Cauchy relations in linear elasticity: Algebraic and physics aspects

TL;DR

This paper analyzes the algebraic and physical aspects of Cauchy relations in linear elasticity, presenting invariant formulations, decompositions, and applications to material physics and wave propagation.

Contribution

It introduces an invariant, group-theoretic framework for Cauchy relations, including partial relations and applications to material physics and wave analysis.

Findings

Invariant formulation of Cauchy relations under transformation groups

Decomposition into Cauchy and non-Cauchy contributions in elasticity

Identification of wave properties independent of non-Cauchy effects

Abstract

The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure's deviation from the basic physical assumptions of Cauchy's model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke's law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters.