Electrical analogous in viscoelasticity Guido Ala a,⇑ , Mario Di Paola b , Elisa Francomano c , Yan Li d , Francesco P. Pinnola b a Università degli Studi di Palermo, Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici (DEIM), Viale delle Scienze Ed.9, I-90128 Palermo, Italy b Università degli Studi di Palermo, Dipartimento di Ingegneria Civile, Ambientale ed Aerospaziale, dei Materiali (DICAM), Viale delle Scienze Ed.8, I-90128 Palermo, Italy c Università degli Studi di Palermo, Dipartimento di Ingegneria Chimica, Gestionale, Informatica e Meccanica (DICGIM), Viale delle Scienze Ed.6, I-90128 Palermo, Italy d School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, PR China article info Article history: Received 24 July 2013 Received in revised form 30 September 2013 Accepted 11 November 2013 Available online 23 November 2013 Keywords: Fractional calculus Viscoelastic models Fractional capacitor Eigenvalues analysis abstract In this paper, electrical analogous models of fractional hereditary materials are introduced. Based on recent works by the authors, mechanical models of materials viscoelasticity behavior are firstly approached by using fractional mathematical operators. Viscoelastic models have elastic and viscous components which are obtained by combining springs and dashpots. Various arrangements of these elements can be used, and all of these visco- elastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed mod- els are validated by using modal analysis. Moreover, a comparison with numerical exper- iments based on finite difference time domain method shows that, for long time simulations, the correct time behavior can be obtained only with modal analysis. The use of electrical analogous in viscoelasticity can better reveal the real behavior of fractional hereditary materials. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction In the last few decades, fractional calculus has attracted a great interest in various scientific areas including physics and engineering [1–9]. Particularly, in the area of viscoelasticity a significant effort has been done in describing more closely the behavior of materials by using fractional mathematical models. Moreover, the analogy between viscoelastic and electrical constitutive equations is well-known so that, in spite of different physical meanings, the widely used Maxwell model, Kelvin–Voigt model, and Standard Linear Solid Model can be applied to predict a circuit behavior as well [10]. Besides, allow for the time varying distributions of elements, a series of generalized models are proposed in either canonical structure or ladder networks [11,12], such as the Maxwell–Wiechert model. All the above mentioned viscoelastic models have elastic and viscous components which are combined of springs and dashpots. The only difference is the arrangement of these elements, and all of these viscoelastic models can be equivalently modeled as electrical circuits, where the spring and dashpot are anal- ogous to the capacitance and resistance respectively [13–15]. Nevertheless, compare to two viscoelastic elements, there are four passive electrical elements including resistor, capacitor, inductor and the recently find memristor [16,17]. Thus, although the circuits of LC, RC, RL, etc. can be transformed in some circumstances, it is still reasonable to expect that there are far more new properties included in the electrical models that are formulated by using the same structure in viscoelastic 1007-5704/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2013.11.007 ⇑ Corresponding author. Tel.: +39 091 23860288; fax: +39 091 488452. E-mail addresses: guido.ala@unipa.it (G. Ala), mario.dipaola@unipa.it (M. Di Paola), elisa.francomano@unipa.it (E. Francomano), liyan.sdu@gmail.com (Y. Li), francesco.pinnola@unipa.it (F.P. Pinnola). Commun Nonlinear Sci Numer Simulat 19 (2014) 2513–2527 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns