Differential geometric formulation of the Cauchy Navier equations

TL;DR

This paper reformulates fundamental equations of linear elasticity, including stress, strain, and the Cauchy Navier equations, using accessible differential geometric tools to enhance understanding and application in elasticity theory.

Contribution

It introduces a simplified differential geometric formulation of elasticity equations, making advanced geometric methods more accessible to researchers in the field.

Findings

Reformulation of stress and strain tensors using differential forms

Expression of Cauchy Navier equations in differential geometric language

Simplified approach compared to existing geometric reformulations

Abstract

The paper presents a reformulation of some of the most basic entities and equations of linear elasticity - the stress and strain tensor, the Cauchy Navier equilibrium equations, material equations for linear isotropic bodies - in a modern differential geometric language using differential forms and lie derivatives. Similar steps have been done successfully in general relativity, quantum physics and electrodynamics and are of great use in those fields. In Elasticity Theory, however, such a modern differential geometric approach is much less common. Furthermore, existing reformulations demand a vast knowledge of differential geometry, including nonstandard entities such as vector valued differential forms and the like. This paper presents a less general but more easily accessible approach to using modern differential geometry in elasticity theory than those published up to now.