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 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 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, and Diffusion MRI.
Hooke's law has a symmetric fourth-order stiffness tensor with 81 components (3×3×3×3), 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 6×6 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).
See also
- Vectorization (mathematics)
- Hooke's law
- Linear_elasticity#Anisotropic_homogeneous_media