Computational Modelling for Description of Rubber-Like Materials with Permanent Deformation under Cyclic Loading

Rubber-like materials exhibit a strongly non-linear behaviour characterised by large strains and a non-linear stress-strain response. When a rubber specimen is subjected to cyclic loading, stress-softening phenomena (the Mullins effect) are also observed. Moreover, carbon-filled rubber specimens, in general, do not return to their initial state after loading and subsequent unloading, but exhibit a residual strain or permanent deformation. The permanent deformation combined with stress-softening effects in rubber-like materials results in complex mechanical behaviour and modelling is still in an initial stage. Many researchers pursue their study in this field. Representative works include (1996), Miehe and (2000) and (2004). Both experimental data and constitutive models can be found in the above-cited works. However, these constitutive models either can not describe the evolution of rubber softening and permanent deformation or have too many adjustable parameters and functions, which can only be determined arbitrarily to some degree. For instance, (2004) proposed a model to capture the Mullins effect and residual strain effects with the inclusion of two variables in the energy function, but they can not describe the evolution of stress softening. We propose a specific form of the pseudo-elastic energy function to represent cyclic loading for incompressible, isotropic materials with stress softening and residual strain. The essence of the pseudo-elasticity theory is that material behaviour in the primary loading path is described by a common elastic strain energy density function W(F), and in unloading, reloading or secondary unloading paths by a different strain energy density function. The switch between strain energy functions is controlled by the incorporation of the damage variable and its function term into the strain energy function. An extra term is added to describe the permanent deformation. The finite element implementation of the proposed model with constitutive elastic law is carried out in this paper. All parameters in the proposed model and elastic law can be easily estimated based on experimental data. The numerical analyses show that the numerical results are in good agreement with experimental data.