The mathematical analysis of contact problems, with or without friction, is an area where progress depends heavily on the integration of pure and applied mathematics. This book presents the state of the art in the mathematical analysis of unilateral contact problems with friction, along with a major part of the analysis of dynamic contact problems without friction.
Much of this monograph emerged from the authors' research activities over the past 10 years and deals with an approach proven fruitful in many situations. Starting from thin estimates of possible solutions, this approach is based on an approximation of the problem and the proof of a moderate partial regularity of the solution to the approximate problem. This in turn makes use of the shift (or translation) technique - an important yet often overlooked tool for contact problems and other nonlinear problems with limited regularity. The authors pay careful attention to quantification and precise results to get optimal bounds in sufficient conditions for existence theorems.
Unilateral Contact Problems: Variational Methods and Existence Theorems a valuable resource for scientists involved in the analysis of contact problems and for engineers working on the numerical approximation of contact problems. Self-contained and thoroughly up to date, it presents a complete collection of the available results and techniques for the analysis of unilateral contact problems and builds the background required for further research on more complex problems in this area. — Features — · Provides a comprehensive overview of the mathematical analysis of unilateral contact problems with friction and dynamic contact problems without friction · Extends the available literature well beyond the simpler, often static, and directly solvable problems typically addressed · Includes a review of the relevant results in nonlinear analysis and the theory of function spaces, with emphasis on interpolation, imbedding, and trace theorems · Quantifies precisely all important constants arising in existence theorems and provides precise formulae or graphs for required conditions on coefficients of friction — Contents — PREFACE INTRODUCTION Notations Linear Elasticity Formulation of Contact Problems Variational Principles in Mechanics The Static Contact Problem Geometry of Domains The Method of Tangential Translations BACKGROUND Fixed Point Theorems Some General Remarks Crash Course in Interpolation Besov and Lizorkin-Triebel Spaces The Potential Spaces Vector-Valued Sobolev and Besov Spaces Extensions and Traces Spaces on Domains STATIC AND QUASISTATIC CONTACT PROBLEMS Coercive Static Case Semicoercive Contact Problem Contact Problems for Two Bodies Quasistatic Contact Problem DYNAMIC CONTACT PROBLEMS A Short Survey About Results for Elastic Materials Results for Materials With Singular Memory Viscoelastic Membranes Problems With Given Friction DYNAMIC CONTACT PROBLEMS WITH COULOMB FRICTION Solvability of Frictional Contact Problems Anisotropic Material Isotropic Material Thermo-Viscoelastic Problems BIBLIOGRAPHY LIST OF NOTATION SUBJECT INDEX