New Representations for the Curvature Tensor of a Surface with Application to Theories of Elastic Shells

Abstract

AbstractConsider two points $P$ P and $Q$ Q on a surface. Modulo rotations about the normal vector to the surface at $P$ P and the normal vector to the surface at $Q$ Q , a rotation can be defined that maps the unit normal vector to the surface at $Q$ Q to the corresponding unit normal vector at $P$ P . With the help of Weingarten’s formulae, new representations are established for the components of the curvature tensor of a surface and the associated mean and Gaussian curvatures in terms of components of a pair of vectors associated with the rotation. The formulae are shown to be helpful in demonstrating how different strain measures for Kirchhoff-Love shell theory are equivalent.