Editing Voigt notation (section)

{{Short description|Mathematical Concept}}

In [[mathematics]], '''Voigt notation''' or '''Voigt form''' in [[multilinear algebra]] is a way to represent a [[symmetric tensor]] by reducing its order.<ref name = "Voigt">{{Cite book
  | title = Lehrbuch der Kristallphysik
  | author = Woldemar Voigt
  | year = 1910
  | publisher = Teubner, Leipzig
  | url = https://archive.org/details/bub_gb_SvPPAAAAMAAJ
  | access-date = November 29, 2016
}}</ref> 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<ref name="Helbig">{{Cite book
  | author = Klaus Helbig
  | title = Foundations of anisotropy for exploration seismics
  | publisher = Pergamon
  | year = 1994
  | isbn = 0-08-037224-4
}}</ref> 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. The notation is named after physicists [[Woldemar Voigt]]<ref name="Voigt"/> & [[John Nye (scientist)]].

For example, a 2&times;2 symmetric tensor ''X'' has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus its rank can be reduced by expressressing it as a vector without loss of information:

<math display="block"> X = \begin{bmatrix} x_{11} & x_{12} \\ x_{12} & x_{22} \end{bmatrix} = \begin{bmatrix} x_{1 1} \\ x_{2 2} \\ x_{1 2} \end{bmatrix}.</math>

Voigt notation is used in [[materials science]] to simplify the representation of the rank-2 stress and strain tensors, and fourth-rank stiffness and compliance tensors.

The 3&times;3 [[Cauchy stress tensor|stress]] and [[Strain_(mechanics)#Strain_tensor|strain]] tensors in their full forms can be written as:
:<math>\boldsymbol{\sigma}=
\begin{bmatrix}
  \sigma_{11} & \sigma_{12} & \sigma_{13} \\
  \sigma_{21} & \sigma_{22} & \sigma_{23} \\
  \sigma_{31} & \sigma_{32} & \sigma_{33}
\end{bmatrix} \quad </math> and <math> \quad \boldsymbol{\varepsilon}=
\begin{bmatrix}
  \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
  \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
  \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
\end{bmatrix} </math>.

Voigt notation then utilises the symmetry of these matrices (<math>\sigma_{12} = \sigma_{21} </math> and so on) to express them instead as a 6&times;1 vector:

:<math>\underline\sigma = \begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} := \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{13} \\ \sigma_{12} \end{bmatrix} \quad
</math> and <math> \quad \underline\varepsilon = \begin{bmatrix} \varepsilon_1 \\ \varepsilon_2 \\ \varepsilon_3 \\ \varepsilon_4 \\ \varepsilon_5 \\ \varepsilon_6 \end{bmatrix} := \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \gamma_{23} \\ \gamma_{13} \\ \gamma_{12} \end{bmatrix}
</math>

where <math>\gamma_{12}=2\varepsilon_{12}</math>, <math>\gamma_{23} = 2\varepsilon_{23}</math>, and <math>\gamma_{13} = 2\varepsilon_{13}</math> are the engineering shear strains.

The benefit of using different representations for stress and strain is that the scalar invariance
<math display="block"> \boldsymbol{\sigma}\cdot\boldsymbol{\varepsilon} = \sigma_{ij}\varepsilon_{ij} = \underline\sigma \cdot \underline\varepsilon
</math>
is preserved.

This notation now allows the three-dimensional symmetric fourth-order [[Hooke%27s_law#Matrix_representation_(stiffness_tensor)|stiffness]], <math>C</math>, and compliance, <math>S</math>, tensors to be reduced to 6&times;6 matrices:

<math display="block"> C_{ijkl}  \Rightarrow C_{\alpha \beta} = \begin{bmatrix}
 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{bmatrix}.</math>