The robust structural design relies on the accuracy of the response analysis to different loadings. Strut-and-tie models and continuum models are often used for the design of masonry structures but they are generally not able to simulate detailed damage and progressive collapse. To this regard, this paper presents a new and robust interfacial constitutive law, which couples the damage and plastic deformation with fracture energy-based softening rules, for the discrete finite element modelling (DFEM) of masonry structures. The model has 14 required parameters and 2 optional parameters of which calibration methods using the standard test data and physical rules are provided in details. It was demonstrated that the comprehensive mechanical behaviour including the pressure-dependent fracture, wearing-off of the friction, dilatation, stiffness degradation, shear retention, and component disintegration can be well captured. The model validation studies were performed with various material and structural tests including shear-compression and uniaxial tension tests on the joints, and horizontal loading and shake-table tests on large-scale masonry structures. Good agreements between the simulated and experimental results could be observed. Comparing to conventional modelling approaches, the DFEM with the interfacial constitutive law can also simply the meshing work and better control the mesh quality.