Reynolds pressure and relaxation in a sheared granular system

Onsager variational principle for granular fluids

Granular fluids, as defined by a collection of moving solid particles, is a paradigm of a dissipative system out of equilibrium. Inelastic collisions between particles is the source of dissipation, and is the origin of a transition from a gas to a liquidlike state. This transition can be triggered by an increase of the solid fraction. Moreover, in compartmentalized systems, this condensation is driving the entire granular fluid into a Maxwell demon phenomenon, localizing most of the grains into a specific compartment. Classical approaches fail to capture these phenomena, thus motivating many experimental and numerical works. Herein, we demonstrate that the Onsager variational principle is able to predict accurately the coexistence of gas-liquid states in granular systems, opening ways to model other phenomena observed in such dissipative systems like segregation or the jamming transition.

Granular solids transmit stress as two-phase composites

A basic problem in the science of realistic granular matter is the plethora of heuristic models of the stress field in the absence of a first-principles theory. Such a theory is formulated here, based on the idea that static granular assemblies can be regarded as two-phase composites. A thought experiment is described, demonstrating that the state of such materials can be varied continuously from marginal stability, via a two-phase granular assembly, then porous structure, and finally be made perfectly elastic. For completeness, I review briefly the condition for marginal stability in infinitely large assemblies. The general solution for the stress equations in d=2 is reviewed in detail and shown to be consistent with the two-phase idea. A method for identifying the phases of finite regions in larger systems is constructed, providing a stability parameter that quantifies the "proximity" to the marginally stable state. The difficulty involved in deriving stress fields in such composites is a unique constraint on the boundary between phases, and, to highlight it, a simple case of a stack of plates of alternating phase is solved explicitly. An effective medium approximation, which satisfies this constraint, is then developed and analyzed in detail. This approach forms a basis for the extension of the stress theory to general granular solids that are not marginally stable or at the yield threshold.

Complex Memory Formation in Frictional Granular Media

Using numerical simulations it is shown that a jammed, random pack of soft frictional grains can store an arbitrary waveform that is applied as a small time-dependent shear while the system is slowly compressed. When the system is decompressed at a later time, an approximation of the input waveform is recalled in time-reversed order as shear stresses on the system boundaries. This effect depends on friction between the grains, and is independent of some aspects of the friction model. This type of memory could potentially be observable in other types of random media that form internal contacts when compressed.

Quantitative Understanding of the Onset of Dense Granular Flows

The question of when and how dense granular materials start to flow under stress, despite many industrial and geophysical applications, remains largely unresolved. We develop and test a simple equation for the onset of quasistatic flows of granular materials which is based on the frictional aging of the granular packing. The result is a nonmonotonic stress-strain relation which-akin to classical friction-is independent of the shear rate. This relation suffices to understand the quasistatic deformations of aging granular media, and its solid-to-liquid transition. Our results also elucidate the (flow) history dependence of the mechanical properties, and the sensitivity to the initial preparation of granular media.

Development of a biaxial apparatus for jamming profiles of photoelastic granular media

We describe a two-dimensional biaxial apparatus that is used to conduct the experimental study of the jamming of granular media. The setup is designed based on the photoelastic imaging technique, which allows us to detect force-bearing contacts among particles, calculate the pressure on each particle according to the mean squared intensity gradient method, and compute contact forces on each particle [T. S. Majmudar and R. P. Behringer, Nature 435, 1079-1082 (2005)]. Particles float in a density-matched solution to avoid basal friction during experiments. We can compress (uniaxially or biaxially) or shear the granular system by an entangled comb geometry by moving the paired boundary walls independently. A novel design for the corner of each pair of perpendicular walls is described, which allows for independent motion. We control the system using a Raspberry Pi with Python code. Three typical experiments are described briefly. Furthermore, more complicated experiment protocols can be implemented to achieve specific granular materials research goals.

Coupled Dynamical Phase Transitions in Driven Disk Packings

Under the influence of oscillatory shear, a monolayer of frictional granular disks exhibits two dynamical phase transitions: a transition from an initially disordered state to an ordered crystalline state and a dynamic active-absorbing phase transition. Although there is no reason a priori for these to be at the same critical point, they are. The transitions may also be characterized by the disk trajectories, which are nontrivial loops breaking time-reversal invariance.

PhotoElasticFinger: Robot Tactile Fingertip Based on Photoelastic Effect

The sense of touch is fundamental for a one-to-one mapping between the environment and a robot that physically interacts with the environment. Herein, we describe a tactile fingertip design that can robustly detect interaction forces given data collected from a camera. This design is based on the photoelastic effect observed in silicone matter. Under the force applied to the silicone rubber, owing to the stress-induced birefringence, the light propagating within the silicone rubber is subjected to the angular phase shift, where the latter is proportional to the increase in the image brightness in the camera frames. We present the calibration and test results of the photoelastic sensor design on a bench using a robot arm and with a certified industrial force torque sensor. We also discuss the applications of this sensor design and its potential relationship with human mechano-transduction receptors. We achieved a force sensing range of up to 8 N with a force resolution of around 0.5 N. The photoelastic tactile fingertip is suitable for robot grasping and might lead to further progress in robust tactile sensing.

Memory Formation in Adaptive Networks

The continuous adaptation of networks like our vasculature ensures optimal network performance when challenged with changing loads. Here, we show that adaptation dynamics allow a network to memorize the position of an applied load within its network morphology. We identify that the irreversible dynamics of vanishing network links encode memory. Our analytical theory successfully predicts the role of all system parameters during memory formation, including parameter values which prevent memory formation. We thus provide analytical insight on the theory of memory formation in disordered systems.

On intermittency in sheared granular systems

We consider a system of granular particles, modeled by two dimensional frictional soft elastic disks, that is exposed to externally applied time-dependent shear stress in a planar Couette geometry. We concentrate on the external forcing that produces intermittent dynamics of stick-slip type. In this regime, the top wall remains almost at rest until the applied stress becomes sufficiently large, and then it slips. We focus on the evolution of the system as it approaches a slip event. Our main finding is that there are two distinct groups of measures describing system behavior before a slip event. The first group consists of global measures defined as system-wide averages at a fixed time. Typical examples of measures in this group are averages of the normal or tangent forces acting between the particles, system size and number of contacts between the particles. These measures do not seem to be sensitive to an approaching slip event. On average, they tend to increase linearly with the force pulling the spring. The second group consists of the time-dependent measures that quantify the evolution of the system on a micro (particle) or mesoscale. Measures in this group first quantify the temporal differences between two states and only then aggregate them to a single number. For example, Wasserstein distance quantitatively measures the changes of the force network as it evolves in time while the number of broken contacts quantifies the evolution of the contact network. The behavior of the measures in the second group changes dramatically before a slip event starts. They increase rapidly as a slip event approaches, indicating a significant increase in fluctuations of the system before a slip event is triggered.

Criticality and marginal stability of the shear jamming transition of frictionless soft spheres

We study numerically the critical behavior and marginal stability of the shear jamming transition for frictionless soft spheres, observed to occur over a finite range of densities, associated with isotropic jamming for densities above the minimum jamming (J-point) density. Several quantities are shown to scale near the shear jamming point in the same way as the isotropic jamming point. We compute the exponents associated with the small force distribution and the interparticle gap distribution and show that the corresponding exponents are consistent with the marginal stability condition observed for isotropic jamming and with predictions of the mean-field theory of jamming in hard spheres.

Understanding slow compression and decompression of frictionless soft granular matter by network analysis

We consider dense granular systems in three spatial dimensions exposed to slow compression and decompression, below, during, above and well above jamming. The evolution of granular systems under slow deformation is non-trivial and involves smooth, continuous, reversible (de)compression periods, interrupted by fast, discontinuous, irreversible transition events. These events are often, but not always, associated with rearrangements of particles and of the contact network. How many particles are involved in these transitions between two states can range from few to almost all in the system. An analysis of the force network that is built on top of the contact network is carried out using the tools of persistent homology. Results involve the observation that kinetic energy is correlated with the intensity of rearrangements, while the evolution of global mechanical measures, such as pressure, is strongly correlated with the evolution of the topological measures quantifying loops in the force network. Surprisingly, some transitions are clearly detected by persistent homology even though motion/rearrangement of particles is much weaker, i.e., much harder to detect or, in some cases, not observed at all.

Rotational diffusion and rotational correlations in frictional amorphous disk packings under shear

We show here that rotations of round particles in amorphous disk packing reveal various nontrivial microscopic features when the packing is close to rigidification. We analyze experimental measurements on disk packing subjected to simple shear deformation with various inter-particle friction coefficients and across a range of volume fractions where the system is known to stiffen. The analysis of measurements indicates that shear induces diffusive microrotation, that can be both enhanced and suppressed depending upon the volume fraction as well as the inter-particle friction. Rotations also display persistent anticorrelated motion. Spatial correlations in microrotation are observed to be directly correlated with system pressure. These observations point towards the broader mechanical relevance of collective dynamics in the rotational degree of freedom of particles.

Strain dependent vorticity in sheared granular media

Displacement fields in sheared particle packings often display vortex-like structures that reveal essential features about the mechanical state of the collection of particles. There are several metrics to quantify these flow field features, yet extracting such quantitative metrics from flow field or particle tracking data involves making numerous choices on the time and length scales over which to average. Here we employ a much used experimental data set on sheared disk packings to explore how such arbitrary data mining choices affect the obtained results. We focus on calculating the strain dependent vorticity, as this metric is a differential method hence potentially sensitive to the way it is computed. We find that the total surface area with an absolute vorticity above a certain threshold approaches a plateau value as shear progresses. This plateau value exhibits a non-monotonic dependence on packing fraction. We also show which range of choices yields results that can support an analysis method independent, physical interpretation of the flow field data.

Dilatancy, shear jamming, and a generalized jamming phase diagram of frictionless sphere packings

Granular packings display the remarkable phenomenon of dilatancy, wherein their volume increases upon shear deformation. Conventional wisdom and previous results suggest that dilatancy, also being the related phenomenon of shear-induced jamming, requires frictional interactions. Here, we show that the occurrence of isotropic jamming densities φ_j above the minimal density (or the J-point density) φ_J leads both to the emergence of shear-induced jamming and dilatancy in frictionless packings. Under constant pressure shear, the system evolves into a steady-state at sufficiently large strains, whose density only depends on the pressure and is insensitive to the initial jamming density φ_j. In the limit of vanishing pressure, the steady-state exhibits critical behavior at φ_J. While packings with different φ_j values display equivalent scaling properties under compression, they exhibit striking differences in rheological behaviour under shear. The yield stress under constant volume shear increases discontinuously with density when φ_j > φ_J, contrary to the continuous behaviour in generic packings that jam at φ_J. Our results thus lead to a more coherent, generalised picture of jamming in frictionless packings, which also have important implications on how dilatancy is understood in the context of frictional granular matter.

How does static granular matter re-arrange for different isotropic strain rate?

The question of how soft granular matter, or dense amorphous systems, re-arrange their microstructure under isotropic compression and de-compression, at different strain rates, will be answered by particle simulations of frictionless model systems in a periodic three-dimensional cuboid. Starting compression below jamming, the systems experience the well known jamming transition, with characteristic evolutions of the state variables elastic energy, elastic stress, coordination number, and elastic moduli. For large strain rates, kinetic energy comes into play and the evolution is more dynamic. In contrast, at extremely slow deformation, the system relaxes to hyper-elastic states, with well-defined elastic moduli, in static equilibrium between irreversible (plastic) re-arrangement events, discrete in time. Small, finite strains explore those reversible (elastic) states, before larger strains push the system into new states, by irreversible, sudden re-arrangements of the micro-structure.

Friction-Controlled Entropy-Stability Competition in Granular Systems

Using cyclic shear to drive a two-dimensional granular system, we determine the structural characteristics for different interparticle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter's effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquidlike rather than glassy.

Sheared Amorphous Packings Display Two Separate Particle Transport Mechanisms

Shearing granular materials induces nonaffine displacements. Such nonaffine displacements have been studied extensively, and are known to correlate with plasticity and other mechanical features of amorphous packings. A well known example is shear transformation zones as captured by the local deviation from affine deformation, D_{min}^{2}, and their relevance to failure and stress fluctuations. We analyze sheared frictional athermal disc packings and show that there exists at least one additional mesoscopic transport mechanism that superimposes itself on top of local diffusive motion. We evidence this second transport mechanism in a homogeneous system via a diffusion tensor analysis and show that the trace of the diffusion tensor equals the classic D_{min}^{2} when this second mesoscopic transport is corrected for. The new transport mechanism is consistently observed over a wide range of volume fractions and even for particles with different friction coefficients and is consistently observed also upon shear reversal, hinting at its relevance for memory effects.

Unified phase diagram of reversible-irreversible, jamming, and yielding transitions in cyclically sheared soft-sphere packings

Self-organization, and transitions from reversible to irreversible behavior, of interacting particle assemblies driven by externally imposed stresses or deformation is of interest in comprehending diverse phenomena in soft matter. They have been investigated in a wide range of systems, such as colloidal suspensions, glasses, and granular matter. In different density and driving regimes, such behavior is related to yielding of amorphous solids, jamming, memory formation, etc. How these phenomena are related to each other has not, however, been much studied. In order to obtain a unified view of the different regimes of behavior, and transitions between them, we investigate computationally the response of soft-sphere assemblies to athermal cyclic-shear deformation over a wide range of densities and amplitudes of shear deformation. Cyclic-shear deformation induces transitions from reversible to irreversible behavior in both unjammed and jammed soft-sphere packings. Well above the minimum isotropic jamming density ([Formula: see text]), this transition corresponds to yielding. In the vicinity of the jamming point, up to a higher-density limit, we designate [Formula: see text], an unjammed phase emerges between a localized, absorbing phase and a diffusive, irreversible, phase. The emergence of the unjammed phase signals the shifting of the jamming point to higher densities as a result of annealing and opens a window where shear jamming becomes possible for frictionless packings. Below [Formula: see text], two distinct localized states, termed point- and loop-reversible, are observed. We characterize in detail the different regimes and transitions between them and obtain a unified density-shear amplitude phase diagram.

On the Mechanism of Laboratory Earthquake Nucleation Highlighted by Acoustic Emission

Dynamics of granular media is the key to understanding behavior of many natural systems. In this work we concentrate on studying regularities of deformation of a gouge-filled fault. Confined granular layer – model fault – subjected to an external stress may display sudden slip owing to rearrangement of the granular layer. In nature fast slip along a fault results in an earthquake. To understand fault behavior better, we have conducted a comprehensive analysis of acoustic emission (AE) data that accompany stick-slip in granular media. Here we reveal and trace the emergence of two populations of AE. The first one is characterized by a waveform with a harsh onset, while the second one exhibits a gradual amplitude rise and a tremor-like waveform. During a regular stick-slip the statistical properties of the first population remains intact. The second one is very sensitive to alterations of stress conditions, and its scaling parameters correlate with the change of mechanical characteristics of the fault. Probably, AE populations were identified corresponding to two gouge-filled fault subsystems – a load-bearing granular network and an ensemble of relatively unloaded grains in the granular layer. The detected regularities point to a compound self-organization processes in fault zones and suggest that the final stage of earthquake preparation can be revealed in analyzing the scaling characteristics of seismic-acoustic data.

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.

Protocol Dependence and State Variables in the Force-Moment Ensemble

Stress-based ensembles incorporating temperaturelike variables have been proposed as a route to an equation of state for granular materials. To test the efficacy of this approach, we perform experiments on a two-dimensional photoelastic granular system under three loading conditions: uniaxial compression, biaxial compression, and simple shear. From the interparticle forces, we find that the distributions of the normal component of the coarse-grained force-moment tensor are exponential tailed, while the deviatoric component is Gaussian distributed. This implies that the correct stress-based statistical mechanics conserves both the force-moment tensor and the Maxwell-Cremona force-tiling area. As such, two variables of state arise: the tensorial angoricity (α[over ^]) and a new temperaturelike quantity associated with the force-tile area which we name keramicity (κ). Each quantity is observed to be inversely proportional to the global confining pressure; however, only κ exhibits the protocol independence expected of a state variable, while α[over ^] behaves as a variable of process.

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.

Force-chain evolution in a two-dimensional granular packing compacted by vertical tappings

We experimentally study the statistics of force-chain evolution in a vertically-tapped two-dimensional granular packing by using photoelastic disks. In this experiment, the tapped granular packing is gradually compacted. During the compaction, the isotropy of grain configurations is quantified by measuring the deviator anisotropy derived from fabric tensor, and then the evolution of force-chain structure is quantified by measuring the interparticle forces and force-chain orientational order parameter. As packing fraction increases, the interparticle force increases and finally saturates to an asymptotic value. Moreover, the grain configurations and force-chain structures become isotropically random as the tapping-induced compaction proceeds. In contrast, the total length of force chains remains unchanged. From the correlations of those parameters, we find two relations: (i) a positive correlation between the isotropy of grain configurations and the disordering of force-chain orientations, and (ii) a negative correlation between the increasing of interparticle forces and the disordering of force-chain orientations. These relations are universally held regardless of the mode of particle motions with or without convection.

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.

An instrument for studying granular media in low-gravity environment

A new experimental facility has been designed and constructed to study driven granular media in a low-gravity environment. This versatile instrument, fully automatized, with a modular design based on several interchangeable experimental cells, allows us to investigate research topics ranging from dilute to dense regimes of granular media such as granular gas, segregation, convection, sound propagation, jamming, and rheology-all without the disturbance by gravitational stresses active on Earth. Here, we present the main parameters, protocols, and performance characteristics of the instrument. The current scientific objectives are then briefly described and, as a proof of concept, some first selected results obtained in low gravity during parabolic flight campaigns are presented.

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.

Normal Stresses, Contraction, and Stiffening in Sheared Elastic Networks

When elastic solids are sheared, a nonlinear effect named after Poynting gives rise to normal stresses or changes in volume. We provide a novel relation between the Poynting effect and the microscopic Grüneisen parameter, which quantifies how stretching shifts vibrational modes. By applying this relation to random spring networks, a minimal model for, e.g., biopolymer gels and solid foams, we find that networks contract or develop tension because they vibrate faster when stretched. The amplitude of the Poynting effect is sensitive to the network's linear elastic moduli, which can be tuned via its preparation protocol and connectivity. Finally, we show that the Poynting effect can be used to predict the finite strain scale where the material stiffens under shear.

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.

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.

Granular materials flow like complex fluids

Granular materials such as sand, powders and foams are ubiquitous in daily life and in industrial and geotechnical applications. These disordered systems form stable structures when unperturbed, but in the presence of external influences such as tapping or shear they 'relax', becoming fluid in nature. It is often assumed that the relaxation dynamics of granular systems is similar to that of thermal glass-forming systems. However, so far it has not been possible to determine experimentally the dynamic properties of three-dimensional granular systems at the particle level. This lack of experimental data, combined with the fact that the motion of granular particles involves friction (whereas the motion of particles in thermal glass-forming systems does not), means that an accurate description of the relaxation dynamics of granular materials is lacking. Here we use X-ray tomography to determine the microscale relaxation dynamics of hard granular ellipsoids subject to an oscillatory shear. We find that the distribution of the displacements of the ellipsoids is well described by a Gumbel law (which is similar to a Gaussian distribution for small displacements but has a heavier tail for larger displacements), with a shape parameter that is independent of the amplitude of the shear strain and of the time. Despite this universality, the mean squared displacement of an individual ellipsoid follows a power law as a function of time, with an exponent that does depend on the strain amplitude and time. We argue that these results are related to microscale relaxation mechanisms that involve friction and memory effects (whereby the motion of an ellipsoid at a given point in time depends on its previous motion). Our observations demonstrate that, at the particle level, the dynamic behaviour of granular systems is qualitatively different from that of thermal glass-forming systems, and is instead more similar to that of complex fluids. We conclude that granular materials can relax even when the driving strain is weak.

Photoelastic force measurements in granular materials

Photoelastic techniques are used to make both qualitative and quantitative measurements of the forces within idealized granular materials. The method is based on placing a birefringent granular material between a pair of polarizing filters, so that each region of the material rotates the polarization of light according to the amount of local stress. In this review paper, we summarize the past work using the technique, describe the optics underlying the technique, and illustrate how it can be used to quantitatively determine the vector contact forces between particles in a 2D granular system. We provide a description of software resources available to perform this task, as well as key techniques and resources for building an experimental apparatus.

Shear-induced rigidity of frictional particles: Analysis of emergent order in stress space

Solids are distinguished from fluids by their ability to resist shear. In equilibrium systems, the resistance to shear is associated with the emergence of broken translational symmetry as exhibited by a nonuniform density pattern that is persistent, which in turn results from minimizing the free energy. In this work, we focus on a class of systems where this paradigm is challenged. We show that shear-driven jamming in dry granular materials is a collective process controlled by the constraints of mechanical equilibrium. We argue that these constraints can lead to a persistent pattern in a dual space that encodes the statistics of contact forces and the topology of the contact network. The shear-jamming transition is marked by the appearance of this persistent pattern. We investigate the structure and behavior of patterns both in real space and the dual space as the system evolves through the rigidity transition for a range of packing fractions and in two different shear protocols. We show that, in the protocol that creates homogeneous jammed states without shear bands, measures of shear jamming do not depend on strain and packing fraction independently but obey a scaling form with a packing-fraction-dependent characteristic strain that goes to zero at the isotropic jamming point ϕ_{J}. We demonstrate that it is possible to define a protocol-independent order parameter in this dual space, which provides a quantitative measure of the rigidity of shear-jammed states.

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.

Softening of stressed granular packings with resonant sound waves

We perform numerical simulations of a two-dimensional bidisperse granular packing subjected to both a static confining pressure and a sinusoidal dynamic forcing applied by a wall on one edge of the packing. We measure the response experienced by a wall on the opposite edge of the packing and obtain the resonant frequency of the packing as the static or dynamic pressures are varied. Under increasing static pressure, the resonant frequency increases, indicating a velocity increase of elastic waves propagating through the packing. In contrast, when the dynamic amplitude is increased for fixed static pressure, the resonant frequency decreases, indicating a decrease in the wave velocity. This occurs both for compressional and for shear dynamic forcing and is in agreement with experimental results. We find that the average contact number Zc at the resonant frequency decreases with increasing dynamic amplitude, indicating that the elastic softening of the packing is associated with a reduced number of grain-grain contacts through which the elastic waves can travel. We image the excitations created in the packing and show that there are localized disturbances or soft spots that become more prevalent with increasing dynamic amplitude. Our results are in agreement with experiments on glass bead packings and earth materials such as sandstone and granite and may be relevant to the decrease in elastic wave velocities that has been observed to occur near fault zones after strong earthquakes, in surficial sediments during strong ground motion, and in structures during earthquake excitation.

Shear-induced rigidity in athermal materials: A unified statistical framework

Recent studies of athermal systems such as dry grains and dense, non-Brownian suspensions have shown that shear can lead to solidification through the process of shear jamming in grains and discontinuous shear thickening in suspensions. The similarities observed between these two distinct phenomena suggest that the physical processes leading to shear-induced rigidity in athermal materials are universal. We present a nonequilibrium statistical mechanics model, which exhibits the phenomenology of these shear-driven transitions, shear jamming and discontinuous shear thickening, in different regions of the predicted phase diagram. Our analysis identifies the crucial physical processes underlying shear-driven rigidity transitions, and clarifies the distinct roles played by shearing forces and the packing fraction of grains.

Spatiotemporal chaotic unjamming and jamming in granular avalanches

We have investigated the spatiotemporal chaotic dynamics of unjamming and jamming of particles in a model experiment – a rotating drum partially filled with bidisperse disks to create avalanches. The magnitudes of the first Lyapunov vector δu(t) and velocity v(t) of particles are directly measured for the first time to yield insights into their spatial correlation C_δu,v, which is on statistical average slightly larger near the unjamming than the value near the jamming transition. These results are consistent with the recent work of Banigan et al (Nature Phys. 2013), and it is for the first time to validate their theoretical models in a real scenario. v(t) shows rich dynamics: it grows exponentially for unstable particles and keeps increasing despite stochastic interactions; after the maximum, it decays with large fluctuations. Hence the spatiotemporal chaotic dynamics of avalanche particles are entangled, causing temporal correlations of macroscopic quantities of the system. We propose a simple model for these observations.

Following the evolution of hard sphere glasses in infinite dimensions under external perturbations: compression and shear strain

We consider the adiabatic evolution of glassy states under external perturbations. The formalism we use is very general. Here we use it for infinite-dimensional hard spheres where an exact analysis is possible. We consider perturbations of the boundary, i.e., compression or (volume preserving) shear strain, and we compute the response of glassy states to such perturbations: pressure and shear stress. We find that both quantities overshoot before the glass state becomes unstable at a spinodal point where it melts into a liquid (or yields). We also estimate the yield stress of the glass. Finally, we study the stability of the glass basins towards breaking into sub-basins, corresponding to a Gardner transition. We find that close to the dynamical transition, glasses undergo a Gardner transition after an infinitesimal perturbation.

Precisely cyclic sand: self-organization of periodically sheared frictional grains

The disordered static structure and chaotic dynamics of frictional granular matter has occupied scientists for centuries, yet there are few organizational principles or guiding rules for this highly hysteretic, dissipative material. We show that cyclic shear of a granular material leads to dynamic self-organization into several phases with different spatial and temporal order. Using numerical simulations, we present a phase diagram in strain-friction space that shows chaotic dispersion, crystal formation, vortex patterns, and most unusually a disordered phase in which each particle precisely retraces its unique path. However, the system is not reversible. Rather, the trajectory of each particle, and the entire frictional, many-degrees-of-freedom system, organizes itself into a limit cycle absorbing state. Of particular note is that fact that the cyclic states are spatially disordered, whereas the ordered states are chaotic.

Shear modulus and dilatancy softening in granular packings above jamming

We investigate experimentally the mechanical response to shear of a monolayer of bidisperse frictional grains across the jamming transition. We inflate an intruder inside the packing and use photoelasticity and tracking techniques to measure the induced shear strain and stresses at the grain scale. We quantify experimentally the constitutive relations for strain amplitudes as low as 10(-3) and for a range of packing fractions within 2% variation around the jamming transition. At the transition strong nonlinear effects set in: both the shear modulus and the dilatancy shear soften at small strain until a critical strain is reached where effective linearity is recovered. The scaling of the critical strain and the associated critical stresses on the distance to jamming are extracted. We check that the constitutive laws, together with mechanical equilibrium, correctly predict to the observed stress and strain profiles. These profiles exhibit a spatial crossover between an effective linear regime close to the inflater and the truly nonlinear regime away from it. The crossover length diverges at the jamming transition.

Plasticity of a colloidal polycrystal under cyclic shear

We use confocal microscopy and time-resolved light scattering to investigate plasticity in a colloidal polycrystal, following the evolution of the network of grain boundaries as the sample is submitted to thousands of shear deformation cycles. The grain boundary motion is found to be ballistic, with a velocity distribution function exhibiting nontrivial power law tails. The shear-induced dynamics initially slow down, similarly to the aging of the spontaneous dynamics in glassy materials, but eventually reach a steady state. Surprisingly, the crossover time between the initial aging regime and the steady state decreases with increasing probed length scale, hinting at a hierarchical organization of the grain boundary dynamics.

Multiple transient memories in experiments on sheared non-Brownian suspensions

A system with multiple transient memories can remember a set of inputs but subsequently forgets almost all of them, even as they are continually applied. If noise is added, the system can store all memories indefinitely. The phenomenon has recently been predicted for cyclically sheared non-Brownian suspensions. Here we present experiments on such suspensions, finding behavior consistent with multiple transient memories and showing how memories can be stabilized by noise.

Variable-amplitude oscillatory shear response of amorphous materials

Variable-amplitude oscillatory shear tests are emerging as powerful tools to investigate and quantify the nonlinear rheology of amorphous solids, complex fluids, and biological materials. Quite a few recent experimental and atomistic simulation studies demonstrated that at low shear amplitudes, an amorphous solid settles into an amplitude- and initial-conditions-dependent dissipative limit cycle, in which back-and-forth localized particle rearrangements periodically bring the system to the same state. At sufficiently large shear amplitudes, the amorphous system loses memory of the initial conditions, exhibits chaotic particle motions accompanied by diffusive behavior, and settles into a stochastic steady state. The two regimes are separated by a transition amplitude, possibly characterized by some critical-like features. Here we argue that these observations support some of the physical assumptions embodied in the nonequilibrium thermodynamic, internal-variables based, shear-transformation-zone model of amorphous viscoplasticity; most notably that "flow defects" in amorphous solids are characterized by internal states between which they can make transitions, and that structural evolution is driven by dissipation associated with plastic deformation. We present a rather extensive theoretical analysis of the thermodynamic shear-transformation-zone model for a variable-amplitude oscillatory shear protocol, highlighting its success in accounting for various experimental and simulational observations, as well as its limitations. Our results offer a continuum-level theoretical framework for interpreting the variable-amplitude oscillatory shear response of amorphous solids and may promote additional developments.

Force and flow at the onset of drag in plowed granular media

We study the transient drag force FD on a localized intruder in a granular medium composed of spherical glass particles. A flat plate is translated horizontally from rest through the granular medium to observe how FD varies as a function of the medium's initial volume fraction, ϕ. The force response of the granular material differs above and below the granular critical state, ϕc, the volume fraction which corresponds to the onset of grain dilatancy. For ϕ<ϕc FD increases monotonically with displacement and is independent of drag velocity for the range of velocities examined (<10 cm/s). For ϕ>ϕc, FD rapidly rises to a maximum and then decreases over further displacement. The maximum force for ϕ>ϕc increases with increasing drag velocity. In quasi-two-dimensional drag experiments, we use granular particle image velocimetry (PIV) to measure time resolved strain fields associated with the horizontal motion of a plate started from rest. PIV experiments show that the maxima in FD for ϕ>ϕc are associated with maxima in the spatially averaged shear strain field. For ϕ>ϕc the shear strain occurs in a narrow region in front of the plate, a shear band. For ϕ<ϕc the shear strain is not localized, the shear band fluctuates in space and time, and the average shear increases monotonically with displacement. Laser speckle measurements made at the granular surface ahead of the plate reveal that for ϕ<ϕc particles are in motion far from the intruder and shearing region. For ϕ>ϕc, surface particles move only during the formation of the shear band, coincident with the maxima in FD, after which the particles remain immobile until the sheared region reaches the measurement region.

How the ideal jamming point illuminates the world of granular media

The zero temperature properties of frictionless soft spheres near the jamming point have been extensively studied both numerically and theoretically; these studies provide a reliable base for the interpretation of experiments. However, recent work by Ikeda et al. showed that, in a parameter space of the temperature and packing fraction, experiments to date on colloids have been rather far from the theoretical scaling regime. An important question is then whether theoretical results concerning point-J are applicable to any physical/experimental system, including granular media, which we consider here. On the surface, such a-thermal, frictional systems might appear even further from the idealized case of thermal soft spheres. In this work we address this question via experiments on shaken granular materials near jamming. We have systematically investigated such systems over a number of years using hard metallic grains. The important feature of the present work is the use of much softer grains, cut from photoelastic materials, making it possible to determine forces at the grain scale, the details of the contact networks and the motion of individual grains. Using this new type of particle, we first show that the contact network exhibits remarkable dynamics. We find strong heterogeneities, which are maximum at the packing fraction ϕ*, distinct from and smaller than the packing fraction ϕ(†), where the average number of contacts per particle, z, starts to increase. In the limit of zero mechanical excitation, these two packing fractions converge at point J. We also determine dynamics on time scales ranging from a small fraction of the shaking cycle to thousands of cycles. We can then map the observed system behavior onto results from simulations of ideal thermal soft spheres. Our results indicate that the ideal jamming point indeed illuminates the world of granular media.

Mesoscale and macroscale kinetic energy fluxes from granular fabric evolution

Recent advances in high-resolution measurements means it is now possible to identify and track the local "fabric" or contact topology of individual grains in a deforming sand throughout loading history. These provide compelling impetus to the development of methods for inferring changes in the contact forces and energies at multiple spatiotemporal scales, using information on grain contacts alone. Here we develop a surrogate measure of the fluctuating kinetic energy based on changes in the local contact topology of individual grains. We demonstrate the method for dense granular materials under quasistatic biaxial shear. In these systems, the initially stable and solidlike response eventually gives way to liquidlike behavior and global failure. This crossover in mechanical behavior, akin to a phase transition, is marked by bursts of kinetic energy and frictional dissipation. Mechanisms underlying this release of energy include the buckling of major load-bearing structures known as force chains. These columns of grains represent major repositories for stored strain energy. Stored energy initially accumulates at all of the contacts along the force chain, but is released collectively when the chain overloads and buckles. The exact quantification of the buildup and release of energy in force chains, and the manner in which force chain buckling propagates in the sample (i.e., diffuse and systemwide versus localized into shear bands), requires detailed knowledge of contact forces. To date, however, the forces at grain contacts continue to elude measurement in natural granular materials like sand. Here, using data from computer simulations, we show that a proxy for the fluctuating kinetic energy in dense granular materials can be suitably constructed solely from the evolving properties of the grain's local contact topology. Our approach directly relates the evolution of fabric to energy flux and makes possible research into the propagation of failure from measurements of grain contacts in real granular materials.