Short-range depinning in the presence of velocity-weakening

TL;DR

The paper investigates short-range depinning under velocity-weakening, showing that nucleation is triggered by avalanches governed by a critical point at $f_c$ slightly above the flow curve minimum $f_{\min}$. Using a one-dimensional inertial elastic line with quenched disorder and a weak driving spring, the authors connect nucleation length $\ell_c(f)\sim (f-f_c)^{-\nu}$ to avalanche statistics and a pseudo-gap in local yield thresholds, yielding bimodal event-size distributions and large finite-size hysteresis. The work combines rate-and-state concepts with perturbative disorder to predict scaling relations for $\nu$, $\zeta$, and the hysteresis amplitude, and confirms these via numerical simulations with short-range elasticity and quasi-static loading. The findings reveal a dual nature: a first-order-like flowing transition coexisting with a continuous critical point controlling nucleation, leading to strong finite-size effects and armouring, with potential applicability to earthquakes and other inertia-dominated frictional systems. This framework provides a quantitative link between microscopic yield statistics, avalanche dynamics, and macroscopic hysteresis in velocity-weakening regimes.

Abstract

Phenomena including friction and earthquakes are complicated by the joint presence of disorder and non-linear instabilites, such as those triggered by the presence of velocity weakening. In [de Geus and Wyart, Phys. Rev. E 106, 065001 (2022)], we provided a theory for the nucleation of flow and the magnitude of hysteresis, building on recent results on disorder-free systems described by so called rate-and-state descriptions of the frictional interface, and treating disorder perturbatively. This theory was tested for models of frictional interfaces, where long range elastic interactions are present. Here we test it for short-range depinning, and confirm that (i) nucleation is triggered by avalanches, governed by a critical point at some threshold force $f_c$ close to the minimum of the flow curve and that (ii) due to an armouring mechanism by which the elastic manifold displays very little plasticity after a big slip event, very slowly decaying finite size effects dominate the hysteresis magnitude, with an exponent we can relate to other observables.

Figures (16)

• Figure 1: [Sketch] \ref{['fig:nonmonotic:a']} When the flow curve (force $f$vs velocity $v$) is non-monotonic, with a minimum at $f = f_{\min}$, there exists a threshold force $f_c$ beyond which the static phase is unstable to system-spanning events. \ref{['fig:nonmonotic:b']} A finite hysteresis is predicted in the thermodynamic limit, such that a system driven quasi-statically through a weak spring displays stick-slip (by "a weak spring" we mean that $k_f$ of Eq. \ref{['eq:motion']} is small: $k_f \leq \mathcal{O}(1 / L^2$) with $L$ the system size, see text in Sec. \ref{['sec:model']}). Thereby, power law distributed avalanches protect the interface from building up a load $f > f_c$ where it is unstable. After the instability, the interface unloads to $f = f_{\min}$ while slipping (see corresponding cycle in panel \ref{['fig:nonmonotic:a']} with the same colour coding).
• Figure 2: [Sketch] (\ref{['fig:pre:flow']}-\ref{['fig:pre:stick-slip']}) In a quasi-statically loaded finite system with disorder, avalanches occur as the force increases. They nucleate slip once their extension $\ell > \ell_c \sim (f - f_c)^{-\nu}$ if $f > f_c$. The force at which nucleation occurs fluctuates, we denote by $f_s$ its means (and study below how the magnitude of hysteresis $\Delta f=f_s-f_c$ depends on $L$). \ref{['fig:pre:bimodal']} The corresponding distribution of event extensions, $P(\ell)$, during a quasi-static cycle is bimodal, with avalanches up to a scale $\ell_{\max}(f_s)$, and system-spanning events ($\ell = L$). See \ref{['sec:criticality']} for numerical evidence of bimodality. \ref{['fig:pre:ellc']} To study the properties of avalanches in an infinite system, and to quantify $\ell_c$, we trigger avalanches at different forces $f$. The distribution of their linear extension $\ell$ is scale free at $f_c$ while at $f > f_c$ avalanches transition to system-spanning events if $\ell > \ell_c$ (system-spanning events are excluded from the shown distributions).
• Figure 3: Schematic of our model with equation of motion in \ref{['eq:motion']} in dimension $d = 1$ for $L = 5$ particles. The potential energy landscape $U_i$ of each particle $i$ is a sequence of parabolic wells with uniform curvature $\mu = 1$ and random widths (that are quenched). In this example, the landscape of each particle is drawn in a different colour. The particle positions $u_i$ are indicated by the black dots. Their elastic interactions are schematically illustrated by black lines (with dashed lines indicating periodicity). The driving frame is indicated by a green line. It is connected to each particle by a weak spring of stiffness $k_f$ illustrated using a dotted green line.
• Figure 4: Measured flow curve, showing that the force $f$ at the driving frame is a non-monotonic function of the velocity $v$. The measurement of the unstable branch (where $\partial_v f < 0$) is possible by driving with a stiff spring. Also indicated: the critical force, $f_c$, defined the lowest force at which we find system-spanning events.
• Figure 5: Position of the interface $u$ before (black) and after (green) an avalanche triggered at $f \approx f_c$. The total number of fails, $S$, is a proxy for the total displacement of the interface (in gray). The number of particles that fail at least once, $\ell$ (in $d = 1$), is a proxy for the linear extension of the avalanche (annotated).