A regularized damage model is considered named "Graded damage" in which the gradient enhancement has the form of an explicit bound for the spatial gradient of damage. The key features of the proposed approach are demonstrated by computing the analytical solution of two problems that are one-parameter dependent. The first one is the classical one-dimensional damageable rod under tensile load, for which the hardening function is determined based on the equivalence with a given cohesive relationship. The second application is a mode-I delamination problem for which the cohesive law for the interface is formulated starting from the graded damage concept, i.e. by prescribing the shape of damage distribution within the cohesive process zone.