##
**Computational inelasticity.**
*(English)*
Zbl 0934.74003

Interdisciplinary Applied Mathematics. 7. Berlin: Springer. xiv, 392 p. (1998).

This book goes back a long way. These words are the first of the preface of this book. Indeed, the scientific community waited many years for this book to be issued. It was announced by the editor since the early nineties of the last century, if the reviewer remembers it well. Of course, the project was mainly delayed because of the sudden dead of Juan Simo in 1994. The manuscript circulated among collegues since 1986, based on course material of the two authors from Stanford University. The main purpose of the book is the numerical computation of elastoplastic problems. These are always nonlinear and are not even continuously differentiable, which causes the main difficulty for the integration. On the other hand, such problems are of supreme importance for many technical applications. So there is a great need for efficient and stable algorithms.

The authors enter into this complex field step-by-step and very systematically. They start with the one-dimensional case, which is already difficult enough. Then the authors pass over to the fully three-dimensional case, but first for small deformations. Here they can already demonstrate such usefull algorithms as the return mappings (e.g. the radial return, the closest point mapping, and the cutting plane algorithm). It is well known that the authors contributed a lot of original research work in this field. The operator splitting methodology is described, as well as different linearizations (consistent stiffness matrix).

In a third step towards geometrically nonlinear theory, the large deformation continuum mechanics are developed and examplified for elasticity, hyperelasticity, and hypoelasticity. An interesting outline of finite plasticity is then given, mainly based on the multiplicative decomposition. Here, all the aforementioned concepts are again applied. The last chapter on viscoelastic problems completes the book. Again, first the material theory is developed from one dimension to three dimensions, and then the numerical treatment of materials of convolution type is shown.

The whole material is presented in a rigorous mathematical format. Domains of functions are precisely introduced, as well as their regularity requirements. Under this aspect, the book is preferable to most other books in the field. Some, not many fully solved examples illustrate the theory. A notation index, however, would have been very helpful for the reader. Many references to other contributions are given, and the discussion of competing theories in finite plasticity shows that this difficult subject is still far from being standard text book knowledge.

All in all, the book is an important and valuable contribution to a complicated field, and therefore very recommendable.

The authors enter into this complex field step-by-step and very systematically. They start with the one-dimensional case, which is already difficult enough. Then the authors pass over to the fully three-dimensional case, but first for small deformations. Here they can already demonstrate such usefull algorithms as the return mappings (e.g. the radial return, the closest point mapping, and the cutting plane algorithm). It is well known that the authors contributed a lot of original research work in this field. The operator splitting methodology is described, as well as different linearizations (consistent stiffness matrix).

In a third step towards geometrically nonlinear theory, the large deformation continuum mechanics are developed and examplified for elasticity, hyperelasticity, and hypoelasticity. An interesting outline of finite plasticity is then given, mainly based on the multiplicative decomposition. Here, all the aforementioned concepts are again applied. The last chapter on viscoelastic problems completes the book. Again, first the material theory is developed from one dimension to three dimensions, and then the numerical treatment of materials of convolution type is shown.

The whole material is presented in a rigorous mathematical format. Domains of functions are precisely introduced, as well as their regularity requirements. Under this aspect, the book is preferable to most other books in the field. Some, not many fully solved examples illustrate the theory. A notation index, however, would have been very helpful for the reader. Many references to other contributions are given, and the discussion of competing theories in finite plasticity shows that this difficult subject is still far from being standard text book knowledge.

All in all, the book is an important and valuable contribution to a complicated field, and therefore very recommendable.

Reviewer: Albrecht Bertram (Magdeburg)

### MSC:

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

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

74Sxx | Numerical and other methods in solid mechanics |

74Bxx | Elastic materials |

65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |