Introduction to linear shell theory. (English) Zbl 0912.73001

Ciarlet, Philippe G.

Series in Applied Mathematics (Paris) 1. Paris: Gauthier-Villars/ North-Holland (ISBN 2-84299-059-5). iv, 184 p. (1998).

This book is an outcome of an intensive scientific research work and a series of lectures of the author in the mathematical theory of linearly elastic shells. The book contains valuable recent advances, is devoted to the mathematical-theoretical foundations of linear elastic shell theories and addresses the reader who is interested in a deeper mathematical understanding. First, the book provides a consistent and complete analysis of the existence and uniqueness of the two-dimensional linear membrane, flexural, and Koiter’s shell equations. In doing so, the necessary proofs rely in a crucial way on inequalities of Korn’s type on surfaces. Second, the book shows how the method of formal asymptotic expansions, with the thickness as the “small” parameter, provides a very effective strategy for justifying the two fundamental classes of membrane and flexural equations of a linearly elastic shell, thus paving the way for the more advanced justification by means of a convergence analysis, also carefully reviewed at the end of the book. The whole treatment is essentially self-contained. All needed preliminaries from differential geometry are expounded at sufficient length. Particularly important results are clearly marked. In all, the book has a good didactic form.

Reviewer: W.Becker (Siegen)

MSC:

74-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of deformable solids
74K15	Membranes

Keywords:

membrane equations; existence; uniqueness; Koiter’s shell equations; inequalities of Korn’s type on surfaces; method of formal asymptotic expansions; flexural equations; linearly elastic shell; convergence analysis; differential geometry