Molecular chain networks and strain energy functions in rubber elasticity

Abstract

A relatively simple molecular–statistical model for rubber elasticity, similar to the Wang–Guth and the Arruda–Boyce (AB) models, is presented. Discussion of some approximate inverse Langevin functions leads to the selection of a new one as the most attractive balance between accuracy and simplicity. Use of this approximation with the AB model, in particular, yields the logarithmic strain energy introduced by Gent that exhibits limiting chain extensibility. A rather unusual facet of the relationship between the three strain energies is that the new one is the mean of the other two. This article is part of the theme issue ‘Rivlin's legacy in continuum mechanics and applied mathematics’.

Rivlin's legacy in continuum mechanics and applied mathematics

“…The question of a proper approximation for the inverse Langevin function can be bypassed by an approach à la Gent [18], which provides a very good approximation of this special function in several circumstances [19]. Finally, Destrade et al [7] recently proposed a detailed discussion and an updating of the present paper by Carroll [20].…”

Mechanics of pressurized planar hyperelastic membranes

“…The intention of this section is to describe the constitutive model development relevant to the topics discussed in the paper. Constitutive models for hyperelastic materials have progressed along the following avenues: The first appeals to formulations that consider the molecular networks and statistical theories [49][50][51][52][53][54][55][56]. The development of constitutive relationships for rubber-like materials proposed by Mooney [57] assumes incompressibility and isotropy of the material and Hookean behaviour in simple shear, which is applicable up to moderately large strains.…”

Thermomechanical Modeling of Microstructure Evolution Caused by Strain-Induced Crystallization

“…such that the final form of the energy balance is obtained by inserting Equations (45) and (46) into (40)…”

“…where parameter μ is determined from the consistency condition of model (11) with the linear elasticity theory for small strains. According to [29,74], this condition is expressed as…”

“…Alternatively to the Taylor series, a range of Padé approximations can be applied for the numerical evaluation of the inverse Langevin function. These approximations have different degrees of accuracy and complexity as discussed in the review papers by Jedynak [33] and Carroll [11].…”