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

Marc Olive (1) , Boris Kolev (1) , R. Desmorat (1) , Boris Desmorat (2)

1 LMT - Laboratoire de Mécanique et Technologie
2 DALEMBERT - Institut Jean Le Rond d'Alembert

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.

Mots clés: Covariant algebras; Elasticity tensors; Fourth order harmonic tensors; Characterization of symmetry classes; Covariants of tensors

Domaines: Matériaux; Théorie des représentations [math.RT]

Marc Olive, Boris Kolev, R. Desmorat, Boris Desmorat. Characterization of the symmetry class of an Elasticity tensor using polynomial covariants. Mathematics and Mechanics of Solids, 2021, ⟨10.1177/10812865211010885⟩. ⟨hal-01845691v4⟩