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

${\displaystyle W=\sum _{i,j=0}^n{C}_{ij}({\overline{I}}_1-3{)}^i({\overline{I}}_2-3{)}^j+\sum _{k=1}^m{D}_k(J-1{)}^{2k}}$

where

${\displaystyle \begin{array}{rl}{\overline{I}}_1& =J^{-2/3}{I}_1;I_1={\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2;J=det(\mathit{F})\\ {\overline{I}}_2& =J^{-4/3}{I}_2;I_2={\lambda }_1^2{\lambda }_2^2+{\lambda }_2^2{\lambda }_3^2+{\lambda }_3^2{\lambda }_1^2\end{array}}$

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

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

References

See also

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

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