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