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

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.

Automated summary

The study introduces effective conditions to identify the symmetry class of an elasticity tensor and proves that these classes are affine algebraic sets, providing a minimal set of generators. Additionally, an original generalized cross-product is introduced on totally symmetric tensors.