# A Brief Introduction to Mathematical Shell Theory

Research output: Conference Papers › RGC 31A - Invited conference paper (refereed items) › Yes › peer-review

## Author(s)

## Related Research Unit(s)

## Detail(s)

Original language | English |
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Pages | 111-185 |

Publication status | Published - 2008 |

### Course

Title | Classical and Advanced Theories of Thin Structures: Mechanical and Mathematical Aspects |
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Location | |

Place | Italy |

City | Udine |

Period | 5 - 9 June 2006 |

## Link(s)

## Abstract

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 ℝ

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”.

^{2}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”.

## Citation Format(s)

**A Brief Introduction to Mathematical Shell Theory.**/ Ciarlet, Philippe G.

2008. 111-185 Classical and Advanced Theories of Thin Structures: Mechanical and Mathematical Aspects, Udine, Italy.

Research output: Conference Papers › RGC 31A - Invited conference paper (refereed items) › Yes › peer-review