Symmetry classes for even-order tensors

Abstract

We give a complete general answer to the problem, recurrent in continuum mechanics, of determining of the number and type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors, and since then different authors have solved this problem for other kinds of physics, such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All these problems were treated using the same computational method, which, though effective, has the drawback of not providing general results. Furthermore, its complexity increases with the tensorial order. Here we provide general theorems that directly give the desired results for any even-order constitutive tensor. As an illustration of this method, and for the first time, the symmetry classes of all even-order tensors of Mindlin second strain-gradient elasticity are provided.