Symmetry classes for even-order tensors

TL;DR

This paper provides a comprehensive theoretical framework to determine symmetry classes of any even-order tensor in continuum mechanics, overcoming previous computational limitations and illustrating the method with second strain-gradient elasticity tensors.

Contribution

It introduces general theorems that directly identify symmetry classes of even-order tensors, applicable across various physical contexts, including elasticity and strain-gradient elasticity.

Findings

Derived general theorems for symmetry class determination

Applied the method to second strain-gradient elasticity tensors

Provided the first complete classification of these tensors' symmetry classes

Abstract

The purpose of this article is to give a complete and general answer to the recurrent problem in continuum mechanics of the determination of the number and the type of symmetry classes of an even-order tensor space. This kind of investigation was initiated for the space of elasticity tensors. Since then, different authors solved this problem for other kinds of physics such as photoelectricity, piezoelectricity, flexoelectricity, and strain-gradient elasticity. All the aforementioned problems were treated by the same computational method. Although being effective, this method suffers the drawback not to provide general results. And, furthermore, its complexity increases with the tensorial order. In the present contribution, we provide general theorems that directly give the sought 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.