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)\[
\begin{aligned}
\psi \left(\bm{C} \right) &= \psi_{vol}(J) + \psi_{iso}(\bar{\bm{C}})\\
&=\frac{\kappa}{4} \left( J^2 - 1 -2 \log J \right) + \frac{\mu_1}{2} \left( \mathbb{\bar{I}_1} - 3 \right) + \frac{\mu_2}{2} \left( \mathbb{\bar{I}_2} - 3 \right).
\end{aligned}
\]
where \(\kappa\) is the bulk modulus of the material.
The second Piola-Kirchoff tensor is computed via (30)
(58)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{vol}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{vol} + \bm{S}_{iso}\\
&= -p J \bm{C}^{-1} + \mu_1 J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I}_3 - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
where the invariants are defined in (17) and (29) and we have used (32), (31), and (33).
The pressure \(p\) in (58) is defined similar to (45).
Tip
Similar to Neo-Hookean isochoric model (46) we can derive a stable form of (58) as
(59)\[
\begin{aligned}
\bm{S} &= -p J \bm{C}^{-1} + \left( 2 \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{C}^{-1} \bm{E}_{dev} \\
&+ 2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E})\bm{I}_3 - \bm{E} \right) - \frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{E}) + 2 \mathbb{{I}_2}(\bm{E})\right) \bm{C}^{-1},
\end{aligned}
\]
where \(\bm{E}_{dev} = \bm{E} - \frac{1}{3}\operatorname{trace} \bm{E} \, \bm{I}_{3} \) 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)\[
\bm{\tau} = \bm{F}\bm{S}\bm{F}^T = -p J \bm{I}_{3} + \mu_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \right)
\]
Tip
In our simulation we use the stable version of Kirchhoff stress tensor (60) as
(61)\[
\begin{aligned}
\bm{\tau} &= -p J \bm{I}_{3} + \left( 2 \mu_1 J^{-2/3} + 4 \mu_2 J^{-4/3} \right) \bm{e}_{dev} \\
&+ 2 \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e})\bm{I}_3 - \bm{e} \right) \bm{b} - \frac{4}{3} \mu_2 J^{-4/3} \left( \mathbb{{I}_1}(\bm{e}) + 2 \mathbb{{I}_2}(\bm{e})\right) \bm{I}_{3},
\end{aligned}
\]
where \(\bm{e}_{dev} = \bm{e} - \frac{1}{3}\operatorname{trace} \bm{e} \, \bm{I}_{3} \) 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)\[
\begin{aligned}
\bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{vol}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_2}} \frac{\partial \mathbb{\bar{I}_2}}{\partial \bm{E}} = \bm{S}_{vol} + \bm{S}_{iso}\\
&= \frac{\kappa}{2} (J^2 -1)\bm{C}^{-1} + \mu_1 J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{I}_3 - \bm{C} - \frac{2}{3}\mathbb{{I}_2}\bm{C}^{-1} \right).
\end{aligned}
\]
and the kirchoff stress
(63)\[
\bm{\tau} = \frac{\kappa}{2} (J^2 -1)\bm{I}_{3} + \mu_1 J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right) + \mu_2 J^{-4/3}\left( \mathbb{{I}_1} \bm{b} - \bm{b}^2 - \frac{2}{3}\mathbb{{I}_2} \right)
\]
Note that the stable form of the above stresses can be derived similar to (59), and (61).