Voight notation

The Voigt notation is hrequantly used to take advantage of the symmetry of the stress tensor to express the stress tensor as a six-dimensional vector of the following form:

$$\tilde{\mbox{\boldmath $\sigma$}}= \left[ \sigma_{x}, \sigma_y, ... ...gma_{xx}, \sigma_{yy}, \sigma_{zz}, \tau_{yz}, \tau_{xz}, \tau_{xy} \right]^T$$

The strain tensor, similar in nature to the stress tensor (both symmetric second-order tensors) can be written in Voight notation as

$$\tilde{\mbox{\boldmath $\varepsilon $}}= \left[ \varepsilon _{x}... ...n _{zz}, 2\varepsilon _{yz}, 2\varepsilon _{xz}, 2\varepsilon _{xy} \right]^T$$

The benefit of using different representations for stress and strain is the scalar invariance

   $\sigma$$$\cdot$$ $\varepsilon$$$=\sigma_{ij}\varepsilon _{ij}=\tilde{\sigma} \tilde{\varepsilon }$$

Similarly, a three-dimensional symmetric fourth-order tensor can be reduced to a [6,6] matrix.