Engineering Online Library

Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis

By: Tod A. Laursen

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Hardcover | 12 May 2003

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476 Pages

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Many physical systems require the description of mechanical interaction across interfaces if they are to be successfully analyzed. Examples in the engineered world range from the design of prosthetics in biomedical engi neering (e. g. , hip replacements); to characterization of the response and durability of head/disk interfaces in computer magnetic storage devices; to development of pneumatic tires with better handling characteristics and increased longevity in automotive engineering; to description of the adhe sion and/or relative slip between concrete and reinforcing steel in structural engineering. Such mechanical interactions, often called contact/impact in teractions, usually necessitate at minimum the determination of areas over which compressive pressures must act to prevent interpenetration of the mechanical entities involved. Depending on the application, frictional be havior, transient interaction of interfaces with their surroundings (e. g. , in termittent stick/slip), thermo-mechanical coupling, interaction with an in tervening lubricant and/or fluid layer, and damage of the interface (i. e. , wear) may also be featured. When taken together (or even separately!), these features have the effect of making the equations of mechanical evolu tion not only highly nonlinear, but highly nonsmooth as well. While many modern engineering simulation packages possess impressive capabilities in the general area of nonlinear mechanics, it can be contended that methodologies typically utilized for contact interactions are relatively immature in comparison to other components of a nonlinear finite element package, such as large deformation kinematics, inelastic material modeling, nonlinear equation solving, or linear solver technology.

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Preface	p. vii
Introduction	p. 1
Scope of this Monograph	p. 2
Useful Background for this Presentation	p. 3
Overview	p. 4
Finite Element Formulations in Nonlinear Solid Mechanics	p. 7
Initial/Boundary Value Problems in the Kinematically Lin-ear Regime	p. 8
Strong Form of the EBVP	p. 9
Weak Form of the IBVP	p. 15
The IBVP in the Finite Strain Case	p. 17
Notation and Problem Formulation	p. 17
Finite Strain Kinematics	p. 18
Stress Definitions Appropriate for Large Deformations	p. 24
Frame Indifference	p. 27
The Strong Form in Finite Strains	p. 31
The Weak Form in Finite Strains	p. 39
Finite Element Discretization	p. 41
Discretized Weak Form; Generation of Discrete Non-linear Equations	p. 43
Discrete Nonlinear Equations for the Kinematically Linear Case	p. 46
Solution Strategies for Spatially Discrete Systems	p. 48
Quasistatics and Incremental Load Methods	p. 48
Dynamics and Global Time Stepping Procedures	p. 50
Local (Constitutive) Time Stepping Procedures	p. 54
Nonlinear Equation Solving	p. 56
Consistent Algorithmic Linearization of Material Re-sponse	p. 61
The Kinematically Linear Contact Problem	p. 69
Strong Forms in Linearized Frictionless Contact	p. 70
The Signorini Problem: Contact with a Rigid Obstacle	p. 70
The Two Body Contact Problem	p. 75
Weak Statements of the Contact Problem	p. 79
Variational Inequalities	p. 81
The Quasistatic Elastic Case: Contact as a Problem of Constrained Optimization	p. 83
Methods of Constraint Enforcement	p. 85
Classical Lagrange Multiplier Methods	p. 85
Penalty Methods	p. 89
Augmented Lagrangian Methods	p. 91
Inclusion of Friction into the Problem Description	p. 94
Friction Kinematics and Traction Measures	p. 94
Unregularized Coulomb Friction Laws	p. 96
Regularization of Friction	p. 98
Variational Statements Including Friction	p. 101
Nonlocal Frictional Descriptions	p. 106
Continuum Mechanics of Large Deformation Contact	p. 109
Two Body Contact Problem Definition	p. 110
Local Momentum Balances	p. 1ll
Initial and Boundary Conditions	p. 112
Contact Constraints in Large Deformations	p. 113
The Gap Function as Defined by Closest Point Projection	p. 113
Frictional Kinematics on Interfaces	p. 116
Frame Indifference of Contact Rate Variables	p. 121
Coulomb Friction in Large Sliding	p. 129
Summary: Strong Form of the Large Deformation Contact Problem	p. 134
Virtual Work Expressions Incorporating Contact	p. 137
Contact Virtual Work: The Contact Integral	p. 139
Linearization of Contact Virtual Work	p. 141
Summary: Weak Form of the Large Deformation Con-tact Problem	p. 144
Finite Element Implementation of Contact Interaction	p. 145
Finite Dimensional Representation of Contact Interaction	p. 147
Contact Surface Discretization	p. 147
Numerical Integration of the Contact Integral	p. 148
Contact Detection (Searching)	p. 152
Time Discretization	p. 158
Global time integration schemes	p. 158
Temporally Discrete Frictional Laws for the Penalty Regularized Case	p. 159
Contact Stiffness and Residual: Penalty Regularized Case	p. 162
Three dimensional matrix expressions	p. 162
Two dimensional matrix expressions	p. 166
Augmented Lagrangian Constraint Enforcement Algorithms	p. 169
Uzawa's Method (Method of Multipliers)	p. 170
Algorithmic Symmetrization Using Augmented La-grangians	p. 174
Augmented Lagrangian Discrete Force and Stiffness Expressions	p. 178
Numerical Examples	p. 180
General Demonstrations of the Computational Frame-work	p. 180
Demonstrations of Augmented Lagrangian Algorith-mic Performance	p. 196
Tribological Complexity in Interface Constitutive Models	p. 211
Rate and State Dependent Friction	p. 212
Motivation	p. 213
One Dimensional Model Development	p. 215
Model Incorporation into Convective Slip Advected Frame	p. 220
Local Time Stepping Algorithm	p. 222
Contact Force Vector and Stiffness Matrix	p. 226
Numerical Examples	p. 227
Thermomechanically Coupled Friction on Interfaces	p. 238
Motivation	p. 239
Thermally Coupled Problem Definition	p. 241
A Thermodynamically Consistent Friction Model	p. 244
Variational Principle and Finite Element Implemen-tation	p. 255
Numerical Examples	p. 269
Thermodynamical Algorithmic Consistency	p. 279
Constitutive Framework for Bulk Continua	p. 280
Thermomechanical Interface Model Framework	p. 283
A Priori Stability Estimates for Dynamic Frictional Contact	p. 286
A New Partitioned Scheme for Thermomechanical Contact	p. 289
Algorithmic Treatment of Contact Conditions According to the Adiabatic Split	p. 291
Energy-Momentum Approaches to Impact Mechanics	p. 295
Energy Stability of Traditional Schemes	p. 297
A Model System	p. 297
The Concept of Energy Stability	p. 299
Influence of Contact Constraints on System Energy	p. 300
Energy-Momentum Methods for Elastodynamics	p. 304
Conservation Laws	p. 305
Conservative Discretization Schemes	p. 309
Energy-Momentum Algorithmic Treatment of Prictionless Impact	p. 312
Discrete Contact Constraints	p. 313
Spatial Discretization and Implementation	p. 316
Numerical Examples	p. 318
Introduction of Frictional and Bulk Dissipation: Energy Con-sistency	p. 325
Coulomb Friction Model Formulation	p. 325
Local Split of the Coulomb Model	p. 331
Algorithmic Formulation	p. 332
Energy Consistent Treatment of Bulk Inelasticity	p. 338
Numerical Examples With Friction and Inelasticity	p. 339
EM Algorithms Involving a Discontinuous Velocity Update	p. 347
Temporally Discontinuous Velocity Update	p. 348
Reexamination of Conservation Conditions	p. 350
Contact Constraints	p. 355
Summary of the Algorithm	p. 357
Numerical Examples	p. 357
Emerging Paradigms for Contact Surface Discretization	p. 369
Contact Smoothing	p. 371
An Alternative Variational Framework	p. 372
Smoothing Strategies in Two Dimensions	p. 374
Smoothing Strategies in Three Dimensions	p. 382
Numerical Examples	p. 390
Mortar-Finite Element Methods for Contact Description	p. 404
Tied Contact and the Role of Mortar Formulations in Convergence	p. 404
A Mortar-Finite Element Formulation of Frictional Contact	p. 416
Numerical Examples of Mortar Treatment of Frictional Contact	p. 425
References	p. 435
Index	p. 451
Table of Contents provided by Publisher. All Rights Reserved.

Engineering Online Library

Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis

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