Editing Voigt notation

{{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>

== Mnemonic rule ==
A simple [[Mnemonic|mnemonic rule]] for memorizing Voigt notation is as follows:

* Write down the second order tensor in matrix form (in the example, the stress tensor)
* Strike out the diagonal
* Continue on the third column
* Go back to the first element along the first row.

Voigt indexes are numbered consecutively from the starting point to the end (in the example, the numbers in blue).

[[File:Voigt notation Mnemonic rule.svg|180px]]

The diagram below also shows the order of the indices:
<math display="block"> 
\begin{matrix}
ij & =\\
\Downarrow & \\
\alpha  & =
\end{matrix}

 \begin{matrix}
11 & 22 & 33 & 23,32 & 13,31 & 12,21 \\
\Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \Downarrow & \\
1  &2 &  3 &  4 &  5 & 6
\end{matrix}</math>

==Mandel notation==
For a symmetric tensor of second rank
<math display="block"> \boldsymbol{\sigma}=
\begin{bmatrix}
  \sigma_{11} & \sigma_{12} & \sigma_{13} \\
  \sigma_{21} & \sigma_{22} & \sigma_{23} \\
  \sigma_{31} & \sigma_{32} & \sigma_{33}
\end{bmatrix}
</math>
only six components are distinct, the three on the diagonal and the others being off-diagonal. 
Thus it can be expressed, in Mandel notation,<ref>{{cite journal
  | title = Généralisation de la théorie de plasticité de WT Koiter
  | author = Jean Mandel
  | journal = International Journal of Solids and Structures
  | volume = 1
  | pages = 273–295
  | date = 1965
  | issue = 3
 | doi=10.1016/0020-7683(65)90034-x}}</ref> as the vector
<math display="block">
\tilde \sigma ^M =
\langle \sigma_{11}, 
\sigma_{22},
\sigma_{33},
\sqrt 2 \sigma_{23},
\sqrt 2 \sigma_{13},
\sqrt 2 \sigma_{12}
\rangle. </math>

The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors,
for example:
<math display="block"> \tilde \sigma : \tilde \sigma = \tilde \sigma^M \cdot \tilde \sigma^M = 
\sigma_{11}^2 +
\sigma_{22}^2 +
\sigma_{33}^2 +
2 \sigma_{23}^2 +
2 \sigma_{13}^2 +
2 \sigma_{12}^2.
</math>

A symmetric tensor of rank four satisfying <math> D_{ijkl} = D_{jikl} </math> and <math> D_{ijkl} = D_{ijlk} </math> has 81 components in three-dimensional space, but only 36 
components are distinct. Thus, in Mandel notation, it can be expressed as
<math display="block"> \tilde D^M =
\begin{pmatrix}
  D_{1111} & D_{1122} & D_{1133}  & \sqrt 2 D_{1123} & \sqrt 2 D_{1113} & \sqrt 2 D_{1112} \\
  D_{2211} & D_{2222} & D_{2233}  & \sqrt 2 D_{2223} & \sqrt 2 D_{2213} & \sqrt 2 D_{2212} \\
  D_{3311} & D_{3322} & D_{3333}  & \sqrt 2 D_{3323} & \sqrt 2 D_{3313} & \sqrt 2 D_{3312} \\
  \sqrt 2 D_{2311} & \sqrt 2 D_{2322} & \sqrt 2 D_{2333}  & 2 D_{2323} & 2 D_{2313} & 2 D_{2312} \\
  \sqrt 2 D_{1311} & \sqrt 2 D_{1322} & \sqrt 2 D_{1333}  & 2 D_{1323} & 2 D_{1313} & 2 D_{1312} \\
  \sqrt 2 D_{1211} & \sqrt 2 D_{1222} & \sqrt 2 D_{1233}  & 2 D_{1223} & 2 D_{1213} & 2 D_{1212} \\
\end{pmatrix}.
</math>

==Applications==
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]],<ref>{{Cite book
  | author1 = O.C. Zienkiewicz
  | author2 = R.L. Taylor
  | author3 = J.Z. Zhu
  | title = The Finite Element Method: Its Basis and Fundamentals
  | year = 2005
  | edition = 6
  | publisher =  Elsevier Butterworth—Heinemann
  | isbn = 978-0-7506-6431-8
}}</ref> and [[Diffusion MRI]].<ref>{{cite book
  | title = Visualization and Processing of Tensor Fields
  | chapter = The Algebra of Fourth-Order Tensors with Application to Diffusion MRI
  | author = Maher Moakher
  | series = Mathematics and Visualization
 | year = 2009
  | pages = 57–80
  | publisher = Springer Berlin Heidelberg
  | doi = 10.1007/978-3-540-88378-4_4
| isbn = 978-3-540-88377-7
 }}</ref>

Hooke's law has a symmetric fourth-order [[stiffness tensor]] with 81 components (3&times;3&times;3&times;3), but because the application of such a rank-4 tensor to a symmetric rank-2 tensor must yield another symmetric rank-2 tensor, not all of the 81 elements are independent. Voigt notation enables such a rank-4 tensor to be ''represented'' by a 6&times;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]]).

A discussion of invariance of Voigt's notation and Mandel's notation can be found in Helnwein (2001).<ref name="Helnwein">{{cite journal
  | title = Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors
  | author = Peter Helnwein
  | journal = Computer Methods in Applied Mechanics and Engineering
  | volume = 190
  | issue = 22–23
  | pages = 2753–2770
  | date = February 16, 2001
  | doi=10.1016/s0045-7825(00)00263-2| bibcode = 2001CMAME.190.2753H
 }}</ref>

==See also==
* [[Vectorization (mathematics)]]
* [[Hooke's law]]
* [[Linear_elasticity#Anisotropic_homogeneous_media]]

==References==
{{reflist}}

{{tensors}}

{{DEFAULTSORT:Voigt Notation}}
[[Category:Tensors]]
[[Category:Mathematical notation]]
[[Category:Solid mechanics]]