Correction to: Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

A Correction to this paper has been published: 10.1007/s00397-021-01283-3

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 [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.

A Generalized Approach to Improve Approximation of Inverse Langevin Function

The inverse Langevin function has a crucial role in different research fields, such as polymer physics, para- or superpara-magnetism materials, molecular dynamics simulations, turbulence modeling, and solar energy conversion. The inverse Langevin function cannot be explicitly derived and thus, its inverse function is usually approximated using rational functions. Here, a generalized approach is proposed that can provide multiple approximation functions with a different degree of complexity/accuracy for the inverse Langevin function. While some special cases of our approach have already been proposed as approximation function, a generic approach to provide a family of solutions to a wide range of accuracy/complexity trade-off problems has not been available so far. By coupling a recurrent procedure with current estimation functions, a hybrid function with adjustable accuracy and complexity is developed. Four different estimation families based four estimation functions are presented here and their relative error is calculated with respect to the exact inverse Langevin function. The level of error for these simple and easy-to-use formulas can be reduced as low as 0.1%.