Polynomial (hyperelastic model)

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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 I1,I2 of the left Cauchy-Green deformation tensor.

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

W=∑i,j=0nCij(I1−3)i(I2−3)j

where Cij are material constants and C00=0.

For compressible materials, a dependence of volume is added

W=∑i,j=0nCij(I¯1−3)i(I¯2−3)j+∑k=1mDk(J−1)2k

where

I¯1=J−2/3I1;I1=λ12+λ22+λ32;J=det⁡(F)I¯2=J−4/3I2;I2=λ12λ22+λ22λ32+λ32λ12

In the limit where C01=C11=0, the polynomial model reduces to the Neo-Hookean solid model. For a compressible Mooney-Rivlin material n=1,C01=C2,C11=0,C10=C1,m=1 and we have

W=C01(I¯2−3)+C10(I¯1−3)+D1(J−1)2

## References

1. ↑ 1.0 1.1 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.

## See also

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

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