Voigt notation

The Voigt notation, named after the physicist Woldemar Voigt, is a shorthand notation for tensors. Based on the index notation of tensors in each case two indexes are " contracted " after a certain rule to an index.

For example, the components of a symmetric tensor of second stage (eg stress tensor ) are normally presented as a 3x3 matrix with 9 components. The tensor has because of its symmetry but only 6 Determination pieces plus 3 equations that describe the symmetry. Then, imagine that a " contraction " as follows

Then can be written as a "vector" of the tensor in this way. This " vector" notation is the Voigt notation of the tensor.

Voigt's notation for tensors of 4th stage

If the components of a tensor 4th stage in the front and rear pair of indices are each symmetric, can the front and rear pair of indices with the same index " contraction " treat like a tensor of second stage. The 81 (3x3 x 3x3) components can then be arranged in a 6x6 matrix. The index, which has arisen out of the front pair of indices, this is the first index of the 6x6 matrix.

Voigt's notation in the theory of elasticity

Voigt's notation of the strain tensor

For the strain tensor is usually a slightly different " contraction " is used, namely

The additional factor of 2 in the last 3 components ensures that the scalar product of voltage in strain in the Voigt notation is equal to the inner tensor in the tensor notation. This product has because of its association with the energy stored elastically a special meaning.

However, this simple relation for the dot product only applies voltage times elongation for other inner products as the square of the voltage, or stretching, and thus not for the usual standard.

Material law

The material law in the linear theory of elasticity is a linear map between strain and stress. In this tensor is a tensor 4th stage, the second stage combines the tensors.

Here the Einstein summation convention. One of these nine equations is, for example,

In the Voigt 's notation, the corresponding figure is a 6x6 matrix.

From the requirement of equivalence of the two spellings, there is the context for the components:

For the notation with 4 indices symmetry is assumed in the first and last two indices, ie. This is possible and common because of the symmetry of the tensors for stress and strain without loss of generality. Because of the existence of a potential is symmetric and the tensor notation applies equivalent, that is.

Advantages, disadvantages, warnings

Benefits

• The Voigt notation is much more compact than the full tensor notation.
• It is evident that a linear material law ( for which the symmetries of C apply ) generally contains 21 independent values ​​(material constants). If C still meets other conditions / symmetries, the number of constants is further reduced.
• The Voigt stiffness matrix can be inverted easily.

Disadvantages / warnings

• There are also other " contraction rules " in use, such as could be, what also affect and would.
• Or are not (neither co- nor contravariant ) vectors. So you do not transform in coordinate changes also as vectors. The same is true for objects in Voigt notation that have multiple indexes.
• Would you, for example, understands " vectors" in Voigt notation as vectors and define the associated vector space a norm as usual, then you would have to find that in general,

• Many sources, only one of two notations is used: either the Voigt notation or more elongated tensor notation, where C has four indices. Authors who use the Voigt notation, surrendering part on the name of Voigt 's symbols by a superscript v or by any other designation. So for example, use the symbol if they think the Voigt 's voltage vector.

Equivalence of the spellings

The Voigt notation is equivalent to the detailed index notation for tensors. More precisely

F here is the free energy, see eg doi: 10.1007/b93853.

One can easily prove the equivalence of both spellings. for example, is

Compliance

Assuming instead of C of the resilience of S according to

And calls to the same symmetries of S that were previously required for C, we arrive at the following representation of the flexibility in Voigt notation shear