On the rest state stability of an objective fractional derivative viscoelastic fluid model

The one-dimensional fractional derivative Maxwell model has been found very useful in modeling the linear viscoelastic response in the glass transition and α-relaxation regions. That motivated further work on generalizations to nonlinear viscoelastic fractional constitutive equations. In the work of Palade et al. [Int. J. Eng. Sci. 37, 315 (1999)], a fully objective constitutive equation for an incompressible fluid—reducible to the linear fractional derivative Maxwell model under the small deformations hypothesis—is given, together with a state of rest stability analysis. In the stability section of the aforementioned work, for some physically aceptable perturbation inputs, it has been shown that no continuously differentiable solutions may be found to the corresponding equations of motion. In this work, we extend the stability analysis framework to L2 Hilbert functional space and prove the existence of solutions to the corresponding equations of motions.