On a new class of non-Gaussian molecular-based constitutive models with limiting chain extensibility for incompressible rubber-like materials

Abstract

In constitutive modelling of rubber-like materials, the strain-hardening effect at large deformations has traditionally been captured successfully by non-Gaussian statistical molecular-based models involving the inverse Langevin function, as well as the phenomenological limiting chain extensibility models. A new model proposed by Anssari-Benam and Bucchi ( Int. J. Non Linear Mech. 2021; 128; 103626. DOI: 10.1016/j.ijnonlinmec.2020.103626), however, has both a direct molecular structural basis and the functional simplicity of the limiting chain extensibility models. Therefore, this model enjoys the benefits of both approaches: mathematical versatility, structural objectivity of the model parameters, and preserving the physical features of the network deformation such as the singularity point. In this paper we present a systematic approach to constructing the general class of this type of model. It will be shown that the response function of this class of models is defined as the [1/1] rational function of [Formula: see text], the first principal invariant of the Cauchy–Green deformation tensor. It will be further demonstrated that the model by Anssari-Benam and Bucchi is a special case within this class as a rounded [3/2] Padé approximant in [Formula: see text] (the chain stretch) of the inverse Langevin function. A similar approach for devising a general [Formula: see text] term as an adjunct to the [Formula: see text] part of the model will also be presented, for applications where the addition of an [Formula: see text] term to the strain energy function improves the fits or is otherwise required. It is concluded that compared with the Gent model, which is a [0/1] rational approximation in [Formula: see text] and has no direct connection to Padé approximations of any order in [Formula: see text], the presented new class of the molecular-based limiting chain extensibility models in general, and the proposed model by Anssari-Benam and Bucchi in specific, are more accurate representations for modelling the strain-hardening behaviour of rubber-like materials in large deformations.

On Modelling Simple Shear for Isotropic Incompressible Rubber-Like Materials

“…Indeed, the higher accuracy of the model predictions of Eq. ( 9) versus the Gent model has been previously demonstrated [26,31,37]. Building on these results, here we wish to specialise the application of this model to the problem of simple shear and demonstrate the various modes of simple shear response that the model is capable of capturing and predicting.…”

“…In comparison with the Gent model, however, the proposed model in Eq. ( 9) is a higher order [1/1] Padé approximation of the inverse Langevin function in terms of I 1 [37] compared with the Gent model, which as shown by Horgan and Saccomandi [38] is of order [0/1]. Indeed, the higher accuracy of the model predictions of Eq.…”

New results in the theory of plane strain flexure of incompressible isotropic hyperelastic materials

“…Subsequently, Horgan [46] demonstrated that this model is directly related to the Gent model, whereby the first derivatives of W with respect to I 1 in these two models differ by just a constant. In comparison with the Gent model, however, the response function of the proposed model in equation (4.1) is a higher-order Padé approximation of the inverse Langevin function [47], i.e. a Padé approximation of order [1/1] versus [0/1] of the Gent model [48].…”

Assessment of a New Isotropic Hyperelastic Constitutive Model for a Range of Rubberlike Materials and Deformations

“…1 are related to those of the Gent model (see also Horgan 33 for a review of the features and applications of the Gent model). However, as analyzed at length by Anssari-Benam, 34 the model in Eq. 1 is a more accurate representation of the inverse Langevin function than that of the Gent model.…”

“…1 is a more accurate representation of the inverse Langevin function than that of the Gent model. This model preserves the point of asymptote of the original inverse Langevin function, 34 whereas the approximant in the Gent model does not. 35 In addition, the approximation used in the model in Eq.…”