Numerical simulation of strong discontinuities by using standard stress-strain constitutive equations including strain-softening is addressed. The concept of strong discontinuity analysis is introduced and driven, as a matter of example, into a standard continuum damage model. Then, the relevant features that make the constitutive equation compatible with the appearance of strong discontinuities are extracted. Those features are used in the design of a specific finite element approach to the strong discontinuity problem which is placed in the framework of the assumed enhanced strain methods. Numerical simulations show that mesh size and mesh alignment dependencies, typical of some continuum approaches, can be removed.