First Basic Problem of Elasticity Theory for a Layer with Cylindrical Cavities Smoothly Contacting Two Cylindrical Bushings

A spatial problem of elasticity is solved for a layer with n longitudinal cylindrical cavities, two of which contain thick-walled pipes in smooth contact with the layer. Stresses are given on the surfaces of the layer, the inner surfaces of the pipes, and the cavities. All canonical surfaces do not intersect each other. The material of the layer and cylindrical pipes is homogeneous and isotropic. An analytical and numerical calculation method, which assumes the fulfillment of statics conditions (for the first basic problem of elasticity theory) and is based on the Lamé equation, is proposed. The basic solutions of the Lamé equation are taken in a form that makes it possible to obtain an exact solution for a separate boundary surface in each separate coordinate system. The basic solutions in these coordinate systems (Cartesian for the layer and local cylindrical for the cylindrical inhomogeneities) are interconnected through the mathematical framework of the generalized Fourier method. The fulfillment of boundary conditions on the upper and lower surfaces of the layer, on the inner surfaces of pipes, on cylindrical cavities, as well as the consideration of interface conditions, create an infinite system of integro-algebraic equations, which is reduced to an infinite linear one. In the numerical study, the reduction method is applied to the resulting infinite linear algebraic system of equations. The solution of the system of equations gives the values of the unknown functions. Numerical calculations have shown the rapid convergence of approximate solutions to the exact one. The numerical analysis of the stressed state of the layer and thick-walled pipes showed that the use of polyamide bushings has almost no effect on the stress-strain state of the structure (compared to their absence), the use of steel bushings reduces the stress in the body of the layer in the areas of their location, redistributing the stress to the bushings themselves. The proposed solution method makes it possible to obtain the stress-strain state of structures containing cylindrical cavities and bushings, and the numerical analysis allows to assess the influence of the material on the values of stress distribution in the structures of machines and mechanisms at the design stage.