Avalanche statistics at the yielding transition of amorphous solids: universality in elastoplastic models

Ezequiel Ferrero

In the last years, several groups have described the yielding phenomenon in the deformation of amorphous materials from a statistical physics point of view. To that end, coarse-grained approaches to amorphous solids were introduced,
the so-called elasto-plastic models (EPM) [1].

In this talk, I will focus on the statistics of avalanches produced by the characteristic stick-slip behavior close to the yielding transition, enquiring into its common properties among different EPM proposals.

More in depth:

I will present in particular the less studied case of EPMs with stress-dependent transition rates for local yielding [2], which help us to see how "dynamical" exponents -those related to the driving speed- may depend on the model details while universality stands more robust for ``static'' critical exponents. On the way, the current understanding of yielding from mean-field descriptions and comparison with the depinning transition of a driven elastic line in random media, will be briefly discussed [3].

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 [2]: 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 $P(0)\sim L^{-d\phi}$ controlling the extremal statistics [4]. 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 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 [2,3,4].


[1] A. Nicolas, E.E. Ferrero, K. Martens, J.-L. Barrat
Deformation and flow of amorphous solids: a review of mesoscale elastoplastic models,
Rev. Mod. Phys. 90 (2018), 045006.

[2] Ezequiel E Ferrero and Eduardo A Jagla,
Criticality in elastoplastic models of amorphous solids with stress-dependent yielding rates,
Soft Matter 15 (2019), 9041--9055.

[3] Ezequiel E Ferrero and Eduardo A Jagla,
Elastic Interfaces on Disordered Substrates: From Mean-Field Depinning to Yielding,
Phys. Rev. Lett. 123 (2019), 218002.

[4] Ezequiel E Ferrero and Eduardo A Jagla,
Properties of the density of shear transformations in driven amorphous solids,
arXiv : 2009.08519 (2020)