# 4th order tensor rotation - sources to refer

I am trying to model a linear elastic material in Abaqus using 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 recap 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.

## 1 Answer

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, 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.

Suppose that you have a rotation matrix

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

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

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

• Thank you for the response. This is really helpful. I appreciate your help. – Sagar Shah Jul 24 at 13:13