The purpose of this textbook is to make graduate students and nonexperts in nonlinear elasticity familiar with recent developments in this field, in particular, by addressing the use of variational methods and by illustrating some of the most important results in nonlinear elasticity which can be obtained with the help of this technique. On the other hand, the style of the book makes it readable also for physicists or engineers who are interested in a better mathematical understanding of nonlinear variational techniques and their applications to elasticity theory.
The book is devided into five chapters, starting with an introduction (chapter 1) into the concepts of elastic materials and the underlying equations leading to a variational formulation of the basic problems. Chapter 2 is devoted to the fundamental notion of quasiconvexity which is the key ingredient for proving existence theorems in the calculus of variations. Another aspect discussed here is the relation between quasiconvexity and Young measures. In chapter 3 the author investigates hyperelastic materials and proves the most important existence theorems for polyconvex energies. Chapter 4 is entitled “Rank-one convexity and microstructure”, and the final chapter 5 collects some technical background material like an existence theorem for Young measures.
The style of the exposition is very clear, and the author tries to include all important definitions and concepts which makes the book self-contained. Moreover, the author succeeds in motivating the theoretical concepts by a number of examples. For readers who want to deepen their knowledge and to see some complete proofs, a list of 216 references is included.