Many numerical algorithms use spectral decompositions to compute material
behavior.
Infinitesimal line segments in the material and spatial configurations are
related by
![{\displaystyle d\mathbf {x} ={\boldsymbol {F}}\cdot d{\boldsymbol {X}}={\boldsymbol {R}}\cdot ({\boldsymbol {U}}\cdot d{\boldsymbol {X}})={\boldsymbol {V}}\cdot ({\boldsymbol {R}}\cdot d{\boldsymbol {X}})~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75891ab47eecad1ff84e29f1abd2034bd4e1b028)
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
![{\displaystyle {\boldsymbol {V}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\cdot {\boldsymbol {R}}^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed40f4739b066d6f36d21f5953fe1203a939260)
Let the spectral decomposition of
be
![{\displaystyle {\boldsymbol {U}}=\sum _{i=1}^{3}\lambda _{i}~{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1949bbb3297c67259c8bf8c9dc3e979ee7b05b07)
and the spectral decomposition of
be
![{\displaystyle {\boldsymbol {V}}=\sum _{i=1}^{3}{\hat {\lambda }}_{i}~\mathbf {n} _{i}\otimes \mathbf {n} _{i}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0a9dcce380c65aafcb298d3b84b7dc806eccb60)
Then
![{\displaystyle {\boldsymbol {V}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\cdot {\boldsymbol {R}}^{T}=\sum _{i=1}^{3}\lambda _{i}~{\boldsymbol {R}}\cdot ({\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i})\cdot {\boldsymbol {R}}^{T}=\sum _{i=1}^{3}\lambda _{i}~({\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i})\otimes ({\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d963de6ce7d5a5dd943e7b5549eadd82a475d8fc)
Therefore the uniqueness of the spectral decomposition implies that
![{\displaystyle \lambda _{i}={\hat {\lambda }}_{i}\quad {\text{and}}\quad \mathbf {n} _{i}={\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae544f07a1fd5150fc5eefea218fc2a1dd96da01)
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
![{\displaystyle {\boldsymbol {F}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df6ae1194b9caf72e0281dc709ab8181921992dc)
In terms of the spectral decomposition of
we have
![{\displaystyle {\boldsymbol {F}}=\sum _{i=1}^{3}\lambda _{i}~{\boldsymbol {R}}\cdot ({\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i})=\sum _{i=1}^{3}\lambda _{i}~({\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i})\otimes {\boldsymbol {N}}_{i}=\sum _{i=1}^{3}\lambda _{i}~\mathbf {n} _{i}\otimes {\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6524ba83c0348f3b73759f444a489194d1c216)
Therefore the spectral decomposition of
can be written as
![{\displaystyle {\boldsymbol {F}}=\sum _{i=1}^{3}\lambda _{i}~\mathbf {n} _{i}\otimes {\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aaa2c7437147c47feee14b8189728fc1e97488e)
Let us now see what effect the deformation gradient has when it is applied
to the eigenvector
.
We have
![{\displaystyle {\boldsymbol {F}}\cdot {\boldsymbol {N}}_{i}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\cdot {\boldsymbol {N}}_{i}={\boldsymbol {R}}\cdot \left(\sum _{j=1}^{3}\lambda _{j}~{\boldsymbol {N}}_{j}\otimes {\boldsymbol {N}}_{j}\right)\cdot {\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/839c596bacb5da05eb4df3f52c88372f9dae32c6)
From the definition of the dyadic product
![{\displaystyle (\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {w} =(\mathbf {w} \cdot \mathbf {v} )~\mathbf {u} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb2d87bc97c3eaa616bfd1b738ee1a21e1807ec)
Since the eigenvectors are orthonormal, we have
![{\displaystyle ({\boldsymbol {N}}_{j}\otimes {\boldsymbol {N}}_{j})\cdot {\boldsymbol {N}}_{i}={\begin{cases}0&{\mbox{if}}~i\neq j\\{\boldsymbol {N}}_{i}&{\mbox{if}}~i=j\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/210fb748ea04e56815a867137fb910e270e0af6a)
Therefore,
![{\displaystyle \left(\sum _{j=1}^{3}\lambda _{j}~{\boldsymbol {N}}_{j}\otimes {\boldsymbol {N}}_{j}\right)\cdot {\boldsymbol {N}}_{i}=\lambda _{i}~{\boldsymbol {N}}_{i}{\text{no sum on}}~i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7800795f14b9ddaffa79bbb71595d6489878b486)
That leads to
![{\displaystyle {\boldsymbol {F}}\cdot {\boldsymbol {N}}_{i}=\lambda _{i}~({\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i})=\lambda _{i}~\mathbf {n} _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c778999146bdf3064c8077efb736cb74e20ab4bc)
So the effect of
on
is to stretch the vector by
and to rotate it to the new orientation
.
We can also show that
![{\displaystyle {\boldsymbol {F}}^{-T}\cdot {\boldsymbol {N}}_{i}={\cfrac {1}{\lambda _{i}}}~\mathbf {n} _{i}~;~~{\boldsymbol {F}}^{T}\cdot \mathbf {n} _{i}=\lambda _{i}~{\boldsymbol {N}}_{i}~;~~{\boldsymbol {F}}^{-1}\cdot \mathbf {n} _{i}={\cfrac {1}{\lambda _{i}}}~{\boldsymbol {N}}_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c64e31fdd4bd455fb16c90dac5cb51bcfd799f)
Recall that the Lagrangian Green strain and its Eulerian counterpart are
defined as
![{\displaystyle {\boldsymbol {E}}={\frac {1}{2}}~({\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}-{\boldsymbol {\mathit {1}}})~;~~{\boldsymbol {e}}={\frac {1}{2}}~({\boldsymbol {\mathit {1}}}-\left({\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}\right)^{-1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a34d7955c324ee8b61fe47fa215baf1c83a96785)
Now,
![{\displaystyle {\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}={\boldsymbol {U}}\cdot {\boldsymbol {R}}^{T}\cdot {\boldsymbol {R}}\cdot {\boldsymbol {U}}={\boldsymbol {U}}^{2}~;~~{\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}={\boldsymbol {V}}\cdot {\boldsymbol {R}}\cdot {\boldsymbol {R}}^{T}\cdot {\boldsymbol {V}}={\boldsymbol {V}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d18fadb9c4a57026b4fcecabb5bacc0b70a3432)
Therefore we can write
![{\displaystyle {\boldsymbol {E}}={\frac {1}{2}}~({\boldsymbol {U}}^{2}-{\boldsymbol {\mathit {1}}})~;~~{\boldsymbol {e}}={\frac {1}{2}}~({\boldsymbol {\mathit {1}}}-{\boldsymbol {V}}^{-2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33624d6951adee7adc9f7fd9d85ddd989088318)
Hence the spectral decompositions of these strain tensors are
![{\displaystyle {\boldsymbol {E}}=\sum _{i=1}^{3}{\frac {1}{2}}(\lambda _{i}^{2}-1)~{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}~;~~\mathbf {e} =\sum _{i=1}^{3}{\frac {1}{2}}\left(1-{\cfrac {1}{\lambda _{i}^{2}}}\right)~\mathbf {n} _{i}\otimes \mathbf {n} _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f65bef3a16b5a9e73c79f5ed465f92b9311043fc)
We can generalize these strain measures by defining strains as
![{\displaystyle {\boldsymbol {E}}^{(n)}={\cfrac {1}{n}}~({\boldsymbol {U}}^{n}-{\boldsymbol {\mathit {1}}})~;~~{\boldsymbol {e}}^{(n)}={\cfrac {1}{n}}~({\boldsymbol {\mathit {1}}}-{\boldsymbol {V}}^{-n})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f3d6c84355d66bc1a41b2ccf820286fac3b4bd5)
The spectral decomposition is
![{\displaystyle {\boldsymbol {E}}^{(n)}=\sum _{i=1}^{3}{\cfrac {1}{n}}(\lambda _{i}^{n}-1)~{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}~;~~\mathbf {e} ^{(n)}=\sum _{i=1}^{3}{\cfrac {1}{n}}\left(1-{\cfrac {1}{\lambda _{i}^{n}}}\right)~\mathbf {n} _{i}\otimes \mathbf {n} _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/005410211ff414dc7a812627138e3011c0a07c72)
Clearly, the usual Green strains are obtained when
.
A strain measure that is commonly used is the logarithmic strain measure. This
strain measure is obtained when we have
. Thus
![{\displaystyle {\boldsymbol {E}}^{(0)}=\ln({\boldsymbol {U}})~;~~{\boldsymbol {e}}^{(0)}=\ln({\boldsymbol {V}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e34550f1fca2253ccd525e77a29b07fa611a149)
The spectral decomposition is
![{\displaystyle {\boldsymbol {E}}^{(0)}=\sum _{i=1}^{3}\ln \lambda _{i}~{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}~;~~\mathbf {e} ^{(0)}=\sum _{i=1}^{3}\ln \lambda _{i}~\mathbf {n} _{i}\otimes \mathbf {n} _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/838126eefcce00c6bbcb148c332bec7108780bf7)