This paper proposes modelling concepts to represent the Mullins effect, rate dependence and the Payne effect. As a fundamental concept, we split the stress into the sum of an equilibrium stress and a rate-dependent overstress. There are two different approaches for representing the overstress: the first one is based on the multiplicative splitting of the deformation gradient into elastic and inelastic partsand, secondly, there is the functional representation with convolution integrals. In order to model the Mullins effect we introduce a damage variable depending on the maximum deformation reached in the process history. To model the Payne effect we introduce an intrinsic time variable which is a functional of deformation history, specify an evolution law driven by a fractional time rate of deformation and calculate analytical formulae for the storage and loss moduli. Finally, it is shown that both the frequency and amplitude dependence are well represented.