An Introduction to Shell Theory

Philippe G. Ciarlet (1) , Cristinel Mardare (2)

1 HKU - Department of Mathematics
2 LJLL - Laboratoire Jacques-Louis Lions

These notes are intended to provide a thorough introduction to the mathematical theory of elastic shells. The main objective of shell theory is to predict the stress and the displacement arising in an elastic shell in response to given forces. Such a prediction is made either by solving a system of partial differential equations or by minimizing a functional, which may be defined either over a three-dimensional set or over a two-dimensional set, depending on whether the shell is viewed in its reference configuration as a three-dimensional or as a two-dimensional body (the latter being an abstract idealization of the physical shell when its thickness is “small”). The first part of these notes is devoted to the three-dimensional theory of elastic bodies, from which the three-dimensional theory of shells is obtained simply by replacing the reference configuration of a general body with that of a shell. The particular shape of the reference configuration of the shell does not play any role in this theory. The second part is devoted to the two-dimensional theory of elastic shells. In contrast to the three-dimensional theory, this theory is specific to shells, since it essentially depends on the geometry of the reference configuration of a shell.

Mots clés: Elasticity theory; Shell theory

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Philippe G. Ciarlet, Cristinel Mardare. An Introduction to Shell Theory. Differential Geometry: Theory and Applications, 9, pp.94-184, 2008, ⟨10.1142/9789812771476_0002⟩. ⟨hal-01077558⟩