Variational Inequalities With Applications

by Sofonea, Mircea; Matei, Andaluzia

• ISBN13: 9780387874593
• ISBN10: 0387874593
• Format: Hardcover
• Copyright: 2009-02-15
• Publisher: Springer Verlag

Summary

This book is motivated by stimulating problems in contact mechanics, emphasizing antiplane frictional contact with linearly elastic and viscoelastic materials. It focuses on the essentials with respect to the qualitative aspects of several classes of variational inequalities (VIs). Clearly presented, easy to follow, and well-referenced, this work treats almost entirely VIs of the second kind, with much of the material being state-of-the-art.Applied mathematicians and advanced graduate students wishing to enter the field of VIs would benefit from this work as it sets out in detail basic features and results in the mathematical theory of contact mechanics. Researchers interested in applications of numerical analysis pertaining to VIs would also find the work useful. Assuming a reasonable knowledge of functional analysis, this volume is a must for graduate students, practitioners, and engineers engaged in contact mechanics.

Table of Contents

Series Preface	p. vii
Preface	p. ix
List of Symbols	p. xvii
Background on Functional Analysis
Preliminaries	p. 3
Linear Operators on Normed Spaces	p. 3
Duality and Weak Convergence	p. 7
Hilbert Spaces	p. 10
Miscellaneous Results	p. 12
Function Spaces	p. 21
The Spaces <$>C^m(\overline \Omega)<$> and <$>L^p (\Omega)<$>	p. 21
Sobolev Spaces	p. 24
Equivalent Norms on the Space H1 (¿)	p. 29
Spaces of Vector-valued Functions	p. 32
Bibliographical Notes	p. 39
Variational Inequalities
Elliptic Variational Inequalities	p. 43
A Basic Existence and Uniqueness Result	p. 43
Convergence Results	p. 48
Elliptic Quasivariational Inequalities	p. 51
Time-dependent Elliptic Variational and Quasivariational Inequalities	p. 56
Evolutionary Variational Inequalities with Viscosity	p. 61
A Basic Existence and Uniqueness Result	p. 61
A Convergence Result	p. 65
Evolutionary Quasivariational Inequalities with Viscosity	p. 67
History-dependent Evolutionary Variational Inequalities with Viscosity	p. 70
Evolutionary Variational Inequalities	p. 75
A First Existence and Uniqueness Result	p. 75
Regularization	p. 88
A Convergence Result	p. 94
Evolutionary Quasivariational Inequalities	p. 96
Volterra-type Variational Inequalities	p. 109
Volterra-type Elliptic Variational Inequalities	p. 109
Volterra-type Evolutionary Variational Inequalities	p. 112
Convergence Results	p. 117
Bibliographical Notes	p. 125
Background on Contact Mechanics
Modeling of Contact Processes	p. 129
Physical Setting	p. 129
Constitutive Laws	p. 132
Contact Conditions	p. 138
Coulomb's Law of Dry Friction	p. 140
Regularized Friction Laws	p. 143
Antiplane Shear	p. 147
Basic Assumptions and Equations	p. 147
A Function Space for Antiplane Problems	p. 152
An Elastic Antiplane Boundary Value Problem	p. 154
Frictional Contact Conditions	p. 157
Antiplane Models for Pre-stressed Cylinders	p. 162
Bibliographical Notes	p. 167
Antiplane Frictional Contact Problems
Elastic Problems	p. 171
Static Frictional Problems	p. 171
A Static Slip-dependent Frictional Problem	p. 181
Quasistatic Frictional Problems	p. 183
A Quasistatic Slip-dependent Frictional Problem	p. 186
Viscoelastic Problems with Short Memory	p. 191
Problems with Tresca and Regularized Friction	p. 191
Approach to Elasticity	p. 195
Slip- and Slip Rate-dependent Frictional Problems	p. 196
Total Slip- and Total Slip Rate-dependent Frictional Problems	p. 199
Viscoelastic Problems with Long Memory	p. 203
Static Frictional Problems	p. 203
Quasistatic Frictional Problems	p. 207
Approach to Elasticity	p. 210
Bibliographical Notes	p. 215
References	p. 219
Index	p. 227
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