as well as relationship (220), we have

where Φ∞ = Φ(∞).

If as a relaxation kernel we use the fractional-exponential function, then K∞ = 1, and

Thus, formulas (201) and (222) define the limiting locations of the stationary shock wave profile.

8.Application of Rabotnov’s Operators in Problems of Impact Response of Thin Structures

The approach existing for studying the impact response of viscoelastic engineering structures is considered below in this Section by the example of the dynamic response of a viscoelastic Bernoulli-Euler beam, which Young’s modulus is the time-dependent operator and the bulk modulus is considered to be constant, transversely impacted by an elastic sphere [67].

Following Timoshenko [80] who considered the problem on a transverse impact of an elastic sphere upon an elastic Bernoulli-Euler beam, let us formulate the problem on the transverse impact response of a viscoelastic beam, the viscoelastic features of which are described by a certain viscoelastic operator. In this case, the equations of motion of an elastic spherical impactor of radius R and the viscoelastic beam of length L have, respectively, the form

where m is the mass of the sphere, w2 is the displacement of the sphere, P (t) is the contact

force, 1 is the displacement of the beam at the contact point, e is a certain viscoelas- w (x, t) E

tic operator, I is the moment of inertia of the beam’s cross section, A is the beam’s cross sectional area, % is its density, δ(x − L2 ) is the Dirac delta-function, x is the longitudinal coordinate, and an overdot denotes partial time-derivative.

Equations (223) and (224) are subjected to the following initial conditions:

where V0 is the initial velocity of the impactor at the moment of impact.

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Substituting the solution for a simply-supported Bernoulli-Euler beam

each of which describes force driven vibrations of the viscoelastic oscillator, where

and E1 is the elastic modulus.

For those operators e, which will be considered below, the Green function n for

E G (t) each oscillator from (228) has the form [67]

where the index n indicates the ordinal number of the oscillator, and all values entering in (229) have the same structure and the same physical meaning as the corresponding values in Eq. (162) discussed in Sect. 6.

Thus, from (229) it is seen that the Green function possesses two terms, one of which, A0n(t), describes the drift of the equilibrium position and is represented by the integral involving the distribution function of dynamic and rheological parameters, while the other term is the product of two time-dependent functions, exponent and sine, and it describes damped vibrations around the drifting equilibrium position, where An is the amplitude, αn is the damping coefficient, and ωn and ϕn are the frequency and phase, respectively. The first term is governed by an improper integral taken along two sides of the cut along the negative real semi-axis of the complex plane (see Fig. 2), while the second term is determined for each n by two complex conjugate roots of the characteristic equation, which locate in the left half-plane of the complex plane.

Knowing the Green functions, the solution of Eq. (224) takes the form

Let us introduce the value characterizing the relative approach of the sphere and beam, i.e., penetration of the beam by the sphere, is

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is the operator involving the geometry of the impactor and the elastic features of the impactor and viscoelastic features of the target, which are described due to the Volterra correspondence principle by the operator E

ν2 and E2 are constant Poisson’s coefficient and Young’s modulus, respectively, for the

elastic sphere (impactor), while 1 and e1 are the operators for the viscoelastic beam (tar-

νe E

get).

Further in order to obtain the integro-differential equation for the values y(t) and P (t),

it is necessary to assign the form of the operator e1.

E

Phillips and Calvit [45] were probably the first to investigate the response of a viscoelastic infinitely extended plate to impact of a rigid sphere. They used the Hertz’s contact law in its hereditary form [45]. This problem is an immediate extension of Zener’s approach [90] for the dynamic rigid spherical-indenter problem for the case of a thin elastic plate [61]. Assuming a falling body to be rigid, the function k (233) in the formula for the contact force in the Hertzian contact law (232) has been represented in [45] by the integral operator with a kernel of heredity based on the constant mechanical loss tangent model.

Ingman and Suzdalnitsky [29] considered dynamic response of a circular viscoelastic plate, viscoelastic features of which are described using the fractional derivative KelvinVoigt model, subjected to the impact of a falling elastic sphere following Timoshenko approach [80] but without considering viscoelastic features of the target in the Hertz’s contact law. The authors of [29] referring to [92] wrote that

“experimental investigations with a steel sphere and a circular composite barrier showed that Hertz’s law is applicable not only in the case of static contact of elastic bodies, but also in problems of response of composite plates to a low-velocity impact,”

and made the conclusion that this argument could justify the choice of Eq. (232) with a constant stiffness coefficient k. However, firstly, the problems of viscoelasticity have not been considered in [92] (only targets made of elastic isotropic or anisotropic materials with and without account for plastic features in the contact region were utilized in this collective monograph), and secondly, Hunter [28] and Phillips and Calvit [45] showed that the local viscoelastic damping associated with deformation in the contact zone is essential at least on the first stage of active loading, since its ignorance in reducing plate impact data results in large error in the calculation of loss tangent and other characteristic values.

Following Rossikhin and Shitikova [67], we will use the governing equations of the viscoelastic material of the target in terms of Rabotnov’s fractional exponential operators

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utilizing the algebra of such operators presented in Sect 4.

Now we write Eq. (228) in terms of the Green function Gn(t) using operator (235) as

the operator e. As a result we obtain

E

and the inversion of the expression (238) on the first sheet of the Riemannian surface via Mellin-Fourier inversion formula (150), results in the desired relationship (229), where the function A0n(t) is defined according to

Z ∞

A0n(t) = τ −1Bn(τ, æn)e−t/τ dτ, (239)

0

with the following distribution of the relaxation-retardation parameters:

where

θ∞n(τ ) = τ 2Ω2∞n + 1, θ0n(τ ) = τ 2Ω20n + 1,

while the amplitude and phase of vibrations have, respectively, the following form:

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