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 ${\displaystyle I_{1},I_{2}}$ of the left Cauchy-Green deformation tensor.

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The strain energy density function for the polynomial model is

${\displaystyle W=\sum _{i,j=0}^{n}C_{ij}(I_{1}-3)^{i}(I_{2}-3)^{j}}$

where ${\displaystyle C_{ij}}$ are material constants and ${\displaystyle C_{00}=0}$.

For compressible materials, a dependence of volume is added

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

where

${\displaystyle {\begin{aligned}{\bar {I}}_{1}&=J^{-2/3}~I_{1}~;~~I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}~;~~J=\det({\boldsymbol {F}})\\{\bar {I}}_{2}&=J^{-4/3}~I_{2}~;~~I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\end{aligned}}}$

In the limit where ${\displaystyle C_{01}=C_{11}=0}$, the polynomial model reduces to the Neo-Hookean solid model. For a compressible Mooney-Rivlin material ${\displaystyle n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1}$ and we have

${\displaystyle W=C_{01}~({\bar {I}}_{2}-3)+C_{10}~({\bar {I}}_{1}-3)+{\frac {1}{D_{1}}}~(J-1)^{2}}$