Time-response functions of fractional derivative rheological models
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ORIGINAL CONTRIBUTION
Time-response functions of fractional derivative rheological models Nicos Makris1,2
· Eleftheria Efthymiou1
Received: 26 February 2020 / Revised: 28 August 2020 / Accepted: 10 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In view of the increasing attention to the time responses of complex fluids described by power-laws in association with the need to capture inertia effects that manifest in high-frequency microrheology, we compute the five basic time-response functions of in-series or in-parallel connections of two elementary fractional derivative elements known as the Scott-Blair (springpot) element. The order of fractional differentiation in each Scott-Blair element is allowed to exceed unity reaching values up to 2 and at this limit-case the Scott-Blair element becomes an inerter—a mechanical analogue of the electric capacitor that its output force is proportional only to the relative acceleration of its end-nodes. With this generalization, inertia effects may be captured beyond the traditional viscoelastic behavior. In addition to the relaxation moduli and the creep compliances, we compute closed-form expressions of the memory functions, impulse fluidities (impulse response functions) and impulse strain-rate response functions of the generalized fractional derivative Maxwell fluid, the generalized fractional derivative Kelvin-Voigt element and their special cases that have been implemented in the literature. Central to these calculations is the fractional derivative of the Dirac delta function which makes possible the extraction of singularities embedded in the fractional derivatives of the two-parameter Mittag-Leffler function that emerges invariably in the time-response functions of fractional derivative rheological models. Keywords Non-integer differentiation · Viscoelasticity · Microrheology · Inertia effects · Inerter · Generalized functions
Introduction Phenomenological constitutive models containing differential operators of non-integer order (fractional derivative models) have been proposed in mechanics, geosciences, electrical networks and biology over the last decades (Gemant 1936, 1938; Scott Blair 1944, 1947; Scott Blair et al. 1947; Scott Blair and Caffyn 1949; Caputo and Mainardi 1971; Rabotnov 1980; Bagley and Torvik 1983a, 1983b; Koeller 1984; Koh and Kelly 1990; Friedrich 1991; Gl¨ockle and Nonnenmacher 1991, 1994; Makris and Constantinou 1991; Schiessel et al. 1995; Makris 1997a; Gorenflo and Mainardi 1997; Challamel et al. 2013; Atanackovic et al. 2015; Westerlund and Ekstam 1994;
Nicos Makris
[email protected] 1
Dept. of Civil and Environmental Engineering, Southern Methodist University, Dallas, TX, 75276, USA
2
Office of Theoretical and Applied Mechanics, Academy of Athens, Athens 10679, Greece
Lorenzo and Hartley 2002; Suki et al. 1994; Puig-deMorales-Marinkovic et al. 2007; and references reported therein). Given that fractional derivative operators are linear differential operators, the time-dependent behavior of me
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