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.

本数据集包含采用沃伊格符号（Voigt notation）表征的高阶非线性各向异性弹性常数的符号解析表达式：其中线性二阶弹性常数为$c_{ij}$，3至6阶非线性常数依次为$c_{ijk}$、$c_{ijkl}$、$c_{ijklm}$、$c_{ijklmn}$，各指标$i,j,k,l,m,n$的取值范围为1至6，且仅保留满足$ileq jleq kleq lleq mleq n$的独立分量。

本数据集包含以下内容：
附录A：全部32种晶类的弹性常数对称性关系（按弹性行为可归为11种独特晶类），以及2种非晶对称性：各向同性与横向各向同性（圆柱对称）。
附录B：将一般各向异性弹性常数旋转平均至上述任意一种对称性（将沃伊格平均推广至非线性情形以及除各向同性外的目标对称性），该方法可通过以高对称性近似低对称性材料来减少独立常数的数量。
附录C：坐标系任意旋转下的弹性常数变换，其中旋转矩阵$R$的元素为$R_{ij}$（$i,j=1,2,3$）。

版本2更新日志：
在版本1中，附录B内四方晶系-II类对称性的平均表达式采用了非标准晶体取向（相对于标准取向绕Z轴旋转了$pi/4$），现已修正；而附录A中对应的独立常数与对称性关系仍保持正确。