Characterization of the symmetry class of an Elasticity tensor using polynomial covariants

TL;DR

This paper develops polynomial covariant-based conditions to identify the symmetry class of elasticity tensors, revealing algebraic structures and providing minimal generating sets for related covariant and invariant algebras.

Contribution

It introduces a novel polynomial covariant framework for classifying elasticity tensor symmetries and derives minimal generating sets for their covariant and invariant algebras.

Findings

Symmetry classes form affine algebraic sets.

Produced a minimal set of 70 generators for the covariant algebra.

Developed a minimal set of 294 generators for the invariant algebra.

Abstract

We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.