Stress state analysis of semi-infinite layer with cylindrical hinged joint

A spatial problem of elasticity theory has been solved to determine the stress-strain state of a semi-infinite elastic layer containing a cylindrical cavity and an elastic cylindrical pipe (hinged joint). Both cylindrical objects are located parallel to the horizontal boundaries of the layer along the z-axis. The layer is additionally limited by the vertical plane z=0, which causes complex spatial edge effects. A distinctive feature of the proposed mathematical model is an original approach to satisfying the conditions at the vertical end of the layer. Instead of directly considering a semi-bounded body, modeling is applied by specifying even or odd (according to the z-coordinate) loads on the horizontal surfaces of an infinite layer. This approach makes it possible to effectively simulate the conditions of a "smooth wall" or "free end." According to the problem statement, normal displacements and tangential stresses are specified on the surface of the cavity and the inner surface of the pipe, and the conditions of smooth contact (equality of normal stresses and displacements in the absence of tangential stresses) are implemented on the surface of the pipe conjugation with the layer. The generalized Fourier method is used to solve the boundary value problem. The displacement vectors for the layer and the pipe are sought as a superposition of the basis solutions of the Lamé equations in Cartesian and local cylindrical coordinate systems. Using addition theorems for the basis solutions, satisfying the boundary conditions is reduced to solving a connected infinite system of linear algebraic equations with respect to unknown spectral and amplitude coefficients. The constructed analytical and numerical apparatus allows us to study with high accuracy the concentration and mutual influence of stresses near the hinge joint and cavity, taking into account the presence of a vertical boundary. The results obtained are important for assessing the strength and reliability of aerospace components and machine-building structures with cylindrical inclusions.