A comparison of limited-stretch models of rubber elasticity

In this paper we describe various limited-stretch models of non-linear rubber elasticity, each dependent on only the first invariant of the left Cauchy-Green strain tensor and having only two independent material constants. The models are described as limited-stretch, or restricted elastic, because the strain energy and stress response become infinite at a finite value of the first invariant. These models describe well the limited stretch of the polymer chains of which rubber is composed. We discuss Gent's model which is the simplest limited-stretch model and agrees well with experiment. Various statistical models are then described: the one-chain, three-chain, four-chain and Arruda-Boyce eight-chain models, all of which involve the inverse Langevin function. A numerical comparison between the three-chain and eight-chain models is provided. Next, we compare various models which involve approximations to the inverse Langevin function with the exact inverse Langevin function of the eight-chain model. A new approximate model is proposed that is as simple as Cohen's original model but significantly more accurate. We show that effectively the eight-chain model may be regarded as a linear combination of the neo-Hookean and Gent models. Treloar's model is shown to have about half the percentage error of our new model but it is much more complicated. For completeness a modified Treloar model is introduced but this is only slightly more accurate than Treloar's original model. For the deformations of uniaxial tension, biaxial tension, pure shear and simple shear we compare the accuracy of these models, and that of Puso, with the eight-chain model by means of graphs and a table. Our approximations compare extremely well with models frequently used and described in the literature, having the smallest mean percentage error over most of the range of the argument

A cyclic stress softening model for the Mullins effect

AbstractIn this paper the inelastic features of stress relaxation, hysteresis and residual strain are combined with the Arruda–Boyce eight-chain model of elasticity, in order to develop a model that is capable of describing the Mullins effect for cyclic stress-softening of an incompressible hyperelastic material, in particular a carbon-filled rubber vulcanizate. We have been unable to identify in the literature any other model that takes into consideration all the above inelastic features of the cyclic stress-softening of carbon-filled rubber. Our model compares favourably with experimental data and gives a good description of stress-softening, hysteresis, stress relaxation, residual strain and creep of residual strain

A model for the Mullins effect during multicyclic equibiaxial loading

In this paper, we derive a model to describe the cyclic stress softening of a carbon-filled rubber vulcanizate through multiple stress–strain cycles with increasing values of the maximum strain, specializing to equibiaxial loading. Since the carbon-filled rubber vulcanizate is initially isotropic, we can show that following initial equibiaxial loading the material becomes transversely isotropic with preferred direction orthogonal to the plane defined by the equibiaxial loading. This is an example of strain-induced anisotropy. Accordingly, we derive nonlinear transversely isotropic models for the elastic response, stress relaxation, residual strain and creep of residual strain in order to model accurately the inelastic features associated with cyclic stress softening. These ideas are then combined with a transversely isotropic version of the Arruda–Boyce eight-chain model to develop a constitutive relation for the cyclic stress softening of a carbon-filled rubber vulcanizate. The model developed includes the effects of hysteresis, stress relaxation, residual strain and creep of residual strain. The model is found to compare extremely well with experimental data