Nonlinear problems of elasticity.

*(English)*Zbl 0820.73002
Applied Mathematical Sciences. 107. New York, NY: Springer-Verlag. xviii, 750 p. (1995).

There are hundreds of books on elastic rods, bars, plates, and shells, but only few, if any, that introduce this topic with mathematical rigor and precision. Antman deserves the honor to have undertaken this difficult task to “lay down a general theory for each kind of elastic body, carefully formulate specific problems, introduce the pertinent mathematical methods (in as unobtrusive a way as possible), and then conduct rigorous analyses of the problem” (from the introduction). The outcome is a voluminous book of more than 700 pages that carefully goes through all related topics of elastic structures and surely satisfies these demands.

In the last part of the book an introduction to three-dimensional problems is announced. Here, however, the author does not restrict himself to elasticity, but also non-elastic materials such as plastic ones are considered. The “general” constitutive equation is given in form of a history functional. Nowadays such a presentation does not have much more than a historical relevance as it excludes most of the materials with (non-fading) memory such as the plastic ones. Therefore, in the chapter on finite plasticity, the author abruptly switches over to an internal variable representation, which is more appropriate and practical to describe a theory of Naghdi’s type. This part of the book is much less convincing than the elastic parts and could have left out as being beyond the title of the book.

As a whole, however, the book is a rich source of numerous interesting considerations and, therefore, highly recommendable.

In the last part of the book an introduction to three-dimensional problems is announced. Here, however, the author does not restrict himself to elasticity, but also non-elastic materials such as plastic ones are considered. The “general” constitutive equation is given in form of a history functional. Nowadays such a presentation does not have much more than a historical relevance as it excludes most of the materials with (non-fading) memory such as the plastic ones. Therefore, in the chapter on finite plasticity, the author abruptly switches over to an internal variable representation, which is more appropriate and practical to describe a theory of Naghdi’s type. This part of the book is much less convincing than the elastic parts and could have left out as being beyond the title of the book.

As a whole, however, the book is a rich source of numerous interesting considerations and, therefore, highly recommendable.

Reviewer: A.Bertram (Berlin)

##### MSC:

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

74B20 | Nonlinear elasticity |

74A20 | Theory of constitutive functions in solid mechanics |

35Q72 | Other PDE from mechanics (MSC2000) |