The fractional derivative Kelvin-Voigt and standard linear solid models were first proposed by Shermergor [76] and Meshkov [38] in 1966 and 1967, respectively, and then independently but at a somewhat later time by Caputo [11] and Caputo and Mainardi [14], respectively. The early applications of these models were made by Caputo [11] in 1967 and Caputo and Mainardi [14, 15] in 1971 for solving the problems dealing with geophysics. In 1974, the fractional derivative Kelvin-Voigt model was utilized by Caputo [12] to study vibrations of an infinite viscoelastic layer.

In the late 1970s, an investigation on fractional derivative models and their application to the problems of structural dynamics was initiated in the USA by Torvik and Bagley. Their first paper in the field [4] dated to 1979 was devoted to modeling an elastomer damper as a fractional derivative Kelvin-Voigt oscillator. During the next decade, these authors contributed significantly to incorporate the fractional derivative standard linear solid model into the numerical procedures for investigating viscoelastically damped structures [5, 83, 84].

Thus, from the papers cited in the forth column of Table 1 it is clearly evident what a tremendous work was carried out in the late 60’s and 70’s by Russian and Italian researchers in the application of fractional calculus viscoelastic models (formulated earlier or in the same period using the two approaches as shown in the first three columns of Table 1) for solving dynamic problems in the mechanics of solids and geophysics.

2.Rabotnov’s Fractional Operators and Main Formulas of Algebra of Fractional Operators

We shall proceed our presentation in this Section from the analysis of the rheological model, which is called as the fractional derivative standard linear solid model

where σ is the stress, is the strain, τ and τσ are the relaxation and retardation times, respectively, E0 and E∞ are the relaxed (prolonged modulus of elasticity, or the rubbery modulus) and nonrelaxed (instantaneous modulus of elasticity, or the glassy modulus) magnitudes of the elastic modulus, respectively.

Meshkov [38] was the first to derive formula (5) in 1967.

Expressing σ in terms of ε or ε in terms of σ from (5) with due account for (6) yields

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Relationships (7) and (8) involve one and the same operator which will be denoted further as 3γ (τiγ ), i.e.,

Operator (10) was introduced into consideration by Yu. N. Rabotnov in a little bit another form [46].

In order to obtain operator (10) in another form, we multiply the numerator and denom-

If we suppose that the right part of formula (12) is the sum of an infinite decreasing geometrical progression, the denominator of which is equal to d = −Iγ τi−γ , then 3γ (τiγ ) could be represented as

Formulas (16) and (17) are Boltzmann-Volterra relationships with weakly singular kernels of heredity 3γ (−t/τi ), which attenuate at t → ∞, in so doing resolvent kernels occur to be the same. Only exponential kernels possess this feature, and the kernels (15) go over into exponential kernels at γ = 1, i.e.,

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These properties of 3γ (−t/τi)-function allowed Rabotnov to call it as a fractional

In conclusion of this Section we present the important formula for the multiplication of the operators

It should be noted that formula (21) does not coincide with that presented in Rabotnov [48] (see formula (5.10) there), since we use the dimensionless operator 3γ (τiγ ), while the dimensional operator was utilized in [48].

3.Generalized Rabotnov Operators

i=1

where mj , tj (j = 1, 2, ..., n) and ni , τiγ (i = 1, 2, ..., n) are constants.

Since operators (19) and (20) are reciprocal, i.e., E J = J E = 1, then the following

Assume that the constants ni and τiγ (i = 1, ..., n) are known, and it is necessary to determine the constants mj and tγj (j = 1, ..., n). Using formula (21), let us rewrite (25) in the form

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or in another form

Equation (31) possesses n real positive roots if τi−γ > 0 and ni > 0.

If we suppose that τk−+1γ > τk−γ , then zeros of the function F1(x) are located between

The plot of the function F1(x) for n = 2 is presented in Fig. 1a, where x = t−γ .

The set of equalities (29) represents a system of n linear equations for n unknown coefficients mj . Assign that

where y = τ −γ .

Function F2(y) (33) possesses n zeros at y = τi−γ , while at y → t−j γ it tends to +∞ or −∞ depending on the sign of mj and the direction of the transition to the limit. Since the

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Figure 1. Functions F1(x) and F2(y).

points τi−γ are located between the points t−j γ , except the point τ1−γ (see formula (32)), then the plot for the function F2(y) for n = 2 should be as that in Fig. 1b, but this is possible only when all mj are positive.

At n = 1, from (27) and (28), (19) and (20) we obtain

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Note that the signs of the values a1, a2, b1, and b2 could be established with help of

which are positive as well.

If, vice versa, it is necessary to determine the values of τi−γ and ni, then using Eqs. (29) and (30), we have

The solution of (46) gives us the values of τ1−γ and τ2−γ , and then substituting τ1−γ and τ2−γ in (45), we find n1 and n2.

Let us present here two more useful formulas, which are the immediate generalization of formulas (36).

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