The benefit of fractional derivatives in modelling the dynamics of filler-reinforced rubber under large strains: a comparison with the Maxwell-element approach

Abstract

The dynamic properties of rubber-like materials are characterised by a significant dependence on the predeformation and the frequency. The focus of this paper is to represent the frequency and predeformation dependent dynamic behaviour of a carbon-black filled SBR rubber with 40 phr amount of filler using the concept of fractional derivatives. Thus, we introduce a constitutive approach of finite fractional viscoelasticity which is suitable to approximate the dynamic material properties with respect to the storage and the loss modulus. The constitutive approach is based on a proposal of [18] which was modified by a deformation dependent relaxation function in a previous work [46] to represent the dependence of the dynamic modulus on the predeformation and the frequency. The constitutive approach in [46] is based on the classical theory of finite viscoelasticity and formulated in the frequency domain. In this work, the approach of [46] will be extended by the concept of fractional derivatives and compared to the classical one. Thus, the classical and the extended fractional constitutive models are firstly introduced and the complex modulus tensors of both models are derived. It should be mentioned that both constitutive approaches are firstly formulated in the time domain. This formulation is necessary to satisfy the thermodynamical consistency. In order to conduct vibration analyses of elastomer structures with high computational efficiency, the equations are then transferred to the frequency domain. To this end, the constitutive model is geometrically linearised in the neighbourhood of a large and temporally constant predeformation. The incremental strain tensor varies harmonically and its amplitude has to be small. Furthermore, parameter identification of both approaches is done on the basis of quasi-static and dynamic investigations of the carbon-black filled SBR rubber. The numerical results of the parameter identification of the classical and the fractional model are compared to each other with respect to the number of necessary material parameters and the quality of the approximation. Finally, the numerical implementation of the frequency domain formulation into the finite element code MSC Marc on the basis of the proposal of [28] will be presented.