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Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity
Weimin Han, University of Iowa, Iowa City, IA, and Mircea Sofonea, Université de Perpignan, France
A co-publication of the AMS and International Press.
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AMS/IP Studies in Advanced Mathematics
2002; 442 pp; hardcover
Volume: 30
ISBN-10: 0-8218-3192-5
ISBN-13: 978-0-8218-3192-2
List Price: US$91
Member Price: US$72.80
Order Code: AMSIP/30
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Phenomena of contact between deformable bodies or between deformable and rigid bodies abound in industry and in everyday life. A few simple examples are brake pads with wheels, tires on roads, and pistons with skirts. Common industrial processes such as metal forming and metal extrusion involve contact evolutions. Because of the importance of contact processes in structural and mechanical systems, considerable effort has been put into modeling and numerical simulations.

This book introduces readers to a mathematical theory of contact problems involving deformable bodies. It covers mechanical modeling, mathematical formulations, variational analysis, and the numerical solution of the associated formulations. The authors give a complete treatment of some contact problems by presenting arguments and results in modeling, analysis, and numerical simulations.

Variational analysis of the models includes existence and uniqueness results of weak solutions, as well as results of continuous dependence of the solution on the data and parameters. Also discussed are links between different mechanical models.

In carrying out the variational analysis, the authors systematically use results on elliptic and evolutionary variational inequalities, convex analysis, nonlinear equations with monotone operators, and fixed points of operators.

Prerequisites include basic functional analysis, variational formulations of partial differential equation problems, and numerical approximations. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics.

Titles in this series are co-published with International Press, Cambridge, MA.

Readership

Graduate students and research mathematicians interested in applied mathematics, computational mathematics, and computational mechanics.

Reviews

"A need for a clear unified representation of the many results has been recognized ... This need is, in part, fulfilled by this monograph, which presents in a unified way a comprehensive collection of very recent results on models dealing with quasistatic contact between viscoelastic or viscoplastic bodies. This makes the results easy to access and study."

-- Mathematical Reviews

"The book deserves recommendation to all who are interested in the theory and applications of evolutionary variational inequalities."

-- Zentralblatt MATH

Table of Contents

Nonlinear variational problems and numerical approximation
  • Preliminaries of functional analysis
  • Function spaces and their properties
  • Introduction to finite difference and finite element approximations
  • Variational inequalities
Mathematical modelling in contact mechanics
  • Preliminaries of contact mechanics of continua
  • Constitutive relations in solid mechanics
  • Background on variational and numerical analysis in contact mechanics
  • Contact problems in elasticity
Contact problems in viscoelasticity
  • A frictionless contact problem
  • Bilateral contact with slip dependent friction
  • Frictional contact with normal compliance
  • Frictional contact with normal damped response
  • Other viscoelastic contact problems
Contact problems in visocplasticity
  • A Signorini contact problem
  • Frictionless contact with dissipative potential
  • Frictionless contact between two viscoplastic bodies
  • Bilateral contact with Tresca's friction law
  • Other viscoelastic contact problems
  • Bibliography
  • Index
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