Quasi-spherical coordinate systems for three-dimensional stress singularity eigenproblems

To avoid the inherent singularity of conventional spherical coordinates at their poles, quasi-spherical coordinate systems are developed. Using these systems, a finite element procedure is developed to determine the displacement eigensolutions at three-dimensional vertices in which the displacement and stress are proportional to the (λ+1)th and λth powers of the distance from the vertices, respectively. The resulting global equation is a second-order characteristic matrix equation. Several demonstrating problems are investigated. Satisfactory results are obtained. Unlike the previous attempts by singular transformation technique, the present predictions are insensitive to the numerical integration order.