ANALYSIS OF THE STRESS STATE OF A LAYER WITH TWO CYLINDRICAL ELASTIC INCLUSIONS AND MIXED BOUNDARY CONDITIONS

The spatial problem of the elasticity theory is studied for a layer with two infinite circular solid cylindrical inclusions that are parallel to each other and to the layer boundaries. The physical characteristics of the layer and the inclusions are different from each other and they are homogeneous, isotropic materials. The spatial function of stresses is given at the upper boundary, and the function of displacements is given at the lower layer boundary. Circular cylindrical elastic inclusions are rigidly connected to the layer. It is necessary to determine the stress-strain state of the composite body. The problem solution is based on the generalized Fourier method, which uses special formulas for the transition between the basic solutions of the Lamé equation in different coordinate systems. Thus, the layer is considered in the Cartesian coordinate system, the inclusions – in the local cylindrical ones. Satisfying the boundary and conjugation conditions, systems of infinite integro-algebraic equations were obtained, which were subsequently reduced to linear algebraic ones. The resulting infinite system is solved by the reduction method. After determining the unknowns, it is possible to find the stress values at any point of the elastic composite body. In numerical studies, a comparative analysis of the stress state in the layer and on the surfaces of inclusions at different distances between them is carried out. The analysis showed that when the inclusions approach each other, the stress state in the layer practically does not change. However, its significant change is observed in the bodies of inclusions, so with dense reinforcement ((R 1 + R 2 ) / L > 0.5), it is necessary to take into account the distances between the reinforcing fibers. At stress values from 0 to 1 and the order of the system of equations m=10, the accuracy of meeting the boundary conditions was 10 -4 . With an increase in the system order, the accuracy of meeting the boundary conditions will increase. The given analytical-numerical solution can be used for high-precision determination of the stress-strain state of the given type of problems, and also as a reference for problems based on numerical methods.


Introduction
In various industries, one has to face the design of parts and structures from composite materials, which often are a layer with longitudinal reinforcement, where, based on engineering calculations, it is necessary to place reinforcement elements at a close distance to each other. In view of this, it is important to understand how the distance between the longitudinal rods affects the stress-strain state in the zone of contact between the layer and the reinforcement. To solve these problems, it is possible to conduct experiments [1,2] or use a numericalexperimental approach [3][4][5][6][7], perceiving the composite as a single layer. Another approach is numericalanalytical, in which the composite body is perceived as a component of the intersection of several elements. However, this requires an efficient and highly accurate method of the stress-strain state calculation. Thus, in paper [8] the two-dimensional boundary value problem of diffraction of symmetric normal longitudinal shear waves for a layer with a cylindrical cavity or inclusion was solved using the method of images, and in papers [9][10][11][12], stationary wave diffraction problems of stress determination for a layer with a cylindrical cavity or inclusion were solved on the basis of the Fourier series decomposition method.
However, for spatial models with a large number of boundary surfaces and high accuracy of determining the stress state, the numerical-analytical generalized Fourier method [13] is most suitable. On the basis of this method, the problems for a half-space with a cavity or an inclusion [14][15][16], a layer with a cavity or inclusions [17][18][19][20], a cylinder with cylindrical cavities or inclusions [21][22][23][24], as well as for a half-space with a cavity coupled to a layer [25] are considered.
The generalized Fourier method is also used in the paper during research studies.

Problem statement
In an elastic homogeneous layer, there are two cylindrical inclusions with radii R p made of materials different from the layer material and placed parallel to its boundaries.
The inclusions are considered in local cylindrical coordinate systems (ρ p , φ p , z), the layer -in the Cartesian coordinate system (x, y, z), combined with the coordinate system of the inclusion with the number p=1. The boundaries of the layer are located at a distance of y=h and y=-h (Fig. 1).

Fig. 1. A layer with two cylindrical inclusions
It is necessary to find a solution to the Lamé equation, provided that stresses are given at the upper layer boundary , displacements are given on the lower one conjugation conditions are given on the boundaries of the contact of the layer and inclusions is stress operator; known functions, which are considered as rapidly decreasing from the origin along the axis z and x.

Solution method
The basic solutions of the Lamé equation for Cartesian and cylindrical coordinate systems are chosen in the form [11]: are inner and outer solutions of the Lamé equation for the cylinder, respectively;     are Lamé equation solutions for the layer σ is Poisson's ratio. The problem solution is given in the form are basic solutions given by the formulas (4), and unknown functions are necessary to be found from the boundary conditions (3) and conjugation conditions (1) and (2).
The formulas [17] are used for a transition between coordinate systems: -for a transition from solutions where pq  is the angle between the axis x p and the line segment.
To meet the boundary conditions at the upper layer boundary, vector (5)  with the help of transition formulas (7). Taking this into account, the first three equations (one for each projection) with 12 unknowns are obtained To meet the boundary conditions at the lower layer boundary, vector (5)   . Thus, all the unknowns of the expressions (5) and (6) will be found.
The solutions of the infinite system by the reduction method given in the paper showed its convergence, which satisfies the boundary conditions with high accuracy.
On the upper layer boundary, stresses are given in the form of a wave , there are no displacements on the lower one The infinite system was truncated by parameter m. At L 12 =30 mm, parameter m=6, at L 12 =25 mm, parameter m=10.
The calculation of integrals is performed by the quadrature Filon's rule (for oscillating functions) and Simpson's rule (for functions without oscillations). The accuracy of meeting the boundary conditions at the values of m and the specified geometric parameters is 10 -4 .
When the inclusions approach each other, the stress state in the layer almost does not change. The difference is observed only in the inclusion bodies. Fig. 2 shows stress graphs σ φ and σ z on the surface of the second inclusion at z=0 in N/mm 2 .
When the second cylindrical inclusion approaches the first one, stresses σ φ and σ z on its surface increase. Moreover, the stresses values σ φ increase significantly (Fig. 2, a), especially in the upper part of the inclusion, where they are maximum. In general, the increase in the inclusion stress, as it approaches the place of the stress dislocation, is natural. Fig. 3 shows graphs of stresses σ φ and σ z on the surface of the first inclusion at z=0 in N/mm 2 .
When the second cylindrical inclusion approaches the first one, the stresses σ φ on the surface of the first cylinder increase (Fig. 3, a), and the stresses σ z slightly decrease (Fig. 3, b), being redistributed to the second cylinder.

Conclusions
Based on the generalized Fourier method, a method for solving the third basic spatial problem of the elasticity theory for a layer with two longitudinal circular cylindrical inclusions is proposed. The problem is reduced to an infinite system of linear equations, which allows the reduction method to be applied to it. Numerical studies give grounds for asserting that its solution can be found with any accuracy by the given method, which is confirmed by the high accuracy of the boundary conditions.
The solution method can be used in the design of composite materials, the calculation scheme of which is a reinforced layer with specified boundary conditions in the form of stress on the upper layer boundary and displacements on the lower layer boundary.
The given comparative analysis shows that the convergence of reinforcement elements affects their stress state, in particular, the stresses σ φ and σ z .
The given solution method allows to obtain the stress-strain state for a layer with only two longitudinal circular cylindrical inclusions. For further development of this method, the number of inclusions can be increased to three or more. For this, it is necessary to change this algorithm by working out the connections between the two shifted coordinate systems, as well as between their basic solutions.