Voigt notation

In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.

For example, a 2×2 symmetric tensor, X has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector \langle x_{1 1}, x_{2 2}, x_{1 2}\rangle.

As another example:

\tilde\epsilon= \begin{pmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \end{pmatrix}

In Voigt notation it is simplified to a 6 dimensional vector:

\tilde\epsilon= (\epsilon_{xx}, \epsilon_{yy}, \epsilon_{zz}, \epsilon_{yz},\epsilon_{xz},\epsilon_{xy}).

Likewise, a three-dimensional fourth-order tensor can be reduced to a 6×6 matrix.

The notation is named after physicist Woldemar Voigt. It is useful in calculations involving the generalized Hooke's law as well as finite element analysis. Hooke's law has a symmetric fourth order stiffness tensor with 81 components (3x3x3x3). Voigt notation enables that to be simplified to a 6x6 matrix. However Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. Kelvin notation is another way to represent symmetric tensors that does preserve the sum of the squares.

See also: Voigt notation, Finite element analysis, Hooke's law, Mathematics, Multilinear algebra, Symmetric tensor, Woldemar Voigt