Erratum: Obtaining self-similar scalings in focusing flows [Phys. Rev. E 92, 043016 (2015)].

This corrects the article DOI: 10.1103/PhysRevE.92.043016.

Shear-Jammed, Fragile, and Steady States in Homogeneously Strained Granular Materials.

We study the jamming phase diagram of sheared granular material using a novel Couette shear setup with a multiring bottom. The setup uses small basal friction forces to apply a volume-conserving linear shear with no shear band to a granular system composed of frictional photoelastic discs. The setup can generate arbitrarily large shear strain due to its circular geometry, and the shear direction can be reversed, allowing us to measure a feature that distinguishes shear-jammed from fragile states. We report systematic measurements of the stress, strain, and contact network structure at phase boundaries that have been difficult to access by traditional experimental techniques, including the yield stress curve and the jamming curve close to ϕ_{SJ}≈0.75, the smallest packing fraction supporting a shear-jammed state. We observe fragile states created under large shear strain over a range of ϕ<ϕ_{SJ}. We also find a transition in the character of the quasistatic steady flow centered around ϕ_{SJ} on the yield curve as a function of packing fraction. Near ϕ_{SJ}, the average contact number, fabric anisotropy, and nonrattler fraction all show a change of slope. Above ϕ_{F}≈0.7 the steady flow shows measurable deviations from the basal linear shear profile, and above ϕ_{b}≈0.78 the flow is localized in a shear band.

Dynamics of a grain-scale intruder in a two-dimensional granular medium with and without basal friction.

We report on a series of experiments in which a grain-sized intruder is pushed by a spring through a two-dimensional granular material composed of photoelastic disks in a Couette geometry. We study the intruder dynamics as a function of packing fraction for two types of supporting substrates: A frictional glass plate and a layer of water for which basal friction forces are negligible. We observe two dynamical regimes: Intermittent flow, in which the intruder moves freely most of the time but occasionally gets stuck, and stick-slip dynamics, in which the intruder advances via a sequence of distinct, rapid events. When basal friction is present, we observe a smooth crossover between the two regimes as a function of packing fraction, and we find that reducing the interparticle friction coefficient causes the stick-slip regime to shift to higher packing fractions. When basal friction is eliminated, we observe intermittent flow at all accessible packing fractions. For all cases, we present results for the statistics of stick events, the intruder velocity, and the force exerted on the intruder by the grains. Our results indicate the qualitative importance of basal friction at high packing fractions and suggest a possible connection between intruder dynamics in a static material and clogging dynamics in granular flows.

Seismicity in sheared granular matter.

We report on experiments investigating the dynamics of a slider that is pulled by a spring across a granular medium consisting of a vertical layer of photoelastic disks. The motion proceeds through a sequence of discrete events, analogous to seismic shocks, in which elastic energy stored in the spring is rapidly released. We measure the statistics of several properties of the individual events: the energy loss in the spring, the duration of the movement, and the temporal profile of the slider motion. We also study certain conditional probabilities and the statistics of mainshock-aftershock sequences. At low driving rates, we observe crackling with Omori-Utsu, Båth, and waiting time laws similar to those observed in seismic dynamics. At higher driving rates, where the sequence of events shows strong periodicity, we observe scaling laws and asymmetrical event shapes that are clearly distinguishable from those in the crackling regime.

Crackling to periodic dynamics in granular media.

We study the local and global dynamics of sheared granular materials in a stick-slip experiment, using a slider and a spring. The system crackles, with intermittent slip avalanches, or exhibits irregular or periodic dynamics, depending on the shear rate and loading stiffness. The global force on the slider during shearing captures the transitions from the crackling to the periodic regime. We deduce a dynamic phase diagram as a function of the shear rate and the loading stiffness and associated scaling laws. Using photoelastic particles, we also capture the grain-scale stress evolution, and investigate the microscopic behavior in the different regimes.

Discontinuous rate-stiffening in a granular composite modeled after cornstarch and water.

Cornstarch in water exhibits impact-activated solidification (IAS) and strong discontinuous shear thickening, with "shear jamming". However, these phenomena are absent in cornstarch in ethanol. Here we show that cornstarch granules swell under ambient conditions. We postulate that this granule swelling is linked to an interparticle force scale that introduces a discontinuous rate-dependence to the generation of stable contacts between granules. We studied this force scale by coating sand with ~ 2 µm-thick polydimethysiloxane, creating a material that exhibits a similar IAS and discontinuous deformation rate-stiffening despite being a granular composite, not a suspension. This result suggests rate-dependence can be tuned by coating granular materials, introducing an interparticle force scale from rate-dependent properties present in the coating material. Our work provides insights into the unique behavior of cornstarch in water, bridges our understanding of suspensions and dry granular materials, and introduces a method to make discontinuous rate-dependent materials without suspending particles.

The physics of jamming for granular materials: a review.

Granular materials consist of macroscopic grains, interacting via contact forces, and unaffected by thermal fluctuations. They are one of a class systems that undergo jamming, i.e. a transition between fluid-like and disordered solid-like states. Roughly twenty years ago, proposals by Cates et al for the shear response of colloidal systems and by Liu and Nagel, for a universal jamming diagram in a parameter space of packing fraction, ϕ, shear stress, τ, and temperature, T raised key questions. Contemporaneously, experiments by Howell et al and numerical simulations by Radjai et al and by Luding et al helped provide a starting point to explore key insights into jamming for dry, cohesionless, granular materials. A recent experimental observation by Bi et al is that frictional granular materials have a a re-entrant region in their jamming diagram. In a range of ϕ, applying shear strain, γ, from an initially force/stress free state leads to fragile (in the sense of Cates et al), then anisotropic shear jammed states. Shear jamming at fixed ϕ is presumably conjugate to Reynolds dilatancy, involving dilation under shear against deformable boundaries. Numerical studies by Radjai and Roux showed that Reynolds dilatancy does not occur for frictionless systems. Recent numerical studies by several groups show that shear jamming occurs for finite, but not infinite, systems of frictionless grains. Shear jamming does not lead to known ordering in position space, but Sarkar et al showed that ordering occurs in a space of force tiles. Experimental studies seeking to understand random loose and random close packings (rlp and rcp) and dating back to Bernal have probed granular packings and their response to shear and intruder motion. These studies suggest that rlp's are anisotropic and shear-jammed-like, whereas rcp's are likely isotropically jammed states. Jammed states are inherently static, but the jamming diagram may provide a context for understanding rheology, i.e. dynamic shear in a variety of systems that include granular materials and suspensions.

Sinking in a bed of grains activated by shearing.

We show how a weak force f enables intruder motion through dense granular materials subject to external mechanical excitations, in the present case, stepwise shearing. A force acts on a Teflon disk in a two-dimensional system of photoelastic disks. This force is much smaller than the smallest force needed to move the disk without any external excitation. In a cycle, the material plus intruder are sheared quasistatically from γ=0 to γ_{max}, and then backwards to γ=0. During various cycle phases, fragile and jammed states form. Net intruder motion δ occurs during fragile periods generated by shear reversals. δ per cycle, e.g., the quasistatic rate c, is constant, linearly dependent on γ_{max} and f. It vanishes as c∝(ϕ_{c}-ϕ)^{a}, with a≃3 and ϕ_{c}≃ϕ_{J}, reflecting the stiffening of granular systems under shear [J. Ren, J. A. Dijksman, and R. P. Behringer, Phys. Rev. Lett. 110, 018302 (2013)]PRLTAO0031-900710.1103/PhysRevLett.110.018302 as ϕ→ϕ_{J}. The intruder motion induces large-scale grain circulation. In the intruder frame, this motion is a granular analog to fluid flow past a cylinder, where f is the drag force exerted by the flow.

Microscopic Origins of Shear Jamming for 2D Frictional Grains.

Shear jamming (SJ) occurs for frictional granular materials with packing fractions ϕ in ϕ_{S}<ϕ<ϕ_{J}^{0}, when the material is subject to shear strain γ starting from a force-free state. Here, ϕ_{J}^{µ} is the isotropic jamming point for particles with a friction coefficient µ. SJ states have mechanically stable anisotropic force networks, e.g., force chains. Here, we investigate the origins of SJ by considering small-scale structures-trimers and branches-whose response to shear leads to SJ. Trimers are any three grains where the two outer grains contact a center one. Branches occur where three or more quasilinear force chain segments intersect. Certain trimers respond to shear by compressing and bending; bending is a nonlinear symmetry-breaking process that can push particles in the dilation direction faster than the affine dilation. We identify these structures in physical experiments on systems of two-dimensional frictional discs, and verify their role in SJ. Trimer bending and branch creation both increase Z above Z_{iso}≃3 needed for jamming 2D frictional grains, and grow the strong force network, leading to SJ.

Characterizing granular networks using topological metrics.

We carry out a direct comparison of experimental and numerical realizations of the exact same granular system as it undergoes shear jamming. We adjust the numerical methods used to optimally represent the experimental settings and outcomes up to microscopic contact force dynamics. Measures presented here range from microscopic through mesoscopic to systemwide characteristics of the system. Topological properties of the mesoscopic force networks provide a key link between microscales and macroscales. We report two main findings: (1) The number of particles in the packing that have at least two contacts is a good predictor for the mechanical state of the system, regardless of strain history and packing density. All measures explored in both experiments and numerics, including stress-tensor-derived measures and contact numbers depend in a universal manner on the fraction of nonrattler particles, f_{NR}. (2) The force network topology also tends to show this universality, yet the shape of the master curve depends much more on the details of the numerical simulations. In particular we show that adding force noise to the numerical data set can significantly alter the topological features in the data. We conclude that both f_{NR} and topological metrics are useful measures to consider when quantifying the state of a granular system.

Force and Mass Dynamics in Non-Newtonian Suspensions.

Above a certain solid fraction, dense granular suspensions in water exhibit non-Newtonian behavior, including impact-activated solidification. Although it has been suggested that solidification depends on boundary interactions, quantitative experiments on the boundary forces have not been reported. Using high-speed video, tracer particles, and photoelastic boundaries, we determine the impactor kinematics and the magnitude and timings of impactor-driven events in the body and at the boundaries of cornstarch suspensions. We observe mass shocks in the suspension during impact. The shock front dynamics are strongly correlated to those of the intruder. However, the total momentum associated with this shock never approaches the initial impactor momentum. We also observe a faster second front associated with the propagation of pressure to the boundaries of the suspension. The two fronts depend differently on the initial impactor speed v_{0} and the suspension packing fraction. The speed of the pressure wave is at least an order of magnitude smaller than (linear) ultrasound speeds obtained for much higher frequencies, pointing to complex amplitude and frequency response of cornstarch suspensions to compressive strains.

Refractive index matched scanning and detection of soft particles.

We describe here how to apply the three-dimensional imaging technique of refractive index matched scanning to hydrogel spheres. Hydrogels are water based materials with a low refractive index, which allows for index matching with water-based solvent mixtures. We discuss here various experimental techniques required to handle specifically hydrogel spheres as opposed to other transparent materials. The deformability of hydrogel spheres makes their identification in three-dimensional images non-trivial. We will also discuss numerical techniques that can be used in general to detect contacting, non-spherical particles in a three-dimensional image. The experimental and numerical techniques presented here give experimental access to the stress tensor of a packing of deformed particles.

Collisional model of energy dissipation in three-dimensional granular impact.

We study the dynamic process occurring when a granular assembly is displaced by a solid impactor. The momentum transfer from the impactor to the target is shown to occur through sporadic, normal collisions of high force carrying grains at the intruder surface. We therefore describe the stopping force of the impact through a collisional-based model. To verify the model in impact experiments, we determine the forces acting on an intruder decelerating through a dense granular medium by using high-speed imaging of its trajectory. By varying the intruder shape and granular target, intruder-grain interactions are inferred from the consequent path. As a result, we connect the drag to the effect of intruder shape and grain density based on a proposed collisional model.

Local and global avalanches in a two-dimensional sheared granular medium.

We present the experimental and numerical studies of a two-dimensional sheared amorphous material composed of bidisperse photoelastic disks. We analyze the statistics of avalanches during shear including the local and global fluctuations in energy and changes in particle positions and orientations. We find scale-free distributions for these global and local avalanches denoted by power laws whose cutoffs vary with interparticle friction and packing fraction. Different exponents are found for these power laws depending on the quantity from which variations are extracted. An asymmetry in time of the avalanche shapes is evidenced along with the fact that avalanches are mainly triggered by the shear bands. A simple relation independent of the intensity is found between the number of local avalanches and the global avalanches they form. We also compare these experimental and numerical results for both local and global fluctuations to predictions from mean-field and depinning theories.

Steady flow dynamics during granular impact.

We study experimentally and computationally the dynamics of granular flow during impacts where intruders strike a collection of disks from above. In the regime where granular force dynamics are much more rapid than the intruder motion, we find that the particle flow near the intruder is proportional to the instantaneous intruder speed; it is essentially constant when normalized by that speed. The granular flow is nearly divergence free and remains in balance with the intruder, despite the latter's rapid deceleration. Simulations indicate that this observation is insensitive to grain properties, which can be explained by the separation of time scales between intergrain force dynamics and intruder dynamics. Assuming there is a comparable separation of time scales, we expect that our results are applicable to a broad class of dynamic or transient granular flows. Our results suggest that descriptions of static-in-time granular flows might be extended or modified to describe these dynamic flows. Additionally, we find that accurate grain-grain interactions are not necessary to correctly capture the granular flow in this regime.

Protocol dependence of the jamming transition.

We propose a theoretical framework for predicting the protocol dependence of the jamming transition for frictionless spherical particles that interact via repulsive contact forces. We study isostatic jammed disk packings obtained via two protocols: isotropic compression and simple shear. We show that for frictionless systems, all jammed packings can be obtained via either protocol. However, the probability to obtain a particular jammed packing depends on the packing-generation protocol. We predict the average shear strain required to jam initially unjammed isotropically compressed packings from the density of jammed packings, shape of their basins of attraction, and path traversed in configuration space. We compare our predictions to simulations of shear strain-induced jamming and find quantitative agreement. We also show that the packing fraction range, over which shear strain-induced jamming occurs, tends to zero in the large system limit for frictionless packings with overdamped dynamics.

Obtaining self-similar scalings in focusing flows.

The surface structure of converging thin fluid films displays self-similar behavior, as was shown in the work by Diez et al. [Q. Appl. Math. 210, 155 (1990)]. Extracting the related similarity scaling exponents from either numerical or experimental data is nontrivial. Here we provide two such methods. We apply them to experimental and numerical data on converging fluid films driven by both surface tension and gravitational forcing. In the limit of pure gravitational driving, we recover Diez' semianalytic result, but our methods also allow us to explore the entire regime of mixed capillary and gravitational driving, up to entirely surface-tension-driven flows. We find scaling forms of smoothly varying exponents up to surprisingly small Bond numbers. Our experimental results are in reasonable agreement with our numerical simulations, which confirm theoretically obtained relations between the scaling exponents.

Stress Relaxation for Granular Materials near Jamming under Cyclic Compression.

We have explored isotropically jammed states of semi-2D granular materials through cyclic compression. In each compression cycle, systems of either identical ellipses or bidisperse disks transition between jammed and unjammed states. We determine the evolution of the average pressure P and structure through consecutive jammed states. We observe a transition point ϕ_{m} above which P persists over many cycles; below ϕ_{m}, P relaxes slowly. The relaxation time scale associated with P increases with packing fraction, while the relaxation time scale for collective particle motion remains constant. The collective motion of the ellipses is hindered compared to disks because of the rotational constraints on elliptical particles.

Universal Quake Statistics: From Compressed Nanocrystals to Earthquakes.

Slowly-compressed single crystals, bulk metallic glasses (BMGs), rocks, granular materials, and the earth all deform via intermittent slips or "quakes". We find that although these systems span 12 decades in length scale, they all show the same scaling behavior for their slip size distributions and other statistical properties. Remarkably, the size distributions follow the same power law multiplied with the same exponential cutoff. The cutoff grows with applied force for materials spanning length scales from nanometers to kilometers. The tuneability of the cutoff with stress reflects "tuned critical" behavior, rather than self-organized criticality (SOC), which would imply stress-independence. A simple mean field model for avalanches of slipping weak spots explains the agreement across scales. It predicts the observed slip-size distributions and the observed stress-dependent cutoff function. The results enable extrapolations from one scale to another, and from one force to another, across different materials and structures, from nanocrystals to earthquakes.

Network flow model of force transmission in unbonded and bonded granular media.

An established aspect of force transmission in quasistatic deformation of granular media is the existence of a dual network of strongly versus weakly loaded particles. Despite significant interest, the regulation of strong and weak forces through the contact network remains poorly understood. We examine this aspect of force transmission using data on microstructural fabric from: (I) three-dimensional discrete element models of grain agglomerates of bonded subspheres constructed from in situ synchrotron microtomography images of silica sand grains under unconfined compression and (II) two-dimensional assemblies of unbonded photoelastic circular disks submitted to biaxial compression under constant volume. We model force transmission as a network flow and solve the maximum flow-minimum cost (MFMC) problem, the solution to which yields a percolating subnetwork of contacts that transmits the "maximum flow" (i.e., the highest units of force) at "least cost" (i.e., the dissipated energy from such transmission). We find the MFMC describes a two-tier hierarchical architecture. At the local level, it encapsulates intraconnections between particles in individual force chains and in their conjoined 3-cycles, with the most common configuration having at least one force chain contact experiencing frustrated rotation. At the global level, the MFMC encapsulates interconnections between force chains. The MFMC can be used to predict most of the force chain particles without need for any information on contact forces, thereby suggesting the network flow framework may have potential broad utility in the modeling of force transmission in unbonded and bonded granular media.