Nonlinear finite elements/Deviatoric and volumetric stress

The Cauchy stress can be additively decomposed as

${\displaystyle {\boldsymbol {\sigma }}=\mathbf {s} -p~{\boldsymbol {\mathit {1}}}}$

where ${\displaystyle \mathbf {s} }$ is the deviatoric stress and ${\displaystyle p}$ is the pressure and

${\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~{\text{tr}}({\boldsymbol {\sigma }})=-{\frac {1}{3}}~{\boldsymbol {\sigma }}:{\boldsymbol {\mathit {1}}}\\\mathbf {s} &={\boldsymbol {\sigma }}+p~{\boldsymbol {\mathit {1}}}~;~~\mathbf {s}$ :{\boldsymbol {\mathit {1}}}={\text{tr}}(\mathbf {s} )=0\end{aligned}}}

In index notation,

${\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~\sigma _{ii}\\s_{ij}&=\sigma _{ij}-{\frac {1}{3}}~\sigma _{kk}~\delta _{ij}\end{aligned}}}$

The second Piola-Kirchhoff stress can be decomposed into volumetric and distortional parts as

${\displaystyle {\boldsymbol {S}}={\boldsymbol {S}}'-p~J~{\boldsymbol {C}}^{-1}}$

where

${\displaystyle {\begin{aligned}p&=-{\frac {1}{3}}~J^{-1}~{\boldsymbol {S}}:{\boldsymbol {C}}\\{\boldsymbol {S}}'&=J~{\boldsymbol {F}}^{-1}\cdot \mathbf {s} \cdot {\boldsymbol {F}}^{-T}~;~~{\boldsymbol {S}}':{\boldsymbol {C}}&=0\end{aligned}}}$