CLASSICAL PROBLEM ABOUT AN ELASTIC SPHERE WITH A SPHERICAL INCLUSION
A. G. Nikolaev, Mariia Viktorivna Skitska · Bulletin of the National Technical University KhPI Series Mathematical modeling in engineering and technologies · 2025
For the first time, an exact analytically justified solution of the second axisymmetric boundary value problem of the theory of elasticity in the general formulation for a sphere with a concentric spherical inclusion has been obtained using the conventional Fourier method. In the scientific works of the classics of natural science of the 19th and 20th centuries, M. G. Lame, W. Thomson, C. Somigliana, V. Cerruti, B. G. Galerkin, G. Fichera, A. I. Lurie, E. Strenberg, and A. F. Ulitko, elastic problems for a solid sphere, space with a spherical cavity, and a sphere with a concentric spherical cavity were solved in various formulations. But even these problems were not strictly justified. The problem considered in this report is much more complex, since it is associated with the conjugation of displacement and stress fields at the inclusion boundary. That's probably why it wasn't considered before. The justification for solving such a problem and establishing its solvability class using the usual Fourier method is based on the analysis of a solvable alge- braic system of the sixth order with coefficients that depend on five independent continuous parameters and one discrete one. The general solution of the problem is given in the form of series in terms of axisymmetric vector basis solutions of the Lamé equation for a sphere, constructed by the authors in one of the previous articles. After transitioning to stresses and satisfying the boundary conditions, a resolving system of the above form is obtained. When analyzing the system, a lower estimate of the modulus of its determinant was first found, from which follows not only the unique solvability of the system, but also estimates of the solutions of the system itself. In estimating the determinant, a new classical inequality was proven for one con- tinuous and one discrete parameter, previously unknown to the authors. The next step was to prove a theorem about the conditions that must be im- posed on the vector of the external load applied to the surface of the sphere, which ensure the existence of a solution to the problem in a certain class of functions. In the numerical implementation of the solution to the problem, two types of loads on the outer surface of the sphere were considered, which satisfy the equilibrium condition. A computer experiment was conducted with three types of materials for a ball and an inclusion: steel, brass and alu- minum. Graphs of normal and tangential stresses on the surface of the inclusion were obtained, and their parametric analysis was performed depending on the geometric and mechanical parameters. The practical convergence of the method was investigated.