Models of fractional viscous stresses for incompressible materials

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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. Here we study shearing motions, observing that some models are more convenient to use than others for the analysis of shear creep and relaxation. Moreover, we investigate the issues of material frame-indifference and thermodynamic consistency for these models and find that on these bases, some are physically unacceptable. Finally, we compute the incremental stresses due to small-amplitude wave propagation in a deformed material, with a view to establish acousto-elastic formulas for prospective experimental calibrations.