On the spectra of composite materials with different dissipation models

. The article deals with the issues of qualitative behavior and methods for calculating the spectra of natural vibrations of a layered composite consisting of elastic and viscoelastic phases with dissipation. As a viscoelastic phase, it is proposed to consider a viscoelastic material with exponential aftereffect kernels, with aftereffect kernels in the form of Rabotnov functions, a material with Kelvin-Voigt friction and a material with fractional Kelvin-Voigt friction. A general scheme is given for studying the qualitative properties of the spectra of one-dimensional oscillations and a method for calculating one-dimensional oscillations, which consists in reducing the spectral problem to solving algebraic equations. The question of the convergence of natural frequencies of oscillations of composite samples to the natural frequencies of averaged boundary value problems, which are non-self-adjoint, is discussed.


Introduction
In connection with various issues of the theory and practical use of composite materials, the question arises about the qualitative properties of layered composites, in which one of the phases is an elastic material, and the other phase is a material with pronounced dissipation due to internal friction.Previously, composite materials consisting of elastic phases with different elastic moduli were studied in detail [1,2].Such materials can be used, for example, as soundproofing materials.In this case, the rheological models of a material with internal dissipation can be very different, so the question arises about the adequacy of one or another model.Verification and calibration (selection of parameters) of models, of course, should be carried out on the basis of experiment.Thus, it is possible to compare experimentally obtained frequencies of natural oscillations of a sample from a composite material with frequencies obtained by calculation on the basis of one or another model of energy dissipation inside one of the phases, as well as on the basis of the use of welldeveloped methods for constructing effective (or averaged) characteristics of composite materials with a periodic structure.
In this paper, we consider a layered composite consisting of two phases: an elastic and a viscoelastic one, and several rheological models are used for the viscoelastic phase.These are a viscoelastic material with an exponential relaxation kernel, a material with a relaxation kernel in the form of the Rabotnov e-function, a material in which the Kelvin-Voigt friction is present, as well as a material with fractional Kelvin-Voigt friction, which is given by the convolution of the Rabotnov e-function with a term corresponding to instantaneous friction Kelvin-Voigt.Such media were considered in works [3][4][5].Kelvin-Voigt fluids can also be classified as media of this type, in which there is not only a term corresponding to the instantaneous Kelvin-Voigt friction, but also the time derivative of the term corresponding to the Kelvin-Voigt friction [6].
In this work, we consider natural vibrations of a sample made of a given composite when the movement occurs in one direction, strictly perpendicular to the material layers.In this case, all functions included in the equation will depend only on time and one spatial variable.
For a sample of a composite material with a layered structure, we first apply the previously developed methods of homogenization theory [1][2][3] , which allow us to reduce the problem to a sample of a homogeneous but anisotropic material, while materials corresponding to the original layers can be isotropic.To analyze the spectrum of the resulting averaged model, the simplifying assumption mentioned above is used -the spectrum of material vibrations is considered in only one direction -the direction of the vertical axis.Further, we will see that the mentioned spectra will coincide with the spectra of problems similar to the Sturm-Liouville spectrum with a nonlinear occurrence of the spectral parameter.In the next section, we indicate the main types of such one-dimensional spectral problems on the interval [0, ]  with a nonlinear occurrence of the spectral parameter.Below, problems of this type will be obtained using the averaging method and taking into account the assumption that the considered motion is one-dimensional.
Previously, the question of the vibration spectra of elastic composites was considered in [1].In this monograph, averaged (effective) spectral problems are constructed and the convergence of the eigenvalues of the problems for composite samples to the eigenvalues for averaged systems is studied.Convergence is proved for the case when both phases are elastic with different elastic characteristics.In the work [7][8][9][10] and in the present work, the material from the elastic and viscoelastic phases is considered.The issue of convergence for these cases has not been studied much, the technique used in [1] cannot be used directly because the problems are not self-adjoint.This is an interesting topic for further research.The asymptotic behavior of the spectra of layered structures upon averaging is considered in the following papers [12][13][14][15][16].  n  (Fig. 2).

Fig. 2.
Frequency spectrum of a string with friction against a medium.

String with long-term aftereffect with one exponent in the kernel
This is the case of string vibrations with long-term aftereffect corresponding to the wellknown Gurtin-Pipkin equation [11] with one exponent in the kernel.For the spectral parameter p, the spectrum figure, can be obtained by simply squaring the spectral pattern for the q parameter.q n q c q   

Frequency spectrum in the case of fractional Kelvin-Voigt friction
( 1, 2, ) n  (Fig. 5).Such a spectrum corresponds to fractional Kelvin-Voigt friction as a model for the viscoelastic phase.For the spectral parameter p, the spectrum pattern can also be obtained by squaring.In this case, some subset of the spectrum will be located on the right halfand these eigenvalues should be discarded as not corresponding to the physical meaning of the original problem.

One-Dimensional Natural Vibrations of a Composite with Elastic and Dissipative Elements
The constitutive relation for the considered two-phase medium has the form In (2) ( ) and t is time variable, s is a phase number 1, 2 s  .For phase number (1) : Phase number ( 2) is purely elastic, and for it we have [ ( ) As usual, we denote here by , (2)   (2) (2) (2) 33 33 ( 2 )e      .
According to the well-known homogenization procedure, to construct the constitutive relation of the averaged material, one must take the reciprocal values of the modules in each of the two phases, average them and take the reciprocal value.

2 1 )
Qualitative properties of the spectra of generalized problems of the Sturm-Liouville type 2.1 String in purely elastic case2 u is the displacement of the points of the string from the equilibrium position in the vertical direction.After applying the Laplace transform in time .It is the purely elastic case.

Fig. 7 .
Fig. 7. Frequency spectrum in the case of classical model of Kelvin-Voigt friction.

k and 2 k
Lame parameters for each layer, by 1 the regular part of the bulk and the shear relaxation respectively, by ij  Kronecker symbol.Suppose that the amplitude of a bulk relaxation kernel is proportional to the amplitude of the shear relaxation kernel.We also assume that the movement occurs only in the direction of the axis 3 Ox .In (4) and (5) we perform the Laplace transform with respect to the variable t .Taking into account equality 1 ://doi.org/10.1051/e3sconf/202341001004 E3S Web of Conferences 410, 01004 (2023) (2)s the thickness of the layers with number (1), and 1 h  is the thickness of layers numbered as(2).https