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We present and review several models of fractional viscous stresses from theliterature, which generalise classical viscosity theories to fractional ordersby replacing total strain derivatives in time with fractional time derivatives.We also briefly introduce Prony-type approximations of these theories. Here weinvestigate the issues of material frame-indifference and thermodynamicconsistency for these models and find that on these bases, some are physicallyunacceptable. Next we study elementary shearing and tensile motions, observingthat some models are more convenient to use than others for the analysis ofcreep and relaxation. Finally, we compute the incremental stresses due tosmall-amplitude wave propagation in a deformed material, with a view toestablish acousto-elastic formulas for prospective experimental calibrations.

Models of fractional viscous stresses for incompressible materials