3.3.2.2 Transformation Operator

The transformation operator describing three successive rotations with Euler's Angles $\alpha$, $\beta$ and $\gamma$ is given as a product of three rotations:

$$\hat{U}(\alpha,\beta,\gamma)=\hat{U}_{z}(\alpha)\hat{U}_{y^{'}}(\beta)\hat{U}_{z^{'}}(\gamma),$$ (3.40)

where the unitary operators $\hat{U}_{z}(\alpha)$, $\hat{U}_{y^{'}}(\beta)$ and $\hat{U}_{z^{'}}(\gamma)$ are given through the expressions:

$$\hat{U}_{z}(\alpha)=\begin{pmatrix}\cos\alpha & -\sin\alpha & 0\\ \sin\alpha & \cos\alpha & 0\\ 0 & 0 & 1\end{pmatrix},$$ (3.41)

$$\hat{U}_{y^{'}}(\beta)=\begin{pmatrix}\cos\beta & 0 & \sin\beta\\ 0 & 1 & 0\\ -\sin\beta & 0 & \cos\beta\end{pmatrix},$$ (3.42)

$$\hat{U}_{z^{'}}(\gamma)=\begin{pmatrix}\cos\gamma & -\sin\gamma & 0\\ \sin\gamma & \cos\gamma & 0\\ 0 & 0 & 1\end{pmatrix}.$$ (3.43)

Thus the transformation operator (3.40) takes the form:

$$\small \hat{U}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha\cos... ...sin\beta\\ -\sin\beta\cos\gamma & \sin\beta\sin\gamma & \cos\beta\end{pmatrix}.$$ (3.44)

Due to the symmetry property (3.32) the transformed strain tensor will not depend on $\gamma$. So $\gamma$ is arbitrary and can be set to zero. The transformation operator takes the final form:

$$\hat{U}(\alpha,\beta)=\begin{pmatrix}\cos\alpha\cos\beta & -\sin\... ...a & \cos\alpha & \sin\alpha\sin\beta\\ -\sin\beta & 0 & \cos\beta\end{pmatrix}.$$ (3.45)

S. Smirnov: