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

E. E. Ferrero; E. A. Jagla

doi:10.1039/c9sm01073d

paper

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

arXiv:1905.05610 · doi:10.1039/c9sm01073d

Abstract

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 $P (S) \sim S^{- τ_{S}} f (S / L^{d_{f}})$ and the exponents describing the density of sites at the verge of yielding, which we find to be of the form $P (x) ≃ P (0) + x^{θ}$ 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 $β$ ( $γ ˙ \sim (σ - σ_{c})^{β}$ ) enters in this category, and is seen to differ in $\frac{1}{2}$ between the two yielding rate cases. We give analytical support to this numerical observation by calculating the exponent variation in the Hébraud-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.

17 pages, 11 figures, 2 tables

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

Abstract

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