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Concept
Notation de Voigt
Résumé
In mathematics, Voigt notation or Voigt form in multilinear algebra is a way to represent a symmetric tensor by reducing its order. 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 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
As another example:
The stress tensor (in matrix notation) is given as
In Voigt notation it is simplified to a 6-dimensional vector:
The strain tensor, similar in nature to the stress tensor—both are symmetric second-order tensors --, is given in matrix form as
Its representation in Voigt notation is
where , , and are engineering shear strains.
The benefit of using different representations for stress and strain is that the scalar invariance
is preserved.
Likewise, a three-dimensional symmetric fourth-order tensor can be reduced to a 6×6 matrix.
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).
For a symmetric tensor of second rank
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
The main advantage of Mandel notation is to allow the use of the same conventional operations used with vectors,
for example:
A symmetric tensor of rank four satisfying and has 81 components in three-dimensional space, but only 36
components are distinct.
Source officielle
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