Mathematical elasticity. Volume I: Three-dimensional elasticity.

*(English)*Zbl 0648.73014
Studies in Mathematics and its Applications, 20. Amsterdam etc.: North- Holland. xi, 451 p. $ 107.25; Dfl. 220.00 (1988).

The title of this book “Mathematical Elasticity” is short but very explicit! Here, we just review the first volume which is more concerned with the nonlinear three-dimensional elasticity. Later, a second volume will develop lower-dimensional theories of plates and rods.

The book lies on the following considerations: i) the linear models of elasticity have a limited range of applicability and, in practice, they have to be more and more replaced by nonlinear models; ii) the mathematical analysis of nonlinear three-dimensional elasticity has become the object of a considerable renewed interest during the past decades; iii) these studies give an excellent motivation for improving knowledge of basic mathematical techniques of analysis and functional analysis.

The contents of this book can be analyzed as follows: Part A: Description of three-dimensional elasticity. Chapter 1 is mainly devoted to the geometry of deformations of an elastic body. It also includes a review of basic results of differential and integral calculus which are subsequently widely used. Chapter 2 states the equations of equilibrium over the deformed configuration and over the reference configuration. Likewise, the corresponding variational formulations, i.e., the principles of virtual work, are given over both configurations. Chapter 3 is concerned with elastic materials and their constitutive equations. In particular, the cases of isotropic and possibly homogeneous materials are considered. This chapter ends by discussing St. Venant-Kirchhoff materials.

Chapter 4 considers an important class of elastic materials, the hyperelastic materials for which there exists a stored energy function. When these materials are loaded by conservative forces, the elastic boundary value problem is formally equivalent to finding the stationary point of the total energy functional. Different expressions of this energy functional are given according to complementary assumptions as isotropy, homogeneity. Finally the fundamental notion of polyconvex stored energy functions, introduced by J. M. Ball [Arnce of infinitely many nonzero time-periodic solutions is proven. As the restoring superlinear force is monotone and odd the Ljusternik- Schnirelman theory may be used to get an approximate solution of the problem. The technique of the proof is applicable to the sublinear case as well.

The book lies on the following considerations: i) the linear models of elasticity have a limited range of applicability and, in practice, they have to be more and more replaced by nonlinear models; ii) the mathematical analysis of nonlinear three-dimensional elasticity has become the object of a considerable renewed interest during the past decades; iii) these studies give an excellent motivation for improving knowledge of basic mathematical techniques of analysis and functional analysis.

The contents of this book can be analyzed as follows: Part A: Description of three-dimensional elasticity. Chapter 1 is mainly devoted to the geometry of deformations of an elastic body. It also includes a review of basic results of differential and integral calculus which are subsequently widely used. Chapter 2 states the equations of equilibrium over the deformed configuration and over the reference configuration. Likewise, the corresponding variational formulations, i.e., the principles of virtual work, are given over both configurations. Chapter 3 is concerned with elastic materials and their constitutive equations. In particular, the cases of isotropic and possibly homogeneous materials are considered. This chapter ends by discussing St. Venant-Kirchhoff materials.

Chapter 4 considers an important class of elastic materials, the hyperelastic materials for which there exists a stored energy function. When these materials are loaded by conservative forces, the elastic boundary value problem is formally equivalent to finding the stationary point of the total energy functional. Different expressions of this energy functional are given according to complementary assumptions as isotropy, homogeneity. Finally the fundamental notion of polyconvex stored energy functions, introduced by J. M. Ball [Arnce of infinitely many nonzero time-periodic solutions is proven. As the restoring superlinear force is monotone and odd the Ljusternik- Schnirelman theory may be used to get an approximate solution of the problem. The technique of the proof is applicable to the sublinear case as well.

Reviewer: W.Schnell

##### MSC:

74B20 | Nonlinear elasticity |

74Axx | Generalities, axiomatics, foundations of continuum mechanics of solids |

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

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

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