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**Foundations of the numerical analysis of plasticity.**
*(English)*
Zbl 0596.73015

North-Holland Mathematics Studies, 107. Lecture Notes in Numerical and Applied Analysis, Vol. 7. Amsterdam-New York-Oxford: North-Holland Publishing Company; Tokyo: Kinokuniya Company. XI, 249 p. $ 55.50; Dfl. 150.00 (1985).

In this book, the author makes a clear presentation of some mechanical models of plasticity. The constitutive laws considered here covers the cases of kinematic or isotropic hardening with a Von Mises elasticity convex. The models are analyzed from a mathematical point of view; the phenomena of plastic yielding and elastic unloading are clearly presented and formulated in the case of an elastoplastic spring with one degree of freedom, then generalized to the case of n degrees of freedom, then to the case of a two-dimensional continuous system under the assumption of small deformations and displacements. Existence results are given for the dynamic and for the quasi-static problems corresponding to a discrete or a continuous two-dimensional system. The technique used here is based on the continuation of solutions which are assumed to be regular enough for ”small time” and in a small neighbourhood of any change of state time. Solutions of the continuous systems considered here are obtained by taking the limit of finite element approximation solutions.

A practical method of resolution based on an explicit scheme is presented and partially analyzed; but the author does not give any numerical result. The techniques developed previously are extended to the case of an elastoplastic beam model taking into account a geometrical nonlinearity for which the author gives an existence result.

A practical method of resolution based on an explicit scheme is presented and partially analyzed; but the author does not give any numerical result. The techniques developed previously are extended to the case of an elastoplastic beam model taking into account a geometrical nonlinearity for which the author gives an existence result.

Reviewer: T.Hadhri

### MSC:

74H99 | Dynamical problems in solid mechanics |

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

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

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

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

74S99 | Numerical and other methods in solid mechanics |

74-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids |

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

35B45 | A priori estimates in context of PDEs |

35B60 | Continuation and prolongation of solutions to PDEs |