Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 460852, 20 pages
Research Article

The Solution of Mitchell's Problem for the Elastic Infinite Cone with a Spherical Crack

Institute of Mathematics, Economics and Mechanics, Odessa Mechnikov University, Odessa 65044, Ukraine

Received 10 November 2009; Accepted 1 April 2010

Academic Editor: Francesco Pellicano

Copyright © 2010 G. Ya. Popov and N. D. Vaysfel'd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The new problem about the stress concentration around a spherical crack inside of an elastic cone is solved for the point tensile force enclosed to a cone's edge. The constructed discontinuous solutions of the equilibrium equations have allowed to express the displacements and stress in a cone through their jumps and the jumps of their normal derivatives across the crack's surface. The application of the integral transformation method under the generalized scheme has reduced the problem solving to the solving of the integrodifferential equation system with regard to the displacements' jumps. This system was solved approximately by the orthogonal polynomial method. The use of this method has allowed to take into consideration the order of the solution's singularities at the ends of an integral interval. The correlation between the crack's geometrical parameters, its distance from an edge, and the SIF values is established after the numerical analysis. The limit of the proposed method applicability is specified.