Gent

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 Pade 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.

Abstract: In 1996, Alan Gent published a short paper that proposed the use of a very simple two parameter phenomenological constitutive model for hyperelastic isotropic incompressible materials. The model is empirical but has the advantages of mathematical simplicity, reflects the severe strain-stiffening at large strains observed experimentally, reduces to the classic neo-Hookean model for small strains and involves just two material parameters namely the shear modulus for infinitesimal deformations and a parameter that measures a maximum allowable value of strain. The model reflects the limiting chain extensibility characteristic of non-Gaussian molecular models for rubber. Here we review some of the numerous developments, extensions and widespread applications that have resulted from that groundbreaking paper not only in rubber elasticity but also in the area of biomechanics of soft biomaterials. The Gent model is remarkably robust: its mathematical simplicity combined with physical basis has ensured that it has reached status as a fundamental canonical phenomenological constitutive model for hyperelastic materials.

Abstract: Within the theory of isothermal isotropic non-linear elasticity, the selection of the appropriate form for the strain energy function W in terms of the strain invariants is still an issue. The purpose of this paper is to introduce ideas and techniques which it is hoped will contribute to the task of finding an appropriate form for the strain energy. Three principal ideas are developed in this paper. Firstly, not all of invariant-space corresponds to real deformations. Constitutive equations only need to match real behaviour over a restricted part of invariant space, called the Attainable Region, bounded by states of deformation corresponding to uniaxial and equi-biaxial extension. Secondly, examples are given of how to exploit the fact that the Attainable Region is restricted. Mapping a deformation onto this region allows visualization of how close the deformation is to the well-understood uniaxial, equi-biaxial and simple shear deformations, and how this varies in space or time. Thirdly, acceptable strain invariants do not have to be obviously symmetric functions of the principal stretches. The ordered principal stretches are themselves invariants, and explicit unique algebraic expressions can be given through which the greatest, middle or least stretch can be calculated in terms of the usual invariants. Thus invariants can be chosen which are apparently non-symmetric functions of the ordered stretches.

Abstract: The invariants in the K-BKZ constitutive equation for an incompressible viscoelastic fluid are usually taken to be the trace of the Finger strain tensor and its inverse. The basis for this choice of invariants is not derived from the K-BKZ theory, but rather is due to the perception that this is the most natural choice. Research into using other sets of invariants in the K-BKZ equation, such as the principal stretches or the eigenvalues of the Finger strain tensor (i.e., the squares of the principal stretches) is relatively new. We attempt here to derive a K-BKZ equation based on the squares of the principal stretches that models the behavior of a low-density polyethylene melt in simple shear and uniaxial elongational deformation. In doing so, two assumptions are made as to the form of the strain-dependent energy function: first, that there is a function f(q) such that the energy function can be written as the sum of f(q i ),i = 1, 2, 3, where the q i 'sare the squares of the principal stretches, and second that f is a power law. We find that the K-BKZ equation resulting from these two assumptions is inadequate to describe both the shear and elongational behavior of our material and we conclude that the second of the above assumptions is not valid. Further investigation, including predictions of the second normal stress difference and some finite element calculations reveals that the first assumption is also invalid for our material.