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Many numerical algorithms use spectral decompositions to compute material behavior.
Infinitesimal line segments in the material and spatial configurations are related by
So the sequence of operations may be either considered as a stretch of in the material configuration followed by a rotation or a rotation followed by a stretch.
Also note that
Let the spectral decomposition of be
and the spectral decomposition of be
Then
Therefore the uniqueness of the spectral decomposition implies that
The left stretch () is also called the spatial stretch tensor while the right stretch () is called the material stretch tensor.
The deformation gradient is given by
In terms of the spectral decomposition of we have
Therefore the spectral decomposition of can be written as
Let us now see what effect the deformation gradient has when it is applied to the eigenvector .
We have
From the definition of the dyadic product
Since the eigenvectors are orthonormal, we have
Therefore,
That leads to
So the effect of on is to stretch the vector by λ_{i} and to rotate it to the new orientation .
We can also show that
Recall that the Lagrangian Green strain and its Eulerian counterpart are defined as
Now,
Therefore we can write
Hence the spectral decompositions of these strain tensors are
We can generalize these strain measures by defining strains as
The spectral decomposition is
Clearly, the usual Green strains are obtained when n = 2.
A strain measure that is commonly used is the logarithmic strain measure. This strain measure is obtained when we have . Thus
The spectral decomposition is
