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

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 Ι₁, 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 λ_c (the chain stretch) of the inverse Langevin function. A similar approach for devising a general Ι₂ term as an adjunct to the Ι₁ part of the model will also be presented, for applications where the addition of an Ι₂ 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 Ι₁ and has no direct connection to Padé approximations of any order in λ_c, 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.