On the complex singularities of the inverse Langevin function

Abstract

We study the inverse Langevin function L −1 (x) because of its importance in modelling limited-stretch elasticity where the stress and strain energy become infinite as a certain maximum strain is approached, modelled here by x → 1. The only real singularities of the inverse Langevin function L −1 (x) are two simple poles at x = ±1 and we see how to remove their effects either multiplicatively or additively. In addition, we find that L −1 (x) has an infinity of complex singularities. Examination of the Taylor series about the origin of L −1 (x) shows that the four complex singularities nearest the origin are equidistant from the origin and have the same strength; we develop a new algorithm for finding these four complex singularities. Graphical illustration seems to point to these complex singularities being of a square root nature. An exact analysis then proves these are square root branch points. which has a removable singularity at y = 0 and is defined on the domain −∞ < y < ∞. All the above models, however, involve the inverse Langevin function defined byThese are both odd functions though on physical grounds y and x may be restricted to be positive. It is the property L −1 (x) → ∞ as x → 1 that makes the inverse Langevin function so useful in modelling limited-stretch elasticity. The Arruda & Boyce [1] eight-chain model has proved to be the most successful of these models, both theoretically and experimentally. However, Beatty [2] has shown that quite remarkably the Arruda & Boyce [1] stress response holds in general in isotropic nonlinear elasticity for a full network model of arbitrarily orientated molecular chains. Thus the eight-chain cell structure is unnecessary and so the inverse Langevin function has an important role to play in the theory of the finite isotropic elasticity of rubber and rubber-like materials.The inverse Langevin function cannot be expressed in closed form and so many approximations to it have been devised and applied to many models of rubber elasticity. Perhaps the simplest is to consider the Taylor series but this does not converge over the whole domain of definition −1 < x < 1 of the inverse Langevin function, see [12] and [11]. Cohen [4] derived an approximation based upon a [3/2] Padé approximant of the inverse Langevin function which has been widely used and is fairly accurate over the whole domain of definition. Many more accurate approximations have been devised, see for example, [5,14,15,16,18,20,21,23,24]. However, rather than finding further approximations to the inverse Langevin function, we emphasise in this paper how to find the approximate positions of its singularities in the complex plane. We are also able to find exactly the positions and nature of these singularities.This paper is structured as follows. In Section 2 we give a brief account of non-linear isotropic elasticity theory as applied to the limited-stretch theory of elasticity of, for example, [1] and [2]. In Section 3 we define and discuss the Langevin and inverse Langevin functions, giving Taylor series fo...

A comprehensive study of the mathematical methods used to approximate the inverse Langevin function

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