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**Models and analysis of quasistatic contact. Variational methods.**
*(English)*
Zbl 1069.74001

Lecture Notes in Physics 655. Berlin: Springer (ISBN 3-540-22915-9/hbk). xi, 262 p. (2004).

The aim of this important book is the study of problems in contact mechanics whose variational forms are inequalities, expressing the principle of virtual power in its inequality form. This book introduces readers to a mathematical theory of contact problem involving deformable bodies. It covers mechanical modeling, mathematical formulations and variational analysis of multidimensional quasistatic contact problems for deformable bodies within the framework of small deformations. The authors give mathematical models for various processes involved in contact. Variational analysis of the models includes existence and uniqueness of weak solutions, as well as results on continuous dependence of solutions on data and parameters.

The book is divided into three parts. The first part provides a detailed introduction to models involving friction, heat generation and thermal effects, wear, adhesion and damage. The second part presents a mathematical analysis of the many models of practical interest and demonstrates close interaction and cross-fertilization between contact mechanics and the theory of variational inequalities. The third part is devoted to reviews of further results. Many references to current research are given, and open problems are discussed. The book gives a rigorous analysis of quasistatic contact problem in elasticity, viscoelasticity and viscoplasticity for a class of elliptic and evolutionary variational inequalities. The book is self-contained.

The work is intended for a wide audience: this would include specialists in contact processes in structural and mechanical systems who wish to know more about the mathematical theory, as well as those with a background in the mathematical sciences who seek a self-contained account of mathematical theory of contact mechanics. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics.

The book is divided into three parts. The first part provides a detailed introduction to models involving friction, heat generation and thermal effects, wear, adhesion and damage. The second part presents a mathematical analysis of the many models of practical interest and demonstrates close interaction and cross-fertilization between contact mechanics and the theory of variational inequalities. The third part is devoted to reviews of further results. Many references to current research are given, and open problems are discussed. The book gives a rigorous analysis of quasistatic contact problem in elasticity, viscoelasticity and viscoplasticity for a class of elliptic and evolutionary variational inequalities. The book is self-contained.

The work is intended for a wide audience: this would include specialists in contact processes in structural and mechanical systems who wish to know more about the mathematical theory, as well as those with a background in the mathematical sciences who seek a self-contained account of mathematical theory of contact mechanics. The text is suitable for graduate students and researchers in applied mathematics, computational mathematics, and computational mechanics.

Reviewer: J. Lovíšek (Bratislava)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74M15 | Contact in solid mechanics |

74M10 | Friction in solid mechanics |

74F05 | Thermal effects in solid mechanics |

49J40 | Variational inequalities |