Ductile and brittle yielding of athermal amorphous solids: a mean-field paradigm beyond the random field Ising model

Abstract

Developing a unified theory describing both ductile and brittle yielding constitutes a fundamental challenge of non-equilibrium statistical physics. Recently, it has been proposed that the nature of the yielding transition is controlled by physics akin to that of the quasistatically driven Random field Ising model (RFIM), which has served as the paradigm for understanding the effect of quenched disorder in slowly driven systems with short-ranged interactions. However, this theoretical picture neglects both the dynamics of, and the elasticity-induced long-ranged interactions between, the mesoscopic material constituents. Here, we address these two aspects and provide a unified theory building on the H\'ebraud-Lequeux elastoplastic description. The first aspect is crucial to understanding the competition between the imposed deformation rate and the finite timescale of plastic rearrangements: we provide a dynamical description of the macroscopic stress drop and predictions for the divergence of the peak susceptibility with inverse shear rate. The second is essential in order to capture properly the behaviour in the limit of quasistatic driving, where avalanches of plasticity diverge with system size at any value of the strain. We fully characterise the avalanche behaviour, which is radically different to that of the RFIM. In the quasistatic, infinite size limit, we find that both models have mean-field Landau exponents, obscuring the effect of the interactions. We show, however, that the latter profoundly affect the behaviour of finite systems approaching the spinodal-like brittle yield point, where we recover qualitatively the finite-size trends found in particle simulations, and modify the nature of the random critical point separating ductile and brittle yielding, where we predict critical behaviour on top of the marginality present at any value of the strain.