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The numerical prediction of crack paths in brittle or quasi-brittle solids isdifficult from a computational point of view, and non-uniqueness of the solution canoccur. To solve such a problem, several computational techniques have been proposedsuch as ad hoc remeshing strategies, strain softening in the context of plasticity,discontinuous finite elements. In the present paper, a continuum finite element (FE)formulation to model the discontinuity of the displacement field in fracture occurring inbrittle or quasi-brittle solids is proposed. A homogeneous discontinuity is assumed toexist in a cracked finite element, and a new simple stress-based implementation of thedisplacement discontinuity is introduced by an appropriate stress field relaxation inorder to simulate the mechanical effects of the crack. The model requires the definitionof crack-bridging stress laws. Simple 2D fracture problems are solved to investigatesome computational aspects of the proposed algorithm, such as the mesh independence.Finally, the developed numerical model is used to simulate experimental results.