Hyperelastic material - Compressible Mooney-Rivlin

The Mooney-Rivlin strain energy function is expressed by

$$\rho_0 \psi = C_1(\bar{I}_1-3) + C_2(\bar{I}_2-3) + \frac{1}{2}K ({\rm ln} J)^2$$ (11)

where $C_1$ and $C_2$ are material constants, $K$ is the bulk modulus, $J$ is the Jacobian (determinant of the deformation gradient, corresponding to the ratio of the current and initial volume), $\bar{I}_1 = J^{-\frac{2}{3}} I_1$, $\bar{I}_2 = J^{-\frac{2}{3}} I_2$, where $I_1$ and $I_2$ are the first and the second principal invariants of the right Cauchy-Green deformation tensor $C$. Compressible neo-Hookean material model is obtained by setting $C_2 = 0$. Then stress-strain law can be derived from (11) as

$$\mbox{\boldmath$P$} = \rho_0 \frac{\partial \psi}{\partial \mbox{\boldmath$F$}} = C_1... ...l\bar{I}_2}{\partial\mbox{\boldmath$F$}} + K {\rm ln}J \mbox{\boldmath$F$}^{-T}$$ (12)

where $P$ is the first Piola-Kirchhoff stress,

$$\frac{\partial \bar{I}_1}{\partial\mbox{\boldmath$F$}} = \frac{2}... ...\frac{2}{3}}}\mbox{\boldmath$F$} - \frac{2}{3}\bar{I}_1\mbox{\boldmath$F$}^{-T}$$ (13)

and

$$\frac{\partial \bar{I}_2}{\partial\mbox{\boldmath$F$}} = 2\bar{I}... ...th$F$}^{-T}-\frac{2}{J^\frac{4}{3}}\mbox{\boldmath$F$}\cdot \mbox{\boldmath$C$}$$ (14)

The first elasticity tensor is derived as

$$A_{ijkl} = \frac{\partial P_{ij}}{\partial F_{kl}} = C_1 A^1_{ijkl} + C_2 A^2_{ijkl} + K(F_{ji}^{-1}F_{lk}^{-1} - {\rm ln}J F_{jk}^{-1}F_{li}^{-1})$$ (15)

where

$$A^1_{ijkl} = \frac{2}{3} J^{-\frac{2}{3}}\left[3\delta_{ik}\delta... ...lk}^{-1}F_{ij}+\frac{2}{3}I_1 F_{ji}^{-1}F_{lk}^{-1}-2F_{ji}^{-1}F_{kl} \right]$$ (16)

and

$$A^2_{ijkl} = 2 J^{-\frac{4}{3}}\left[ I_1 \delta_{ik} \delta_{jl}... ...-\frac{8}{9}I_2 F_{ji}^{-1}F_{lk}^{-1}-\frac{4}{3}I_1 F_{ji}^{-1}F_{kl} \right.$$

$$\left. + \frac{4}{3} F_{kn}C_{nl}F_{ji}^{-1} +\frac{2}{3}I_2 F_{l... ..._{im}C_{mj} - \delta_{ik}C_{lj} - F_{il}F_{kj} + F_{km}F_{im}\delta_{jl}\right]$$ (17)

The model description and parameters are summarized in Tab. 5.

Table 5: Compressible Mooney-Rivlin - summary.

Description	Mooney-Rivlin
Record Format	MooneyRivlin (in) # d(rn) # K(rn) # C1(rn) # C2(rn) #
Parameters	- material number
	- d material density
	- K bulk modulus
	- C1 material constant
	- C2 material constant
Supported modes	3dMat, PlaneStrain