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{aligned}{\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{aligned}}

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}