Advanced elasticity/Incompressible hyperelastic material

Incompressible hyperelastic materials
For an incompressible material $$J := \det\boldsymbol{F} = 1$$. The incompressibility constraint is therefore $$J-1= 0$$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:

W = W(\boldsymbol{F}) - p~(J-1) $$ where the hydrostatic pressure $$p$$ functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola-Kirchhoff stress now becomes

\boldsymbol{P}=-p~\boldsymbol{F}^{-T}+\frac{\partial W}{\partial \boldsymbol{F}} = -p~\boldsymbol{F}^{-T} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}} = -p~\boldsymbol{F}^{-T} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} ~. $$ This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy Stress tensor which is given by

\boldsymbol{\sigma}=\boldsymbol{P}\cdot\boldsymbol{F}^T= -p~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^T = -p~\boldsymbol{\mathit{1}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^T = -p~\boldsymbol{\mathit{1}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T ~. $$ For incompressible isotropic hyperelastic materials, the strain energy density function is $$W(\boldsymbol{F})=\hat{W}(I_1,I_2)$$. The Cauchy stress is then given by

\boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + 2\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] $$