An original perspective on variable-order fractional operators for viscoelastic materials

This work deals with viscoelastic constitutive models involving variable-order fractional operators. There exist two main fractional models in the literature representing the stress-strain relation of viscoelastic materials with time-varying mechanical properties. Here, their features are analyzed and the physical assumptions involved are critically discussed. Specifically, it is shown that only one of these fractional models seems to be actually meaningful in the context of viscoelasticity. Then, a novel formulation is discussed, still in the context of variable-order fractional calculus, to effectively compute the strain response of a viscoelastic material with time-dependent mechanical properties due to any stress input. The proposed formulation exhibits a clear physical meaning and is proved to rely on a consistent application of the Boltzmann superposition principle to a fictitious system that is considered, in some sense, equivalent to the original viscoelastic material under study. The main novelty of the paper is to show that the proposed formulation is strictly related to the meaningful fractional model existing in the literature to which, therefore, a sound mechanical meaning can be given.