Properties of the density of shear transformations in driven amorphous solids

TL;DR

This paper investigates how the average strain load between avalanches in amorphous solids depends on system size, revealing that the distribution of local instabilities tends to a finite limit at small values, challenging previous assumptions.

Contribution

The study provides a theoretical and simulation-based analysis showing that the distribution of local stability thresholds in amorphous solids saturates at small values, affecting size-dependent yielding behavior.

Findings

Distribution P(x) tends to a finite limit as x approaches zero.

Finite-size effects influence the average strain load and related scalings.

The assumed power-law form P(x) ~ x^θ is not accurate at low x.

Abstract

The strain load $\Delta\gamma$ that triggers consecutive avalanches is a key observable in the slow deformation of amorphous solids. Its temporally averaged value $\langle \Delta\gamma \rangle$ displays a non-trivial system-size dependence that constitutes one of the distinguishing features of the yielding transition. Details of this dependence are not yet fully understood. We address this problem by means of theoretical analysis and simulations of elastoplastic models for amorphous solids. An accurate determination of the size dependence of $\langle \Delta\gamma \rangle$ leads to a precise evaluation of the steady-state distribution of local distances to instability $x$. We find that the usually assumed form $P(x)\sim x^\theta$ (with $\theta$ being the so-called pseudo-gap exponent) is not accurate at low $x$ and that in general $P(x)$ tends to a system-size-dependent \textit{finite} limit as $x\to 0$. We work out the consequences of this finite-size dependence standing on exact results for random-walks and disclosing an alternative interpretation of the mechanical noise felt by a reference site. We test our predictions in two- and three-dimensional elastoplastic models, showing the crucial influence of the saturation of $P(x)$ at small $x$ on the size dependence of $\langle \Delta\gamma \rangle$ and related scalings.