We construct a mean-field elastoplastic description of the dynamics of amorphous solids under arbitrary time-dependent perturbations, building on the work of Lin and Wyart [Phys. Rev. X 6, 011005 (2016)] for steady shear. Local stresses are driven by power-law distributed mechanical noise from yield events throughout the material, in contrast to the well-studied Hébraud–Lequeux model where the noise is Gaussian. We first use a mapping to a mean first passage time problem to study the phase diagram in the absence of shear, which shows a transition between an arrested and a fluid state. We then introduce a boundary layer scaling technique for low yield rate regimes, which we first apply to study the scaling of the steady state yield rate on approaching the arrest transition. These scalings are further developed to study the aging behavior in the glassy regime for different values of the exponent μ characterizing the mechanical noise spectrum. We find that the yield rate decays as a power-law for 1 < μ < 2, a stretched exponential for μ = 1, and an exponential for μ < 1, reflecting the relative importance of far-field and near-field events as the range of the stress propagator is varied. A comparison of the mean-field predictions with aging simulations of a lattice elastoplastic model shows excellent quantitative agreement, up to a simple rescaling of time.
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We consider here a full local stress relaxation. The post-yield local stress may however also be modeled as drawn from a distribution of residual stress other than a Dirac function at zero. Previous studies have shown this not to change the behaviour qualitatively.38
This will change quantitative aspects such as the location of the phase diagram; however, the scaling forms below are determined by the asymptotic power-law regime for small |δσ|, making the precise form of the upper cutoff irrelevant.
The exponent μ/2 corresponds to the pseudogap exponent in Ref. 7.
Although we have not studied in detail the aging of asymmetric distributions in the current model, for the HL model,15 the same asymptotic decay and scalings are shown to hold also for the asymmetric case under generic assumptions.
For lattices of different sizes, the large δσ details of the list of propagator elements are identical. In order to run Gillespie simulations with an arbitrary number of sites, we proceed by extrapolating the small δσ power law in order to obtain the desired number of propagator elements.
