An Introduction to Differential Geometry with Applications to Elasticity

1. Front Matter

   Pages i-2

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2. Three-Dimensional Differential Geometry

   Pages 3-52

3. Differential Geometry of Surfaces

   Pages 53-102

4. Applications to Three-Dimensional Elasticity in Curvilinear Coordinates

   Pages 103-145

5. Applications to Shell Theory

   Pages 147-193

6. Back Matter

   Pages 195-209

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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].

• Gaussian curvature
• curvature
• curvilinear coordinates
• differential geometry
• differential geometry of surfaces
• elasticity theory
• shell theory
• surface theory
• partial differential equations

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)

• City University of Hong Kong, Hong Kong

  Philippe G. Ciarlet

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