Investigation of the second main problem of elasticity for a layer with n cylindrical inclusions

When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy. 
In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed. 
The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10-2 for values from 0 to 1. 
Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions. 
In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared. The result has shown that close placement of reinforcement bars increases stresses  on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. 
The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.


Introduction
Determination of the stress-strain state of a body is the main task in the design of various kinds of structures, buildings, and underground communications. When designing the listed objects, computational schemes, where a layer densely reinforced with longitudinal bars becomes the main design element, are often encountered. At present, the mutual influence of these bars has been little studied, and there is no technique for high-precision solution of such computational schemes at all.
However, there are works close to the task at hand in the scientific literature. Thus, in [1 -4], problems are solved for a layer with cavities perpendicular to its boundaries.
Stationary problems of diffraction of elastic waves with cylindrical cavities parallel to the layer boundaries are considered in [5 -7], where the generalized Fourier method is used in combination with the image method.
For elastic bodies with several limiting surfaces, the generalized Fourier method [8], which is based on the addition theorems for the basic solutions of Lame's equation, is used.
Using this method, the following problems have been solved: for a layer with a spherical cavity [9], for a half-space with longitudinal cylindrical cavities [10 -11], for a cylinder with cylindrical cavities and inclusions [12 -16]. Problems for a layer with one or several cylindrical cavities are considered in [17 -20], for a layer with an elastic inclusion-in [21,22].
For a layer with several cylindrical inclusions and displacements specified on limiting surfaces, there are no ready-made algorithms in the spatial version, so the problem of calculating such problems is urgent.
The purpose of this work is: -development of an analytical-numerical method for calculating the second main problem of elasticity (displacements are specified on all limiting surfaces) for a layer with N cylindrical inclusions that are parallel to the surfaces of the layer and to each other; -to analyze the influence of inclusions on the stress state of the layer, as well as on each other.

Problem statement
There are N circular cylindrical inclusions with radius R p in an elastic homogeneous layer (Fig. 1).

Fig. 1. Layer with cylindrical inclusions
The inclusion will be considered in local cylindrical coordinate systems (ρ p , φ p , z), the layer will be considered in the Cartesian coordinate system (x, y, z) connected to the inclusion coordinate system with the number p = 1. The layer boundaries are located at the distance y = h and y = -h .
It is necessary to find a solution of Lame's equation provided that displacements are specified at the boundaries of the layer: known functions, which will be considered to be as rapidly decreasing to zero at long distances from the origin along the x and z coordinates. At the boundaries of the contact between the layer and the inclusions, the specified matching conditions where p U  -displacements in cylindrical inclusion with number p; ( ) We represent the solution to the problem in the form: of the cylindrical coordinate system: -the angle between the x p axis and the segment . The distance and angle between the parallel displaced cavities shall be calculated by the formulas: In order to fulfill the boundary conditions at the layer boundaries, we are rewritten the vectors The infinite system of equations was reduced to a finite one-m=6. The integrals are calculated using the Philo quadrature formulas (for oscillating functions) and Simpson (for functions without oscillations). The accuracy of the boundary conditions at the specified values of geometric parameters is 10 -2 .
In comparison, the problem for a layer with one cylindrical inclusion has been calculated [22]. Figure 2 shows the stress state on the cylindrical mating surface of the layer and the first inclusion (in the body of the layer), at z = 0, MPa. With an increase in the number of longitudinal bars, the maximum stresses in the layer body decrease slightly (Fig. 2).
The maximum stresses occur in the upper part of the layer. In the lower part of the layer, the stresses are relatively small; therefore, they are not shown in Figure 2.
The stresses in the body of inclusions are shown in Figure 3. The maximum negative stresses   (Fig. 3 (a)) arise at 4 /    and in the upper part of the inclusion.
The largest stresses z  are positive, the maximum values of which arise in the lower part of the inclusion at 4 / 6   (Fig. 3 (b)). Negative stress values .

Conclusions
Based on the generalized Fourier method, an analytical-numerical algorithm for calculating the second main spatial problem of elasticity for the layer with N cylindrical inclusions that are parallel to the layer surfaces has been developed. The problem is reduced to a set of infinite systems of linear algebraic equations.
On the strength of numerical studies of the algebraic system, it can be argued that the solution of this system can be found with any degree of accuracy by the reduction method. This is confirmed by the high accuracy of fulfilling the boundary conditions.
A comparative analysis of the layer stress state with one and three inclusions has been carried out. When the inclusions are located close, an increase in stresses   in their bodies is revealed. Further studies in this direction are relevant for a layer lying on an elastic foundation (conjugate with a half-space) and having several longitudinal cylindrical inclusions. Мірошніков В.Ю., Денисова Т. В.

INVESTIGATION OF THE SECOND MAIN PROBLEM OF ELASTICITY FOR A LAYER WITH N CYLINDRICAL INCLUSIONS
When designing structures in the form of a reinforced layer, one has to deal with the situation when the reinforcement bars are located close to each other. In this case, their influence on each other increases. In order to obtain the stress-strain state in the contact zone of the layer and the inclusion, it is necessary to have a method that would allow obtaining a result with high accuracy.
In this work, an analytical-numerical approach to solving the spatial problem of the theory of elasticity for a layer with a given number of longitudinal cylindrical inclusions and displacements given at the boundaries of the layer has been proposed.
The solution of the problem has been obtained by the generalized Fourier method with respect to the system of Lame's equation in local cylindrical coordinates associated with inclusions and Cartesian coordinates associated with layer boundaries. Infinite systems of linear algebraic equations obtained by satisfying the boundary conditions and conjugation conditions of a layer with inclusions have been solved by the reduction method. As a result, displacements and stresses have been obtained at different points of the considered medium. When the order of the system of equations is 6, the accuracy of fulfilling the boundary conditions was 10 -2 for values from 0 to 1.
Numerical studies of the algebraic system of equations give grounds to assert that its solution can be found with any degree of accuracy by the reduction method, which is confirmed by the high accuracy of fulfilling the boundary conditions.
In the numerical analysis, variants of the layer with 1 and 3 inclusions have been compared.
The result has shown that close placement of reinforcement bars increases stresses   on the surface of these inclusions. The values of stresses on the layer contact surfaces with inclusions have also been obtained. The proposed solution algorithm can be used in the design of structures, the computational scheme of which is the layer with longitudinal cylindrical inclusions and displacements specified at the layer boundaries.