Bridging the Gap Between Avalanche Relaxation and Yielding Rheology

TL;DR

The paper addresses how avalanche relaxation dynamics relate to yielding rheology in amorphous materials under passive and active driving. It introduces the Controlled Relaxation Time Model (CRTM), built on an athermal quasistatic framework, to tune the relaxation time $t_r$ and access avalanche behavior across quasistatic and dynamic regimes, with avalanches characterized by $T \sim l^z$ and analyzed through finite-size scaling and spatial correlations. The authors find that passive (SS) driving obeys a scaling relation $\nu/\beta = 1/(\delta+z)$ linking avalanche statistics to rheology, whereas active (SRF) driving exhibits significant deviations, indicating missing ingredients in the yielding description. CRTM provides a unified framework bridging quasistatic and dynamic yielding and clarifies how relaxation dynamics shape flow, offering a flexible approach to study yielding across diverse passive and active amorphous systems.

Abstract

The yielding transition in amorphous materials, whether driven passively (simple shear) or actively, remains a fundamental open question in soft matter physics. While avalanche statistics at the critical point have been extensively studied, the emergence of the dynamic regime at yielding and the steady-state flow properties remain poorly understood. In particular, the significant variability observed in flow curves across different systems lacks a clear explanation. We determine, for the first time, the relationship between avalanche duration and size across the yielding transition, revealing how it evolves from quasistatic to dynamic flow regimes. This precise measurement is made using the Controlled Relaxation Time Model (CRTM), a new simulation framework that treats the relaxation time as a tunable parameter. CRTM reproduces known results in both limits and enables a direct analysis of the change of regime between them. Applying the model to different microscopic dynamics, we find that the existing scaling relation connecting critical exponents under flow holds for passive systems. However, active systems exhibit significant deviations, suggesting a missing ingredient in the current understanding of yielding.

Figures (3)

• Figure 1: Deformation protocols and relaxation framework for yielding dynamics. Schematic representation of the deformation scenarios: (a) Simple shear (SS), where the system is subjected to a velocity profile. (b) Self-random force (SRF), where each particle experiences a force $f$ applied along a fixed random direction $\hat{n}^{\mathrm{rnd}}_i$, which remains constant over time (infinite persistence). As in SS, when $f \!> \!f_c$, the system fails to reach mechanical equilibrium and continuously transitions between non-equilibrium states. (c) Stress–strain response obtained for different imposed strain rates $\dot{\gamma}$. At low $\dot{\gamma}$, the system exhibits intermittent stress drops associated with plastic avalanches. In this limit, close to the quasistatic regime, plastic events are triggered by a deformation increment $\delta\gamma$ and characterized by the resulting stress drop $\delta\sigma$, whereas higher rates lead to smoother flow curves. (d) Spatial map of the particle velocity magnitude $|\vec{\nu_i}|$ during one of these plastic events, showing a localized region of intense motion that propagates through the system, illustrating the avalanche dynamics. (e) Diagram illustrating the residual force factor $\lambda_F$, which quantifies the degree of mechanical equilibration as the ratio between the mean residual $\langle |\vec{F_i}| \rangle$, and contact force magnitudes ${\langle f_{ij} \rangle}$. Panels (f) and (g) display $\sigma$ vs. $\gamma$ curves under CRTM for shear deformation with relaxation times $t_r$ either sufficient or insufficient for the redistribution of stress to restore mechanical equilibrium. (f)$t_r$ sufficient: The athermal quasistatic limit is reached. The system starts in a mechanically equilibrated configuration (A, E). Each step applies an affine deformation, inducing interparticle forces and generating macroscopic shear stress $\sigma$ (B, D). Relaxation restores mechanical equilibrium (C, F), and the process repeats. (g)$t_r$ insufficient: Starting from near mechanical equilibrium (A, D), affine deformation (B, E) is followed by incomplete relaxation, leaving residual elastic forces (C, F) before the next deformation step.
• Figure 2: Crossover from quasistatic to dynamic behavior via residual force statistics. Probability distributions $P(\lambda_F)$ for (a) SRF and (b) SS, shown for different relaxation times $t_r$. Vertical lines indicate, from left to right in each panel, the median $\tilde{\lambda_F}$ and mean $\langle \lambda_F \rangle$ for (a)$t_r = 1.4 \times 10 ^ 4$ and (b)$t_r = 6.7 \times 10 ^ 4$. The final pair of lines corresponds to the case where both values coincide, for (a)$t_r = 1.3 \times 10 ^ 3$ and (b)$t_r = 2.3 \times 10 ^ 4$. At low $t_r$, the distribution displays a single symmetric peak with $\tilde{\lambda_F} \! \approx\! \langle \lambda_F \rangle$. As $t_r$ increases, the distribution broadens and shifts, eventually developing a new dominant peak accompanied by a long tail. In this transitional regime, the median and mean differ ($\tilde{\lambda_F} \!\neq \!\langle \lambda_F \rangle$). Simulations were performed with $N = 4096$ particles.
• Figure 3: Flow curves, spatial correlations, and dynamic scaling across the yielding crossover. Flow curves and spatial correlation functions obtained from CRTM simulations. Panels (a) and (b): Flow curves for SRF and SS, respectively. Red markers correspond to the Steepest Descent relaxation method, which recovers the expected Herschel-Bulkley exponents CD1: $\beta = 1.6$ for SRF and $\beta = 2.4$ for SS. Green markers show results using the Conjugate Gradient method, yielding lower effective exponents: $\beta = 1.0$ for SRF and $\beta = 1.3$ for SS. Black lines represent best-fit power-law behaviors. Simulations were performed with $N = 4096$ particles. Panels (c) and (d): Rescaled spatial correlation function $G_2(x)$ for SRF and SS, respectively, evaluated at various relaxation times $t_r$ using the Steepest Descent method. Insets display the unscaled data. The method yields $\nu / \beta$ values close to the expected ones CD1: $\nu / \beta = 0.15$ for SRF and $\nu / \beta = 0.28$ for SS. All reported $\nu / \beta$ values correspond to the best fit, with values within $\pm 0.03$ remaining within a reasonable range. In panel (d), we include an additional curve with $t_r = 5.4 \times 10^3$, corresponding to a regime already entering the dynamic–quasi-static crossover. This is reflected in its inability to collapse with the other curves. Simulations were performed with $N = 8192$. Panels (e) and (f) show finite-size collapsed $M_{\lambda_F}$ as a function of relaxation time $t_r$ for SRF and SS respectively, using the Steepest Descent relaxation method. The insets display the raw data before rescaling. The extracted dynamic exponents are $z \!=\! 1.9$ for SRF and $z \!=\! 2.3$ for SS. All reported $z$ values correspond to the best fit, with values within $\pm 0.1$ remaining within a reasonable range.