Modelling large crack propagation: from gradient damage to cohesive zone models

Continuum Damage Mechanics and cohesive zone models are both prone to model large crack propagation inside quasi-brittle materials. Comparing the advantages of the formulations, the latter could be advantageously applied in cases where the crack path is known a priori while the former implicitly encompasses crack path prediction but requires more complex computations involving nonlocal interactions. In order to assess the acceptability of such a hierarchy for industrial studies, it is necessary that the predictions coincide quantitatively. Starting from a gradient damage model in a one-dimensional context, a cohesive law is derived as the asymptotic response of the damage model for vanishing nonlocal length scale. The cohesive law is hence independent of the nonlocal length scale, which is consistent with the fact that it ignores details characteristic of the best-estimate damage approach. Besides, the existence of such a limit ensures that the damage model is not much sensitive to a small nonlocal length scale, which then appears rather as a numerical regularisation parameter. A numerical comparison between the damage model and its asymptotic cohesive law is then carried out for large bi-dimensional crack propagation. The computed responses remain close to each other although some small discrepancies arise probably related to the damage spread resulting from the stress distribution in the vicinity of the crack tip: the hierarchy strategy is thus validated.