НАПРУЖЕНО-ДЕФОРМОВАНИЙ СТАН ШАРУ З ЦИЛІНДРИЧНИМИ ПОРОЖНИНАМИ ПІД ДІЄЮ ПРОСТОРОВО РОЗПОДІЛЕНИХ НАВАНТАЖЕНЬ НЕСКІНЧЕНОЇ ДОВЖИНИ

Abstract

The subject of this article is the stress and strain fields in an isotropic elastic layer in contact with rigid supports and weakened by a system of longitudinal cylindrical cavities. The goal of this study is to mathematically model the stress-strain state of an elastic layer with a specified number of cylindrical cavities; to develop a mathematical model and improve an analytical-numerical approach based on the generalized Fourier method for the exact determination of stress fields in an elastic layer with a specified number of cylindrical cavities under the action of infinitely distributed spatial loads. Tasks: to formulate a boundary value problem of elasticity theory for a multiconnected layer considering loads that do not decay at infinity; to construct a general solution of the Lamé equation by the superposition method using the auxiliary problem introduction algorithm; to reduce the boundary value problem to solving a quasi-regular infinite system of linear algebraic equations; to perform a numerical analysis for a layer with three cavities and to evaluate the effect of concentrators on stress distribution. The generalized Fourier method is applied to a layer in the Cartesian coordinate system and cavities in local cylindrical coordinate systems. The scientific novelty lies in the creation of an improved approach that combines the generalized Fourier method with the auxiliary problem algorithm. For the first time, the stress distribution and stress concentration patterns in the vicinity of holes for spatially distributed loads acting on an infinite length have been established. This made it possible to obtain the following results: correctly take into account loads that do not have a periodic structure and do not attenuate at infinity; for an aluminum layer with three cavities, accurate pictures of the distribution of stress tensor components were obtained and zones of their maximum concentration were determined; analysis of the stress state showed that ignoring the mutual influence of cutouts leads to an underestimation of the calculated stresses by more than 20%. Conclusions: The study's results explain the complex interactions among general layer bending, local deformations, and stress-field interference from neighboring concentrators. The significant asymmetry of the stress state confirms the presence of bending moments, and the rapid attenuation of disturbances is consistent with Saint-Venant’s principle. The practical value of the work lies in the potential to use the obtained data to design critical components of aviation technology (integral wing panels, floor power profiles) to increase reliability and optimize weight