Constants of anisotropic elasticity up to the 6th order of nonlinearity: rotational transformations, symmetries, and averages for all crystal classes, isotropy, and transverse isotropy

Symbolically computed expressions for high-order nonlinear anisotropic elastic constants in the Voigt notation (linear 2nd-order constants cij, and non-linear orders from 3 to 6: cijk, cijkl, cijklm, cijklmn, where i,j,k,l,m,n = 1,...,6, and only unique components with i<=j<=k<=l<=m<=n are considered).

The dataset consists of:
Appendix A: Symmetry relationships between the constants for all 32 crystal classes (falling into only 11 distinctive classes by elastic behavior), as well as for 2 non-crystalline symmetries: isotropic and transversely isotropic (cylindrical).
Appendix B: Rotational averages of the generally anisotropic constants to any of the symmetries mentioned above (a generalization of the Voigt average to nonlinearity and to goal symmetries other than isotropic). This can be used to reduce the number of constants by approximating low-symmetry materials with higher symmetry.
Appendix C: Transformations of the constants under arbitrary rotation of the system of coordinates with the rotation matrix R with elements Rij, where i,j = 1,2,3.

Version 2 changelog:
In Version 1, expressions for averages for the Tetragonal-II symmetry in Appendix B were for a non-standard crystal orientation (rotated by pi/4 about Z-axis from the standard orientation, now corrected), while the corresponding independent constants and symmetry relations in Appendix A remain correct.

本数据集包含以下内容：

附录A：针对所有32种晶体类别的常数之间的对称关系（这些类别仅根据弹性行为划分为11个独特的类别），以及针对两种非晶体对称性：各向同性和横观各向同性（圆柱形）。

附录B：将一般各向异性常数的旋转平均应用于上述任何一种对称性（Voigt平均的推广，用于非线性和除各向同性之外的其他目标对称性）。这可以通过将低对称性材料近似为高对称性材料来减少常数的数量。

附录C：在系统坐标的任意旋转下，常数的变换，该旋转由旋转矩阵R及其元素Rij（其中i,j = 1,2,3）表示。

版本2更新日志：在版本1中，附录B中针对四方-II对称性的平均值表达式针对的是非标准晶体取向（从标准取向沿Z轴旋转π/4，现已更正），而相应的独立常数和对称关系在附录A中保持正确。