A Pad� approximant to the inverse Langevin function

1991 ◽  
Vol 30 (3) ◽  
pp. 270-273 ◽  
Author(s):  
A. Cohen
Keyword(s):  
Langevin Function ◽  
Inverse Langevin Function

A generalized approach to generate optimized approximations of the inverse Langevin function

2018 ◽  
Vol 24 (7) ◽  
pp. 2047-2059 ◽  
Author(s):  
Vahid Morovati ◽  
Hamid Mohammadi ◽  
Roozbeh Dargazany
Keyword(s):  
Langevin Function ◽  
Inverse Langevin Function

Improvement of HIA model and the impact of the inverse Langevin function proposed by Marchi and Arruda

2020 ◽  
Vol 60 (1) ◽  
pp. 49-58
Author(s):  
Moustapha Mohamadou Bachirou ◽  
Bale Baidi Blaise ◽  
Kaoutoing Maxime Dawoua ◽  
Gambo Betchewe ◽  
Tibi Beda
Keyword(s):  
Langevin Function ◽  
Inverse Langevin Function ◽  
The Impact

A new based error approach to approximate the inverse langevin function

2014 ◽  
Vol 53 (8) ◽  
pp. 585-591 ◽  
Author(s):  
Alain Nkenfack Nguessong ◽  
Tibi Beda ◽  
François Peyraut
Keyword(s):  
Langevin Function ◽  
Inverse Langevin Function

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

2021 ◽  
pp. 108128652110010
Author(s):  
Afshin Anssari-Benam
Keyword(s):  
Strain Hardening ◽  
Large Deformations ◽  
Structural Basis ◽  
Deformation Tensor ◽  
Model Parameters ◽  
New Class ◽  
Langevin Function ◽  
Inverse Langevin Function ◽  
Limiting Chain Extensibility ◽  
Non Gaussian

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.

Generalized error-minimizing, rational inverse Langevin approximations

2018 ◽  
Vol 24 (6) ◽  
pp. 1630-1647 ◽  
Author(s):  
Benjamin C Marchi ◽  
Ellen M Arruda
Keyword(s):  
Asymptotic Behavior ◽  
Statistical Mechanics ◽  
Relative Error ◽  
Network Models ◽  
Rubber Elasticity ◽  
Maximum Relative Error ◽  
Function Form ◽  
Langevin Function ◽  
Inverse Langevin Function ◽  
Finite Domains

The Langevin function is a non-invertible function, whose inverse commonly appears in statistical mechanics problems, particularly network models of rubber elasticity. This non-invertibility often results in the use of approximations. Owing to the prevalence of the inverse Langevin function, numerous forms of approximate have been proposed. Rational approximates are often employed because of their ability to admit asymptotic behavior in finite domains, similar to the exact inverse Langevin function. Despite the desired asymptotics of rational approximates, there is unavoidable error associated with the approximate within its domain. In this work, an error-minimizing approach for determining specific forms of rational approximates is generalized to approximates of arbitrary numerator and denominator orders. By expanding to general orders of rational approximates, the best approximate can be selected for an application based on either maximum relative error or function form considerations.

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