Solution of the First Fundamental Problem of Elasticity Theory for a Space with a Paraboloidal Notch and a Spherical Cavity

Abstract

The paper is devoted to the development of an analytical method for solving spatial problems of elasticity theory in domains with complex geometric shapes. The application of curvilinear coordinate systems – paraboloidal and spherical – to the investigation of the stress-strain state of elastic bodies is considered. The relevance of this research is determined by the necessity of mathematical modeling of the behavior of structures containing stress concentrators in the form of paraboloidal notches and spherical cavities, which are frequently encountered in engineering practice. Basic solutions of the Lamé vector equation describing the equilibrium of an isotropic elastic body are obtained in paraboloidal and spherical coordinates. Based on the proposed addition theorems, the relations between the basic solutions of the Lamé equation in paraboloidal and spherical coordinate systems are established. These relations allow for the transition from one representation to another when specifying boundary conditions on surfaces of various geometric shapes. The method proposed in the paper is applied to solve the first fundamental boundary value problem for an elastic space with a paraboloidal notch and a spherical cavity. The case where the paraboloidal surface is free from load and hydrostatic pressure acts on the spherical cavity is considered. The solution of the boundary value problem reduces to an infinite system of linear algebraic equations with respect to the unknown expansion coefficients. It is shown that the matrix coefficients of this system possess the property of exponential decay, which significantly simplifies the numerical implementation of the algorithm. A rigorous mathematical investigation of the properties of the resulting infinite system of equations is carried out. The complete continuity of the system operator in the Hilbert space of numerical sequences is proved under certain restrictions on the geometric parameters of the problem. The conditions for quasi-regularity and complete regularity of the infinite system are established, which proves the existence and uniqueness of the solution, as well as the possibility of obtaining it by the reduction method with any preassigned accuracy. The obtained results can be used for calculating the stress-strain state of structural elements of complex shape and are of interest to specialists in the fields of mechanics of deformable solids, applied mathematics, and computational methods.