ISBN: 3540429069
TITLE: Computational Contact and Impact Mechanics
AUTHOR: Laursen
TOC:

Preface vii
1 Introduction 1
1.1 Scope of this Monograph 2
1.2 Useful Background for this Presentation 3
1.3 Overview 4
2 Finite Element Formulations in Nonlinear Solid Mechanics 7
2.1 Initial/Boundary Value Problems in the Kinematically Linear Regime 8
2.1.1 Strong Form of the IBVP 9
2.1.2 Weak Form of the IBVP 15
2.2 The IBVP in the Finite Strain Case 17
2.2.1 Notation and Problem Formulation 17
2.2.2 Finite Strain Kinematics 18
2.2.3 Stress Definitions Appropriate for Large Deformations 24
2.2.4 Frame Indifference 27
2.2.5 The Strong Form in Finite Strains 31
2.2.6 The Weak Form in Finite Strains 39
2.3 Finite Element Discretization 41
2.3.1 Discretized Weak Form; Generation of Discrete Non-linear Equations 43
2.3.2 Discrete Nonlinear Equations for the Kinematically Linear Case 46
2.4 Solution Strategies for Spatially Discrete Systems 48
2.4.1 Quasistatics and Incremental Load Methods 48
2.4.2 Dynamics and Global Time Stepping Procedures 50
2.4.3 Local (Constitutive) Time Stepping Procedures 54
2.4.4 Nonlinear Equation Solving 56
2.4.5 Consistent Algorithmic Linearization of Material Response 61
3 The Kinematically Linear Contact Problem 69
3.1 Strong Forms in Linearized Frictionless Contact 70
3.1.1 The Signorini Problem: Contact with a Rigid Obstacle 70
3.1.2 The Two Body Contact Problem 75
3.2 Weak Statements of the Contact Problem 79
3.2.1 Variational Inequalities 81
3.2.2 The Quasistatic Elastic Case: Contact as a Problem of Constrained Optimization 83
3.3 Methods of Constraint Enforcement 85
3.3.1 Classical Lagrange Multiplier Methods 85
3.3.2 Penalty Methods 89
3.3.3 Augmented Lagrangian Methods 91
3.4 Inclusion of Friction into the Problem Description 94
3.4.1 Friction Kinematics and Traction Measures 94
3.4.2 Unregularized Coulomb Friction Laws 96
3.4.3 Regularization of Friction 98
3.4.4 Variational Statements Including Friction 101
3.4.5 Nonlocal Frictional Descriptions 106
4 Continuum Mechanics of Large Deformation Contact 109
4.1 Two Body Contact Problem Definition 110
4.1.1 Local Momentum Balances 111
4.1.2 Initial and Boundary Conditions 112
4.2 Contact Constraints in Large Deformations 113
4.2.1 The Gap Function as Defined by Closest Point Projection 113
4.2.2 Frictional Kinematics on Interfaces 116
4.2.3 Frame Indifference of Contact Rate Variables 121
4.2.4 Coulomb Friction in Large Sliding 129
4.3 Summary: Strong Form of the Large Deformation Contact Problem 134
4.4 Virtual Work Expressions Incorporating Contact 137
4.4.1 Contact Virtual Work: The Contact Integral 139
4.4.2 Linearization of Contact Virtual Work 141
4.4.3 Summary: Weak Form of the Large Deformation Contact Problem 144
5 Finite Element Implementation of Contact Interaction 145
5.1 Finite Dimensional Representation of Contact Interaction 147
5.1.1 Contact Surface Discretization 147
5.1.2 Numerical Integration of the Contact Integral 148
5.1.3 Contact Detection (Searching) 152
5.2 Time Discretization 158
5.2.1 Global time integration schemes 158
5.2.2 Temporally Discrete Frictional Laws for the Penalty Regularized Case 159
5.3 Contact Stiffness and Residual: Penalty Regularized Case 162
5.3.1 Three dimensional matrix expressions 162
5.3.2 Two dimensional matrix expressions 166
5.4 Augmented Lagrangian Constraint Enforcement Algorithms 169
5.4.1 Uzawa's Method (Method of Multipliers) 170
5.4.2 Algorithmic Symmetrization Using Augmented Lagrangians 174
5.4.3 Augmented Lagrangian Discrete Force and Stiffness Expressions 178
5.5 Numerical Examples 180
5.5.1 General Demonstrations of the Computational Framework 180
5.5.2 Demonstrations of Augmented Lagrangian Algorithmic Performance 196
6 Tribological Complexity in Interface Constitutive Models 211
6.1 Rate and State Dependent Friction 212
6.1.1 Motivation 213
6.1.2 One Dimensional Model Development 215
6.1.3 Model Incorporation into Convective Slip Advected Frame 220
6.1.4 Local Time Stepping Algorithm 222
6.1.5 Contact Force Vector and Stiffness Matrix 226
6.1.6 Numerical Examples 227
6.2 Thermomechanically Coupled Friction on Interfaces 238
6.2.1 Motivation 239
6.2.2 Thermally Coupled Problem Definition 241
6.2.3 A Thermodynamically Consistent Friction Model 244
6.2.4 Variational Principle and Finite Element Implementation 255
6.2.5 Numerical Examples 269
6.3 Thermodynamical Algorithmic Consistency 279
6.3.1 Constitutive Framework for Bulk Continua 280
6.3.2 Thermomechanical Interface Model Framework 283
6.3.3 A Priori Stability Estimates for Dynamic Frictional Contact 286
6.3.4 A New Partitioned Scheme for Thermomechanical Contact 289
6.3.5 Algorithmic Treatment of Contact Conditions According to the Adiabatic Split 291
7 Energy-Momentum Approaches to Impact Mechanics 295
7.1 Energy Stability of Traditional Schemes 297
7.1.1 A Model System 297
7.1.2 The Concept of Energy Stability 299
7.1.3 Influence of Contact Constraints on System Energy 300
7.2 Energy-Momentum Methods for Elastodynamics 304
7.2.1 Conservation Laws 305
7.2.2 Conservative Discretization Schemes 309
7.3 Energy-Momentum Algorithmic Treatment of Frictionless Impact 312
7.3.1 Discrete Contact Constraints 313
7.3.2 Spatial Discretization and Implementation 316
7.3.3 Numerical Examples 318
7.4 Introduction of Frictional and Bulk Dissipation: Energy Consistency 325
7.4.1 Coulomb Friction Model Formulation 325
7.4.2 Local Split of the Coulomb Model 331
7.4.3 Algorithmic Formulation 332
7.4.4 Energy Consistent Treatment of Bulk Inelasticity 338
7.4.5 Numerical Examples With Friction and Inelasticity 339
7.5 EM Algorithms Involving a Discontinuous Velocity Update 347
7.5.1 Temporally Discontinuous Velocity Update 348
7.5.2 Reexamination of Conservation Conditions 350
7.5.3 Contact Constraints 355
7.5.4 Summary of the Algorithm 357
7.5.5 Numerical Examples 357
8 Emerging Paradigms for Contact Surface Discretization 369
8.1 Contact Smoothing 371
8.1.1 An Alternative Variational Framework 372
8.1.2 Smoothing Strategies in Two Dimensions 374
8.1.3 Smoothing Strategies in Three Dimensions 382
8.1.4 Numerical Examples 390
8.2 Mortar-Finite Element Methods for Contact Description 404
8.2.1 Tied Contact and the Bole of Mortar Formulations in Convergence 404
8.2.2 A Mortar-Finite Element Formulation of Frictional Contact 416
8.2.3 Numerical Examples of Mortar Treatment of Frictional Contact 425
References 435
Index 451
END
