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

This book, which has only one very distant forerunner authored by David A. Hills with David Nowell, represents a very big step that is the quantification of these problems and represents the twenty-five years’ worth of work which have gone on at Oxford since the first book on the subject. Fatigue (popularly ‘metal fatigue’) is the primary failure mode of all machines, engines, transmissions and indeed almost all mechanical devices. The propagation of cracks is well understood and is treated in the subject Fracture Mechanics. By contrast, the nucleation of cracks is very hard to quantify and this remains the case with so-called ‘free initiation’ and, to a lesser extent, at cracks nucleated from stress raising features. But the third form of nucleation, where cracks start from the edges of rubbing components, that is, at joints, is potentially a very much better-defined environment, and therefore, the problem is amendable to attack by applied mechanics and experiment. The contents are of value both to those embarking on research on the subject and to practitioner in industry. Preface 6 Acknowledgements 8 Contents 9 1 Some Fundamentals 12 1.1 Fretting in Practice 12 1.2 Basics, Equilibrium and `Coupling' 14 1.3 Friction 16 1.4 Contact Requirements 17 1.5 Classes of Contact 17 1.6 Methods of Solution 22 1.7 Shakedown 23 1.8 Three-Dimensional Aspects 23 2 Plane Elasticity and Half-Plane Contacts 25 2.1 Airy Stress Functions and the Half-Plane 25 2.2 Integral Equation Formulation 29 2.3 Solution 31 2.3.1 Cauchy Equations of the First Kind 32 2.3.2 Cauchy Equations of the Second Kind 34 2.3.3 Numerical Solutions 34 2.4 Mossakovskii–Barber Procedure 35 2.4.1 Normal Loading Problem 35 2.4.2 Application of Shear Force 38 2.4.3 Influence of Bulk Tension 39 2.4.4 Application of a Moment 41 2.5 Conditions for Full Stick 43 2.6 Solutions Based on Dislocations 44 2.7 Summary 46 3 Williams' Solution 48 3.1 Introduction 48 3.2 Anti-Plane Loading 54 3.3 General Loading 56 3.4 Crack Tip and Incomplete Contact-Edge Solution 57 3.5 Bonded Wedges 57 3.6 Sliding Wedges 59 3.7 Summary 61 4 Half-Plane Partial Slip Contact Problems 63 4.1 Introduction 63 4.2 The Normal Load Problem for Asymmetrical Contacts 65 4.3 The Sequence of Loading 67 4.3.1 Sequential Loading (Constant Normal Load) 68 4.3.2 Proportional Loading 71 4.3.3 Two-Stage Proportional Loading 73 4.3.4 Application to a Hertzian Geometry 74 4.4 The Effect of Differential Bulk Tension 77 4.4.1 Tangential Load and Moderate Differential Bulk Tension 77 4.4.2 Bulk Tension Dominated Partial Slip Problems 79 4.5 Periodic Loading 79 4.6 More General Loading Scenarios 82 4.7 General Cyclic Proportional Loading 85 4.7.1 The Permanent Stick Zone 86 4.7.2 Mapping Between the Normal and Tangential Problems 90 4.7.3 Application to the Tilted Shallow Wedge 90 4.8 Partial Slip Solutions Based on Dislocations 92 4.8.1 The Insertion of Glide Dislocations 93 4.8.2 Solution (Constant Normal Load) 95 4.8.3 Shear Force Only 97 4.8.4 Differential Tension Only 97 4.8.5 Cyclic Loading (Constant Normal Load) 98 4.8.6 Application to a Hertzian Geometry 102 4.8.7 Cyclic Loading (Varying Normal Load) 106 4.9 Anti-Plane Formulation 110 4.10 Influence of a Screw Dislocation 110 4.11 Correction to a Fully Adhered Solution 111 4.12 Application to a Hertz' Problem 112 4.13 Summary 114 5 Complete Contacts and Their Behaviour 115 5.1 Introduction 115 5.2 General Frictional Response—Square Contacting Element 119 5.3 Finite Slip Zones 125 5.4 Sliding Asymptote (Bilateral) 127 6 Representation of Half-Plane Contact Edge Behaviour by Asymptotes 132 6.1 Introduction 132 6.2 Basic Solution to the Normal Problem 133 6.3 Basic Solution to the Tangential Problem 142 6.4 Partial Slip Under Constant Normal Load 143 6.5 Partial Slip Under Varying Normal Load 145 6.5.1 Asymptotic Description for Steady-State Problems 145 6.6 Summary 147 7 Crack Propagation, Nucleation and Nucleation Modelling 149 7.1 Introduction 149 7.2 Notch and Critical Distance Methods 150 7.3 Critical Plane Methods 152 7.4 Short Crack Methods 154 7.5 Wear and Corrosion 156 8 Experiments to Measure Fretting Fatigue Strength 157 8.1 Fundamental and Historic Considerations 157 8.2 Single Actuator Experimental Apparatus 160 8.3 Two Actuator Experimental Apparatus 162 8.4 Further Developments 164 8.5 Concluding Remarks 166 9 Fretting Strength 167 9.1 Introduction 167 9.2 Asymptotic Representation 168 9.3 Frictional and Plastic Slip 171 9.4 Experimental Data 172 9.5 Interpreting the Measured Fretting Fatigue Strength 174 9.6 The Asymptotic Descriptions of Contact Edges 180 9.7 Stress Intensity Factor Calibration for Short Cracks 184 9.7.1 Adhered Contact Edge 185 9.7.2 Slipping Contact Edge 186 9.7.3 Worn-Away Contact Edge 188 9.8 Cracks at Complete Contact Edges 190 9.8.1 Application of Calibration of Contact Edge Short Crack Stress Intensity Factors 192 9.9 Calibration Against Finite Contacts 193 Appendix A Plane Contacts: Mathematical Techniques 196 A.1 Complex Variable Preliminaries 196 A.1.1 Cauchy's Formula 197 A.1.2 The Hölder Condition 198 A.1.3 The Principal Value of a Singular Integral 198 A.1.4 The Plemelj–Sokhotski Formulae 199 A.1.5 Index 200 A.2 The Riemann–Hilbert Problem for a Closed Contour with Continuous Coefficients 201 A.2.1 Homogeneous Problem 201 A.2.1.1 κ= 0 201 A.2.1.2 κ> 0 202 A.2.2 Inhomogeneous Problem 203 A.2.3 A Note on Discontinuous α(x) and g(x) 204 A.3 Riemann–Hilbert Problems in Half-Plane Theory 204 A.3.1 Inverting Singular Integral Equations 207 A.3.1.1 Singular Integral Equations of the First Kind 207 A.3.1.2 Singular Integral Equations of the Second Kind 209 Appendix B The International Fretting Fatigue Symposia 212 Appendix C Fretting Fatigue Strength: Experimental Data 214 Appendix References 226 226