We propose a general and efficient method to derive various minimal representations of material tensors or pseudotensors for crystals. By a minimal representation we mean one that pertains to a specific Cartesian coordinate system under which the number of independent components in the representation is the smallest possible. The proposed method is based on the harmonic and Cartan decompositions and, in particular, on the introduction of orthonormal irreducible basis tensors in the chosen harmonic decomposition. For crystals with non-trivial point group symmetry, we demonstrate by examples how deriving restrictions imposed by symmetry groups (e.g., \(C_{2}\), \(C_{s}\), \(C_{3}\), etc.) whose symmetry elements do not completely specify a coordinate system could possibly miss the minimal representations, and how the Cartan decomposition of SO(3)-invariant irreducible tensor spaces could lead to coordinate systems under which the representations are minimal. For triclinic materials, and for material tensors and pseudotensors which observe a sufficient condition given herein, we describe a procedure to obtain a coordinate system under which the explicit minimal representation has its number of independent components reduced by three as compared with the representation with respect to an arbitrary coordinate system.

我们提出了一种通用且有效的方法来推导晶体的材料张量或伪张量的各种最小表示。我们所说的最小表示是指属于特定笛卡尔坐标系的表示，在该坐标系下，表示中独立分量的数量尽可能少。所提出的方法基于调和和 Cartan 分解，特别是在所选调和分解中引入标准正交不可约基张量。对于具有非平凡点群对称性的晶体，我们通过示例演示如何推导对称群施加的限制（例如，\(C_{2}\)、\(C_{s}\)、\(C_{3}\)等），其对称元素未完全指定坐标系可能会错过最小表示，以及 SO(3) 不变不可约张量空间的 Cartan 分解如何导致表示最小的坐标系。对于三斜材料，以及满足此处给定充分条件的材料张量和伪张量，我们描述了获得坐标系的过程，在该坐标系下，与关于任意坐标系。