4th order tensor rotation - sources to refer

There are two main ways to write stress/strain tensors as 6 components vectors:

• Voigt notation, that is the most common; and

• Mandel-Kelvin notation, that has the advantage of writing stress and strains in the same way, so their rotations are done via the same $6times 6$ matrices.

A reference that I consider good for Voigt's notation is Auld's book (Vol. 1, Ch. 3, D) and Mehrabadi and Cowin's paper describes the rotation matrix for Mandel-Kelvin notation. In general, I would suggest that you use a CAS for your calculation since they can get long really fast. I developed a Python package and the developing version has these matrices in there.

References

• Auld, B. A. (1973). Acoustic fields and waves in solids. Рипол Классик.

• Bower, Allan F. Applied mechanics of solids. CRC press, 2009. Ch. 3.

• Carcione, J. M. (2007). Wave fields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media. Elsevier.

• Mehrabadi, Morteza M., y Stephen C. Cowin. 1990. “Eigentensors of Linear Anisotropic Elastic Materials”. The Quarterly Journal of Mechanics and Applied Mathematics 43(1):15–41.

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Suppose that you have a rotation matrix

$$[Q] = begin{bmatrix} Q_{xx} &Q_{xy} &Q_{xz}\ Q_{yz} &Q_{yy} &Q_{yz}\ Q_{zx} &Q_{zy} &Q_{zz} end{bmatrix}, ,$$

and you are using the following Voigt notation order: $xx$, $yy$, $zz$, $yz$, $xz$, $xy$.

You can form the (Bond-like) rotation matrices that are described below.

Voigt notation

Rotation of stresses

$$[M] =begin{bmatrix} Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & 2 Q_{xy} Q_{xz} & 2 Q_{xx} Q_{xz} & 2 Q_{xx} Q_{xy}\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & 2 Q_{yy} Q_{yz} & 2 Q_{yx} Q_{yz} & 2 Q_{yx} Q_{yy}\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & 2 Q_{zy} Q_{zz} & 2 Q_{zx} Q_{zz} & 2 Q_{zx} Q_{zy}\ Q_{yx} Q_{zx} & Q_{yy} Q_{zy} & Q_{yz} Q_{zz} & Q_{yy} Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\ Q_{xx} Q_{zx} & Q_{xy} Q_{zy} & Q_{xz} Q_{zz} & Q_{xy} Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\ Q_{xx} Q_{yx} & Q_{xy} Q_{yy} & Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} end{bmatrix}$$

The rotation of a stiffness tensor in Voigt's notation is done via

$$[C'] = [M] [C] [M^T], .$$

Rotation of strains

$$[N] = begin{bmatrix}Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & Q_{xy} Q_{xz} & Q_{xx} Q_{xz} & Q_{xx} Q_{xy}\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & Q_{yy} Q_{yz} & Q_{yx} Q_{yz} & Q_{yx} Q_{yy}\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & Q_{zy} Q_{zz} & Q_{zx} Q_{zz} & Q_{zx} Q_{zy}\ 2 Q_{yx} Q_{zx} & 2 Q_{yy} Q_{zy} & 2 Q_{yz} Q_{zz} & Q_{yy} Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\ 2 Q_{xx} Q_{zx} & 2 Q_{xy} Q_{zy} & 2 Q_{xz} Q_{zz} & Q_{xy} Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\ 2 Q_{xx} Q_{yx} & 2 Q_{xy} Q_{yy} & 2 Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} end{bmatrix}$$

The rotation of a compliance tensor in Voigt's notation is done via

$$[C'] = [N] [C] [N^T], .$$

Mandel-Kelvin notation

$$[M] = begin{bmatrix} Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & sqrt{2} Q_{xy} Q_{xz} & sqrt{2} Q_{xx} Q_{xz} & sqrt{2} Q_{xx} Q_{xy}\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & sqrt{2} Q_{yy} Q_{yz} & sqrt{2} Q_{yx} Q_{yz} & sqrt{2} Q_{yx} Q_{yy}\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & sqrt{2} Q_{zy} Q_{zz} & sqrt{2} Q_{zx} Q_{zz} & sqrt{2} Q_{zx} Q_{zy}\ sqrt{2}Q_{yx} Q_{zx} & sqrt{2}Q_{yy} Q_{zy} & sqrt{2}Q_{yz} Q_{zz} & Q_{yy}Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\ sqrt{2}Q_{xx} Q_{zx} & sqrt{2}Q_{xy} Q_{zy} & sqrt{2}Q_{xz} Q_{zz} & Q_{xy}Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\ sqrt{2}Q_{xx} Q_{yx} & sqrt{2}Q_{xy} Q_{yy} & sqrt{2}Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} end{bmatrix}$$

In this case, you can rotate stiffness and compliance tensors with

$$[A'] = [M] [A] [M^T], .$$