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

Abstract

The paper develops an analytical method for solving elasticity problems in bodies with complex geometries using paraboloidal and spherical coordinates. Addition theorems relate basic solutions of the Lamé equation, enabling the treatment of various boundary surfaces. The method is applied to a body with a paraboloidal notch and a spherical cavity, leading to an infinite system of linear equations with rapidly decaying coefficients. The results ensure existence, uniqueness, and numerical solvability, and are relevant for calculating stress-strain states in complex structural elements.