Mathematical problems in elasticity and homogenization. (English) Zbl 0768.73003

Studies in Mathematics and its Applications. 26. Amsterdam etc.: North- Holland. xiii, 398 p. (1992).
This monograph is based on the research of the authors over the last ten years. It deals with the homogenization of partial differential operators. The homogenization theory was stimulated by problems arising in mechanics and physics for media with a high density of strong heterogeneities. The description of finite volumes of so complicated media is practically possible at the macroscopic level only when an equivalent continuous medium can be defined. The main problem consists in constructing an effective medium i.e. in defining the so-called homogenized system with slowly varying coefficients and finding its solutions which approximate the solutions of the given system describing a strongly non-homogeneous medium.
This book deals with homogenization problems in linear elasticity. In Chapter I are collected the results concerning the system of linear elasticity, which are used throughout the book. Particularly, proofs of Korn’s inequalities and Saint-Venant principle are given. Chapter II is devoted to the homogenization of boundary value problems for linear elastostatics with rapidly oscillating coefficients in domains which may be perforated. A detailed consideration is given to structures which may be non periodic and to stratified media.
The theory of free vibrations of strongly non-homogeneous elastic bodies is the main subject of Chapter III. These problems are not adequately represented in the existing monographs. First section proves theorems on the convergence of eigenvalues and eigenvectors of a sequence of self- adjoint operators depending on a parameter and defined on different Hilbert spaces depending also on that parameter. On the basis of these theorems the homogenization of eigenvalues and eigenfunctions of the boundary value problems considered in Chapter II is performed. This method is also applied to some other similar problems.
With the exception of some well-known facts from functional analysis and the theory of partial differential equations, all results in this book are given detailed mathematical proof.
Reviewer: Th. Lévy (Paris)


74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74E05 Inhomogeneity in solid mechanics
74B05 Classical linear elasticity
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure