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 $I_1$, 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 ${\lambda }_c$ (the chain stretch) of the inverse Langevin function. A similar approach for devising a general $I_2$ term as an adjunct to the $I_1$ part of the model will also be presented, for applications where the addition of an $I_2$ 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 $I_1$ and has no direct connection to Padé approximations of any order in ${\lambda }_c$, 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.

在橡胶样材料的本构模型中，传统上已成功地通过非高斯统计分子基模型（包括逆朗芬蛋白功能）以及现象学极限链可扩展性模型成功捕获了大变形下的应变硬化效应。Anssari-Benam和Bucchi（Int。J.Nonlinear Mech。2021年；128; 103626. DOI：10.1016 / j.ijnonlinmec.2020.103626），但是，它既具有直接的分子结构基础，又具有极限链可扩展性模型的功能简单性。因此，该模型具有两种方法的优点：数学通用性，模型参数的结构客观性以及保留网络变形的物理特征（例如奇异点）。在本文中，我们提出了一种系统的方法来构造这种类型的模型的通用类。将显示此类模型的响应函数定义为的[1/1]有理函数。${\mathrm{一世}}_{\mathrm{1个}}$，柯西-格林变形张量的第一个主要不变式。将进一步证明，Anssari-Benam和Bucchi的模型是该类中的一种特例，它是舍入的[3/2]Padé近似值${\lambda }_C$Langevin函数的反（链拉伸）。设计一般的类似方法${\mathrm{一世}}_{\mathrm{2个}}$ 术语作为 ${\mathrm{一世}}_{\mathrm{1个}}$ 该模型的一部分也将介绍，适用于添加了 ${\mathrm{一世}}_{\mathrm{2个}}$应变能函数的一个术语会改善拟合度，或者是其他要求。得出的结论是，与Gent模型相比，它是[0/1]的有理逼近${\mathrm{一世}}_{\mathrm{1个}}$ 并与任何阶次的Padé逼近没有直接关系 ${\lambda }_C$，一般来说，提出的新一类基于分子的极限链可扩展性模型，特别是由Anssari-Benam和Bucchi提出的模型，是对橡胶类材料在大变形中的应变硬化行为进行建模的更精确表示。