Ogden hyperelastic model

Example

The incompressible Ogden model is the isotropic hyperelastic material whose strain energy density function is $$W = \sum_{p=1}^{N} \frac{\mu_p}{\alpha_p}\left(\lambda_1^{\alpha_p} + \lambda_2^{\alpha_p} + \lambda_3^{\alpha_p} - 3\right),$$ subject to $\lambda_1 \lambda_2 \lambda_3 = 1$. Here $\lambda_1, \lambda_2, \lambda_3$ are the principal stretches, the square roots of the eigenvalues of $F^T F$, and $N \ge 1$, $\mu_p$, $\alpha_p \ne 0$ are material constants. $W$ is a symmetric function of the principal stretches, so it defines an isotropic stored energy, and $W = 0$ when all stretches are $1$. The shear modulus at small strain is $\mu = \tfrac12 \sum_p \mu_p \alpha_p$. Check of two special cases: $N = 1$, $\alpha_1 = 2$ gives $\frac{\mu_1}{2}(I_1 - 3)$, the neo-Hookean solid. $N = 2$, $\alpha_1 = 2$, $\alpha_2 = -2$ gives the Mooney–Rivlin solid, because $\lambda_1^{-2} + \lambda_2^{-2} + \lambda_3^{-2} = I_2$ when $\lambda_1\lambda_2\lambda_3 = 1$.