The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation

TL;DR

This paper explores the unique and irreducible symmetric decomposition of the elasticity tensor in linear anisotropic elasticity, revealing its advantages over traditional decompositions and applying it to wave propagation and material properties.

Contribution

It introduces the SA-decomposition of the elasticity tensor, proving its uniqueness and irreducibility, and demonstrates its applications in understanding Cauchy relations, null Lagrangians, and wave behavior.

Findings

SA-decomposition is unique and irreducible under the general linear group.

The decomposition separates Cauchy and non-Cauchy parts, clarifying wave propagation.

A class of anisotropic media with pure polarization directions is proposed.

Abstract

In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the SA-decomposition are discussed: the Cauchy relations (vanishing of A), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the SA-decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.