Polynomial hyperelastic model

The  polynomial hyperelastic material model is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the form of a polynomial in the two invariants $$I_1,I_2$$ of the left Cauchy-Green deformation tensor.

The strain energy density function for the polynomial model is

W = \sum_{i,j=0}^n C_{ij} (I_1 - 3)^i (I_2 - 3)^j $$ where $$C_{ij}$$ are material constants and $$C_{00}=0$$.

For compressible materials, a dependence of volume is added

W = \sum_{i,j=0}^n C_{ij} (\bar{I}_1 - 3)^i (\bar{I}_2 - 3)^j + \sum_{k=1}^m \frac{1}{D_{k}}(J-1)^{2k} $$ where

\begin{align} \bar{I}_1 & = J^{-2/3}~I_1 ~; I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~; J = \det(\boldsymbol{F}) \\ \bar{I}_2 & = J^{-4/3}~I_2 ~; I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2 \end{align} $$

In the limit where $$C_{01}=C_{11}=0$$, the polynomial model reduces to the Neo-Hookean solid model. For a compressible Mooney-Rivlin material $$n = 1, C_{01} = C_2, C_{11} = 0, C_{10} = C_1, m=1$$ and we have

W = C_{01}~(\bar{I}_2 - 3) + C_{10}~(\bar{I}_1 - 3) + \frac{1}{D_1}~(J-1)^2 $$