On the study of some geological structure models

. In this work, we study problems of harmonic vibrations of extended elastic structures with a coating. The studied boundary value problems can have different physical interpretations; in this case, they are considered as models of geological structures. We used two different coating models. The equation of a bending plate was chosen to describe one of them. The second type of coating is not a traditionally studied stringer or overlay, since it is characterized by several parameters of a two-dimensional deformable object. Further we have conducted a study of the properties for the substrate/coating system. Where numerical analysis showed us that the difference between the zero and pole curves for an elastic medium with a coating from the dispersion curves for an elastic medium without a coating is stronger the greater the thickness of the coating plate and the denser its material. The conclusions are valid for both plate models used. The models of lithospheric structures considered in this work make it possible to study the features of wave propagation in a three-dimensional deformable foundation with a two-dimensional coating, not only for continuous coatings, but also in the presence of vertical defects in them, as well as heterogeneous composite coatings. The presented approach can be extended to problems for heterogeneous substrates.


Introduction
A large number of works are devoted to the problems of elasticity for plates and materials with coatings in various formulations [1][2][3][4][5][6][7][8].Models for the transmission of vibration and static loads by coatings are aimed at solving problems of civil engineering, seismology, etc., since the determination of the components for the stress-strain state in thin-walled elements and structures in contact with them is of great scientific and practical interest.
The study of regional patterns in the spatiotemporal development of seismic processes requires the active use of physical and mathematical models that reflect the structural characteristics of the geophysical environment, during the modeling of which it is necessary to take into account the natural layering of some geological materials.
In this work, we study problems of harmonic vibrations of extended elastic structures with a coating.The studied boundary value problems can have different physical interpretations; in this case, they are considered as models of geological structures.We used two different coating models.The equation of a bending plate was chosen to describe one of them [1].The second type of coating is not a traditionally studied stringer or overlay, since it is characterized by several parameters of a two-dimensional deformable object.To mathematically describe its dynamics, we used linearized equations of steady-state oscillations for plates [2].

Materials and methods
In this work we consider models of coating/substrate systems.An elastic plate (or system of plates) with thickness-averaged parameters (coating) is in interaction with an infinite elastic layer (substrate).We study system oscillations resulting from harmonic (with frequency ) load.
The source of vibration excitation is a harmonic load localized in the area  on the surface of the coating/substrate system.The coordinate plane x 1 Оx 2 is connected with the middle surface of the coating.We assume the full adhesion of the coating to the underlying foundation.

Coating models under consideration
We consider a coating model described by the bending plate equation [1].For a continuous coating, the equation is as follows: Here the plate rigidity is Another model is described by several components of a two-dimensional deformable object.To estimate the stress concentration in lithospheric plates, we can limit ourselves to linearized equations of plate motion [2].The linearized equations of plate vibration relative to the displacement amplitudes of the midplane , after the separation of the time factor will take the form [9]  is matrix differential operator with components: , h is the thickness of covering, ,  are the module of shift and the Poisson ratio accordingly,  is material density, , u x x are displacements of points of a median surface in the

Model of the foundation
The choice of the model for the foundation can be different: a deformable half-space (homogeneous, stratified), a layer, a package of layers, a block medium, including one containing internal defects, etc.At the same time, taking into account the complex properties of the foundation (heterogeneity, anisotropy, etc.) even within the framework of the linear theory of elasticity leads to an increase in mathematical difficulties in the study of these problems.
In the work, a deformable base in the form of a homogeneous elastic layer with material constants is described by a model of the linear theory of elasticity.In the case of steady-state harmonic oscillations, the displacements of the foundation points satisfy the system of Lame equations 2 0 The conditions for ideal contact between the coating and the substrate represent the conditions of continuity for stresses and displacements in the area of their contact.

Solution method
Displacements for the upper face   For substrates of finite thickness, the elements of the Green's matrix symbol , ,0 ,  KK     are meromorphic functions that do not have branch points.The contours j    1, 2 j  deviate from the real axis only when going around the poles of   12 , K  , according to the principle of limiting absorption [10].

Relationship between stress and displacement vectors in
Here, the Fourier transforms of the values denoted by the corresponding lowercase letters are denoted in capital letters.Applying the Fourier transform in horizontal coordinates to (1) and ( 2), we obtain an ordinary differential equation (ODE) for ( 1) The elements R have the form: From the conditions of ideal contact and representation (3) for the surface displacements of the system, as was done in [11], we obtain the functional matrix relations

Results and Discussion
For both coating models, we conducted a study of the properties for the substrate/coating system.The calculation results are presented for the second coating model.Figure 1 illustrates the real poles location of the elements * K for an elastic layer rigidly coupled to a non-deformable foundation, with and without a coating.The black curves correspond to the uncoated layer, the red curves correspond to the coated layer with a thickness of 0,01, and the green curves correspond to the thickness of 0,1.The following calculated values were taken: for the layer -


. All values are reduced to the layer parameters The thicker the coating, the more the pole curves for the plate-reinforced medium deviate from the foundation poles.From the results of computational experiments, we can draw the following conclusions: the difference between the zero and pole curves for the coating-substrate system from the corresponding curves of the substrate is bigger, the thicker the coating and the denser its material.As the structure of the substrate becomes more complex, the number of zeros and E3S Web of Conferences 463, 03011 (2023) EESTE2023 https://doi.org/10.1051/e3sconf/202346303011poles of the matrix elements increases.The conclusions are valid for both plate models.When considering only vertical surface loads, as the vibration frequency increases, the curves for the first and second plate models all deviate more and more from each other.

Conclusion
The models of lithospheric structures considered in this work make it possible to study the features of wave propagation in a three-dimensional deformable foundation with a twodimensional coating, not only for continuous coatings, but also in the presence of vertical defects in them, as well as heterogeneous composite coatings.The results can find applications not only in seismology (when studying induced regional seismicity, geological mapping, when detailing the tectonic structures of the region), but also in materials science when studying the properties of materials with coatings.The approach considered in this work extends to similar problems for substrates of different modules.

Figure 2 (
a and b) shows the zero curves for the elements in the form of an elastic layer.

Figure 3 (KE3SFig. 1 .Fig. 2 . 11 K
Fig. 1.Poles curves for the matrix * K elements and the foundation in the form of an elastic layer.

Figure 3 (
Figure 3(b) illustrates the poles (blue lines) and zeros (red lines) for the same matrix element for a two-layer coated stack, where we added a bottom layer with parameters:2