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

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

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

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

For compressible materials, a dependence of volume is added

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

${\displaystyle {\begin{aligned}{\bar {I}}_{1}&=J^{-2/3}~I_{1}~;~~I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}~;~~J=\det({\boldsymbol {F}})\\{\bar {I}}_{2}&=J^{-4/3}~I_{2}~;~~I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\end{aligned}}}${\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 ${\displaystyle C_{01}=C_{11}=0}${\displaystyle C_{01}=C_{11}=0}, the polynomial model reduces to the Neo-Hookean solid model. For a compressible Mooney-Rivlin material ${\displaystyle n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1}${\displaystyle n=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},m=1} and we have

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

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

• Continuum mechanics
• Non-Newtonian fluids
• Rubber properties
• Solid mechanics

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