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I am trying to model a linear elastic material in Abaqus using a UMAT. For my application, I need to rotate the 6x6 compliance matrix for a given set of eigenvectors (or a rotation matrix). I came across a thread titled "debugging a rotation matrix for elastic constants" where this theory was explained in very good detail.
I was wondering if there are any good sources out there that show the actual matrix rotation process either in a computational sense or just a theoretical representation.
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 $6\times 6$ 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.
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., Stephen C. Cowin. 1990. “Eigentensors of Linear Anisotropic Elastic Materials”. The Quarterly Journal of Mechanics and Applied Mathematics 43(1):15–41.
Suppose that you have a rotation matrix
$$[Q] = \begin{bmatrix} Q_{xx} &Q_{xy} &Q_{xz}\\ Q_{yx} &Q_{yy} &Q_{yz}\\ Q_{zx} &Q_{zy} &Q_{zz} \end{bmatrix}\, ,$$
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.
$$[M] =\begin{bmatrix} Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & 2 Q_{xy} Q_{xz} & 2 Q_{xx} Q_{xz} & 2 Q_{xx} Q_{xy}\\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & 2 Q_{yy} Q_{yz} & 2 Q_{yx} Q_{yz} & 2 Q_{yx} Q_{yy}\\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & 2 Q_{zy} Q_{zz} & 2 Q_{zx} Q_{zz} & 2 Q_{zx} Q_{zy}\\ Q_{yx} Q_{zx} & Q_{yy} Q_{zy} & Q_{yz} Q_{zz} & Q_{yy} Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\\ Q_{xx} Q_{zx} & Q_{xy} Q_{zy} & Q_{xz} Q_{zz} & Q_{xy} Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\\ Q_{xx} Q_{yx} & Q_{xy} Q_{yy} & Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} \end{bmatrix}$$
The rotation of a stiffness tensor in Voigt's notation is done via
$$[C'] = [M] [C] [M^T]\, .$$
$$[N] = \begin{bmatrix}Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & Q_{xy} Q_{xz} & Q_{xx} Q_{xz} & Q_{xx} Q_{xy}\\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & Q_{yy} Q_{yz} & Q_{yx} Q_{yz} & Q_{yx} Q_{yy}\\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & Q_{zy} Q_{zz} & Q_{zx} Q_{zz} & Q_{zx} Q_{zy}\\ 2 Q_{yx} Q_{zx} & 2 Q_{yy} Q_{zy} & 2 Q_{yz} Q_{zz} & Q_{yy} Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\\ 2 Q_{xx} Q_{zx} & 2 Q_{xy} Q_{zy} & 2 Q_{xz} Q_{zz} & Q_{xy} Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\\ 2 Q_{xx} Q_{yx} & 2 Q_{xy} Q_{yy} & 2 Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} \end{bmatrix}$$
The rotation of a compliance tensor in Voigt's notation is done via
$$[C'] = [N] [C] [N^T]\, .$$
$$[M] = \begin{bmatrix} Q_{xx}^{2} & Q_{xy}^{2} & Q_{xz}^{2} & \sqrt{2} Q_{xy} Q_{xz} & \sqrt{2} Q_{xx} Q_{xz} & \sqrt{2} Q_{xx} Q_{xy}\\ Q_{yx}^{2} & Q_{yy}^{2} & Q_{yz}^{2} & \sqrt{2} Q_{yy} Q_{yz} & \sqrt{2} Q_{yx} Q_{yz} & \sqrt{2} Q_{yx} Q_{yy}\\ Q_{zx}^{2} & Q_{zy}^{2} & Q_{zz}^{2} & \sqrt{2} Q_{zy} Q_{zz} & \sqrt{2} Q_{zx} Q_{zz} & \sqrt{2} Q_{zx} Q_{zy}\\ \sqrt{2}Q_{yx} Q_{zx} & \sqrt{2}Q_{yy} Q_{zy} & \sqrt{2}Q_{yz} Q_{zz} & Q_{yy}Q_{zz} + Q_{yz} Q_{zy} & Q_{yx} Q_{zz} + Q_{yz} Q_{zx} & Q_{yx} Q_{zy} + Q_{yy} Q_{zx}\\ \sqrt{2}Q_{xx} Q_{zx} & \sqrt{2}Q_{xy} Q_{zy} & \sqrt{2}Q_{xz} Q_{zz} & Q_{xy}Q_{zz} + Q_{xz} Q_{zy} & Q_{xx} Q_{zz} + Q_{xz} Q_{zx} & Q_{xx} Q_{zy} + Q_{xy} Q_{zx}\\ \sqrt{2}Q_{xx} Q_{yx} & \sqrt{2}Q_{xy} Q_{yy} & \sqrt{2}Q_{xz} Q_{yz} & Q_{xy} Q_{yz} + Q_{xz} Q_{yy} & Q_{xx} Q_{yz} + Q_{xz} Q_{yx} & Q_{xx} Q_{yy} + Q_{xy} Q_{yx} \end{bmatrix}$$
In this case, you can rotate stiffness and compliance tensors with
$$[A'] = [M] [A] [M^T]\, .$$
