Mechanics of Fretting and Fretting Fatigue (Solid Mechanics and Its Applications, 266) 3030707458, 9783030707453

Table of contents :
Preface
Acknowledgements
Contents
1 Some Fundamentals
1.1 Fretting in Practice
1.2 Basics, Equilibrium and `Coupling'
1.3 Friction
1.4 Contact Requirements
1.5 Classes of Contact
1.6 Methods of Solution
1.7 Shakedown
1.8 Three-Dimensional Aspects
2 Plane Elasticity and Half-Plane Contacts
2.1 Airy Stress Functions and the Half-Plane
2.2 Integral Equation Formulation
2.3 Solution
2.3.1 Cauchy Equations of the First Kind
2.3.2 Cauchy Equations of the Second Kind
2.3.3 Numerical Solutions
2.4 Mossakovskii–Barber Procedure
2.4.1 Normal Loading Problem
2.4.2 Application of Shear Force
2.4.3 Influence of Bulk Tension
2.4.4 Application of a Moment
2.5 Conditions for Full Stick
2.6 Solutions Based on Dislocations
2.7 Summary
3 Williams' Solution
3.1 Introduction
3.2 Anti-Plane Loading
3.3 General Loading
3.4 Crack Tip and Incomplete Contact-Edge Solution
3.5 Bonded Wedges
3.6 Sliding Wedges
3.7 Summary
4 Half-Plane Partial Slip Contact Problems
4.1 Introduction
4.2 The Normal Load Problem for Asymmetrical Contacts
4.3 The Sequence of Loading
4.3.1 Sequential Loading (Constant Normal Load)
4.3.2 Proportional Loading
4.3.3 Two-Stage Proportional Loading
4.3.4 Application to a Hertzian Geometry
4.4 The Effect of Differential Bulk Tension
4.4.1 Tangential Load and Moderate Differential Bulk Tension
4.4.2 Bulk Tension Dominated Partial Slip Problems
4.5 Periodic Loading
4.6 More General Loading Scenarios
4.7 General Cyclic Proportional Loading
4.7.1 The Permanent Stick Zone
4.7.2 Mapping Between the Normal and Tangential Problems
4.7.3 Application to the Tilted Shallow Wedge
4.8 Partial Slip Solutions Based on Dislocations
4.8.1 The Insertion of Glide Dislocations
4.8.2 Solution (Constant Normal Load)
4.8.3 Shear Force Only
4.8.4 Differential Tension Only
4.8.5 Cyclic Loading (Constant Normal Load)
4.8.6 Application to a Hertzian Geometry
4.8.7 Cyclic Loading (Varying Normal Load)
4.9 Anti-Plane Formulation
4.10 Influence of a Screw Dislocation
4.11 Correction to a Fully Adhered Solution
4.12 Application to a Hertz' Problem
4.13 Summary
5 Complete Contacts and Their Behaviour
5.1 Introduction
5.2 General Frictional Response—Square Contacting Element
5.3 Finite Slip Zones
5.4 Sliding Asymptote (Bilateral)
6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes
6.1 Introduction
6.2 Basic Solution to the Normal Problem
6.3 Basic Solution to the Tangential Problem
6.4 Partial Slip Under Constant Normal Load
6.5 Partial Slip Under Varying Normal Load
6.5.1 Asymptotic Description for Steady-State Problems
6.6 Summary
7 Crack Propagation, Nucleation and Nucleation Modelling
7.1 Introduction
7.2 Notch and Critical Distance Methods
7.3 Critical Plane Methods
7.4 Short Crack Methods
7.5 Wear and Corrosion
8 Experiments to Measure Fretting Fatigue Strength
8.1 Fundamental and Historic Considerations
8.2 Single Actuator Experimental Apparatus
8.3 Two Actuator Experimental Apparatus
8.4 Further Developments
8.5 Concluding Remarks
9 Fretting Strength
9.1 Introduction
9.2 Asymptotic Representation
9.3 Frictional and Plastic Slip
9.4 Experimental Data
9.5 Interpreting the Measured Fretting Fatigue Strength
9.6 The Asymptotic Descriptions of Contact Edges
9.7 Stress Intensity Factor Calibration for Short Cracks
9.7.1 Adhered Contact Edge
9.7.2 Slipping Contact Edge
9.7.3 Worn-Away Contact Edge
9.8 Cracks at Complete Contact Edges
9.8.1 Application of Calibration of Contact Edge Short Crack Stress Intensity Factors
9.9 Calibration Against Finite Contacts
Appendix A Plane Contacts: Mathematical Techniques
A.1 Complex Variable Preliminaries
A.1.1 Cauchy's Formula
A.1.2 The Hölder Condition
A.1.3 The Principal Value of a Singular Integral
A.1.4 The Plemelj–Sokhotski Formulae
A.1.5 Index
A.2 The Riemann–Hilbert Problem for a Closed Contour with Continuous Coefficients
A.2.1 Homogeneous Problem
A.2.1.1 κ= 0
A.2.1.2 κ> 0
A.2.2 Inhomogeneous Problem
A.2.3 A Note on Discontinuous α(x) and g(x)
A.3 Riemann–Hilbert Problems in Half-Plane Theory
A.3.1 Inverting Singular Integral Equations
A.3.1.1 Singular Integral Equations of the First Kind
A.3.1.2 Singular Integral Equations of the Second Kind
Appendix B The International Fretting Fatigue Symposia
Appendix C Fretting Fatigue Strength: Experimental Data
Appendix References

Citation preview

Solid Mechanics and Its Applications

David A. Hills Hendrik N. Andresen

Mechanics of Fretting and Fretting Fatigue

Solid Mechanics and Its Applications Founding Editor G. M. L. Gladwell

Volume 266

Series Editors J. R. Barber, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, USA Anders Klarbring, Mechanical Engineering, Linköping University, Linköping, Sweden

The fundamental questions arising in mechanics are: Why?, How?, and How much? The aim of this series is to provide lucid accounts written by authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations of solids and structures; dynamical systems and chaos; the theories of elasticity, plasticity and viscoelasticity; composite materials; rods, beams, shells and membranes; structural control and stability; soils, rocks and geomechanics; fracture; tribology; experimental mechanics; biomechanics and machine design. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of the field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. Springer and Professors Barber and Klarbring welcome book ideas from authors. Potential authors who wish to submit a book proposal should contact Dr. Mayra Castro, Senior Editor, Springer Heidelberg, Germany, email: [email protected] Indexed by SCOPUS, Ei Compendex, EBSCO Discovery Service, OCLC, ProQuest Summon, Google Scholar and SpringerLink.

More information about this series at http://www.springer.com/series/6557

David A. Hills · Hendrik N. Andresen

Mechanics of Fretting and Fretting Fatigue

David A. Hills Department of Engineering Science University of Oxford Oxford, UK

Hendrik N. Andresen Department of Engineering Science University of Oxford Oxford, UK

ISSN 0925-0042 ISSN 2214-7764 (electronic) Solid Mechanics and Its Applications ISBN 978-3-030-70745-3 ISBN 978-3-030-70746-0 (eBook) https://doi.org/10.1007/978-3-030-70746-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

Fatigue, popularly metal fatigue, is a primary cause of failure of structural engineering components. Cracks may freely initiate, but this is rare unless the metal (alloy) is of poor quality. Stress concentrations are known to promote nucleation, and one particularly insidious form of concentration is that occurring at the edges of a contact. It is almost impossible to eliminate this effect, and conditions may be exacerbated by a small amount of local slip—fretting. This book treats the analysis of contacts and, in particular, contact edges. It describes methods of finding how much the contact edge slips, for all classes of contact and for a wide range of different loading conditions. It also records the developing sets of test apparatus we have used to determine the fretting fatigue strength of various materials under very well-controlled conditions using purpose-built servo-hydraulic test apparatus. A key feature of the book is the development of methods of matching the condition in the laboratory test with a wide range of practically occurring geometries. The intention is to be able to make use of the results of tests applicable to an ever-wider range of problems and to reveal material properties. This is done by looking at various single-term representations of the pressure and shear stress at the contact edges and quantifying the contact-edge properties in much the same that stress intensity factors quantify the crack tip stress. An over-riding ambition is to unify the quantification of incomplete, complete and other classes of contact edge, but this ambition has yet to be fulfilled. The small Department of Engineering Science at Oxford has a tradition of imbuing in its pupils examination of phenomena at their most fundamental and one of us (DAH) as an undergraduate had lectures from Professor John J. O’Connor on contact mechanics when he had just undertaken some very careful experiments on fretting, and before he went on to develop the artificial ‘Oxford’ knee. It was he who also guided the initial research in 1984 when work on fretting was resumed, and this book represents the current state of development after 36 years of effort. It reports progress well beyond that set out in its predecessor (Hills & Nowell 1994) which was based substantially on half-plane analysis at a constant normal load. This period has also seen the establishment of a regular, small symposium on the subject, with regular

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contributions from a number of people, some of whom have been regulars since the very first one. Oxford, UK Berlin, Germany January 2021

David A. Hills Hendrik N. Andresen

Acknowledgements

The new theoretical results in this book represent the efforts of a number of research students in the Department of Engineering Science of Oxford University, including Daniele Dini, Chris Churchman, Rob Flicek, Rodolfo Fleury, Rob Paynter and many others. Particular thanks should go to Saravanan Karuppanan who generously provided verbatim results for Williams’ eigenvectors, Nils Cwiekala who permitted his latest results for contact-edge crack solutions to be included as soon as they were available and Daniel Riddoch for rigorously checking some of the results for strain nuclei. Jim Barber has been an ever-present encouraging, unfailingly generous guiding hand and has, for thirty-five years, steered one of us (DAH), and we thank him warmly. Recently, rigour has been added by Matthew Moore who has been instrumental in developing new solutions for singular integral equations, and who is responsible for the whole of Appendix A, for which we thank him. The book also makes use of experimental results found both within the department, principally by David Nowell and also by Daniele Dini. Lastly, the seminal work of John O’Connor, who taught one of us (DAH) and whose prescient experimental work and analytical ideas are the forerunners of a number of ideas here, is also warmly thanked.

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Contents

1 Some Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fretting in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basics, Equilibrium and ‘Coupling’ . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Contact Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Classes of Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Methods of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Shakedown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Three-Dimensional Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 5 6 6 11 12 12

2 Plane Elasticity and Half-Plane Contacts . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Airy Stress Functions and the Half-Plane . . . . . . . . . . . . . . . . . . . . . 2.2 Integral Equation Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Cauchy Equations of the First Kind . . . . . . . . . . . . . . . . . . . . 2.3.2 Cauchy Equations of the Second Kind . . . . . . . . . . . . . . . . . 2.3.3 Numerical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Mossakovskii–Barber Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Normal Loading Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Application of Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Influence of Bulk Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Application of a Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Conditions for Full Stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Solutions Based on Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 19 21 22 24 24 25 25 28 29 31 33 34 36

3 Williams’ Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Anti-Plane Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 General Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Crack Tip and Incomplete Contact-Edge Solution . . . . . . . . . . . . . .

39 39 45 47 48

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3.5 3.6 3.7

Bonded Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sliding Wedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48 50 52

4 Half-Plane Partial Slip Contact Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Normal Load Problem for Asymmetrical Contacts . . . . . . . . . . 4.3 The Sequence of Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Sequential Loading (Constant Normal Load) . . . . . . . . . . . . 4.3.2 Proportional Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Two-Stage Proportional Loading . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Application to a Hertzian Geometry . . . . . . . . . . . . . . . . . . . 4.4 The Effect of Differential Bulk Tension . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Tangential Load and Moderate Differential Bulk Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Bulk Tension Dominated Partial Slip Problems . . . . . . . . . . 4.5 Periodic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 More General Loading Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 General Cyclic Proportional Loading . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 The Permanent Stick Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Mapping Between the Normal and Tangential Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Application to the Tilted Shallow Wedge . . . . . . . . . . . . . . . 4.8 Partial Slip Solutions Based on Dislocations . . . . . . . . . . . . . . . . . . . 4.8.1 The Insertion of Glide Dislocations . . . . . . . . . . . . . . . . . . . . 4.8.2 Solution (Constant Normal Load) . . . . . . . . . . . . . . . . . . . . . 4.8.3 Shear Force Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Differential Tension Only . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5 Cyclic Loading (Constant Normal Load) . . . . . . . . . . . . . . . 4.8.6 Application to a Hertzian Geometry . . . . . . . . . . . . . . . . . . . 4.8.7 Cyclic Loading (Varying Normal Load) . . . . . . . . . . . . . . . . 4.9 Anti-Plane Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Influence of a Screw Dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Correction to a Fully Adhered Solution . . . . . . . . . . . . . . . . . . . . . . . 4.12 Application to a Hertz’ Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55 55 57 59 60 63 65 66 69

82 82 84 85 87 89 89 90 94 98 102 102 103 104 106

5 Complete Contacts and Their Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 General Frictional Response—Square Contacting Element . . . . . . 5.3 Finite Slip Zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Sliding Asymptote (Bilateral) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 107 111 117 119

69 71 71 74 77 78

Contents

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6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Basic Solution to the Normal Problem . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Basic Solution to the Tangential Problem . . . . . . . . . . . . . . . . . . . . . 6.4 Partial Slip Under Constant Normal Load . . . . . . . . . . . . . . . . . . . . . 6.5 Partial Slip Under Varying Normal Load . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Asymptotic Description for Steady-State Problems . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 125 126 135 136 138 138 140

7 Crack Propagation, Nucleation and Nucleation Modelling . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Notch and Critical Distance Methods . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Critical Plane Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Short Crack Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Wear and Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 144 146 148 150

8 Experiments to Measure Fretting Fatigue Strength . . . . . . . . . . . . . . . . 8.1 Fundamental and Historic Considerations . . . . . . . . . . . . . . . . . . . . . 8.2 Single Actuator Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . 8.3 Two Actuator Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . 8.4 Further Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151 151 154 156 158 160

9 Fretting Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Asymptotic Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Frictional and Plastic Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Interpreting the Measured Fretting Fatigue Strength . . . . . . . . . . . . 9.6 The Asymptotic Descriptions of Contact Edges . . . . . . . . . . . . . . . . 9.7 Stress Intensity Factor Calibration for Short Cracks . . . . . . . . . . . . 9.7.1 Adhered Contact Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2 Slipping Contact Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3 Worn-Away Contact Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8 Cracks at Complete Contact Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.8.1 Application of Calibration of Contact Edge Short Crack Stress Intensity Factors . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Calibration Against Finite Contacts . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 162 165 166 168 174 178 179 180 182 184 186 187

Appendix A: Plane Contacts: Mathematical Techniques . . . . . . . . . . . . . . . 191 Appendix B: The International Fretting Fatigue Symposia . . . . . . . . . . . . 207 Appendix C: Fretting Fatigue Strength: Experimental Data . . . . . . . . . . . 209 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Chapter 1

Some Fundamentals

1.1 Fretting in Practice Sliding contacts in engineering are always, properly, lubricated in some way, and therefore little damage should ensue. But when we have notionally stationary contacts, which may be bolted or established by some other methods, such as the generation of centrifugal loads in a gas turbine, or hydraulic action in a wellhead locking segment, secondary, fluctuating loads may cause tiny amounts of differential movement, or borrowing a word with an associated common usage ‘fretting’. It is hard to quantify exactly what amplitude of movement is defined under this heading. It ranges from that which is barely detectable (a few microns) up to say 100 µm or thereabouts. Beyond that, we would think of macroscopic sliding. At the lower end of the scale, the contact would, necessarily, be in ‘partial slip’, that is, parts of the contact would be adhered whilst others were slipping. As we will be restricting our attention to components which are elastic, the uniaxial yield strain of common metals is of order 10−3 , so if an extreme edge of a contact is stuck and the size of contact is, say, 10 mm, it follows that the maximum theoretical slip displacement experienced at the far edge in partial slip would be 20 µm. So, much of fretting involves slip displacements controlled by the elasticity of the contact itself, and the effects of local friction, but some forms may involve very small amounts of rigid body motion or ‘sliding’. Figure 1.1 shows the very common example of a dovetail root contact between a disc and a blade in a gas turbine. Centrifugal forces form the contact and expand the disc. Superimposed is a fluctuating load acting on the blade, for example induced by an unsteady gas stream or the excitation of an eigenfrequency. Often, these contacts are found in a state of ‘partial slip’ for long periods during operation, and fretting damage is accumulated in the regions of repetitive relative movement between blade and disc. Although there are many types of contacts experiencing fretting, this is a common practical example studied, as fretting is known to be a dominant failure mode for dovetail root contacts in gas turbines. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. A. Hills and H. N. Andresen, Mechanics of Fretting and Fretting Fatigue, Solid Mechanics and Its Applications 266, https://doi.org/10.1007/978-3-030-70746-0_1

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1 Some Fundamentals

Dovetail Root Joint Blade

Stick

Stick Slip & Fretting Damage

Disk

Fig. 1.1 Illustration of a dovetail root contact between disc and blade in a gas turbine with attendant fretting damage

In its essence, this book is about the mechanics of fretting rather than the materialsrelated aspects. Fretting, by itself, often just produces wear debris in the form of ‘cocoa’, but it also produces conditions ripe for the nucleation of fatigue cracks, and hence the term ‘fretting fatigue’. It is still uncertain whether slip itself encourages or suppresses crack nucleation, but the stress concentration at contact edges is relaxed by its presence whilst at the same time it causes surface modification. The mechanics of the nucleation process itself will vary from material to material and no attempt will be made to model the micromechanics of it here. We do note, though, that cracks that nucleate under notch conditions have more tightly grouped lives than those which are ‘freely initiated’, and that those which nucleate from fretting contacts seem to have still less ‘spread’ in terms of lives experienced. The subject was studied principally as a ‘materials’ subject until the 1960s, and a pioneer of this aspect was R.B. Waterhouse, who wrote or edited two books on the problem; first on corrosion accelerated fretting (Waterhouse 1972), and then a collected monograph on fretting itself (Waterhouse 1981). The latter includes two chapters by J. J. O’Connor who conducted prescient experiments at Oxford in the late 1960s and early 1970s, culminating in three important DPhil theses—Wright (1970), Lau (1972), Bramhall (1973)— and it is on these foundations that the current authors build. The forerunner text to this Hills and Nowell (1994) looked at principally ‘half-

1.1 Fretting in Practice

3

plane’ problems with constant normal load, and in this text, we shall describe recent developments in our understanding of the subject, including a much greater treatment of partial slip problems, further insight into complete contacts and the development of much more versatile experimental test equipment. The two books by Barber in this same series—Elasticity (Barber 2010) and Contact Mechanics (Barber 2018)—provide a deeper understanding of the solid mechanics foundations of solutions presented here and should be read in conjunction with it. Where possible, we will adhere to the same notation and symbols to provide an element of continuity.

1.2 Basics, Equilibrium and ‘Coupling’ Many contact problems encountered will have a very complex geometry, so that the state of stress they endure must, inevitably, be solved by a numerical method—usually the finite element method—and the software within the commercial packages available is improving all the time. But, even so, commercial programmes often include linearisation to improve convergence, such as permitting some slip between surfaces which are intended to be perfectly stuck together, or unexpected ‘improvements’ such as allowances for rotation of elements of material which will give differences from a solution obtained using conventional linear elastic theory, and where no rotation of material is taken into account. Before making a finite element model, useful properties of the solution may be anticipated from considerations of the geometry of the problem and its taxonomy; see Sect. 1.5. Consider the problem of two elastic blocks, made from the same material and of the same thickness, h, pressed together by a normal force, P, alone Fig. 1.2. The contact is symmetrical with respect to x, so that the contact pressure, p(x), will also be symmetrical with respect to x as, indeed, will both direct components of stress, at all points, in each body, i.e.

Fig. 1.2 Two elastic blocks pressed together by a normal force, P, and subject to a shear force, Q, and moment, M = 2Qh

σx x (x, y) = σx x (−x, y) ,

(1.1)

σ yy (x, y) = σ yy (−x, y) ,

(1.2) y

P Q 1

h

M = 2Qh b

-b a

-a h Q

P

2

x

4

1 Some Fundamentals

whilst the shear stress is antisymmetric σx y (x, y) = −σx y (−x, y) .

(1.3)

If we require points on the centreline (x = 0) not to displace sideways, then we can see that the x-direction displacement of points generally is antisymmetric, i.e. u(x, y) = −u(−x, y) .

(1.4)

This applies to the surface of the bodies, and we note that, unless the two bodies are actually geometrically the same, i.e. a = b in Fig. 1.2, if the interfaces were frictionless, surface particles will displace laterally by different amounts, i.e. u 1 (x, 0) = u 2 (x, 0) and, in fact, we shall see later that for the geometry shown u 1 (x, 0) > u 2 (x, 0), (x > 0) so that the smaller body will tend to ‘spread’ more than the wider one. This will be resisted by friction unless the interface is perfectly lubricated, and will therefore cause shear tractions to be generated. So, the application of a normal load alone generates both contact pressure and interfacial shear, and such a problem is said to be coupled. In fact, most contact problems will be coupled unless (a) they are geometrically similar1 and (b) they are made from the same material.2 In contact problems which are coupled, the application of a shear force also causes a re-distribution of the contact pressure; that is coupling acts both ways. Note that the modification of contact pressure associated with the change in normal surface displacement arising from surface shear tractions is usually small compared with the shear traction modification arising from normal loads. Neglect of the former is often called the ‘Goodman approximation’, after Lawrence Goodman, late Professor of Civil Engineering, University of Minnesota (Goodman 1962). Suppose that shear forces, Q, are now applied to the top and bottom of the contacting pair. Of course, the assembly, as a whole, is no longer in rotational equilibrium, and so an external anticlockwise moment of magnitude 2Qh must now be applied. This could be done in any number of different ways. For example, it could be applied to the upper body 1 alone and, provided that the lower body did not rotate about its left-hand corner, this would be perfectly acceptable. Another possibility would be to apply an anticlockwise moment M = Qh to each body, and, if this was done, it would follow that the shear forces applied would be statically equivalent to ones exerted in the plane of the contact; in this case, the contact pressure would remain symmetrical, although the contact pressure distribution would change slightly (unless a = b). In experiments, we will often use the technique of applying shear forces remotely but imposing also, an external moment to render them equivalent to forces applied through the plane of contact. In this way, a rocking moment which would cause the problem to become inherently unsymmetrical is avoided and, at the same time, the impact of a very locally applied external force on the contact is also averted.

1 Or 2 Or

may both be represented by half-planes, see Chap. 3. certain other special combinations of materials, see Chap. 3.

1.3 Friction

5

1.3 Friction The law of friction most commonly used has been in existence for hundreds of years and, although there promises, at any moment, to be refinements, the oldest form, associated with the names of Amontons and Coulomb, remains the most used and quite acceptable for most engineering purposes, but probably not on the micro- or nano-scales. The macroscopically determined ratio between the shear force, Q, and the normal force, P, on a sliding block is given the symbol f , the coefficient of friction. It is apparently independent of the shape of the block, and the value of f is substantially independent of the normal load. Particles on the surfaces of two components pressed together may be in one of two states; they may be stuck together, or they may be slipping. A contact where at least some of the interface is stuck cannot be sliding. In fact, most of this book is concerned with the study of contacts where at least part of the contact is slipping for some of the time. We define the shear traction, q(x) ≡ τx y (x, 0) and contact pressure, p(x) ≡ −σ yy (x, 0). In a region of contact that is slipping, the ratio of shear traction to normal pressure |q(x)|/ p(x) is equal to the coefficient of friction, f , that is the point-wise value and the global value are assumed to be the same. When a plane contact (and so far, we have discussed only two-dimensional or plane contacts) is in partial slip, there are two sets of conditions which hold in the slip and stick regions of contact: • Stick region—here, the local shear traction must be lower in magnitude than the contact pressure multiplied by the coefficient of friction, i.e. |q(x)| < f p(x), and the relative slip displacement of pairs of contacting points is preserved at its current value, i.e. u 1 (x) − u 2 (x) = constant, or u˙ (x) = 0, where u˙ (x) = u˙1 (x) − u˙2 (x), and where the dot,˙, denotes the derivative with respect to time. • Slip region—the magnitude of the shear traction is limited to the product of the coefficient of friction and the contact pressure, so that q(x) = ± f p(x) and the choice of sign must be consistent with the change in slip of surface points, so that it opposes the motion, and hence sgn (q (x)) = − sgn (u˙ (x)). The presence of friction in a solid mechanics problem, rather like plasticity, makes the problem inherently non-linear, even if it were linear in the first place (as we will see, doubling the load on contacting cylinders does not double the contact pressure and hence, nor does it double the magnitude of the state of stress). Problems which appear linear (see stationary, i.e. complete contacts and common-edge contacts, Sect. 1.5) are no longer linear when the condition for the onset of slip has been reached, and superposition principles do not, generally, apply. Also, the state of the contact interface, that is, the distribution of regions of stick and slip, do not depend just on where the problem lies in, say, P − Q space (Sect. 4.3), but how that point was reached. We shall see, later on, that it is quite possible to have the same set of applied forces present on a contacting pair and that, in one case, the contact will be wholly adhered whilst in another, there will be significant areas of slip, that is, the solution is loading path-dependent.

6

1 Some Fundamentals

1.4 Contact Requirements We have dealt with the concept of point-wise friction, first, because that naturally arises in any contact, but we have not looked at the conditions for contact themselves. In the case of a problem such as that shown in Fig. 1.2, it hardly seemed necessary, because it seems pretty clear that, as the front face of body 1 is flat over the contact length (−a < x < a) and the front face of the lower body is flat over a greater distance, contact will be over the full front face; that is, the size of the contact is defined by body 1 alone.3 However, it is not clear in all cases how big the contact will be. Consider the contact problem shown in Fig. 1.3a, where two elastically similar convex bodies are being pressed together. It is easiest (but not necessary) to think of them as two large equal cylinders and it will be shown in Chap. 2 that the usual solution has a semi-elliptical contact pressure in form, so that it falls smoothly to zero at the edges, where the bodies now have a common tangent. What would have happened if we had chosen the contact width (correctly, 2a) wrongly, and would the solution found have had any practical relevance? If the contact width had been chosen too big, Fig. 1.3b, the direct interfacial traction would have become singularly tensile at the edges, effectively ‘pulling’ the two bodies together.4 On the other hand, if the contact width had been chosen too small, Fig. 1.3c, material exterior to the intended contact would have interpenetrated. The formal requirements of conventional contacts (where tension is not supported) are called the Signorini conditions and are given by the following statements: • Within the contact—the contact pressure, p(x), must be everywhere positive (compressive) and the separation between the contacting faces, g(x), must be zero. • Outside the contact—the contact pressure must vanish and the separation or gap between the contacting faces must be positive, g(x) > 0.

1.5 Classes of Contact We have already seen that there are at least two fundamental kinds of contact which might arise; first ones where the contacting surfaces of the two bodies conform over a contour extending across the full width of the contact-defining body (body 1, Fig. 1.4a), and further in the other body. In these cases, the size of the contact is determined by an abrupt change in the front-face profile of the contact-defining body. These are called complete contacts, and are exemplified by the fundamental problem of an elastic block pressed into an elastically similar half-plane shown also 3 In

fact, the material will ‘lap around’ the corners onto the side faces of the upper body; at least that is the implication of linear elastic theory. The phenomenon will not be pursued here but the interested reader may look up details in Churchman et al. (2006). 4 This is precisely what happens in the presence of cohesive or adhesive forces, important in very small contacts; see Johnson et al. (1971).

1.5 Classes of Contact

7

a)

P

p(x) g(x) > 0

b)

P

x

a

-a

c)

x

a

-a

P

g(x) < 0 -a a

x

Fig. 1.3 Incomplete contact in which its size is decided by the Signorini inequalities; a the correct solution, b the contact is too large, so that the contact pressure becomes tensile at the edges and c the contact is too small, thereby implying interpenetration

in Fig. 1.2. These are not the kinds of contact usually met with first in textbooks on contact mechanics, e.g. Johnson (1985), Hills et al. (1993), Jäger (2004), because they are not capable of analysis, at least completely, in a closed form. Consider the problem shown in Fig. 1.2 from a different angle. Let the lower body be very large and therefore capable of being thought of as a half-plane, so the problem may be treated fairly simply (see Chap. 2). However, the upper body, with sides that must remain traction-free, would require a formulation for an elastic rectangle, and that is only available in series form Khadem and O’Connor (1969), and even then only for the cases of a frictionless or an adhered interface. But the target of this book is partial slip and fretting, and fretting damage starts almost invariably at the edges, so that a really high quality local analytical solution is needed. This may be achieved by using a formulation for wedges, for which Chap. 3 provides the basics. There it is shown that it is useful to ‘zoom in’ to one of the corners, and assume, usually,

8

1 Some Fundamentals

a)

b)

p(s) ~ s

p(s) ~ s P

P

s

s

c)

d)

p(s) ~ s

p(s) ~ s

P

P

s

s

e)

f)

p(s) ~ s

p(s) ~ s

P

P

s s

Fig. 1.4 Classes of contacts a a complete contact formed by a square block indenting a half-plane, b an incomplete contact formed by two cylinders pressed together, c an almost conforming contact formed by a cylinder in a hole, d an incomplete contact formed by a rounded punch with a narrow flat face, e a receding contact formed by an elastic strip resting on a half-plane, f a common-edge contact formed by two square blocks pressed together

that the interface is adhered. Note that, as shown above, the coupling will certainly be present because of the difference in the domain shapes, but, in addition, if there is an elastic mismatch, further coupling may arise, and this will depend on the two so-called Dundurs’ parameters (Dundurs 1969b), the two combinations of elastic constants, α and β, which are defined as follows: (κ1 + 1)/μ1 − (κ2 + 1)/μ2 (κ1 + 1)/μ1 + (κ2 + 1)/μ2 (κ1 − 1)/μ1 − (κ2 − 1)/μ2 β= (κ1 + 1)/μ1 + (κ2 + 1)/μ2 α=

(1.5) (1.6)

1.5 Classes of Contact

9

where κi is Kolosov’s constant, and μi the modulus of the rigidity of body i. In plane strain, κ = 3 − 4ν, where ν is Poisson’s ratio. In Chap. 5, we will show that the edges of the contact will slip if the coefficient of friction is below a critical value or, if it is higher than that figure, the whole contact will adhere under a wide range of loads (not just normal). For any loading which maintains intimate contact near the edge, if we took a local coordinate, s, measured positive inwards from either edge of contact, we would find that the contact pressure nearby varied as p (s) ∼ s λ−1 , where the value of λ is a function of the interior contact angle at the edge in question. If the contact edge is slipping, a similar asymptotic procedure may be followed and, in these cases, the value of λ depends also on the value of f . For a wide range of conditions, λ < 1, so that the contact pressure is locally power-order singular. The second kind of contact is exemplified by the sketch in Fig. 1.4b. It occurs when two convex bodies are pressed together, so that, when the normal load is infinitesimal, they touch at a point (or line, strictly, for a plane problem). These contacts are usually called incomplete. A more general kind is shown in Fig. 1.4c which shows an elastic cylinder pressed against the surface of a cylindrical hole in a large plate, whose diameter is not much larger than that of the cylinder. Characteristics, generally, of problems of this kind include the observation that the contact will get monotonically bigger as the contact load is increased. When both bodies are convex (or the radii of curvature differ by a significant amount), so that the width of contact is small compared with either radius, the contact continues to grow with the applied load until some limit of elasticity theory is reached, e.g. the neglect of element rotation, or the infinitesimal nature of strains, or the material yields. In the case of the cylinder in the almost conforming hole (where the difference between the radius of the cylinder and the radius of the hole is much smaller than either, 2(Ro −Ri )/Ro +Ri  1), the application of an increasing force causes the contact size to increase (Fig. 1.4c), until a characteristic angle is reached (Ciavarella et al. 2006; Decuzzi and Ciavarella 2001a, b; Lau 1964). A second property is that, at least locally near the contact edge, the contacting bodies behave like half-planes. Anticipating a result which will be found in Chap. 2, where half-plane problems are treated extensively, the near edge contact pressure is given by p (s) ∼ s 1/2 , i.e. the contact pressure adjacent to the edge is square root bounded in character, provided only that the bodies are made from the same material, and that the Signorini conditions are satisfied. If the contact does not ‘conform’, so that the contact surface is approximately plane, both bodies may be idealised by (i.e. represented using an elasticity formulation appropriate to) half-planes; see Chap. 2. As well as cylinders, problems involving very shallow wedges may be treated this way and certain other problems, such as the perhaps rather contrived one of a shallow wedge and a cylinder, or possibly the case of a punch having a narrow flat face but with significant regions of rounding on either side (Fig. 1.4d). Because the same basic stiffness properties are present in each body (those of a half-plane), the contact is always uncoupled provided only that the bodies are made from the same material. If they are elastically dissimilar, there is a very reduced dependency on the combination of elastic constants; Dundurs’ first constant, α, has no effect—only β.

10

1 Some Fundamentals

Two further kinds of behaviour may arise. The first is shown in Fig. 1.4e where a layer is placed on the surface of an elastically similar half-plane and then a line force applied. Before the normal load is exerted, contact occurs everywhere between the two bodies but, upon the application of an infinitesimal normal load, it ‘snaps’ to a contact size weakly dependent on the coefficient of friction but essentially the width is a particular fraction of the layer thickness. As the normal load is increased, the angle of lift-off is increased but the contact size remains the same so that, to this extent, the problem becomes linear. The contact pressure near the edge again varies in a square root bounded manner. A practical problem with modelling receding contacts by the finite element method is the snap-off of contact at an infinitesimal normal load. This produces great difficulties in achieving convergence, and the practical way around this is to model the contact in a slightly contrived way. We know roughly where the separation point will be, and so model a small ‘step’ in the layer which does not affect its overall elastic properties but which maintains a small separation from the substrate, even at zero load. The location of the step is chosen so as to be just ‘outboard’ of the separation position which will naturally establish itself. Lastly, in Fig. 1.4f, we show the problems of two equal elastically similar square blocks pressed together (and then subjected to shear). The edge of the contact is defined by both bodies simultaneously and, in the problem shown, is rather like a beam with axial compression and with a shear force. If we zoom in and restrict our attention to an edge of contact, we see that there are neither singularities nor must the only non-zero component of stress, the direct stress parallel with the free surface, vanish and, as the latter constitutes the contact pressure just inside the contact, we see that p (s) ∼ s 0 . So, of all the forms of contact we can have, this is the only one where the contact pressure may be finite at all points along the contact patch. An additional observation is that the contact is stationary (size independent of contact load) in the cases of complete and common-edge contacts, that incomplete contacts advance with applied normal load and that whilst some receding contacts do just that when the normal load is applied; the subsequent application of a shear force will produce a smoothly changing contact size. When modelling stationary contact problems by the finite element method, providing that the loading trajectory is simple, a very practical approach is to start off by modelling the two bodies as if they were glued together, as a monolith, and determining the state of stress. A simple example is shown in Fig. 1.2. The contacting blocks should be ‘sewn together’ along the line [−a a]. Two checks then need to be made to test the validity of this approach: • First—is the value of the direct stress across the trace line of the interface σ yy (x) < 0, over −a < x < a? If so there is full contact and the Signorini conditions are satisfied.   • Second—plot the ratio σx y /−σ yy (x), over −a < x < a. Find the maximum value of this quantity. Providing that the coefficient of friction, f , is at least as big as this, the contact will be wholly stuck. For the example problem, we might run the finite element programme for the geometry defined as a monolith for two load cases—first with Q = 0 and then with

1.5 Classes of Contact

11

P = 0. The finite element output would then reveal the four dimensionless quantities σ yy (x, 0)a/P, σx y (x, 0)a/P, σ yy (x, 0)a/Q, and σx y (x, 0)a/Q from which it would be easy to determine the value of Q/P at which separation occurs and, for smaller values of this quantity, the necessary coefficient of friction to keep the contacting pair fully adhered. Use will be made of these ideas in Chap. 5.

1.6 Methods of Solution Most readers of this book will use the finite element method to model problems. It has the fantastic advantage over all analytical methods that even the most complex geometry may be modelled with ease. And that versatility is not something we shall overlook. But, what this text will provide is a means of anticipating what the finite element programme should reveal. It is all too easy to make an incorrect inference from finite element output because the solution is not as fully converged, as had been believed. A comparison of standard cases shows this up really well and a procedure currently being advocated is to ‘reduce’ the finite element matrix so that all interior node solutions are explicitly inverted, and then the contact inequalities, see Sects. 1.2 and 1.3, are explicitly met using MATLABTM or some equivalent numerical handling software. These permit the inequalities to be applied exactly, and for multiple load cases to be handled with only a single finite element ‘run’ (Beisheim et al. 2018; Thaitirarot et al. 2014). Finite element methods are relatively poor when there are (a) inequalities, as just discussed, which are often linearised out; (b) singularities, which almost always arise in complete contacts; (c) sudden changes in contact size, which is what occurs in the class of receding contact described in Sect. 1.5. So, a good understanding of what is happening when these arise, and even better, an asymptotic form which may be inserted to sharpen up the output are what this book provides. Other general numerical schemes, such as the boundary element method, remain less popular and will not be looked at here. Later chapters introduce formulations for the whole contact based on half-plane theory and for contact-edge solutions based on wedge theory. A further possibility which has been used quite extensively as an approach where accuracy is required but the geometry is relatively simple is to model the contacting pair as a monolith, and then to introduce dislocation arrays so as to permit (separately) slip and separation. Comninou (1976) and Dundurs (1969b) were pioneers of this approach which gives valuable information, very detailed, about the size and evolution of slip zones. They used the approach exclusively where the separation region was finite and bordered by stuck, closed contacts, i.e. effectively continuous material. Recently, we have extended the ideas to model receding contacts where separation and slip extend to infinitely remote points by using pre-formed dislocation arrays to represent the behaviour at infinity, and which effectively becomes a hybrid eigenstrain/distributed dislocation approach.

12

1 Some Fundamentals

1.7 Shakedown The concept of shakedown has entered few undergraduate textbooks. Its traditional application is to cyclically loaded structures where the severity of loading is such that part of them go into a plastic state during the first application of load. As the load is removed, residual stresses develop, and they are invariably such that, when the load is re-applied, the resultant state of stress is milder (meaning less likely to reach the yield condition) than before. Over a number of cycles, the residual state of stress will evolve and, in some cases, the resultant state of stress at maximum applied load is elastic everywhere, and so reversible. When this is the case, the applied loads are said to lie within the ‘shakedown limit’. By analogy with the plastic case, the same nomenclature has been borrowed and applied to frictional contacts. When slip occurs, there is cleaning of the surfaces which increases the coefficient of friction. This is analogous to work hardening in the case of plastic cyclic loading. But, in addition, when slip occurs and the applied load is removed, often the particles which have slipped do not return to their original position but, instead, lock-in a residual displacement and with it, an interfacial shear traction distribution. This is such that, upon re-application of the load, the tendency to slip again is reduced. In the case of incomplete contacts, the formal ‘shakedown limit’ that is, the applied shear force which may be applied before slip starts, is zero but it will be shown in Chap. 5 that the extent of slip is governed by the range of shear force and that the steady-state slip zone size is half of that occurring in the first half cycle—so some shaking down has certainly occurred. Barber has recently shown (Barber et al. 2008) that whilst in uncoupled contacts if shakedown can occur, it certainly will (cf. Melan’s theorem in plasticity) in the case of coupled contacts; shake down, if theoretically achievable, may occur only if the starting conditions, i.e. the initial locked in tractions, if any, are suitable. Additional research efforts regarding shakedown have been published in Klarbring et al. (2007), Young et al. (2008), and Yong and Barber (2011a, b).

1.8 Three-Dimensional Aspects A book strictly on contact mechanics might be expected to have a number of threedimensional problems presented—at least axi-symmetric ones and probably other cases. But this book is concerned, predominantly, with edge effects, and it has also an ambition to rationalise our way of viewing the behaviour of contacts at their edges. We may expect the contact disc for a spherical Hertzian contact suffering partial slip to consist of a central stick region surrounded by an (approximately) constant width annulus of interfacial slip, aligned (again approximately) with the direction of the applied shear force, Q; see Fig. 1.5. With the local axis set at position θ as shown, the shear traction, τ Q , may be resolved (as a vector), into an in-plane component, τx y (the y-direction points out of the plane of the paper), and an anti-plane component, τ yz .

1.8 Three-Dimensional Aspects

13

Fig. 1.5 Contact disc for a spherical Hertzian contact suffering partial slip

stick region Q z y

x

To handle this arrangement, we shall, where appropriate, look at two-dimensional anti-plane loading of problems as well as in-plane. It is perhaps worth remarking that, in contact problems, the anti-plane form cannot exist on its own—in this example problem, the contact pressure, for all values of θ , is −σ yy and the corresponding displacement is u y . On θ = 0, π (in the direction of the applied shear force), the shear traction is purely in-plane (τx y ), whilst on the perpendicular plane, θ = ±π/2, it is purely anti-plane (τ yz ). The latter has no x direction displacement and so, even if the contact bodies are elastically dissimilar, it produces no relative change in normal displacement, and therefore no coupling. It follows that, in three-dimensional problems between elastically dissimilar bodies, the two Dundurs constants α and β remain sufficient to quantify coupling effects, and it is only the in-plane component of shear traction that contributes.

Chapter 2

Plane Elasticity and Half-Plane Contacts

2.1 Airy Stress Functions and the Half-Plane The basis of fundamental contact solutions should be ‘exact’ elasticity solutions, where we sacrifice the realism of details of the geometry of a real contact by replacing them with something quite idealised, but then obtain a solution that is precise. The first idealisation we shall make is to make the problems ‘plane’ (or anti-plane in some cases) and this simply means that, if the plane of the analysis is x-y, there are no gradients of any components of stress or strain with respect to z. The standard limiting plane cases are, first, plane stress, where all stress components which incorporate a subscript z are assumed to vanish everywhere. Clearly, there are no tractions on the free surfaces of the component, but is it realistic to assume that they are zero throughout? Plane stress is assumed to hold when the component is ‘thin’ by which we mean that its dimension in the z-direction is small compared with the smallest salient in-plane dimension, and it is therefore very unlikely to be of relevance in most contact problems where, for stability, (the contact pressure is always compressive, of course) components are always ‘thick’. Further, plane stress is an approximate solution in cases where the sum of the in-plane direct stresses varies more quickly than linearly (Barber 2010, p. 41) which it certainly does in contact problems. The second plane form of solution, plane strain, occurs when the object studied is long in the z-direction compared with all in-plane dimensions, so that all slices must continue to have flat parallel faces after the load is applied for them to fit together without voids or overlap. We normally assume that the axial strain is zero with the possibility of adding a uniform force (and possibly bending moments) in the most general case (see Barber 2010, pp. 38–39), so that the free end surfaces are, indeed, traction-free. Most contact problems, if geometrically two-dimensional, are likely to correspond to the plane strain limit, with a transition region near to the ends to move to a traction-free state on the end faces.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. A. Hills and H. N. Andresen, Mechanics of Fretting and Fretting Fatigue, Solid Mechanics and Its Applications 266, https://doi.org/10.1007/978-3-030-70746-0_2

15

16

2 Plane Elasticity and Half-Plane Contacts

The second level of approximation concerns the domain shape in which the model is to be built. We will be concerned, for example, with the contact of cylinders and shallow wedges, and we would like to represent each contacting body as a halfplane, that is, we assume that the stiffness of the material in the neighbourhood of the contact behaves as though the surface is (approximately) flat and all other faces are remote. For this to apply to cylinders pressed together, the contact size must be small compared with the radius of the smaller. This is partly to ensure that the material on either side of the contact is adequately represented and also, as we shall see, so that the circular form of the surface may, over the contact, be sufficiently well approximated by a parabola. In the case of a wedge, the requirement is that the contact be shallow (meaning an exterior wedge angle of not more than about 10◦ again partly so that material on either side of the contact is adequately represented by a half-plane, and partly so that the rotation of surface elements is sufficiently small to be neglected Sackfield et al. 2005). This sweeping simplification was first made in the 1880s (Hertz 1881) when Hertz considered contacts having an axi-symmetric form (spherical contact) and it has proved to be an inspired idealisation, and where liberties may often be taken. The method used to develop our most basic solutions is the application of Airy stress functions; see Barber (2010), p. 46. It is not possible to derive the basic results fully, for which Barber’s book should be consulted. For our purposes, cylindrical coordinates are appropriate to construct the most fundamental solutions needed here (Flamant’s solution for a tip-loaded wedge, specialised to a half-plane). The relationships between the stress function, φ, and the stress components are defined by the following derivatives in the absence of body forces, such as gravity. 1 ∂ 2φ 1 ∂φ + 2 2 , r ∂r r ∂θ ∂ 2φ = , ∂r 2 1 ∂φ 1 ∂ 2φ + 2 . =− r ∂r ∂θ r ∂θ

σrr =

(2.1)

σθθ

(2.2)

σr θ

(2.3)

With these definitions, the field equilibrium equations are automatically satisfied. If we find the corresponding strains using Hooke’s Law, for plane deformation, and substitute them into the compatibility field equations, we realise that there is not a free choice of φ, and it must satisfy the following so-called biharmonic equation.  ∇ 4φ =

1 ∂2 ∂2 1 ∂ + 2 2 + 2 ∂r r ∂r r ∂θ

2 φ=0.

(2.4)

It is not possible to infer what potential function might be appropriate to any particular geometry, that is, we cannot deduce what stress function is needed for any particular boundary value problem, and we must use the experience of known functions which have been explored to guide us in our choice. The starting point

2.1 Airy Stress Functions and the Half-Plane

a)

17

b)

y,v x,u Q

P

P r

v(x)

rr

c)

d)

Q (1-2 )(1+ ) E

P

Q u(x)

Fig. 2.1 Half-plane (a) subject to a normal force, P, and shear force Q, (b) displacement in ydirection due to the normal force, (c) displacement in x-direction due to the normal force and (d) displacement in y-direction due to the shear force

for our analysis is the application of a normal and shear pair of line forces, (P, Q), applied at the origin located on the surface of the half-plane, Fig. 2.1a. We choose the stress function φ=−

rθ (P sin θ + Q cos θ ) . π

(2.5)

If we substitute this function into Eqs. (2.1–2.3), the following state of stress is revealed: 2 (P cos θ − Q sin θ ) , πr = σr θ = 0 everywhere.

σrr = −

(2.6)

σθθ

(2.7)

This very unusual state of stress satisfies the boundary conditions that, on θ = ±π/2, the tractions σr θ and σθθ must vanish. And, if we draw a semi-circle centred on the origin, and consider an element of material lying on it, the only non-zero component of stress we observe is in the radial direction, directed towards the point of application of the force: the contribution from the normal force has its maximum value beneath the load (θ = 0) and decreases harmonically towards zero at the surface (θ = ±π/2) whilst that associated with the shear force has its maximum compressive value ahead of the force (θ = −π/2) and falls to zero beneath the load and has its maximum tensile value on the surface immediately behind the load (θ = π/2). We

18

2 Plane Elasticity and Half-Plane Contacts

can easily convert these results to use in a Cartesian axis set, and superpose them to find the stress state under distributed tractions, but far more important, for the time being, is to find the corresponding surface strains and then displacements. Almost all contact problems are in a state of plane strain so, for simplicity, that will be assumed from now onwards, and it follows directly that σzz = ν (σrr + σθθ ) ,

(2.8)

and hence, as only one in-plane direct stress is non-zero, we arrive at   2  σrr  1 − ν2 = − 1 − ν 2 (P cos θ − Q sin θ ) , E Eπr 2 ν (1 + ν) (P cos θ − Q sin θ ) , = Eπr

εrr = εθθ

(2.9) (2.10)

where E is Young’s modulus and ν Poisson’s ratio. Integration, followed by the elimination of arbitrary functions of integration gives the following expressions for the displacement, u(x) and v(x), when specialised to the surface:   2 1 − ν2 (1 − 2ν) (1 + ν) sgn (x) + Q ln |x| + A, u (x) = −P 2E πE  2 1 − ν2 (1 − 2ν) (1 + ν) v (x) = −P ln |x| − Q sgn (x) + B , πE 2E

(2.11) (2.12)

where sgn (x) is the signum function (= 1 if x > 0, and = −1 if x < 0) and A and B are constants of integration. It is worth remarking on the physical implications of these. First, the normal load, P, gives rise to a surface normal displacement, v(x), which shows a logarithmic singularity beneath the point of application of the load, but it also has the effect of drawing all points towards the origin, on each side, by a constant value (so material adjacent to the load, in this mathematically contrived problem, must overlap); see Fig. 2.1b, c. Secondly, in the case of the tangential load, the relevant functions are exchanged, so that material is displaced in the x-direction, u(x), by an amount that varies logarithmically with distance, but, at the same time, material ahead of the crack is depressed by a constant amount and that behind it is elevated by a constant amount leaving a step of height Q(1 − 2ν)(1 + ν)/E at the point of application of the load; see Fig. 2.1d. We conclude that the normal load has an influence on the surface normal displacement everywhere, whereas the shearing force produces a changing surface normal displacement only immediately beneath it, and nowhere else. It should also be noted that the displacements include arbitrary constants of integration which can be fixed only by choosing an arbitrary datum depth at which the displacements are assumed to vanish. To make progress, we need to find the effects of distributed tractions and to remove the irritating datum reference for the displacements.

2.2 Integral Equation Formulation

19

2.2 Integral Equation Formulation To formulate a general solution, the derivatives of the surface displacements are taken, and we differentiate Eqs. (2.11) and (2.12) with respect to x to give the derivatives dv/dx , the surface slope and the surface strain ε du x x ≡ /dx . Note that if an element of shear force dQ is smeared over a small distance dx (so that the shear traction is given by q(x) = dQ/dx ), this produces a surface slope at this point of magnitude dv/dx = q(1−2ν)(1+ν)/E . Therefore, in plane strain,

εx x

  2 1 − ν2 p (ξ ) dξ dv (1 − 2ν) (1 + ν) =− − q (x) , dx πE E contact x − ξ

(2.13)

  2 1 − ν2 q (ξ ) dξ du (1 − 2ν) (1 + ν) =− p(x) + , ≡ dx E πE contact x − ξ

(2.14)

where p(x) = d P/dx and is compressive when positive. Note that this means that the surface tractions are related by p(x) = − σ yy (x, 0) ,

(2.15)

q(x) = σx y (x, 0) .

(2.16)

These integral equations may be used to solve a range of contact problems, where both bodies are elastic (though not necessarily made from the same material). The general form of the problem to be solved is shown schematically in Fig. 2.2. The main addition needed to the formulation developed is to adopt a common coordinate set so that the one we have employed already continues to apply to body 1, but changes in signs arise when we look at the effects of the tractions on body 2. Careful consideration shows that, in the coordinate set shown in Fig. 2.2a, the surface displacement gradients in body 2 are given by   2 1 − ν2 p (ξ ) dξ dv2 (1 − 2ν) (1 + ν) = − q (x) , dx πE E contact x − ξ εx x,2 ≡

  2 1 − ν2 q (ξ ) dξ du 2 (1 − 2ν) (1 + ν) =− p(x) − . dx E πE contact x − ξ

(2.17)

(2.18)

If we think, first, about the normal indentation problem, we first assume that the bodies may interpenetrate each other, by an amount g(x), Fig. 2.2b. We then apply contact pressure (and shearing tractions may also arise which, if the contacting bodies are made from different material, will themselves change the profile) to push the surfaces back to a common line, Fig. 2.2c. Note, though, that the contact length will not be the same as the length over which interpenetration occurs, and is determined by the Signorini conditions developed in Chap. 1. Also, although we could have formulated the whole problem using the displacements themselves, this would have

20

2 Plane Elasticity and Half-Plane Contacts

a) P y, p(x)

body 2

Q

x body 1

b)

c) P

body22 body

body 2

g(x)

x

body 1 body 1

Fig. 2.2 Two contacting half-planes when (a) subject to a normal force, P, including the resulting pressure distribution, p(x), and shear force, Q, (b) an illustration of the gap function, g(x), and (c) displacements necessary to avoid penetration of the contacting bodies

meant that the datum depth had to be retained, and a better method is to work in terms of the surface slopes and their change, so that the requirement that the deformed surfaces maintain intimate contact is dv1 dv2 dg (x) = − , dx dx dx

(2.19)

where g(x) is the overlapping profile, and hence E ∗ dg 1 = 2 dx π

 contact

p (ξ ) dξ − βq (x) . x −ξ

(2.20)

Here, we have introduced the composite plane strain stiffness, E ∗ , defined by

where

1 1 1 = ∗+ ∗ ∗ E E1 E2

(2.21)

1 − νi2 1 i = 1, 2 , ∗ = Ei Ei

(2.22)

2.2 Integral Equation Formulation

21

and we repeat Dundurs’ second elastic mismatch parameter, β,1 for the reader’s convenience   E ∗ (1 − 2ν1 ) (1 + ν1 ) (1 − 2ν2 ) (1 + ν2 ) . (2.23) − β= 2 E1 E2 In simple contact problems where it is primarily the contact law and pressure distribution which are to be determined, this equation is probably all that is needed, but in problems where we want to know about the surface slip displacement (as we usually do in fretting problems), this is given by h (x) ≡ u 1 (x) − u 2 (x). We will then also need to make use of the following equation, which links this quantity to the surface tractions.  E∗ q (ξ ) dξ E ∗ dh 1 ≡ + βp (x) . (2.24) (εx x1 (x) − εx x2 (x)) = 2 dx 2 π contact x − ξ We have already described the phenomenon of coupling, and here it is evident that the domains are the same, and elastic mismatch is its sole origin for half-plane geometries—the term βq(x) in Eq. (2.20). It is clear that the second term will be zero if either (a) the coefficient of friction is zero so that the shear tractions vanish anyway or (b) Dundurs’ second constant, β, vanishes—usually because the materials from which the contacting components are made are elastically similar. If β is non-zero, it will be found that modest changes to the pressure distribution arise.

2.3 Solution When a general contact is formed, where we know the contacting profile and hence g(x), and where there is also both elastic mismatch so β = 0, and the coefficient of friction is finite, the effect of a normal load alone will be to give rise to a contact which will be composed of a mixture of stick and slip zones. The solution to two, coupled singular integral equations with the Cauchy (1/(x−ξ ))) kernels, Eqs. (2.20), and (2.24), is needed, and this will be very complicated. The reader is referred to the classic extended paper by Erdogan et al. (1973) for detailed discussions of how to treat these cases numerically and very efficiently. Synopses of the analytical, exact inversion process for the simpler cases are given in Hills and Nowell (1994), based on the classical process devised by Hilbert, and originally described very fully in one of Muskhelishvili’s classic books (Muskhelishvili 1953b), but this is not an easy read. Under conditions of sliding, when at all points along the interface q(x) = f p(x), the appearance of Eq. (2.20) takes on a much simpler form, viz.

1 See

also Chap. 3 and Sect. 3.5 for further notes on Dundurs’ constants.

22

2 Plane Elasticity and Half-Plane Contacts

1 E ∗ dg = 2 dx π

 contact

p (ξ ) dξ − fβp (x) . x −ξ

(2.25)

Here, the unknown function, p(x), appears both on its own and within the integral, and this is known as a Cauchy integral equation of the second kind. We will now simplify the problem even further by assuming that f and/or β is zero so that Eq. (2.20) becomes a Cauchy equation of the first kind and we will now look at elements of its inversion.

2.3.1 Cauchy Equations of the First Kind The equation to be solved has the following form: E ∗ dg 1 F(x) ≡ = 2 dx π



a −a

p (ξ ) dξ x −ξ

−a < x a

(2.41)

P y x

26

2 Plane Elasticity and Half-Plane Contacts

Fig. 2.4 Single indentation by a flat punch over contact [−a a]

P

f0(x;a)

p(x)

-a

h0(x;a) a

The corresponding surface displacement may be found from Eq. (2.13), under conditions of plane strain and with no shear traction present. When using this equation, it is important to understand that, when |x| < a, the observation point and a point within the integrand may approach each other, the integrand includes a logarithmic singularity, and is said to be ‘Cauchy’. When |x| > a, this cannot happen and the integral is then ‘regular’. This distinction needs to be recalled when evaluating the integral, using either tables of standard integrals or algebraic manipulators. The surface slope is given by   4 1 − ν2 dv = h 0 (x; a) , dx E

(2.42)

where  h 0 (x; a) =

0 √ −sgn(x)/ x 2 −a 2

−a < x < a . |x| > a

(2.43)

The contact pressure and surface normal displacement are shown in Fig. 2.4. So, we know the contact pressure which corresponds to a constant surface normal displacement, and also the external profile. The load needed to achieve this is found from equilibrium  P=

a

−a

p(x)dx = π .

(2.44)

This gives us a tool for solving normal contact problems (between elastically similar bodies) which provides an alternative to the singular integral equation approach described above. We imagine that we have already pressed two bodies together, and increase the load by a small amount, d P, which gives rise to a corresponding change in half-width, ds. We introduce a new, geometry-dependent weighting function, F(s), at present undefined, and integrate up Eqs. (2.40) and (2.41) with the result that, when the contact length is a, the pressure distribution is given by 

a

p(x; a) = 0

F(s) f 0 (x; s)ds

(2.45)

2.4 Mossakovskii–Barber Procedure

27

and gives rise to a surface slope (note that there is a contribution from the integrand only when the observation point, x, lies outside the current contact half-width, s,)  x  4 1 − ν2 dv = F(s)h 0 (x; s) ds dx E 0

0 a

(2.64)

So, in the spirit of the weighted superposition procedure carried out in the previous two sections, we now introduce a density function G 1 (s), and hence the total strain difference between the surfaces, due to both the applied tensions and the presence of the interfacial shear traction, is given by εx x (x, a) =

σ0 4 + ∗ E∗ E



a

G 1 (s)h 1 (x; s) ds

0

f

Q| < Q2

C PY

P

Note that Eq. (4.37) is essentially the same result as the one found in Sect. 4.3.1, only that here we add an additional corrective term which is associated with the initial proportional loading path under full stick, λI p(x, PY ). In other words, in the case of sequential loading under a constant normal load, the initial loading trajectory has gradient naught (and therefore the additional corrective is zero), while here we have considered a gradient 0 < λI < f . It is easy to see that this results in a larger value for PY compared to sequential loading under constant normal load, compare Eqs. (4.24) and (4.38), so that the new stick zone half-width, b = a(PY ), will consequently be larger as well. An interesting aspect of the two-stage loading problem described is the form the shear traction distribution takes under fully adhered conditions, i.e. when the coefficient of friction is large enough to inhibit all slip in both the first and the second phase of the loading. The shear traction distribution corresponding to point C would remain unchanged, of course. However, if λII < f , the shear traction distribution along the load trajectory C–2 will change from the one given in Eq. (4.37) to a fully adhered shear traction distribution, given by  q(x) =

λI p(x, PC ) + λII ( p(x, P) − p(x, PC )) λII p(x, P)

|x| < aC , aC < |x| < a

(4.39)

where aC = a(PC ) is the contact half-width corresponding to the normal load at load point C. In the following example, we will develop a geometry-specific solution for the different loading scenarios and we will illustrate the four possible shear traction distributions.

4.3.4 Application to a Hertzian Geometry As mentioned in the introduction to this chapter, it does not come as a surprise that most early partial slip contact problems were associated with the Hertzian geometry. By applying the contact problem solution first solved by Hertz, we hope to provide complementary insight into the two general solutions developed in Sects. 4.3.1 and 4.3.2 for sequential and proportional loading, as shown in Fig. 4.3.

4.3 The Sequence of Loading

67

y

Fig. 4.6 Two contacting Hertzian cylinders with respective radii R1 and R2 subject to normal and shear loading

P p(x)

R1

Q a

-a

x

R2

First, consider the case of a plane Hertzian contact, as shown in Fig. 4.6, between similar materials in which the normal force, P, induces the typical elliptic normal traction over the contact [−a a] so that Hertz (1881)  p(x, P) = p0

x2 2P and a(P) = 1 − 2 where p0 = a πa



4P R , π E∗

(4.40)

where 1/R = 1/R1 + 1/R2 is the equivalent curvature of the contacting bodies. Suppose that while holding the normal force constant at value P1 , we now apply a monotonically increasing shear force, Q, insufficient to cause full sliding. From Sect. 4.3.1, we know that, given the coefficient of friction, f , is finite, regions of slip and regions of stick develop. As the normal pressure distribution is symmetrical and falls smoothly to zero at the edges, we expect the point-wise condition for slip, q(x) = f p(x), to be first fulfilled at the outer ends of the contact interface, b < |x| < a, and the stick region to be positioned centrally, |x| < b. Applying the simple result given in Eq. 4.21 and using Barber’s notation, we find the resulting shear traction as (Eq. (4.23) repeated)  q(x) =

f [ p(x, P) − p(x, PY )] f p(x, P)

|x| < b , b < |x| < a

(4.41)

where PY = f P−Q/ f is the normal force associated with corrective shear traction required to establish the stick zone of extent [−b b]. For a P Q-problem, it is possible to find the value of PY through a geometrical construction as shown in Fig. 4.4, where f = Q/P−PY . It follows that the stick zone half-width, b, is conveniently found by determining the contact half-width corresponding to PY  b = a(PY ) =

4PY R . π E∗

(4.42)

Figure 4.7 shows an illustration of the superposition of shear tractions and the corresponding relative surface strains for sequential loading. Note that the resulting relative surface strains must be zero in the stick region [−b b].

68

4 Half-Plane Partial Slip Contact Problems

Turning to the second loading trajectory illustrated in Fig. 4.3b, we notice that normal and shear load are applied simultaneously and grow in proportion, d P/P = dQ/Q , so that Q2 < f . (4.43) λ= P2 Under this condition, no slip will develop and the contact is to remain fully adhered throughout the entire loading trajectory. From Sect. 4.3.2, we know that the shear traction distribution at any point along the trajectory will be given by q(x) = λ p(x, P) .

(4.44)

The remaining loading scenario we wish to consider in this example section is depicted in Fig. 4.3c. The two-stage proportional loading trajectory begins under a fully adhered condition until load point C is reached. This results in qualitatively the same shear traction as given above in Eq. (4.44). Once we pass load point C, the condition for full stick is violated if λII > f and some slip will occur at the edges of the contact so that the shear traction distribution at load point 2 is given by Eq. (4.34). If the coefficient of friction has a value such that λII < f , no slip is expected to develop and the shear traction distribution at load point 2 is given by Eq. (4.37). Note that in comparison with the case that λII > f , the value of the coefficient of friction has to be increased for all slip to inhibited, and in Fig. 4.8, this is evident in the comparison between the four different shear tractions arising depending on the loading trajectory for a Hertzian contact. If λII corresponds to a vertical loading path, the coefficient of friction has to be infinite to inhibit all slip and, given  a λI = 0, a singular shear traction distribution arises. Note also that the integral Q = −a q(x) dx q(x)

slip

-a

f p(x,P) f p(x,PY) q (x) = f p(x,P) _ f p(x,PY) slip

stick

-b

b

a

x xx [f

p(x, P )] p(x,PY)] xx [q(x)] xx [f

xx

-a

a

x

Fig. 4.7 Illustration of shear tractions and relative surface strains for a Hertzian contact subject to a monotonically increasing shear force with constant normal load

4.3 The Sequence of Loading

69

q ( x) q ( x) q ( x) q ( x)

q( x )

a

-a

sequential loading, see Fig 4.3 a) proportional loading, see Fig 4.3 b) two-stage proportional loading [

II

> f ], see Fig 4.3 c)

two-stage proportional loading [

II

< f ], see Fig 4.3 c)

x

Fig. 4.8 Comparison of four possible shear traction distributions under sequential loading and linear and bi-linear proportional loading at load point 2 in P Q-space; see Fig. 4.3

must yield the same result for each shear traction distribution as the end points of the loading sequences and the values of (Q 2 , P2 ) are the same in all four cases, i.e. the area under the shear traction distributions is the same—only the loading path differs.

4.4 The Effect of Differential Bulk Tension So far, we have only looked at problems involving normal and shear loads, neglecting the effect of remote tensions. Bulk tensions are stresses arising remotely from the contact interface in one or both contacting bodies. For example, in a gas turbine engine, it is the centrifugal loading of the disc and blades as the assembly rotates and, albeit being small in magnitude compared with the centrifugal contribution, vibrations are an additional source. These remote stresses cause differential tensions between the contacting bodies thereby exciting interfacial shear tractions. This mismatch in remote stresses and hence strains will cause an additional term to appear in Eq. (4.17) for the relative surface strain over the whole contact εx x =

2 π E∗



a

−a

σ q(ξ )dξ + x −ξ 2E ∗

−a ≤ x ≤a ,

(4.45)

where the differential tension is σ = σA − σB ; see Fig. 4.1.

4.4.1 Tangential Load and Moderate Differential Bulk Tension Fretting fatigue usually takes place when an underlying oscillation of a stress field propels a crack that has been nucleated in the slip zones of a contact interface. A common fretting fatigue experimental configuration involves two pads that are pressed onto the bottom and top of a fretting fatigue specimen, which is then subject to a combination of oscillating shear and bulk stresses; see Fig. 4.9. In the case of the

70

4 Half-Plane Partial Slip Contact Problems

Fig. 4.9 Typical set up for a fretting fatigue experiment

P B

Q

B

Q P

experiment, it will be apparent that the bulk stress is present only in the dog bone, but not in the pads. This induces a corresponding strain absent from the pads which affects the partial slip behaviour (Nowell and Hills 1987), given the loading is such that the fully adhered condition is violated; see Sect. 2.5. In the context of frictional half-plane problems, it is tempting to assume that the Jäger–Ciavarella theorem still applies. A differential tension does not affect overall tangential equilibrium due to its skew symmetry; however, when a shear load and a differential bulk tension are present simultaneously, the latter will add to the shear traction at one end of the contact interface and subtract from the shear traction at the other end. Superimposing a corrective shear traction over the stick region works under this mixed-loading as long as the differential tension present is moderate so that the direction of slip is the same at both ends, Fig. 4.10a. Note that the most obvious consequence of any mixed-loading will be that the stick zone is no longer centrally positioned, hence the stick zone is now spanning [−m n], where extent and eccentricity depend on the magnitudes of shear and bulk tension. Arguments analogous to those made in Sect. 2.3 and 4.3.1 lead to  − f

 n  σ q (ξ )dξ 2 dg +α − = dx 2E ∗ π E ∗ −m x − ξ

−m ≤ x ≤n ,

(4.46)

where the contacting half-planes’ profile has relative gradient dg/dx and tilt angle α. With this modification (the tilt), the corrective shear traction q  (x) remains a scaled and shifted form of the normal traction. Exploiting tangential equilibrium, Eq. (4.22), together with the consistency condition to the solution of Eq. (4.46) 

n −m

πσ dξ = −π α + w(ξ, m, n) 2 f E∗ dg/dξ

(4.47)

reveal the extent and position of the stick zone. Note that for geometries incapable of supporting a moment (in particular, the Hertzian geometry), the angle of tilt α → 0. In that case, the tangential problem becomes uncoupled, meaning the extent of the stick zone is merely dependent on the applied shear force, while the differential bulk tension determines its eccentricity.

4.4 The Effect of Differential Bulk Tension

a)

b)

q(x)

-a

slip

stick -m

n

a

q(x) f p(x)

f p(x)

f p(x)

slip

71

x

slip -a

slip

stick

-m

n

a

x

-f p(x)

Fig. 4.10 Shear tractions for mixed-loading for a moderate differential bulk tension and b large differential bulk tension

4.4.2 Bulk Tension Dominated Partial Slip Problems In the case of large differential tension, Fig. 4.10b, the direction of slip is reversed at one end of the contact. The Jäger–Ciavarella theorem hinges upon the superposition of a full sliding shear traction and a further scaled/shifted form. This solution becomes inapplicable once the slipping traction is not of the same sign at both ends of the contact. A method that overcomes this limitation and which allows us to solve problems regardless of the magnitude of differential tension is based on dislocations; see Sect. 2.6. The solution utilises a strain correction to the fully adhered interface in order to introduce slip instead of a traction correction to the full sliding case in order to introduce stick. The mathematical framework for finding the corrective strain and the slip–stick boundaries in partial slip based on using dislocations as strain nuclei is given in Sect. 4.8.

4.5 Periodic Loading So far all tangential problems discussed in this chapter have been subject to a monotonically increasing shear force. For fretting fatigue, however, it is important to look at problems in which the loads change in a periodic manner as this is a) how loads change in most practical engineering applications and b) what nurtures the accumulation of surface damage and the eventual nucleation of cracks in partial slip problems, and it was Mindlin and Deresiewicz who first looked at periodic loading for contacts (Mindlin and Deresiewicz 1953). Suppose we have a symmetrical half-plane contact under constant normal load and the shear force is cycled between the limits Q 2 and Q 1 where Fig. 4.11a shows the shear load in marching-in-time sense, where the maximum of the load cycle is reached at Q 2 and the minimum at Q 1 . In this illustration, we will assume that Q 1 = −Q 2 , which is no general restriction. In Fig. 4.11b, we see the same load history in P Q-space which will allow us to visualise some of the findings we present in the following.

72

4 Half-Plane Partial Slip Contact Problems

b)

Q

Q Q

=

fP

a)

2 2+

t

0 periodic load cycle

O

0

P

1 (P,Q ) 1

Q

1

periodic load cycle

2+ 2 (P,Q2)

= -f P

Fig. 4.11 Periodic loading of a shear force in a a time-history plot and b in P Q-space

From Sect. 4.3.1, we know the developing shear tractions distribution, see Eq. (4.23), and the ensuing slip–stick pattern, b < |x| < a, up to point 2. Figure 4.12a illustrates the shear tractions we can expect for a Hertzian contact at the maximum point of the load cycle. So far we have not discussed what happens when the shear load is reduced infinitesimally from its maximum, Q 2 , at point 2, say to point 2+ . The shear tractions and corresponding tangential displacement reduce infinitesimally so that the rate of change of relative displacement is now opposite in sign to that occurring during the increase of the shear load, 0 → 2. If we remind ourselves of the conditions for slipping in a point-wise sense, Eqs. (4.1) and (4.2), we notice that the latter, the requirement for the shear tractions in the slip zones to oppose relative velocity, becomes violated. It follows that relative motion of surface particles comes to a halt and the requirement for a fully adhered state, Eq. (4.3), is fulfilled so that instantaneous stick must occur everywhere within the contact interface, Fig. 4.12b. A further reduction in shear force leads to the onset of reverse slip at the edges of the contact as surface particles that have slipped in one direction during the loading phase 0–2 start to move in the opposite direction in order to allow for the conservation of material. In these new slip zones, say b < |x| < a, the shear traction will now have changed from q(x) = f p(x, P) to q(x) = − f p(x, P); see Fig. 4.12c. If the reduction, or change, in shear load from its maximum value is given by Q = Q 2 − Q and as we wish to find the extent of the emerging slip–stick pattern, an additional corrective traction distribution must therefore be applied over the new stick region, |x| ≤ b . By analogy with the results of Sect. 4.3.1, it is possible to write down overall tangential equilibrium at any point during the reversal of the shear load, 2–1, as  a Q = Q 2 − Q = q(x) dx = Q 2 − 2 f P + 2 f PS , (4.48) −a

4.5 Periodic Loading

a) q(x)

-b

73

b)

Q = Q2

q(x)

x

b

c) q(x)

Q = Q2 - Q

-b

Q = Q2 +

d) q(x)

Q = Q2 - Q = 0

Q| < Q 2 -b

,

b -b

e) q(x)

Q| = Q 2

,

b

-b

x

Q = Q2 - Q

,

b -b

b

f) q(x)

Q| > Q 2 , -b -b

bb

, x

x

b

, -b = -b

, x

Q = Q2 - Q = Q 1 Q| = Q 2 - Q 1 , b=b

x

Fig. 4.12 The evolution of shear tractions occurring during the loading cycle shown in Fig. 4.11 for a Hertzian example geometry

where PS corresponds to the new corrective shear traction, p(x, PS ), which is applied over the stick region. The new corrective force can simply be given as PS =

2 f P − Q , 2f

(4.49)

so that the new stick zone half-width, b , is found by evaluating the contact law a(PS ). Compare, now, Eq. (4.24) with Eq. (4.49) and note the factor of 2 which occurs because the corrective term must cancel the relative displacement occurring when the slip zone traction change by 2 f p(x, P) instead of f p(x, P) during the reloading phase, O–2. For a P Q-problem, the corrective term can be found from a geometrical construction, Fig. 4.13, in which the load PS corresponds to the intersection of a line of gradient f starting at load point 2 and a line of gradient − f starting at the instantaneous load point. The total shear tractions distribution is therefore given by

4 Half-Plane Partial Slip Contact Problems

=

fP

Q 2 (P,Q2) Q| < Q2

f

-f

Fig. 4.13 The visual construction to find the corrective force PS in P Q-space for periodic loading

Q

74

O

0 PY

PS

P

Q

1 (P,Q ) 1

= -f P

⎧ ⎨ f [− p(x, P) + 2 p(x, PS ) − p(x, PY )] q(x) = f [− p(x, P) + 2 p(x, PS )] ⎩ − f p(x, P)

|x| < b b < |x| < b . b < |x| < a

(4.50)

Note that when the shear  a force has been reduced by the magnitude of Q 2 so that the net shear force Q = −a q(x) dx = 0 we observe residual shear tractions; see Fig. 4.12d. Figure 4.12e shows the qualitative behaviour of a Hertzian contact once the net shear force of the load cycle under consideration is negative as the shear force is reduced further. Once the shear force is fully reversed Q = Q 1 , the new and old stick region coincide and b = b, Fig. 4.12f. The findings presented apply to shear load cycles under a constant normal force. The findings further apply when the shear load is not fully reversing, i.e. Q 2 = −Q 1 , with a non-zero mean shear load. The stick zone would still shakedown to its final extent, [−b b], but with a ‘kink’ in the locked-in shear tractions in the permanent stick zone. Furthermore, it is of interest to understand what happens when the normal load is varying as well and this is what we shall look at next in its most general form. Later in this chapter, we will look in more detail at problems in which additionally a (varying) differential tension and moment are present.

4.6 More General Loading Scenarios It is possible to extend the superposition exploited by the Jäger–Ciavarella theorem, with some modification, to more general loading scenarios. Consider the shaft of a motor with an eccentric weight attached rotating at an angular speed, ω, as depicted in Fig. 4.14. The assembly rests on a plane so that the contact interface may be idealised using half-plane principles. The eccentric weight will cause the normal and shear load to change from its respective mean value as the shaft rotates, so that after an initial loading trajectory, O–A, a cyclic loading phase is entered and the loads can be given as

4.6 More General Loading Scenarios

75

Fig. 4.14 Motor with eccentric shaft oscillating harmonically and half-plane idealisation of its supporting contact

P (t) Q (t) contact width

P(t) = P0 + Pa sin (ωt) ,

(4.51)

Q(t) = Q 0 + Q a sin (ωt + ϕ) ,

(4.52)

where ϕ is a phase shift. The entire loading trajectory can be tracked out in P Q-space; see Fig. 4.15. Note that the phase shift determines the shape of the cyclic loading loop and if, for some reason, ϕ = 0, the loop collapses to a straight line, which is a special case considered separately in Sect. 4.7. P0 and Q 0 are the mean values of the load, and Pa and Q a are the respective normal and shear load amplitudes. Major progress in solving problems of this type has been made by Barber et al. (2011). The approach taken tracks out the partial slip behaviour in marching-in-time sense, so that the shear tractions, as well as the position and extent of the stick zone, can be found at any point in the load cycle. Note first that the contact becomes fully adhered if the condition for full stick is fulfilled, that is the normal load is increasing so that d P/dt > 0 and the shear force is incrementally increased but by a relatively small amount so that |dQ| < f . dP

(4.53)

Since the initial loading path O–A satisfies the criterion (4.53) in the given example, the shear traction at any point between O and A will be of the form (4.34). Now we enter the cyclic loading at point A, and start by investigating the shear tractions up to point B. We find the answer by summing the shear tractions under conditions of full stick up to point B 4

4 This

is the incremental form of Eq. (4.35)

76

4 Half-Plane Partial Slip Contact Problems

Q

Q

=

fP

Fig. 4.15 Illustration of a generic loading trajectory in P Q-space with points T and Y constructed

(P0,Q0)

f

X

T B

-f

Y

o

 q B (x) = q A (x) +

PB

p  (x)

PA

dQ d P, dP

A

P

(4.54)

where p  (x) = ∂ p(x,P) . Once we pass point B, the condition for full stick is violated, ∂P dQ/d P > f , and a region of frictional slip develops. At point X , we express the shear tractions as a superposition employing the ideas developed by Jäger and Ciavarella (1997; 1998). Note that shear tractions and attendant slip displacements have been locked-in during the fully stuck loading phase, O–B, and we need to add an as yet unknown contribution to maintain stick over a half-width, given by a(PY ), so that the shear traction distribution is  PY dQ d P. (4.55) p  (x) q X (x) = f p(x, PX ) − f p(x, PY ) + dP 0 Compare now Eqs. (4.55) and (4.37) and note that we subtract the corrective term, f p(x, PY ), from the full sliding tractions at point X , f p(x, PX ), and add the lockedin shear tractions from an earlier point, Y . The corrective force, PY , is located by the imposition of tangential equilibrium, and gives f =

Q X − QY . PX − PY

(4.56)

It is therefore found by drawing a line of gradient f in P Q-space, Fig. 4.15, and finding where the line intersects the initial loading curve. These results have strong similarities with those explored in Sect. 4.3.2, adding-in the complication of nonlinearly changing loads. This allows the user to track out the complete picture of the ensuring slip–stick pattern and shear tractions, and Barber et al. (2011) and Hills et al. (2011) should be consulted for details of the method. After the first cycle of loading, a steady-state cycle is entered. Most systems will operate in the steady-state for the majority of their service lives, and a complete marching-in-time solution might not be needed. We will make a concise statement

4.6 More General Loading Scenarios

77

about steady-state loading which will prove very useful in the next section, Sect. 4.7. One thing that emerges from the steady-state solution is that after completion of the first cycle, there is a part of the contact that never experiences any relative motion during steady-state loading. This part of the contact is the so-called permanent stick zone spanning [−m m] for a P Q partial slip problem. We can find this region by drawing lines of gradient ± f which are tangential to the envelope of loading, and these define a point, which in analogy to Sect. 4.5, we shall denote point T , as shown in Fig. 4.15.

4.7 General Cyclic Proportional Loading Consider a dovetail root contact as shown in Fig. 4.16. As the gas turbine starts rotating, the dovetail geometry experiences centrifugal and expansion forces (Fc and T , respectively) so that disk and blade come into contact. These external loads constitute the primary source of local contact loads along the two established contact flanks, and which can be assumed to stay constant for the largest part of a flight cycle (~engine speed in cruise). Superimposed is an additional load, Fv , due to vibration, which may be induced by an unsteady air stream. This secondary load imposes many thousands of cycles of load change per major cycle and causes the local contact loads, i.e. normal load, P, moment, M, shear load, Q, and differential bulk tension, σ = σA − σB to oscillate in a cyclic manner. As the secondary external load excites the cyclic load components solitarily, the local contact loads must change in phase; hence the change in load is proportional. With the phase lag being zero, the typical

B

x

y

B

A B

A A

T

Fig. 4.16 Illustration of a dovetail root contact with the resulting contact loads

T

78

Q, M, cyc le

2

stea dystat e

Fig. 4.17 Illustration of a steady-state cycle in load space for variations of normal load, P, shear load,Q, moment, M, and differential bulk tension, σ

4 Half-Plane Partial Slip Contact Problems

0

se ha tp n e 1 i ns tra

P

Q, M,

P

steady-state cycle collapses to a straight line as depicted in Fig. 4.17, in which 0 marks the mean point of the cycle, and 1 and 2 mark the minimum and the maximum points of the cycle. If the contact size and range of load components ( P, Q, M, σ ) are such that the condition for full stick is violated at either end of the contact, see Eq. (2.84), a partial slip problem arises. Because an assembly might suffer many thousands of cycles of minor load per major cycle, the problem we set ourselves is to establish the steady-state solution to the problem as succinctly as possible. Finding the coordinates of the contact and the coordinates of the permanent stick zone is enough to determine the maximum extent of the slip zones. Note that while the transient phase does affect the locked-in shear tractions in the permanent stick zone, it does not affect the maximum extent and position of the slip zones during the steady-state. In other words, the permanent stick zone’s size and position do not matter on which path the steady-state cycle is entered.

4.7.1 The Permanent Stick Zone From Sect. 4.6, we know that in cyclic partial slip problems some part of the contact interface experiences no relative movement at any point during the steady-state cycle. In a problem restricted to normal and shear loads, finding this permanent stick zone is relatively straightforward, as shown through a geometrical construction in P Qspace; see Fig. 4.15. However, here we look at more complex loading scenarios involving all loads varying, including a moment and differential bulk tension. If the contact interface is capable of supporting a moment, the effects of the moment on the normal problem might manifest themselves in an asymmetrical pressure distribution and/or in an asymmetrical contact extent spanning [−a c]; see Sect. 4.2. However, the effects on the tangential problem are far from obvious and will be covered in what follows.

4.7 General Cyclic Proportional Loading

x transient phase

79

contact edge forward slip

steady-state cycle

c2 c1

reverse slip

n

permanent stick

0

-m

2

1

permanent stick zone

temporary stick

2

1

2

position in load cycle

-a1 -a2 Fig. 4.18 Illustration of an evolving steady-state cycle in a marching-in-time sense with the outer parts of the contact experiencing reciprocating slip and stick and the inner part being permanently stuck

In problems that can be approximated by half-planes, the contacting bodies might also be loaded by remote stresses σA and σB at infinity as illustrated in Fig. 4.1, resulting in a differential bulk tension, σ = σA − σB . In this approach, we shall assume that the differential tension is moderate, meaning the direction of slip is the same at both ends of the contact. However, the direction of slip is reversed when going from load point 2 to 1 compared with the forward slip when going from 1 to 2 in order to allow for the conservation of material in the slip zones. As indicated in Fig. 4.18, the extent of the slip zones reaches a maximum as the end points of the load cycle are approached. Changes in normal load result in a movement of the contact edge throughout the steady-state cycle resulting in particles moving back into their original position outside of the instantaneous interface during load reversal. We argue that any fretting damage experienced during the transient stage can be neglected as the far more significant number of cycles is accumulated in the steady-state in most engineering applications. In fact, it is possible to write down the formulation for the permanent stick zone such that only the steady-state loading is required to be known, more specifically the loads present at the ends of steady-state cycle, 1 and 2 Fig. 4.17. Once the position and extent of the permanent stick zone as well as the evolution of the contact size are revealed, we hold the information necessary for a fretting fatigue experiment to be set up. The maximum extent of the slip zones is effectively established and this is what is intended to be matched in a laboratory experiment with the goal of determining the fretting fatigue strength of a prototype. As shown in Fig. 4.18, the slip zones reach their maximum as the end points of the loading cycle are approached and the remaining stick zone is what we call the permanent stick zone, [−m, n]. Particles inside the permanent stick zone do not experience any relative movement during steady-state loading. Conversely, particles within the interface, but outside of the permanent stick zone are subject to reciprocating slip.

80

4 Half-Plane Partial Slip Contact Problems

At load reversal, the interface becomes instantaneously stuck as the relative movement comes to a halt. Depending on the position in the load cycle, particles in the slip zones then start to move in a forward or reverse direction starting from the edges of the c1 ] contact patch, where we denote the minimum extent of the contact patch [−a1 c2 ], where −a2 < −a1 < −m < n < c1 < c2 . and the maximum [−a2 We begin developing the solution by writing down the difference in surface strain parallel with the interface, εx x,1 , just before load point 1 (P1 , Q 1 , M1 , σ1 ) is reached and the slip zone is at its maximum extent. Once more, we superimpose the full sliding tractions, − f p1 (x), over the whole contact [−a1 , c1 ] with an as yet unknown corrective term, q1∗ (x), over the permanent stick region [−m, n]. Thus, εx x,1

  c1   n ∗ f p1 (ξ )dξ q1 (ξ )dξ 2 π = − + + σ1 . π E∗ ξ −x 4 −a1 −m ξ − x

(4.57)

Just before the other end of the steady-state cycle is reached at load point 2 (P2 , Q 2 , M2 , σ2 ), a very similar expression for the difference in direct strain, εx x,2 , can be stated. Note that the direction of slip is reversed and the full sliding traction is now of opposite sign. We obtain εx x,2 =



2 π E∗

c2

−a2

f p2 (ξ )dξ + ξ −x



n −m

π q2∗ (ξ )dξ + σ2 ξ −x 4

 .

(4.58)

Now, let us utilise the physical information we have about the permanent stick region. We know that in this region, the difference between the surface strains must remain unchanged, i.e. εx x,1 = εx x,2 , − m < x < n

(4.59)

and hence, using Eqs. (4.57) and (4.58), we find  −

c1 −a1

f p1 (ξ )dξ − ξ −x



c2

−a2

π f p2 (ξ )dξ − (σ2 − σ1 ) = ξ −x 4



n −m

∗  q2 − q1∗ dξ , (4.60) ξ −x

where x ∈ (−m, n). Making use of the relation between normal tractions and the gradient of the surface profile, see Eq. (4.5), we re-write this equation in the form 1 σ + f E α0 − = fE dx 4 π ∗ dg

∗

where α0 =



α1 + α2 2

n

−m

∗  q2 − q1∗ (ξ )dξ , − m < x < n, ξ −x

and

σ = σ2 − σ1

(4.61)

(4.62)

are, respectively, the average angle of tilt and the range of differential tension. Taking a closer look at Eq. (4.62), we note that the left-hand side can be restated as including

4.7 General Cyclic Proportional Loading

81

one term depending on the gradient of surface profile, dg/dx , and the remaining term being constant,   σ dg . + α0 − dx 4 f E∗ As indicated in Sect. 4.2, there are significant parallels between the normal solution to an asymmetrical contact problem and the corresponding tangential problem, which facilitate the evaluation of the problem. For example, the inversion of equation (4.61), bounded at both ends, is given by Hills et al. (1993) f E∗ [q2∗ − q1∗ ](x) = w(x, m, n) π





dg/dξ

n

+ α0 −

σ 4 f E∗

 dξ

w(ξ, m, n)(ξ − x)

−m

,

(4.63)

and arguments exactly parallel to those in Sect. 4.2 lead to the results 

n −m

dg/dξ

dξ π σ = −π α0 + , w(ξ, m, n) 4 f E∗

(4.64)

and [q2∗

−

q1∗ ](x)

f E∗ = w(x, m, n) π



dg/dξ

n −m

dξ , − m ≤ x ≤ n . (4.65) w(ξ, m, n)(ξ − x)

The resultant corrective shear forces are given by Q i∗ =



qi∗ (x)dx ; i = 1, 2 ,

(4.66)

so that tangential equilibrium is imposed by setting Q 1 = − f P1 + Q ∗1 ;

Q 2 = f P2 + Q ∗2 .

(4.67)

The range of shear force, Q, is given by Q = Q 2 − Q 1 = 2 f P0 + Q ∗ , where P0 =

P1 + P2 2

(4.68)

(4.69)

is the mean normal load and Q ∗ =



n

−m

[q2∗ − q1∗ ](x)dx .

(4.70)

82

4 Half-Plane Partial Slip Contact Problems

4.7.2 Mapping Between the Normal and Tangential Problems It is not necessary to evaluate the inverted integral equation, see Eq. (4.65), or any other of the equations formalised in the section above in order to determine the extent and position of the permanent stick zone. We noticed that there are striking parallels between the normal solution to asymmetrical contact problems, i.e. those capable of supporting a moment, and the tangential solution developed above. These can be utilised to define a succinct mapping between normal and tangential problems, where we replace the parameters of the normal solution as follows: [−a

c] → [−m n] 1 [q ∗ − q1∗ ](x) p(x) → − 2f 2 Q , P → P0 − 2f σ α → α0 − 4 f E∗

(4.71)

mutatis mutandis. The details of this mapping imply that we have found expressions for the contact coordinates [−a c], and the pressure distribution, p(x), for arbitrary values of P and α. If the appropriate changes are made using the range of tangential loads present in the steady-state cycle, as suggested in Eq. (4.71), the solution to the tangential problem is readily known. In many cases, the average angle of tilt, α0 , is not a desired input to the problem, but the moment, M, will be a known quantity, and a preliminary stage of the analysis will involve the determination of the corresponding tilt angles α1 and α2 . We refer to the original publications Andresen et al. (2020a, b) for full details of the procedure described. In the following section, we will illustrate it by way of a simple example, applied to the normal solution of the tilted shallow wedge.

4.7.3 Application to the Tilted Shallow Wedge In this example, we first present the closed-form solution for a tilted shallow wedge of apex angle (π − 2φ) where 0 < φ 1. The angle of tilt, 0 < α 1, is measured anti-clockwise from the unrotated state, Fig. 4.19. We assume the internal wedge angle to be large enough for the half-plane assumption to hold. Suppose now that contact is established by applying a normal force, P, and a moment, M, where the normal load acts through the vertex. The normal solution was first published by Sackfield et al. (2005), where the pressure distribution can be given in closed-form

4.7 General Cyclic Proportional Loading

83

P

B

M

B

c

Q

A

A

Fig. 4.19 A tilted wedge contact subject to a normal load, moment and shear load with remote bulk tensions arising in each half-plane

√  √  γ + (γ a − x)/(x + a)  E ∗φ  ,  p(x) = log  √ √ π γ − (aγ − x)/(x + a) 

(4.72)

where γ is given by Moore and Hills (2021)     −1 πα πα γ = 1 − sin 1 + sin . 2φ 2φ

(4.73)

In Moore and Hills (2021), it was shown that the angle of tilt can be expressed in terms of normal load, P, and moment, M, given by   2E ∗ φ M 2φ α=− . arcsin  π 4(E ∗ φ M)2 + P 4

(4.74)

Note that the symmetry of the problem makes it possible to connect the left-hand and right-hand contact coordinates, so that c = γa ,

(4.75)

where a is found by imposing normal equilibrium  P=

c

−a

 p(x) dx = E ∗ φ γ a 2 ,

(4.76)

to give the following two explicit expressions for the contact coordinates  P a= ∗ E φ

1 ; γ

c=

P √ γ . E ∗φ

(4.77)

84

4 Half-Plane Partial Slip Contact Problems

Evaluating rotational equilibrium results in an expression for the moment, M, which is  c E ∗φ √ M= p(x)x dx = γ (1 − γ )a 2 . (4.78) 4 −a Now, the normal solution outlined above is enough to write down readily the solution to the tangential problem. The mapping given in Sect. 4.7.2 states that by , we find substituting α → α0 − 4 σ f E∗  ⎞⎞ ⎛  ⎞⎞−1 ⎛  ⎛  σ π α0 − 4 σ − π α 0 ∗ ∗ fE 4fE ⎠⎠ ⎝1 + sin ⎝ ⎠⎠ , (4.79) γt = ⎝1 − sin ⎝ 2φ 2φ ⎛

where α0 is the average angle of tilt and σ is the range of differential bulk tension over the course of the steady-state cycle; see Eq. (4.62). For any partial slip–slip steady-state problem, the zone of permanent stick is expected to ensue spanning [−m n]. The coordinates can be determined immediately by employing the mapping, Eq. (4.71), into Eq. (4.77), giving 1 m= ∗ E φ



Q P0 − 2f



1 ; γt

1 n= ∗ E φ



Q P0 − 2f



√ γt ,

(4.80)

where P0 is the mean normal load and Q denotes the range of the shear load in the steady state. Eq. (4.80) is the key expression in the tangential problem as it defines the extent and position of the permanent stick zone. Note that m and n depend on P0 , α0 (M0 , P0 ), Q and σ but do not depend on the range of the normal load and moment, P and M. The latter two parameters, however, affect the contact extent at the minimum and maximum of the steady-state cycle, [−a(Pi , Mi ), c(Pi , Mi )] , i = 1, 2, thereby effectively establishing the maximum and minimum extent of the slip zones at either end of the interface. Lastly, the mapping, Eq. (4.71), provides means of determining the change of corrective tractions in the permanent stick zone readily, given by [q2∗ − q1∗ ](x) = −

√  √ 2 f E ∗ φ  1/γt − (a−x)/(x+γt a)  . ln  √ √ 1/γt + (a−x)/(x+γt a)  π

(4.81)

4.8 Partial Slip Solutions Based on Dislocations All of the partial slip solutions considered so far in this chapter have used a combination of the fully sliding traction distribution together with a superposed ‘corrective’ distribution in the stick region to enable the relative surface strains there to be maintained during any change of load. It has been shown that the Jäger–Ciavarella

4.8 Partial Slip Solutions Based on Dislocations

85

principle which demonstrates that the corrective traction is always a scaled and possibly shifted form of the contact pressure distribution is of very wide applicability, well beyond that originally conceived. It may be applied to transient problems and to steady-state periodically loaded problems, and for quite general variations in the applied load—changes in shear force, normal load, differential bulk tension and even applied moment may all be catered for. These results are very powerful and of general applicability, but they do have a limitation, and this is that the slip zones must be of the same sign so that the underlying sliding shear traction distribution does, indeed, apply, and this was the reason for introducing the concept of ‘moderate’ tension; see Sect. 4.4.1. If the slip zones are of opposite sign, it is possible to use a formulation based on a shear traction distribution limited by friction and of opposing signs in the left- and right-hand sides of the contact, together with a corrective traction in the central stick region (Nowell and Hills 1987), but the integral equation in terms of the corrective shear must be solved numerically, and an alternative approach is preferable.

4.8.1 The Insertion of Glide Dislocations Instead of thinking of the solution to the problem as one where the slipping condition is automatically satisfied, and looking for a solution for the corrective shear, it is possible to make the opposite assumption, viz. that full stick persists over the contact region, thereby automatically satisfying the condition in the stick region, and then to insert regions of slip, d, and the best way to do this is to use a distribution of glide dislocations, Fig. 4.20. In Chap. 2, we derived the relationship between the surface tangential displacement, u(x), and shear traction distribution, q(x), when two elastically similar half-planes are joined over the interval [−a a]. This is given by (Eq. (4.17) repeated)  a 2 q (ξ ) dξ du = , (4.82) ∗ dx π E −a c − ξ and its inversion, when a ‘singular both ends’ solution is sought, implying that the ends of the interval are known and fixed, is given by Hills et al. (1993) q(x) =

1

√ π a2 − x 2



E∗ C+ 2



a

−a

  a 2 − ξ 2 u  (ξ ) dξ |x| < a. ξ −x

(4.83)

If the surfaces are everywhere glued together except at one point, c, where there is a jump in displacement, bx , the surface displacement gradient is given by du = bx (c) δ (x − c) |x| < a, dx where δ(•) is Dirac’s delta function and hence Eq. (4.83) becomes

(4.84)

86

4 Half-Plane Partial Slip Contact Problems y z r

c

d

slip

x

stick

contact width

Fig. 4.20 The insertion of an array of glide dislocations with the intention of correcting the tractions in the slip zone, d, while keeping the stick zone displacement at zero and the insertion of a single glide dislocation at point r resulting in a shear stress, measured at coordinate z

q(x) =

E ∗ bx (c) √ 2π a 2 − x 2

√ a 2 − c2 ξ −x

|x| < a,

(4.85)

where the rigid body term, C, has been dropped. Therefore, if a distribution of dislocations, which extends to the edges of the contact is needed, of density Bx (x) = dbx/dx , the shear traction induced is given by q(x) =

E∗



√ 2π a 2 − x 2

a −a

 a 2 − ξ 2 Bx (ξ ) dξ ξ −x

|x| < a.

(4.86)

In instances where the dislocation distribution is present only within the contact, and is non-zero over an interval [−b b], a bounded-both-ends distribution will be required and this may be found from the appropriate inversion of the integral equation with the result √  Bx (ξ ) dξ E ∗ b2 − x 2 b |x| < b < a.  (4.87) q(x) = 2π b2 − ξ 2 (ξ − x) −b Lastly, the partial slip solutions derived for asymptotic representations of an incomplete contact edge may be derived using a dislocation-based solution, and these take on a particularly simple form. For example, if we look at the solution for a glide dislocation near to the left-hand contact edge in Fig. 4.20, and have new coordinates measured from that point, the shear stress induced at point z by a dislocation located at r is given by q (z) =

E ∗ bx (r ) 2π (r − z)



r r, z > 0. z

(4.88)

In principle, the dislocation kernel might be used to provide a correction, to achieve a region of stick, to the full slip solution, just as a shear traction might be used to achieve a region of slip in a fully stuck solution, and all these possibilities are explored in Hills et al. (2018), but it seems natural to use either a traction correction

4.8 Partial Slip Solutions Based on Dislocations a)

b)

y

adhered condition

q(x) =

duA duB dx dx = 0 |x| < a

87

Q (a 2- x 2)1/2

y q(x) =

a

-a

x

x(

A B) (a 2- x 2)1/2

a

-a

Fig. 4.21 Shear tractions arising along an established interface, [−a conditions

x

a], under fully adhered

to the full sliding solution or a dislocation correction to a fully stuck solution. The former has been done in Sects. 4.3–4.7, and here we shall consider dislocation-based corrections superimposed on the full stick solution.

4.8.2 Solution (Constant Normal Load) In order to develop a complete solution for partial slip problems under constant normal load, let us begin by looking at the shear tractions arising under fully adhered conditions. Figure 4.21a depicts the symmetrical singular shear tractions arising along an established interface, [−a a], due to a shear load, and the skew-symmetric singular shear traction arising due to a differential tension, σ = σA − σB ; see Fig. 4.21b. When both tangential loads are superimposed, we can state the fully stuck tractions as Q σx + √ |x| < a . (4.89) qst (x) = √ 2 2 π a −x 4 a2 − x 2 We initially inhibit all slip by assuming the coefficient of friction, f , to be infinitely high. For slip to occur, however, this assumption needs relaxation where the slipping traction is of the form qsl (x) = f p(x) ∀ x ∈ slip zones, where p(x) is the normal pressure. This requires a corrective traction, qc , to be introduced in order to bring down the singular fully stuck tractions to the desired slipping tractions in the regions of slip but to leave the fully adhered state, du/dx = 0, in the stick region unchanged, i.e. the corrective traction can be given as  qc (x) = qsl (x) − qst (x) = f p(x) −

Q σx + √ √ 2 2 π a −x 4 a2 − x 2

 ,

(4.90)

where x ∈ slip zones. This correction in the as yet unknown slip zones is achieved by inserting an array of glide dislocations of an as yet unknown density Bx (x), where this density is equivalent to a direct strain resulting in the desired corrective traction

88

4 Half-Plane Partial Slip Contact Problems

in the regions of slip. Suppose now, we wish to investigate a partial slip problem subject to a constant normal load and mixed tangential loading, where we denote the stick region [−m n]. Using Eq. (4.87), we may express the corrective traction as qc (x) =



E∗

√ 2π a 2 − x 2

−m

−a



a

+ n

 a 2 − ξ 2 Bx (ξ ) dξ ξ −x   a 2 − ξ 2 Bx (ξ ) dξ , ξ −x

(4.91)

where x ∈ slip zones. Moore et al. (2018) inverted the above equation to find the dislocation density, given as Bx (x) = ±

  −m qc (ξ ) dξ 2  (x − n)(x + m) − √ E ∗π (ξ − n)(ξ + m)(ξ − x) −a   a qc (ξ ) dξ , (4.92) + √ (ξ − n)(ξ + m)(ξ − x) n

and note that we choose the upper sign when −a ≤ x < −m and the lower sign when n < x ≤ a. From the dislocation density, one can also derive, directly, the slip displacement, u(x). So for example we have that 

x

u(x) =

Bx (ξ ) dξ , n < x ≤ a.

(4.93)

n

The inversion of Eq. (4.91) results in two indispensable consistency conditions which determine the desired position of the slip stick boundaries, [−m n], 

−m

−a  −m −a

 a qc (ξ ) dξ qc (ξ ) dξ − =0 , √ √ (ξ − n)(ξ + m) (ξ − n)(ξ + m) n  a qc (ξ ) ξ dξ qc (ξ ) ξ dξ − =0 . √ √ (ξ − n)(ξ + m) (ξ − n)(ξ + m) n

(4.94) (4.95)

Recall that we know the form of the corrective traction, qc (x), and substituting Eq. (4.90) into the integrals simplifies the inverted integral equation, Eq. (4.92), as well as the side conditions, Eqs. (4.94), (4.95), which give Moore et al. (2018) 

−m

−a  −m −a

 a ± f p (ξ ) dξ ∓ f p (ξ ) dξ σ + =π , √ 4 (ξ − n)(ξ + m) (ξ − n)(ξ + m) n  a ± f p (ξ ) ξ dξ ∓ f p (ξ ) ξ dξ σ (n − m) −Q. + =π √ √ 8 (ξ − n)(ξ + m) (ξ − n)(ξ + m) n √

(4.96) (4.97)

4.8 Partial Slip Solutions Based on Dislocations

89

4.8.3 Shear Force Only Let us consider the application of a shear force only, leading to a loading scenario very similar to the one considered in Sect. 4.3.1. The absence of any differential tension or moment results in a symmetry of pressure distribution and slip zone extents about the centre axis of the contact, [−a a], i.e. n = m = b. This leads to a number of simplifications, starting with the fully adhered tractions in the presence of a shear load only qst (x) =

Q √ π a2 − x 2

|x| ≤ a .

(4.98)

As the slip direction is the same at both ends of the contact, the dislocation density is an odd function of x, hence Bx (x) = −Bx (−x), giving Moore et al. (2018)  4f Bx (x) = ∗ sgn(x) x 2 − b2 E π



a b

ξ p(ξ ) dξ  . 2 ξ − b2 (ξ 2 − x 2 )

(4.99)

The side condition, Eq. (4.96), is automatically satisfied as σ = 0, and the remaining side condition, see Eq. (4.97), may be written as 

a b

p (ξ ) ξ dξ Q  . = 2 2 2f ξ −n

(4.100)

Evaluating Eq. (4.100) (for the particular pressure distribution, p(x), under consideration) reveals an expression equivalent to the tangential equilibrium expression we exploited earlier when we employed the Jäger–Ciavarella principle in Sect. 4.3.1. Either route is sufficient to determine the slip–stick transition point, b.

4.8.4 Differential Tension Only Before proceeding to more complicated loading sequences, we will first consider the extension of the method to problems involving differential tension only. This constitutes a loading scenario we were unable to treat using the superposition principle by Jäger and Ciavarella and the dislocations method remedies this. We note that the symmetry of the problem exploited above is lost, but the skew symmetry of the arising shear tractions allows us to make appropriate simplifications. The slip zones are still of equal size at either end of the contact so that n = m = b and the slipping traction will be of opposite sign at either end of the contact, so that we start by giving the corrective traction as

90

4 Half-Plane Partial Slip Contact Problems

xσ qc (x) = sgn(x) f p(x) − √ 4 a2 − x 2

b ≤ |x| ≤ a .

(4.101)

The dislocation density must be even to account for the opposing slip directions, hence Bx (x) = Bx (−x), yielding Moore et al. (2018) Bx (x) =

 4f sgn(x) x 2 − b2 ∗ E π



a b

x p(ξ ) dξ  . ξ 2 − b2 (ξ 2 − x 2 )

(4.102)

This time, the consistency condition given in Eq. (4.97) is automatically satisfied. As σ = 0, we write the consistency condition given in Eq. (4.96) as 

a b

p (ξ ) dξ σπ  . = 8f (ξ 2 − b2

(4.103)

Again, this side condition must be attached to a particular pressure distribution, p(x), in order to arrive at an explicit expression for the slip–stick boundary, b. Note that, as the equilibrium condition is automatically satisfied when the differential tension is the sole source of the shear tractions arising along the interface, i.e. Q = contact q(x)dx = 0, Eq. (4.103) is the only choice for finding a closed-form solution to the problem posed.

4.8.5 Cyclic Loading (Constant Normal Load) As emphasised and explored in Sects. 4.5, 4.6 and 4.7, in most practical problems, the tangential loads change in a cyclic manner. We have found that the Jäger–Ciavarella theorem allows us to tackle a large number of loading scenarios, excluding those that involve large differential tension. In this section, we wish to approach the problem of cyclic tangential loading under constant normal load, using the solution framework of glide dislocations developed above to overcome this limitation. Similar to the approach taken in Sect. 4.7, we will develop the solution around the maximum and minimum points of a steady-state cycle, skipping the transient cycle. A marching-intime solution might be undertaken to investigate the transient problem, but focussing on the steady-state solution seems sensible as in most engineering applications a significantly larger number of minor cycles is accumulated per major cycle. The steady state oscillates between two values (Q 1 , Q 2 ), and, at the same time, the differential tension fluctuates between two values (σ1 , σ2 ), see Fig. 4.22, while the normal load, P, is held constant. This problem was first treated using dislocations in Andresen et al. (2020) and the objective was to develop a closed-form steady-state solution under contact normal load that provides the same answers as those presented in Andresen et al. (2020a) for cases in which the differential tension is moderate, but extents the results to cases where the differential tension is large.

4.8 Partial Slip Solutions Based on Dislocations

Q 2 (P;Q stea dy s tate

Fig. 4.22 The steady-state cycle in σ -Q-space under a constant normal load P

91

0 t sien tran O`

1 (P;Q

, ) Q

, )

As in Sect. 4.8.2, we start with a fully adhered interface. A change in shear force, Q, and differential tension, σ , is imposed, when moving from the minimum load state to the maximum, Fig. 4.22. Under fully adhered conditions, the corresponding change in shear tractions is given by q(x) =

x σ Q + √ . √ π a2 − x 2 4 a2 − x 2

(4.104)

Suppose now, the loading is close to load state 1 and has just reversed by a small increment and is now heading towards load state 2, locking in q1+ (x). As the other end of the load cycle is approached, we can express the shear tractions by Q x σ + √ √ 2 2 π a −x 4 a2 − x 2   −m a 2 − ξ 2 BxL (ξ ) E∗ dξ + √ ξ −x 2π a 2 − x 2 −a   a 2 a − ξ 2 BxL (ξ ) + dξ , ξ −x n

q2− (x) = q1+ (x) +

(4.105)

where x ∈ (−a, a) and BxL (x) is the density of dislocations arising during the loading phase in order to correct the fully adhered tractions and allow for slip in the regions [−a − m] and [n a]. When the loading increment changes sign at load point 2 the shear tractions q2+ (x) become locked-in and as we approach point 1, we state Q x σ − √ √ 2 2 π a −x 4 a2 − x 2   −m a 2 − ξ 2 BxU (ξ ) E∗ dξ + √ ξ −x 2π a 2 − x 2 −a   a 2 a − ξ 2 BxU (ξ ) + dξ , ξ −x n

q1− (x) = q2+ (x) −

(4.106)

92

4 Half-Plane Partial Slip Contact Problems

where x ∈ (−a, a) and BxU (x) is the density of dislocations arising during the unloading phase. Subtracting Eq. (4.106) from Eq. (4.105) yields the following integral: 2 Q 2x σ + √ q2− (x) − q1− (x) = q1+ (x) − q2+ (x) + √ π a2 − x 2 4 a2 − x 2   −m a 2 − ξ 2 BxL (ξ ) E∗ dξ + √ ξ −x 2π a 2 − x 2 −a  a 2 a − ξ 2 BxL (ξ ) + dξ ξ −x n  −m  2 a − ξ 2 BxU (ξ ) − dξ ξ −x −a   a 2 a − ξ 2 BxU (ξ ) dξ , − ξ −x n

(4.107)

where x ∈ (−a, a). In the regions of slip, we expect the shear tractions to take their limiting value, so that q2− (x) = q2+ (x) = f p(x) sgn(x)

− a ≤ x ≤ −m and n ≤ x ≤ a

(4.108)

and q1− (x) = q1+ (x) = − f p(x) sgn(x)

− a ≤ x ≤ −m and n ≤ x ≤ a , (4.109)

where we omit the sgn(•) functions when we solve a loading scenario under moderate tension, Fig. 4.10a, and include them when the differential tension is large, Fig. 4.10b. In cyclic partial slip problems, particles that have translated relative to their counterparts during a half-cycle must translate back into their original position during unloading in order to conserve material within the interface. We know the maximum extent of slip penetration is reached near the end points of the load cycle. This requires the relative slip displacement within the slip zones to be equal in magnitude, but opposing in the sign at load points, 1− , 2− . The slip displacement can be found from the dislocation density, so that, for example, in the right-hand slip region, the slip displacement at point x, u L (x), during the loading phase is given by  u (x) = L

n

x

BxL (ξ ) dξ

n σthreshold , this linear relationship is replaced by the highly coupled non-linear behaviour inherent in Eqs. (4.124) and (4.125) and they must be treated numerically in order to determine the extent and position of the permanent stick zone. The most sensible way to illustrate these results is to keep the change in shear load constant, while plotting the corresponding steady-state solutions, i.e. −a ≤ −m < n ≤ a, for different values of change in differential tension, σ . By doing so the transition between both branches of the solution and their individual behaviour becomes most obvious, as observable in Fig. 4.24. ,

98

4 Half-Plane Partial Slip Contact Problems

4.8.7 Cyclic Loading (Varying Normal Load) In many engineering applications, the cyclic loading stems from a solitary remote source which affects all local contact loads, including the normal load so that the loads present change in phase in a cyclic manner; see Fig. 4.25. This means the normal load varies proportionally with the tangential loads and for reasons which will become clear, we continue to assume the contact to be symmetrical, precluding the presence of a moment, so that dQ dσA dσB dP = = = . P Q σA σB

(4.126)

The starting point for all partial slip solutions employing the idea of inserting glide dislocations in the slip zones is a fully adhered contact interface. When the normal load is kept constant, the arising shear traction distribution is singular; see Eq. (4.89). However, from Sect. 4.3.2 we know that under proportional loading, the shear tractions arising under fully adhered conditions are bounded. Our ambition is to find, first, the change in fully adhered shear tractions when moving from the minimum point of the steady-state cycle, 1, [P1 , Q 1 , σ1 , a1 ], to the maximum point 2 [P2 , Q 2 , σ2 , a2 ], where (P2 > P1 ), Fig. 4.25. Suppose the normal load changes by a small amount, d P, so that the change in contact pressure, d p(x), is given by Barber et al. (2011) dP , (4.127) d p(x) = √ π a2 − x 2 where a is the instantaneous contact half-width. If a simultaneous small change in shear load, dQ, and differential tension, dσ , is applied, the change in shear tractions, dq(x), is given by dQ xdσ dq(x) = √ + √ . (4.128) 2 2 π a −x 4 a2 − x 2 In Sect. 2.5 we used these results to find the minimum coefficient of friction needed to maintain a fully adhered contact interface during finite positive load changes, i.e.

Q,

2 (P , Q

t sien tran

, )

stead y stat e

Fig. 4.25 Two-dimensional illustration of a P-Q-σ load space

0

1 (P , Q

, )

P

4.8 Partial Slip Solutions Based on Dislocations

99

| Q| π a | σ | + < f . P 4 P

(4.129)

We require the finite linear change in normal and shear loads, when moving from load state 1 → 2, to be Q2 − Q1 Q =λ= , (4.130) P P2 − P1 and

σ 2η = , P (a2 + a1 )

where η=

(4.131)

(a2 + a1 ) (σ2 − σ1 ) . 2 (P2 − P1 )

(4.132)

Now, let us apply the Mossakovskii–Barber procedure which was introduced in Chap. 2 considering a sequence of flat punches each adding the infinitesimal pressure change given by Eq. 4.127, so that the overall change in normal traction, p(x), moving from state 1 → 2, is given by Barber et al. (2011) p(x) =

1 π



P2 P1

dP = p(x, P2 ) − p(x, P1 ) . √ a2 − x 2

(4.133)

If we replace the change in normal load, P, with the expressions given in Eqs. (4.130 and 4.131), we find the change in shear traction to be a scaled form of the change in normal tractions over the steps 1 → 2 and 2 → 1, i.e. 

ηxπ q(x) = [ p(x, P2 ) − p(x, P1 )] λ + 2 (a2 + a1 )

 .

(4.134)

We now know the change in shear tractions under fully adhered conditions when the normal load is varying cyclically as well. This allows us to make the necessary adjustments to the formulation presented in Sect. 4.8.5 in order to find the two consistency conditions necessary to determine the permanent stick zone, [−m n]. We again start to develop the solution by stating the tractions in steady-state. Suppose we are near load point 1, Fig. 4.25, and the loads have just reached the turning point, meaning the interface becomes instantaneously stuck, locking in the shear traction q1+ (x). Since the loads change linearly when going from 1 → 2 and as we approach load point 2, the shear traction distribution is given by

100

4 Half-Plane Partial Slip Contact Problems

  ηxπ q2− (x) =q1+ (x) + [ p(x, P2 ) − p(x, P1 )] λ + 2 (a2 + a1 )  ⎡  −m a 2 − ξ 2 B L (ξ ) x 2 E∗ ⎣  + dξ ξ −x −a2 2π a 2 − x 2 2



a2

+ n

 a22 − ξ 2 BxL (ξ ) ξ −x

⎤

dξ ⎦ ,

(4.135)

where −a2 < x < a2 and BxL (x) is the as yet unknown dislocation density to be distributed over the slip regions during this loading phase. Now, as the loads are reversed for unloading, the interface becomes fully adhered once more, locking in the shear tractions q2 (x)+ . As we approach point 1, we can write q1− (x)

=q2+ (x)

  ηxπ − [ p(x, P2 ) − p(x, P1 )] λ + 2 (a2 + a1 )  ⎡  −m a 2 − ξ 2 B U (ξ ) x 1 E∗ ⎣  + dξ ξ − x −a2 2π a12 − x 2  ⎤  a2 a 2 − ξ 2 B U (ξ ) x 1 dξ ⎦ , + ξ −x n

(4.136)

where −a2 < x < a2 and BxU (x) is the dislocation density in this unloading phase. The glide dislocation distributions over the slip intervals physically represent the slip displacements there, and the shear tractions ratio remains constant outside the intervals stated. We know that, in the slip regions, the shear tractions are equal to their limiting value, i.e. q1 (x) = f p(x, P1 ) sgn(x)

− a1 ≤ x ≤ −m

n ≤ x ≤ a1 ,

(4.137)

where we omit the sgn (•) function when the direction of slip is the same at both ends of the contact and we include the function when the direction is of opposite sign at one end of the contact. For |x| > a1 , the shear stress vanishes. At load state 2, we write q2 (x) = − f p(x, P2 ) sgn(x)

− a2 ≤ x ≤ −m

n ≤ x ≤ a2 ,

(4.138)

and again the sgn (•) is omitted when the direction of slip is the same at both ends of the contact and included when the direction is of opposite sign at one end of the contact. Again, for |x| > a2 , the shear stress must vanish. We assert that the

4.8 Partial Slip Solutions Based on Dislocations

101

maximum extent of slip penetration is the same at points 1− , 2− by employing the principle of conservation of material. It follows that the slip–stick boundaries are the same at each end of the loading cycle, i.e. there is a permanent stick zone [−m n]. An additional consequence concerns the change in the slip displacement. Particles which have moved in one direction during the loading phase 1–2 will move back into their original position during the unloading phase 2–1. If we examine the right-hand slip region, we see that the slip displacement at point x, u(x) during unloading is given by  x

u(x) = n

n < x < a2 ,

BxU (ξ ) dξ

(4.139)

and we infer that the dislocation density during loading has the same magnitude but opposite sign, so that BxL (x) = −BxU (x)

− a2 ≤ x ≤ −m

and

n ≤ x ≤ a2 .

(4.140)

Note that, in the part of the interface which is touching permanently [−a1 < x < −m] and [n < x < a1 ], u(x) represents the relative displacement, while in the interval of the contact that is changing size, [a2 < |x| < a1 ], it continues to represent the tangential displacement of particles. In other words, while some particles move back into their original position while being part of the contact, others move back into their original position outside of the instantaneous contact extent. Using this physical insight, we may focus on evaluating the integral equation at the maximum load point 2, and re-write Eq. (4.135) as    ηxπ ηxπ + p(x, P1 ) f sgn(x) − λ − = p(x, P2 ) f sgn(x) + λ + 2 (a2 + a1 ) 2 (a2 + a1 )   ⎤ ⎡  a2 a 2 − ξ 2 B L (ξ )  −m a 2 − ξ 2 B L (ξ ) x x 2 2 E∗ ⎣  dξ + dξ ⎦ ξ − x ξ − x 2 2 n −a 2 2π a2 − x 

− a2 < x < −m

n ≤ x ≤ a2 .

and

(4.141)

The primary unknown is the dislocation density BxL (x), and this can be found by inverting the integral equation in the standard way. This is needed if the slip displacements are to be evaluated, but we are interested in the extent of the permanent stick zone, so that we pay close attention to the side conditions which must be enforced. From Sect. 4.8.5, we are already familiar with the nature of the consistency conditions and a bounded-both-ends solution of Eq. (4.141) leads to  0=

−m

−a1

qc (ξ )ξ k dξ − √ (ξ − n)(ξ + m)



a1 n

qc (ξ )ξ k dξ , √ (ξ − n)(ξ + m)

(4.142)

where k = 0, 1. The corrective traction, qc (x), is equal to the left-hand side of Eq. (4.141) and consists of terms which are the contact pressure distributions together

102

4 Half-Plane Partial Slip Contact Problems

with products of those distributions and x. When the solution is attached to a specific geometry, a numerical implementation of Eqs. (4.142) is needed to determine the integral limits, m and n.

4.9 Anti-Plane Formulation Some progress can be made towards a solution for three-dimensional problems by combining plane and anti-plane solutions. Let us start off with some basic equations for in-plane analysis. In anti-plane strain (Barber 2010), pp. 230–2, a force R applied to the surface of a half-plane and coming out of the page produces a surface displacement, w(x), in the same direction of magnitude w (x) = −R

ln |x| , πμ

(4.143)

where μ is the modulus of rigidity, and so 1 dw =− dx πμ

 contact

r (ξ ) dξ . x −ξ

(4.144)

Further, if we have a pair of elastically similar half-spaces brought into contact, an interfacial anti-plane shear traction distribution, r (x), will correspond to a relative anti-plane slip displacement given by d 2 (w1 − w2 ) = − dx πμ



r (ξ ) dξ . x −ξ

(4.145)

4.10 Influence of a Screw Dislocation Two elastically similar half-planes (y > 0 and y < 0), having a modulus of rigidity μ are bonded together over the interval [−a a]. There is a relative slip displacement between the two surfaces, in the z-direction, w(x). The interfacial shear traction is related to the slip displacement by 2 dw =− dx πμ



a −a

r (ξ ) dξ . ξ −x

(4.146)

As the extent of contact is determined by the Signorini inequalities, a ‘boundedboth-ends’ solution is needed to this equation. It is given by

4.10 Influence of a Screw Dislocation

103

√   w (ξ ) dξ μ a2 − x 2 a  r (x) = . 2π a 2 − ξ 2 (ξ − x) −a

(4.147)

Now let us suppose that the surface irregularity is simply a screw dislocation of magnitude bz and located at point c. The anti-plane shear traction developed is given by  √  a2 − x 2 μbz (c) μ a 2 − x 2 a bz (c)δ (ξ − c) dξ  , = r (x) = 2π 2π (c − x) a 2 − c2 a 2 − ξ 2 (ξ − x) −a

(4.148)

and a distribution of dislocations of density Bz (x) = dbz /dx will produce a shear traction distribution given by √  Bz (ξ ) dξ μ a2 − x 2 a  . r (x) = 2 2π a − ξ 2 (ξ − x) −a

(4.149)

The inversion of the equation when the end points are fixed( and hence the solution singular), is r (x) =



1

√ π a2 − x 2

μ C− 2



a

−a

  a 2 − ξ 2 w (ξ )dξ x −ξ

− a < x < a . (4.150)

Suppose that we set the slip displacement to be zero everywhere except at point c where we insert a dislocation of the Burgers vector bz , so that w (x) = bx δ(x − c), where δ(•) is Dirac’s delta function, and hence  a  2 μbx a − ξ 2 δ(ξ − c)dξ r (x) = − √ x −ξ 2π a 2 − x 2 −a √ μbx a2 − x 2 =− −a < x λI . Now apply just a small shear force, Q, to the square block, and which corresponds to small changes K I0 , K II0 in the generalised stress intensity factors, referred to the contact interface. The traction distributions adjacent to the trailing edge, well within the slip zone (s  c) are now given by p(s) = K s s λs −1 + K I0 s λI −1 + K II0 s λII −1

(5.16)

116

5 Complete Contacts and Their Behaviour

Fig. 5.7 Eigenvalues for the quarter plane in contact with an elastically similar half-plane: slipping eigenvalue, λs , as a function of the coefficient of friction, f . The adhered eigenvalues ,λ I and λII , are shown as dashed lines

q(s) = K s s λs −1 + K I0 grI θ s λI −1 + K II0 grIIθ s λII −1 . .

(5.17)

The first term in Eq. (5.16) is positive, that is, there is pressure adjacent to the contact edge, associated with the slipping eigensolution. But the second term, which is more strongly singular, is given by the top right term in the matrix in Eq. (5.12), K I0 a λI −1 = 0.179



Q 2a



and is negative. It follows that there must be separation at the contact edge and the extent of the trailing edge separation length may be estimated from Eq. (5.16), neglecting the K II0 term, and looking for violations of a positive contact pressure which occurs at p(ds ) = 0, where  ds =

K I0 Ks

1 λs −λ

I

.

In order to make practical use of this equation, we need to estimate the calibration for K s at the end of the normal loading phase. Its definition is given by K s = p(s)s 1−λs s → 0 and may be found from finite element analysis output (Churchman and Hills 2006a). When this is done, it is straightforward to estimate both the extent of separation and also the implied length of the reverse slip region, cs , again based on violations of the slipping law, Eqs. (5.16), (5.17), and is given by

5.2 General Frictional Response—Square Contacting Element

117

Fig. 5.8 Exponent controlling the extent of separation and slip at the trailing edge of a square block as a function of the coefficient of friction, f

 cs =

f − grI θ 2f

1  λs −λ I

K I0 KIs

1 λs −λ

I

.

(5.18)

The results are shown in Fig. 5.8 and indicate just how close to a fully stuck contact the configuration is. This is the principal result of this piece of analysis, viz. that when sharp edges are present defining the edges of a contact, it is either truly fully stuck up to the contact edges, if f > grI θ , or, even when this condition does not hold, the cyclic edge slip/separation regions, for realistic values of applied load are extremely small, and are likely, in reality often to be within the limits of any edge radii actually present and ignored in the idealised problem.

5.3 Finite Slip Zones In the preceding section, the size of the slip zone has been estimated by assuming that the whole contact sticks to the very edge, and finding regions where there is a violation of the slip or separation with slip condition. This underestimates the size of the slip zones developed, and by an unquantified amount. One way to model the problem accurately very efficiently is to distribute an array of glide dislocations along the contact interface to represent the slip developed Churchman and Hills (2006b). The starting point for the calculation is, therefore, to establish the state

118

5 Complete Contacts and Their Behaviour

of stress developed by a glide dislocation of strength bx located at point ξ , on the tractions developed at point x, where both dislocation and observation point lie on the contact line, and (x, ξ ) are measured from the contact edge, Fig. 5.9 and are given in numerical form in Churchman and Hills (2006b), and see also Churchman (2006). They could also be found analytically as the domain in which they exist is the relatively simple one of a semi-infinite wedge (whose angle is, of course, the sum of the angle in the contact-defining body plus π radians in the half-plane so, for the special case of a square block on a half-plane, as studied in detail here, a total angle of 3π/2 radians). When the contact is formed by exerting a normal force, the direct and shear tractions arising along the interface adjacent to the contact edge (N (x), S(x), respectively) are, therefore, given by N (x) = K I0 x λI −1 + K II0 x λII −1 +

2μ π (κ + 1)



c

Bx (ξ ) Fx yy (x, ξ ) dξ

(5.19)

0

S(x) =K I0 x λI −1 grI θ + K II0 x λII −1 grIIθ +  

c 1 2μ + Fx x y (x, ξ ) dξ Bx (ξ ) + π (κ + 1) 0 x −ξ

(5.20)

where the dislocation density Bx (x) = dbx /dx and Fxi y (x, ξ ) are the bounded contribution to the traction σi y (x) due to a unit glide dislocation present at point ξ , (Churchman and Hills 2006b). The slip zone is of length c, and the friction law is S(x) = − f N (x)

0≤x ≤c.

(5.21)

The integral equation to be solved is, therefore x λI −1 + f + + f + d0  

c 1 + f Fx yy (x, ξ ) + Fx x y (x, ξ ) dξ (5.22) Bx (ξ ) + x −ξ π (κ + 1) K I0 d0λI −1 0



grI θ





x d0 2μ

λI −1



grI θ





This equation may be solved numerically using the well-established procedures based on Chebyshev polynomials. However, in this instance, the required behaviour of the unknown function (the dislocation density) does not quite follow the usual pattern. We expect the dislocation density to be square-root bounded as the stick– slip transition point is approached, i.e. x → c, but at the corner we expect σi j ∼ x λs −1 . We, therefore, choose a fundamental function which shows the correct square-root bounded behaviour at the stick–slip transition point but which is square-root singular at the contact edge. This means that there will be no side condition available to determine the value of c, so we have to rely on using the inequalities defining slip, viz. |S (x)| < − f N (x) x >c (5.23)

5.3 Finite Slip Zones

119

Fig. 5.9 Glide dislocation of strength bx located at point ξ

c

x

and sgn(h(x)) = sgn(S(x))

where

x 0 )

2

1

and σr θ = − f σθθ .

(5.28)

These are conventional restrictions for frictional contact problems. The Signorini conditions are not encompassed here with the implication that bodies will remain in contact even if the normal tractions become tensile, σθθ > 0, see Chap. 1 Sect. 1.4. We further note that the coefficient of friction, f , is a signed parameter. If the normal tractions are compressive, a negative value of f suggests the kind of slip which is experienced at a leading edge, Fig. 5.10, whilst a positive value of f implies the kind of slip seen at the trailing edge of contact. For asymptotic expansions of the fully adhered problem, the pioneering solution of Williams (1952) can be employed, capturing the state of stress near the apex in the form of σi j ∼ r λ−1 f i j (θ ).

(5.29)

Revisiting the characteristic Eq. (3.65), the value of λ may be determined for the case of ψ2 = π/2, giving    λπ sin2 − λ2 cos [λπ ] + 0.5 sin2 [λπ ] + f λ(1 + λ) sin [λπ ] = 0 . (5.30) 2 Figure 5.11 shows the roots of Eq. (5.30) plotted against the coefficient of friction, f . This plot exposes several features of the solution which are worthy of comment. Let us assume, first, that normal tractions remain compressive so that σθθ < 0. If the contact is frictionless λs ≈ 0.77 and for f < 0 the solution applies to the leading edge. If f > 0 , the trailing edge is considered. Figure 5.11 displays three distinct regions; each exhibiting a different behaviour which we will discus in detail in the following: In region I, f < 1/π , the state of stress varies as r λs −1 , where r is a polar coordinate measured from the contact edge. The lowest root is always less than unity, so that the dominant behaviour of the solution is singular in nature. It follows that the contact pressure may be written in the form

5.4 Sliding Asymptote (Bilateral)

121

Fig. 5.11 The slipping eigenvalues, λs as a function of the coefficient of friction, f . The grey lines on the right represent the real and imaginary parts

p(r ) = −σθθ (r, 0) = K s r λs −1

(5.31)

where K s is the generalised stress intensity factor. It can be found by collocating the solution to whatever finite problem is being studied. If f = 1/π the tractions become bounded. One root is always λ = 1 but the corresponding eigenvector is of zero amplitude. In region II, 1/π < f < 0.392, there are two wholly real roots in the interval 1 < λ < 2, and these correspond to a local pressure distribution which is bounded but where d p/dr → ∞ as r → 0. In this region, the contact pressure may be written in the form (5.32) p(r ) = −σθθ (r, 0) = K b1 r λb1 −1 + K b2 r λb2 −1 , where K b 1 and K b 2 are the power order bounded stress intensity factors. If the coefficient of friction exceeds 0.392, region III, the lowest wholly real root lies in the interval 2 < λ < 3, so that, as r → 0, not only does the pressure fall to zero, but also d p/dr → 0. Therefore, the appropriate eigenvalue in this region is a complex one, whose real branch originates at the point where the locus folds back, see Fig. 5.11, and an imaginary part also develops. The contact pressure in this region may be written in the form   r , p(r ) = −σθθ (r, 0) = K c r ζ −1 sin η ln ro

1 < ζ < 2, 0 < η < 1 , (5.33)

122

5 Complete Contacts and Their Behaviour

where K c is the stress intensity factor, ζ and η are the real and imaginary parts of the eigenvalue respectively, and ro denotes the position from the contact edge where the argument of the logarithm passes through unity. It is clear that, if σθθ < 0 when r > ro , the contact pressure must become tensile close to the contact corner, i.e. when r < ro . The solutions in regions I and II are physically reasonable, but that in region III clearly does not satisfy the normal Signorini boundary conditions, and a correction to the Comninou solution is needed in order to permit the wedge and half-plane to separate in a region where there is implied interfacial tension. It will be shown later how the separation condition can be restored within the asymptote. When this happens the contact pressure falls smoothly to zero and is square-root bounded. The general state of stress in Region III can be expressed in the following form: σ jk (r, θ ) = H rˆ ζ +iη−1 f jk (θ )

  = H rˆ ζ −1 exp iη ln rˆ f jk (θ ) exp iφ jk (θ ) ,

(5.34)

where H is an arbitrary multiplier, rˆ√is a normalised coordinate (implicitly rˆ = r/ao ), corresponding to ao is an arbitrary length scale, i = −1, f jk (θ ) is the eigenvector

the complex eigenvalue (ζ + iη) with magnitude f jk (θ ) and phase φ jk (θ ) when written in complex polar form. The solution of the characteristic equation for f > 0.392 has complex conjugate pairs which are simultaneously excited, so



   σ jk (r, θ ) = H rˆ ζ −1 f jk (θ ) exp i η ln rˆ + φ jk (θ )     + exp −i η ln rˆ + φ jk (θ )

  = 2H rˆ ζ −1 f jk (θ ) cos η ln rˆ + φ jk (θ ) ,

(5.35)

where it has been noted that the conjugate eigenvectors have the same magnitude but opposite sign phase. The σθθ stress along the interface, from Eq. (5.35), can be expressed as  σθθ (x, 0) = 2H | f θθ (0)|

x ao

ζ −1

  x π + φθθ (0) + . sin η ln ao 2

(5.36)

If we define σθθ stress along the interface in a finite problem as σθθ (x, 0) = −K c x

ζ −1

  x , sin η ln xo

(5.37)

we can find the multiplier H and the ratio x o /ao by comparing Eqs. (5.36) and (5.37) giving −K c H= (5.38) a ζ −1 , 2 | f θθ (0)| o

5.4 Sliding Asymptote (Bilateral)

xo = exp ao

123



−(φθθ (0) + π/2) η

.

(5.39)

Therefore, the general state of stress incorporating a length scale, a, taken from any finite problem under consideration may be written as σ jk (r, θ ) = po



−K c a ζ −1 po



  f jk (θ )  r ζ −1 r a + φ jk (θ ) , (5.40) cos η ln | f θθ (0)| a a ao

where a is contact half-width, po = P/2a is the applied normal mean pressure and a/ao is a constant which scales the semi-infinite asymptote into the finite problem. This scaling ratio, a/ao is found by the following relationship: a xo /ao = , ao xo /a

(5.41)

where x o /ao is from Eq. (5.39) and x o /a is found from a model of the finite problem with bilateral boundary conditions (Karuppanan 2007). See Flicek et al. (2015) for a discussion of the transient problem describing the transition from normal load only to full sliding of a square block pressed onto an elastically similar half-plane.

Chapter 6

Representation of Half-Plane Contact Edge Behaviour by Asymptotes

6.1 Introduction When we were looking at complete contacts (Chap. 5), the lack of an elasticity formulation for the contact as a whole meant that, of necessity, we had to concentrate our attention on the contact edges where a partly closed-form solution in the form of Williams’ analysis (Chap. 3) was available for us to build on. Turning back to halfplane problems, our ability to analyse the contact as a whole, by various methods outlined in Chap. 2, and developed in Chap. 4, seems to render a simplified study of the problem, looking in detail only at the edges, unnecessary, but this is not so, and we have three reasons for wanting to develop an asymptotic framework for these problems. The first is that we would like to able to characterise the contact edge state of stress in a way which (a) permits us to compare the edges of geometrically different contact with each other, in terms of the local state of stress, and (b) from this, to build up a rigorous framework enabling contact edge crack nucleation conditions to be probed under laboratory conditions, and hence to establish a database for crack nucleation lives, as well as ‘below threshold’ infinite life conditions to be found. Secondly, although, in Chap. 4, we were able to solve partial slip problems for a wide range of conditions the solution set is not quite complete, and there are some problems, particularly where the contact is unsymmetrical, either through the contact-defining body’s general form, or because of the application of a moment, where a comprehensive solution is impossible. Under these conditions, a reduced analysis where each contact edge is studied separately, in the form of an asymptote, becomes very attractive. Lastly, if we approach the edge of a half-plane contact, so that the far edge falls well outside our field of view, and we further assume, for the time being, that the two bodies are ‘locked’ together, by tessellations of the surface on a very local scale which effectively provide the effects of infinite friction, the problem becomes like that of an infinite crack, albeit one in which the mode I loading is compressive, see Fig. 6.1. We, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. A. Hills and H. N. Andresen, Mechanics of Fretting and Fretting Fatigue, Solid Mechanics and Its Applications 266, https://doi.org/10.1007/978-3-030-70746-0_6

125

126

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

Fig. 6.1 Half-plane problem subject to normal load, P, and shear load, Q, with a particular focus on the contact edge

KI

ij

~ s

K II

s

P

K II

Q

KI

Q

x

P

therefore, have also to imagine that there is still a small gap exterior to the contact, but not sufficiently big to impair our idealisation of the bodies as half-planes. This idea is also implicit in the conventional half-plane formulation. We can then use the Williams eigensolution, developed in detail in Chap. 3, to describe the contact edge fields (see Sect. 3.4). If we continue consideration of the mode I loading for a moment, and we restrict ourselves to problems of engineering dimensions so that no cohesive tractions arise, the singular term (K I in fracture mechanics notation) must vanish. The next term, the so-called T-stress in fracture mechanics notation, may have relevance to partial slip considerations or modification of any local process zone, but, as it acts parallel with the surface, it has no relevance to the contact problem itself, and it is the third term in the expansion, where √ the characteristic form of the state of stress is that it varies in the form σi j ∼ s, where s is measured from the contact edge, which is important. At the edge of all incomplete contacts, of whatever geometric form, the stress components including, in particular, the contact pressure must vary in this way. On the other hand, because of the tessellations we have inserted, locking the bodies together in a shear sense, the excitation of shear along the interface will be fully resisted and result in a square-root singular stress distribution, antisymmetric with respect to the contact interface line if it is caused by a shear force alone, and be given by the same solution as so-called ‘mode II’ crack tip loading.

6.2 Basic Solution to the Normal Problem Half-plane problems which are uncoupled may be solved consecutively —meaning that we may solve for the form of the contact pressure and contact law first, without studying the tangential loading problem. On the other hand, when we come to solve the shear loading problem, although the problem is uncoupled to the extent that the contact pressure remains unmodified, regardless of the development of interfacial shear tractions, the shear traction distribution itself does depend on the loading sequence (Sect. 4.3).

6.2 Basic Solution to the Normal Problem

127

KII s LI s s=a+x

-a

d

P

A

A

Q a

-a Q

P

B

x B

Fig. 6.2 Representation of the contact edge behaviour for a half-plane partial slip problem by asymptotes

Consider a symmetrical half-plane contact problem subject to normal and tangential loading as depicted in Fig. 6.2. First, we wish to approximate the normal tractions in terms of an asymptotic multiplier. So, when the contact is formed the contact pressure distribution adjacent to the contact edge is given by √ p(s) = L I s

(6.1)

where s is a coordinate measured positive inwards from the contact edge, and L I is given by1 p(s) L I = lim √ . (6.2) s→0 s One way of finding it, therefore, for contact problems in which the contact pressure is explicitly known, is to make a simple transformation of coordinates to the contact edge, conducting a series expansion, and taking the lead term. So, for example, if we have a Hertzian contact where we may write the contact pressure in the form  p(x) = p0 1 −

 x 2 a

,

(6.3)

make the change of coordinate x = −a + s, and hence if s  a  p(s)  p0

2s , a

1 In many of the papers leading up to this monograph we used the symbol

(6.4)

K N for this quantity. Here we adopt a notation which is designed to show consistency with the Williams solution, and hence alignment with complete contact problems.

128

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes



and therefore L I = p0

2 , a

(6.5)

or this can be re-written in many alternative forms. Another way of arriving at the calibration is to make use of the Barber-Mossakovskii formulation (Sect. 2.4) which is particularly well suited, at any rate when the contact is symmetrical. Refer to Chap. 2 for details of the foundations of the calculation, and note that the pressure distribution caused by a set of punches of density F(u) is given by 

a

p(x, a) =

F(u) f 0 (x, u) du ,

(6.6)

−a < x a

(6.7)

0

where f 0 (x, a) =

√ 1 a 2 −x 2

and the normal load on the punch is given by  P(a) = π

a

F(u) du .

(6.8)

0

If we differentiate the last equation with respect to a we find that F(a) =

1 dP . π da

(6.9)

If, now, we take our definition of L I , but refer the pressure distribution to the contact overall, and focus on the end x → a, i.e. √ p(x) = L I a − x ,

(6.10)

and differentiate this equation through, also, with respect to a which gives

so that

∂ p (x, a) LI ∂ LI √ a−x+ √ , = ∂a ∂a 2 a−x

(6.11)

√ ∂ p (x, a) . L I = lim− 2 a − x x→a ∂a

(6.12)

Now differentiate Eq. (6.6) through with respect to a. The term involving the derivative of the integrand vanishes and we are left only with the term involving the integrand evaluated at the upper limit, that is (using Eq. (6.9))

6.2 Basic Solution to the Normal Problem

Thus

129

∂ p (x, a) dP 1 = F(a) f 0 (x, a) = √ . ∂a π a 2 − x 2 da

(6.13)

√ dP 1 , LI = 2 a − x √ π (a − x) (a + x)) da

(6.14)

and taking the limit x → a we finally arrive at 1 LI = π



2 dP . a da

(6.15)

As is evident from Eq. (6.15), L I is dependent on the instantaneous contact law, and the instantaneous contact half-width, a. Table 6.1 shows the calibration of L I for three common symmetrical contact geometries subject to a normal force, P. Recent work by Moore (Moore and Hills 2021) extended the Barber-Mossakovskii method to asymmetrical contacts spanning [−a c] in the presence of a moment, M, see Fig. 6.3. Remedying the previous restriction to symmetrical indenters requires us to express the left-hand contact coordinate as a function of the right-hand coordinate, i.e. c = c(a). Similar to the symmetrical case, Eq. (6.6), the normal traction, p(x, c), is given by a series of flat punches of yet unknown density, F(u)

d P/da ,

 p(x, c) =

c

F(s)h(x, s) ds ,

(6.16)

0

Table 6.1 Contact law, P(a), and asymptotic multiplier for the normal tractions, L I , for a Hertzian cylinder, a shallow wedge and a flat and rounded punch P πa 2 E ∗ P(a) = , R p(s) = LI s 4R dP πa E ∗ = , s = x+a da 2R a -a ∗ E a 2 . LI = 2R a ∗ P P(a) = E φa , dP p(s) = LI s = E ∗φ , da  s = x+a E ∗φ 2 a -a LI = . π a 

 P b E∗  2 P(a) = b a − b2 − a 2 arccos , p(s) = LI s 2R a  R ∗ dP E b s = x+a , = a arccos a -a da R a 2b b E∗ √ LI = 2a arccos . πR a

130

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes KII t

KII s L I,a s s=a+x

-a

t= c - x

M P

d

A

d

c

P

B

c

A

Q -a Q

L I,c t

x B

M

Fig. 6.3 Representation of the contact edge behaviour for a half-plane partial slip problem by asymptotes

where h(x, c) corresponds to the singular pressure distribution caused by a single flat punch, that is ⎧ 1 ⎨√ for − a(c) < x < c (a(c) + x)(c − x) . h(x, c) = ⎩ 0 otherwise

(6.17)

Normal equilibrium is imposed as  P=

c

p(s, c) ds .

(6.18)

1 dP . π dc

(6.19)

−a(c)

Differentiating with respect to c yields F(c) =

Then, Moore and Hills (2021) expand Eq. (6.16) close to the edges of the contact and show that, near end x → c  1 p≈ π

dP √ 4 1 c − x as c − x → 0, so that L I,c = c + a(c) dc π

 dP 4 , c + a(c) dc (6.20)

whilst at end x → −a  1 p≈ πa (c)

dP  4 a(c) + x as a(c) + x → 0 , c + a(c) dc

(6.21)

6.2 Basic Solution to the Normal Problem

where a (c) =

da dc

131

is the total derivative of a with respect to c so that  L I,a

1 = πa (c)

dP 4 . c + a(c) dc

(6.22)

This general result can be attached to any half-plane contact problem for which the contact law is known. For example, in Sect. 4.7.3, we introduced the closed-form solution to the contact problem between a half-plane and a tilted shallow wedge which yielded two explicit equations for the contact coordinates  P a= ∗ E φ connected by

1 ; γ

c=

P √ γ , E ∗φ

    −1 πα πα γ = 1 − sin 1 + sin . 2φ 2φ

(6.23)

(6.24)

This allows us to find the total derivatives dP √ = E ∗φ γ , dc

da 1 = ; dc γ

(6.25)

so that the asymptotic multiplier for the normal tractions at the left-hand contact edge of a tilted shallow wedge, Fig. 6.4, can be given in explicit form as L I,a

E ∗φ = π



4γ , a+c

(6.26)

which can be readily connected to the other end of the contact by L I,c =

L I,a . γ

(6.27)

Whilst the method for finding L I is easy to follow for simple geometries, like the wedge, finding the total derivatives da/dc and d P/dc becomes progressively more complicated for most other geometries within the half-plane idealisation. The authors hope that symbolic manipulators will make the mechanics of differentiation easier and more reliable, and so make the complicated results easier to obtain. A much simpler general expression for the asymptotic multiplier can be found if the contact coordinates at the ends of an asymmetrical interface, [−a c], are assumed to be independent. For normal contact problems, such as those solved in Chaps. 2 and 4, the starting point of the solution is a singular integral equation that connects the surface gradient of the contact interface, g (x), to the pressure distribution

132

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

p(s) = LI,a s

s = x+a p(x) p(s)

P M

c

Fig. 6.4 A tilted wedge contact subject to a normal load and moment with its asymptotic representation of the normal tractions at the left-hand contact edge

 p(x) = χ

c −a

g (s) ds , √ (a + s)(c − s)(s − x)

(6.28)

√

for notational convenience. The corresponding consistency where χ = E 4π(a+x)(c−x) (1−ν 2 ) condition connects the contact coordinates a and c implicitly or explicitly. The asymptotic description of the near edge normal traction can be found by shifting the coordinate system to either end of the contact and set √ √ p(x) = L I,c c − x + O( c − x) as x → c− , √ √ p(x) = L I,a x + a + O( x + a) as x → −a + .

(6.29) (6.30)

One can readily apply the method of Gakhov (1966) to show that the Cauchy principal value integral is bounded as x approaches either endpoint, and an asymptotic expansion of the integral reveals that √ L I,c = χ a + c L I,a

√ = χ a+c



c



−a c −a



2g (c) g (c) 1 g (s) ds + ,(6.31) − √ √ (c − s)3/2 a+c a+c s+a 

g (s) 2g (a) 1 g (−a) ds − .(6.32) − √ √ (s + a)3/2 a+c c−s a+c

Normal equilibrium can be imposed as  P=

c −a

 p(x) dx = π χ

 (a − c) g (s) s+ ds. √ 2 (c − s)(s + a) −a c

(6.33)

6.2 Basic Solution to the Normal Problem

133

Note that normally we would cite the consistency condition to remove the constant term in the brackets in the integrand, but in the approach taken here, we want to assume (6.28) and (6.33) are true and forget a and c are not independent.  −π χ c 2 −a  −π χ c = 2 −a

P=

−2s + (c − a) g (s) ds √ (c − s)(s + a)  d  (c − s)(s + a) g (s) ds. ds

(6.34)

Hence, by parts  P = πχ

c

−a

 g (s) (c − s)(s + a) ds.

(6.35)

Now, suppose a and c are treated as independent variables so that taking partial derivatives gives  ∂P π χ c g (s) √ = s + a ds. (6.36) √ ∂c 2 −a c − s We wish to integrate by parts again, but we must be careful since the integrand is already square-root singular at c. So let us split the range of integration as follows: ∂P πχ πχ = (I1 + I2 ) = ∂c 2 2



c−

−a

g (s) √ s + a ds √ c−s

 c− g (s) √ s + a ds , + √ c−s c−

(6.37)

where 0 <  1. We will be considering this in the limit that → 0, but must keep it small but finite for now. Considering I2 first, make the change of variable s = c − S, then we have  I2 = 0



g (c − S) √ a + c − S dS. √ S

(6.38)

Now, since is small, S is small, so Taylor expanding the integrand we get to leading-order  g (c) √ 2g (c) √ a + c dS = √ I2 ≈

. (6.39) √ a+c S 0 Thus, I2 → 0 as → 0. Now consider I1 . We have, by parts √  √  s+a g (c) a + c 1 c− 1 ds. (6.40) g (s) + − I1 = √ √ 2 −a (c − s)3/2

(c − s)(s + a)

134

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

Combining the brackets in the integral √  g (s) g (c) a + c (a + c) c−

− I1 = ds. √ √ 2

(c − s)3/2 s + a −a

(6.41)

To deal with the (·)−3/2 unboundedness of the integrand at c, we now add and subtract the singular part I1 =

√  g (s) g (c) a + c (a + c) c−

− √ √ 2

(c − s)3/2 s + a −a g (c) g (c) −√ + ds, √ a + c(c − s)3/2 a + c(c − s)3/2

(6.42)

so that integrating the final term in the integral, gives √  g (s) g (c) a + c (a + c) c−

− I1 = √ √ 2

(c − s)3/2 s + a −a √  2 g (c) 2 g (c) a + c , −√ − ds − √ √ 2

a + c(c − s)3/2 a+c

(6.43)

√ The 1/ terms cancel and so taking the limit as → 0 gives I1 =

(a + c) 2



c −a

g (c) 2g (c) g (s) ds + − √ √ a+c (c − s)3/2 a + c s + a(c − s)3/2

(6.44)

(where the minus sign outside the √ integral has gone inside). But, the term inside the square brackets is exactly L I,c /(χ a + c), so that √ πχ π a+c ∂P = I1 = L I,c . ∂c 2 4

(6.45)

A similar analysis holds for the other end of the interface, so that the asymptotic multipliers can be given in the form L I,c L I,a

 ∂P 4 1 2 , = π a + c ∂c  ∂P 4 1 2 . = π a + c ∂a

(6.46) (6.47)

A geometry in which the assumption of independence between a and c seems to hold up well is the tilted flat but rounded punch as the flat portion, say 2b, is generally far larger than the difference of the contact half-width, 2b a − c.

6.3 Basic Solution to the Tangential Problem

135

6.3 Basic Solution to the Tangential Problem We turn, now, to the shear asymptote. The singular form of the shear traction will occur only when the contact is formed first and held at a constant normal load, and then at least one of the two quantities exciting shear tractions, the shear force, Q, or the bulk tension, σ , is imposed, Fig. 6.5. The Barber-Mossakovskii ideas, see Hills et al. (2011), and Chap. 2, Eqs. (2.79) and (2.80), tell us immediately that the shear tractions developed under fully adhered conditions are given by q(x) =

σx Q + √ . √ π a2 − x 2 4 a2 − x 2

(6.48)

Note that this distribution is developed regardless of form of the contact profile. We again move the coordinate set to the contact edge by making the substitution x = −a + s and note that, if s  a K II q(s) = √ , s where

Q σ K II = √ − 4 π 2a

(6.49) 

a . 2

(6.50)

The finite contact is formed from two half-planes, effectively joined over the length [−a a]. As the contact pressure represents continuous direct traction, and under conditions of full stick, both the shear tractions and displacements are continuous, the domain is like an infinite plane with two inserted semi-infinite cracks representing the region of separation. If the point of observation is very close to the contact edge, the effect of the far region of separation is small and the environment is equivalent to a wedge of internal angle 2π radians or, in other words, a crack tip. a)

q(x) =

-a

b)

y Q (a 2- x 2)1/2

y q(x) =

a

x

-a

(

B)x (a 2- x 2)1/2

A

a

x

Fig. 6.5 Shear tractions under fully adhered conditions in the presence of a a shear force, Q and b a differential bulk tension, σ = σA − σB

136

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

Therefore, a contact edge where the normal load is kept constant but the shear oscillates means that the reversing state of stress (in a spatial sense) is the same as a crack experiencing reversing mode II loading, but where the crack is necessarily guided to extend along the contact interface (θ = 0 line) until it branches or forms an approximate mode I extension. This is presumably the kind of interpretation which Giannakopoulos had in mind in his prescient papers (Lindley et al. 1998; Giannakopoulos et al. 2000). There is, of course, one important difference between the two problems; looked at from the point of view of slip relieving the singularity we see that in the case of the crack tip only plastic slip (the process zone) occurs whereas in the case of the contact edge frictional and possibly plastic slip will occur. Note, too, that there will be some frictional shaking down so that, in the steady state, the frictional slip is always fully reversing regardless of whether the applied load is fully reversing.

6.4 Partial Slip Under Constant Normal Load The question arises of whether or not it is possible to solve the problem of partial slip within the framework on the asymptotic formulation. The motivation for this is partly one of simplicity; if it is possible then the results will be of a completely general nature, and permit the extent of slip to be found independently of the geometry of the problem (which enter the asymptotes via the calibration for L I , Eq. (6.15), or the mix of tension and shear exerted (which enter the problem via the calibration of K II , Eq. (6.50)). Just as the partial slip problems described in Chap. 4 may be treated either based on a correction to full slip or a correction to full adhesion, the same routes may be followed here. So, if we consider the right-hand edge of the contact and adopt, for the time being, a coordinate set fixed to the centre of the contact, we know from application of the Jäger-Ciavarella theorem (see Chap. 4), and a restriction of the contact pressure distribution to those described by the asymptotic form (Eq. (6.10)), that the shear traction in the neighbourhood may be written down in the form  √ q(x) = L I a − x − L˜ I a˜ − x , f

(6.51)

where a is the contact half-width and L I quantifies the local contact pressure, and  L˜ I , a˜ are the multiplier on the corrective term and coordinate of the stick–slip transition point, respectively. When the observation point is remote from the contact edge the slip zone becomes insignificant, and the shear traction distribution approaches that of an adhered contact, so that Eq. (6.49) applies. If we let the size of the slip zone be d, and move the origin to the contact edge so that s = a − x, we may re-write Eq. (6.51) as   L˜ I √ q(s) √ = s− s−d , (6.52) f LI LI

6.4 Partial Slip Under Constant Normal Load

137

and if we look at observation points remote from the contact edge so that Eq. (6.52) may be re-written as

s d

1

        √ q(s) √ L˜ I d L˜ I d + . . .) . (6.53) = s 1− 1− (1 −  s 1− f LI LI s LI 2s If we further assume that L I  L˜ I and compare Eqs. (6.49) and (6.53), we see that the size of the slip zone is given approximately by fd K II . = LI 2

(6.54)

In Dini et al. (2005), we go on to look at cyclic loading and show that, in the steady state, the slip zone is controlled by the range of mode II loading and is half the size it shows in the transient loading, i.e. K II = fd . LI

(6.55)

A solution starting from the full-stick shear traction distribution is also possible and requires a knowledge of the state of stress induced by a glide dislocation on the centreline of a semi-infinite crack, Fig. 6.6. If r , z are coordinates measured positive inwards from the contact edge the shear traction, qc (z), induced along the centreline (the contact interface) by a distribution of glide dislocations of density Bx (r ) is E∗ qc (z) = √ 2π z



d 0

√ r Bx (r ) dr , r −z

(6.56)

and hence we require to find Bz (r ) such that √ K II √ + qc (z) = L I z 0 < z < d . z

Fig. 6.6 Insertion of a glide dislocation on the centreline of a semi-infinite crack

(6.57)

z d

r

138

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

The solution to this integral equation is Bx (z) =

f LI √ d−z , E∗

(6.58)

together with a side condition which provides Eq. (6.54) directly.

6.5 Partial Slip Under Varying Normal Load Whilst controlling the sequence of load application, as well as keeping the normal load constant throughout periodic loading is easy to implement in laboratory environments, most real engineering applications will exhibit a cyclically varying normal load and moment. When it comes to an asymptotic approach, the varying normal load problem differentiates itself from the constant normal load approach in two important observations; (a) the contact edge will move throughout the cycle and (b) the maximum extent of the slip zone size is different at either end of the load cycle precisely because the minimum size of the stick zone is fixed. In Chap. 4, we presented closed-form solutions for the steady-state solution to half-plane contact problems under a varying tangential and normal load. The dislocations method of Sect. 4.8 allowed us even to treat problems involving large differential tension, a load case in which the celebrated Jäger-Ciavarella theorem becomes inapplicable when seeking closed-form solutions. Adopting the asymptotic framework to varying normal load problems not only provides a succinct and easy-to-apply point of comparison for cases where an analytical solution is available, but also enables the user to treat problems where the presence of large differential tension together with a varying moment prohibit a mathematically exact solution.

6.5.1 Asymptotic Description for Steady-State Problems Consider a generic steady-state loading scenario as illustrated in Fig. 6.7. From Chap. 4, we know that the transient phase does not affect the steady-state slip–stick pattern. Barber et al. (2011) demonstrated how a closed-form solution to the partial slip problem is straightforward when the loading is purely due to a varying normal and shear load (P, Q), but intractable in the case of large differential tension in the presence of a varying moment. Here, we shall assume that the variation in the steady cycle is such that all loads change in phase, collapsing the steady-state cycle to a straight line in load space. Let us begin developing the solution by writing down the shear tractions as a superposition of scaled forms of the full sliding shear traction in an asymptotic manner. If we expand the shear traction distribution at the left-hand end of the contact in a series at load point 1 and take the leading-order terms, we obtain a superposition of four bounded terms, giving Fleury et al. (2017)

6.5 Partial Slip Under Varying Normal Load

Q, M, cyc le

2

Q, M,

stea dystat e

Fig. 6.7 Cyclic loading with in phase changes in all load components

139

0

tra

o

e

has

tp

en nsi

1

P

P

  √ q1 (x) L II,1 x + a1 − d1 + f L I,1 x + a1 − 2 f L I,1 x1 + a1 − d1 +  (6.59) f L I,1 x + a2 − d2 ; where d1 and a1 are the slip zone size and contact size at the minimum load point, and d2 and a2 are the slip zone size and contact size at the maximum steady-state loading point. L II is an as yet unknown asymptotic bounded multiplier for the lockedin shear tractions so that generally the locked-in shear tractions near the left-hand contact edge are of the bounded form √ q(x) L II x + a ,

(6.60)

and where L II is a scaled form of the normal traction multiplier, L I . The latter may be obtained using the methods outlined in Sect. 6.2. Further, we can write down the asymptotic approximation of the shear traction distribution at the maximum of the loading cycle as   √ q2 (x)  L II,2 x + a2 − d2 − f L I,2 x + a2 + f L I,2 x + a2 − d2 .

(6.61)

Similar arguments to those made in Sect. 6.4 allow us to compare the change in shear traction, q(x) = q1 (x) − q2 (x) over the course of the cycle with a change in shear tractions in the absence of slip (Dini et al. 2005). To make this comparison, we assume the change in L I and L II to be negligible throughout the steady state. Thus  √ q(x)  L II,1 x + a1 − d1 + f L I,1 x + a1   − 2 f L I,1 x + a1 − d1 + f L I,1 x + a2 − d2     √ − L II,2 x + a2 − d2 − f L I,2 x + a2 + f L I,2 x + a2 − d2 . (6.62) If we now take the limits (L II,1 , L II,2 ) → L II and (L I,1 , L I,2 ) → L I , Eq. (6.62) simplifies to

140

6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes

d1

a

1

-a1 permanent stick zone

-a2 2

d2

Fig. 6.8 Slip-stick pattern at the left-hand contact edge for a contact under varying contact size and the minimum and maximum extent of the slip zones at either end of the load cycle

q(x)  f L I

√

 x + a1 − 2 x + a1 − d1    √ + x + a2 − d2 + x + a2 − x + a2 − d2 .

(6.63)

Now we expand the above expression at (x + a1 ) → ∞. This is distant enough for the effect of the slip zones on the shear traction distribution in the stick zone to be negligible, thereby enabling a comparison against a singular traction multiplier   d2 a + ... + √ q(x)  f L I 2 √ 2 x + a1 2 x + a1 K II =√ , x + a1

(6.64) (6.65)

where a = |a2 − a1 | is the change in contact size over one half loading cycle at the end of the contact under consideration and where 0 ≤ a d)

On the line θ = 0 which is both the possible line of frictional slip and the line taken to characterise plastic slip, r p (0), we find 2r p (0) σ y2 K II2

=6

(9.9)

to be independent of the transverse constraint, because a state of pure shear exists. Under oscillatory loading of magnitude K II some shakedown will reduce the implied size of the shear stress by a factor of 4 (or we might think of replacing σ y by 2σ y to represent the range of stress which might be accommodated, to give r p (0) =

3 4



K II σy

2 .

(9.10)

Considering the approximation of the steady-state slip zones size, Eq. (9.4), the relative plastic and frictional slip extent is given by rp 3 f K II L I = . d 4σ y2

(9.11)

If r p /d > 1 the plastic zone envelops any slipping region of the interface, and so the frictional slip may have only a mild effect, see Fig. 9.2. Under these conditions, a fretting contact edge will behave like a crack tip.

9.4 Experimental Data In Chap. 8, we reviewed different apparatus developed to investigate fretting fatigue. Some were single actuator test machines, see Fig. 8.4, others used two actuators to induce and control the loads present in the specimen; compare Figs. 8.6 and 8.9.

9.4 Experimental Data

167

Regardless of the exact experimental set up used, the parameters involved in describing the experimental results can commonly be summarised as follows: (a) The material under consideration. For metals relevant mechanical properties include the Young’s modulus, E, Poisson’s ratio, ν, and the yield stress, σ y , of the material used. (b) The coefficient of friction, f , is often considered a constant material property for metal to metal contacts, determined by a different set of experiments. (c) The pad geometry used. In the case of a Hertzian pad the radius, R, is sufficient, for flat and rounded pads the half-width of the flat portion, b, is an additional parameter. (d) The mean normal load, Pmean , and its range, P, (for half-plane problems given in force per unit length), and the peak normal pressure, p0 . (e) The contact extent, for symmetrical contacts given by the half-width, a, depends on the applied normal load, material, and contact geometry. (f) The applied tangential loads, i.e. the mean shear load, Q mean , and its range, Q, as well as the mean bulk stress in the specimen, σmean , together with the range, σ . (g) The measured fretting fatigue life, Nf , sometimes divided into a nucleation and propagation stage, where the transition point has been determined by applying Paris’ law or similar modelling techniques. We wish to review five experimental data sets, three conducted by the Oxford laboratory (Nowell 1988; Dini 2003; Blades et al. 2020), one at Purdue University (Szolwinski and Farris 1998), and one at Ghent University (Hojjati-Talemi et al. 2014). These all constitute reliable, carefully derived, dependable results. We acknowledge the authors’ efforts and will use their data in the subsequent sections of this chapter. This analysis would not be possible without the painstaking efforts of these we have just cited. The apparatus used to obtain the first data set was introduced in Sect. 8.2, when Nowell confirmed the detrimental effect of slip on the fretting fatigue strength and the dependence of the fretting fatigue strength on, prima facie, contact size. In his experiments, Nowell used the very common aluminium alloy HE15-TF, see Table 9.1, and a Hertzian contact geometry. He varied the contact size whilst keeping the peak normal pressure constant and found that a larger contact extent led to a shorter fatigue

Table 9.1 Material parameters for the Aluminium and Titanium alloys under consideration, including the coefficient of friction Material data Aluminium alloy

HE15-TF

(Nowell)

E (GPa)

ν (−)

σ y (MPa)

f (−)

74

0.33

400

0.7

AL2024-T351

(Szolwinski and Ferris)

74

0.33

310

0.7

AL2024-T3

(Hojjati-Talemi et al.)

72

0.33

383

0.7

Titanium alloy

Ti6Al4V

(Dini)

123

0.33

880

0.7

Steel alloy

AISI8630

(Blades)

205

0.33

600

0.7

168

9 Fretting Strength

life. The original test data Nowell obtained contains the results of 36 experiments and can be found in Appendix C. The simple calculations needed to obtain the additional information about the asymptotic multipliers, as well as the slip zone size and plastic radius, were carried out using the procedures outlined in Sects. 9.2 and 9.3. Some years later, Dini conducted a set of experiments to investigate the fretting fatigue behaviour of the titanium alloy Ti6Al4V in conjunction with different symmetrical flat and rounded pads (Dini 2003). The flat portion was left unchanged with half-width b = 4 mm, and the end radii were varied in order to mimic different aerospace applications and to determine their fretting fatigue strength. All 69 experiments were conducted under closely controlled conditions of partial slip with the normal load kept constant. Very recently, Blades re-purposed the two-actuator test machine shown in Fig. 8.9 to investigate a very common steel alloy AISI8630 under the conditions of fretting (Blades et al. 2020). The test data Blades generated stands out from all other experimental setups considered in this chapter. The indenting pads utilised were subject to significantly higher normal pressure gradients near the contact edge, i.e. L I is at least one order of magnitude larger than the stress gradients calculated for all other data sets of this chapter. The higher normal loads applied to the flat and rounded geometry under investigation led to a near edge stress state likely to be relieved by plastic deformation. We will make use of this special characteristic to calculate the plastic radius near the contact edge and compare the results against the approximated slip zone size for all data sets. At Purdue University, Szolwinski and Ferris attempted to predict the fretting fatigue strength of the aluminium alloy AL 2024-T351 utilising multiaxial fatigue analysis. They performed all their 37 experiments under conditions of partial slip and used cylindrical pads to establish an interface with their fatigue specimen and details of their analysis can be found in Szolwinski and Farris (1998). The final fretting fatigue data set stems from Hojjati-Talemi et al. (2014), who focussed on investigating the transition between crack nucleation and propagation. The material under consideration was the aluminium alloy AL2024-T3 and, once more, Hertzian-type pads were used for the experiments. In what follows, we wish to collapse this vast parameter space by employing the asymptotic methods developed and re-interpret the fretting fatigue data summarised above. The results for all additional calculations needed in this approach have been added to the data tables presented in Appendix C.

9.5 Interpreting the Measured Fretting Fatigue Strength We begin to interpret the experimental data by assessing the dominant damage mechanism. In Sect. 9.3, two ways of dissipating energy were described, thereby relieving the stress state near the contact edge. Applying this simple pair of calculations to the wide range of tests which have been published sheds light on the underlying mechanism, i.e. whether the stress state near the contact edge is relieved primarily by

9.5 Interpreting the Measured Fretting Fatigue Strength

169

rp [ m] 1000 Blades (Steel) Dini (Titanium) Nowell (Aluminium) Szolwinski (Aluminium)

100

Hojjati-Talemi (Aluminium) Linear (rp/d=1)

10

1 1

10

100

1000

d [ m]

Fig. 9.3 Plastic radii as a function of the slip zone size for different sets of experimental data

frictional slip, thereby making fretting the dominant damage mechanism, or whether the stress state is such that the effects of plasticity have to be taken into account, see Eq. (9.11). The results of this preliminary step are shown in Fig. 9.3. From this angle, it becomes clear that in the vast majority of problems the frictional slip zone is very much bigger than the plastic slip zone, or what we might also think of as a process zone from which cracks nucleate. But, there is one set of tests, having intended application to the fretting fatigue strength of well head riser couplings conducted on steel, where the plastic zone size dwarfs the frictional slip size (Blades et al. 2020). In these cases, we conclude that the contact edge behaviour is very much more like that of notch fatigue, as the interfacial slip implied may never occur, but be wrapped up in the plastic deformation actually present. Further consideration of these points is given in Truelove and Hills (2021). With the exception of the experiments by Blades set aside, in the remaining data sets frictional slip is likely to be present, and this, together with the reversing K II field, will cause conditions ripe for the nucleation of cracks. We argue that capturing the zones of fretting damage is a necessary step in determining the fretting fatigue strength experimentally. As indicated by Eq. (9.4), the slip zone size is a result of the ratio between tangential and normal stress contributions, assuming the coefficient of friction is a finite and constant material property. We note that the majority of a component’s total life is dictated by the nucleation phase (Nowell 1988; Szolwinski and Farris 1998), so that we will refrain from assessing the propagation stage which can be examined by classical fracture mechanics. We hypothesise, however, that it is not the slip zone size alone which determines the fretting fatigue strength, but also both the normal and tangential stress components play a role, and the slip zone size is merely a consequence of the presence of the two. It is the intention of this section to demonstrate that the use of a L I -K II parameter space

170

9 Fretting Strength

a) 800

1/2

1/2

High L I - 743 MPa/mm - 352 MPa/mm 1/2

Middle L I - 311 MPa/mm - 229 MPa/mm

600

Low L I - 223 MPa/mm - 169 MPa/mm

1/2

1/2

1/2

L I [MPa/mm ]

1/2

700

500 400 300 200 100 0

Test Number (sorted from high L I to low L I )

b) 140

1/2

KII [MPa mm ]

120 100 80 60 40 20 0

Test Number (sorted from high L I to low L I ) Fig. 9.4 Available range of a L I and b K II values for the pooled data set of Aluminium alloys

is sufficient to characterise the fretting fatigue strength of a material, independent of other geometrical factors. The elastic material properties of the aluminium alloys investigated are similar enough to pool the data points for further investigations, leaving us with one large set of experimental results for aluminium alloys. We begin to visualise the two-parameter space, first, by looking at the specific values of the asymptotic multipliers. Figure 9.4 shows the available range of calculated L I and K II values based on the pooled data set for Aluminium alloys. Our intention is to demonstrate that both asymptotic multipliers have their distinct influence on the outcome of an experiment and are indeed sufficient to characterise the fretting fatigue strength of the material under consideration. As indicated in Fig. 9.4a, we divide the data set into three groups of equal size, e.g. high, middle and low L I . The corresponding range of K II values is

9.5 Interpreting the Measured Fretting Fatigue Strength

171

given in Fig. 9.4b. We note that for our purposes the ideal data set would have had a fixed range of K II values, with three distinct values of L I tested along with the complete range of K II , making it easier to distinguish the individual effects of the two asymptotic multipliers. However, the nature of the data reflects the originally different intent behind the experiments and this is expected. Eq. (9.4) suggests that there is an infinite amount of combinations of K II and L I giving rise to the same slip zone size. Consequently, we shall see that the slip zone size on its own is insufficient to characterise the fretting fatigue strength of a specimen. Two experiments can be subject to the same size of slip zone, yet the outcome of the test (i.e. the total number of cycles to failure) might be a completely different one. If, on the other hand, the effects of the tangential and normal contribution are separated, a clearer picture starts to emerge. In Fig. 9.5, we see the slip zone size, d, plotted against the total life, Nf , in a logarithmic manner. Run outs (Nf > 107 ) are indicated by an arrow to the right. On its own, no salient direct correlation can be observed between the slip zone size and the number of cycles to failure. However, when the data is grouped into high, middle and low L I , it becomes obvious that the life achieved in a test extends as the size of the slip zone decreases. This is emphasised by the logarithmic fit added for each group of L I . Recalling Eq. (9.4), a decrease in slip zone size, and therefore, a longer life is due to a smaller value of K II (given L I and the coefficient of friction are held constant, of course). We further note that

High L I

1200

Middle L I Low L I Log. (High L I )

1000

KII

Log. (Middle L I ) Log. (Low L I )

Slip Zone Size [ m]

800

7

Run Out (>10 )

600

KII

400

200

KII LI

0 1.E+05

1.E+06

1.E+07

Number of Cycles to Failure, Nf

Fig. 9.5 Total life of Aluminium alloy experiments plotted against the slip zone size, data grouped in terms of L I to show dependence on both, the tangential asymptotic multiplier, K II , and the normal traction multiplier, L I

172

9 Fretting Strength

a) 1/2

High L I - 3050 MPa/mm 3500

1/ 2

1/ 2

1/2

L I [MPa/mm ]

3000

1/2

Middle L I - 2355 MPa/mm - 2163 MPa/mm 1/ 2

Low L I - 2163 MPa/mm - 1500 MPa/mm

2500 2000 1500 1000 500 0

Test Number (sorted from high L I to low L I )

b) 200

1/2

K II [MPa mm ]

180 160 140 120 100 80 60 40 20

7

Run Out (>10 )

0

Test Number (sorted from high L I to low L I ) Fig. 9.6 Available range of a L I and b K II values for the data set for Ti6Al4V

the higher the value of the normal traction multiplier, L I , the more sensitive does the fatigue strength become to changes in the slip zone size. We now turn to the other experimental data set in which frictional slip is believed to be the cause for crack initiation. In the tests conducted by Dini, the fretting fatigue strength of the titanium alloy Ti6Al4V was investigated (Dini 2003) and in order to strengthen the L I -K T approach introduced, we carry out the same steps as in the data analysis as outlined above. The range of L I and K II values calculated from the original test data is illustrated in Fig. 9.6. It is worth pointing out that the nature of the experimental data is such that only five different values of L I were tested. This has the benefit that the effect of K II is expected to be a little clearer as the L I values

9.5 Interpreting the Measured Fretting Fatigue Strength

173 High L I Middle L I Low L I Log. (High L I )

140

120

Log. (Middle L I ) Log. (Low L I )

K II

7

Slip Zone Size [ m]

Run Out (>10 )

100

K II

80

60

K II

40

LI

20

0 1.E+04

1.E+05

1.E+06

1.E+07

Number of Cycles to Failure, N f

Fig. 9.7 Total life of Titanium alloy experiments plotted against the slip zone size, data grouped in terms of L I to show dependence on both, the tangential asymptotic multiplier, K II , and the normal traction multiplier, L I

are closer together in their respective groups, see Fig. 9.6a. Additionally, we indicate those data points which led to run outs. The outcome is such that tests which were subject to a lower K II had a higher probability of running out than others, whereas the value of L I appears to have only a secondary effect on the fretting fatigue strength when only run outs are considered. It is natural to ask whether it might be possible to obtain a threshold value for K II for the material under consideration, below which the fretting fatigue life is always infinite? Moving on to the depiction of the actual fretting fatigue strength, we again use the slip zone size, d, as the guiding parameter to visualise the data, see Fig. 9.7. The effect of K II is indicated by a logarithmic fit for each of the three L I groups. It is reassuring to observe a similar trend as we saw in the data for aluminium alloys; higher values of K II not only lead to a larger slip zone size, but also have a detrimental effect on the life. The higher the value of L I , the shallower is the gradient of the logarithmic fit, which consequently increases the sensitivity of the achievable life in the respective L I group. The fretting fatigue data presented in this section highlights the potential that lies within the asymptotic description and visualisation of fretting problems. The obvious advantages include a much simpler initial analysis of the frictional contact behaviour and a significantly reduced parameter space when the problem is treated in terms of

174

9 Fretting Strength

asymptotic multipliers, L I and K II . Additionally, it is possible to characterise the fretting fatigue behaviour of a material in a comprehensible manner which makes possible a comparison of experimental results across different designs. An observation we made in Fig. 9.6b is the value of K II associated with run outs and this prompts further investigation as there is the possibility of a material-dependent critical value for K II above which the fretting fatigue strength becomes finite. However, these results are limited to incomplete contacts under constant normal load and although we have put some effort into understanding frictional slip and how it is linked to the crack nucleation stage, there are other aspects which need further attention, for instance, the effects of a varying normal load or wear in the slip zones and their effect on the stress field. Furthermore, once a crack has grown to a certain length, there is a transition from crack nucleation to the propagation stage and only the first stage is controlled by the near contact edge stress field (L I and K II ), whereas during crack propagation these asymptotic multipliers will need to be calibrated to the subsurface stress field controlled by the remote tension, σ .

9.6 The Asymptotic Descriptions of Contact Edges A central theme of this book has been to show both how complete contact edges can be treated using asymptotic wedge theory, and how the same principles may be applied to the edges of incomplete (half-plane) contacts with the possibility of a fairly straightforward extension to receding contacts. Here we wish to contrast the results for the two classes of the problem and have composed a set of twelve scenarios we wish to highlight, see Fig. 9.8. The contacting components are assumed to have the same elastic constants so that, when considering the two bodies combined in the sense that the interface is in intimate contact everywhere and both components of displacement are continuous across the interface, the two components considered together may be thought of as a solitary, monolithic, homogeneous entity. Note that, for each class of contact, we start off with the assumption that the interface is tessellated at a very fine scale, providing an infinite coefficient of friction, so that shear tractions of any magnitude may be resisted without slip taking place. This assumption is relaxed at the second stage. In the case of incomplete contacts, the William’s eigensolution for a ‘wedge’ of included angle 2π radians has properties absent from the solutions for smaller wedge angles, see Fig. 9.8a. In particular, the two solution sets—one symmetrical and the other antisymmetrical—have matched pairs of eigenvalues leading to a state of stress which vary like r λ−1 , where λ − 1 = − 21 , 0, 21 , 1, . . . and, when the asymptote is positioned at the contact edge the wedge bisector lies along the line of the contact. The antisymmetric solution has no direct stresses along that line, including the component corresponding to contact pressure, σθθ , and the symmetric solution has no shear stress along that same line, so that the two solutions uncouple, as expected, along the contact interface. The application of a normal load or moment to the contact,

9.6 The Asymptotic Descriptions of Contact Edges

175

b)

a)

c)

LI K II slip

stick

e)

d)

worn area

f)

LI K II slip

l

stick

worn area

l

l

i)

h)

g) KI K II

slip

stick

l)

k)

j)

worn area

KI K II slip

l

l

stick

worn area

l

Fig. 9.8 Comparison between possible cases for incomplete (a–c) and complete (g–i) contacts under mode I and II loading, viz. a fully stuck interface, partial slip, and partially worn in the slip zones. In cases (d–f) and (j–l), a crack of length l, growing perpendicular to the interface, was added to the problem

176 Fig. 9.9 Two bonded half-planes each subject to a remote tension, where in a these are the unequal tensions, σA and σB , which can be decomposed into two equal tensions giving rise to a symmetrical eigensolution as shown in b and equal and opposite tensions giving rise to the antisymmetric eigensolution shown in c

9 Fretting Strength a) A

B

= b)

12

A

B

(sym.) 12

A

B

+

c) 12

A

B

(anti-sym.) 12

A

B

therefore, induces only the symmetric eigensolution, in fact, only those terms which are odd multiples of 21 , and as no shear tractions are induced, there is no tendency for the bodies to slip at any point; an infinitesimal coefficient of friction would, therefore, suffice to inhibit all slip for this problem. The application of a shear force induces only the antisymmetric solution, and therefore, the contact pressure distribution remains unchanged, so that the normal and shear solutions are fully uncoupled, as speculated. In the case of the application of remote tensions parallel with the free surface, a little care is needed. Half-plane problems may have tensions present in both bodies, Fig. 9.9a, and it is appropriate to decompose them into a sum and difference, so that Fig. 9.9b shows an equal tension present in both bodies which excites a symmetrical eigensolution and Fig. 9.9c which shows an equal and opposite tension in each body which excites an antisymmetric eigensolution. Note that the equal-tension case implies, at its most simple, a tension which is uniform with depth but, as the eigensolutions admit power order variations with depth, also satisfy the elastic field equations— one body might have a constant depth tension in it, whilst the other had one varying linearly with depth, for example, as long as the surface values were the same. In the case of complete contacts, the total wedge angle formed from the half-plane and the second body, will always be less than 2π radians, so that the wedge bisector, where the eigensolutions uncouple, will not lie along the contact interface, see Fig. 9.8g. There will be symmetrical and unsymmetrical solutions, but the eigenvalue of the dominant symmetric solution will always be lower than that of the dominant antisymmetrical solution, and each solution will give rise to non-zero values of both components of traction along the line of the contact interface. In this sense, then, the solution is inherently coupled, regardless of the form of loading of any finite body into which the solution may be collocated. This property was brought out in Chap. 5 Sect. 5.1, and, as the symmetric eigensolution is always more strongly sin-

9.6 The Asymptotic Descriptions of Contact Edges

177

gular than the antisymmetric solution, it follows that, at points sufficiently close to the corner, the spatial variation of stresses is fixed to within a multiplicative constant. This explains why a coefficient of friction of at least the traction ratio implied by the symmetric eigenvector is required to inhibit all slip. This aspect of coupling can be very significant; as we have shown for the case of a quarter plane pressed onto a half-plane the traction ratio, q/p, at the contact edge is about 54%. In addition, when the solution is fitted into a finite body, the application of a normal load will induce both components of traction along the bisector of the combined bodies, and therefore, both eigensolutions will be excited. Equally, the applications of a moment, shear force, or tension in the half-plane will all induce both components of traction, and therefore, influence the magnitude of each eigensolution. The multipliers on each solution are linearly related to the components of load (P, Q, M, σ ). It is important to note that both eigensolutions are present at any general load state, and that as the load set changes it merely moves the value of the characteristic internal length dimension (called d0 in earlier chapters), so that when r < d0 the very near contact edge is certainly controlled by the symmetric solution alone, provided that the sign of the multiplier is one which implies intimate contact (positive contact pressure), and therefore, the satisfaction of the Signorini conditions. Formally, if there is separation, the contact is no longer complete, even though, superficially it may appear so. We turn, now, to the behaviour of the contact edge when there is slipping. The Williams solution is no longer appropriate; it does not include a line along which the tractions are geared together by the coefficient of friction, and so we must start again with a two-wedge problem (where one wedge is a half-plane), and establish continuity of displacement in the normal direction across the joint but also demand that the traction ratio is a given (signed) value, the coefficient of friction, f . The outer faces of the wedges remain traction free, and this is the basis of the BogyComninou-Theocaris formulation, described in Chap. 3 Sect. 3.6. In the case of the incomplete contact, the slipping eigensolution relates to two half-plane joined, as described, over a half-line (the slipping interface). This has not yet been treated in detail, because it has not been needed. We expect that it will n reveal that there are several valid solutions where the tractions vary like r 2 , and of most relevance will be the square-root bounded solution. Note that in this case, exceptionally, we expect that the solution will display a lack of coupling along the wedge centreline, and therefore, solutions can be found for either sign of f , i.e. that intimate contact will be maintained for slipping in either a leading or trailing edge sense, without a modification of the eigenvalue. In the case of complete contacts, which have already been investigated in some depth, the above remark does not apply, and therefore, because the shear and direct tractions are related by the equation q = f p, one sign of f (negative in our usual convention) corresponds to leading edge behaviour in the presence of intimate contact (positive, compressive pressure) whilst a positive value corresponds to trailing edge behaviour. If the eigenvalue corresponding to a slipping eigensolution is λ0 , then λI < λ0 < λII (Karuppanan 2007).

178

9 Fretting Strength

9.7 Stress Intensity Factor Calibration for Short Cracks Here we will look at the results for crack tip stress intensity factors for short cracks growing normal to the surface at the contact edges, first for incomplete contacts, Fig. 9.8d. The word ‘short’ does not have a unique meaning; in the case of fatigue crack propagation, ‘short’ means that the crack length is less than the critical distance, using denoted a0 (see Sect. 7.4) defining the nominal boundary between ‘long’ cracks where the crack tip stress intensity factor certainly controls the rate of propagation, and the long crack threshold, K 0 , holds. In this section, it has a different interpretation; we mean that the crack is sufficiently short for the stress field it experiences to be controlled by the contact edge stress intensity factors L I , K II , see Fig. 9.10a, together with any stress acting parallel with the free surface, that is remote bulk tension. With these assumptions, it means that the crack exists in an environment which may be thought of as either: (a) an infinite plane together with a semi-infinite ‘slit’ which represents the region external to the contacting bodies, the contacting interface is initially adhered, and dislocation solutions in this same domain may be used to permit both interfacial slip and crack slip and opening. This formulation may be used to allow for the influence of the presence of the crack on contact behaviour. Or, (b) a half-plane may be used with given surface tractions and solutions for edge dislocations in a half-plane used to represent crack opening and slip displacement. Here we will use the latter. The approximation involved will be extremely small, Fig. 9.10a. For details of the method behind using dislocations to insert a crack in an intact body, see Hills et al. (1996). The essential idea is to distribute edge dislocations, (a) climb in character, b y , and (b) glide in character, bx , to represent, respectively, the opening and shear displacements between the crack faces, Fig. 9.10b. The key is to obtain, first, the solution for a dislocation present in the domain being studied (here a half-plane) and, if the point where the state of stress to be evaluated lies on a line through the dislocation and perpendicular to the free surface the kernel takes on a particularly simple form, viz.

b)

a)

c)

KII r

L

fL r

r

fL r

r r d

y

l

l x

r d fL

r d

Fig. 9.10 Representation of the contact end with a the near edge asymptotic normal and shear tractions, b the insertion of a crack of length l perpendicular to the interface, and c the superposition of the asymptotic tractions for full sliding and a corrective term

9.7 Stress Intensity Factor Calibration for Short Cracks

179

  1 1 2ξ 4ξ 2 Ebi (ξ, 0)

− − σi y (x, 0) = + x +ξ (x + ξ )2 (x + ξ )3 4π 1 − ν 2 x − ξ Eb y (ξ, 0)

F (x, ξ ) i = x, y ≡ (9.12) 4π 1 − ν 2 where bi (ξ ) is the Burgers vector of the dislocation. Note that there is no coupling, i.e. the climb dislocation induces no shear traction and the glide dislocation induces no direct traction. If the tractions induced along the line θ = −π/2, y = 0 in the absence of the crack are  σi y (x, 0) and Bi (x) = dbi /dx represents the dislocation density, the total direct traction, N (x), and total shear traction, S(x), are given by N (x) =  σ yy (x, 0) +

E

4π 1 − ν 2



l

F(x, ξ )B y (ξ ) dξ

(9.13)

F(x, ξ )Bx (ξ ) dξ ,

(9.14)

0

and E

S(x) =  σx y (x, 0) + 4π 1 − ν 2



l

0

where l is the length of the crack, over which dislocations are distributed. If the crack is open, its faces must be traction free so that we may write N (x) = S(x) = 0

0 a. Hence, a further application of the Plemelj–Sokhotski formulae on (A.60) combined with (A.57) gives √  a P0 f (t) a 2 − t 2 − dt + √ g(x) = − √ t−x π a 2 − x 2 −a a2 − x 2 1

(A.61)

for −a < x < a. Similarly, the left-unbounded, right-bounded solution is given by φ(z) = −

1 2πi

so that 1 g(x) = − π





  z − a a a + t f (t) − dt , z + a −a a − t t − z

(A.62)

  a − x a a + t f (t) − dt a + x −a a − t t − x

(A.63)

for −a < x < a. The left-bounded, right-unbounded solution is given by 1 φ(z) = 2πi Therefore 1 g(x) = − π for −a < x < a.





  z + a a a − t f (t) − dt . z − a −a a + t t − z

(A.64)

  a + x a a − t f (t) − dt a − x −a a + t t − x

(A.65)

204

Appendix A: Plane Contacts: Mathematical Techniques

Table A.1 Reference table of inversions of singular integrals of the first kind (A.55) for different end conditions. Note in particular the consistency condition that appears in the bounded-both-ends solution. P0 is an arbitrary constant Type Solution √  a f (t) a 2 − t 2 1 P0 Unbounded-both-ends g(x) = − √ − dt + √ 2 2 2 t − x π a − x −a a − x2 1 a + x a a − t f (t) Left-bounded, right-unbounded g(x) = − − dt π  a − x −a  a + t t − x  a a + t f (t) 1 a−x Left-unbounded, right-bounded g(x) = − − dt π√ a + x −a a − t t − x  a a2 − x 2 f (t) g(x) = − dt − √ 2 π a − t 2 (t − x) −a  a Bounded-both-ends f (t) 0=− √ dt a2 − t 2 −a

Finally, the bounded-both-ends solution is given by √

so that

 z2 − a2 a f (t) − √ dt , φ(z) = − 2 2πi a − t 2 (t − z) −a

(A.66)

√  a2 − x 2 a f (t) − √ g(x) = − dt . 2 π a − t 2 (t − x) −a

(A.67)

This solution is only valid provided that the consistency condition  a f (t) − √ dt = 0 a2 − t 2 −a

(A.68)

is satisfied. Note that in the case of a symmetric indenter for the example described above, this will trivially be satisfied. However, in general this may not be the case and may, for example, give a condition for the contact size, a. We summarise these results for the reader’s convenience in Table A.1.

A.3.1.2

Singular Integral Equations of the Second Kind

A further important singular integral equation of relevance to contact mechanics is the so-called singular integral equation of the second kind, which for a constant γ takes the form  g(t) 1 dt (A.69) f (x) = γ g(x) + πi L t − x for x ∈ L, and we shall again take to be L = (−a, a).

Appendix A: Plane Contacts: Mathematical Techniques

205

We again appeal to the Cauchy-type integral (A.56), which, alongside the Plemelj– Sokhotski formulae (A.16)–(A.17) allows us to recast (A.69) as a Riemann–Hilbert problem of the form  φ+ (x) +

1−γ 1+γ

 φ− (x) =

f (x) 1+γ

(A.70)

for x ∈ L. This is again of the form (A.39) with α = (γ − 1)/(γ + 1), so that we take   γ −1 1 β= ln , (A.71) 2πi γ +1 where the branch of the logarithm is chosen so that 0 < Re(β) < 1. By following the same procedure as in the previous section, the unbounded-bothends solution is given by (z − a)β−1 (z + a)−β φ(z) = − 2πi(1 + γ )eiβπ



a −a

1 f (t) dt (a − t)β−1 (a + t)−β t − x

+ P0 (z − a)β−1 (z + a)−β

(A.72)

for some arbitrary constant P0 . The branch cuts are again taken along L. A further application of the Plemelj–Sokhotski formulae yields  1 f (t) γ f (x) (a − x)β−1 (a + x)−β a g(x) = 2 − dt β−1 (a + t)−β t − x γ −1 (γ 2 − 1)πi (a − t) −a (A.73) + P¯0 (a − x)β−1 (x + a)−β , where P¯0 is a new constant. To find the other three forms of possible solutions, the methodology is exactly the same as in the previous sections and is left as an exercise to the reader. A summary of the four possible solutions is presented in Table A.2.

206

Appendix A: Plane Contacts: Mathematical Techniques

Table A.2 Reference table of inversions of singular integrals of the second kind (A.69) for different end conditions, where β is given by (A.71). Again, the bounded-both-ends solution contains a consistency condition, whilst P¯0 is an arbitrary constant Type Unbounded-both-ends

Left-bounded, right-unbounded

Left-unbounded, right-bounded

Bounded-both-ends

Solution γ f (x) g(x) = P¯0 (a − x)β−1 (x + a)−β + 2 γ −1  (a − x)β−1 (a + x)−β a f (t) − dt (γ 2 − 1)πi −a (a − t)β−1 (a + t)−β (t − x) γ f (x) g(x) = 2 γ −1  f (t) (a − x)β−1 (a + x)1−β a dt − 2 (γ − 1)πi −a (a − t)β−1 (a + t)1−β (t − x) γ f (x) g(x) = 2 γ −1  f (t) (a − x)β (a + x)−β a − dt (γ 2 − 1)πi −a (a − t)β (a + t)−β (t − x) γ f (x) g(x) = 2 γ −1  (a − x)β (a + x)1−β a f (t) dt − 2 −a (a − t)β (a + t)1−β (t − x)  a (γ − 1)πi f (t) 0 = dt −a (a − t)β (a + t)1−β

Appendix B

The International Fretting Fatigue Symposia

[ISFF0] Standardisation of Fretting Fatigue Methods and Equipment, San Antonio, Texas, 12-13 November 1990. Proceedings published as ASTM, STP1159, (same title) ed H.M. Attia and R.B. Waterhouse, ASTM, Race Street, Philadelphia, PA, USA, (1992). [ISFF1] Fretting Fatigue, conference held in Sheffield, UK (1993) under the auspices of ESIS, proceedings have the same name, ed. R.B. Waterhouse and T.C. Lindley, pub. John Wiley, (1994). [ISFF2] Fretting Fatigue: Current Technology and Practices, University of Utah, Salt Lake City, Utah, 31 August - 2 September 1998. Proceedings published as ASTM, STP 1367, (same title) ed D.W. Hoeppner, V. Chandrasekaran and C. B. Elliott III, ASTM, Race Street, Philadelphia, PA, USA, (2000). [ISFF3] Third International Symposium on Fretting Fatigue, Nagaoka, Japan, May 15 -18, 2001. Proceedings published as ASTM, STP1425, Fretting fatigue, advances in basic understanding and applications, ed E. Kinyon, D.W. Hoeppner, Y. Mutoh, ASTM, Race Street, Philadelphia, PA, USA, (2003). [ISFF4] Fourth International Symposium on Fretting Fatigue held in the Laboratoire de Tribologie et Dynamique des Systemes, Lyon, France, May 26 - 28, 2004. Proceedings published as a special issue of Tribology International, ed P. Kapsa, L. Vincent, V. Fridici, (2006), 39, 10. [ISFF5] Fifth International Symposium on Fretting Fatigue held in the Aerospace Manufacturing Technology Centre, Montreal, April 21-23 2007. Proceedings published as a special issue of Tribology International, ed H. Attia, M. Meshreki, (2009), 42, 9. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. A. Hills and H. N. Andresen, Mechanics of Fretting and Fretting Fatigue, Solid Mechanics and Its Applications 266, https://doi.org/10.1007/978-3-030-70746-0

207

208

Appendix B: The International Fretting Fatigue Symposia

[ISFF6] Sixth International Symposium on Fretting (Fatigue) held in Chengdu, China, April 19-21, 2010. Proceedings published as a special issue of Tribology International, ed Z.-R. Zhou, M.H. Zhu, (2011), 44, 11. [ISFF7] Seventh International Symposium on Fretting Fatigue held in Christ Church Oxford 8th–11th April 2013 published as a special issue of Tribology International, ed D. Nowell and D.A. Hills, (2014), 76, 1–141(format change). [ISFF8] Eighth International Symposium on Fretting Fatigue held at the University of Brasília, Brazil, 17–20th April 2016, published as a special issue of Tribology International, ed J Aarujo, F.C. de Castro, E.N. Mamiya, T.C. Doca, (2017), 108, 1–201. [ISFF9] Ninth International Symposium on Fretting Fatigue held at the University of Seville1st - 3rd April 2019, published as a special issue of Tribology International, ed J. Vázquez, J. Domínguez, (2020), July 2020, 147, Editorial doi:106279

Appendix C

Fretting Fatigue Strength: Experimental Data

See Tables C.1, C.2, C.3, C.4, C.5, C.6.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 D. A. Hills and H. N. Andresen, Mechanics of Fretting and Fretting Fatigue, Solid Mechanics and Its Applications 266, https://doi.org/10.1007/978-3-030-70746-0

209

R [mm]

13

25

38

50

75

100

125

150

13

25

50

75

100

125

150

13

25

38

50

75

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0.54

0.36

0.27

0.18

0.09

1.08

0.9

0.72

0.54

0.36

0.18

0.09

1.14

0.95

0.76

0.57

0.38

0.28

0.19

0.1

a [mm]

143

143

143

143

143

143

143

143

143

143

143

143

157

157

157

157

157

157

157

157

p0 [MPa]

121

81

61

40

20

243

202

162

121

81

40

20

281

234

187

141

94

69

47

25

P [N/mm]

0.45

0.45

0.45

0.45

0.45

0.24

0.24

0.24

0.24

0.24

0.24

0.24

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

Q/ P [-]

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

σ mean [MPa]

109

73

55

36

18

116

97

78

58

39

19

10

253

211

169

127

84

62

42

22

Q [N/mm]

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

185

σ [MPa]

288

352

407

498

705

203

223

249

288

352

498

705

209

229

256

296

362

414

512

743

LI [MPa/mm1/2 ]

58

47

41

33

23

59

54

48

42

34

24

17

88

81

72

62

51

44

36

26

K II LHS [MPamm1/2 ]

8.00E+05

1.50E+06

4.04E+06

1.00E+07

1.00E+07

1.28E+06

1.22E+06

5.06E+06

1.00E+07

1.00E+07

1.00E+07

1.00E+07

6.70E+05

7.30E+05

8.50E+05

6.70E+05

1.29E+06

1.00E+07

1.00E+07

1.00E+07

N f [-]

267

178

133

89

44

389

324

259

194

130

65

32

564

470

376

282

188

141

94

47

d LHS [mm]

(continued)

0.06

0.06

0.06

0.06

0.06

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.06

0.06

0.06

0.06

0.06

0.06

0.06

0.07

r p/d LHS [-]

Table C.1 A summary of parameters and the observed number of cycles for fretting fatigue experiments employing Hertzian pads of radius R, conducted by Nowell (1988) for the aluminium alloy HE15-TF

210 Appendix C: Fretting Fatigue Strength: Experimental Data

R [mm]

100

125

150

13

25

50

75

100

125

25

38

50

75

100

125

150

No.

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

0.85

0.71

0.57

0.42

0.28

0.21

0.14

0.9

0.72

0.54

0.36

0.18

0.09

1.08

0.9

0.72

a [mm]

Table C.1 (continued)

120

120

120

120

120

120

120

143

143

143

143

143

143

143

143

143

p0 [MPa]

160

134

107

79

53

40

26

202

162

121

81

40

20

243

202

162

P [N/mm]

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

0.45

Q/ P [-]

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

σ mean [MPa]

144

120

97

71

48

36

24

182

146

109

73

36

18

218

182

146

Q [N/mm]

240

240

240

240

240

240

240

154

154

154

154

154

154

185

185

185

σ [MPa]

180

198

222

254

311

359

439

223

249

288

352

498

705

203

223

249

LI [MPa/mm1/2 ]

74

68

61

52

43

37

30

69

62

54

44

31

22

81

74

66

K II LHS [MPamm1/2 ]

1.23E+06

1.57E+06

1.00E+07

1.00E+07

1.00E+07

1.00E+07

1.00E+07

1.02E+06

1.42E+06

1.20E+06

1.00E+07

1.00E+07

1.00E+07

1.08E+06

1.24E+06

6.10E+05

N f [-]

549

458

366

275

183

137

92

413

331

248

165

83

41

533

444

356

d LHS [mm]

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.06

0.06

0.06

r p/d LHS [-]

Appendix C: Fretting Fatigue Strength: Experimental Data 211

R [mm]

127

127

127

121

121

229

229

127

178

178

127

127

127

178

229

178

229

127

178

178

229

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2

1.77

1.77

1.28

1.88

1.75

1.88

1.53

1.51

1.51

1.3

1.66

1.66

1.4

1.75

1.76

1.37

1.21

1.31

1.24

1.54

a [mm]

177

201

201

204

167

200

166

174

240

240

207

189

189

223

155

156

231

203

208

198

246

p0 [MPa]

557

558

558

410

494

551

490

418

569

569

423

493

494

490

427

429

497

384

427

385

595

P [N/mm]

0.24

0.21

0.21

0.52

0.32

0.34

0.32

0.38

0.31

0.31

0.35

0.27

0.27

0.23

0.37

0.43

0.31

0.35

0.31

0.28

0.22

Q/ P [-]

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

σ mean [MPa]

267

234

234

426

316

374

314

318

353

353

296

266

267

226

316

369

308

269

265

216

262

Q [N/mm]

164

170

170

232

171

226

194

172

204

204

177

200

200

170

226

223

221

201

221

169

221

σ [MPa]

182

220

220

262

176

218

176

204

284

284

264

213

213

274

170

170

284

267

265

258

287

LI [MPa/mm1/2 ]

83

80

80

131

93

117

99

95

109

109

94

92

92

78

107

115

105

94

97

77

96

K II LHS [MPamm1/2 ]

7.47E+05

7.49E+05

6.65E+05

4.65E+05

8.57E+05

4.56E+05

7.39E+05

5.83E+05

3.38E+05

5.45E+05

5.64E+05

4.34E+05

3.50E+05

6.68E+05

2.50E+05

2.38E+05

2.17E+05

2.41E+05

2.41E+05

4.22E+05

3.14E+05

N f [-]

707

558

558

770

815

821

861

718

589

589

548

666

666

440

966

1040

568

542

562

459

514

d LHS [mm]

(continued)

0.08

0.09

0.09

0.17

0.08

0.13

0.09

0.10

0.16

0.16

0.13

0.10

0.10

0.11

0.09

0.10

0.15

0.13

0.13

0.10

0.14

r p/d LHS [-]

Table C.2 A summary of parameters and the observed number of cycles for fretting fatigue experiments using Hertzian pads of radius R, conducted by Szolwinski et al. (Szolwinski and Farris 1998) for the aluminiuma alloy AL2024-T351

212 Appendix C: Fretting Fatigue Strength: Experimental Data

R [mm]

229

127

229

229

178

229

127

229

127

229

178

178

178

178

127

127

No.

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

1.5

1.31

1.53

1.69

1.53

1.65

2.01

1.4

1.87

1.49

1.99

1.79

1.74

1.73

1.4

2

a [mm]

Table C.2 (continued)

239

209

175

192

174

188

178

224

166

238

176

204

154

153

223

177

p0 [MPa]

560

430

421

509

419

486

563

494

487

557

551

571

419

417

489

557

P [N/mm]

0.27

0.33

0.38

0.34

0.36

0.27

0.24

0.36

0.33

0.27

0.34

0.31

0.26

0.31

0.35

0.25

Q/ P [-]

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

σ mean [MPa]

303

284

320

346

302

263

270

356

322

301

375

354

218

258

343

278

Q [N/mm]

171

194

221

213

195

169

196

196

222

218

219

199

166

162

218

164

σ [MPa]

284

265

204

215

204

212

182

274

176

283

181

221

169

169

274

182

LI [MPa/mm1/2 ]

93

95

107

109

97

85

92

109

107

102

114

107

76

82

111

85

K II LHS [MPamm1/2 ]

3.82E+05

3.12E+05

3.31E+05

2.26E+05

4.60E+05

6.21E+05

4.63E+05

4.64E+05

4.80E+05

2.54E+05

3.21E+05

5.52E+05

7.68E+05

8.67E+05

3.03E+05

7.30E+05

N f [-]

503

552

803

780

734

613

778

611

933

558

972

742

689

746

623

722

d LHS [mm]

0.13

0.13

0.11

0.12

0.10

0.09

0.08

0.15

0.09

0.15

0.11

0.12

0.07

0.07

0.15

0.08

r p/d LHS [-]

Appendix C: Fretting Fatigue Strength: Experimental Data 213

R [mm]

50

50

50

50

50

50

50

50

50

No.

1

2

3

4

5

6

7

8

9

0.46

0.46

0.46

0.46

0.46

0.46

0.46

0.46

0.46

a [mm]

136

136

136

136

136

136

136

136

136

p0 [MPa]

100

100

100

100

100

100

100

100

100

P [N/mm]

0.59

0.49

0.59

0.61

0.36

0.29

0.41

0.34

0.29

Q/ P [-]

121

121

113

105

88

74

74

63

55

σ mean [MPa]

160

133

160

166

98

79

111

92

79

Q [N/mm]

198

198

185

171

144

122

122

104

90

σ [MPa]

388

388

388

388

388

388

388

388

388

LI [MPa/mm1/2 ]

77

68

75

75

50

41

51

43

37

K II LHS [MPamm1/2 ]

8.66E+04

9.96E+04

1.15E+05

1.42E+05

2.46E+05

3.58E+05

4.20E+05

1.11E+06

1.41E+06

N f [-]

304

269

298

299

197

161

204

170

146

d LHS [mm]

0.10

0.09

0.10

0.10

0.06

0.05

0.07

0.06

0.05

r p/d LHS [-]

Table C.3 A summary of parameters and the observed number of cycles for fretting fatigue experiments using Hertzian pads of radius R, conducted by Hojati-Talemi et al. (Hojjati-Talemi et al. 2014) for the aluminium alloy AL2024-T3

214 Appendix C: Fretting Fatigue Strength: Experimental Data

R [mm]

8.2

4.9

4.9

4.9

4.9

4.9

4.9

4.9

4.9

4.9

4.9

13

13

13

13

13

13

13

13

13

8.2

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

4.19

4.34

4.34

4.34

4.34

4.25

4.25

4.25

4.25

4.25

4.13

4.13

4.13

4.13

4.13

4.13

4.13

4.13

4.13

4.13

4.19

a [mm]

565

665

665

665

665

478

478

478

478

478

665

665

665

665

665

665

665

665

665

665

565

p0 [MPa]

1200

1950

1950

1950

1950

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

P [N/mm]

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

Q/ P [-]

400

210

233

327

400

327

233

280

400

187

163

210

280

280

280

280

280

280

400

400

400

σ mean [MPa]

312

507

507

507

507

312

312

312

312

312

312

312

312

312

312

312

312

312

312

312

312

Q [N/mm]

229

120

133

187

229

187

133

160

229

107

93

120

160

160

160

160

160

160

160

229

229

σ [MPa]

2163

1867

1867

1867

1867

1575

1575

1575

1575

1575

3050

3050

3050

3050

3050

3050

3050

3050

3050

3050

2163

LI [MPa/mm1/2 ]

117

99

104

123

139

102

83

92

117

73

68

78

92

92

92

92

92

92

92

117

117

K II LHS [MPamm1/2 ]

6.12E+04

1.45E+06

8.19E+05

2.61E+05

6.23E+04

1.83E+05

1.00E+07

4.53E+05

8.81E+04

1.00E+07

1.00E+07

1.00E+07

2.11E+05

3.04E+05

2.15E+05

4.74E+05

3.45E+05

2.19E+05

1.23E+05

6.09E+04

5.56E+04

N f [-]

83

82

86

102

115

100

81

90

115

71

34

39

46

46

46

46

46

46

46

59

83

d LHS [mm]

(continued)

0.2

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

r p/d LHS [-]

Table C.4 A summary of parameters and the observed number of cycles for fretting fatigue experiments using symmetrical flat and rounded pads with flat portion 2b = 8 mm and radius R, conducted by Dini (Dini 2003) for the titanium alloy Ti6Al4V—part A

Appendix C: Fretting Fatigue Strength: Experimental Data 215

R [mm]

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

No.

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

4.22

4.22

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

a [mm]

Table C.4 (continued)

665

665

565

565

565

565

565

565

565

565

565

565

565

565

565

565

p0 [MPa]

1550

1550

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

1200

P [N/mm]

0.13

0.13

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.26

0.13

0.13

0.13

0.13

Q/ P [-]

327

373

233

93

140

187

210

373

327

280

233

257

233

257

280

327

σ mean [MPa]

403

403

624

624

624

624

624

624

624

624

624

624

312

312

312

312

Q [N/mm]

187

213

133

53

80

107

120

213

187

160

133

147

133

147

160

187

σ [MPa]

2355

2355

2163

2163

2163

2163

2163

2163

2163

2163

2163

2163

2163

2163

2163

2163

LI [MPa/mm1/2 ]

112

122

117

88

98

107

112

146

136

127

117

122

83

87

92

102

K II LHS [MPamm1/2 ]

1.43E+05

1.04E+05

2.69E+05

1.00E+07

2.07E+06

5.89E+05

4.29E+05

6.81E+04

1.25E+05

1.65E+05

7.54E+05

2.02E+05

1.00E+07

1.00E+07

2.31E+05

1.54E+05

N f [-]

73

79

83

63

69

76

80

104

97

90

83

87

59

62

66

72

d LHS [mm]

0.2

0.2

0.2

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.2

0.2

0.1

0.1

0.1

0.1

r p/d LHS [-]

216 Appendix C: Fretting Fatigue Strength: Experimental Data

R [mm]

8.2

8.2

8.2

8.2

8.2

13

13

13

4.9

4.9

4.9

4.9

4.9

4.9

13

13

No.

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

4.34

4.34

4.13

4.13

4.13

4.13

4.13

4.13

4.34

4.34

4.25

4.22

4.22

4.22

4.22

4.22

a [mm]

665

665

665

665

665

665

665

665

665

665

478

665

665

665

665

665

p0 [MPa]

1950

1950

1200

1200

1200

1200

1200

1200

1950

1950

1200

1550

1550

1550

1550

1550

P [N/mm]

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

0.13

Q/ P [-]

187

187

350

327

233

233

257

280

257

280

257

210

280

233

210

257

σ mean [MPa]

507

507

312

312

312

312

312

312

507

507

312

403

403

403

403

403

Q [N/mm]

107

107

200

187

133

133

147

160

147

160

147

120

160

133

120

147

σ [MPa]

1867

1867

3050

3050

3050

3050

3050

3050

1867

1867

1575

2355

2355

2355

2355

2355

LI [MPa/mm1/2 ]

94

94

106

102

82

82

87

92

109

114

87

88

102

93

88

97

K II LHS [MPamm1/2 ]

7.80E+05

4.98E+05

7.68E+04

1.21E+05

1.00E+07

4.17E+05

7.15E+05

2.31E+05

2.63E+05

2.17E+05

5.04E+06

1.00E+07

5.53E+05

6.17E+05

1.00E+07

2.81E+05

N f [-]

77

77

54

51

42

42

44

46

90

94

85

57

67

60

57

64

d LHS [mm]

(continued)

0.11

0.11

0.20

0.20

0.16

0.16

0.17

0.18

0.13

0.13

0.09

0.13

0.15

0.14

0.13

0.14

r p/d LHS [-]

Table C.5 A summary of parameters and the observed number of cycles for fretting fatigue experiments using symmetrical flat and rounded pads with flat portion 2b = 8 mm and radius R, conducted by Dini (Dini 2003) for the titanium alloy Ti6Al4V—part B

Appendix C: Fretting Fatigue Strength: Experimental Data 217

R [mm]

13

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

8.2

No.

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

4.19

4.19

4.19

4.19

4.22

4.22

4.22

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.19

4.34

a [mm]

Table C.5 (continued)

565

565

565

565

665

665

665

565

565

565

565

565

565

565

565

665

p0 [MPa]

1200

1200

1200

1200

1550

1550

1550

1200

1200

1200

1200

1200

1200

1200

1200

1950

P [N/mm]

0.26

0.26

0.26

0.26

0.13

0.13

0.13

0.39

0.13

0.13

0.39

0.39

0.39

0.39

0.39

0.13

Q/ P [-]

560

513

490

467

560

513

490

93

187

147

140

187

233

280

327

163

σ mean [MPa]

624

624

624

624

403

403

403

936

312

312

936

936

936

936

936

507

Q [N/mm]

320

293

280

267

320

293

280

53

1307

1026

80

107

133

160

187

93

σ [MPa]

2163

2163

2163

2163

2355

2355

2355

2163

2163

2163

2163

2163

2163

2163

2163

1867

LI [MPa/mm1/2 ]

184

175

170

165

160

151

146

122

507

406

132

142

151

161

170

89

K II LHS [MPamm1/2 ]

2.82E+04

3.43E+04

3.96E+04

4.06E+04

2.51E+04

3.64E+04

4.05E+04

6.01E+06

1.17E+04

3.08E+04

1.17E+06

4.16E+05

2.77E+05

1.40E+05

8.92E+04

1.00E+07

N f [-]

131

124

121

117

105

98

95

87

361

288

94

101

108

114

121

73

d LHS [mm]

0.25

0.24

0.23

0.22

0.24

0.22

0.22

0.17

0.69

0.55

0.18

0.19

0.21

0.22

0.23

0.10

r p/d LHS [-]

218 Appendix C: Fretting Fatigue Strength: Experimental Data

R [mm]

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

No.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

7.60

7.55

7.55

7.60

7.60

7.60

7.60

7.57

7.57

7.60

7.60

7.57

7.57

7.60

7.60

7.60

7.60

7.60

7.60

7.60

7.60

7.60

a [mm]

2867

1358

1358

2981

2981

2981

2981

2084

2084

2867

2867

1971

1971

2981

2981

2981

2981

2981

2981

2981

2981

2981

p0 [MPa]

6145

2000

2000

6516

6516

6516

6516

3805

3805

6145

6145

3500

3500

6516

6516

6516

6516

6516

6516

6516

6516

6516

P [N/mm]

0.03

0.09

0.09

0.02

0.03

0.03

0.02

0.09

0.09

0.06

0.06

0.10

0.10

0.03

0.05

0.03

0.03

0.05

0.06

0.06

0.06

0.06

Q/ P [-]

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

166

σ mean [MPa]

350

350

350

200

350

350

200

720

720

720

720

720

720

350

600

350

450

600

720

720

720

720

Q [N/mm]

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

271

σ [MPa]

15122

10404

10404

15420

15420

15420

15420

12890

12890

15122

15122

12537

12537

15420

15420

15420

15420

15420

15420

15420

15420

15420

LI [MPa/mm1/2 ]

161

160

160

148

161

161

148

191

191

191

191

191

191

161

181

161

169

181

191

191

191

191

K II LHS [MPamm1/2 ]

3.48E+05

2.82E+05

9.00E+05

7.96E+05

6.71E+05

5.05E+05

2.22E+05

2.46E+05

3.62E+05

4.34E+05

2.39E+05

2.11E+05

6.65E+05

3.98E+05

6.05E+05

5.09E+05

4.25E+05

2.97E+05

4.30E+05

2.85E+05

3.56E+05

2.97E+05

N f [-]

15

22

22

14

15

15

14

21

21

18

18

22

22

15

17

15

16

17

18

18

18

18

d LHS [mm]

(continued)

3.54

2.43

2.43

3.34

3.61

3.61

3.34

3.59

3.59

4.21

4.21

3.49

3.49

3.61

4.07

3.61

3.80

4.07

4.29

4.29

4.29

4.29

r p/d LHS [-]

Table C.6 A summary of parameters and the observed number of cycles for fretting fatigue experiments using symmetrical flat and rounded pads with flat portion 2b = 15 mm and radius R, conducted by Blades (Blades et al. 2020) for steel AISI8630

Appendix C: Fretting Fatigue Strength: Experimental Data 219

R [mm]

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

1.5

No.

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

7.59

7.58

7.58

7.59

7.59

7.59

7.58

7.58

7.58

7.58

7.60

7.60

7.60

7.60

7.60

7.60

7.55

7.55

7.60

a [mm]

Table C.6 (continued)

2663

2330

2330

2499

2499

2499

2154

2154

2272

2272

2981

2981

2981

2981

2981

2981

1313

1313

2867

p0 [MPa]

5500

4500

4500

5000

5000

5000

4000

4000

4333

4333

6516

6516

6516

6516

6516

6516

1903

1903

6145

P [N/mm]

0.03

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.09

0.09

0.06

0.06

0.06

0.06

0.09

0.09

0.03

Q/ P [-]

166

166

166

166

166

166

166

166

166

166

110

110

138

138

110

110

166

166

166

σ mean [MPa]

350

350

350

350

350

350

350

350

350

350

1200

1200

720

720

720

720

350

350

350

Q [N/mm]

271

271

271

271

271

271

271

271

271

271

180

180

225

225

180

180

271

271

271

σ [MPa]

14573

13631

13631

14118

14118

14118

13107

13107

13461

13461

15420

15420

15420

15420

15420

15420

10232

10232

15122

LI [MPa/mm1/2 ]

161

161

161

161

161

161

160

160

161

161

186

186

168

168

147

147

160

160

161

K II LHS [MPamm1/2 ]

4.80E+05

4.76E+05

4.02E+05

3.86E+05

3.45E+05

6.02E+05

5.79E+05

4.01E+05

5.39E+05

4.91E+05

6.63E+05

7.20E+05

8.50E+05

2.14E+06

1.35E+06

2.99E+05

2.56E+05

7.12E+05

8.09E+05

N f [-]

16

17

17

16

16

16

17

17

17

17

17

17

16

16

14

14

22

22

15

d LHS [mm]

3.41

3.19

3.19

3.31

3.31

3.31

3.07

3.07

3.15

3.15

4.18

4.18

3.79

3.79

3.29

3.29

2.39

2.39

3.54

r p/d LHS [-]

220 Appendix C: Fretting Fatigue Strength: Experimental Data

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