Three-Dimensional Elastic Bodies in Rolling Contact Book

Preface XIII

Introduction XV

Notation XXI

Chapter 1 The Rolling Contact Problem 1

1.1 Statement of the problem 4

1.2 Mathematical modeling of the contact formation 6

1.3 Mathematical modeling of the slip 10

1.4 Mathematical modeling of friction 18

1.5 The complete boundary conditions 20

1.6 The half-space approximation 22

1.6.1 Many geometries are elastically alike 23

1.6.2 A(x,y) may be calculated exactly

1.6.3 Quasiidentity is common in half-space problems 24

1.6.3.1 The Panagiotopoulos process 24

1.6.3.2 An Alternative to the Panagiotopoulos process (KOMBI) 24

1.6.3.3 The Johnson process 25

1.6.3.4 Symmetry and quasiidentity 25

1.6.3.5 Mindlin's method 28

1.6.4 Exact three-dimensional solutions of contact problems 28

1.7 Boundary conditions for some applications 28

1.7.1 The Hertz problem 28

1.7.2 Frictionless or quasiidentical contact formation for concentrated or semi-concentrated non-Hertzian contact 35

1.7.3 Frictional boundary conditions for bodies of revolution with the axes almost in one plane 41

1.7.3.1 Concentrated, e.g. Hertzian, geometry 44

1.7.3.2 A ball rolling in a conforming groove 45

Chapter 2 Review 47

2.1 Frictionless contact 48

2.1.1 Element methods 49

2.1.1.1 Fridman and Chernina 50

2.1.1.2 Later authors 51

2.1.1.3 Influence Function Methods for the half-space: choice of elements 52

2.1.1.4 The accuracy of the elements 56

2.1.1.5 Conclusion 59

2.2 Elastic rolling contact 59

2.2.1 Carter and Fromm 59

2.2.2 The no-slip theory of rolling contact 64

2.2.2.1 Comparison of Johnson's spin theory with the exact values 66

2.2.2.2 Comparison of Vermeulen and Johnson's no-spin theory with the exact values 67

2.2.2.3 Calculation of the exact values of the C_ij by separating the variables in Laplace's equation 68

2.2.2.4 Calculation on the basis of a generalisation of Galin's Theorem 69

2.2.2.5 Strip theory/line contact theory 71

2.2.2.6 IF methods for the half-space 73

2.2.3 Nonlinear, finite friction rolling contact 74

2.2.3.1 Johnson and Vermeulen-Johnson 74

2.2.3.2 Strip theory 78

2.2.3.3 Simplified theory 80

2.2.3.4 The first exact theory 82

2.2.3.5 A linear programming method for the two-dimensional case 84

2.2.3.6 Generalisation of the method of Sec. 2.2.3.5 to the three-dimensional case 94

2.2.3.7 Duvaut-Lions based methods 95

Chapter 3 The Simplified Theory of Contact 99

3.1 Recapitulation of the linear theory of elasticity 100

3.2 The thin elastic layer 101

3.3 Validation by frictionless contact 103

3.3.1 Comparison with the theory of Meijers 103

3.3.2 Comparison with the Hertz theory 107

3.3.3 Conclusion 112

3.4 Frictional compression 112

3.5 The FASTSIM algorithm 117

3.6 The shift 119

3.6.1 ψ & = 0, w = (L₁,0)^T, elliptic contact 120

3.6.2 ψ = L₁, w = 0 122

3.7 Steady state rolling contact 122

3.7.1 The full adhesion solution 123

3.7.2 Finite friction coefficient 126

3.8 Transient rolling contact 133

3.9 An alternative method to find the L_i 133

3.10 Conclusion of tangential simplified theory 134

Chapter 4 Variational and Numerical Theory of Contact 137

4.1 The principle of virtual work and its dual for contact problems 138

4.1.1 Virtual work 138

4.1.2 Complementary virtual work 144

4.2 Application to elasticity 148

4.2.1 Minimality of the potential energy, maximality of the complementary energy, and uniqueness of the solution 150

4.2.2 The case δg ≠ 0 154

4.2.3 Existence-uniqueness theory 156

4.2.4 Surface mechanical principles 157

4.2.5 Complementary energy or potential energy in numerical work? 158

4.3 Implementation 158

4.3.1 The basic algorithm 160

4.3.2 Discretisation of the contact problem 168

4.3.3 The algorithm of 4.3.1 applied to half-space problems 172

4.3.4 Steady state rolling, elastic and viscoelastic 181

4.3.5 Prescription of total force components 181

4.3.6 Sensitivities 182

4.3.7 Calculation of the influence numbers in a half-space 183

4.3.8 The subsurface elastic field in a half-space 184

4.3.9 Note on the generalisation to non-concentrated contacts 184

Chapter 5 Results 185

5.1 The normal contact problem 186

5.1.1 Validation (normal contact) 188

5.1.2 New results achieved by RNJLK and CC 193

5.2 Quasiidentical frictional contact problems 202

5.2.1 Validation 203

5.2.1.1 The Cattaneo shift 203

5.2.1.2 The Mindlin shift 205

5.2.1.3 The creepage and spin coefficients for steady state rolling 206

5.2.1.4 The theory of Vermeulen-Johnson on steady state rolling and its generalizations 207

5.2.1.5 The Vermeulen-Johnson theory and its generalisations: Validation 213

5.2.1.6 Brickie's experiments compared with CONTACT and FASTSIM 214

5.2.2 New results in Hertzian frictional rolling contact 215

5.2.2.1 The total tangential force 216

5.2.2.2 The areas of adhesion and slip 218

5.2.2.3 Surface tractions 219

5.2.2.4 Subsurface stresses 221

5.2.2.5 Transient rolling contact 224

5.2.2.6 Some remarks on corrugation 229

5.3 Non-quasiidentical frictional contact problems 231

5.3.1 Validation 231

5.3.2 New results 233

5.3.2.1 Unloading the Spence compression 233

5.3.2.2 Transition from the Spence compression to steady state rolling 235

Chapter 6 Conclusion 237

Appendix A The basic equations of the linear theory of elasticity 239

Appendix B Some notions of mathematical programming 245

Appendix C Numerical calculation of the elastic field in a half-space 255

Appendix D Three-dimensional viscoelastic bodies in steady state frictional rolling contact with generalisation to contact perturbations 265

Appendix E Tables 285

Bibliography 295

Index 307