Analysis and Approximation of Contact Problems with Adhesion or Damage
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Book Description
Research into contact problems continues to produce a rapidly growing body of knowledge. Recognizing the need for a single, concise source of information on models and analysis of contact problems, accomplished experts Sofonea, Han, and Shillor carefully selected several models and thoroughly study them in Analysis and Approximation of Contact Problems with Adhesion or Damage. The book describes very recent models of contact processes with adhesion or damage along with their mathematical formulations, variational analysis, and numerical analysis.
Following an introduction to modeling and functional and numerical analysis, the book devotes individual chapters to models involving adhesion and material damage, respectively, with each chapter exploring a particular model. For each model, the authors provide a variational formulation and establish the existence and uniqueness of a weak solution. They study a fully discrete approximation scheme that uses the finite element method to discretize the spatial domain and finite differences for the time derivatives. The final chapter summarizes the results, presents bibliographic comments, and considers future directions in the field.
Employing recent results on elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators, and fixed points of operators, Analysis and Approximation of Contact Problems with Adhesion or Damage places these important tools and results at your fingertips in a unified, accessible reference.
Table of Contents
Preface
List of Symbols
Modeling and Mathematical Background
Basic Equations and Boundary Conditions
Physical Setting and Evolution Equations
Boundary Conditions
Contact Processes with Adhesion
Constitutive Equations with Damage
Preliminaries on Functional Analysis
Function Spaces and Their Properties
Elements of Nonlinear Analysis
Standard Results on Variational Inequalities and Evolution Equations
Elementary Inequalities
Preliminaries on Numerical Analysis
Finite Difference and Finite Element Discretizations
Approximation of Displacements and Velocities
Estimates on the Discretization of Adhesion Evolution
Estimates on the Discretization of Damage Evolution
Estimates on the Discretization of Viscoelastic Constitutive Law
Estimates on the Discretization of Viscoplastic Constitutive Law
Frictionless Contact Problems with Adhesion
Quasistatic Viscoelastic Contact with Adhesion
Problem Statement
Existence and uniqueness
Continuous Dependence on the Data
Spatially Semidiscrete Numerical Approximation
Fully Discrete Numerical Approximation
Dynamic Viscoelastic Contact with Adhesion
Problem Statement
Existence and Uniqueness
Fully Discrete Numerical Approximation
Quasistatic Viscoplastic Contact with Adhesion
Problem Statement
Existence and Uniqueness for the Signorini Problem
Numerical Approximation for the Signorini Problem
Existence and Uniqueness for the Problem with Normal Compliance
Numerical Approximation of the Problem with Normal Compliance
Relation between the Signorini and Normal Compliance Problems
Contact Problems with Damage
Quasistatic Viscoelastic Contact with Damage
Problem Statement
Existence and Uniqueness
Fully Discrete Numerical Approximation
Dynamic Viscoelastic Contact with Damage
Problem Statement
Existence and Uniqueness
Fully Discrete Numerical Approximation
Quasistatic Viscoplastic Contact with Damage
Problem Statement
Existence and Uniqueness for the Signorini Problem
Numerical Approximation for the Signorini Problem
Existence and Uniqueness for the Problem with Normal Compliance
Numerical Approximation of the Problem with Normal Compliance
Relation between the Signorini and Normal Compliance Problems
Notes, Comments, and Conclusions
Bibliographical Notes, Problems for Future Research, and Conclusions
Bibliographical Notes
Problems for Future Research
Conclusions
References
Index
Reviews
“This book summarizes and completes the work of the authors on the topic of dynamic and quasistatic contact problems with adhesion or damage of viscoelastic structures in recent years. Different models involving adhesion and material damages are presented with both the theoretical result (existence and uniqueness of a weak solution) and the numerical analysis result (optimal convergence of discrete approximation by finite element methods) in a unified framework. The book is well presented and easy to read.”
— Yves Renard, (Villeurbanne), in Mathematical Reviews, Issue 2007f gt; A Seminal Contribution to the Field by Renowned Researchers