Mooney-Rivlin model

The Mooney-Rivlin hyperelastic constitutive model.^1,2

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The extra modulus ${\mu }_m$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The Mooney-Rivlin model reduces to the Neo-Hookean model when ${\mu }_m\to 0$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{\mu -{\mu }_m}{2}\left[\mathrm{tr}({\mathbf{B}}^{\ast })-3\right]+\frac{{\mu }_m}{2}\left[I_2({\mathbf{B}}^{\ast })-3\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{\mu -{\mu }_m}{J}{{\mathbf{B}}^{\ast }}^{\prime }-\frac{{\mu }_m}{J}{\left({\mathbf{B}}^{\ast -1}\right)}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{\mu -{\mu }_m}{J^{5/3}}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\\ & -\frac{{\mu }_m}{J}\left[\frac{2}{3}{B}_{ij}^{\ast -1}{F}_{kL}^{-T}-B_{ik}^{\ast -1}{F}_{jL}^{-T}-B_{ik}^{\ast -1}{F}_{iL}^{-T}+\frac{2}{3}{\delta }_{ij}{\left(B_{km}^{\ast -1}\right)}^{\prime }{F}_{mL}^{-T}-{\left(B_{ij}^{\ast -1}\right)}^{\prime }{F}_{kL}^{-T}\right]\end{array}$$

1. M. Mooney, J. Appl. Phys. 11, 582 (1940). ↩

2. R.S. Rivlin, Philos. Trans. R. Soc. London, Ser. A 241, 379 (1948). ↩