Polynomial hyperelastic model

{{#invoke:sidebar|collapsible | name = Continuum mechanics | class = plainlist | pretitle = Part of a series on | title = Continuum mechanics |listtitlestyle = background:transparent;border-top:1px solid #aaa;text-align:center; | expanded = Solid mechanics | image = $J=-D\frac{d\phi }{dx}$ | caption = Fick's laws of diffusion

| list1name = laws | list1title = Laws | list1 =

  Template:Sidebar

| list2name = solid | list2title = Solid mechanics | list2 = Template:Startflatlist

• Deformation
• Elasticity
  • linear
• Plasticity
• Hooke's law
• Stress
• Strain
  • Finite strain
  • Infinitesimal strain
• Compatibility
• Bending
• Contact mechanics
  • frictional
• Material failure theory
• Fracture mechanics

| list3name = fluid | list3title = Fluid mechanics | list3 =

  Template:Sidebar

| list4name = rheology | list4title = Rheology | list4 =

  Template:Sidebar

| list5name = scientists | list5title = Scientists | list5 = Template:Startflatlist

• Bernoulli
• Boyle
• Cauchy
• Charles
• Euler
• Fick
• Gay-Lussac
• Graham
• Hooke
• Newton
• Navier
• Noll
• Pascal
• Stokes
• Truesdell

}} 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

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

where

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

In the limit where $C_{01}=C_{11}=0$, 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

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

See also

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