Mooney–Rivlin solid

Example

The incompressible Mooney–Rivlin solid is the hyperelastic material with strain energy density function $$W = C_{10}(I_1 - 3) + C_{01}(I_2 - 3),$$ subject to $\det F = 1$. Here $F$ is the deformation gradient, $C = F^T F$ is the right Cauchy–Green tensor, $I_1 = \tr C$ and $I_2 = \tfrac12\left((\tr C)^2 - \tr(C^2)\right)$, and $C_{10}, C_{01}$ are material constants. Check that it is hyperelastic: $W$ is a differentiable function of $F$ through $C$, so the stress $P = \partial W / \partial F - pF^{-T}$ is defined by a stored energy. At $F = I$ one has $I_1 = I_2 = 3$, so $W(I) = 0$. Linearising at $F = I$ gives the shear modulus $\mu = 2(C_{10} + C_{01})$. With $C_{01} = 0$ it reduces to the neo-Hookean solid. In terms of the principal stretches, $I_1 = \lambda_1^2 + \lambda_2^2 + \lambda_3^2$ and, when $\lambda_1\lambda_2\lambda_3 = 1$, $I_2 = \lambda_1^{-2} + \lambda_2^{-2} + \lambda_3^{-2}$.