# Voigt notation

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Voigt notation is useful to understand the 4th-rank tensors of geophysics, for example the elastic stiffness tensor ${\displaystyle \{C_{ijkl}\}}$, needed for wave propagation, or the elastic compliance tensor ${\displaystyle \{S_{ijkl}\}}$, needed for geomechanics. These tensors are hard to understand intuitively, since one cannot write them down on paper. (If you use ${\displaystyle i}$ to index the rows, and ${\displaystyle j}$ to index the columns, where do you put the indices ${\displaystyle k}$ and ${\displaystyle l}$ ?) But, Woldemar Voigt (1850-1919) realized that because stress and strain are symmetric tensors, and because the order of the index pairs ${\displaystyle ij}$ and ${\displaystyle kl}$ can be interchanged, all the information in these 4th-rank tensors is contained in symmetric 2nd-rank matrices (which can be written on paper!). (This transform, called Voigt notation, should be used for displays (to aid intuition), not for calculation, since the matrices do not obey tensor algebra.)

For example, the stiffness tensor ${\displaystyle \{C_{ijkl}\}}$ can be represented as the stiffness matrix ${\displaystyle \{C_{\alpha \beta }\}}$:

${\displaystyle \{C_{ijkl}\}}$ => ${\displaystyle \{C_{\alpha \beta }\}={\begin{pmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&c_{66}\\\end{pmatrix}}}$

A common convention is to use Latin indices ${\displaystyle i}$ (1...3) for the tensors, and Greek indices ${\displaystyle \alpha }$ (1...6) for the matrices. The mapping of the tensor indices ${\displaystyle ij}$ to the matrix indices ${\displaystyle \alpha }$ is:

${\displaystyle 11}$ => ${\displaystyle 1}$

${\displaystyle 22}$ => ${\displaystyle 2}$

${\displaystyle 33}$ => ${\displaystyle 3}$

${\displaystyle 23}$ = ${\displaystyle 32}$ => ${\displaystyle 4}$

${\displaystyle 13}$ = ${\displaystyle 31}$ => ${\displaystyle 5}$

${\displaystyle 12}$ = ${\displaystyle 21}$ => ${\displaystyle 6}$

Hence, the compliance matrix above can be written as:

${\displaystyle \{C_{\alpha \beta }\}={\begin{pmatrix}C_{1111}&C_{1122}&C_{1133}&C_{1123}&C_{1113}&C_{1112}\\C_{1122}&C_{2222}&C_{2233}&C_{2223}&C_{2213}&C_{2212}\\C_{1133}&C_{2233}&C_{3333}&C_{3323}&C_{3313}&C_{3312}\\C_{1123}&C_{2223}&C_{3323}&C_{2323}&C_{2313}&C_{2312}\\C_{1113}&C_{2213}&C_{3313}&C_{2313}&C_{2323}&C_{2312}\\C_{1112}&C_{2212}&C_{3312}&C_{2312}&C_{1312}&c_{1212}\\\end{pmatrix}}}$

(The compliances use a modification of this Voigt notation.)