to be sure, but it was coined also to characterize the notion that information on the surface may be obtained analytically and rigorously without the encumbrance of the entire solution. This may be achieved by employing various transform methods. It now embraces continuum treatment of surface, surface layer, and interface phenomena.

There are ten chapters and an appendix in the book. The first chapter provides a cursory review of irreversible thermodynamics which serves as a hub for interacting sets of field and constitutive equations. The appendix surveys singular integral equations as they may impinge on the subject matter. The remaining chapters are devoted to selected models and solu­tions which are considered basic to aspects of surface studies and descrip­tions of key experimental results. These fall, respectively, under categories on heat conduction in solids, interface temperatures, classical elasticity, thermoelasticity, viscoelasticity, perfect plasticity, rough surfaces, chemi­cal effects, and applications. Smooth surfaces are used where permissible and nonsmooth surfaces used where necessary. Emphasis has been given to surfaces of bodies which involve motion and entail interaction of field, for example, thermoplastic.

While simplicity is sought, mathematical complexity has not been avoided when the situation so demands. I have used the material in a series of lec­tures bearing the same name given as a course on special topics in mechan­ics at Rensselaer Polytechnic Institute.

I am greatly indebted to former students who attended the lectures and made useful comments on the text material during its development. Also, particular thanks are due to those who have graciously read the manuscript and have offered many suggestions: Professor H. Blok, Director, Insti­tute of Tribomechanics, University of Technology, Delft, Holland; Professor R. Courtel, Directeur de Recherches au Centre National de la Recherche Scientifique and Directeur, Laboratoire Mecanique des Surfaces, l'Institut Superior des Materiaux et de la Construction Mecanique, France; Profes­sor D. Dowson, Director, Institute of Tribology and Co-Director, Bioengi­neering Group for the Study of Human Joints, University of Leeds, Eng­land; Professor M.D. Hersey, Brown University; Professor W.F. Hughes, Carnegie-Mellon University; P.M. Ku, Director, Department of Fluids and Lubrication Technology, Southwest Research Institute; Dr. W.R. Osgood, Washington, D.C.; Professor E.A. Saibel, Carnegie-Mellon University; Pro­fessor M.C. Shaw, Head, Department of Mechanical Engineering and Di­rector, Processing Research Institute, Carnegie-Mellon University; and Dr.

Preface to Surface Mechanics xi

D. Tabor, Head, Department of Surface Physics, Cavendish Laboratory, University of Cambridge, England.

During the last year of the preparation of this book, I was a 1970 National Science Foundation Senior Postdoctoral Fellow.

To Miss J .E. Doocey, I wish to express my special thanks for an excellent job of technical typing. Thanks are due also to Mr. P.S. Kounas for a careful job reading of the proof.

Troy, New York April 1972

Frederick Fongsun Ling

Contents

Series Preface v

Preface to the Second Edition vii

Preface to Surface Mechanics ix

Credits for Illustrations and Tables xix

1 Introduction 1 1.1 Balance of Momentum 1 1.2 Energy Balance. . . . 2 1.3 Entropy . . . . . . . . 3 1.4 Constitutive Relationship and the Energy Equation for an

Elastic Solid. . . . . . . . . . . . . . . . . . . . . . 5 1.5 Constitutive Relationship-Heat Conduction . . . 6 1.6 Constitutive Relationship-Other Diffusion Types 7 1. 7 Constitutive Relationship-Linearly Viscous Fluid 7 1.8 Constitutive Relationship-Perfectly Plastic Bodies. 7 1.9 Constitutive Relationship-Viscoelastic Bodies . . . 8 1.10 Constitutive Relationship-Maxwellian Dielectric. . 8 1.11 Field Equations of Classical Electromagnetic Theory 9

2 Surface Temperatures in Moving Bodies 2.1 Introduction ................ .

11 11

xiv Contents

2.2 Instantaneous Point Heat Source-Infinite Medium . . . . 13 2.3 Instantaneous Point Heat Source-Semi-infinite Medium . 14 2.4 Continuous Point Heat Source-Semi-infinite Medium 15 2.5 Partition of Heat-Circular Area of Contact. . . . . . 20 2.6 Line Heat Source on the Surface of a

Semi-infinite Medium ............ . 2.7 Instantaneous Point Heat Source-Moving

2.8

2.9 2.10

2.11

2.12 2.13 2.14

2.15

2.16

2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24

Semi-infinite Medium .......... . Continuous Point Heat Source-Moving Semi-infinite Medium .......... . Distributive Heat Source-Moving Semi-infinite Medium. Partition of Heat in Sliding Contact of Two Large Bodies ... . . . . . . . . . . . . Continuous Line Heat Source-Moving Semi-infinite Medium ................... . Continuous Strip Source-Moving Semi-infinite Medium Blok's Approximate Method for Partition of Heat. Fourier Series, Fourier Integrals, and the Fourier Transform ............ ......... . Continuous Strip Heat Source-Moving Semi-infinite Medium. . . . . . . . . . . . . . . . . . . . . . Continuous Strip Heat Source-Moving Layered Semi-infinite Medium ............... . Finite Fourier Transform. . . . . . . . . . . . . . Continuous Arc Heat Source-Rotating Circular Disk Legendre Polynomial . . . Legendre Series . . . . . . . . . . . . . . . . . Legendre Transform Pair. . . . . . . . . . . . Stationary Sphere Heated Over Its Polar Cap Fourier Cosine Transform . . . . . . . . . . . Effects of Temperature-Dependent Thermal Properties

Exercises ......................... .

3 Stress and Deformation Fields in Half-Space and Layered Elastic Media 3.1 Introduction ..................... . 3.2 Stress-Strain Relations for a Linear Isotropic Elastic

Solid ....................... . 3.3 Navier Equations ................ . 3.4 Fundamental Potential Functions for Problems

of Elastostatics . . . . . . . . . . . . . . . . 3.5 The Kelvin Problem ............ . 3.6 Stress Field and Displacement Field for the

Kelvin Problem . . . . . . . . . . . . . . . .

21

22

23 25

28

31 33 37

38

41

45 50 52 55 57 57 59 61 62 67

69 69

70 70

71 71

75

Contents xv

3.7 Concentrated Force Acting Vertically on an Elastic Half-Space ............. .

3.8 Concentrated Force Acting Tangentially on an Elastic Half-Space ... .

3.9 The Boussinesq Problem ............ . 3.10 The Cerruti Problem .............. . 3.11 Distributive Normal Load on the Surface of an

Elastic Half-Space ............... . 3.12 Distributive Tangential Load on the Surface of an

Elastic Half-Space ................. . 3.13 The Flamant Problem ............... . 3.14 An Elastic Half-Space Subjected to a Distributive Load Over

79

80 81 87

89

92 93

a Strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.15 Double Fourier Integral ................. .98 3.16 Distributive Normal Load-Double Transform Method. 99 3.17 Distributive Normal Load-Transform Method 105 3.18 Moving Distributive Normal Load on an

Elastic Half-Space ................ 111 3.19 Distributive Normal Load on a Hollow Cylinder . 117 3.20 Indentation of an Elastic Half-Space by a Rigid

Smooth Body . . . . . . . . . . . . . . . . . . . . 127 3.21 Indentation by a Rigid Flat-Ended Smooth Cylinder 131 3.22 Indentation by an Axisymmetric Smooth Rigid Indenter 132 3.23 Formulas for Indentation by an Axisymmetric

Rigid Indenter ................... 139 3.24 Indentation by a Rigid Conical Smooth Indenter 141 3.25 Indentation by a Smooth Rigid Sphere . . . . . . 142 3.26 Indentation by an Axisymmetric Rigid Indenter . 145 3.27 Indentation by a Plane-Ended Cylindrical Indenter 151 3.28 Indentation of an Elastic Layer by a Rigid

Spherical Indenter .............. 152 3.29 Indentation of an Elastic Half-Space by an

Elastic Sphere. 154 Exercises ..... .

4 Thermoelasticity 4.1 Introduction.. 4.2 Semi-infinite Solid 4.3 Effect of Inertia . . 4.4 Effect of Thermoelastic Coupling 4.5 Convective Half-Space with Moving Heat Source-

2D Case ..................... . 4.6 Elastic Half-Space with Moving Heat Source-

3D Case ..................... .

157

163 163 163 166 169

171

176

XVI Contents

5 Viscoelasticity 179 5.1 Introduction. · . 179 5.2 Models. · . 179 5.3 Elasticity-Viscoelasticity Analogy. 181 5.4 Other Representations of Mechanical Properties . 182 5.5 Influence of Temperature on Viscoelastic Behavior 183 5.6 Example of the Elasticity-Viscoelasticity Analogy 186 5.7 Response of a Viscoelastic Half-Space to Moving Loads. 188 5.8 Deformation of a Soft Layer Material Under a

Moving Load · .. 191 5.9 Viscoelastic Layer on an Elastic Half-Space 193 5.10 Multilayered Viscoelastic Media Under a Moving Load 200

6 Perfect Plasticity 201 6.1 Introduction. · .. 201 6.2 Slip-Line Theory · . 201 6.3 Stress Field in a Semi-infinite Solid Under a Lubricated

Flat Punch · .. 206 6.4 Stress Field in a Truncated Wedge Under a Lubricated

Flat Punch · . 207 6.5 Stress Field in a Wedge Under Lateral Pressure . 207 6.6 Compression of a Wedge by a Flat Die . 208 6.7 Sliding of a Wedge Under a Flat Die Under Load 210 6.8 Indentation of a Semi-infinite Solid by a

Lubricated Wedge · .. 211 6.9 A Friction Model · ... 212 6.10 Friction of Ploughing by Rigid Asperities 213 6.11 Different Regimes of Friction and Wear 214 6.12 Indentation of Sandwich Metal Strips Between Flat Dies 217 6.13 Oblique Impact of a Hard Ball Against a Ductile Solid 219 6.14 Slip-line Field of the Rolling Contact Problem at

High Loads · . 221 6.15 Indentation of a Semi-infinite Solid by a Cylinder. 223 6.16 Flattening of Circular Cylinder by a Lubricated Die 224 6.17 Indentation of a Semi-infinite Solid by a Lubricated

Spherical Die · . 225 6.18 Indentation of a Semi-infinite Solid by the End of a

Lubricated Cylinder 226 6.19 Indentation of a Semi-infinite Solid by a Lubricated

Truncated Cone . · . 227

7 Rough Surfaces 229 7.1 Introduction. · . 229 7.2 Bearing Area Curves . 230 7.3 Profilometric Representation of Surfaces 231

Contents xvii

7.4 Characterization of Surfaces by Autocorrelation Functions . . . . . . . . . . . . . . . . . 233

7.5 Characterization of Surfaces by Actual Area of Contact 234 7.6 Characterization of Surfaces by Compliance . . . . 236 7.7 Characterization of Surfaces by Fractal Geometry. 244 7.8 Some Studies Involving Surface Textures. . . . . . 247

8 Applications 251 8.1 Introduction........................ 251 8.2 On Blok's Conjecture .................. 251 8.3 Interface Temperature Between a Rotating Disc and a

Ring Sector . . . . . . . . . . . . . . . . . . . . . . . . 254 8.4 Surface Temperature on a Fast-Moving Half-Space Due to a

Unit Heat Source Over a Rectangular Area . . . . . . . 258 8.5 Temperature at the Sliding Interface for a Coated Body 259 8.6 Coated Medium with Random Uniform

Coating Thickness . . . . . . . . . . . . . . . . . . 260 8.7 Rotating Cylinder Subject to Surface Heating and

Convective Cooling. . . . . . . . . . . . . . . . 262 8.8 Rotating Layered Circular Cylinder Subject to

Surface Heating. . . . . . . . . . . . . . . . . . 263 8.9 Rotating Layered Sphere Subject to Surface Heating 267 8.10 Transient Case of a Moving Semi-infinite Body 270 8.11 General Three-Dimensional Transient Case of a

Cylindrical Rod . . . . . . . . . . . . . . . . . . 271 8.12 Surface Temperature on the Truncated Face of a Cone 274 8.13 Disc with Normal Edge Load . . . . . . . . . . 278 8.14 Disc with Tangential Edge Load ........ 279 8.15 Semi-infinite Circular Cylinder with End Load 280 8.16 Edge Effect on the Hertz Solution. . . . . . 287 8.17 Compliance of Elastic Bodies in Contact. . . . 288 8.18 Microslips Between Contacting Paraboloids . . 291 8.19 Contact Pressures as an Elastic Roller Crosses a Scratch. 293 8.20 Layered Disc with Normal Edge Load .. . . . . . 294 8.21 Layered Elastic System Under a Moving Load. . . 297 8.22 Stress at the Interface of a Layered Elastic System 300 8.23 Indentation of Anisotropic Materials . . . 303 8.24 Contact Stress of a Layered Transversely

Isotropic Half-Space . . . . . . . . . . . . 303 8.25 On Elastohydrodynamic Lubrication . . . 307 8.26 Thermal Stresses in an Elastic Half-Space with a Moving

Heat Source. . . . . . . . . . . . . . . . . . . . . . . . . . . 311 8.27 Thermoelastic Solutions for a Half-Space with a Moving

Heat Source. . . . . . . . . . . . . . . . . . . . 314 8.28 Thermomechanical Cracking in Layered Media . . . . . . . 316

xviii Contents

8.29 On Deformation Friction. . . . . . . . . . . . . . . . . 319 8.30 On Two-Dimensional Rolling in Viscoelastic Material. 320 8.31 Contact Problems in Linear Theory Viscoelasticity . 323 8.32 Thermal Softening Mechanism of Lubricated Brakes

and Clutches . . . . . . . . . . . . . . . . . 331 8.33 Plastic Shakedown in Rolling Contact .. . 335 8.34 Soft Metals in Static and Dynamic Loading 336 8.35 Friction Under Metal-Working Processes. 347 8.36 Contact Between Rough Surfaces with

Longitudinal Texture . . . . . . . . . . . . 352 8.37 Transient Temperatures in the Vicinity of an

Asperity Contact . . . . . . . . . . . . . . 353 8.38 Normal Impact Model of Rough Surfaces. 358

Appendix 361 A.l Singular Integral Equations . . . . . . . . . . 361 A.2 Abel's Integral Equation of the First Kind . . 361 A.3 Abel's Integral Equation of the Second Kind. 362 AA Carleman's Integral Equation . . . 363 A.5 Other Singular Integral Equations 364 A.6 Fredholm Integral Equations 370

References 371

Index 387

Credits for Illustrations and Tables

The authors wish to thank the following publishers for permission to use the illustrations listed:

Figure 2.13 (p. 27): H. Blok. Theoretical studies of temperature rise at surfaces of actual contact under oiliness lubricating conditions. Proceedings of General Discussion of Lubrication and Lubricants, The Institution of Mechanical Engineers, London, United Kingdom, pages 225-235, 1937.

Figures 2.25 (p. 45), 2.26 (p. 49), and 2.27 (p. 50): F. F. Ling and C. F. Yang. Surface temperatures of moving layered composites. Surface Mechanics-A Symposium Volume, F. F. Ling, (Ed.), American Society of Mechanical Engineers, New York, New York, pages 164-176, 1969.

Figures 5.3 (p. 194) and 5.4 (p. 197): S. K. Batra and F. F. Ling. Deforma­tional friction of a viscoelastic layered system under moving load. American Society of Lubrication Engineers Transactions, 10:294-301, 1967.

Figures 6.2 (p. 206), 6.3, and 6.4 (p. 207): L. Prandtl. Uber die Harte plastisher Korper. Gottingen Nachricht, Mathematik-Physik, 1920:74-85, 1920.

Figure 6.8 (p. 211): R. Hill, E. H. Lee, and S. J. Tupper. The theory of wedge indentation of ductile materials. Proceedings of the Royal Society, A188:273-289, 1945.

Figures 6.9 (p. 212) and 6.10 (p. 213): B. Avitzur, C. K. Huang, and Y. D. Zhu. A friction model based on the upper-bound approach to the ridge and sublayer deformations. Wear, 95:59-77, 1984.

Figures 6.13 and 6.14 (p. 215), 6.15 (p. 216), and 6.16 (p. 6.16): J. M. Challen and P. L. B. Oxley. An explanation of the different regimes of

xx Credits for Illustrations and Tables

friction and wear using asperity deformation models. Wear, 53:229-243, 1972.

Figures 6.17 and 6.18 (p. 218): W. H. Harden and A. S. Weinstein. Slip­line solution for plane-strain indentation of sandwich metal strips between flat dies. Journal of Engineering Materials and Technology, Transactions of the American Society of Mechanical Engineers, 96:182-189, 1974.

Figures 6.19 (p. 219), 6.20, and 6.21 (p. 220): I. M. Hutchings, N. H. Macmillan, and D. G. Rickerby. Further studies of the oblique impact of a hard sphere against a ductile solid. International Journal of Mechanical Science, 23:639-646, 1981.

Figures 6.22 (p. 221), 6.23, and 6.24 (p. 222): H. Petryk. Slip-line field analysis of the rolling contact problem at high loads. International Journal of Mechanical Science, 16:75-82, 1974.

Figures 6.25 (p. 223) and 6.26 (p. 224): V. V. Sokolovskii. Teoriia Plas­tichnosti. Moscow, Izd-vo AN SSSR, 1949.

Figure 6.27 (p. 225): A. In. Ishlinskii. Osesimmetrichnaia zadacha teorii plastichnosti i proba brinella. Prikladu. matem. i mechanika, 8, Moscow, Izd-vo AN SSSR, 1944.

Figure 6.28 (p. 226): R. T. Shield. On the plastic flow of metals under conditions of axial symmetry. Proceedings of the Royal Society, A233:267-286, 1955.

Figures 6.29 (p. 227) and 6.30 (p. 228): V. N. Marochkin. The limit­ing plastic state in indenting and compressing a truncated cone. Friction and Wear in Machinery, 13:79-131, 1959. (Translated from Russian by the American Society of Mechanical Engineers.)

Figures 7.2 (p. 231) and 7.8 (p. 235): I. V. Kragelsky and N. B. Demkin. Contact area of rough surfaces. Wear, 3:170-187, 1960.

Figures 7.9 (p. 238), 7.10 (p. 239), 7.11 (p. 240), 7.12 (p. 242), 7.13 (p. 243), and 7.14 (p. 244): F. F. Ling. On asperity distributions of metalic surfaces. Journal of Applied Physics, 29:1168-1174, 1958.

Figures 7.15 and 7.16 (p. 245), 7.17 (p. 246), and 7.18 (p. 247): F. F. Ling. Fractals, engineering surfaces and tribology. Wear, 136:141-156, 1990.

Figure 7.19 (p. 248): J. D. Cogdell, M. C. Dawson, F. F. Ling, and S. F. Murray. Surface texture effects in thin film lubrication of steel by silicons. American Society of Lubrication Engineers Transactions, 30:141-148, 1986.

Figure 7.20 (p. 249): W. Holzhauer and F. F. Ling. In-situ SEM study of boundary lubricated contacts. Tribology Transactions, 31:359-368, 1987.

Figures 8.1 and 8.2 (p. 253): F. F. Ling. A quasi-iterative method for computing interface temperature distributions. Zeitschrijt fur Angewandte Mathematik und Physik, X:461-474, 1959.

Figures 8.11 and 8.12 (p. 275): S. L. Pu. Surface temperature on a trun­cated right circular cone. Proceedings of the 5th Midwestern Conference on Mechanics, 1966.

Credits for Illustrations and Tables XXl

Figures 8.13 (p. 278) and 8.14 (p. 279): C. W. Ng. Green's function of radial displacement in a circular disc due to unit normal and tangential loads. Wear, 7:344~353, 1964.

Figures 8.15 (p. 280),8.16 (p. 284), 8.17, and 8.18 (p. 287): E. A. Wilson and F. F. Ling. Surface displacements on the end of an elastic cylinder. AFML-TR-68-37, 1968.

Figure 8.28 (p. 297): Y. C. Hsu and F. F. Ling. Shear stresses in a layered elastic system under a moving load. Recent Advances in Engineering Science, II:323~351, 1965.

Figures 8.29 (p. 301), 8.30, and 8.31 (p. 302): R. F. Maye and F. F. Ling. On the solution of the ring for cosserat materials. Developments in Mechanics, 4:151~169, 1967.

Figures 8.32 and 8.33 (p. 304), and 8.34 and 8.35 (p. 305), 8.36 and 8.37 (p. 306), 8.38 and 8.39 (p. 307), and 8.40 (p. 308): W. T. Chen. Stresses in some anisotropic materials due to indentation and sliding. International Journal of Solids and Structures, 5:191~214, 1969.

Figures 8.53 (p. 320), 8.54, and 8.55 (p. 321): D. G. Flom. Rolling friction of polymeric materials-elastomers. Journal of Applied Physics, 31:306~314, 1960.

Figures 8.56 (p. 322), 8.57, and 8.58 (p. 323): A. A. Elsharkawy. A nu­merical solution for dry contact between two viscoelastic rollers. Tribology Transactions, 39:627-635, 1996.

Figure 8.61 (p. 332): K. Y. Li and F. F. Ling. The sliding of copper-based sintered material against steel in paraffinic mineral oil. Wear, 15:249~256, 1970.

Figure 8.62 (p. 332), 8.63 and 8.64 (p. 333), and 8.65 (p. 334): F. F. Ling and C. C. Yang. Temperature distribution in a semi-infinite solid under a fast-moving arbitrary heat source. International Journal of Heat and Mass Transfer, 14:199~206, 1971.

Figure 8.66 (p. 334): V. C. Mow and H. S. Cheng. Thermal stresses in an elastic half space associated with an arbitrarily distributed moving heat source. Zeitschrijt fur Angewandte Mathematik und Physik, XVIII:500~509, 1967.

Figures 8.67 (p. 335) and 8.68 (p. 336): J. E. Merwin and K. L. Johnson. An analysis of plastic deformation in rolling contact. Proceedings of the Institute of Mechanical Engineers, 177:676~685, 1963.

Figures 8.78 (p. 347), 8.79 (p. 348), and 8.80 (p. 351): F. F. Ling and M. B. Peterson. Friction and lubrication in metalworking processes. Pro­ceedings of the International Conference on Manufacturing Technology, American Society of Tool and Manufacturing Engineers, pages 1181~1192, 1967.

Figures 8.81 (p. 352) and 8.82 (p. 353): H. Aramaki, H. S. Cheng, and Y. W. Chung. The contact between rough surfaces with longitudi­nal texture-part II: flash temperature. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 115:425~431, 1993.

xxii Credits for Illustrations and Tables

Acknowledgment is also made to the American Society of Mechanical En­gineers for the use of the following figures:

Figure 2.17 (p. 31): H. H. H. Shu, E. W. Gaylord, and W. F. Hughes. The relation between the rubbing interface temperature distribution and dynamic thermocouple temperature. Transaction of the American Society of Mechanical Engineers, 86:417-422, 1964.

Figures 2.28 (p. 52), 8.3 (p. 254), and 8.4 (p. 257): F. F. Ling and C. W. Ng. On temperatures at the interfaces of bodies in sliding contact. In Pro­ceedings of the Fourth U.S. National Congress of Allied Mechanics, pages 1343-1349. American Society of Mechanical Engineers, 1962.

Figure 4.2 (p. 167): W. Sternberg and J. G. Chakravorty. On inertia effects in a transient thermoelastic problem. Journal of Applied Mechanics, 26:503-509, 1959.

Figure 5.1 (p. 184): E. Sternberg. On transient thermal stresses in linear viscoelasticity. Proceedings of the Third U.S. National Congress of Applied Mechanics, pages 673-683, 1958.

Figure 5.2 (p. 192): G. R. Abrahamson and J. N. Goodier. The hump deformation preceding a moving load on a layer of soft material. Journal of Applied Mechanics, 28:608-610, 1961.

Figures 5.5 and 5.6 (p. 199), and 5.7 (p. 200): 1. Goryacheva, F. Sadeghi, and D. A. Nickel. Internal stresses in contact of a rough body and a vis­coelastic layered semi-infinite plane. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 118:131-136, 1996.

Figure 6.5 (p. 208): R. Hill. Some special problems of indentation and compression in plasticity. Proceedings of the 7th International Congress of Applied Mechanics, 1:365-377, 1949.

Figure 6.6 (p. 209): F. F. Ling. Some factors influencing the area-load characteristics of semi-smooth surfaces under "static" loading. Transac­tions of the American Society of Mechanical Engineers, 80:1113-1120,1958.

Figure 6.7 (p. 210): C. H. Popelar. On the basic equation of junction growth. Journal of Applied Mechanics, 36:132-133, 1969.

Figure 6.11 (p. 214): A. Azarkhin and O. Richmond. On the friction of ploughing by rigid asperities in the presence of straining upper-bound method. Journal of Tribology, Transactions of the American Society of Me­chanical Engineers, 112:324-329, 1990.

Figure 6.12 (p. 214): V. Bhargava, G. T. Hahn, and C. A. Rubin. An elastic-plastic finite element model of rolling contact part 2: Analysis of repeated contacts. Journal of Applied Mechanics, 52:75-82, 1985.

Figure 7.1 (p. 230): E. J. Abbott and F. A. Firestone. Specifying surface quality. Mechanical Engineering, 55:569-572, 1933.

Figures 7.3 and 7.4 (p. 232): J. Wallach. Surface topography description and measurement. In Surface Mechanics, a Symposium Volume, pages 1-23. American Society of Mechanical Engineers, 1969.

Credits for Illustrations and Tables xxiii

Figures 7.5 and 7.6 (p. 232), and 7.7 (p. 233): J. B. P. Williamson, J. Pullen, and R. T. Hunt. The shape of solid surfaces. Surface Mechanics, a symposium volume, pages 24-35, 1969. American Society of Mechanical Engineers.

Figure 8.5 (p. 259): X. Tian and F. E. Kennedy. Temperature rise at the sliding contact interface for a coated semi-infinite body. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 115:1-9, 1993.

Figures 8.6 (p. 260), and 8.7 (p. 261): J.-C. Liu and F. D. Ju. Asperity excited temperature field in a coated medium with a random uniform coat­ing thickness. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 111:129-135, 1989.

Figure 8.8 (p. 262): B. Gecim and W. O. Winer. Steady temperature in a rotating cylinder subject to surface heating and convective cooling. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 106:120-127, 1984.

Figure 8.9 (p. 266): B. Gecim and W. O. Winer. Effect of a surface film on the surface temperature of a rotating cylinder. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 108:92-97, 1986.

Figures 8.19 (p. 288), 8.20, 8.21 (p. 289), and 8.22 (p. 290): R. D. Mindlin. Compliance of elastic bodies in contact. Journal of Applied Mechanics, 16:259-268, 1949.

Figures 8.23 (p. 291), 8.24 and 8.25 (p. 292): V. C. Mow, P. L. Chow, and F. F. Ling. Microslip between contacting p&faboloids. Journal of Applied Mechanics, 34:321-328, 1967.

Figure 8.26 (p. 293): J. A. Greenwood. Contact pressures as an elastic roller crosses a scratch. Journal of Applied Mechanics, 64:425-427, 1997.

Figure 8.27 (p. 294): S. D. Beck and F. F. Ling. Stresses in layered disc. Proceedings of the F~fth U. S. National Congress of Applied Mechanics, page 244, 1966.

Figure 8.28 (p. 297): Y. C. Hsu and F. F. Ling. Shear stresses in a layered elastic system under a moving load. Recent Advances in Engineering Science, 11:323-351, 1965.

Figures 8.41 (p. 308), 8.42 and 8.43 (p. 309): C. H. Kuo and L. M. Keer. Contact stress analysis of a layered transversely isotropic half space. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 114:254-263, 1992.

Figures 8.44 (p. 310) and 8.45 (p. 313): H. S. Cheng. A refined solution to the thermal-elastohydrodynamic lubrication of rolling and sliding cylinders. American Society of Lubrication Engineers Transactions, 8:397-410, 1965.

Figure 8.46 (p. 314): M. D. Bryant. Thermoelastic solutions for thermal distributions moving over half space surfaces and application to the moving heat source. Journal of Applied Mechanics, 55:87-92, 1988.

xxiv Credits for Illustrations and Tables

Figure 8.47 (p. 315): Y. Yu and T. N. Farris. FFT thermoelastic solution for moving heat sources. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 119:156-162, 1997.

Figures 8.48 and 8.49 (p. 317): F. D. Ju and T. Y. Chen. Thermome­chanical cracking in layered media for moving friction load. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 106:513-518, 1984.

Figures 8.50 and 8.51 (p. 318), and 8.52 (p. 319): T. Y. Chen and F. D. Ju. Friction-induced thermo-mechanical cracking in a coated medium with a near surface cavity. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 111:270-277, 1989.

Figures 8.59 (p. 324) and 8.60 (p. 329): T. C. T. Ting. Contact problems in the linear theory ofviscoelastidty. Journal of Applied Mechanics, 35:248-254, 1968.

Figures 8.69 (p. 337), 8.70 (p. 338), 8.71 (p. 340), 8.72 (p. 341), 8.73 (p. 342), and 8.74 (p. 343): C. H. Yew and W. Goldsmith. Stress distribu­tions in soft metals due to static and dynamic loading by a steel sphere. Journal of Applied Mechanics, 31:635-646, 1964.

Figures 8.75 (p. 344), 8.76 (p. 345), and 8.77 (p. 346): E. R. Kral, K. Komvoploulos, and D. B. Bogy. Finite element analysis of repeated indentation of an elastic-plastic layered medium by a rigid sphere, part 1: surface results. Journal of Applied Mechanics, 62:20-28, 1995.

Figures 8.83 (p. 354), 8.84 (p. 355), 8.85 (p. 356), and 8.86 (p. 357): B. Gedm and W. O. Winer. Transient temperatures in the vicinity of an asperity contact. Journal of Tribology, Transactions of the American Soci­ety of Mechanical Engineers, 107:333-342, 1985.

Figures 8.87 (p. 358), 8.88 and 8.89 (p. 360): W. R. Chang and F. F. Ling. Normal impact model of rough surfaces. Journal of Tribology, Transactions of the American Society of Mechanical Engineers, 114:430-437, 1992.