Session A43: Avalanches in Granular and Other Particle-based Materials I

Critical Dynamics Near the Erosion Onset

Erosion shapes the Earth's landscape. Experiments reveal that there is an erosion threshold of a granular bed sheared by a viscous fluid. The granular particles start to flow when the shearing force is above the threshold $\theta_c$. Near $\theta_c$, the particle flux grows linearly $J\sim\theta-\theta_c$. The stationary state is reached after a transient time $t_{\rm conv}$, which diverges as $t_{\rm conv}\sim|\theta-\theta_c|^{-z}$. We theoretically study this dynamical transition by introducing a model capturing both the drainage effect of the disordered landscape and the interactions among the granular particles. Based on the model, we make the first time quantitative testable predictions for the drainage pattern -- how the granular flux is spatially distributed and correlated. Our model enables us to rationalize the critical dynamics of erosion, which may also apply to the plastic depinning transition of vortex lattices in dirty superconductors.   

Critical scaling with strain rate in overdamped sheared disordered solids

In the limit of quasistatic shear, disordered solids demonstrate non-equilibrium critical behavior including power-law distributions of avalanches \footnote{K. M. Salerno and M. O. Robbins, Phys. Rev. E \bf{77}, 062206 (2013)}. Using molecular dynamics simulations of 2D and 3D overdamped binary LJ glasses, we explore the critical behavior in the limit of finite strain rate. We use finite-size scaling to find the critical exponents characterizing shear stress, kinetic energy, and measures of temporal and spatial correlations. The shear stress of the system rises as a power $\beta$ of the strain rate. Larger system size extends this power law to lower rates. This behavior is governed by a power law drop of the dynamic correlation length with increasing shear stress defined by the exponent $\nu$. This finite-size effect also impacts the scaling of the RMS kinetic energy with strain rate as avalanches begin nucleating simultaneously leading to continuous deformation of the solid. As system size increases, avalanches begin overlapping at lower rates. The correlation function of non-affine displacement exhibits novel anisotropic power law scaling with the magnitude of the wave vector. Its strain rate dependence is used to determine the scaling of the dynamic correlation length.   

Scaling theory of the process zone of quasibrittle materials: an avalanche crossover analysis

We present progress towards a natural theory of the process zone surrounding cracks in quasibrittle materials using renormalization group methods. Quasibrittle or disordered brittle materials like concrete evade usual fracture analysis because of strong finite-size effects and a large disordered process zone. Unlike metals, where the process zone is relatively small and dominated by plasticity, microcracking relieves stress around the tip of quasibrittle cracks, a process that is not well understood. Recently, a scaling crossover theory was developed by Sethna and Shekhawat to explain the influence of finite size on the fracture mechanism and avalanche precursors. We extend this theory to model the scaling of stress and distribution of microcracking in the process zone.   

Avalanches and local force evolution in a granular stick-slip experiment

We carry out experiments to characterize stick-slip for granular materials. In our experiment, a constant speed stage pulls a slider which rests on a vertical bed of circular photoelastic particles in a 2D system. The stage is connected to the slider by a spring. We measure the force on the spring by a force sensor attached to the spring. The distributions of energy release and time duration of avalanches during slip obey power laws. We analyze the power spectrum of the force signal to understand the effect of the loading speed and of the spring stiffness on the statistical behavior of the system. From a more local point of view and by using a high speed camera and the photoelastic properties of our particles, we characterize the internal granular structure during avalanches. By image processing and analyzing the skeleton of force network inside the media, we try to understand the flow of particles and evolution of force chains inside the media and during avalanches.   

Avalanche-like fluidization of a non-Brownian particle gel

We report on the fluidization dynamics of an attractive gel composed of non-Brownian particles made of fused silica colloids. Extensive rheology coupled to ultrasonic velocimetry allows us to characterize the global stress response together with the local dynamics of the gel during shear startup experiments. In practice, after being rejuvenated by a preshear, the gel is left to age during a time $t_w$ before being submitted to a constant shear rate $\dot \gamma$. We investigate in detail the effects of both $t_w$ and $\dot \gamma$ on the fluidization dynamics and build a detailed state diagram of the gel response to shear startup flows. The gel may either display transient shear banding towards complete fluidization, or steady-state shear banding. In the former case, we unravel that the progressive fluidization occurs by successive steps that appear as peaks on the global stress relaxation signal. Flow imaging reveals that the shear band grows up to complete fluidization of the material by sudden avalanche-like events which are distributed heterogeneously along the vorticity direction and correlated to large peaks in the slip velocity at the moving wall.   

Spatiotemporal stick-slip phenomena in a coupled continuum-granular system

In sheared granular media, stick-slip behavior is ubiquitous, especially at very small shear rates and weak drive coupling. The resulting slips are characteristic of natural phenomena such as earthquakes and well as being a delicate probe of the collective dynamics of the granular system. In that spirit, we developed a laboratory experiment consisting of sheared elastic plates separated by a narrow gap filled with quasi-two-dimensional granular material (bi-dispersed nylon rods) . We directly determine the spatial and temporal distributions of strain displacements of the elastic continuum over 200 spatial points located adjacent to the gap. Slip events can be divided into large system-spanning events and spatially distributed smaller events. The small events have a probability distribution of event moment consistent with an $M^{-3/2}$ power law scaling and a Poisson distributed recurrence time distribution. Large events have a broad, log-normal moment distribution and a mean repetition time. As the applied normal force increases, there are fractionally more (less) large (small) events, and the large-event moment distribution broadens. The magnitude of the slip motion of the plates is well correlated with the root-mean-square displacements of the granular matter. Our results are consistent with mean field descriptions of statistical models of earthquakes and avalanches. We further explore the high-speed dynamics of system events and also discuss the effective granular friction of the sheared layer. We find that large events result from stored elastic energy in the plates in this coupled granular-continuum system.   

Tuning Parameters and Scaling For Avalanches On A Slowly-Driven Conical Bead Pile with Cohesion

Slip avalanches on a slowly driven pile are investigated experimentally using a 3D conical pile built from uniform 3~mm steel beads. Beads are added to the pile by dropping them onto the apex one at a time; avalanches are measured through changes in pile mass. We investigate the dynamic response of the pile by recording avalanches from the pile over the course of tens of thousands of bead drops. The statistical properties of the avalanches, including probability of particular avalanche sizes and the time between avalanches of given size, are well-characterized by universal power laws and scaling functions. By adding a uniform magnetic field, we may systematically vary the cohesion between the beads and tune the critical behavior of the system. As the cohesion increases we observe an increase in both size and number for very large avalanches and decreases in the mid-size avalanches, causing a deviation from the power law. A full study of the effect of cohesion on the size and time distributions is in process, combining the experimental results with predictions from an analytical mean-field model [Dahmen, Nat Phys 7, 554 (2011)].   

Universality and depinning models for plastic yield in amorphous materials.

Plastic yield in amorphous materials occurs as a result of complex collective dynamics of local reorganizations, which gives rise to rich phenomena such as strain localization, intermittent dynamics and power-law distributed avalanches. While such systems have received considerable attention, both theoretical and experimental, controversy remains over the nature of the yielding transition. We present a new fully-tensorial coarsegrained model in 2D and 3D, and demonstrate that the exponents describing avalanche distributions are universal under a variety of loading conditions, system dimensionality and size, and boundary conditions. Our results show that while depinning-type models in general are apt to describe the system, mean field depinning models are not.   

Atomic-scale reversibility in sheared glasses

Systems become irreversible on a macroscopic scale when they are sheared beyond the yield strain and begin flowing. Using computer simulations of oscillatory shear, we investigate atomic scale reversibility. We employ molecular dynamics simulations to cool binary Lennard-Jones liquids to zero temperature over a wide range of cooling rates. We then apply oscillatory quasistatic shear at constant pressure to the zero-temperature glasses and identify neighbor-switching atomic rearrangement events. We determine the critical strain $\gamma^*$, beyond which atoms in the system do not return to their original positions upon reversing the strain. We show that for more slowly cooled glasses, the average potential energy is lower and the typical size of atomic rearrangements is smaller, which correlates with larger $\gamma^*$. Finally, we connect atomic- and macro-scale reversibility by determining the number of and correlations between the atomic rearrangements that occur as the system reaches the yield strain.   

Multiscale minimal modeling of microscale crystal plasticity: Finite-size scaling and stochastic plastic flow

We investigate the multiscale description from continuum to discrete modeling of crystal plasticity in the context of a minimal model. We develop a continuum plasticity description of discrete edge dislocations moving athermally in a single slip system; Our continuum modeling not only matches the statistical behavior of the model, but also the onset of emergent length scales as load increases. We perform quasistatic stress-controlled simulations of our continuum model and compare it with the corresponding discrete dislocation dynamics model, which describes crystal plasticity at a smaller spatiotemporal discretization scale. We investigate the properties of strain bursts (dislocation avalanches) occurring during plastic deformation, as well as the onset of a dislocation patterning lengthscale, and compare in detail the continuum and discrete descriptions. Our approach provides a pathway to multiscale modeling of complex, multi-slip and three dimensional crystal plasticity.   

The effectiveness of mean-field theory for avalanche distributions

We explore the mean-field theory of the pseudogap found in avalanche systems with long-range anisotropic interactions using analytical and numerical tools. The pseudogap in the density of low-stability states emerges from the competition between stabilizing interactions between spins in an avalanche and the destabilizing random movement towards the threshold caused by anisotropic couplings. Pazmandi et al. have shown that for the Sherrington-Kirkpatrick model, the pseudogap scales linearly and produces a distribution of avalanche sizes with exponent t=1 in contrast with that predicted from RFIM t=3/2. Lin et al. have argued that the scaling exponent ? of the pseudogap depends on the tail of the distribution of couplings and on non-universal values like the strain rate and the magnitude of the coupling strength. Yet others have argued that the relationship between the pseudogap scaling and the distribution of avalanche sizes is dependent on dynamical details. Despite the theoretical arguments, the class of RFIM mean-field models is surprisingly good at predicting the distribution of avalanche sizes in a variety of different magnetic systems. We investigate these differences with a combination of theory and simulation.   

Elasto-plastic automata with realistic near field interactions: avalanches and diffusion

We present results on an elasto-plastic automaton model of an athermal amorphous solid under shear. We study four different variants of the model with two different loading geometries and two different stochastic prescriptions (random yield thresholds or random strain amplitudes). We perform a finite size scaling analysis for the avalanche size distribution and single-site displacement and strain statistics. The avalanche size distribution in all four cases is inconsistent with mean-field depinning results. For three of the four variants, the distribution is consistent with previous results from atomistic simulations and other related elasto-plastic models. The fourth seems to exhibit different scaling properties and may lie in a different universality class. The mean-squared displacement exhibits a pronounced dependence on the microscopic ingredients of the model and is completely non-universal. These results show that while certain microscopic ingredients of the model may be irrelevant for the individual avalanches, they may exhibit a profound impact on long-time correlations and long-lived shear localization.