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**Unilateral contact problems. Variational methods and existence theorems.**
*(English)*
Zbl 1079.74003

Pure and Applied Mathematics (Boca Raton) 270. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-57444-629-0/hbk). x, 398 p. (2005).

This research monograph is devoted to rigorous mathematical analysis of contact problems with friction and of dynamic contact problems without friction. It reflects in major parts the authors’ research activities over the last years which led to the present self-contained exposition of this interesting field.

The book is organized as follows: in chapter 1 the general framework of the problems under consideration is outlined together with a list of useful links to related works and to the historical development. After this the reader is made familiar with the basic concepts of linear elasticity, and a precise formulation of contact problems is given. Moreover, this introduction gives a short outline of classical variational principles in mechanics and finally concentrates on geometric properties of domains in Euclidean space. Chapter 2 presents on approximately 100 pages various mathematical tools from functional analysis and from the theory of function spaces (e.g. Besov and Lizorkin-Triebel spaces), which are needed for formulation and proofs of an adequate existence theory. Chapter 3 addresses static and quasistatic contact problems: in the first part, an existence proof for the static problem is given via the penalty method, in the second part this approach is extended to semicoercive contact problems, the third part contains contact problem for two bodies, and in the fourth section of chapter 3 the quasistatic contact problem is discussed. Chapter 4 contains a summary of various results on contact problems (without friction and with given friction) for several viscoelastic constitutive laws and contact conditions formulated in velocities or in displacements. The last chapter contains results on dynamic contact problems with contact conditions formulated in velocities and with Coulomb law of friction. This chapter finishes with contact problems coupled with heat diffusion and with the generation of heat by friction.

The book contains many useful references and is written at a level making the material accessible for scientist working in fields like partial differential equations, calculus of variations or numerical analysis. It can also serve for experts as a starting point for further research on more complex problems in this area. The book may also used for giving lectures or organize seminars for graduate students having a good background in analysis.

The book is organized as follows: in chapter 1 the general framework of the problems under consideration is outlined together with a list of useful links to related works and to the historical development. After this the reader is made familiar with the basic concepts of linear elasticity, and a precise formulation of contact problems is given. Moreover, this introduction gives a short outline of classical variational principles in mechanics and finally concentrates on geometric properties of domains in Euclidean space. Chapter 2 presents on approximately 100 pages various mathematical tools from functional analysis and from the theory of function spaces (e.g. Besov and Lizorkin-Triebel spaces), which are needed for formulation and proofs of an adequate existence theory. Chapter 3 addresses static and quasistatic contact problems: in the first part, an existence proof for the static problem is given via the penalty method, in the second part this approach is extended to semicoercive contact problems, the third part contains contact problem for two bodies, and in the fourth section of chapter 3 the quasistatic contact problem is discussed. Chapter 4 contains a summary of various results on contact problems (without friction and with given friction) for several viscoelastic constitutive laws and contact conditions formulated in velocities or in displacements. The last chapter contains results on dynamic contact problems with contact conditions formulated in velocities and with Coulomb law of friction. This chapter finishes with contact problems coupled with heat diffusion and with the generation of heat by friction.

The book contains many useful references and is written at a level making the material accessible for scientist working in fields like partial differential equations, calculus of variations or numerical analysis. It can also serve for experts as a starting point for further research on more complex problems in this area. The book may also used for giving lectures or organize seminars for graduate students having a good background in analysis.

Reviewer: Martin Fuchs (Saarbrücken)

### MSC:

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

74M15 | Contact in solid mechanics |

74M10 | Friction in solid mechanics |

74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |

74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |

49J40 | Variational inequalities |