Fractional characteristic times and dissipated energy in fractional linear viscoelasticity

• Published in: Communications in nonlinear science & numerical simulation 2016-08, Vol.37, p.14-30
• Main Authors: Colinas-Armijo, Natalia, Di Paola, Mario, Pinnola, Francesco P.
• Format: Article
• Language: English
• Subjects: Derivatives; Differential equations; Dissipated energy; Dissipation; Equivalence; Excitation; Fractional calculus in linear viscoelasticity; Fractional creep and relaxation times; Loss and storage modulus; Mathematical models; Stress-strain relationships; Viscoelasticity
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.01.003

•Viscoelasticity is modeled through fractional stress–strain relationships.•Elastic and viscous contribution in fractional viscoelasticity cannot be dissociated.•Fractional characteristic times defined to isolate elastic and viscous contribution.•Classical models and dissipated energy to calculate fractional characteristic times. In fractional viscoelasticity the stress–strain relation is a differential equation with non-integer operators (derivative or integral). Such constitutive law is able to describe the mechanical behavior of several materials, but when fractional operators appear, the elastic and the viscous contribution are inseparable and the characteristic times (relaxation and retardation time) cannot be defined. This paper aims to provide an approach to separate the elastic and the viscous phase in the fractional stress–strain relation with the aid of an equivalent classical model (Kelvin–Voigt or Maxwell). For such equivalent model the parameters are selected by an optimization procedure. Once the parameters of the equivalent model are defined, characteristic times of fractional viscoelasticity are readily defined as ratio between viscosity and stiffness. In the numerical applications, three kinds of different excitations are considered, that is, harmonic, periodic, and pseudo-stochastic. It is shown that, for any periodic excitation, the equivalent models have some important features: (i) the dissipated energy per cycle at steady-state coincides with the Staverman–Schwarzl formulation of the fractional model, (ii) the elastic and the viscous coefficients of the equivalent model are strictly related to the storage and the loss modulus, respectively.