Mooney-Rivlin Isochoric-split
The Mooney-Rivlin decoupled strain energy function is in terms of \(J\) and modified invariants \(\mathbb{\bar{I}}_1, \mathbb{\bar{I}}_2\) (37)
(73)\[
\begin{aligned}
\psi \left(\bm{C} \right) &= \psi_{\text{vol}}(J) + \psi_{\text{iso}}(\bar{\bm{C}})\\
&=\bulk \, V(J) + \frac{\secondlame_1}{2} \left( \mathbb{\bar{I}_1} - 3 \right) + \frac{\secondlame_2}{2} \left( \mathbb{\bar{I}_2} - 3 \right).
\end{aligned}
\]
where \(\bulk\) is the bulk modulus of the material and \(V(J)\) is a strictly convex function.
The second Piola-Kirchoff tensor is computed via (38)
(74)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{\text{vol}}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{\text{vol}} + \bm{S}_{\text{iso}}\\
&= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \secondlame_1 J^{-2/3}\left( \bm{I} - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \secondlame_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I} - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
where the invariants are defined in (25) and (37) and we have used (40), (39), and (41).
The pressure-like variable \(p\) in (74) is defined similar to (61).
Tip
Similar to Neo-Hookean isochoric model (63) we can derive a stable form of (74) as
(75)\[
\begin{aligned}
\bm{S} &= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \left( 2 \secondlame_1 J^{-2/3} + 4 \secondlame_2 J^{-4/3} \right) \bm{C}^{-1} \bm{E}_{\text{dev}} \\
&+ 2 \secondlame_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E})\bm{I} - \bm{E} \right) - \frac{4}{3} \secondlame_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E}) + 2 \mathbb{{I}_2}(\bm{E})\right) \bm{C}^{-1},
\end{aligned}
\]
where \(\bm{E}_{\text{dev}} = \bm{E} - \frac{1}{3}\trace \bm{E} \, \bm{I} \) is the deviatoric part of Green-Lagrange strain tensor and \(\mathbb{{I}_1}(\bm{E}), \mathbb{{I}_2}(\bm{E})\) are first and second invariants of the tensor \(\bm{E}\).
The isochoric Mooney-Rivlin stress relation can be represented in current configuration by pushing forward (74) using (22)
(76)\[
\bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \left(\bulk_p J \, V' - p \, J \right) \bm{I} + \secondlame_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I} \right) + \secondlame_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \bm{I} \right)
\]
Tip
In our simulation we use the stable version of Kirchhoff stress tensor (76) as
(77)\[
\begin{aligned}
\bm{\tau} &= \left(\bulk_p J \, V' - p \, J \right) \bm{I} + \left( 2 \secondlame_1 J^{-2/3} + 4 \secondlame_2 J^{-4/3} \right) \bm{e}_{\text{dev}} \\
&+ 2 \secondlame_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e})\bm{I} - \bm{e} \right) \bm{b} - \frac{4}{3} \secondlame_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e}) + 2 \mathbb{{I}_2}(\bm{e})\right) \bm{I},
\end{aligned}
\]
where \(\bm{e}_{\text{dev}} = \bm{e} - \frac{1}{3}\trace \bm{e} \, \bm{I} \) is the deviatoric part of Green-Euler strain tensor and \(\mathbb{{I}_1}(\bm{e}), \mathbb{{I}_2}(\bm{e})\) are first and second invariants of the tensor \(\bm{e}\).
Note
We can solve the isochoric model with single field displacement if the material is not incompressible which leads to the second Piola-Kirchoff tensor
(78)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{\text{vol}}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{\text{vol}} + \bm{S}_{\text{iso}}\\
&= \bulk J \, V' \,\bm{C}^{-1} + \secondlame_1 J^{-2/3}\left( \bm{I} - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \secondlame_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I} - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
and the kirchoff stress
(79)\[
\bm{\tau} = \bulk J \, V' \, \bm{I} + \secondlame_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I} \right) + \secondlame_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \bm{I} \right)
\]
Note that the stable form of the above stresses can be derived similar to (75), and (77).