The linear Elasticity complex: a natural formulation

TL;DR

This paper reformulates the Elasticity complex using a generalized differential complex, providing new formulas and operators to recover displacement and stress potentials in elasticity theory.

Contribution

It introduces a natural modification of the de Rham complex to incorporate tensor symmetry, enabling new formulations in elasticity and stress analysis.

Findings

Reformulation of the Elasticity complex using Dubois-Violette-Henneaux complex.

An integrating formula to recover displacement from strain.

Introduction of a Hodge star operator and dual complex for stress potentials.

Abstract

We reformulate the Elasticity complex and Saint-Venant's compatibility condition using the generalized differential complex of Dubois-Violette-Henneaux. This is just a slight and natural modification of the de Rham complex to take account of the index symmetry of the tensors involved. An integrating formula to recover the displacement from the strain and similar to the Poincar{\'e} formula is provided. Finally, a Hodge star operator and a dual complex is introduced, which allows to recover stress potentials in dimensions 2 and 3.