Voigt notation

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In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. There are a few variants and associated names for this idea: Mandel notation, Mandel-Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig (1994) of old ideas of Lord Kelvin. The differences here lie in certain weights attached to the selected entries of the tensor. Nomenclature may vary according to what is traditional in the field of application.

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.

[edit] Mandel notation

For a symmetric tensor of second rank

$\tilde\sigma= \begin{pmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{pmatrix}$

only six components are distinct, the three on the diagonal and the other being off-diagonal. Thus it can be expressed, in Mandel notation, as the vector

$\tilde \sigma ^M= \langle \sigma_{11}, \sigma_{22}, \sigma_{33}, \sqrt 2 \sigma_{12}, \sqrt 2 \sigma_{23}, \sqrt 2 \sigma_{13} \rangle$

The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example:

$\tilde \sigma : \tilde \sigma = \tilde \sigma^M \cdot \tilde \sigma^M = \sigma_{11}^2 + \sigma_{22}^2 + \sigma_{33}^2 + 2 \sigma_{12}^2+ 2 \sigma_{23}^2+ 2 \sigma_{13}^2$

A symmetric tensor of rank four satisfying $D i j k l = D j i k l$ and $D i j k l = D i j l k$ has 81 components in three-dimensional space, but only 36 components are distinct. Thus, in Mandel notation, it can be expressed as

$\tilde D^M= \begin{pmatrix} D_{1111} & D_{1122} & D_{1133} & \sqrt 2 D_{1112} & \sqrt 2 D_{1123} & \sqrt 2 D_{1113} \\ D_{2211} & D_{2222} & D_{2233} & \sqrt 2 D_{2212} & \sqrt 2 D_{2223} & \sqrt 2 D_{2213} \\ D_{3311} & D_{3322} & D_{3333} & \sqrt 2 D_{3312} & \sqrt 2 D_{3323} & \sqrt 2 D_{3313} \\ \sqrt 2 D_{1211} & \sqrt 2 D_{1222} & \sqrt 2 D_{1233} & 2 D_{1212} & 2 D_{1223} & 2 D_{1213} \\ \sqrt 2 D_{2311} & \sqrt 2 D_{2322} & \sqrt 2 D_{2333} & 2 D_{2312} & 2 D_{2323} & 2 D_{2313} \\ \sqrt 2 D_{1311} & \sqrt 2 D_{1322} & \sqrt 2 D_{1333} & 2 D_{1312} & 2 D_{1323} & 2 D_{1313} \\ \end{pmatrix}$

[edit] Applications

The notation is named after physicist Woldemar Voigt. It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalized Hooke's law, as well as finite element analysis.

Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3). Voigt notation enables this to be simplified to a 6×6 matrix. However, Voigt's form does not preserve the sum of the squares, which in the case of Hooke's law has geometric significance. This explains why weights are introduced (to make the mapping an isometry).