On anisotropic polynomial relations for the elasticity tensor

Nicolas Auffray (1) , Boris Kolev (2) , Michel Petitot (3, 4)

1 MSME - Laboratoire de Modélisation et Simulation Multi Echelle
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités
3 LIFL - Laboratoire d'Informatique Fondamentale de Lille
4 CALFOR - Calcul Formel

In this paper, we explore new conditions for an elasticity tensor to belong to a given symmetry class. Our goal is to propose an alternative approach to the identification problem of the symmetry class, based on polynomial invariants and covariants of the elasticity tensor C, rather than on spectral properties of the Kelvin representation. We compute a set of algebraic relations which describe precisely the orthotropic ([D2]), trigonal ([D3]), tetragonal ([D4]), transverse isotropic ([SO(2)]) and cubic ([O]) symmetry classes in H4, the higher irreducible component in the decomposition of Ela. We provide a bifurcation diagram which describes how one "travel" in H4 from a given isotropy class to another. Finally, we study the link between these polynomial invariants and those obtained as the coefficients of the characteristic or the Betten polynomials.We show, in particular, that the Betten invariants do not separate the orbits of the elasticity tensors.

Domaines: Mécanique [physics.med-ph]

Nicolas Auffray, Boris Kolev, Michel Petitot. On anisotropic polynomial relations for the elasticity tensor. Journal of Elasticity, 2014, 115 (1), pp.77-103. ⟨10.1007/s10659-013-9448-z⟩. ⟨hal-00827948⟩