1956–57 Beşiktaş J.K. season

Template:Continuum mechanics 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 $I_{1}, I_{2}$ of the left Cauchy-Green deformation tensor.

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

W = \sum_{i, j = 0}^{n} C_{i j} (I_{1} - 3)^{i} (I_{2} - 3)^{j}

where $C_{i j}$ are material constants and $C_{00} = 0$ .

For compressible materials, a dependence of volume is added

W = \sum_{i, j = 0}^{n} C_{i j} ({\bar{I}}_{1} - 3)^{i} ({\bar{I}}_{2} - 3)^{j} + \sum_{k = 1}^{m} D_{k} (J - 1)^{2 k}

where

\begin{matrix} {\bar{I}}_{1} & = J^{- 2 / 3} I_{1}; I_{1} = λ_{1}^{2} + λ_{2}^{2} + λ_{3}^{2}; J = det (𝑭) \\ {\bar{I}}_{2} & = J^{- 4 / 3} I_{2}; I_{2} = λ_{1}^{2} λ_{2}^{2} + λ_{2}^{2} λ_{3}^{2} + λ_{3}^{2} λ_{1}^{2} \end{matrix}

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) + D_{1} (J - 1)^{2}

References

1 2 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.

1956–57 Beşiktaş J.K. season

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1956–57 Beşiktaş J.K. season

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