On the representation of fourth and higher order anisotropic elasticity tensors in generalized continuum models

TL;DR

This paper reviews the classification and explicit computation of fourth-order anisotropic elasticity tensors, extending to non-symmetric matrices and providing general forms for various material symmetries in generalized continuum models.

Contribution

It introduces explicit formulas and classifications for fourth-order elasticity tensors, including non-symmetric cases, in generalized continuum models, expanding prior classical elasticity results.

Findings

Explicit tensor forms for orthotropic, isotropic, cubic, and transversely isotropic materials.

Extension of tensor classification to non-symmetric matrices.

Detailed calculations and examples provided.

Abstract

The classification of all fourth-order anisotropic tensor classes for classical linear elasticity is well known. In this article, we review the related problem of explicitly computing the dimension and the expressions of the elements belonging to these classes, and we extend this computation to fourth-order elasticity tensors acting on non-symmetric matrices. These tensors naturally appear in generalized continuum models. Based on tensor symmetrization, we provide the most general forms of these tensors for orthotropic, transversely isotropic, cubic, and isotropic materials. We present a self-contained discussion and provide detailed calculations for simple examples.