Talk:Coordinate transformations

=Do not use the Voigt notation !=

The projection of second order tensor in the orthonormal base is not the Voigt one. I think it's a better idea to introduce real base.



E_{1}=e_1 \otimes e_1 $$



E_{2}=e_2 \otimes e_2 $$



E_{3}=e_3 \otimes e_3 $$



E_{4}=\frac{1}{\sqrt{2}}(e_2 \otimes e_3 + e_3 \otimes e_2) $$



E_{5}=\frac{1}{\sqrt{2}}( e_3 \otimes e_1 + e_1 \otimes e_3) $$



E_{6}=\frac{1}{\sqrt{2}}( e_1 \otimes e_2 + e_2 \otimes e_1) $$

The stress and strain tensors are now defined by :



\left \{\sigma \right \} = \left \{ \begin{align} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sqrt{2}\sigma_{23} \\ \sqrt{2}\sigma_{31} \\ \sqrt{2}\sigma_{12} \\ \end{align} \right \} $$ and

\left \{\varepsilon \right \} = \left \{ \begin{align} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \sqrt{2}\varepsilon_{23} \\ \sqrt{2}\varepsilon_{31} \\ \sqrt{2}\varepsilon_{12} \\ \end{align} \right \} $$

Then once constructs the bound matrix in the orthonormal base $$ E_{i} \otimes E_{j}$$



\left [ \hat{R}(\theta) \right ]= \left [ \begin{align} R_{11}^2 & R_{12}^2 & R_{13}^2 & \sqrt{2}R_{12}R_{13} & \sqrt{2}R_{11}R_{13} & \sqrt{2}R_{11}R_{12}\\ R_{21}^2 & R_{22}^2 & R_{23}^2 & \sqrt{2}R_{22}R_{23} & \sqrt{2}R_{21}R_{23} & \sqrt{2}R_{22}R_{21}\\ R_{31}^2 & R_{32}^2 & R_{33}^2 & \sqrt{2}R_{33}R_{32} & \sqrt{2}R_{33}R_{31} & \sqrt{2}R_{31}R_{32}\\ \sqrt{2}R_{21}R_{31} & \sqrt{2}R_{22}R_{32} & \sqrt{2}R_{23}R_{33} & R_{22}R_{32}+R_{23}R_{32} & R_{21}R_{33}+R_{31}R_{23} & R_{21}R_{32}+R_{31}R_{22}\\ \sqrt{2}R_{11}R_{31} & \sqrt{2}R_{12}R_{32} & \sqrt{2}R_{13}R_{23} & R_{12}R_{23}+R_{32}R_{13} & R_{11}R_{33}+R_{13}R_{31} & R_{11}R_{32}+R_{31}R_{12}\\ \sqrt{2}R_{11}R_{21} & \sqrt{2}R_{12}R_{22} & \sqrt{2}R_{13}R_{23} & R_{12}R_{23}+R_{22}R_{13} & R_{11}R_{23}+R_{21}R_{13} & R_{11}R_{22}+R_{21}R_{12}\\ \end{align} \right ] $$

with

$$ \left [ R(\theta) \right ] $$ the rotation matrix in $$ e_{i} \otimes e_{j}$$ base.

Example


\left [ R(\theta) \right ]= \left [ \begin{matrix} 1 & 0 & 0 \\ 0 & cos \theta & sin \theta \\ 0 & -sin \theta& cos \theta \end{matrix} \right ] $$ is the rotation along the axis $$e_1$$ in the :$$e_i \otimes e_j$$ base

The associated rotation in the $$E_i \otimes E_j$$ base is :



\left [ \hat{R}(\theta) \right ]= \left [ \begin{matrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & cos^2 \theta & sin^2 \theta & \sqrt{2} sin \theta cos \theta & 0 & 0 \\ 0 & sin^2 \theta & cos^2 \theta & -\sqrt{2} sin \theta cos \theta & 0 & 0 \\ 0 & - \sqrt{2} sin \theta cos \theta & \sqrt{2} sin \theta cos \theta  & cos^2 \theta - sin^2 \theta & 0 \\ 0 & 0 & 0 & 0 & cos \theta & -sin \theta \\ 0 & 0 & 0 & 0 & sin \theta & cos \theta \\ \end{matrix} \right ] $$

The rotation of a second order tensor is now defined by :
 * $$ \left \{ \sigma(\theta) \right \} = {\left [ \hat{R}(\theta) \right ]}^T \left \{ \sigma \right \} $$

=Four order tensor=

The élasticity tensor $$C_{ijkl}$$ in the :$$e_i \otimes e_j \otimes e_k \otimes e_l$$ is defined in the  :$$E_i\otimes E_j$$ by



\left [ \overline{C} \right ] = \left[\begin{align} C_{1111} & C_{1122} & C_{1133} & \sqrt{2}C_{1123} & \sqrt{2}C_{1131} & \sqrt{2}C_{1112} \\ C_{1122} & C_{2222} & C_{2233} & \sqrt{2}C_{2223} & \sqrt{2}C_{2231} & \sqrt{2}C_{2212} \\ C_{1133} & C_{2233} & C_{3333} & \sqrt{2}C_{3323} & \sqrt{2}C_{3331} & \sqrt{2}C_{3312} \\ \sqrt{2}C_{1123} & \sqrt{2}C_{2223} &  \sqrt{2}C_{2333} & 2C_{2323} & 2C_{2331} & 2C_{2312} \\ \sqrt{2}C_{1131} & \sqrt{2}C_{2231} &  \sqrt{2}C_{3331} & 2C_{2331} & 2C_{3131} & 2C_{3112} \\ \sqrt{2}C_{1112} & \sqrt{2}C_{2212} &  \sqrt{2}C_{3312} & 2C_{2312} & 2C_{3112} & 2C_{1212} \end{align}\right] $$

and is rotated by:



{\left [ \overline{C} (\theta) \right ]}_g = {\left [ \hat{R}(\theta) \right ]}^T \left [ \overline{C} \right ]\left [ \hat{R}(\theta) \right ] $$