Voigt notation

In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order.[1] 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[2] 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.

For example, a 2×2 symmetric tensor X has only three distinct elements, the two on the diagonal and the other being off-diagonal. Thus it can be expressed as the vector

\( \langle x_{{11}},x_{{22}},x_{{12}}\rangle . \)

As another example:

The stress tensor (in matrix notation) is given as

\( {\boldsymbol {\sigma }}=\left[{{\begin{matrix}\sigma _{{xx}}&\sigma _{{xy}}&\sigma _{{xz}}\\\sigma _{{yx}}&\sigma _{{yy}}&\sigma _{{yz}}\\\sigma _{{zx}}&\sigma _{{zy}}&\sigma _{{zz}}\end{matrix}}}\right]. \)

In Voigt notation it is simplified to a 6-dimensional vector:

\( {\tilde \sigma }=(\sigma _{{xx}},\sigma _{{yy}},\sigma _{{zz}},\sigma _{{yz}},\sigma _{{xz}},\sigma _{{xy}})\equiv (\sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}). \)

The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix form as

\( {\boldsymbol {\epsilon }}=\left[{{\begin{matrix}\epsilon _{{xx}}&\epsilon _{{xy}}&\epsilon _{{xz}}\\\epsilon _{{yx}}&\epsilon _{{yy}}&\epsilon _{{yz}}\\\epsilon _{{zx}}&\epsilon _{{zy}}&\epsilon _{{zz}}\end{matrix}}}\right]. \)

Its representation in Voigt notation is

\( {\tilde \epsilon }=(\epsilon _{{xx}},\epsilon _{{yy}},\epsilon _{{zz}},\gamma _{{yz}},\gamma _{{xz}},\gamma _{{xy}})\equiv (\epsilon _{1},\epsilon _{2},\epsilon _{3},\epsilon _{4},\epsilon _{5},\epsilon _{6}), \)

where \( \gamma _{{xy}}=2\epsilon _{{xy}} \) , \( \gamma _{{yz}}=2\epsilon _{{yz}} \) , and \( \gamma _{{zx}}=2\epsilon _{{zx}} \) are engineering shear strains.

The benefit of using different representations for stress and strain is that the scalar invariance

\({\boldsymbol {\sigma }}\cdot {\boldsymbol {\epsilon }}=\sigma _{{ij}}\epsilon _{{ij}}={\tilde \sigma }\cdot {\tilde \epsilon } \)

is preserved.

Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix.

Mnemonic rule

A simple 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).

Voigt notation Mnemonic rule.svg
Mandel notation

For a symmetric tensor of second rank

\( {\boldsymbol {\sigma }}=\left[{{\begin{matrix}\sigma _{{11}}&\sigma _{{12}}&\sigma _{{13}}\\\sigma _{{21}}&\sigma _{{22}}&\sigma _{{23}}\\\sigma _{{31}}&\sigma _{{32}}&\sigma _{{33}}\end{matrix}}}\right] \)

only six components are distinct, the three on the diagonal and the others being off-diagonal. Thus it can be expressed, in Mandel notation[3], as the vector

\( {\tilde \sigma }^{M}=\langle \sigma _{{11}},\sigma _{{22}},\sigma _{{33}},{\sqrt 2}\sigma _{{23}},{\sqrt 2}\sigma _{{13}},{\sqrt 2}\sigma _{{12}}\rangle . \)

The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example:

\( {\tilde \sigma }:{\tilde \sigma }={\tilde \sigma }^{M}\cdot {\tilde \sigma }^{M}=\sigma _{{11}}^{2}+\sigma _{{22}}^{2}+\sigma _{{33}}^{2}+2\sigma _{{23}}^{2}+2\sigma _{{13}}^{2}+2\sigma _{{12}}^{2}. \)

A symmetric tensor of rank four satisfying \( D_{{ijkl}}=D_{{jikl}} \) and \( D_{{ijkl}}=D_{{ijlk}} \) has 81 components in three-dimensional space, but only 36 components are distinct. Thus, in Mandel notation, it can be expressed as

\( {\tilde D}^{M}={\begin{pmatrix}D_{{1111}}&D_{{1122}}&D_{{1133}}&{\sqrt 2}D_{{1123}}&{\sqrt 2}D_{{1113}}&{\sqrt 2}D_{{1112}}\\D_{{2211}}&D_{{2222}}&D_{{2233}}&{\sqrt 2}D_{{2223}}&{\sqrt 2}D_{{2213}}&{\sqrt 2}D_{{2212}}\\D_{{3311}}&D_{{3322}}&D_{{3333}}&{\sqrt 2}D_{{3323}}&{\sqrt 2}D_{{3313}}&{\sqrt 2}D_{{3312}}\\{\sqrt 2}D_{{2311}}&{\sqrt 2}D_{{2322}}&{\sqrt 2}D_{{2333}}&2D_{{2323}}&2D_{{2313}}&2D_{{2312}}\\{\sqrt 2}D_{{1311}}&{\sqrt 2}D_{{1322}}&{\sqrt 2}D_{{1333}}&2D_{{1323}}&2D_{{1313}}&2D_{{1312}}\\{\sqrt 2}D_{{1211}}&{\sqrt 2}D_{{1222}}&{\sqrt 2}D_{{1233}}&2D_{{1223}}&2D_{{1213}}&2D_{{1212}}\\\end{pmatrix}}. \)

Applications

The notation is named after physicist Woldemar Voigt & John Nye (scientist). 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,[4] and Diffusion MRI.[5]

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).[6]
References

Woldemar Voigt (1910). Lehrbuch der kristallphysik. Teubner, Leipzig. Retrieved November 29, 2016.
Klaus Helbig (1994). Foundations of anisotropy for exploration seismics. Pergamon. ISBN 0-08-037224-4.
Jean Mandel (1965). "Généralisation de la théorie de plasticité de WT Koiter". International Journal of Solids and Structures. 1 (3): 273–295. doi:10.1016/0020-7683(65)90034-x.
O.C. Zienkiewicz; R.L. Taylor; J.Z. Zhu (2005). The Finite Element Method: Its Basis and Fundamentals (6 ed.). Elsevier Butterworth—Heinemann. ISBN 978-0-7506-6431-8.
Maher Moakher (2009). "The Algebra of Fourth-Order Tensors with Application to Diffusion MRI". Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer Berlin Heidelberg. pp. 57–80. doi:10.1007/978-3-540-88378-4_4. ISBN 978-3-540-88377-7.

Peter Helnwein (February 16, 2001). "Some Remarks on the Compressed Matrix Representation of Symmetric Second-Order and Fourth-Order Tensors". Computer Methods in Applied Mechanics and Engineering. 190 (22–23): 2753–2770. Bibcode:2001CMAME.190.2753H. doi:10.1016/s0045-7825(00)00263-2.