Many numerical algorithms use spectral decompositions to compute material
behavior.
Spectral decompositions of stretch tensors
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Infinitesimal line segments in the material and spatial configurations are
related by
d x = F ⋅ d X = R ⋅ ( U ⋅ d X ) = V ⋅ ( R ⋅ d X ) . {\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}})~.} 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
V = R ⋅ U ⋅ R T . {\displaystyle {\boldsymbol {V}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}\cdot {\boldsymbol {R}}^{T}~.} Let the spectral decomposition of U {\displaystyle {\boldsymbol {U}}} be
U = ∑ i = 1 3 λ i N i ⊗ N i {\displaystyle {\boldsymbol {U}}=\sum _{i=1}^{3}\lambda _{i}~{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}} and the spectral decomposition of V {\displaystyle {\boldsymbol {V}}} be
V = ∑ i = 1 3 λ ^ i n i ⊗ n i . {\displaystyle {\boldsymbol {V}}=\sum _{i=1}^{3}{\hat {\lambda }}_{i}~\mathbf {n} _{i}\otimes \mathbf {n} _{i}~.} Then
V = R ⋅ U ⋅ R T = ∑ i = 1 3 λ i R ⋅ ( N i ⊗ N i ) ⋅ R T = ∑ i = 1 3 λ i ( R ⋅ N i ) ⊗ ( R ⋅ N i ) {\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})} Therefore the uniqueness of the spectral decomposition implies that
λ i = λ ^ i and n i = R ⋅ N i {\displaystyle \lambda _{i}={\hat {\lambda }}_{i}\quad {\text{and}}\quad \mathbf {n} _{i}={\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i}} The left stretch (V {\displaystyle {\boldsymbol {V}}} ) is also called the spatial stretch tensor while
the right stretch (U {\displaystyle {\boldsymbol {U}}} ) is called the material stretch tensor.
Spectral decompositions of deformation gradient
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The deformation gradient is given by
F = R ⋅ U {\displaystyle {\boldsymbol {F}}={\boldsymbol {R}}\cdot {\boldsymbol {U}}} In terms of the spectral decomposition of U {\displaystyle {\boldsymbol {U}}} we have
F = ∑ i = 1 3 λ i R ⋅ ( N i ⊗ N i ) = ∑ i = 1 3 λ i ( R ⋅ N i ) ⊗ N i = ∑ i = 1 3 λ i n i ⊗ N i {\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}} Therefore the spectral decomposition of F {\displaystyle {\boldsymbol {F}}} can be written as
F = ∑ i = 1 3 λ i n i ⊗ N i {\displaystyle {\boldsymbol {F}}=\sum _{i=1}^{3}\lambda _{i}~\mathbf {n} _{i}\otimes {\boldsymbol {N}}_{i}} Let us now see what effect the deformation gradient has when it is applied
to the eigenvector N i {\displaystyle {\boldsymbol {N}}_{i}} .
We have
F ⋅ N i = R ⋅ U ⋅ N i = R ⋅ ( ∑ j = 1 3 λ j N j ⊗ N j ) ⋅ N i {\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}} From the definition of the dyadic product
( u ⊗ v ) ⋅ w = ( w ⋅ v ) u {\displaystyle (\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {w} =(\mathbf {w} \cdot \mathbf {v} )~\mathbf {u} } Since the eigenvectors are orthonormal, we have
( N j ⊗ N j ) ⋅ N i = { 0 if i ≠ j N i if i = j {\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}}} Therefore,
( ∑ j = 1 3 λ j N j ⊗ N j ) ⋅ N i = λ i N i no sum on i {\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} That leads to
F ⋅ N i = λ i ( R ⋅ N i ) = λ i n i {\displaystyle {\boldsymbol {F}}\cdot {\boldsymbol {N}}_{i}=\lambda _{i}~({\boldsymbol {R}}\cdot {\boldsymbol {N}}_{i})=\lambda _{i}~\mathbf {n} _{i}} So the effect of F {\displaystyle {\boldsymbol {F}}} on N i {\displaystyle {\boldsymbol {N}}_{i}} is to stretch the vector by λ i {\displaystyle \lambda _{i}}
and to rotate it to the new orientation n i {\displaystyle \mathbf {n} _{i}} .
We can also show that
F − T ⋅ N i = 1 λ i n i ; F T ⋅ n i = λ i N i ; F − 1 ⋅ n i = 1 λ i N i {\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}} Spectral decompositions of strains
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Recall that the Lagrangian Green strain and its Eulerian counterpart are
defined as
E = 1 2 ( F T ⋅ F − 1 ) ; e = 1 2 ( 1 − ( F ⋅ F T ) − 1 ) {\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})} Now,
F T ⋅ F = U ⋅ R T ⋅ R ⋅ U = U 2 ; F ⋅ F T = V ⋅ R ⋅ R T ⋅ V = V 2 {\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}} Therefore we can write
E = 1 2 ( U 2 − 1 ) ; e = 1 2 ( 1 − V − 2 ) {\displaystyle {\boldsymbol {E}}={\frac {1}{2}}~({\boldsymbol {U}}^{2}-{\boldsymbol {\mathit {1}}})~;~~{\boldsymbol {e}}={\frac {1}{2}}~({\boldsymbol {\mathit {1}}}-{\boldsymbol {V}}^{-2})} Hence the spectral decompositions of these strain tensors are
E = ∑ i = 1 3 1 2 ( λ i 2 − 1 ) N i ⊗ N i ; e = ∑ i = 1 3 1 2 ( 1 − 1 λ i 2 ) n i ⊗ n i {\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}} Generalized strain measures
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We can generalize these strain measures by defining strains as
E ( n ) = 1 n ( U n − 1 ) ; e ( n ) = 1 n ( 1 − V − n ) {\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})} The spectral decomposition is
E ( n ) = ∑ i = 1 3 1 n ( λ i n − 1 ) N i ⊗ N i ; e ( n ) = ∑ i = 1 3 1 n ( 1 − 1 λ i n ) n i ⊗ n i {\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}} Clearly, the usual Green strains are obtained when n = 2 {\displaystyle n=2} .
Logarithmic strain measure
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A strain measure that is commonly used is the logarithmic strain measure. This
strain measure is obtained when we have n → 0 {\displaystyle n\rightarrow 0} . Thus
E ( 0 ) = ln ( U ) ; e ( 0 ) = ln ( V ) {\displaystyle {\boldsymbol {E}}^{(0)}=\ln({\boldsymbol {U}})~;~~{\boldsymbol {e}}^{(0)}=\ln({\boldsymbol {V}})} The spectral decomposition is
E ( 0 ) = ∑ i = 1 3 ln λ i N i ⊗ N i ; e ( 0 ) = ∑ i = 1 3 ln λ i n i ⊗ n i {\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}}