Communications in Nonlinear Science and Numerical Simulation
Electrical analogous in viscoelasticity
Introduction
In the last few decades, fractional calculus has attracted a great interest in various scientific areas including physics and engineering [1], [2], [3], [4], [5], [6], [7], [8], [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 analogous to the capacitance and resistance respectively [13], [14], [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 models. Particularly, the introduction of the fractional elements [18] and power-law phenomena cannot only extend the above discussions but also better reveal the real physical world such as the mechanical model of fractional hereditary materials [19] and the Abel’s singular problem [20]. In the paper, mechanical models of viscoelasticity behavior are firstly approached by using fractional operators, based on recent works by the authors [19], [21], [22]. Then, electrical analogous models are introduced in order to obtain electrical equivalent circuits useful to predict the behavior of fractional hereditary materials in an easy way. The validity of the proposed models is demonstrated by using modal analysis. Moreover, the comparison with numerical experiments based on finite difference time domain (FDTD) method shows that, for long time simulations, the correct time behavior can be obtained only with modal analysis.
Section snippets
Mechanical models of fractional viscoelasticity
Many materials, like rubbers, polymers, bones, bitumen and so on, show a viscoelastic mechanical behavior; moreover also biological tissues have viscoelastic properties [23], [24], [25], [26], [27], [28]. Viscoelasticity is the property of such materials that exhibit at the same time elastic and viscous behavior. The elastic behavior is typical of simple solid materials in which the strain history is linked by the stress history through a proportional relation as shown in Eq. (1):
Fractional capacitor
In electrical networks models, the capacitors play a very important role since they model the conservative part of the electric field effects. The well-known Curie’s law [48] reveals an empiric relation between the current related to the applied voltage (being the unit step function):while is a constant depending on the physical characteristics of the capacitor. Both and can be obtained by experimental data and subsequent best fitting
Electrical equivalent circuit of fractional capacitor ()
In this section electrical circuit models of fractional element whose constitutive law is expressed by Eq. (14), is presented for . Firstly, let consider the electrical circuit shown in Fig. 3. A longitudinal pure resistor with a per-unit length resistance r and a transversal pure capacitor with a per-unit length capacitance c, constitute the elementary cell of a transmission line model along the x abscissa. By using the Kirchhoff voltage and current laws the following coupled partial
Discretization
Electrical circuit of Fig. 3 can be discretized by considering small abscissa intervals , as shown in Fig. 5. Let denote with the source voltage and with its current; with the current flowing in the longitudinal resistors , and with the nodal voltages which are the same of that applied to the transversal capacitors , except for the last one. By using the constitutive relations of the lumped elements and the Kirchhoff laws, and by omitting the temporal
Numerical examples
In this section numerical examples are presented in order to show the accuracy of the discretized model. At first, for the discretized circuit shown in Fig. 5, the current response to a unit step voltage is considered. In this case, the exact solution of Eq. (13) is the ERF described in Eq. (12). Different comparisons have been carried out. In Fig. 11 the comparison between discrete circuit results, with 2000 eigenvalues, , and ERF function behavior described in Eq. (12), is
Conclusions
Electrical analogous models of fractional hereditary materials have been introduced. At first, mechanical models of materials viscoelasticity behavior have been approached by using fractional calculus. By combining springs and dashpots, different viscoelastic models have been obtained. These models have been equivalently modeled as electrical circuits, where the spring and dashpot are analogous to the capacitance and resistance, respectively. The proposed models have been validated by using
Acknowledgment
Support by Universita’ degli Studi di Palermo is gratefully acknowledged.
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