View source for Physics:Polynomial (hyperelastic model)

{{Continuum mechanics|cTopic=[[Physics:Solid mechanics|Solid mechanics]]}}
The ''' polynomial''' [[Physics:Hyperelastic material|hyperelastic material]] model <ref name=Rivlin/> is a phenomenological model of [[Physics:Rubber elasticity|rubber elasticity]].  In this model, the [[Physics:Strain energy density function|strain energy density function]] is of the form of a polynomial in the two invariants <math>I_1,I_2</math> of the left Cauchy-Green deformation tensor.

The strain energy density function for the polynomial model is <ref name=Rivlin>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.</ref>
:<math>
  W = \sum_{i,j=0}^n C_{ij} (I_1 - 3)^i (I_2 - 3)^j
</math>
where <math>C_{ij}</math> are material constants and <math>C_{00}=0</math>.

For compressible materials, a dependence of volume is added
:<math>
  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}
</math>
where
:<math>
  \begin{align}
    \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{align}
 </math>

In the limit where <math>C_{01}=C_{11}=0</math>, the polynomial model reduces to the [[Physics:Neo-Hookean solid|Neo-Hookean solid]] model. For a compressible [[Physics:Mooney–Rivlin solid|Mooney-Rivlin material]] <math>n = 1, C_{01} = C_2, C_{11} = 0, C_{10} = C_1, m=1</math> and we have
:<math>
   W = C_{01}~(\bar{I}_2 - 3) + C_{10}~(\bar{I}_1 - 3) + D_1~(J-1)^2
 </math>

==References==
<references/>

== See also ==
* [[Physics:Hyperelastic material|Hyperelastic material]]
* [[Physics:Strain energy density function|Strain energy density function]]
* Mooney-Rivlin solid
* [[Physics:Finite strain theory|Finite strain theory]]
* [[Physics:Stress measures|Stress measures]]

[[Category:Continuum mechanics]]
[[Category:Non-Newtonian fluids]]
[[Category:Rubber properties]]
[[Category:Solid mechanics]]

{{Sourceattribution|Polynomial (hyperelastic model)|1}}