Continuum mechanics/Spectral decomposition

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Spectral decompositions[edit]

Many numerical algorithms use spectral decompositions to compute material behavior.

Spectral decompositions of stretch tensors[edit]

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.

Spectral decompositions of deformation gradient[edit]

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 and to rotate it to the new orientation .

We can also show that

Spectral decompositions of strains[edit]

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

Generalized strain measures[edit]

We can generalize these strain measures by defining strains as

The spectral decomposition is

Clearly, the usual Green strains are obtained when .

Logarithmic strain measure[edit]

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