We obtain a cohesive fracture model as a Γ-limit of scalar damage models in which the elastic coefficient is computed from the damage variable v through a function f_k of the form f_k (v) = min{1, ε_k^{1/2} f (v)}, with f diverging for v close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening s at small values of s and has a finite limit as s → ∞. If the function f is allowed to depend on the index k, for specific choices we recover in the limit Dugdale’s and Griffith’s fracture models, and models with surface energy density having a power-law growth at small openings.