Lectures on three-dimensional elasticity. Notes by S. Kesavan.

*(English)*Zbl 0542.73046The book consists of two parts. The first one contains a review of rational elasticity and hyperelasticity with emphasis on the constitutive relations. Unfortunately, the author does not use the same notation in the second part, but changes the symbols in the middle of the book.

Several examples of failure of uniqueness of the solution of different types of boundary-value-problems motivate the reader to study the second part, a description of the ”state of the art” of existence and uniqueness theorems in finite elasticity. At first, the implicit function theorem is used to prove uniqueness for the pure displacement-boundary-condition problem if only the loads are ”small enough”. The discretization with respect to the applied forces (semi-discrete incremental method) is an effective method to solve boundary-value-problems similar to the method of Euler for approximating ordinary differential equations. The convergence of this method is shown for solutions ”near zero”. The last sections of the treatise are related to the existence of solutions by minimizing the energy using the concept of polyconvexity introduced by J. Ball [see: Finite elasticity, Proc. IUTAM Symp., Bethlehem/PA 1980, 1-12 (1982; Zbl 0518.73031)].

Presenting the state of the art of a subject being in rapid progress, the book includes critical remarks, open problems, and an up-to-date list of references. Throughout the analysis the author uses accurate and advanced mathematical tools and rigorous proofs of his theorems.

Several examples of failure of uniqueness of the solution of different types of boundary-value-problems motivate the reader to study the second part, a description of the ”state of the art” of existence and uniqueness theorems in finite elasticity. At first, the implicit function theorem is used to prove uniqueness for the pure displacement-boundary-condition problem if only the loads are ”small enough”. The discretization with respect to the applied forces (semi-discrete incremental method) is an effective method to solve boundary-value-problems similar to the method of Euler for approximating ordinary differential equations. The convergence of this method is shown for solutions ”near zero”. The last sections of the treatise are related to the existence of solutions by minimizing the energy using the concept of polyconvexity introduced by J. Ball [see: Finite elasticity, Proc. IUTAM Symp., Bethlehem/PA 1980, 1-12 (1982; Zbl 0518.73031)].

Presenting the state of the art of a subject being in rapid progress, the book includes critical remarks, open problems, and an up-to-date list of references. Throughout the analysis the author uses accurate and advanced mathematical tools and rigorous proofs of his theorems.

Reviewer: A.Bertram

##### MSC:

74B20 | Nonlinear elasticity |

74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |

74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |

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

74B99 | Elastic materials |

74H99 | Dynamical problems in solid mechanics |

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