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.