The Elastic Theory of Shells using Geometric Algebra

TL;DR

This paper introduces a new derivation of shell elasticity theory using Geometric algebra, enabling coordinate-free expressions and clarifying previous misconceptions about angular velocity and strain in linearized models.

Contribution

It provides a novel, coordinate-independent formulation of shell elasticity theory using Geometric algebra, clarifying previous assumptions and incorporating prior strain effects.

Findings

Component-free, coordinate-independent equations derived

Clarification of angular velocity and moments in shell theory

Inclusion of prior strain in linearized shell models

Abstract

We present a novel derivation of the elastic theory of shells. We use the language of Geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearised theory, clarification of previous coordinate conventions which have been the cause of confusion, is provided, and the introduction of prior strain into the linearised theory of shells is made possible.