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An Introduction to Differential Geometry with Applications to Elasticity,Used
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curvilinear coordinates. This treatment includes in particular a direct proof of the threedimensional 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 twodimensional, 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 twodimensional 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 threedimensional 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 NorthHolland, 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].
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