# Neo-Hookean Isochoric-split¶

The Neo-Hookean decoupled strain energy function is in terms of $$J$$ and modified invariant $$\mathbb{\bar{I}}_1$$

(59)\begin{aligned} \psi \left(\bm{C} \right) &= \psi_{\text{vol}}(J) + \psi_{\text{iso}}(\bar{\bm{C}})\\ &=\bulk \, V(J) + \frac \secondlame 2 \left( \mathbb{\bar{I}_1} - 3 \right). \end{aligned}

where $$\bulk$$ is the bulk modulus of the material and $$V(J)$$ is a strictly convex function.

One approach to obtain mixed formulation is to consider the hydrostatic pressure $$-\frac{\trace \bm \sigma}{3}$$ as a variable, where in terms of energy is

(60)\begin{aligned} p_{\text{hyd}} = - \frac{\trace \bm \sigma}{3} &= -\frac{1}{3J} \trace (\bm F \bm S \bm{F}^T) = -\frac{1}{3J} \trace (\bm S \bm F^T \bm F) = -\frac{1}{3J} \trace (\bm S \bm C) \\ &= -\frac{1}{3J} \trace \left( \frac{\partial \psi_{\text{vol}}}{\partial J} ~ J \bm{I} + \sum_{i=1}^3 \frac{\partial \psi_{\text{iso}}}{\partial \mathbb{\bar{I}}_i} \frac{\partial \mathbb{\bar{I}}_i}{\partial \bm E} \bm C \right) \\ &= - \frac{\partial \psi_{\text{vol}}}{\partial J} = -\bulk \frac{\partial V(J)}{\partial J} = -\bulk V', \end{aligned}

where we have used $$\trace (\bm A \bm B) = \trace (\bm B \bm A)$$ and

\begin{aligned} \trace \left( \frac{\partial \mathbb{\bar{I}}_1}{\partial \bm E} \bm C \right) &= 2 J^{-2/3} \trace\left( \bm C -\frac{1}{3} \mathbb{I}_1 \bm{I} \right) = 0, \\ \trace \left( \frac{\partial \mathbb{\bar{I}}_2}{\partial \bm E} \bm C \right) &= 2 J^{-4/3} \trace \left( \mathbb{I}_1 \bm C - \bm C^2 - \frac{2}{3} \mathbb{I}_2 \bm{I} \right) = 0, \\ \trace \left( \frac{\partial \mathbb{\bar{I}}_3}{\partial \bm E} \bm C \right) &= 0. \end{aligned}

However, we define a pressure-like variable similar to mixed linear elasticity (48) in terms of bulk and primal bulk (49) moduli

(61)$p = - (\bulk - \bulk_p) \frac{\partial V}{\partial J} = p_{\text{hyd}} + \bulk_p \frac{\partial V}{\partial J},$

and considered it as independent field variable in our mixed hyperelastic formulation.

The second Piola-Kirchoff tensor is computed via (38)

(62)\begin{aligned} \bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \underbrace{\frac{\partial \psi_{\text{vol}}}{\partial J}}_{-p_{\text{hyd}}} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} =\bm{S}_{\text{vol}} + \bm{S}_{\text{iso}}\\ &= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \secondlame J^{-2/3}\left( \bm{I} - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right). \end{aligned}

where the invariants are defined in (25) and (37) and we have used (40), (39), and (41).

Tip

In our simulation we use the stable version of (62) as

(63)$\bm{S} = \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + 2 \secondlame J^{-2/3} \bm{C}^{-1} \bm{E}_{\text{dev}},$

where $$\bm{E}_{\text{dev}} = \bm{E} - \frac{1}{3}\trace \bm{E} \, \bm{I}$$ is the deviatoric part of Green-Lagrange strain tensor.

The isochoric Neo-Hookean stress relation can be represented in current configuration by pushing forward (62) using (22)

(64)$\bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \left(\bulk_p J \, V' - p \, J \right) \bm{I} + \secondlame J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I} \right).$

Tip

In our simulation we use the stable version of Kirchhoff stress tensor (64) as

(65)$\bm{\tau} = \left(\bulk_p J \, V' - p \, J \right) \bm{I} + 2 \secondlame J^{-2/3} \bm{e}_{\text{dev}}$

where $$\bm{e}_{\text{dev}} = \bm{e} - \frac{1}{3}\trace \bm{e} \, \bm{I}$$ is the deviatoric part of Green-Euler strain tensor

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

(66)\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}} =\bm{S}_{\text{vol}} + \bm{S}_{\text{iso}}\\ &= \bulk J \, V' \, \bm{C}^{-1} + \secondlame J^{-2/3}\left( \bm{I} - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right). \end{aligned}

and the kirchoff stress

(67)$\bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \bulk J \, V' \, \bm{I} + \secondlame J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I} \right).$

Note that the stable form of the above stresses can be derived similar to (63), and (65).