Eigenfunction expansion for penny-shaped and circumferential cracks

Abstract

Cracks that are developed in pipes, pressure vessels and circular bars due to improper welding are often idealized to be circumferential, penny-shaped or flat toroidal. Based on the three-dimensional Navier's equations in cylindrical coordinates, we derive the general displacement functions for both the penny-shaped and circumferential cracks by eigenfunction expansion method. We find that only those terms of the first order displacement and stress are similar to the case of three-dimensional straight plane cracks and that the higher order terms are coupled through the equilibrium equations. The strength of coupling depends on the ratio r̄ = r/a where r is the radius normal to the crack front and 2a is the diameter of the circular crack tip. It decreases with r̄.