Deviatoric and volumetric stress
edit
Often it is convenient to decompose the stress tensor into volumetric and
deviatoric (distortional) parts. Applications of such decompositions can be
found in metal plasticity, soil mechanics, and biomechanics.
Decomposition of the Cauchy stress
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The Cauchy stress can be additively decomposed as
σ
=
s
−
p
1
{\displaystyle {\boldsymbol {\sigma }}=\mathbf {s} -p~{\boldsymbol {\mathit {1}}}}
where
s
{\displaystyle \mathbf {s} }
p
{\displaystyle p}
p
=
−
1
3
tr
(
σ
)
=
−
1
3
σ
:
1
s
=
σ
+
p
1
;
s
:
1
=
tr
(
s
)
=
0
{\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~{\text{tr}}({\boldsymbol {\sigma }})=-{\frac {1}{3}}~{\boldsymbol {\sigma }}:{\boldsymbol {\mathit {1}}}\\\mathbf {s} &={\boldsymbol {\sigma }}+p~{\boldsymbol {\mathit {1}}}~;~~\mathbf {s} :{\boldsymbol {\mathit {1}}}={\text{tr}}(\mathbf {s} )=0\end{aligned}}}
In index notation,
p
=
−
1
3
σ
i
i
s
i
j
=
σ
i
j
−
1
3
σ
k
k
δ
i
j
{\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~\sigma _{ii}\\s_{ij}&=\sigma _{ij}-{\frac {1}{3}}~\sigma _{kk}~\delta _{ij}\end{aligned}}}
Decomposition of the 2nd P-K stress
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The second Piola-Kirchhoff stress can be decomposed into volumetric and
distortional parts as
S
=
S
′
−
p
J
C
−
1
{\displaystyle {\boldsymbol {S}}={\boldsymbol {S}}'-p~J~{\boldsymbol {C}}^{-1}}
where
p
=
−
1
3
J
−
1
S
:
C
S
′
=
J
F
−
1
⋅
s
⋅
F
−
T
;
S
′
:
C
=
0
{\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~J^{-1}~{\boldsymbol {S}}:{\boldsymbol {C}}\\{\boldsymbol {S}}'&=J~{\boldsymbol {F}}^{-1}\cdot \mathbf {s} \cdot {\boldsymbol {F}}^{-T}~;~~{\boldsymbol {S}}':{\boldsymbol {C}}&=0\end{aligned}}}
