First Basic Problem of Elasticity Theory for a Composite Layer with Two Thick-Walled Tubes

The spatial problem of elasticity theory for a fibrous composite in the form of a layer with two thick-walled cylindrical tubes is solved. Stresses are given on the flat surfaces of the layer and on the inner surface of the tubes. The solution to the problem is presented in the form of Lamé equations in different coordinate systems, where the layer is considered in a Cartesian system and the tubes – in local cylindrical ones. To combine the basic solutions in different coordinate systems, the generalized Fourier method is used. Satisfying the boundary conditions and conjugation conditions between the layer and the tubes, an infinite system of integro-algebraic equations is formed, which is reduced to linear algebraic equations of the second kind, and the reduction method is applied. After finding the unknowns, it is possible to obtain the stress-strain state at any point of the elastic combined bodies using the generalized Fourier method to the basic solutions of the problem. According to the results of numerical studies, it can be stated that the problem can be solved with a given accuracy, which depends on the order of the system of equations and has a rapid convergence of solutions to the exact one. Numerical analysis of the stressed state was considered with a variation of the distance between the tubes. The graphs of the distribution of internal stresses in the tubes and the layer are obtained. The results show an inverse relationship between the magnitude of stresses and the distance between the tubes. In addition to the absolute value of stresses, changes in the character of the diagrams and the sign are possible. The proposed method of solution can be applied in the design of a layer with tubes. The obtained stress-strain state makes it possible to preliminarily evaluate the geometric parameters of the structure. Further development of the research topic is necessary for a model where tubes are combined with other types of inhomogeneities.