Fractional creep and relaxation models of viscoelastic materials via a non-Newtonian time-varying viscosity: physical interpretation

Abstract

Fractional viscoelastic models have been confirmed to achieve good agreement with experimental data using only a few parameters, in contrast to the classical viscoelastic models in previous studies. With an increasing number of applications, the physical meaning of fractional viscoelastic models has been attracting more attention. This work establishes an equivalent viscoelasticity (including creep and relaxation) between the fractional Maxwell model and the time-varying viscosity Maxwell model to reveal the physical meaning of fractional viscoelastic models. The obtained time-varying viscosity functions are used to interpret the physical meaning of the order of the fractional derivative α from the perspective of rheology. When α changes from 0 to 1, the viscosity functions quantitatively exhibit the transformation of viscoelasticity from elastic solid to Newtonian fluid, which can be considered as an extension of the Deborah number. The infinite viscosity coefficient for α = 0 shows the elastic solid property, while the constant viscosity coefficient for α = 1 exhibits the Newtonian fluid property. The sharply decreasing viscosity coefficient (versus α) near α = 0 indicates that the elastic solid property decays rapidly. In addition, similar viscoelastic responses between the Hausdorff and fractional derivative models are found due to a similar time-varying viscosity.