An Introduction to Differential Geometry with Applications to Elasticity

ISBN-13: 9789048170852 / Angielski / Miękka / 2010 / 210 str.

Philippe G. Ciarlet

Opis

curvilinear coordinates. This treatment includes in particular a direct proof of the three-dimensional Korn inequality in curvilinear coordinates. The fourth and last chapter, which heavily relies on Chapter 2, begins by a detailed description of the nonlinear and linear equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are two-dimensional, in the sense that they are expressed in terms of two curvilinear coordinates used for de?ning the middle surface of the shell. The existence, uniqueness, and regularity of solutions to the linear Koiter equations is then established, thanks this time to a fundamental Korn inequality on a surface and to an in?nit- imal rigid displacement lemma on a surface . This chapter also includes a brief introduction to other two-dimensional shell equations. Interestingly, notions that pertain to di?erential geometry per se, suchas covariant derivatives of tensor ?elds, are also introduced in Chapters 3 and 4, where they appear most naturally in the derivation of the basic boundary value problems of three-dimensional elasticity and shell theory. Occasionally, portions of the material covered here are adapted from - cerpts from my book Mathematical Elasticity, Volume III: Theory of Shells, published in 2000by North-Holland, Amsterdam; in this respect, I am indebted to Arjen Sevenster for his kind permission to rely on such excerpts. Oth- wise, the bulk of this work was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604]."

Recenzje

From the reviews:

"This is a book about differential geometry and elasticity theory also published earlier as journal article. And, indeed it covers both subjects in a coextensive way that can not be found in any other book in the field. ... the list of references containing more than 120 items is representative enough and the interested reader should be able to find them among these." (Ivailo Mladenov, Zentralblatt MATH, Vol. 1100 (2), 2007)

Spis treści

Preface; Chapter 1. Three-dimensional differential geometry: 1.1. Curvilinear coordinates, 1.2. Metric tensor, 1.3. Volume, areas, and lengths in curvilinear coordinates, 1.4. Covariant derivatives of a vector field, 1.5. Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor, 1.6. Existence of an immersion defined on an open set in R3 with a prescribed metric tensor, 1.7. Uniqueness up to isometries of immersions with the same metric tensor, 1.8. Continuity of an immersion as a function of its metric tensor; Chapter 2. Differential geometry of surfaces: 2.1. Curvilinear coordinates on a surface, 2.2. First fundamental form, 2.3. Areas and lengths on a surface, 2.4. Second fundamental form; curvature on a surface, 2.5. Principal curvatures; Gaussian curvature, 2.6. Covariant derivatives of a vector field defined on a surface; the Gauss and Weingarten formulas, 2.7. Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations; Gauss' theorema egregium, 2.8. Existence of a surface with prescribed first and second fundamental forms, 2.9. Uniqueness up to proper isometries of surfaces with the same fundamental forms, 2.10.Continuity of a surface as a function of its fundamental forms; Chapter 3. Applications to three-dimensional elasticity in curvilinear coordinates: 3.1. The equations of nonlinear elasticity in Cartesian coordinates, 3.2. Principle of virtual work in curvilinear coordinates, 3.3. Equations of equilibrium in curvilinear coordinates; covariant derivatives of a tensor field, 3.4. Constitutive equation in curvilinear coordinates, 3.5. The equations of nonlinear elasticity in curvilinear coordinates, 3.6. The equations of linearized elasticity in curvilinear coordinates, 3.7. A fundamental lemma of J.L. Lions, 3.8. Korn's inequalities in curvilinear coordinates, 3.9. Existence and uniqueness theorems in linearizedelasticity in curvilinear coordinates; Chapter 4. Applications to shell theory: 4.1. The nonlinear Koiter shell equations, 4.2. The linear Koiter shell equations, 4.3. Korn’s inequality on a surface, 4.4. Existence and uniqueness theorems for the linear Koiter shell equations; covariant derivatives of a tensor field defined on a surface, 4.5. A brief review of linear shell theories; References; Index.