Yielding, shear banding, and brittle failure of amorphous materials

Widespread processes in nature and technology are governed by the dynamical transition whereby a material in an initially solid-like state, whether soft or hard, then yields. Major unresolved questions concern whether any material will yield smoothly and gradually (``ductile'' behavior) or fail abruptly and catastrophically (``brittle'' behavior); 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 predicted before it happens. We address these questions by studying theoretically the yielding of slowly sheared athermal amorphous materials, within a minimal mesoscopic lattice elastoplastic description. Our contributions are fourfold. First, we elucidate whether yielding will be ductile or brittle, for any given level of sample annealing prior to shear. For highly annealed samples, we find brittle yielding for all samples sizes. For poorly annealed samples we uncover an important dependence on the size of the sample of material being sheared, with ductile yielding for small samples, and brittle yielding only for large system sizes. Second, we show that yielding comprises two distinct stages: a prefailure stage, in which small levels of strain heterogeneity slowly accumulate within the material, followed by a catastrophic brittle failure event, in which a shear band quickly propagates across the sample via a cooperating line of (individually) localized plastic events. Third, we provide an exact expression for the slowly growing level of strain heterogeneity in the prefailure stage, expressed in terms of the macroscopically measured stress-strain curve and the sample size, and in excellent agreement with our simulation results. Fourth, we elucidate the basic mechanism via which a shear band then nucleates, in terms of the onset of cooperativity between plastic events. We furthermore provide an expression for the probability distribution of shear strains at which failure occurs, expressed in terms of the sample size and the disorder inherent in the sample, as determined by the degree of annealing prior to shear.