Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates

We analyze the behavior of different elastoplastic models approaching the yielding transition. We propose two kind of rules for the local yielding events: yielding occurs above the local threshold either at a constant rate or with a rate that increases as the square root of the stress excess. We establish a family of "static" universal critical exponents which do not depend on this dynamic detail of the model rules: in particular, the exponents for the avalanche size distribution $P(S)\sim S^{-\tau_S}f(S/L^{d_f})$ and the exponents describing the density of sites at the verge of yielding, which we find to be of the form $P(x)\simeq P(0) + x^\theta$ with $P(0)\sim L^{-a}$ controlling the extremal statistics. On the other hand, we discuss "dynamical" exponents that are sensitive to the local yielding rule details. We find that, apart form the dynamical exponent $z$ controlling the duration of avalanches, also the flowcurve's (inverse) Herschel-Bulkley exponent $\beta$ ($\dot\gamma\sim(\sigma-\sigma_c)^\beta$) enters in this category, and is seen to differ in $\frac12$ between the two yielding rate cases. We give analytical support to this numerical observation by calculating the exponent variation in the H\'ebraud-Lequeux model and finding an identical shift. We further discuss an alternative mean-field approximation to yielding only based on the so-called Hurst exponent of the accumulated mechanical noise signal, which gives good predictions for the exponents extracted from simulations of fully spatial models.