# 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.

${\displaystyle E_{1}=e_{1}\otimes e_{1}}$
${\displaystyle E_{2}=e_{2}\otimes e_{2}}$
${\displaystyle E_{3}=e_{3}\otimes e_{3}}$
${\displaystyle E_{4}={\frac {1}{\sqrt {2}}}(e_{2}\otimes e_{3}+e_{3}\otimes e_{2})}$
${\displaystyle E_{5}={\frac {1}{\sqrt {2}}}(e_{3}\otimes e_{1}+e_{1}\otimes e_{3})}$
${\displaystyle E_{6}={\frac {1}{\sqrt {2}}}(e_{1}\otimes e_{2}+e_{2}\otimes e_{1})}$

The stress and strain tensors are now defined by :

{\displaystyle \left\{\sigma \right\}=\left\{{\begin{aligned}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\{\sqrt {2}}\sigma _{23}\\{\sqrt {2}}\sigma _{31}\\{\sqrt {2}}\sigma _{12}\\\end{aligned}}\right\}}

and

{\displaystyle \left\{\varepsilon \right\}=\left\{{\begin{aligned}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\{\sqrt {2}}\varepsilon _{23}\\{\sqrt {2}}\varepsilon _{31}\\{\sqrt {2}}\varepsilon _{12}\\\end{aligned}}\right\}}

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

{\displaystyle \left[{\hat {R}}(\theta )\right]=\left[{\begin{aligned}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{aligned}}\right]}

with

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

## Example

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

is the rotation along the axis ${\displaystyle e_{1}}$ in the :${\displaystyle e_{i}\otimes e_{j}}$ base

The associated rotation in the ${\displaystyle E_{i}\otimes E_{j}}$ base is :

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

The rotation of a second order tensor is now defined by :

${\displaystyle \left\{\sigma (\theta )\right\}={\left[{\hat {R}}(\theta )\right]}^{T}\left\{\sigma \right\}}$

# Four order tensor

The élasticity tensor ${\displaystyle C_{ijkl}}$ in the :${\displaystyle e_{i}\otimes e_{j}\otimes e_{k}\otimes e_{l}}$ is defined in the :${\displaystyle E_{i}\otimes E_{j}}$ by

{\displaystyle \left[{\overline {C}}\right]=\left[{\begin{aligned}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{aligned}}\right]}

and is rotated by:

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