On the Frequency and Amplitude Dependence of the Payne Effect: Theory and Experiments

Abstract

Abstract In this paper, we develop a physical approach to represent the Payne effect which is observed in filler-reinforced elastomers. The starting point for the constitutive model is the well-known theory of linear viscoelasticity, where the stress is a linear functional of the deformation history. Since the corresponding relations are unable to describe any kind of amplitude dependence, we introduce a nonlinearity into the model. To this end, we replace the physical time, t, by a modified time scale, z, which is a functional of the deformation history. This approach was originally introduced by Valanis in the context of rate-independent plasticity and applied in a modified form; for example, by Haupt and Sedlan to represent process-dependent relaxation properties of rubber. The modified time, z, is a monotonic function of the physical time, t, and can be interpreted as an intrinsic time scale of the material. The rate of this time scale is non-negative and depends on the process history. We propose a constitutive relation for the variable z(t), which is driven by a fractional time derivative of the deformation, calculate analytical expressions for the storage and dissipation moduli and show that such phenomena as the frequency and the amplitude dependence, observed in experiments, are well represented.