Study of the stress–strain state of an eccentric circular ring

This paper investigates analytical solutions to certain two‐dimensional boundary value problems in the theory of classical linear elasticity, formulated in bipolar coordinates for homogeneous isotropic bodies bounded by coordinate curves of the bipolar coordinate system. Specifically, boundary value problems are considered for an eccentric circular ring and an eccentric circular semi‐ring. The equilibrium equations and Hooke's law are expressed in bipolar coordinates, and the plane elasticity problems are solved in terms of stress functions. The main formulas are derived using the stress function expressed in bipolar coordinates, and the method of separation of variables is applied. Boundary conditions on the diametral edge of the eccentric circular semi‐ring, involving symmetry or antisymmetry, allow the extension of the solution across the horizontal axis, thereby yielding solutions for the full eccentric circular ring. Analytical solutions are obtained, numerical results for some boundary value problems are presented, and the corresponding graphs are provided.