1st Edition

Analysis and Approximation of Contact Problems with Adhesion or Damage

By Mircea Sofonea, Weimin Han, Meir Shillor Copyright 2006
    238 Pages
    by Chapman & Hall

    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.

    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

    Biography

    Mircea Sofonea, Weimin Han, Meir Shillor

    “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