Nonlinear finite elements/Kinematics - spectral decomposition

Spectral decompositionsEdit

Many numerical algorithms use spectral decompositions to compute material behavior.

Spectral decompositions of stretch tensorsEdit

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 gradientEdit

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 strainsEdit

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 measuresEdit

We can generalize these strain measures by defining strains as

 

The spectral decomposition is

 

Clearly, the usual Green strains are obtained when  .

Logarithmic strain measureEdit

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