A particular implementation of the hybrid boundary element method is presented for the two dimensional analysis of potential and elasticity problems, which, although general in concept, is suited for fracture mechanics applications. The formulation requires integrations only along the boundary and uses fundamental solutions to interpolate fields in the domain. Generalized Westergaard stress functions, as proposed by Tada et al in 1993, are used as the problem s fundamental solutions. The proposed formulation leads to displacement-based concepts that resemble those presented by Crouch and Starfield, although in a variational framework that leads to matrix equations with sound mechanical meanings. Problems of general topology, such as in the case of unbounded and multiply-connected domains, may be modeled. The formulation, which is directly applicable to notches and generally curved, internal or external cracks, is especially suited for the description of the stress field in the vicinity of crack tips and is an easy means of evaluating stress intensity factors. The plastic phenomenon is taken into account around the crack tip through an iterative process. This thesis focuses on the mathematical fundamentals of the formulation of potential and elasticity problems. Several validating numerical examples are presented.