Mooney-Rivlin Materials#

While the Neo-Hookean model depends on just two scalar invariants, \(\mathbb{I}_1 = \operatorname{trace} \bm{C} = 3 + 2 \operatorname{trace} \bm{E}\) and \(J\), Mooney-Rivlin models depend on the additional invariant, \(\mathbb{I}_2 = \frac 1 2 \left( \mathbb{I}_1^2 - \bm{C} \tcolon \bm{C} \right)\). A coupled Mooney-Rivlin strain energy density (cf. Neo-Hookean (13)) is [Hol00]

(29)#\[ \psi \left( \mathbb{I_1}, \mathbb{I_2}, J \right) = \frac{\lambda}{4} \left( J^2 - 1 -2 \log J \right) - \left( \mu_1 + 2\mu_2 \right) \log J + \frac{\mu_1}{2} \left( \mathbb{I_1} - 3 \right) + \frac{\mu_2}{2} \left( \mathbb{I_2} - 3 \right). \]

We differentiate \(\psi\) as in the Neo-Hookean case (14) to yield the second Piola-Kirchoff tensor,

(30)#\[ \begin{aligned} \bm{S} &= \frac{\lambda}{2} \left( J^2 - 1 \right)\bm{C}^{-1} - \left( \mu_1 + 2\mu_2 \right) \bm{C}^{-1} + \mu_1\bm{I}_3 + \mu_2 \left( \mathbb{I_1} \bm{I}_3 - \bm{C} \right) \\ &= \frac{\lambda}{2} \left( J^2 - 1 \right)\bm{C}^{-1} + \mu_1 \left( \bm{I}_3 - \bm{C}^{-1} \right) + \mu_2 \left( \mathbb{I_1} \bm{I}_3 - 2 \bm{C}^{-1} - \bm{C} \right), \end{aligned} \]

where we have used

(31)#\[ \begin{aligned} \frac{\partial \mathbb{I_1}}{\partial \bm{E}} &= 2 \bm{I}_3, & \frac{\partial \mathbb{I_2}}{\partial \bm{E}} &= 2 \mathbb{I}_1 \bm{I}_3 - 2 \bm{C}, & \frac{\partial \log J}{\partial \bm{E}} &= \bm{C}^{-1}. \end{aligned} \]

This is a common model for vulcanized rubber, with a shear modulus (defined for the small-strain limit) of \(\mu_1 + \mu_2\) that should be significantly smaller than the first Lamé parameter \(\lambda\).

Tip

Similar to the Neo-Hookean materials, the stable form for the Mooney-Rivlin model in initial configuration (30) can be written as

(32)#\[ \bm{S} = \frac{\lambda}{2} \mathtt{J_{-1}} \left(\mathtt{J_{-1}} + 2 \right) \bm{C}^{-1} + 2 \left( \mu_1 + 2\mu_2 \right)\bm{C}^{-1} \bm{E} + 2\mu_2 \left(\operatorname{trace} \left(\bm{E} \right) \bm{I}_3 - \bm{E} \right). \]

The Kirchhoff stress tensor \(\bm{\tau}\) for Mooney-Rivilin model is given by

(33)#\[ \bm{\tau} = \bm{F}\bm{S}\bm{F}^T = \frac{\lambda}{2} \left( J^2 - 1 \right)\bm{I}_{3} + \mu_1 \left( \bm{b} - \bm{I}_3 \right) + \mu_2 \left( \mathbb{I_1} \bm{b} - 2 \bm{I}_{3} - \bm{b}^2 \right). \]

Note that \(\mathbb{I_i}\left(\bm{b}\right) = \mathbb{I_i} \left(\bm{C} \right),\) for \(i=1,2,3\).

Tip

The stable Kirchhoff stress tensor version of (33) is given by

(34)#\[ \bm{\tau} = \frac{\lambda}{2} \mathtt{J_{-1}} \left(\mathtt{J_{-1}} + 2 \right) \bm{I}_{3} + 2 \left( \mu_1 + 2\mu_2 \right)\bm{e} + 2\mu_2 \left(\operatorname{trace} \left(\bm{e}\right) \bm{I}_3 - \bm{e} \right) \bm{b}. \]

where \(\mathtt{J_{-1}}\) is computed by (17)