Application of fractional time derivatives in modeling the finite deformation viscoelastic behavior of carbon-black filled NR and SBR

Abstract Mechanical behavior of elastomers includes nonlinear and time dependent phenomena such as creep and relaxation where due to low stiffness, these materials usually undergo large deformations. The classical approaches of viscoelasticity,i.e. differential constitutive models and hereditary integrals have been widely used to model the viscoelastic behavior of these materials even at finite deformations. However, from the viewpoint of computational analysis, the integral based methods are often time and memory consuming and the differential methods usually need a large number of material parameters whose identification is a real challenge. A potential solution is to introduce fractional order derivatives instead of regular differential operators in the standard spring-dashpot mechanical models which remarkably reduces the number of required material parameters. In the present paper, relaxation behavior of NR and SBR is investigated in finite strains. Due to the remarkable effects of filler particles and deformation rate on the mechanical properties of elastomers, compounds of each material with different amounts of filler content are investigated at two strain rates. A main interest is to examine efficiency of the fractional derivative models in modeling the finite deformation relaxation behavior of elastomers with different filler contents. The other goal is to investigate the effect of loading rate on the obtained model parameters. It is observed that, the fractional Maxwell chain is able to well fit the stress relaxation response of all compounds with only three parameters. Analyzing the obtained parameters and the experimental data, it is concluded that, order of the associated Maxwell element is not much affected with the filler content. Seemingly, it depends on the matrix material and more noticeably on the deformation rate. Moreover, as much as the deformation rate, the associated relaxation time is low. Decrease of the relaxation time with the deformation rate is more evident for compounds of SBR compared with NR compounds.