Discrete element numerical simulations of unsteady, homogeneous shear flows have been performed by instantly applying a constant shear rate to a random, static, isotropic assembly of identical, soft, frictional spheres at either zero or finite pressure by keeping constant the solid volume fraction until the steady state is reached. If the system is slowly sheared, or, equivalently, if the particles are sufficiently rigid, the granular material exhibits either large or small fluctuations in the evolving pressure, depending whether the average number of contacts per particle (coordination number) is less or larger than a critical value. The amplitude of the pressure fluctuations is rate-dependent when the coordination number is less than the critical and rate-independent otherwise, signatures of fluid-like and solid-like behaviour, respectively. The same critical coordination number has been previously found to represent the minimum value at which rate-independent components of the stresses develop in steady, simple shearing and the jamming transition in isotropic random packings. The observed complex behaviour of the measured pressure in the fluid-solid transition clearly suggests the need for incorporating in a nontrivial way the coordination number, the solid volume fraction, the particle stiffness and the intensity of the particle agitation in constitutive models for the onset and the arrest of granular flows.
We experimentally study the transition from steady flow to unsteady flow in a quasi-2D granular heap when small amounts of water are added to monodisperse glass spheres. Particles flow uniformly down both sides of the heap for low water content, but unsteady flow occurs as the water content increases. The unsteady flow mode consists of a non-depositing downslope avalanche and an upslope propagating granular jump. The transition from steady to unsteady flow occurs when the slope exceeds a critical angle as a result of water-induced cohesion. Under unsteady flow conditions, the deposited heap co…
The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first "kinetic" stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady "hydrodynamic" stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-…
The time dependent Ginzburg-Landau equation for a two-dimensional granular shear flow is numerically solved, where we study both the transient dynamics and the steady state of the order parameter. The structural changes of the numerical solutions are qualitatively similar to the shear bands observed in the discrete element method (DEM) simulation of the two-dimensional granular shear flow.
Diverging correlation lengths on either side of the jamming transition are used to formulate a rheological model of granular shear flow, based on the propagation of stress through force chain networks. The model predicts three distinct flow regimes, characterized by the shear rate dependence of the stress tensor, that have been observed in both simulations and experiments. The boundaries separating the flow regimes are quantitatively determined and testable. In the limit of jammed granular solids, the model predicts the observed anomalous scaling of the shear modulus and a new relation for the…
We numerically study the linear response of two-dimensional frictional granular materials under oscillatory shear. The storage modulus $G'$ and the loss modulus $G''$ in the zero strain rate limit depend on the initial strain amplitude of the oscillatory shear before measurement. The shear jammed state (satisfying $G'>0$) can be observed at an amplitude greater than a critical initial strain amplitude. The fragile state is defined by the emergence of liquid-like and solid-like states depending on the form of the initial shear. In this state, the observed $G'$ after the reduction of the strain …
We analyse the flow curves of a two-dimensional assembly of granular particles which are interacting via frictional contact forces. For packing fractions slightly below jamming, the fluid undergoes a large scale instability, implying a range of stress and strainrates where no stationary flow can exist. Whereas small systems were shown previously to exhibit hysteretic jumps between the low and high stress branches, large systems exhibit continuous shear thickening arising from averaging unsteady, spatially heterogeneous flows. The observed large scale patterns as well as their dynamics are foun…
We study strain localization in slow shear flow focusing on layered granular materials. A heretofore unknown effect is presented here. We show that shear zones are refracted at material interfaces in analogy with refraction of light beams in optics. This phenomenon can be obtained as a consequence of a recent variational model of shear zones. The predictions of the model are tested and confirmed by 3D discrete element simulations. We found that shear zones follow Snell's law of light refraction.
Weakly nonlinear analysis of a two dimensional sheared granular flow is carried out under the Lees-Edwards boundary condition. We derive the time dependent Ginzburg-Landau (TDGL) equation of a disturbance amplitude starting from a set of granular hydrodynamic equations and discuss the bifurcation of the steady amplitude in the hydrodynamic limit.
We experimentally investigate shear thickening in dense granular suspensions under oscillatory shear. Directly imaging the suspension-air interface, we observe dilation beyond a critical strain $γ_c$ and the end of shear thickening as the maximum confining stress is reached and the contact line moves. Analyzing the shear profile, we extract the viscosity contributions due to hydrodynamics $η_μ$, dilation $η_c$ and sedimentation $η_g$. While $η_g$ governs the shear thinning regime, $η_μ$ and $η_c$ together determine the shear thickening behavior. As the suspending liquid's viscosity varies from…
The uniform longitudinal flow is characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$ is the strain rate, a uniform density $n(t)\propto a(t)$, and a uniform granular temperature $T(t)$. Direct simulation Monte Carlo solutions of the Boltzmann equation for inelastic hard spheres are presented for three (one positive and two negative) representative values of the initial strain rate $a_0$. Starting from different initial conditions, the temporal evolution of the reduced strain rate $a^*\propto a_0/\sqrt{T}$, the non-Newtonian viscosity, the secon…
We examine two basic assumptions of kinetic theory-- binary collisions and molecular chaos-- using numerical simulations of sheared granular materials. We investigate a wide range of densities and restitution coefficients and demonstrate that kinetic theory breaks down at large density and small restitution coefficients. In the regimes where kinetic theory fails, there is an associated emergence of clusters of spatially correlated grains.
We investigate the onset of irreversibility in a dense granular medium subjected to cyclic shear in a split-bottom geometry. To probe the micro and mesoscale we image bead trajectories in 3D throughout a series of shear strain oscillations. Though beads lose and regain contact with neighbors during a cycle, the topology of the contact network exhibits reversible properties for small oscillation amplitudes. With increasing reversal amplitude a transition to an irreversible diffusive regime occurs.
We discuss the uniform shear flow of a fluidized granular bed composed of monodisperse Hertzian spheres. Considering high densities around the glass transition density of inelastic Hertzian spheres, we report kinetic theory expressions for the Newtonian viscosity as well as the Bagnold coefficient. We discuss the dependence of the transport coefficients on density and coefficient of restitution.
We discuss the advantages and results of using a mixing-length, compressible model to account for shear banding behaviour in granular flow. We formulate a general approach based on two function of the solid fraction to be determined. Studying the vertical chute flow, we show that shear band thickness is always independent from flowrate in the quasistatic limit, for Coulomb wall boundary conditions. The effect of bin width is addressed using the functions developed by Pouliquen and coworkers, predicting a linear dependence of shear band thickness by channel width, while literature reports contr…
We extend a recent theory for the rheology of frictionless granular materials [K. Suzuki and H. Hayakawa, Phys. Rev. Lett. 2015, 115, 098001] to the case of frictional disks in two dimensions. Employing a frictional contact model for molecular dynamics simulations, we derive difference equations of the shear stress, the granular temperature, and the spin temperature from the generalized Green-Kubo formula, where all the terms are given by microscopic expressions. The numerical solutions of the difference equations not only describe the flow curve, but also reproduce the hysteresis of shear str…
In this study, we numerically investigated the mechanical responses and trajectories of frictional granular particles under oscillatory shear in the reversible phase where particle trajectories form closed loops below the yielding point. When the friction coefficient is small, the storage modulus exhibits softening, and the loss modulus remains finite in the quasi-static limit. As the friction coefficient increases, the softening and residual loss modulus are suppressed. The storage and loss moduli satisfy scaling laws if they are plotted as functions of the areas of the loop trajectories divi…
The evolution of granular shear flow is investigated as a function of height in a split-bottom Couette cell. Using particle tracking, magnetic-resonance imaging, and large-scale simulations we find a transition in the nature of the shear as a characteristic height $H^*$ is exceeded. Below $H^*$ there is a central stationary core; above $H^*$ we observe the onset of additional axial shear associated with torsional failure. Radial and axial shear profiles are qualitatively different: the radial extent is wide and increases with height while the axial width remains narrow and fixed.
We perform numerical simulations of a two-dimensional frictional granular system under oscillatory shear confined by constant pressure. We found that the system undergoes dilatancy as the strain increases. We confirmed that compaction also takes place at an intermediate strain amplitude for a small mutual friction coefficient between particles. We also found that compaction depends on the confinement pressure while dilatancy little depends on the pressure.
Slow and dense granular flows often exhibit narrow shear bands, making them ill-suited for a continuum description. However, smooth granular flows have been shown to occur in specific geometries such as linear shear in the absence of gravity, slow inclined plane flows and, recently, flows in split-bottom Couette geometries. The wide shear regions in these systems should be amenable to a continuum description, and the theoretical challenge lies in finding constitutive relations between the internal stresses and the flow field. We propose a set of testable constitutive assumptions, including rat…
We study the emergence of shear thickening in dense suspensions of non-Brownian particles. We combine local velocity and concentration measurements using Magnetic Resonance Imaging with macroscopic rheometry experiments. In steady state, we observe that the material is heterogeneous, and we find that that the local rheology presents a continuous transition at low shear rate from a viscous to a shear thickening, Bagnoldian, behavior with shear stresses proportional to the shear rate squared, as predicted by a scaling analysis. We show that the heterogeneity results from an unexpectedly fast mig…
Herein, we numerically study the rheology of a two-dimensional frictional granular system confined by constant pressure under oscillatory shear. Several scaling laws for the storage and loss moduli against the scaled strain amplitude have been found. The scaling laws in plastic regime for large strain amplitude can be understood by the angular distributions of the contact force. The scaling exponents are estimated by considering the physical mechanism.
Refraction and deflection of shear zones in layered granular materials was studied experimentally and numerically. We show, that (i) according to a recent theoretical prediction [T. Unger, Phys. Rev. Lett. 98, 018301 (2007)] shear zones refract in layered systems in analogy with light refraction, (ii) zone refraction obeys Snell's law known from geometric optics and (iii) under natural pressure conditions (i.e. in the presence of gravity) the zone can also be deflected by the interface so that the deformation of the high friction material is avoided.
In this letter, we discuss how flow inhomogeneity affects the self-diffusion behavior in granular flows. Whereas self-diffusion scalings have been well characterized in the past for homogeneous shearing, the effect of shear localization and nonlocality of the flow has not been studied. We therefore present measurements of self-diffusion coefficients in discrete numerical simulations of steady, inhomogeneous, and collisional shearing flows of nearly identical, frictional, and inelastic spheres. We focus on a wide range of dense solid volume fractions, that correspond to geophysical and industri…
We present numerical results on spontaneous symmetry breaking strain localization in axisymmetric triaxial shear tests of granular materials. We simulated shear band formation using three-dimensional Distinct Element Method with spherical particles. We demonstrate that the local shear intensity, the angular velocity of the grains, the coordination number, and the local void ratio are correlated and any of them can be used to identify shear bands, however the latter two are less sensitive. The calculated shear band morphologies are in good agreement with those found experimentally. We show that…
We perform numerical simulations to examine particle diffusion at steady shear in a model granular material in two dimensions at the jamming density and zero temperature. We confirm findings by others that the diffusion constant depends on shear rate as $D\sim\dotγ^{q_D}$ with $q_D<1$, and set out to determine a relation between $q_D$ and other exponents that characterize the jamming transition. We then examine the the velocity auto-correlation function, note that it is governed by two processes with different time scales, and identify a new fundamental exponent, $λ$, that characterizes an alg…
Using particle trajectory data obtained from x-ray tomography, we determine two kinds of effective temperatures in a cyclically sheared granular system. The first one is obtained from the fluctuation-dissipation theorem which relates the diffusion and mobility of lighter tracer particles immersed in the system. The second is the Edwards compactivity defined via the packing volume fluctuations. We find robust excellent agreement between these two temperatures, independent of the type of the tracers, cyclic shear amplitudes, and particle surface roughness. We further elucidate that in granular s…
The transition between shear-flowing and shear-arrested states of frictional granular matter is studied using constant-stress discrete element simulations. By subjecting a dilute system of frictional grains to a constant external shear stress and pressure, friction-dependent critical shear stress and density are clearly identified with both exhibiting a crossover between low and high friction. The critical shear stress bifurcates two nonequilibrium steady states: (i) steady state shear flow characterized by a constant deformation rate, and (ii) shear arrest characterized by temporally decaying…
This paper is devoted to the basic question of what factors determine the strain field in a quasi static granular flow. It is shown that using stress - strain rate relations is not the proper way to understand quasi static rheology. An alternative approach is discussed where the local deformation is regarded as the cause of deformation in the vicinity. We suggest a continuum model where the local shear strain is proportional to its Laplacian and the proportionality factor is determined by the local stress. The predicted behavior is tested in a three dimensional shear cell by means of computer …
We present experimental results on dense corn-starch suspensions as examples of non-Brownian, nearly-hard particles that undergo continuous and discontinuous shear thickening (CST and DST) at intermediate and high densities respectively. Our results offer strong support for recent theories involving a stress-dependent effective contact friction among particles. We show however that in the DST regime, where theory might lead one to expect steady-state shear bands oriented layerwise along the vorticity axis, the real flow is unsteady. To explain this, we argue that steady-state banding is generi…
The evolution of the internal granular structure in shear-arrested and shear-flowing states of granular materials is characterized using fabric tensors as descriptors of the internal contact and force networks. When a dilute system of frictional grains is subjected to a constant pressure and shear stress, the bulk stress ratio is well-predicted from the anisotropy of its contact and force networks during transient flow prior to steady shear flow or shear arrest. Although the onset of shear arrest is a stochastic process, the fabric tensors upon arrest are distributed around nearly equal contri…
We investigate the emergence of long-range correlations in granular shear flow. By increasing the density of a simulated granular flow we observe a spontaneous transition from a dilute regime, where interactions are dominated by binary collisions, to a dense regime characterized by large force networks and collective motions. With increasing density, interacting grains tend to form networks of simultaneous contacts due to the dissipative nature of collisions. We quantify the size of these networks by measuring correlations between grain forces and find that there are dramatic changes in the st…
We computationally determine the force on single spherical intruder particles in sheared granular flows as a function of particle size, particle density, shear rate, overburden pressure, and gravitational acceleration. The force scales similarly to, but deviates from, the buoyancy force predicted by Archimedes' principle. The deviation depends only on the intruder to bed particle size ratio, but not the density ratio or flow conditions. We propose a simple force model that successfully predicts whether intruders rise or sink, knowing only the size and density ratios, for a variety of flow conf…
While convective flows are implicated in many granular segregation processes, the associated particle-scale rearrangements are not well understood. A three-dimensional bidisperse mixture segregates under steady shear, but the cyclically driven system either remains mixed or segregates slowly. Individual grain motion shows no signs of particle-scale segregation dynamics that precede bulk segregation. Instead, we find that the transition from non-segregating to segregating flow is accompanied by significantly less reversible particle trajectories, and the emergence of a convective flow field.
The properties of confined granular flows are studied through discrete numerical simulations. Two types of flows with different boundaries are compared: (i) gravity-driven flows topped with a free surface and over a base where erosion balances accretion (ii) shear-driven flows with a constant pressure applied at their top and a bumpy bottom moving at constant velocity. In both cases we observe shear localization over or/and under a creep zone. We show that, although the different boundaries induce different flow properties (e.g. shear localization of transverse velocity profiles), the two type…
We consider dense rapid shear flow of inelastically colliding hard disks. Navier-Stokes granular hydrodynamics is applied accounting for the recent finding \cite{Luding,Khain} that shear viscosity diverges at a lower density than the rest of constitutive relations. New interpolation formulas for constitutive relations between dilute and dense cases are proposed and justified in molecular dynamics (MD) simulations. A linear stability analysis of the uniform shear flow is performed and the full phase diagram is presented. It is shown that when the inelasticity of particle collision becomes large…
Simulation of a system consisted of free particle bouncing on a vertically vibrated based is performed. Two different states, which are steady and unsteady energy transfer state are found. The vibrating based is hold at constant vibration frequency $f = 0.1$ as the vibration amplitude $A$ varied. Sinusoidal form is used. Granular temperature $T_g$ as function of based velocity and coefficient of restitution is used but shown no role in determining energy transfer state of the system. Peak of free particle trajectory $x_m$ around value 20 seperate region of the two states.
Computer simulations of sheared granular fluids, modeled as inelastic hard spheres, are presented which show signs of a uniquely three-dimensional instabilty. In the stable regime, a linear velocity profile, $v_{x}=ay$, with shear rate $a$ is established using Lees-Edwards boundary conditions. In the unstable regime, the velocity profile aquires a dependence on the third dimension of the form $v_{x}=ay+sin(2πz/L)$ in a cubic box with sides of length $L$. An analysis of the linearized Navier-Stokes equations shows the presence of an instability and gives a simple expression for the critical wav…
We present an X-ray tomography study of the segregation mechanisms of tracer particles in a three-dimensional cyclically sheared bi-disperse granular medium. Big tracers are dragged by convection to rise to the top surface and then remain trapped there due to the small downward convection cross-section, which leads to segregation. Additionally, we also find that the local structural up-down asymmetry due to arching effect around big tracers will induce the tracers to have a net upward displacement against its smaller neighbors, which is another mechanism for segregation.
We describe experiments that probe the evolution of shear jammed states, occurring for packing fractions $ϕ_S \leq ϕ\leq ϕ_J$, for frictional granular disks, where above $ϕ_J$ there are no stress-free static states. We use a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This fixed $ϕ$ system exhibits coupling between the shear strain, $γ$, and the pressure, $P$, which we characterize by the `Reynolds pressure', and a `Reynolds coefficient', $R(ϕ) = (\partial ^2 P/\partial γ^2)/2$. $R$ depends only on $ϕ$, and diverges as $R \sim (ϕ_c - ϕ)^α$, where $ϕ…
The shear strength of cohesionless granular materials is generally attributed to the compactness or anisotropy of their microstructure. An open issue is how such compact or anisotropic microstructures, and thus the shear strength, depend on the particle properties. We first recall the role of fabric and force anisotropies with respect to the critical-state shear stress. Then, a model of accessible geometrical states in terms of particle connectivity and contact anisotropy is presented. This model incorporates in a simple way the fact that, due to steric exclusions, the highest levels of connec…
Shear localization in granular materials is studied experimentally and numerically. The system consists of two material layers with different effective frictions. The presence of the material interface leads to a special type of "total internal reflection" of the shear zone. In a wide range of configurations the reflection is characterized by a fixed angle which is analogous to the critical angle of refraction in optics. The zone leaves and reenters the high friction region at this critical angle and in between it stays near the interface in the low friction region. The formalism describing th…
We study the jamming phase diagram of sheared granular material using a novel Couette shear set-up with multi-ring bottom. The set-up 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 set-up 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 b…
Dynamic experiments with Al-W granular/porous composites revealed qualitatively different behavior with respect to shear localization depending on bonding between Al particles. Two-dimensional numerical modeling was used to explore the mesomechanics of the large strain dynamic deformation in Al-W granular/porous composites and explain the experimentally observed differences in shear localization between composites with various mesostructures. Specifically, the bonding between the Al particles, the porosity, the roles of the relative particle sizes of Al and W, the arrangements of the W particl…
Jammed states of frictional granular systems can be induced by shear strain at densities below the isostatic jamming density ($ϕ_c$). It remains unclear, however, how much friction affects this so-called shear-jamming. Friction appears in two ways in this type of experiment: friction between particles, and friction between particles and the base on which they rest. Here, we study how particle-bottom friction, or basal friction, affects shear jamming in quasi-two dimensional experiments. In order to study this issue experimentally, we apply simple shear to a disordered packing of photoelastic d…
Studies of granular materials, both theoretical and experimental, are often restricted to convex grain shapes. We demonstrate that a non-convex grain shape can lead to a qualitatively novel macroscopic dynamics. Spatial crosses (hexapods) are continuously sheared in a split-bottom container. Thereby, they develop a secondary flow profile that is completely opposite to that of rod-shaped or lentil-shaped convex grains in the same geometry. The crosses at the surface migrate towards the rotation center and sink there, mimicking a `reverse Weissenberg effect'. The observed surface flow field sugg…
The shear modulus of jammed frictional granular materials with the harmonic repulsive interaction under an oscillatory shear is numerically investigated. It is confirmed that the storage modulus, the real part of the shear modulus, for frictional grains with sufficiently small strain amplitude $γ_0$ discontinuously emerges at the jamming transition point. The storage modulus for small $γ_0$ differs from that of frictionless grains even in the zero friction limit, while they are almost identical with each other for sufficiently large $γ_0$, where the transition becomes continuous. The stress-st…
Several mechanisms cause a granular material to strengthen over time at low applied stress. The strength is determined from the maximum frictional force F_max experienced by a shearing plate in contact with wet or dry granular material after the layer has been at rest for a waiting time τ. The layer strength increases roughly logarithmically with τ-only- if a shear stress is applied during the waiting time. The mechanisms of strengthening are investigated by sensitive displacement measurements and by imaging of particle motion in the shear zone. Granular matter can strengthen due to a slow shi…
The size-segregation of granular materials, a process colloquially known as the Brazil Nut Effect, has generally been thought to proceed faster the greater the size difference of the particles. We experimentally investigate sheared bidisperse granular materials as a function of the size ratio of the two species, and find that the mixing rate at low confining pressure behaves as expected from percolation-based arguments. However, we also observe an anomalous effect for the re-segregation rates, wherein particles of both dissimilar and similar sizes segregate more slowly than intermediate partic…
We report the results of three-dimensional molecular dynamics simulations of sheared granular system in a Couette geometry.\cite{movies} The simulations use realistic boundary conditions that may be expected in physical experiments. For a range of boundary properties we report velocity and density profiles, as well as forces on the boundaries. In particular, we find that the results for the velocity profiles throughout the shearing cell depend strongly on the interaction of the system particles with the physical boundaries. Even frictional boundaries can allow for significant slippage of the p…
