A Molecular-Statistical Basis for the Gent Constitutive Model of Rubber Elasticity
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Molecular constitutive models for rubber based on non-Gaussian statistics generally involve the inverse Langevin function. Such models are widely used since they successfully capture the typical strain-hardening at large strains. Limiting chain extensibility constitutive models have also been developed on using phenomenological continuum mechanics approaches. One such model, the Gent model for incompressible isotropic hyperelastic materials, is particularly simple. The strain-energy density in the Gent model depends only on the first invariant I 1 of the Cauchy–Green strain tensor, is a simple logarithmic function of I 1 and involves just two material parameters, the shear modulus μ and a parameter J m which measures a limiting value for I 1−3 reflecting limiting chain extensibility. In this note, we show that the Gent phenomenological model is a very accurate approximation to a molecular based stretch averaged full-network model involving the inverse Langevin function. It is shown that the Gent model is closely related to that obtained by using a Padè approximant for this function. The constants μ and J m in the Gent model are given in terms of microscopic properties. Since the Gent model is remarkably simple, and since analytic closed-form solutions to several benchmark boundary-value problems have been obtained recently on using this model, it is thus an attractive alternative to the comparatively complicated molecular models for incompressible rubber involving the inverse Langevin function.
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- 3.M.F. Beatty, Topics in finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues - with examples. Appl. Mech. Rev. 40 (1987) 1699–1733. Reprinted with minor modifications as Introduction to nonlinear elasticity in: M.M. Carroll and M.A. Hayes (eds), Nonlinear Effects in Fluids and Solids. Plenum Press, New York (1996) pp. 16-112.CrossRefGoogle Scholar
- 4.M.F. Beatty, A stretch averaged full-network model for rubber elasticity. J. Elasticity (in press).Google Scholar
- 5.M.C. Boyce, Direct comparison of the Gent and Arruda-Boyce constitutive models of rubber elasticity. Rubber Chemistry Technol. 69 (1996) 781–785.Google Scholar
- 6.M.C. Boyce and E.M. Arruda, Constitutive models of rubber elasticity: A review. Rubber Chemistry Technol. 73 (2000) 504–523.Google Scholar
- 8.B. Erman and J.E. Mark, Structures and Properties of Rubberlike Networks. Oxford Univ. Press, Oxford (1997).Google Scholar
- 11.A.N. Gent, Elastic instabilities of inflated rubber shells. Rubber Chemistry Technol. 72 (1999) 263–268.Google Scholar
- 20.C.O. Horgan and G. Saccomandi, Helical shear for hardening generalized neo-Hookean elastic materials. Math. Mech. Solids 8 (2003) (in press).Google Scholar
- 25.R.W. Ogden, Non-linear Elastic Deformations. Ellis Horwood, Chichester (1984). Reprinted by Dover, New York (1997).Google Scholar
- 27.E. Pucci and G. Saccomandi, A note on the Gent model for rubber-like materials. Rubber Chemistry Technol. 75 (2002) 839–851.Google Scholar
- 29.L.R.G. Treloar, The Physics of Rubber Elasticity, 3rd edn. Oxford Univ. Press, Oxford (1975).Google Scholar