### Voigt Notation for Symmetric Tensors

Voigt notation is a way to represent a symmetric tensor by reducing its order. This is very useful when implementing finite element programs in Fortran. For example, the stress tensor in matrix representation is often given as

\[[ \boldsymbol{\sigma} ]=
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix}\]

In Voigt notation, it is simplified to a vector:

\[[\boldsymbol{\sigma}]= \left[ \sigma_{xx}, \sigma_{yy}, \sigma_{zz},
\sigma_{xy},\sigma_{yz},\sigma_{zx} \right] \equiv \left[ \sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}, \sigma_{31} \right]\]

For 2D problems, the stress tensor in matrix representation is simply

\[[ \boldsymbol{\sigma}]=
\begin{bmatrix}
\sigma_{xx} & \sigma_{xy} \\
\sigma_{yx} & \sigma_{yy}
\end{bmatrix}\]

In Voigt notation, it becomes:

\[[ \boldsymbol{\sigma} ] = \left[ \sigma_{xx}, \sigma_{yy}, \sigma_{xy} \right] \equiv \left[ \sigma_{11}, \sigma_{22}, \sigma_{12} \right]\]

Note that the Voigt notation used in Abaqus software is slightly different. We will cover that later.