This paper presents a Finite Element model for the modeling of the failure of heterogeneous material at the meso-scale. This model is cast into the framework of the Enhanced Finite Element Method (E-FEM). Two kinds of enhancement are performed: (1) in the displace-ment field (strong discontinuity approach) in order to take into account micro-cracks, (2) in the strain field (weak discontinuity) in order to take into account heterogeneities without any mesh adaptation. Mechanical applications (uniaxial tension and compression loading, non-proportional loading) are performed in the context of cementitious materials such as concrete. We show the capability of the model to represent some of the main features of such materials observed at macro-scale.