Voigt notation

Voigt's notation : The components of a symmetrical matrix are noted as six components of a column matrix. On the left the standard notation of a symmetrical matrix, on the right Voigt's notation.

The Voigt notation , named after the physicist Woldemar Voigt , is an abbreviated mathematical notation for certain mathematical functions (symmetrical tensors ) that map a certain number of vectors to a numerical value. Based on the index notation for tensors, 2 indices are "pulled together" to form one index according to a certain rule. A second order tensor often has 9 components in use cases, which can be summarized in a 3 × 3 matrix:

${\displaystyle {\begin{aligned}{\begin{bmatrix}\sigma _{ij}\end{bmatrix}}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\\\end{bmatrix}}\end{aligned}}}$

A symmetric tensor also has 9 components - but only 6 determiners, so that one can write more briefly:

${\displaystyle {\begin{aligned}{\begin{bmatrix}\sigma _{ij}\end{bmatrix}}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\&\sigma _{22}&\sigma _{23}\\{\text{sym}}&&\sigma _{33}\\\end{bmatrix}}\end{aligned}}}$

Instead of being arranged in a square 3 × 3 matrix, the 6 determination pieces can also be arranged in a 6 × 1 column matrix (column vector). While the elements of the 3 × 3 matrix are identified by two indices, the elements of the 6 × 1 column matrix are identified by exactly one index - so that the way in which the indices are "contracted" has to be defined. In the picture on the right you can see the most frequently used assignment ("contraction" rule) between the indices of the 6 × 1 column vector and the indices of the 3 × 3 matrix. ${\displaystyle \sigma _{11},\sigma _{12},\sigma _{13},\sigma _{22},\sigma _{23},\sigma _{33}}$

The combination of the 6 determinants of a symmetrical tensor into a 6 × 1 column vector using a “contraction” rule is called Voigt's notation (of the components) of the tensor.

Voigt's notation in elasticity theory

Stress tensor and strain tensor

For the stress tensor one defines:

${\displaystyle {\begin{aligned}{\begin{bmatrix}\sigma _{ij}\end{bmatrix}}={\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\&\sigma _{22}&\sigma _{23}\\{\text{sym}}&&\sigma _{33}\\\end{bmatrix}}\longrightarrow &{\begin{bmatrix}\sigma _{\alpha }\end{bmatrix}}^{\text{V}}={\begin{bmatrix}\color {red}{\sigma _{1}}\\\color {red}{\sigma _{2}}\\\color {red}{\sigma _{3}}\\\color {blue}{\sigma _{4}}\\\color {blue}{\sigma _{5}}\\\color {blue}{\sigma _{6}}\\\end{bmatrix}}^{\text{V}}:={\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{23}\\\sigma _{13}\\\sigma _{12}\\\end{bmatrix}}\end{aligned}}}$

The 6 × 1 Voigt matrix is ​​indicated here in the article by a superscript V, and the components of the Voigt column vector have only one index. These features show whether Voigt notation or classical notation is used for a quantity. The components of the stress tensor have two indices in the classical tensor notation, which are combined in the matrix . The number of determination pieces is 6 because of the symmetry, namely . In Voigt notation, these determinants are arranged in a column vector and can therefore be addressed by just one index. The 6 components of Voigt's column vector, namely , are defined according to the last equation (rule of "contraction"). ${\displaystyle {\begin{bmatrix}\sigma _{ij}\end{bmatrix}}}$ ${\displaystyle \sigma _{11},\sigma _{12},\sigma _{13},\sigma _{22},\sigma _{23},\sigma _{33}}$ ${\displaystyle \sigma _{1},\sigma _{2},\sigma _{3},\sigma _{4},\sigma _{5},\sigma _{6}}$

A slightly different "contraction" is used for the strain tensor, namely:

${\displaystyle {\begin{aligned}{\begin{bmatrix}\varepsilon _{ij}\end{bmatrix}}={\begin{bmatrix}\varepsilon _{11}&\varepsilon _{12}&\varepsilon _{13}\\&\varepsilon _{22}&\varepsilon _{23}\\{\text{sym}}&&\varepsilon _{33}\\\end{bmatrix}}\longrightarrow &{\begin{bmatrix}\varepsilon _{\alpha }\end{bmatrix}}^{\text{V}}={\begin{bmatrix}\color {red}{\varepsilon _{1}}\\\color {red}{\varepsilon _{2}}\\\color {red}{\varepsilon _{3}}\\\color {blue}{\varepsilon _{4}}\\\color {blue}{\varepsilon _{5}}\\\color {blue}{\varepsilon _{6}}\\\end{bmatrix}}^{\text{V}}:={\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{23}\\2\varepsilon _{13}\\2\varepsilon _{12}\\\end{bmatrix}}\end{aligned}}}$

The factor 2 is new for the last 3 components of the Voigt vector. This factor ensures that:

${\displaystyle \sigma _{\alpha }\varepsilon _{\alpha }=\sigma _{ij}\varepsilon _{ij}=2F}$

Here F is the free energy .

Color coding: Each red Voigt vector component is assigned exactly one tensor component. And exactly two tensor components are assigned to each blue Voigt vector component, so z. B .:

${\displaystyle {\begin{aligned}\color {red}{\varepsilon _{1}}&=\varepsilon _{11}\\\color {blue}{\varepsilon _{5}}&=2\varepsilon _{13}=2\varepsilon _{31}\end{aligned}}}$

Stiffness

If the components of a 4th order tensor in the (i, j) index pair and in the (k, l) index pair are symmetrical, the anterior and posterior index pairs can be treated with the same index "contraction" as with a tensor 2. Step. The 3 × 3 × 3 × 3 = 81 tensor components can then be assigned to a 6 × 6 Voigt matrix. The index resulting from the front index pair becomes the first index of the 6 × 6 matrix, so that: ${\displaystyle C_{ijkl}}$

${\displaystyle {\begin{aligned}{\begin{bmatrix}C_{\alpha \beta }\end{bmatrix}}^{\text{V}}\end{aligned}}={\begin{bmatrix}\color {red}{C_{11}}&\color {red}{C_{12}}&\color {red}{C_{13}}&\color {blue}{C_{14}}&\color {blue}{C_{15}}&\color {blue}{C_{16}}\\\color {red}{C_{21}}&\color {red}{C_{22}}&\color {red}{C_{23}}&\color {blue}{C_{24}}&\color {blue}{C_{25}}&\color {blue}{C_{26}}\\\color {red}{C_{31}}&\color {red}{C_{32}}&\color {red}{C_{33}}&\color {blue}{C_{34}}&\color {blue}{C_{35}}&\color {blue}{C_{36}}\\\color {blue}{C_{41}}&\color {blue}{C_{42}}&\color {blue}{C_{43}}&C_{44}&C_{45}&C_{46}\\\color {blue}{C_{51}}&\color {blue}{C_{52}}&\color {blue}{C_{53}}&C_{54}&C_{55}&C_{56}\\\color {blue}{C_{61}}&\color {blue}{C_{62}}&\color {blue}{C_{63}}&C_{64}&C_{65}&C_{66}\\\end{bmatrix}}^{\text{V}}:={\begin{bmatrix}C_{1111}&C_{1122}&C_{1133}&C_{1123}&C_{1113}&C_{1112}\\C_{2211}&C_{2222}&C_{2233}&C_{2223}&C_{2213}&C_{2212}\\C_{3311}&C_{3322}&C_{3333}&C_{3323}&C_{3313}&C_{3312}\\C_{2311}&C_{2322}&C_{2333}&C_{2323}&C_{2313}&C_{2312}\\C_{1311}&C_{1322}&C_{1333}&C_{1323}&C_{1313}&C_{1312}\\C_{1211}&C_{1222}&C_{1233}&C_{1223}&C_{1213}&C_{1212}\\\end{bmatrix}}}$

Exactly one tensor component is assigned to each red Voigt matrix component. Each blue Voigt matrix component is assigned exactly two tensor components. And exactly four tensor components are assigned to each black Voigt matrix component. E.g .:

${\displaystyle {\begin{aligned}\color {red}{C_{21}}&=C_{2211}\\\color {blue}{C_{26}}&=C_{2212}=C_{2221}\\\color {blue}{C_{62}}&=C_{1222}=C_{2122}\\C_{65}&=C_{1213}=C_{1231}=C_{2113}=C_{2131}\\\end{aligned}}}$

There are 9 red, 18 blue and 9 black (36 total) Voigt matrix components. And all 3 × 3 × 3 × 3 = 81 tensor components are assigned, because:

${\displaystyle {\begin{aligned}\color {red}{9}\color {black}{\cdot 1}+\color {blue}{18}\color {black}{\cdot 2}+9\cdot 4&=\\9+36+36&=81\\\end{aligned}}}$

Material law

The material law in linear elasticity theory is a linear mapping between strain and stress. In the tensor notation, this is a 4th level tensor, which links the 2nd level tensors.

${\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}$

The Einstein sums convention is used here. For example, one of these 9 equations is

${\displaystyle {\begin{aligned}\sigma _{23}&=C_{23kl}\varepsilon _{kl}\\&=C_{2311}\varepsilon _{11}+C_{2312}\varepsilon _{12}+C_{2313}\varepsilon _{13}+C_{2321}\varepsilon _{21}+C_{2322}\varepsilon _{22}+C_{2323}\varepsilon _{23}+C_{2331}\varepsilon _{31}+C_{2332}\varepsilon _{32}+C_{2333}\varepsilon _{33}\end{aligned}}}$

In Voigt's notation, the corresponding figure is a 6 × 6 matrix.

${\displaystyle {\begin{aligned}\sigma _{\alpha }&=C_{\alpha \beta }\varepsilon _{\beta }\\{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\\\end{bmatrix}}^{\text{V}}&={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}\\\end{bmatrix}}^{\text{V}}{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\\\end{bmatrix}}^{\text{V}}\end{aligned}}}$

The relationship for the components results from the requirement of the equivalence of the two notations:

${\displaystyle {\begin{aligned}{\begin{bmatrix}C_{1111}&C_{1122}&C_{1133}&C_{1123}&C_{1113}&C_{1112}\\C_{2211}&C_{2222}&C_{2233}&C_{2223}&C_{2213}&C_{2212}\\C_{3311}&C_{3322}&C_{3333}&C_{3323}&C_{3313}&C_{3312}\\C_{2311}&C_{2322}&C_{2333}&C_{2323}&C_{2313}&C_{2312}\\C_{1311}&C_{1322}&C_{1333}&C_{1323}&C_{1313}&C_{1312}\\C_{1211}&C_{1222}&C_{1233}&C_{1223}&C_{1213}&C_{1212}\\\end{bmatrix}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\C_{21}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\C_{31}&C_{32}&C_{33}&C_{34}&C_{35}&C_{36}\\C_{41}&C_{42}&C_{43}&C_{44}&C_{45}&C_{46}\\C_{51}&C_{52}&C_{53}&C_{54}&C_{55}&C_{56}\\C_{61}&C_{62}&C_{63}&C_{64}&C_{65}&C_{66}\\\end{bmatrix}}^{\text{V}}={\begin{bmatrix}C_{\alpha \beta }\end{bmatrix}}^{\text{V}}\end{aligned}}}$

For the spelling with four indices symmetry is provided in the first and last two indices, ie . Because of the symmetry of the tensors for strain and tension, this is possible and common without loss of generality. Because of the existence of a potential is symmetric, and for the tensor notation it is equivalent that is. I.e. the following applies: ${\displaystyle C_{ijkl}=C_{ijlk}=C_{jikl}}$ ${\displaystyle {\begin{bmatrix}C_{\alpha \beta }\end{bmatrix}}^{\text{V}}}$ ${\displaystyle C_{ijkl}=C_{klij}}$

${\displaystyle {\begin{aligned}{\begin{bmatrix}C_{\alpha \beta }\end{bmatrix}}^{\text{V}}={\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\&&C_{33}&C_{34}&C_{35}&C_{36}\\&&&C_{44}&C_{45}&C_{46}\\&&&&C_{55}&C_{56}\\{\text{sym}}&&&&&C_{66}\\\end{bmatrix}}^{\text{V}}\end{aligned}}}$

indulgence

If one proceeds instead of C from the flexibility S according to

${\displaystyle {\begin{aligned}\varepsilon _{ij}=S_{ijkl}\sigma _{kl}\end{aligned}}}$

and if one demands the same symmetries for S that were previously required for C, then one arrives at the following representation of the compliance in Voigt's notation

${\displaystyle {\begin{aligned}\varepsilon _{\alpha }&=S_{\alpha \beta }\sigma _{\beta }\\{\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\varepsilon _{3}\\\varepsilon _{4}\\\varepsilon _{5}\\\varepsilon _{6}\\\end{bmatrix}}^{\text{V}}&={\begin{bmatrix}S_{1111}&S_{1122}&S_{1133}&2S_{1123}&2S_{1113}&2S_{1112}&\\&S_{2222}&S_{2233}&2S_{2223}&2S_{2213}&2S_{2212}&\\&&S_{3333}&2S_{3323}&2S_{3313}&2S_{3312}&\\&&&4S_{2323}&4S_{2313}&4S_{2312}&\\&{\text{sym}}&&&4S_{1313}&4S_{1312}&\\&&&&&4S_{1212}&\\\end{bmatrix}}{\begin{bmatrix}\sigma _{1}\\\sigma _{2}\\\sigma _{3}\\\sigma _{4}\\\sigma _{5}\\\sigma _{6}\\\end{bmatrix}}^{\text{V}}\end{aligned}}}$

Comparison of the tensor notation with the Voigt notation

Advantages and disadvantages of Voigt notation

The Voigt notation is much more compact than the full tensor notation and the Voigt stiffness matrix can easily be inverted. Furthermore, it is easy to see that a linear material law (for which the symmetries of C apply) generally contains 21 independent values ​​(material constants). If C fulfills further conditions / symmetries, the number of constants is further reduced.

These advantages are offset by a number of disadvantages: Other "contracting rules" are also possible, e.g. As could be: . Voigt's notation is just the most common form. or are not vectors (neither co- nor contravariant). When changing coordinates, they do not transform like vectors. The same applies to objects in Voigt notation that have multiple indices. Would you z. If, for example, the “vectors” in Voigt notation are to be understood as vectors and a norm defined on the associated vector space as usual, then one would have to establish that generally applies ${\displaystyle \varepsilon _{4}:=1\varepsilon _{12}}$ ${\displaystyle {\begin{bmatrix}\sigma _{\alpha }\end{bmatrix}}^{\text{V}}}$ ${\displaystyle {\begin{bmatrix}\varepsilon _{\alpha }\end{bmatrix}}^{\text{V}}}$ ${\displaystyle V^{\text{v}}}$

${\displaystyle {\begin{aligned}\|{\begin{bmatrix}\sigma _{\alpha }\end{bmatrix}}\|_{V^{\text{v}}}\neq \|\sigma _{ij}\|_{V^{3\times 3}}\end{aligned}}}$

where on the right the usual norm on the vector space of the 3 × 3 matrices is meant.

Spelling equivalence

Voigt's notation is equivalent to the detailed index notation for tensors. More precisely:

${\displaystyle \left.{\begin{aligned}\sigma _{\alpha }:=&\dots \\\varepsilon _{\alpha }:=&\dots \\C_{\alpha \beta }:=&\dots \\\sigma _{\alpha }=&C_{\alpha \beta }\varepsilon _{\beta }\\C_{\alpha \beta }=&C_{\beta \alpha }\\\end{aligned}}\right\}\Leftrightarrow {\begin{cases}\sigma _{ij}=C_{ijkl}\varepsilon _{kl}&\\\sigma _{ij}=\sigma _{ji}&\Rightarrow C_{ijkl}=C_{jikl}\qquad {\text{lässt sich o. B. d. A. fordern}}\\F:={\frac {1}{2}}\varepsilon _{ij}C_{ijkl}\varepsilon _{kl}\quad \land \quad \sigma _{mn}={\frac {\partial F}{\partial \varepsilon _{mn}}}&\Rightarrow C_{ijkl}=C_{klij}\\\varepsilon _{ij}=\varepsilon _{ji}&\end{cases}}}$

One can easily show the equivalence of both spellings. z. B. is

${\displaystyle {\begin{aligned}\sigma _{23}&=C_{2311}\varepsilon _{11}+C_{2312}\varepsilon _{12}+C_{2313}\varepsilon _{13}+C_{2321}\varepsilon _{21}+C_{2322}\varepsilon _{22}+C_{2323}\varepsilon _{23}+C_{2331}\varepsilon _{31}+C_{2332}\varepsilon _{32}+C_{2333}\varepsilon _{33}\\&=C_{2311}\varepsilon _{11}+(C_{2312}+C_{2321})\varepsilon _{12}+(C_{2313}+C_{2331})\varepsilon _{13}+(C_{2323}+C_{2332})\varepsilon _{23}+C_{2322}\varepsilon _{22}+C_{2333}\varepsilon _{33}\\&=C_{2311}\varepsilon _{11}+2C_{2312}\varepsilon _{12}+2C_{2313}\varepsilon _{13}+2C_{2323}\varepsilon _{23}+C_{2322}\varepsilon _{22}+C_{2333}\varepsilon _{33}\\&=C_{2311}\varepsilon _{11}+C_{2322}\varepsilon _{22}+C_{2333}\varepsilon _{33}+C_{2323}2\varepsilon _{23}+C_{2313}2\varepsilon _{13}+C_{2312}2\varepsilon _{12}\\&=C_{2311}\varepsilon _{1}+C_{2322}\varepsilon _{2}+C_{2333}\varepsilon _{3}+C_{2323}\varepsilon _{4}+C_{2313}\varepsilon _{5}+C_{2312}\varepsilon _{6}\\&=C_{1123}\varepsilon _{1}+C_{2223}\varepsilon _{2}+C_{3323}\varepsilon _{3}+C_{2323}\varepsilon _{4}+C_{2313}\varepsilon _{5}+C_{2312}\varepsilon _{6}\\&=C_{14}\varepsilon _{1}+C_{24}\varepsilon _{2}+C_{34}\varepsilon _{3}+C_{44}\varepsilon _{4}+C_{45}\varepsilon _{5}+C_{46}\varepsilon _{6}\\&=\sigma _{4}\end{aligned}}}$

Alternative notations

Other “contraction rules” are also possible. E.g. the notation of the components of the stress tensor named after Nye is:

${\displaystyle {\begin{aligned}{\begin{bmatrix}{\sigma _{1}}\\{\sigma _{2}}\\{\sigma _{3}}\\{\sigma _{4}}\\{\sigma _{5}}\\{\sigma _{6}}\\\end{bmatrix}}^{\text{N}}:={\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\\sigma _{12}\\\sigma _{13}\\\sigma _{23}\\\end{bmatrix}}\end{aligned}}}$

And the Nye notation for the components of the strain tensor is:

${\displaystyle {\begin{aligned}{\begin{bmatrix}{\varepsilon _{1}}\\{\varepsilon _{2}}\\{\varepsilon _{3}}\\{\varepsilon _{4}}\\{\varepsilon _{5}}\\{\varepsilon _{6}}\\\end{bmatrix}}^{\text{N}}:={\begin{bmatrix}\varepsilon _{11}\\\varepsilon _{22}\\\varepsilon _{33}\\2\varepsilon _{12}\\2\varepsilon _{13}\\2\varepsilon _{23}\\\end{bmatrix}}\end{aligned}}}$

Other notations are named after Kelvin (1856) and Mandel (1965). The Kelvin-Mandel notation of the stress tensor is:

${\displaystyle {\begin{aligned}{\begin{bmatrix}{\sigma _{1}}\\{\sigma _{2}}\\{\sigma _{3}}\\{\sigma _{4}}\\{\sigma _{5}}\\{\sigma _{6}}\\\end{bmatrix}}^{\text{M}}:={\begin{bmatrix}\sigma _{11}\\\sigma _{22}\\\sigma _{33}\\{\sqrt {2}}\sigma _{23}\\{\sqrt {2}}\sigma _{13}\\{\sqrt {2}}\sigma _{12}\\\end{bmatrix}}\end{aligned}}}$

This notation has the advantage that the associated tensor basis is normalized. For example, the identity applies to this notation

${\displaystyle {\begin{aligned}\|{\begin{bmatrix}\sigma _{\alpha }\end{bmatrix}}\|_{V^{\text{v}}}=\|\sigma _{ij}\|_{V^{3\times 3}}.\end{aligned}}}$

Due to the normalization of the basis, the usual matrix arithmetic operations such as B. the inversion, eigenvalues ​​are transferred to the stiffness and compliance tensors.

See also

More on the special cases of anisotropy and thus on the occupancy of the stiffness matrix / compliance matrix:

• Orthotropy
• Transverse isotropy
• Isotropy

literature

• Woldemar Voigt: Textbook of crystal physics: with exclusion of d. Crystal optics . Teubner, Leipzig a. a. 1910.
• JF Nye: Physical Properties of Crystals: Their Representation by Tensors and Matrices . Oxford University Press, 1985, ISBN 0-19-851165-5 .
• I. Müller, P. Strehlow: Rubber and Rubber Balloons, Paradigms of Thermodynamics (=  Lect. Notes Phys. No. 637 ). 2004, ISBN 978-3-540-20244-8 .

Individual evidence

1. ↑ For more information see z. B. doi: 10.1007 / b93853 .
2. ^ SC Cowin and MM Mehrabadi: The Structure of Linear Anisotropic Elastic Symmetries (1992), J. Mech. Phys. Solids (40) No. 7 pp. 1459-1471.
3. ^ W. Thomson: Elements of a mathematical theory of elasticity (1856), Phil. Trans. R. Soc. (146), 481-498.
4. ^ Jean Mandel: Généralisation de la théorie de plasticité de WT Koiter . In: International Journal of Solids and structures . 1, 1965, pp. 273-295.
5. ^ R. Brannon: Rotation, Reflection, and Frame Changes: Orthogonal tensors in computational engineering mechanics (2018), IOP Publishing Ltd, chapter 26