Polynomial hyperelastic model

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title: "Polynomial hyperelastic model" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["continuum-mechanics", "non-newtonian-fluids", "rubber-properties", "solid-mechanics"] topic_path: "general/continuum-mechanics" source: "https://en.wikipedia.org/wiki/Polynomial_hyperelastic_model" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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} (\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

References

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.

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