Nonlinear finite elements/Deviatoric and volumetric stress

Deviatoric and volumetric stress
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
The Cauchy stress can be additively decomposed as

\boldsymbol{\sigma} = \mathbf{s} - p~\boldsymbol{\mathit{1}} $$ where $$\mathbf{s}$$ is the deviatoric stress and $$p$$ is the pressure and

\begin{align} 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{align} $$ In index notation,

\begin{align} p & = -\frac{1}{3}~\sigma_{ii} \\ s_{ij} & = \sigma_{ij} - \frac{1}{3}~\sigma_{kk}~\delta_{ij} \end{align} $$

Decomposition of the 2nd P-K stress
The second Piola-Kirchhoff stress can be decomposed into volumetric and distortional parts as

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

\begin{align} 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{align} $$