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ABSTRACT: 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.
Journal of Elasticity 04/2012; 68(1):167-176. · 1.11 Impact Factor
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ABSTRACT: Constitutive models are proposed for compressible isotropic hyperelastic materials that reflect limiting chain extensibility. These are generalizations of the model proposed by Gent for incompressible materials. The goal is to understand the effects of limiting chain extensibility when the compressibility of polymeric materials is taken into account. The basic homogeneous deformation of simple tension is considered and simple closed-form relations for the deformation characteristics are obtained for slightly compressible materials. An explicit first-order approximation is obtained for the lateral contraction and for the Poisson function in terms of the axial extension which is shown to be valid for each of two specific compressible versions of the Gent model. One of the main results obtained is that the effect of limiting chain extensibility is to stiffen the material relative to the neo-Hookean compressible case.
Journal of Elasticity 10/2004; 77(2):123-138. · 1.11 Impact Factor
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01/2004;
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ABSTRACT: The purpose of this research is to investigate the basic issues that arise when generalized plane strain deformations are superimposed on anti-plane shear deformations in isotropic incompressible hyperelastic materials. Attention is confined to a subclass of such materials for which the strain-energy density depends only on the first invariant of the strain tensor. The governing equations of equilibrium are a coupled system of three nonlinear partial differential equations for three displacement fields. It is shown that, for general plane domains, this system decouples the plane and anti-plane displacements only for the case of a neo-Hookean material. Even in this case, the stress field involves coupling of both deformations. For generalized neo-Hookean materials, universal relations may be used in some situations to uncouple the governing equations. It is shown that some of the results are also valid for inhomogeneous materials and for elastodynamics.
Journal of Elasticity 01/2003; 73(1):221-235. · 1.11 Impact Factor
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SIAM Journal of Applied Mathematics. 01/2002; 62:1712-1727.
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ABSTRACT: The purpose of this research is to investigate the pure azimuthal shear problem for a circular cylindrical tube composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These involve a strain-energy density which depends only on the "rst invariant of the Cauchy}Green tensor. In the limit as a polymeric chain extensibility tends to in"nity, all of these models reduce to the classical neo-Hookean form. The stress "elds and axial displacements are characterized for each of these models. Explicit closed-form analytic expressions are obtained for two of the strain-energy densities considered. The results are compared with one another and with the predictions of the neo-Hookean model.
International Journal of Non-Linear Mechanics 01/2001; 36:465-475. · 1.21 Impact Factor
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ABSTRACT: The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic
hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at
large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of
the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to
the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form.
In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models
agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material.
In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by
any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies
to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies,
which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which
can be resolved only by large strain experiments.
Journal of Elasticity 04/1999; 56(2):159-170. · 1.11 Impact Factor
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ABSTRACT: The purpose of this research is to investigate the pure axial shear problem for a circular cylindrical tube composed of isotropic
hyperelastic incompressible materials with limiting chain extensibility. Two popular models that account for hardening at
large deformations are examined. These involve a strain-energy density which depends only on the first invariant of the Cauchy–Green
tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean
form. The stress fields and axial displacements are characterized for each of these models. Explicit closed-form analytic
expressions are obtained. The results are compared with one another and with the predictions of the neo-Hookean model.
Journal of Elasticity 04/1999; 57(3):307-319. · 1.11 Impact Factor
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ABSTRACT: We consider an incompressible nonlinearly elastic material in which a matrix is reinforced by strong fibers, for example fibers of nylon or carbon aligned in one family of curves in a rubber matrix. Rather than adopting the constraint of fiber inextensibility as has been previously assumed in the literature, here we develop a theory of fiber-reinforced materials based on the less restrictive idea of limiting fiber extensibility. The motivation for such an approach is provided by recent research on limiting chain extensibility models for rubber. Thus the basic idea of the present paper is simple: we adapt the limiting chain extensibility concept to limiting fiber extensibility so that the usual inextensibility constraint traditionally used is replaced by a unilateral constraint. We use a strain-energy density composed with two terms, the first being associated with the isotropic matrix or base material and the second reflecting the transversely isotropic character of the material due to the uniaxial reinforcement introduced by the fibers. We consider a base neo-Hookean model plus a special term that takes into account the limiting extensibility in the fiber direction. Thus our model introduces an additional parameter, namely that associated with limiting extensibility in the fiber direction, over previously investigated models. The aim of this paper is to investigate the mathematical and mechanical feasibility of this new model and to examine the role played by the extensibility parameter. We examine the response of the proposed models in some basic homogeneous deformations and compare this response to those of standard models for fiber reinforced rubber materials. The role of the strain-stiffening of the fibers in the new models is examined. The enhanced stability of the new models is then illustrated by investigation of cavitation instabilities. One of the motivations for the work is to apply the model to the biomechanics of soft tissues and the potential merits of the proposed models for this purpose are briefly discussed.
Journal of the Mechanics and Physics of Solids.
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ABSTRACT: On using a rational function approximation scheme for the response functions of hyperelastic isotropic materials we propose a new model to describe the mechanical behaviour of atactic polymers. The new model takes into account the finite chain extensibility effects characteristic of this class of materials.
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ABSTRACT: Rubber-like materials and soft tissues exhibit a significant stiffening or hardening in their stress–strain curves at large strains. Considerable progress has been made recently in the phenomenological modeling of this effect within the context of isotropic hyperelasticity. In particular, constitutive models reflecting limiting chain extensibility at the molecular level have been used to accurately capture strain-hardening. Here we generalize such models to isotropic thermoelasticity. We also show that specific non-polynomial strain-energies for both hyperelastic and thermoelastic materials can be obtained on using a modification of a systematic scheme of Rivlin and Signorini. The Rivlin–Signorini method was based on approximation of the strain-energy density function by polynomials whereas here we use the more general class of rational functions to approximate the material response functions. We then propose a simple generalization to thermoelasticity of a constitutive model for incompressible hyperelastic materials reflecting limiting chain extensibility due to Gent (Rubber Chem. Technol. 69 (1996) 59–61). For this new thermoelastic constitutive model we investigate the inhomogeneous deformation problem of axial shear of a circular cylindrical tube.
Journal of the Mechanics and Physics of Solids.
