Elastoplastic description of sudden failure in athermal amorphous materials during quasistatic loading

The response of amorphous materials to an applied strain can be continuous or instead discontinuous if the initial configuration is very stable. We study theoretically how such a stress drop emerges as the system's initial stability is increased. We show that this emergence is well reproduced by elastoplastic models and is predicted by a mean field approximation, where it corresponds to a continuous transition. In the mean field, failure can be forecasted from the avalanche statistics. We show that this is not the case for very stable materials in finite dimensions due to rare weak regions where a shear band nucleates. To understand the nucleation, we build an analogy with fracture mechanics predicting that the critical nucleation radius of a shear band follows a(c) similar to (Sigma - Sigma(b))(-2), where Sigma is the stress and Sigma(b) is the stress that a shear band can carry.