Contact in Structural Mechanics

Anh Le van is Professor of Structural Mechanics in the Faculty of Science and Technology, University of Nantes, France. His research at the Research Institute in Civil and Mechanical Engineering (GeM) focuses on membrane structures and, more specifically, on contact and bifurcation problems in these structures.

Preface ix

1 Introduction to Contact Problems in Structural Mechanics 1

1.1 Solving a contact problem numerically via the penalty method 3

1.2 Numerical solution of a contact problem using the multiplier method 8

1.2.1 Preliminaries: problems with equality constraints 8

1.2.2 Problems with inequality constraints 10

1.3 Numerical solution of a contact problem by the augmented Lagrangian method 15

1.4 Book synopsis 21

2 Contact Kinematics 23

2.1 Motions and strains 23

2.2 Potential contact surfaces 25

2.3 Normal contact kinematics 26

2.4 Variation of kinematic quantities with respect to time 29

2.5 Tangential contact kinematics – Relative velocity 34

3 Sthenics of Contact 37

3.1 Stresses in bodies 37

3.2 Contact stress vector 38

4 The Constitutive Law 39

4.1 Hyperelastic materials 39

4.2 Elastoplastic materials with isotropic hardening 41

5 Contact Laws 45

5.1 Normal contact law 45

5.2 Tangential contact law 47

6 Strong Formulation of the Contact Problem 51

6.1 Field equations 51

6.2 Boundary conditions 52

6.3 Initial conditions 53

6.4 Remarks 53

7 Weak Formulation of the Contact Problem 55

7.1 Transforming the contact laws into equalities 55

7.2 Preliminary ideas for the weak form 59

7.3 Weak form of the contact problem 60

7.4 Equivalence between the strong and the weak forms 62

7.5 Final remarks 66

8 Matrix Equations of the Contact Problem 69

8.1 Introduction 69

8.2 Meshes 70

8.3 Matrix notation in finite elements 72

8.4 The element nodal vectors 73

8.5 Interpolation of positions, displacements and virtual velocities 75

8.5.1 Interpolation on the contactor surface 75

8.5.2 Interpolation on the target surface 75

8.6 Interpolation of multipliers 76

8.6.1 Definition of the vector λ 76

8.6.2 Interpolation of λ 78

8.6.3 Interpolation of λ∗ 78

8.7 Discretization of the element virtual contact power (P∗contact)e(1) 78

8.7.1 Explicit expressions for {Φe(1)contact}, {Φe(2)contact} and {Re(1)Λ } in the three cases: algorithmic gap, algorithmic slip and algorithmic stick 85

8.8 System of matrix equations for the contact problem 88

8.8.1 Global nodal vectors 88

8.8.2 Discretization of the classical terms 90

8.8.3 Assembly of element virtual contact powers 91

8.8.4 System of matrix equations 94

8.9 Abnormal contact stresses 96

8.9.1 First cause of abnormal contact stresses 96

8.9.2 Second cause of abnormal contact stresses 98

8.9.3 Third cause of abnormal contact stresses 98

8.10 Projection calculation: contact detection 99

8.11 Discrete expression of the slip VTΔt 101

8.12 Physical units 106

8.13 Chapter summary 107

9 Solution of the Quasi-static Contact Problem 109

9.1 System of equations for the static contact problem 109

9.2 Incremental loop initialization: the vectors U0, Λ0 111

9.3 Calculation of step n ≥ 1: calculating Un , Λn 111

9.3.1 Principle of the iterative Newton–Raphson scheme 111

9.3.2 Tangent matrix 113

9.3.3 Block matrix inversion 114

9.3.4 Iterative loop initialization: the vectors U0n and Λ0n 115

9.4 Solution algorithm 115

9.5 Calculation method for the tangent matrix 117

9.5.1 Direct method 117

9.5.2 Indirect method 118

9.5.3 Restriction to the contact tangent matrix 121

9.6 Calculation of the contact tangent matrix 123

9.6.1 Variations of the arguments of the functional P∗contact 123

9.6.2 Calculation of the variation δP∗contact 126

9.6.3 Calculation of the variation (δP∗contact)e(1) 127

9.6.4 Discretization of the variation (δP∗contact)e(1) – Element contact tangent matrix [Kecontact] 133

9.6.5 Discretization of the variation δP∗contact – Contact tangent matrix [Kcontact] 135

9.6.6 Explicit expression for the element contact tangent matrix [Kecontact] 138

9.6.7 [Kecontact] in the case of the algorithmic gap at the considered integration point 144

9.6.8 [Kecontact] in the case of algorithmic contact with slip at the considered integration point 144

9.6.9 [Kecontact] in the case of algorithmic contact with stick at the considered integration point 146

9.6.10 Symmetry of the contact tangent matrix [Kcontact] 147

9.7 Particular case of two non-contacting bodies 148

9.8 Particular case of the frictionless problem 149

9.8.1 Algorithmic gap case at the considered integration point 150

9.8.2 Algorithmic contact with slip case at the considered integration point 152

9.9 Solution via the arc-length method 152

9.10 Physical units 154

9.11 Summary of the chapter 155

10 Numerical Examples of Quasi-static Contact 157

10.1 Contact patch test 157

10.2 Hertzian contact problem 159

10.2.1 Frictionless contact case 160

10.2.2 Case of frictional contact with μ = 0.3 163

10.3 Rolling disk 167

10.4 Contact between two beams 171

10.4.1 Dead load 171

10.4.2 Follower load 176

10.5 Contact of two pressurized membranes 176

10.5.1 Centered membranes 180

10.5.2 Membranes Staggered Along X 182

10.6 Extrusion of an elastoplastic cylinder 184

10.7 Interference fit problem 189

10.7.1 Abnormal contact stresses 192

10.7.2 Influence of the mesh 194

10.8 Conclusion 194

11 Solution of the Dynamic Contact Problem 197

11.1 A brief review of the computational methods in dynamic contact 197

11.2 Solution of the dynamic contact problem via Newmark’s algorithm 200

11.2.1 Initializing the incremental loop: the vectors U0 , V0 , A0 and Λ0 202

11.2.2 Calculation for a step n ≥ 1: calculating Un , Vn , An , Λn 202

11.2.3 Initializing the iterative loop: the vectors U0n, V0n, A0n, Λ0n 207

11.3 Solution algorithm 208

11.4 Summary 210

12 Numerical Examples of Dynamic Contact 213

12.1 Impact of two elastic rods 213

12.1.1 Analytical solution 214

12.1.2 Numerical applications 217

12.1.3 Numerical solution 218

12.2 Disk impacting a rigid plane 220

12.2.1 Frictionless case 222

12.2.2 Case with friction μ = 0.3 224

12.3 Disk falling into a funnel 228

12.3.1 Frictionless case 231

12.3.2 Case with friction μ = 0.4 234

12.4 Final remarks 236

Appendix A: Variations of Kinematic Quantities 239

References 247

Index 257