Neo-Hookean Isochoric-split

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

(43)\[ \begin{aligned} \psi \left(\bm{C} \right) &= \psi_{vol}(J) + \psi_{iso}(\bar{\bm{C}})\\ &=\frac{\kappa}{4} \left( J^2 - 1 -2 \log J \right) + \frac \mu 2 \left( \mathbb{\bar{I}_1} - 3 \right). \end{aligned} \]

where \(\kappa\) is the bulk modulus of the material.

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

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

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

The pressure \(p\) in (44) is defined by

(45)\[ p = - \frac{\partial \psi_{vol}}{\partial J} = -\frac{\kappa}{2 J} (J^2 -1). \]

Tip

In our simulation we use the stable version of (44) and (45) as

(46)\[ \bm{S} = -p J \bm{C}^{-1} + 2 \mu J^{-2/3} \bm{C}^{-1} \bm{E}_{dev}, \]

(47)\[ p = -\frac{\kappa}{2 J} \mathtt{J_{-1}} \left(\mathtt{J_{-1}} + 2 \right), \]

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 \(\mathtt{J_{-1}}\) is computed by (38).

The isochoric Neo-Hookean stress relation can be represented in current configuration by pushing forward (44) using (7)

(48)\[ \bm{\tau} = \bm{F}\bm{S}\bm{F}^T = -p J \bm{I}_{3} + \mu J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right). \]

Tip

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

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

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

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

and the kirchoff stress

(51)\[ \bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \frac{\kappa}{2} (J^2 -1)\bm{I}_{3} + \mu J^{-2/3}\left( \bm{b} - \frac{1}{3}\mathbb{{I}_1} \bm{I}_{3} \right). \]

Note that the stable form of the above stresses can be derived similar to (46), (47) and (49).