Voigt notation

Inmathematics,Voigt notationorVoigt forminmultilinear algebrais a way to represent asymmetric tensorby reducing its order.There are a few variants and associated names for this idea:Mandel notation,Mandel–Voigt notationandNye notationare others found.Kelvin notationis a revival by Helbigof old ideas ofLord 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 tensorXhas 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 $${\boldsymbol {\sigma }}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}.$$

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 $${\boldsymbol {\epsilon }}={\begin{bmatrix}\epsilon _{xx}&\epsilon _{xy}&\epsilon _{xz}\\\epsilon _{yx}&\epsilon _{yy}&\epsilon _{yz}\\\epsilon _{zx}&\epsilon _{zy}&\epsilon _{zz}\end{bmatrix}}.$$

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 simplemnemonic rulefor 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).

Mandel notation

For a symmetric tensor of second rank $${\boldsymbol {\sigma }}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}$$ only six components are distinct, the three on the diagonal and the others being off-diagonal. Thus it can be expressed, in Mandel notation,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 physicistWoldemar Voigt&John Nye (scientist).It is useful, for example, in calculations involving constitutive models to simulate materials, such as the generalizedHooke's law,as well asfinite element analysis,andDiffusion MRI.

Hooke's law has a symmetric fourth-orderstiffness tensorwith 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 berepresentedby 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 anisometry).

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