An Introduction to Differential Geometry With Applications to Elasticity

by Ciarlet, Philippe G.

• ISBN13: 9781402042478
• ISBN10: 1402042477
• Format: Hardcover
• Copyright: 2006-06-30
• Publisher: Kluwer Academic Pub

Summary

This book is based on a series of lectures delivered over the years by the author at the University Pierre et Marie Curie in Paris, at the University of Stuttgart, and at City University of Hong Kong. Its two-fold aim is to provide a thorough introduction to the basic theorems of differential geometry and to elasticity in curvilinear coordinates and shell theory.To this end, the fundamental existence and uniqueness theorems are proved in great details. Such theorems include the fundamental theorem of surface theory, which asserts that the Gauss and Codazzi-Mainardi equations are sufficient for the existence of a surface with prescribed fundamental forms, as well as the corresponding rigidity theorem. Recent results, which have not yet appeared in book form are also included, such as the continuity of a surface as a function of its fundamental forms.This book also provides a detailed description of the equations of nonlinear and linearized elasticity in curvilinear coordinates, together with a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The book also includes a detailed description of Koiter's equations for nonlinearly and linearly elastic shells, a complete analysis of the existence, uniqueness, and regularity of the solutions of Koiter's equations in the linear case.The treatment is essentially self-contained and proofs are complete. In particular, no a priori knowledge of diferential geometry or elasticity theory or shell theory is assumed. Another highlight of this book is the focus on the interplay between "theoretical" and "applied" differential geometry. For instance, rather than being introduced in a formal way, covariant derivatives of a tensor field appear in a natural way in the course of the derivation of the basic boundary value problems of nonlinear elasticity in curvilinear coordinates and of shell theory.

Table of Contents

Preface	1	(2)
Three-dimensional differential geometry
Introduction	3	(2)
Curvilinear coordinates	5	(2)
Metric tensor	7	(3)
Volumes, areas, and lengths in curvilinear coordinates	10	(3)
Covariant derivatives of a vector field	13	(5)
Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor	18	(1)
Existence of an immersion defined on an open set in R3 with a prescribed metric tensor	19	(11)
Uniqueness up to isometries of immersions with the same metric tensor	30	(8)
Continuity of an immersion as a function of its metric tensor	38	(17)
Differential geometry of surfaces
Introduction	53	(2)
Curvilinear coordinates on a surface	55	(4)
First fundamental form	59	(2)
Areas and lengths on a surface	61	(2)
Second fundamental form; curvature on a surface	63	(4)
Principal curvatures; Gaussian curvature	67	(6)
Covariant derivatives of a vector field defined on a surface; the Gauß and Weingarten formulas	73	(3)
Necessary conditions satisfied by the first and second fundamental forms: the Gauß and Codazzi-Mainardi equations; Gauß Theorema Egregium	76	(3)
Existence of a surface with prescribed first and second fundamental forms	79	(10)
Uniqueness up to proper isometries of surfaces with the same fundamental forms	89	(5)
Continuity of a surface as a function of its fundamental forms	94	(12)
Applications to three-dimensional elasticity in curvilinear coordinates
Introduction	103	(3)
The equations of nonlinear elasticity in Cartesian coordinates	106	(7)
Principle of virtual work in curvilinear coordinates	113	(8)
Equations of equilibrium in curvilinear coordinates; covariant derivatives of a tensor field	121	(2)
Constitutive equation in curvilinear coordinates	123	(1)
The equations of nonlinear elasticity in curvilinear coordinates	124	(2)
The equations of linearized elasticity in curvilinear coordinates	126	(3)
A fundamental lemma of J.L. Lions	129	(2)
Korn's inequalities in curvilinear coordinates	131	(7)
Existence and uniqueness theorems in linearized elasticity in curvilinear coordinates	138	(11)
Applications to shell theory
Introduction	147	(2)
The nonlinear Koiter shell equations	149	(9)
The linear Koiter shell equations	158	(8)
Korn's inequalities on a surface	166	(13)
Existence and uniqueness theorems for the linear Koiter shell equations; covariant derivatives of a tensor field defined on a surface	179	(8)
A brief review of linear shell theories	187	(8)
References	195	(8)
Index	203