A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

When designing spatial structures, it is necessary to know the stressed-strained state of a body. These problems include the calculation schemes, in which there is a half-space with cylindrical cavities, at the boundaries of which the contact type conditions are assigned. The segment of such problems is not enough researched and requires further attention.The analytical and numerical algorithm for solving a special problem of the theory of elasticity for a half-space with cylindrical cavities was proposed in this paper. Radial displacements and tangential stresses are assigned at the boundaries of cavities, and one of the two types of the boundary conditions – displacement or stress – is assigned at the half-space boundary. Calculations revealed the stressed-strained state of the half-space.Under the fixed geometrical conditions, a numerical analysis of the three variants of the problem, when displacements are assigned at the half-space boundary, and of the tree variants of the problem, when stresses are assigned at the half-space boundary, was conducted. A comparative analysis of the variants with different boundary conditions was carried out.It was found that at the boundary conditions of the contact type, assigned at the boundaries of cylindrical cavities, if the assigned displacement function and the assigned function of stresses are the same, the boundary conditions at the half-space boundary in the form of stresses have more influence of the stressed state that boundary conditions in the form of displacements. It was also established that at different kinds of the assigned boundary conditions (stresses or displacements), stresses o and on the surface of applying such conditions change for the opposite, that is, from stretching to compressing and vice versa.The presented analysis can be used during designing the structures, in the calculation schemes of which there is a half-space boundary with boundary conditions of the contact type, assigned on it, and cylindrical cavities, on the surfaces of which displacements and stresses are assigned.

The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using exact methods. One such model is the problem of a layer with cylindrical embedded supports that are parallel to each other and the layer boundaries. In this work, the supports are represented by cylindrical cavities with zero displacements set on them. The layer is considered in Cartesian coordinates, and the cavities are in cylindrical coordinates. To solve the problem, the Lamé equation is used, where the basic solutions between different coordinate systems are linked using the generalized Fourier method. By satisfying the boundary conditions and linking different coordinate systems, a system of infinite linear algebraic equations is created. For numerical realization, the method of reduction is used to find the unknowns. The numerical analysis has shown that the boundary conditions are fulfilled with high accuracy, and the physical pattern of the stress distribution and the comparison with results of similar studies indicate the accuracy of the obtained results. The proposed method for calculating the stress–strain state can be applied to the calculation of structures whose model is a layer with cylindrical embedded supports. The numerical results of the work make it possible to predetermine the geometric parameters of the model to be designed.