Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models

Viscoelastic materials like amorphous polymers or organic glasses show a complex relaxation behavior in the softening dispersion region, i.e. from glass transition to the α relaxation zone. It is known that a uni-dimensional Maxwell model, modified within the conceptual framework of fractional calculus, has been found to predict experimental data in this range of temperatures. After developing a fully objective constitutive relation for an incompressible fluid, it is shown here that the fractional derivative Maxwell model results from the linearization of this objective equation about the state of rest, when some assumptions about the memory kernels are made. Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. Consequently, it appears that fractional derivative constitutive equations may be used to study a very limited category of flows in rheology rather than the whole spectrum.