Preface to volume I

Abstract

This chapter discusses the mathematical elasticity. A fascinating aspect of three-dimensional elasticity is that, in the course of its study, the need for studying basic mathematical techniques of matrix theory, analysis, and functional analysis is naturally felt. Both common and uncommon results from matrix theory are often needed, such as the polar factorization theorem, or the celebrated Rivlin–Ericksen representation theorem. The understanding of the geometry of deformations relies on a perhaps elementary, but applicable, knowledge of differential geometry. The study of geometrical properties of mappings in ℝ 3 naturally leads to using such basic tools as the invariance of domain theorem or the topological degree, yet these are unfortunately all too often left out from standard analysis courses. Differential calculus in Banach spaces is an indispensable tool which is used in the chapter. The fundamental Cauchy-Lipschitz existence theorem for ordinary differential equations in Banach spaces and the convergence of its approximation by Euler's method are required in the analysis of incremental methods, often used in the numerical approximation of the equations for nonlinearly elastic structures. The chapter also considers the notion of compensated compactness.