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. 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 specially suited for the description of the stress field in the vicinity of crack tips and is an easy means of evaluating stress intensity factors and of checking some basic concepts laid down by Rice in 1968. This dissertation focuses on the mathematical fundamentals of the formulation. Several validating numerical examples are presented.