Polynomial hyperelastic model

{{Continuum mechanics|cTopic=Solid mechanics}}

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 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.

:$$

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} (\bar{I}_1 - 3)^i (\bar{I}_2 - 3)^j + \sum_{k=1}^m \frac{1}{D_{k}}(J-1)^{2k}

where

:$$

\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}

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}~(\bar{I}_2 - 3) + C_{10}~(\bar{I}_1 - 3) + \frac{1}{D_1}~(J-1)^2