Voigt notation

 ( Tensors ) 

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 its rank can be reduced by expressressing it as a vector without loss of information:

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 3×3 stress and strain tensors in their full forms can be written as:

$\backslash boldsymbol\{\backslash sigma\}=$

 \sigma_{11} & \sigma_{12} & \sigma_{13} \\
 \sigma_{21} & \sigma_{22} & \sigma_{23} \\
 \sigma_{31} & \sigma_{32} & \sigma_{33}
     

 \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{13} \\
 \varepsilon_{21} & \varepsilon_{22} & \varepsilon_{23} \\
 \varepsilon_{31} & \varepsilon_{32} & \varepsilon_{33}
     

Voigt notation then utilises the symmetry of these matrices ($\backslash sigma\_\{12\}\; =\; \backslash sigma\_\{21\}$ and so on) to express them instead as a 6×1 vector:

$\backslash underline\backslash sigma\; =\; \backslash begin\{bmatrix\}\; \backslash sigma\_1\; \backslash \backslash \; \backslash sigma\_2\; \backslash \backslash \; \backslash sigma\_3\; \backslash \backslash \; \backslash sigma\_4\; \backslash \backslash \; \backslash sigma\_5\; \backslash \backslash \; \backslash sigma\_6\; \backslash end\{bmatrix\}\; :=\; \backslash begin\{bmatrix\}\backslash sigma\_\{11\}\; \backslash \backslash \; \backslash sigma\_\{22\}\; \backslash \backslash \; \backslash sigma\_\{33\}\; \backslash \backslash \; \backslash sigma\_\{23\}\; \backslash \backslash \; \backslash sigma\_\{13\}\; \backslash \backslash \; \backslash sigma\_\{12\}\; \backslash end\{bmatrix\}\; \backslash quad$

where $\backslash gamma\_\{12\}=2\backslash varepsilon\_\{12\}$, $\backslash gamma\_\{23\}\; =\; 2\backslash varepsilon\_\{23\}$, and $\backslash gamma\_\{13\}\; =\; 2\backslash varepsilon\_\{13\}$ are the engineering shear strains.

The benefit of using different representations for stress and strain is that the scalar invariance $$\backslash boldsymbol\{\backslash sigma\}\backslash cdot\backslash boldsymbol\{\backslash varepsilon\}\; =\; \backslash sigma\_\{ij\}\backslash varepsilon\_\{ij\}\; =\; \backslash underline\backslash sigma\; \backslash cdot\; \backslash underline\backslash varepsilon$$ is preserved.

This notation now allows the three-dimensional symmetric fourth-order stiffness, $C$, and compliance, $S$, tensors to be reduced to 6×6 matrices:

$$C\_\{ijkl\}\; \backslash Rightarrow\; C\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash begin\{bmatrix\}$$

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}
     

• 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).

The diagram below also shows the order of the indices:

\begin{matrix}
     

 \sigma_{11} & \sigma_{12} & \sigma_{13} \\
 \sigma_{21} & \sigma_{22} & \sigma_{23} \\
 \sigma_{31} & \sigma_{32} & \sigma_{33}
     

The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors, for example: $$\backslash tilde\; \backslash sigma\; :\; \backslash tilde\; \backslash sigma\; =\; \backslash tilde\; \backslash sigma^M\; \backslash cdot\; \backslash tilde\; \backslash sigma^M\; =\; \backslash sigma\_\{11\}^2\; +\; \backslash sigma\_\{22\}^2\; +\; \backslash sigma\_\{33\}^2\; +\; 2\; \backslash sigma\_\{23\}^2\; +\; 2\; \backslash sigma\_\{13\}^2\; +\; 2\; \backslash sigma\_\{12\}^2.$$

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 $$\backslash tilde\; D^M\; =\; \backslash begin\{pmatrix\}$$

 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} \\
     

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).

• Vectorization (mathematics)
• Hooke's law
• Linear_elasticity#Anisotropic_homogeneous_media