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.
Steps to reproduce
The Symbolic Package for GNU Octave was utilized.