Validation of the QuasiIncompressibleSecondPiolaKirchhoffOperator
After reviewing the implementation of the local operator, it turns that the formulation that was implemented was not homogeneous. For the pressure part, we had
\int_{\Omega_0} \left( \frac{\partial W^{\text{vol}}}{\partial I_3} + \frac{p}{\kappa} \right) q \, d\Omega = 0,
where dividing by \kappa
is actually wrong and we are missing a term, it should read
\int_{\Omega_0} \left( 2J \frac{\partial W^{\text{vol}}}{\partial I_3} + p \right) q \, d\Omega = 0.
However, this expression for the hydrostatic constraint on p assumes that the hyperelastic potential can be decomposed additively as
W^e = W^{\text{dev}}(J_1, J_2, J_4) + W^{\text{vol}}(J),
which is not always the case. A generic implementation of the constraint would read
\int_{\Omega_0} \left( \frac{\partial W^{e}}{\partial J} + p \right) q \, d\Omega = 0.
If we want to stick to using the standard invariants of the CauchyGreen tensor (and not their reduced counter parts) we have to apply the following chain rule
\int_{\Omega_0} \left( \sum_{i=0}^4 \frac{\partial W^{e}}{\partial I_i}\frac{\partial I_i}{\partial J} + p \right) q \, d\Omega = 0.
The operator (ands its test) will be updated accordingly.