Models of fractional viscous stresses for incompressible materials

We present and review several models of fractional viscous stresses from the literature, which generalise classical viscosity theories to fractional orders by replacing total strain derivatives in time with fractional time derivatives. We also briefly introduce Prony-type approximations of these theories. Here, we investigate the issues of material frame-indifference and thermodynamic consistency for these models and find that on these bases, some are physically unacceptable. Next, we study elementary shearing and tensile motions, observing that some models are more convenient to use than others for the analysis of creep and relaxation. Finally, we compute the incremental stresses due to small-amplitude wave propagation in a deformed material, with a view to establish acoustoelastic formulas for prospective experimental calibrations.