# 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).