Criticality in models for fracture in disordered media

It has been recently noticed that heterogeneous media undergoing a fracturing process display a set of properties characteristic of systems at the critical state. In the present work we focus on the way in which the critical regime is reached. It is possible to define a branching ratio, for the breaking processes in the material, that represents the probability to trigger future breakdowns given an initial failure. This probability takes the value 1 when the system is critical thereby representing a measure of the distance of the system from the critical state. We show that, although the models considered in literature become really critical only in correspondence of the global failure, different dynamical rules may drive the system close to the critical state at different rates, such that the duration of the `quasi-critical' stage largely varies from model to model.