Theory of Affine Shells: Towards Advanced Numerical Approximations

Abstract

The well-known Theory of Shells, a topic of Geometry and Mathematical Physics, has been exposed, through the contributions of many authors, within the framework of Euclidean Geometry, i.e., based on the classical theory of surfaces in three-dimensional space, which is invariant under the
Lie group generated by translations and rotations. On the other hand, we ourselves have already developed and presented an alternative foundation of the theory, invariant under the action of the Unimodular Affine group, i.e., dealing with Affine Surface Geometry. In this paper we analyze exclusively the behavior of physical objects of the shell in the interior, without reference to any boundary conditions at the edge. Our main goal here is to establish a comparison with the results already obtained in the Euclidean Theory of Shells, beginning with some geometrical objects and following, afterwards, through the exposition of some distinguished examples in further papers.