# Mooney-Rivlin Materials While the Neo-Hookean model depends on just two scalar invariants, $\mathbb{I}_1 = \trace \bm{C} = 3 + 2 \trace \bm{E}$ and $J$, Mooney-Rivlin models depend on the additional invariant, $\mathbb{I}_2$ defined in {eq}invariants. A coupled Mooney-Rivlin strain energy density (cf. Neo-Hookean {eq}neo-hookean-energy) is {cite}holzapfel2000nonlinear \begin{aligned} \psi \left( \mathbb{I_1}, \mathbb{I_2}, J \right) &= \firstlame V(J) - \left( \secondlame_1 + 2\secondlame_2 \right) \log J + \frac{\secondlame_1}{2} \left( \mathbb{I_1} - 3 \right) + \frac{\secondlame_2}{2} \left( \mathbb{I_2} - 3 \right), \\ &= \frac{\firstlame}{4} \left( J^2 - 1 -2 \log J \right) - \left( \secondlame_1 + 2\secondlame_2 \right) \log J + \frac{\secondlame_1}{2} \left( \mathbb{I_1} - 3 \right) + \frac{\secondlame_2}{2} \left( \mathbb{I_2} - 3 \right). \end{aligned} (mooney-rivlin-energy_coupled) We differentiate $\psi$ as in the Neo-Hookean case {eq}neo-hookean-stress to yield the second Piola-Kirchoff tensor, \begin{aligned} \bm{S} &= \firstlame J V' \bm{C}^{-1} - \left( \secondlame_1 + 2\secondlame_2 \right) \bm{C}^{-1} + \secondlame_1\bm{I} + \secondlame_2 \left( \mathbb{I_1} \bm{I} - \bm{C} \right) \\ &= \frac{\firstlame}{2} \left( J^2 - 1 \right)\bm{C}^{-1} + \secondlame_1 \left( \bm{I} - \bm{C}^{-1} \right) + \secondlame_2 \left( \mathbb{I_1} \bm{I} - 2 \bm{C}^{-1} - \bm{C} \right), \end{aligned} (mooney-rivlin-stress_coupled) where we have used {eq}invariants-derivative. This is a common model for vulcanized rubber, with a shear modulus (defined for the small-strain limit) of $\secondlame_1 + \secondlame_2$ that should be significantly smaller than the first LamÃ© parameter $\firstlame$. :::{dropdown} Mooney-Rivlin strain energy comparison We apply traction to a block and plot integrated strain energy $\psi$ as a function of the loading parameter. {altair-plot} :hide-code: import altair as alt import pandas as pd def source_path(rel): import os return os.path.join(os.path.dirname(os.environ["DOCUTILSCONFIG"]), rel) nh = pd.read_csv(source_path("doc/data/NH-strain.csv")) nh["model"] = "Neo-Hookean" nh["parameters"] = "E=2.8, nu=0.4" mr = pd.read_csv(source_path("doc/data/MR-strain.csv")) mr["model"] = "Mooney-Rivlin; Neo-Hookean equivalent" mr["parameters"] = "mu_1=1, mu_2=0, nu=.4" mr1 = pd.read_csv(source_path("doc/data/MR-strain1.csv")) mr1["model"] = "Mooney-Rivlin" mr1["parameters"] = "mu_1=0.5, mu_2=0.5, nu=.4" df = pd.concat([nh, mr, mr1]) highlight = alt.selection_point( on = "mouseover", nearest = True, fields=["model", "parameters"], ) base = alt.Chart(df).encode( alt.X("increment"), alt.Y("energy", scale=alt.Scale(type="sqrt")), alt.Color("model"), alt.Tooltip(("model", "parameters")), opacity=alt.condition(highlight, alt.value(1), alt.value(.5)), size=alt.condition(highlight, alt.value(2), alt.value(1)), ) base.mark_point().add_params(highlight) + base.mark_line()  ::: :::{tip} Similar to the Neo-Hookean materials, the stable form for the Mooney-Rivlin model in initial configuration {eq}mooney-rivlin-stress_coupled can be written as $$\bm{S} = \frac{\firstlame}{2} \mathtt{J_{-1}} \left(\mathtt{J_{-1}} + 2 \right) \bm{C}^{-1} + 2 \left( \secondlame_1 + 2\secondlame_2 \right)\bm{C}^{-1} \bm{E} + 2\secondlame_2 \left(\trace \left(\bm{E} \right) \bm{I} - \bm{E} \right).$$ (mooney-rivlin-stress-stable) ::: The Kirchhoff stress tensor $\bm{\tau}$ for Mooney-Rivlin model is given by \begin{aligned} \bm{\tau} = \bm{F}\bm{S}\bm{F}^T &= \firstlame J V' \bm{I} + \secondlame_1 \left( \bm{b} - \bm{I} \right) + \secondlame_2 \left( \mathbb{I_1} \bm{b} - 2 \bm{I} - \bm{b}^2 \right), \\ &= \frac{\firstlame}{2} \left( J^2 - 1 \right)\bm{I} + \secondlame_1 \left( \bm{b} - \bm{I} \right) + \secondlame_2 \left( \mathbb{I_1} \bm{b} - 2 \bm{I} - \bm{b}^2 \right). \end{aligned} (mooney-rivlin-tau) :::{tip} The stable Kirchhoff stress tensor version of {eq}mooney-rivlin-tau is given by $$\bm{\tau} = \frac{\firstlame}{2} \mathtt{J_{-1}} \left(\mathtt{J_{-1}} + 2 \right) \bm{I} + 2 \left( \secondlame_1 + 2\secondlame_2 \right)\bm{e} + 2\secondlame_2 \left(\trace \left(\bm{e}\right) \bm{I} - \bm{e} \right) \bm{b}.$$ (mooney-rivlin-tau-stable) where $\mathtt{J_{-1}}$ is computed by {eq}JM1 :::