In linear elasticity, a fourth-order elasticity (stiffness) tensor of 21 independent components completely describes deformation properties elastic constants of a material. The main goal of the current work is to derive a compact matrix representation of the elasticity tensor that correlates with its intrinsic algebraic properties. Such representation can be useful in design of artificial materials. Owing to Voigt, the elasticity tensor is conventionally represented by a (6 × 6) symmetric matrix. In this paper, we construct two alternative matrix representations that conform with the irreducible decomposition of the elasticity tensor. The 3 × 7 matrix representation is in correspondence with the permutation transformations of indices and with the general linear transformation of the basis. An additional representation of the elasticity tensor by two scalars and three 3 × 3 matrices is suitable to describe 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.

中文翻译：

---

弹性张量对称类的不可约矩阵分辨率

在线性弹性中，21 个独立分量的四阶弹性（刚度）张量完整地描述了材料的变形特性弹性常数。当前工作的主要目标是推导出与其固有代数特性相关的弹性张量的紧凑矩阵表示。这种表示可用于人造材料的设计。由于 Voigt，弹性张量通常由 (6 × 6) 对称矩阵表示。在本文中，我们构造了两个符合弹性张量不可约分解的替代矩阵表示。3 × 7 矩阵表示对应于索引的置换变换和基的一般线性变换。由两个标量和三个 3 × 3 矩阵对弹性张量的附加表示适用于描述旋转变换下的不可约分解。我们以这些紧凑矩阵形式呈现所有晶体系统的弹性张量，并基于这种表示构建层次图。