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**Problèmes mathématiques en plasticité.**
*(French)*
Zbl 0547.73026

Méthodes Mathématiques de l’Informatique, 12. Publié avec le concours du C.N.R.S. Paris: Gauthier-Villars. Bordas. VII, 353 p. FF 230.00 (1983).

The aim of this book is to present any very recent mathematical results concerning a classical problem in the so called elasto-perfect-plastic theory of materials subjected to small deformations (which is the Hencky model). These models lead from the mathematical point of view to some problems of the variational calculus. The most natural functional space in which the problems of the theory of plasticity are well posed seems to be the subspace of B\(D(\Omega)\) with the divergence of the velocity from \(L^ 2(\Omega).\)

Though the mathematical model is apparently very simple, the mathematical apparatus developed in this book is susceptible to be extended to other problems. On the other hand the boundary problem considered here is complete, which means that on a part of the boundary of the given body we know the stresses and on the complementary part we know the displacements. This mathematical problem is very difficult due to the important difficulties introduced by the displacement problem and cannot, until recently, be solved. The method of solving this problem is developed.

In the first chapter the author presents the problem which is the object of the book and makes any explanations on functional analysis, convex analysis and so on. In the second chapter, the space B\(D(\Omega)\) (bounded deformations) which is the key for all the rest of the mathematical analysis and other necessary mathematical entities are introduced.

After this mathematical preliminaries the author introduces and studies the relaxed variational problem which generalizes the Hencky displacement problem. The third chapter is devoted to the study of some asymptotic variational problems which correspond to some plastically imperfect models (such as elastic-viscoplastic and stress hardening models) and to the theory of shells. An extensive list of references completes this very interesting book. The book is to be recommended to engineers, mechanicians, mathematicians and to all research workers which are interested in some mathematical problems which arise from the nonlinear mechanics.

Though the mathematical model is apparently very simple, the mathematical apparatus developed in this book is susceptible to be extended to other problems. On the other hand the boundary problem considered here is complete, which means that on a part of the boundary of the given body we know the stresses and on the complementary part we know the displacements. This mathematical problem is very difficult due to the important difficulties introduced by the displacement problem and cannot, until recently, be solved. The method of solving this problem is developed.

In the first chapter the author presents the problem which is the object of the book and makes any explanations on functional analysis, convex analysis and so on. In the second chapter, the space B\(D(\Omega)\) (bounded deformations) which is the key for all the rest of the mathematical analysis and other necessary mathematical entities are introduced.

After this mathematical preliminaries the author introduces and studies the relaxed variational problem which generalizes the Hencky displacement problem. The third chapter is devoted to the study of some asymptotic variational problems which correspond to some plastically imperfect models (such as elastic-viscoplastic and stress hardening models) and to the theory of shells. An extensive list of references completes this very interesting book. The book is to be recommended to engineers, mechanicians, mathematicians and to all research workers which are interested in some mathematical problems which arise from the nonlinear mechanics.

Reviewer: V.Tigoiu

### MSC:

74C99 | Plastic materials, materials of stress-rate and internal-variable type |

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

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

35G30 | Boundary value problems for nonlinear higher-order PDEs |

35J20 | Variational methods for second-order elliptic equations |

74B20 | Nonlinear elasticity |