Neo-Hookean Isochoric-split

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

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

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

(44)\[ \begin{aligned} p_{\text{hyd}} = - \frac{\trace \bm \sigma}{3} &= -\frac{1}{3J} \trace (\bm F \bm S \bm{F}^T) = -\frac{1}{3J} \trace (\bm S \bm F^T \bm F) = -\frac{1}{3J} \trace (\bm S \bm C) \\ &= -\frac{1}{3J} \trace \left( \frac{\partial \psi_{vol}}{\partial J} ~ J \bm I_3 + \sum_{i=1}^3 \frac{\partial \psi_{iso}}{\partial \mathbb{\bar{I}}_i} \frac{\partial \mathbb{\bar{I}}_i}{\partial \bm E} \bm C \right) \\ &= - \frac{\partial \psi_{vol}}{\partial J} = -\bulk \frac{\partial V(J)}{\partial J} = -\bulk V', \end{aligned} \]

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

\[ \begin{aligned} \trace \left( \frac{\partial \mathbb{\bar{I}}_1}{\partial \bm E} \bm C \right) &= 2 J^{-2/3} \trace\left( \bm C -\frac{1}{3} \mathbb{I}_1 \bm I_3 \right) = 0, \\ \trace \left( \frac{\partial \mathbb{\bar{I}}_2}{\partial \bm E} \bm C \right) &= 2 J^{-4/3} \trace \left( \mathbb{I}_1 \bm C - \bm C^2 - \frac{2}{3} \mathbb{I}_2 \bm I_3 \right) = 0, \\ \trace \left( \frac{\partial \mathbb{\bar{I}}_3}{\partial \bm E} \bm C \right) &= 0. \end{aligned} \]

However, we define a pressure-like variable in terms of bulk and primal bulk moduli

(45)\[ p = - (\bulk - \bulk_p) \frac{\partial V}{\partial J} = p_{\text{hyd}} + \bulk_p \frac{\partial V}{\partial J}, \]

and considered it as independent field variable in our mixed formulation whit \(\bulk_p = 2 \mu (1 + \nu_p) / 3 (1 - 2 \nu_p) \) is the primal portion of the bulk modulus, defined in terms of \(\nu_p\) with \(-1 \leq \nu_p < \nu\).

The second Piola-Kirchoff tensor is computed via (30)

(46)\[ \begin{aligned} \bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \underbrace{\frac{\partial \psi_{vol}}{\partial J}}_{-p_{\text{hyd}}} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} =\bm{S}_{vol} + \bm{S}_{iso}\\ &= \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + \mu J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right). \end{aligned} \]

where the invariants are defined in (17) and (29) and we have used (32), (31), and (33).

Tip

In our simulation we use the stable version of (46) as

(47)\[ \bm{S} = \left(\bulk_p J \, V' - p \, J \right) \bm{C}^{-1} + 2 \mu J^{-2/3} \bm{C}^{-1} \bm{E}_{dev}, \]

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

The isochoric Neo-Hookean stress relation can be represented in current configuration by pushing forward (46) using (7)

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

Tip

In our simulation we use the stable version of Kirchhoff stress tensor (48) as

(49)\[ \bm{\tau} = \left(\bulk_p J \, V' - p \, J \right) \bm{I}_{3} + 2 \mu J^{-2/3} \bm{e}_{dev} \]

where \(\bm{e}_{dev} = \bm{e} - \frac{1}{3}\trace \bm{e} \, \bm{I}_{3} \) 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

(50)\[ \begin{aligned} \bm{S} = \frac{\partial \psi}{\partial \bm{E}} &= \frac{\partial \psi_{vol}}{\partial J} \frac{\partial J}{\partial \bm{E}} + \frac{\partial \psi}{\partial \mathbb{\bar{I}_1}} \frac{\partial \mathbb{\bar{I}_1}}{\partial \bm{E}} =\bm{S}_{vol} + \bm{S}_{iso}\\ &= \bulk J \, V' \, \bm{C}^{-1} + \mu J^{-2/3}\left( \bm{I}_3 - \frac{1}{3}\mathbb{{I}_1}\bm{C}^{-1} \right). \end{aligned} \]

and the kirchoff stress

(51)\[ \bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \bulk J \, V' \, \bm{I}_{3} + \mu J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right). \]

Note that the stable form of the above stresses can be derived similar to (47), and (49).