  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.
