Electro-mechanical analogy for Prabhakar-like fractional viscoelasticity

The aim of this paper is to set up a formal equivalence between a mechanical system and an electrical one. Specifically, we consider the Maxwell-Prabhakar linear viscoelastic model, based on Prabhakar fractional operators. Therefore, we find the analogous expression for the electric current due to a step potential. The expression for the resulting electric current depends on the variable characterizing the viscoelastic model and its behaviour is then discussed with the support of some interesting plots.


Introduction
Fractional calculus have been gaining considerable attention due to its large variety of applications in different branches of mathematical physics.One of the main playgrounds for fractional calculus could surely be considered the linear theory of viscoelasticity [1].
Viscoelastic models are mostly applied for geophysical and mechanical purposes and, in particular, we are going to focus our attention on the so called Maxwell-Prabhakar model [2], which considers the standard Maxwell model of linear viscoelasticity where the ordinary time derivatives are replaced by (regularized) Prabhakar fractional derivatives.
From the general theory of linear viscoelasticity, we can write a linear model in two equivalent forms, namely the creep and the relaxation representations.For both these representations, we associate the so called material functions, G(t) and J(t), called respectively relaxation modulus and creep compliance.It is also important to emphasize that both functions contain all the physical information about the viscoelastic model, and their linear constitutive equation is written in terms of the stress σ(t) and the strain ε(t).
Prabhakar fractional derivative has demonstrated its potential in enhancing the accuracy of not only viscoelastic modeling, regarding the storage and dissipation of energy [3] too, but also in stochastic processes [4] and concerning phenomena of anomalous relaxation in dielectrics [5].
Therefore, after an introduction about the main tools of Prabhakar fractional calculus, presented in Sect.2, in Sect. 3 we set up the electro-mechanical analogy between a viscoelastic and an electrical model.In the following Sect.4, some interesting plots are presented and commented, in order to understand the physical interpretation of the involved quantities.Finally, some conclusions are summed up and discussed in Sect. 5.

Prabhakar fractional calculus
Prabhakar fractional calculus is based on the so called Prabhakar function [6], also known as three-parameters Mittag-Leffler function [7], defined as It is worthy to remark that E γ α,β (z) in ( 1) is an entire function of order ρ = 1/Re(α) and type σ = 1.
The Prabhakar function in (1) satisfies several properties [8], and hereinafter we are going to list the key features useful for the purposes of our work.Firstly, it is remarkable to notice that for γ = 1 we obtain the two-parameters Mittag-Leffler function E 1 α,β (z) = E α,β (z), and for β = γ = 1 we retrieve the original Mittag-Leffler function [9], , while setting all the parameters to the unit, namely α = β = γ, simply recover the classical exponential function Let us define the Prabhakar kernel for ω ∈ C.Then, for a given ω, the derivative of the kernel is written and the expression for e γ α,β (ω, t) in ( 2), in the Laplace domain, reads [10] In addition, the kernel in (2), identifies the Prabhakar integral too.Let us consider a function reminding that ω, α, β, γ ∈ C, with Re(α) > 0, and t ∈ R. From this latter expression in (5), we then define the so called regularized Prabhakar fractional derivative, which is where h = ⌈β⌉ so that f (h) (t) represents the time derivative of order h of the function f (t).
For the purposes of this work, we also remark that the Laplace transform of the regularized Prabhakar derivative in ( 6) is also known and it results

Maxwell-Prabhakar model
As already mentioned in the introductory section, we are going to consider the Maxwell-Prabhakar viscoelastic-like model [2].Taking into account the stress-strain relation for the Maxwell model of linear viscoelasticity, we then replace the time derivatives with (6), obtaining where p and q are real constants.
Employing then the linear theory of viscoelasticity, which presents the following two equivalent forms for the constitutive equation, At this point, applying ( 7) inside (8)and using (9), we obtain the expressions for J and G in the Laplace domain, which can subsequently written in the time domain as follows

Electro-mechanical analogy
In 1956, the studies lead by Gross and Fuoss provided a formal analogy between a viscoelastic system and the description of electrical ladder structures, identifying a formal equivalence between the known quantities characterizing a mechanical system and an electrical system, e. g. [11] and [12].The main results from the arguments presented in this regards identify the following formal equivalence In addition, dealing with a general viscoelastic system, a single electric component would not simply describe its corresponding analogous electrical system.Thus, a viscoelastic system would formally correspond to a class of electrical ladder networks, coming from the formal duality between spring and resistors, dashpot and capacity.
Briefly, from the general non-dimensional stress-strain relation, expressed in the time domain, where the dot represents the time derivative with respect to the argument of G and * is intended as the convolution product, in sense of time Laplace convolution.Implementing the electro-mechanical analogy previously introduced, we find the relation providing a characteristic equation for a certain electrical ladder networks in consistency with a given viscoelastic model determined by the definition of the relaxation modulus G(t), namely Taking into consideration the step potential denoted by the Heaviside step function, namely V (t) = Θ(t), it is straightforward to deduce from (13) the following expression for the current

Numerical Results
In this section, we are going to present some plots describing the behaviour of the function I(t), as written in (14).
The first case analysed is taking into account the so called Havriliak-Negami relaxation model, given by imposing the relation β = αγ, with 0 < α, γ < 1 and ω = −λ, for λ ∈ R + [13].Thus, for a fix value of α and for variable values of γ, we find that the current I(t) has a decreasing behaviour, as shown in Figure 1.Secondly, we consider constant values of β and γ, and we consider α as a varying the parameter, obtaining the results depicted in Figure 2.For the last case, we keep constant values of α and γ, varying β.Again, for the chosen parameters, we find a decreasing behaviour for the current, comparable so with the previous cases, as plotted in Figure 3.

Conclusions and discussion
In conclusion, we have introduced the so called Maxwell-Prabhakar model of linear viscoelasticity and we have employed the electro-mechanical analogy between a viscoelastic and an electrical system, in order to find the expression for the electric current in terms of the quantities which characterize the model.In fact, I(t) in ( 14) depends on the three-parameters Mittag-Leffler functions and, to obtain the figures in Sect.4, we have employed the MATLAB routine developed by R. Garrappa available in [14].
It is particularly interesting to notice the decreasing behaviour of the electric current, especially in the first case analysed in Figure 1 and concerning the Havriliak-Negami.Indeed, we recover a physical relaxation process that is compatible with the results expected from the literature.