A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber

Abstract

Stress softening during initial loading cycles, known as the Mullins effect, and the residual strain upon unloading are not accounted for when the mechanical properties of rubber are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper we first describe some experimental results that illustrate stress softening in particle-reinforced rubber together with associated residual strain effects. In particular, the results show how the stress softening and residual strain change with the magnitude of the applied strain. Then, on the basis of these data a constitutive model is derived to describe this behaviour. The theory of pseudo-elasticity is used for this model, the basis of which is the inclusion of two variables in the energy function in order separately to capture the stress softening and residual strain effects. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is also accounted for in the model by the use of a dissipation function, which evolves with the deformation history.