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Developing a unified theory describing both ductile and brittle yieldingconstitutes a fundamental challenge of non-equilibrium statistical physics.Recently, it has been proposed that the nature of the yielding transition iscontrolled by physics akin to that of the quasistatically driven Random fieldIsing model (RFIM), which has served as the paradigm for understanding theeffect of quenched disorder in slowly driven systems with short-rangedinteractions. However, this theoretical picture neglects both the dynamics of,and the elasticity-induced long-ranged interactions between, the mesoscopicmaterial constituents. Here, we address these two aspects and provide a unifiedtheory building on the H\'ebraud-Lequeux elastoplastic description. The firstaspect is crucial to understanding the competition between the imposeddeformation rate and the finite timescale of plastic rearrangements: we providea dynamical description of the macroscopic stress drop and predictions for thedivergence of the peak susceptibility with inverse shear rate. The second isessential in order to capture properly the behaviour in the limit ofquasistatic driving, where avalanches of plasticity diverge with system size atany value of the strain. We fully characterise the avalanche behaviour, whichis radically different to that of the RFIM. In the quasistatic, infinite sizelimit, we find that both models have mean-field Landau exponents, obscuring theeffect of the interactions. We show, however, that the latter profoundly affectthe behaviour of finite systems approaching the spinodal-like brittle yieldpoint, where we recover qualitatively the finite-size trends found in particlesimulations, and modify the nature of the random critical point separatingductile and brittle yielding, where we predict critical behaviour on top of themarginality present at any value of the strain.

Towards a complete mean-field theory for the ductile and brittle  yielding of amorphous solids: beyond the paradigm of the Ising model in a  random field