The paper focuses on the Continuum Strong Discontinuity Approach (CSDA) to fracture mechanics, and the traction-separation cohesive laws induced from continuum dissipative models as their projections onto the failure interface. They are compared with the cohesive laws commonly used for the fracture simulation in quasi-brittle materials, typically concrete. Emphasis is placed in the analysis of the mechanical stress-strain states induced by the CSDA into the fracture process zone: first when the damage mechanism is initiated and, after, when the cohesive model determines the crack response. The influence of the material parameters, particularly the fracture energy and the initial continuum softening modulus, in the obtained phenomenological responses is also analyzed. Representative numerical solutions of fracture problems are finally presented.