Analysis and Approximation of Contact Problems with Adhesion or Damage: 1st Edition (Hardback) book cover

Analysis and Approximation of Contact Problems with Adhesion or Damage

1st Edition

By Mircea Sofonea, Weimin Han, Meir Shillor

Chapman and Hall/CRC

220 pages

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pub: 2005-09-26
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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.

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

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

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied