A brief introduction to mathematical shell theory

  • Philippe G. Ciarlet
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 503)


In the first chapter, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature and covariant derivatives. We also state the fundamental theorem of surface theory, which asserts that the Gauß and Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields defined in a simply-connected open subset of ℝ2 to be the two fundamental forms of a surface in a three-dimensional Euclidean space. We also state the corresponding rigidity theorem.

The second chapter, which heavily relies on Chapter 1, 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 defining 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 “infinitesimal rigid displacement lemma on a surface”.


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Copyright information

© CISM, Udine 2008

Authors and Affiliations

  • Philippe G. Ciarlet
    • 1
  1. 1.City University of Hong KongHong Kong

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