Dislocation and point-force-based approach to the special Greenʹs Function BEM for elliptic hole and crack problems in two dimensions

In this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special GreenÕs function boundary element method and formulate, as an example, the special GreenÕs function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of SomiglianaÕs identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the inÞnite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisÞed. The GreenÕs functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special GreenÕs function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate.