Mooney-Rivlin Isochoric-split
The Mooney-Rivlin decoupled strain energy function is in terms of \(J\) and modified invariants \(\mathbb{\bar{I}}_1, \mathbb{\bar{I}}_2\) (29)
(57)\[
\begin{aligned}
\psi \left(\bm{C} \right) &= \psi_{vol}(J) + \psi_{iso}(\bar{\bm{C}})\\
&=\bulk \, V(J) + \frac{\mu_1}{2} \left( \mathbb{\bar{I}_1} - 3 \right) + \frac{\mu_2}{2} \left( \mathbb{\bar{I}_2} - 3 \right).
\end{aligned}
\]
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 (30)
(58)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{vol}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{vol} + \bm{S}_{iso}\\
&= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \mu_1 J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I}_3 - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
where the invariants are defined in (17) and (29) and we have used (32), (31), and (33).
The pressure-like variable \(p\) in (58) is defined similar to (45).
Tip
Similar to Neo-Hookean isochoric model (47) we can derive a stable form of (58) as
(59)\[
\begin{aligned}
\bm{S} &= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \left( 2 \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{C}^{-1} \bm{E}_{dev} \\
&+ 2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E})\bm{I}_3 - \bm{E} \right) - \frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E}) + 2 \mathbb{{I}_2}(\bm{E})\right) \bm{C}^{-1},
\end{aligned}
\]
where \(\bm{E}_{dev} = \bm{E} - \frac{1}{3}\trace \bm{E} \, \bm{I}_{3} \) 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 (58) using (7)
(60)\[
\bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \left(\bulk_p J \, V' - p \, J \right) \bm{I}_{3} + \mu_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \right)
\]
Tip
In our simulation we use the stable version of Kirchhoff stress tensor (60) as
(61)\[
\begin{aligned}
\bm{\tau} &= \left(\bulk_p J \, V' - p \, J \right) \bm{I}_{3} + \left( 2 \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{e}_{dev} \\
&+ 2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e})\bm{I}_3 - \bm{e} \right) \bm{b} - \frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e}) + 2 \mathbb{{I}_2}(\bm{e})\right) \bm{I}_{3},
\end{aligned}
\]
where \(\bm{e}_{dev} = \bm{e} - \frac{1}{3}\trace \bm{e} \, \bm{I}_{3} \) 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
(62)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{vol}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{vol} + \bm{S}_{iso}\\
&= \bulk J \, V' \,\bm{C}^{-1} + \mu_1 J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I}_3 - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
and the kirchoff stress
(63)\[
\bm{\tau} = \bulk J \, V' \, \bm{I}_{3} + \mu_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \right)
\]
Note that the stable form of the above stresses can be derived similar to (59), and (61).