  Widespread processes in nature and technology are governed by the dynamical
transition whereby a material in an initially solid-like state then yields
plastically. Major unresolved questions concern whether any material will yield
smoothly and gradually (ductile behaviour) or fail abruptly and
catastrophically (brittle behaviour); the roles of sample annealing, disorder
and shear band formation in the onset of yielding and failure; and, most
importantly from a practical viewpoint, whether any impending catastrophic
failure can be anticipated before it happens. We address these questions by
studying the yielding of slowly sheared athermal amorphous materials, within a
minimal mesoscopic lattice elastoplastic model. Our contributions are fourfold.
First, we elucidate whether yielding will be ductile or brittle, for any given
level of sample annealing. Second, we show that yielding comprises two distinct
stages: a pre-failure stage, in which small levels of strain heterogeneity
slowly accumulate, followed by a catastrophic brittle failure event, in which a
crack quickly propagates across the sample via a cooperating line of plastic
events. Third, we provide an expression for the slowly growing level of strain
heterogeneity in the pre-failure stage, expressed in terms of the macroscopic
stress-strain curve and the sample size, and in excellent agreement with our
simulation results. Fourth, we elucidate the basic mechanism via which a crack
then nucleates and provide an approximate expression for the probability
distribution of shear strains at which failure occurs, as determined by the
disorder inherent in the sample, expressed in terms of a single annealing
parameter, and the system size.
