# Irreducible matrix resolution of the elasticity tensor for symmetry systems

## TL;DR

This paper introduces new matrix representations of the elasticity tensor that align with its irreducible decomposition, providing more accurate and symmetry-conforming descriptions for different crystal systems.

## Contribution

It develops two novel matrix forms of the elasticity tensor that better reflect its irreducible structure and symmetry properties, improving upon classical representations.

## Key findings

- New $3\times7$ matrix representation linked to index permutations
- Three $3\times3$ matrices for irreducible rotation decomposition
- Hierarchy diagrams for all crystal systems based on these representations

## Abstract

In linear elasticity, a fourth order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties of a material. Due to Voigt, this tensor is conventionally represented by a $6\times 6$ symmetric matrix. This classical matrix representation does not conform with the irreducible decomposition of the elasticity tensor. In this paper, we construct two alternative matrix representations. The $3\times 7$ matrix representation is in a correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by three $3\times 3$ matrices is suitable for description the irreducible decomposition under the rotation transformations. We present the elasticity tensor of all crystal systems in these compact matrix forms and construct the hierarchy diagrams based on this representation.