Advanced elasticity/Spectral decomposition

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




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




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




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