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The displacement discontinuity technique is employed for the determination of the critical conditions causing propagation of a preexisting finite fracture that lies close to the free boundary of a semi-infinite medium. The technique is applied by using a suitably adapted commercial indirect boundary element code, permitting the determination of the displacement, strain and stress fields in multiply-connected elastic bodies. The critical conditions are then determined by empoying a suitable failure criterion. The results of the numerical analysis are compared with the respective ones obtained from a series of experiments with pre-cracked specimens simulating the semi-infinite medium. The agreement is proved to be satisfactory.