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**Contact problems in elasticity: A study of variational inequalities and finite element methods.**
*(English)*
Zbl 0685.73002

SIAM Studies in Applied Mathematics, 8. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. viii, 495 p. £82.50 (1988).

This book is entirely devoted to the study of contact problems in elasticity: modeling in terms of variational inequalities, mathematical studies of corresponding equations, approximation of the solutions by using finite element methods and some illustrative examples.

This monograph comprises essentially two parts: i) the first eight chapters consider problems of infinitesimal deformation of linearly elastic bodies for which the contact surface does not present friction. ii) the remaining six chapters study some generalizations which introduce additional nonlinearities to the classical contact problems considered in part i).

Let us give a short summary of the contents of the different chapters. After the introduction (chapter 1), the classical Signorini problem is discussed from mechanical point of view in chapter 2, i.e., the equilibrium of a linearly elastic body in contact with a frictionless rigid foundation. Then chapters 3 and 4, respectively, review minimization theory and convex analysis on the one hand, and finite element approximations on the other hand. The mathematical tools needed to study contact problems in elasticity are given in chapter 5. These tools are used in chapter 6 in order to state several results on existence, uniqueness and approximation of solutions of Signorini problems. Extensions of such studies to incompressible elastic materials and their approximation by mixed finite elements are considered in chapter 7. Some alternate variational principles for Signorini’s problem and various numerical methods for their solution are taken up in chapter 8.

As we have previously mentioned, remaining chapters are devoted to highly nonlinear class of contact problems. Firstly chapter 9 deals with unilateral problems involving nonlinear pseudomonotone operators of fourth order like that modeling von Kármánn plate deformations. The study of contact problems with friction starts in chapter 10 by considering the cases of a contact on a surface on which the normal or tangential stresses are prescribed. Next contact problems with nonclassical friction laws are examined in chapter 11. The cases of contact problems involving finite elastic deformations or large elastoplastic deformations are studied in chapter 12, and some applications to problems of metal forming are described. Then dynamic friction problems are discussed in chapter 13 and rolling contact problems for finite deformations of a rolling cylinder are taken up in chapter 14. Finally the authors conclude by giving comments on other kind of contact problems and they list numerous open problems.

I find this voluminous monograph very attractive. Indeed the authors combine an excellent presentation and a complete treatment of the different kinds of considered problems: description of physical problems including very expressive figures, a thorough mathematical study in particular for the classical contact problems on linearly elastic bodies without friction, approximation by finite element methods and many illustrative numerical examples. I also appreciate the second half of the book (chapters 9 to 14) which discusses several generalizations developed in order to precisely model real life problems such as frictional models, metal forming processes... Good references (unfortunately updated to 1981-1982 only) and index complete the book.

In conclusion, this book gives an excellent example of how mechanics and mathematics can interact in order to produce liable models of real life problems. With no doubt, it will be really helpful to researchers in both fields and to engineers.

This monograph comprises essentially two parts: i) the first eight chapters consider problems of infinitesimal deformation of linearly elastic bodies for which the contact surface does not present friction. ii) the remaining six chapters study some generalizations which introduce additional nonlinearities to the classical contact problems considered in part i).

Let us give a short summary of the contents of the different chapters. After the introduction (chapter 1), the classical Signorini problem is discussed from mechanical point of view in chapter 2, i.e., the equilibrium of a linearly elastic body in contact with a frictionless rigid foundation. Then chapters 3 and 4, respectively, review minimization theory and convex analysis on the one hand, and finite element approximations on the other hand. The mathematical tools needed to study contact problems in elasticity are given in chapter 5. These tools are used in chapter 6 in order to state several results on existence, uniqueness and approximation of solutions of Signorini problems. Extensions of such studies to incompressible elastic materials and their approximation by mixed finite elements are considered in chapter 7. Some alternate variational principles for Signorini’s problem and various numerical methods for their solution are taken up in chapter 8.

As we have previously mentioned, remaining chapters are devoted to highly nonlinear class of contact problems. Firstly chapter 9 deals with unilateral problems involving nonlinear pseudomonotone operators of fourth order like that modeling von Kármánn plate deformations. The study of contact problems with friction starts in chapter 10 by considering the cases of a contact on a surface on which the normal or tangential stresses are prescribed. Next contact problems with nonclassical friction laws are examined in chapter 11. The cases of contact problems involving finite elastic deformations or large elastoplastic deformations are studied in chapter 12, and some applications to problems of metal forming are described. Then dynamic friction problems are discussed in chapter 13 and rolling contact problems for finite deformations of a rolling cylinder are taken up in chapter 14. Finally the authors conclude by giving comments on other kind of contact problems and they list numerous open problems.

I find this voluminous monograph very attractive. Indeed the authors combine an excellent presentation and a complete treatment of the different kinds of considered problems: description of physical problems including very expressive figures, a thorough mathematical study in particular for the classical contact problems on linearly elastic bodies without friction, approximation by finite element methods and many illustrative numerical examples. I also appreciate the second half of the book (chapters 9 to 14) which discusses several generalizations developed in order to precisely model real life problems such as frictional models, metal forming processes... Good references (unfortunately updated to 1981-1982 only) and index complete the book.

In conclusion, this book gives an excellent example of how mechanics and mathematics can interact in order to produce liable models of real life problems. With no doubt, it will be really helpful to researchers in both fields and to engineers.

Reviewer: M.Bernadou

### MSC:

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

74A55 | Theories of friction (tribology) |

74M15 | Contact in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

49J40 | Variational inequalities |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

74S05 | Finite element methods applied to problems in solid mechanics |

74B20 | Nonlinear elasticity |

74C15 | Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) |

74C20 | Large-strain, rate-dependent theories of plasticity |