Initiation of Cracks in Griffith's Theory: An Argument of Continuity in Favor of Global Minimization

The initiation of a crack in a sound body is a real issue in the setting of Griffith's theory of brittle fracture. If one uses the concept of critical energy release rate (Griffith's criterion), it is in general impossible to initiate a crack. On the other hand, if we replace it by a least energy principle (Francfort-Marigo's criterion), it becomes possible to predict the onset of cracking in any circumstance. However this latter criterion can appear too strong. We propose here to reinforce its interest by an argument of continuity. Specifically, we consider the issue of the initiation of a crack at a notch whose angle omega is considered as a parameter. The result predicted by the Griffith criterion is not continuous with respect to omega, since no initiation occurs when omega > 0 while a crack initiates when omega=0. In contrast, the Francfort-Marigo's criterion delivers a response which is continuous with respect to omega, even though the onset of cracking is necessarily brutal when omega > 0. The theoretical analysis is illustrated by numerical computations.