Polynomial (hyperelastic model)

The polynomial hyperelastic material model [1] 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.

The strain energy density function for the polynomial model is [1]

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

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

For compressible materials, a dependence of volume is added

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

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

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

References

1. Rivlin, R. S. and Saunders, D. W., 1951, Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber. Phi. Trans. Royal Soc. London Series A, 243(865), pp. 251-288.

See also

• Hyperelastic material
• Strain energy density function
• Mooney-Rivlin solid
• Finite strain theory
• Stress measures