The paper focuses the numerical modelling of material failure in a strong discontinuity setting using a continuum format. Displacement discontinuities, like fractures, cracks, slip lines, etc., are modelled in a strong discontinuity approach, enriched by a transition from weak to strong discontinuities to get an appropriate representation of the fracture process zone. The introduction of the strong discontinuity kinematics automatically projects any standard dissipative constitutive model, equipped with strain softening, into a discrete traction–separation law that is fulfilled at the discontinuity interface. Numerical issues like a global discontinuity tracking algorithm via a heat conduction-like problem are also presented. Some representative numerical simulations illustrate the performance of the presented approach.