# 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).