Fractional derivative models for viscoelastic materials at finite deformations

Abstract

Fractional derivative models, which are expressed by combining standard dashpots, fractional dashpots and elastic springs in series or parallel, are often utilized to account for the behaviors for viscoelastic materials. Even with the models extended to finite deformation, the precise definition of objective fractional derivative remains challenging. The proposed fractional derivative model is expressed by the combination of an elastic spring in series with two parallel fractional dashpots. We extend the fractional derivative model to finite deformation through a new approach without defining an objective fractional derivative and assuming the decomposition of the deformation rate into the elastic and inelastic parts. This proposed model can be reduced to the Maxwell model for finite deformation. Such reduction results in a model that stands in between the two existing Maxwell models in which the objective rate of the Cauchy stress is taken as the material corotational rate and the relative corotational rate respectively. The proposed model is applied to the simple shear deformation.