Exact mechanical models of fractional hereditary materials

Fractional Viscoelasticity is referred to materials, whose constitutive law involves fractional derivatives of order β R such that 0 β 1. In this paper, two mechanical models with stress-strain relation exactly restituting fractional operators, respectively, in ranges 0 β 1 / 2 and 1 / 2 β 1 are presented. It is shown that, in the former case, the mechanical model is described by an ideal indefinite massless viscous fluid resting on a bed of independent springs (Winkler model), while, in the latter case it is a shear-type indefinite cantilever resting on a bed of independent viscous dashpots. The law of variation of all mechanical characteristics is of power-law type, strictly related to the order of the fractional derivative. Because the critical value 1/2 separates two different behaviors with different mechanical models, we propose to distinguish such different behavior as elasto-viscous case with 0< β <1 / 2 and visco-elastic case for 1 / 2 <β <1. The motivations for this different definitions as well as the comparison with other existing mechanical models available in the literature are presented in the paper