The present work deals with the use of asymptotic numerical methods (ANM) to manage crack onset and crack growth in the framework of Continuum Damage Mechanics (CDM). More specifically, an application of regularization techniques to a 1D cohesive model is proposed. The standard "triangle" damageable elastic model, often used in finite element codes to describe fracture of brittle materials, was chosen. Results associated with load-unload cycle showed that ANM is convenient to take numerically this specific non regular behaviour into account. Moreover, the present paper also shows that the chosen damageable interface model can be introduced in the generalized standard material formalism which unables us to define a complete energy balance associated with the damage process. In such a framework, the new damage state variable is a displacement. Finally, a 1D finite element application to a simple elastic damageable structure is shown to emphasize the potentialities of such an approach.