Editing Voigt notation (section)

==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>