# Computational Biomechanics

Dr. Kewei Li

## Finite Element Basics

### 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.

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