app_gammatensor.tex

WARNING: this is not finished. 

Let us start from 
\[
\vec{\dot{\varepsilon}} = {\bm \Gamma}(\vec\sigma)\cdot \vec\sigma
\]
Note that we will later need $\bm\Gamma^{-1}$ which begs the question 
of it being invertible... 
In \textcite{zien75}, the author states: ``In many forms of the 
visco-plastic law the relationship $\vec{\dot{\varepsilon}} = {\bm \Gamma}(\vec\sigma)\cdot \vec\sigma$ is such that no volumetric strain rate 
exists i.e. the material is incompressible. 
Now ${\bm \Gamma}$ does not posses an inverse''.


%-------------------------------------------
\subsection{Computing the ${\Gamma}$ matrix}


Let us first establish that we can write quite generally in the isotropic case
\[
\frac{\partial Q}{\partial \vec\sigma} = {\bm \Gamma}_0 \cdot \vec{\sigma}
\]
By applying the chain rule we can write
\begin{eqnarray}
\frac{\partial Q}{\partial \vec\sigma} 
&=&
\frac{\partial Q}{\partial {\III}_1(\bm\sigma)} 
\frac{\partial {\III}_1(\bm\sigma)}{\partial \vec\sigma} 
+
\frac{\partial Q}{\partial {\III}_2(\bm\tau)} 
\frac{\partial {\III}_2(\bm\tau)}{\partial \vec\sigma}
+
\frac{\partial Q}{\partial {\III}_3(\bm\tau)} 
\frac{\partial {\III}_3(\bm\tau)}{\partial \vec\sigma} \nonumber\\
&=&\left(
\frac{\partial Q}{\partial {\III}_1(\bm\sigma)} {\bm M}_1(\bm\sigma)
+
\frac{\partial Q}{\partial {\III}_2(\bm\tau)}  {\bm M}_2(\bm\sigma)
+
\frac{\partial Q}{\partial {\III}_3(\bm\tau)}  {\bm M}_3(\bm\sigma)
\right) \cdot \vec\sigma
\end{eqnarray}
All we have to do now is to compute the three symmetric matrices ${\bm M}_{1,2,3}(\bm\sigma)$ which independent of $F$ or $Q$.

%............................
\subsubsection{Computing matrix $M_1$}


\begin{eqnarray}
\frac{\partial {\III}_1(\bm\sigma)}{\partial \vec\sigma} 
&=& \frac{\partial }{\partial \vec\sigma} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz})\nonumber\\
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \sigma_{xx}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) \\
\frac{\partial }{\partial \sigma_{yy}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) \\
\frac{\partial }{\partial \sigma_{zz}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) \\
\frac{\partial }{\partial \sigma_{xy}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) \\
\frac{\partial }{\partial \sigma_{xz}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) \\
\frac{\partial }{\partial \sigma_{yz}} (\sigma_{xx}+\sigma_{yy}+\sigma_{zz}) 
\end{array}
\right) \nonumber\\
&=& \left(
\begin{array}{c}
1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 
\end{array}
\right) \nonumber\\
&=&
\frac{1}{\sigma_{xx}+\sigma_{yy}+\sigma_{zz}}
\left(
\begin{array}{cccccc}
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 
\end{array}
\right) \cdot \vec\sigma \nonumber\\
&=&
\underbrace{
\frac{1}{{\III}_1(\bm\sigma)}
\left(
\begin{array}{cccccc}
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 
\end{array}
\right)}_{{\bm M}_1(\bm\sigma)}
\cdot \vec\sigma
\end{eqnarray}


%.......................................
\subsubsection{Computing matrix $M_2$}


We start from 
\begin{eqnarray}
{\III}_2({\bm \tau})   
%&=&
%\frac{1}{2}(\tau_{xx}^2 + \tau_{yy}^2 + \tau_{zz}^2 ) + \tau_{xy}^2 + %\tau_{xz}^2 + \tau_{yz}^2  \nonumber\\
&=&
\frac{1}{6}\left[(\sigma_{xx}-\sigma_{yy})^2 + (\sigma_{yy}-\sigma_{zz})^2 + (\sigma_{xx}-\sigma_{zz})^2 \right]  + \sigma_{xy}^2 + \sigma_{xz}^2 + \sigma_{yz}^2  
\end{eqnarray}
Then
\begin{eqnarray}
\frac{\partial {\III}_2(\bm\tau)}{\partial \vec\sigma}  
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \sigma_{xx}} {\III}_2(\bm\tau) \\
\frac{\partial }{\partial \sigma_{yy}} {\III}_2(\bm\tau) \\
\frac{\partial }{\partial \sigma_{zz}} {\III}_2(\bm\tau) \\
\frac{\partial }{\partial \sigma_{xy}} {\III}_2(\bm\tau) \\
\frac{\partial }{\partial \sigma_{xz}} {\III}_2(\bm\tau) \\
\frac{\partial }{\partial \sigma_{yz}} {\III}_2(\bm\tau) 
\end{array}
\right) \nonumber\\
&=&
\left(
\begin{array}{c}
\frac16(2 (\sigma_{xx}-\sigma_{yy}) + 2(\sigma_{xx}-\sigma_{zz}) ) \\
\frac16(-2(\sigma_{xx}-\sigma_{yy}) + 2(\sigma_{yy}-\sigma_{zz}) ) \\
\frac16(-2(\sigma_{yy}-\sigma_{zz}) - 2(\sigma_{xx}-\sigma_{zz}) ) \\
2 \sigma_{xy} \\
2 \sigma_{xz} \\
2 \sigma_{yz} 
\end{array}
\right) \nonumber\\
&=&
\left(
\begin{array}{c}
\frac23 \sigma_{xx} -\frac13\sigma_{yy} -\frac13\sigma_{zz}  \\
-\frac23 \sigma_{xx} +\frac43\sigma_{yy} -\frac23\sigma_{zz}  \\
-\frac23\sigma_{xx} -\frac23\sigma_{yy} +\frac43\sigma_{zz} \\
2 \sigma_{xy} \\
2 \sigma_{xz} \\
2 \sigma_{yz} 
\end{array}
\right) \nonumber\\
&=&
\underbrace{
\left(
\begin{array}{cccccc}
2/3 & -1/3 & -1/3 & 0 & 0 & 0 \\
-1/3 & 2/3 & -1/3 & 0 & 0 & 0 \\
-1/3 & -1/3 & 2/3 & 0 & 0 & 0 \\
0 & 0 & 0 & 2 & 0 & 0 \\
0 & 0 & 0 & 0 & 2 & 0 \\
0 & 0 & 0 & 0 & 0 & 2
\end{array}
\right)}_{{\bm M}_2(\bm\sigma)}
\cdot \vec\sigma
\end{eqnarray}
This is the same matrix as in Eq.~(13.11) in \textcite{zien75} (1975).

Another look at it using tensors: we start this time from 
\begin{eqnarray}
{\III}_2({\bm \tau})   
%&=&
%\frac{1}{2}(\tau_{xx}^2 + \tau_{yy}^2 + \tau_{zz}^2 
%+ \tau_{xy}^2 + \tau_{xz}^2 + \tau_{yz}^2 + \tau_{yx}^2 + \tau_{zx}^2 + \tau_{zy}^2) \\
&=&
\frac{1}{6}\left[(\sigma_{xx}-\sigma_{yy})^2 + (\sigma_{yy}-\sigma_{zz})^2 + (\sigma_{xx}-\sigma_{zz})^2 \right]  
+\frac12( \sigma_{xy}^2 + \sigma_{xz}^2 + \sigma_{yz}^2  
+ \sigma_{yx}^2 + \sigma_{zx}^2 + \sigma_{zy}^2) 
\end{eqnarray}
Then
\begin{eqnarray}
\frac{\partial {\III}_2(\bm\tau)}{\partial \bm\sigma}  
&=&
\left(
\begin{array}{ccc}
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{xx}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{xy}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{xz}} \\
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{yx}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{yy}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{yz}} \\
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{zx}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{zy}} &   
\frac{\partial {\III}_2(\bm\tau)}{\partial \sigma_{zz}} 
\end{array}
\right) \nonumber\\
&=&
\left(
\begin{array}{ccc}
\frac23 \sigma_{xx} -\frac13\sigma_{yy} -\frac13\sigma_{zz} & \sigma_{xy}& \sigma_{xz} \\
\sigma_{yx}&-\frac23 \sigma_{xx} +\frac43\sigma_{yy} -\frac23\sigma_{zz} & \sigma_{yz}\\
\sigma_{zx}& \sigma_{zy}& -\frac23\sigma_{xx} -\frac23\sigma_{yy} +\frac43\sigma_{zz} 
\end{array}
\right) \nonumber\\
&=&{\bm\sigma} - \frac13 {\III}_1(\bm\sigma)\bm 1 \nonumber\\
&=& \bm \tau \label{eq:dI2dtau}
\end{eqnarray}









%............................
\subsubsection{Computing matrix $M_3$}

The third invariant proves to be the most annoying:
\begin{eqnarray}
{\III}_3({\bm \tau}) 
&=& \frac13 \sum_{i,j,k} \tau_{ij}\tau_{jk}\tau_{ki} \\
&=& \frac{1}{3} \tau_{xx} (  \tau_{xx}^2 + 3 \tau_{xy}^2 + 3 \tau_{xz}^2  )     \nonumber\\
&+& \frac{1}{3} \tau_{yy} (3 \tau_{xy}^2 +   \tau_{yy}^2 + 3 \tau_{yz}^2  )     \nonumber\\
&+& \frac{1}{3} \tau_{zz} (3 \tau_{xz}^2 + 3 \tau_{yz}^2 +   \tau_{zz}^2)       \nonumber\\
&+& 2 \tau_{xy} \tau_{xz} \tau_{yz}  
\end{eqnarray}
Then 

\begin{eqnarray}
\frac{\partial {\III}_3(\bm\tau)}{\partial \vec\sigma}  
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \sigma_{xx}} {\III}_3(\bm\tau) \\
\frac{\partial }{\partial \sigma_{yy}} {\III}_3(\bm\tau) \\
\frac{\partial }{\partial \sigma_{zz}} {\III}_3(\bm\tau) \\
\frac{\partial }{\partial \sigma_{xy}} {\III}_3(\bm\tau) \\
\frac{\partial }{\partial \sigma_{xz}} {\III}_3(\bm\tau) \\
\frac{\partial }{\partial \sigma_{yz}} {\III}_3(\bm\tau) 
\end{array}
\right) \nonumber\\
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \tau_{xx}} {\III}_3(\bm\tau) 
\frac{\partial \tau_{xx}}{\partial \sigma_{xx}}\\
\frac{\partial }{\partial \tau_{yy}} {\III}_3(\bm\tau) 
\frac{\partial \tau_{yy}}{\partial \sigma_{yy}}\\
\frac{\partial }{\partial \tau_{zz}} {\III}_3(\bm\tau)
\frac{\partial \tau_{zz}}{\partial \sigma_{zz}}\\
\frac{\partial }{\partial \tau_{xy}} {\III}_3(\bm\tau)
\frac{\partial \tau_{xy}}{\partial \sigma_{xy}}\\
\frac{\partial }{\partial \tau_{xz}} {\III}_3(\bm\tau)
\frac{\partial \tau_{xz}}{\partial \sigma_{xz}}\\
\frac{\partial }{\partial \tau_{yz}} {\III}_3(\bm\tau)
\frac{\partial \tau_{yz}}{\partial \sigma_{yz}}\\
\end{array}
\right) \nonumber\\
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \tau_{xx}} {\III}_3(\bm\tau) \frac23 \\
\frac{\partial }{\partial \tau_{yy}} {\III}_3(\bm\tau) \frac23 \\
\frac{\partial }{\partial \tau_{zz}} {\III}_3(\bm\tau) \frac23 \\
\frac{\partial }{\partial \tau_{xy}} {\III}_3(\bm\tau)  1\\
\frac{\partial }{\partial \tau_{xz}} {\III}_3(\bm\tau)  1\\
\frac{\partial }{\partial \tau_{yz}} {\III}_3(\bm\tau)  1\\
\end{array}
\right) \nonumber\\
&=& \left(
\begin{array}{c}
\frac{\partial }{\partial \tau_{xx}} {\III}_3(\bm\tau) \frac23 \\
\frac{\partial }{\partial \tau_{yy}} {\III}_3(\bm\tau) \frac23 \\
\frac{\partial }{\partial \tau_{zz}} {\III}_3(\bm\tau) \frac23 \\
2\tau_{xx}\tau_{xy} +2 \tau_{yy}\tau_{xy} + 2 \tau_{xz}\tau_{yz} \\
2\tau_{xx}\tau_{xz} +2 \tau_{zz}\tau_{xz} + 2 \tau_{xy}\tau_{yz} \\
2\tau_{yy}\tau_{yz} +2 \tau_{zz}\tau_{yz} + 2 \tau_{xy}\tau{xz} 
\end{array}
\right) \\
&=& \left(
\begin{array}{c}
(\tau_{xx}^2 +  \tau_{xy}^2 +  \tau_{xz}^2 ) \frac23 \\
(\tau_{xy}^2 +  \tau_{yy}^2 +  \tau_{yz}^2 ) \frac23 \\
(\tau_{xz}^2 +  \tau_{yz}^2 +  \tau_{zz}^2 ) \frac23 \\
2\tau_{xx}\tau_{xy} +2 \tau_{yy}\tau_{xy} + 2 \tau_{xz}\tau_{yz} \\
2\tau_{xx}\tau_{xz} +2 \tau_{zz}\tau_{xz} + 2 \tau_{xy}\tau_{yz} \\
2\tau_{yy}\tau_{yz} +2 \tau_{zz}\tau_{yz} + 2 \tau_{xy}\tau{xz} 
\end{array}
\right) 
\end{eqnarray}

We have ${\bm\tau}=\bm\sigma -\frac13 {\III}_1(\bm\sigma) {\bm 1}$
so 
\begin{align}
\tau_{xx} &= \sigma_{xx}-\frac13{\III}_1 & \Rightarrow 
\tau_{xx}^2 &= (\sigma_{xx}-\frac13{\III}_1)^2
=\sigma_{xx}^2 -\frac23 \sigma_{xx} {\III}_1 + \frac19{\III}_1^2
\\
\tau_{yy} &= \sigma_{yy}-\frac13{\III}_1 & \Rightarrow 
\tau_{yy}^2 &= (\sigma_{yy}-\frac13{\III}_1)^2
=\sigma_{yy}^2 -\frac23 \sigma_{yy} {\III}_1 + \frac19{\III}_1^2
\\
\tau_{zz} &= \sigma_{zz}-\frac13{\III}_1 & \Rightarrow
\tau_{zz}^2 &= (\sigma_{zz}-\frac13{\III}_1)^2
=\sigma_{zz}^2 -\frac23 \sigma_{zz} {\III}_1 + \frac19{\III}_1^2
\end{align}

Finally
\begin{eqnarray}
\frac{\partial {\III}_3(\bm\tau)}{\partial \vec\sigma}  
&=& \left(
\begin{array}{c}
(\sigma_{xx}^2 -\frac23 \sigma_{xx} {\III}_1 + \frac19{\III}_1^2 +  \sigma_{xy}^2 +  \sigma_{xz}^2 ) \frac23 \\
(\sigma_{xy}^2 +  \sigma_{yy}^2 -\frac23 \sigma_{yy} {\III}_1 + \frac19{\III}_1^2 +  \sigma_{yz}^2 ) \frac23 \\
(\sigma_{xz}^2 +  \sigma_{yz}^2 + \sigma_{zz}^2 -\frac23 \sigma_{zz} {\III}_1 + \frac19{\III}_1^2 ) \frac23 \\
2(\sigma_{xx}-\frac13{\III}_1)\sigma_{xy} +2 (\sigma_{yy}-\frac13{\III}_1)\sigma_{xy} + 2 \sigma_{xz}\sigma_{yz} \\
2(\sigma_{xx}-\frac13{\III}_1)\sigma_{xz} +2 (\sigma_{zz}-\frac13{\III}_1)\sigma_{xz} + 2 \sigma_{xy}\sigma_{yz} \\
2(\sigma_{yy}-\frac13{\III}_1)\sigma_{yz} +2 (\sigma_{zz}-\frac13{\III}_1)\sigma_{yz} + 2 \sigma_{xy}\sigma{xz} 
\end{array}
\right) \\
&=& \left(
\begin{array}{c}
(\sigma_{xx}^2 -\frac23 \sigma_{xx} {\III}_1  +  \sigma_{xy}^2 +  \sigma_{xz}^2 ) \frac23 \\
(\sigma_{xy}^2 +  \sigma_{yy}^2 -\frac23 \sigma_{yy} {\III}_1  +  \sigma_{yz}^2 ) \frac23 \\
(\sigma_{xz}^2 +  \sigma_{yz}^2 + \sigma_{zz}^2 -\frac23 \sigma_{zz} {\III}_1  ) \frac23 \\
2\sigma_{xx}\sigma_{xy} +2 \sigma_{yy}\sigma_{xy} + 2 \sigma_{xz}\sigma_{yz} \\
2\sigma_{xx}\sigma_{xz} +2 \sigma_{zz}\sigma_{xz} + 2 \sigma_{xy}\sigma_{yz} \\
2\sigma_{yy}\sigma_{yz} +2 \sigma_{zz}\sigma_{yz} + 2 \sigma_{xy}\sigma{xz} 
\end{array}
\right) 
+
\left(
\begin{array}{c}
\frac19{\III}_1^2 \frac23\\
\frac19{\III}_1^2 \frac23\\
\frac19{\III}_1^2 \frac23\\
-\frac43{\III}_1 \sigma_{xy} \\
-\frac43{\III}_1 \sigma_{xz} \\
-\frac43{\III}_1 \sigma_{yz} \\
\end{array}
\right) \nonumber\\
&=&
\left(
\begin{array}{c}
(\sigma_{xx}^2 -\frac23 \sigma_{xx} {\III}_1  +  \sigma_{xy}^2 +  \sigma_{xz}^2 ) \frac23 \\
(\sigma_{xy}^2 +  \sigma_{yy}^2 -\frac23 \sigma_{yy} {\III}_1  +  \sigma_{yz}^2 ) \frac23 \\
(\sigma_{xz}^2 +  \sigma_{yz}^2 + \sigma_{zz}^2 -\frac23 \sigma_{zz} {\III}_1  ) \frac23 \\
2(\sigma_{xx}+\sigma_{yy})\sigma_{xy} + 2 \sigma_{xz}\sigma_{yz} \\
2(\sigma_{xx}+\sigma_{zz})\sigma_{xz} + 2 \sigma_{xy}\sigma_{yz} \\
2(\sigma_{yy}+\sigma_{zz})\sigma_{yz} + 2 \sigma_{xy}\sigma{xz} 
\end{array}
\right) 
+
\frac{2}{3}{\III}_1
\left(
\begin{array}{c}
\frac19{\III}_1 \\
\frac19{\III}_1 \\
\frac19{\III}_1 \\
-2\sigma_{xy} \\
-2\sigma_{xz} \\
-2\sigma_{yz} \\
\end{array}
\right) \\
&=&
\left(
\begin{array}{c}
(\sigma_{xx}^2 -\frac23 \sigma_{xx} {\III}_1  +  \sigma_{xy}^2 +  \sigma_{xz}^2 ) \frac23 \\
(\sigma_{xy}^2 +  \sigma_{yy}^2 -\frac23 \sigma_{yy} {\III}_1  +  \sigma_{yz}^2 ) \frac23 \\
(\sigma_{xz}^2 +  \sigma_{yz}^2 + \sigma_{zz}^2 -\frac23 \sigma_{zz} {\III}_1  ) \frac23 \\
2({\III}_1-\sigma_{zz})\sigma_{xy} + 2 \sigma_{xz}\sigma_{yz} \\
2({\III}_1-\sigma_{yy})\sigma_{xz} + 2 \sigma_{xy}\sigma_{yz} \\
2({\III}_1-\sigma_{xx})\sigma_{yz} + 2 \sigma_{xy}\sigma{xz} 
\end{array}
\right) 
+
\frac{2}{3}{\III}_1
\left(
\begin{array}{cccccc}
\frac13 \sigma_{xx} & \frac13 \sigma_{yy} & \frac13 \sigma_{zz} && \\
\frac13 \sigma_{xx} & \frac13 \sigma_{yy} & \frac13 \sigma_{zz} && \\
\frac13 \sigma_{xx} & \frac13 \sigma_{yy} & \frac13 \sigma_{zz} && \\
 &&&-2 \\
&&&&-2 \\
&&&&&-2 \\
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
\sigma_{xx}\\
\sigma_{yy}\\
\sigma_{zz}\\
\sigma_{xy}\\
\sigma_{xz}\\
\sigma_{yz}
\end{array}
\right)
\end{eqnarray}




FINISH!!!!! not complicated but no rush. Also try to see whether it matches
table I of \textcite{zico74} - note the different ordering of terms in vector
\begin{eqnarray}
&&
\left(
\begin{array}{cccccc}
\frac13 \sigma_{xx} & \frac13\sigma_{zz} & \frac13\sigma_{yy} & 
-\frac23 \sigma_{yz} & \frac13\sigma_{xz} & \frac13\sigma_{xy}\\
\frac13\sigma_{zz} & \frac13\sigma_{yy} & \frac13 \sigma_{xx} &
\frac13 \sigma_{yz} & -\frac23\sigma_{xz} & \frac13\sigma_{xy}\\
\frac13\sigma_{yy} & \frac13 \sigma_{xx} & \frac13\sigma_{zz} &
\frac13 \sigma_{yz} & \frac13\sigma_{xz} & -\frac23\sigma_{xy}\\
-\frac23 \sigma_{yz} & \frac13 \sigma_{yz} & \frac13 \sigma_{yz}&
-\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\frac13\sigma_{xz} & -\frac23\sigma_{xz} & \frac13\sigma_{xz} &
\sigma_{xy} & -\sigma_{yy} & \sigma_{yz} \\
\frac13\sigma_{xy} & \frac13\sigma_{xy} & -\frac23\sigma_{xy} &
\sigma_{xz} & \sigma_{yz} & -\sigma_{zz}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
\sigma_{xx}\\
\sigma_{yy}\\
\sigma_{zz}\\
\sigma_{yz}\\
\sigma_{xz}\\
\sigma_{xy}
\end{array}
\right)\\
&=&
\frac13
\left(
\begin{array}{cccccc}
 \sigma_{xx} & \sigma_{zz} & \sigma_{yy} & 
-2 \sigma_{yz} & \sigma_{xz} & \sigma_{xy}\\
\sigma_{zz} & \sigma_{yy} &  \sigma_{xx} &
 \sigma_{yz} & -2\sigma_{xz} & \sigma_{xy}\\
\sigma_{yy} & \sigma_{xx} & \sigma_{zz} &
 \sigma_{yz} & \sigma_{xz} & -2\sigma_{xy}\\
-2 \sigma_{yz} & \sigma_{yz} &  \sigma_{yz} &
-3\sigma_{xx} & 3\sigma_{xy} & 3\sigma_{xz} \\
\sigma_{xz} & -2\sigma_{xz} & \sigma_{xz} &
3\sigma_{xy} & -3\sigma_{yy} & 3\sigma_{yz} \\
\sigma_{xy} & \sigma_{xy} & -2\sigma_{xy} &
3\sigma_{xz} & 3\sigma_{yz} & -3\sigma_{zz}
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
\sigma_{xx}\\
\sigma_{yy}\\
\sigma_{zz}\\
\sigma_{yz}\\
\sigma_{xz}\\
\sigma_{xy}
\end{array}
\right)\\
&=&
\left(
\begin{array}{c}
\sigma_{xx}^2+2\sigma_{yy}\sigma_{zz}
-\sigma_{xy}\sigma_{yz}+\sigma_{xz}^2\\
\sigma_{yy}^2+2\sigma_{xx}\sigma_{zz}
+2\sigma_{xy}\sigma_{yz}-2\sigma_{xz}^2\\
\sigma_{zz}^2+2\sigma_{xx}\sigma_{yy}
-\sigma_{xy}\sigma_{yz}+\sigma_{xz}^2\\
(-2\sigma_{xx}+\sigma_{yy}+\sigma_{zz})\sigma_{yz} 
-3\sigma_{xx}\sigma_{xy}
+3\sigma_{xy}\sigma_{xz}
+3\sigma_{xz}\sigma_{yz}\\
(\sigma_{xx}-2\sigma_{yy}+\sigma_{zz})\sigma_{xz} 
+3\sigma_{xy}^2
-3\sigma_{yy}\sigma_{xz}
+3\sigma_{yz}^2\\
(\sigma_{xx}+\sigma_{yy}-2\sigma_{zz})\sigma_{xy} \\

+3\sigma_{xy}\sigma_{xz}
+3\sigma_{xz}\sigma_{xz}
-3\sigma_{xz}\sigma_{yz}\\


\end{array}
\right)
\end{eqnarray}