We have made experimental observations of the force networks within a two-dimensional granular silo similar to the classical system of Janssen. Models like that of Janssen predict that pressure within a silo saturates with depth as the result of vertical forces being redirected to the walls of the silo where they can then be carried by friction. By averaging ensembles of experimentally-obtained force networks in different ways, we compare the observed behavior with various predictions for granular silos. We identify several differences between the mean behavior in our system and that predicted by Janssen-like models: We find that the redirection parameter describing how the force network transfers vertical forces to the walls varies with depth. We find that changes in the preparation of the material can cause the pressure within the silo to either saturate or to continue building with depth. Most strikingly, we observe a non-linear response to overloads applied to the top of the material in the silo. For larger overloads we observe the previously reported "giant overshoot" effect where overload pressure decays only after an initial increase [G. Ovarlez et al., Phys. Rev. E 67, 060302(R) (2003)]. For smaller overloads we find that additional pressure propagates to great depth. This effect depends on the particle stiffness, as given for instance by the Young's modulus, E, of the material from which the particles are made. Important measures include E, the unscreened hydrostatic pressure, and the applied load. These experiments suggest that when the load and the particle weight are comparable, particle elasticity acts to stabilize the force network, allowing non-linear network effects to be seen in the mean behavior.
Acceleration and sound measurements during granular discharge from silos are used to show that silo music is a sound resonance produced by silo quake. The latter is produced by stick-slip friction between the wall and the granular material in tall narrow silos. For the discharge rates studied, the occurrence and frequency of flow pulsations are determined primarily by the surface properties of the granular material and the silo wall. The measurements show that the pulsating motion of the granular material drives the oscillatory motion of the silo and the occurrence of silo quake does not requi…
We report experimental observations of sound waves in a granular material during resonant silo discharge called silo music. The grain motion was tracked by high speed imaging while the resonance of the silo was detected by accelerometers and acoustic methods. The grains do not oscillate in phase at neighboring vertical locations, but information propagates upward in this system in the form of sound waves. We show that the wave velocity is not constant throughout the silo, but considerably increases towards the lower end of the system, suggesting increased pressure in this region, where the flo…
This paper is aimed at pursuing a recent discussion about the comparison between Self-Organised Criticality, the jamming process and the percolation theory in the problem of a silo discharge [I. Zuriguel, A. Garcimartin, D. Maza, L.A.Pugnaloni, J.M.Pastor, "Jamming during the discharge of granular matter from a silo", Phys.Rev.E 71, 051303 (2005)]. Statistics of blocking a silo is investigated from different models: percolation, self organised criticality..
The side-wall pressure distribution in static silos is an important and open problem related to storage safety of granular materials. Referring to Lvin's theory [Powder Technology 4, 280 (1971)] and Rodolfo's 2D numerical result [Phys. Rev. E 97.1(2018)], a new theoretical model of side wall pressure is established by assuming the friction coefficient as well as the ratio of horizontal to vertical stress changing with depth position in silo. Furthermore, the transition of stress state between hydrostatic-like and Janssen-like is systematically investigated by our model and experiment, and fina…
We use Topological Data Analysis to study the post buckling behavior of laboratory scale cylindrical silos under gravity driven granular discharges. Thin walled silos buckle during the discharge if the initial height of the granular column is large enough. The deformation of the silo is reversible as long as the filling height does not exceed a critical value, $L_c$. Beyond this threshold the deformation becomes permanent and the silo often collapses. We study the dynamics of reversible and irreversible deformation processes, varying the initial filling height around $L_c$. We find that all re…
Resistance force exerted on an obstacle in a gravity-driven slow granular silo flow is studied by experiments and numerical simulations. In a two-dimensional granular silo, an obstacle is placed just above the exit. Then, steady discharge flow is made and its flow rate can be controlled by the width of exit and the position of obstacle. During the discharge of particles, flow rate and resistance force exerting on the obstacle are measured. Using the obtained data, a dimensionless number characterizing the force balance in granular flow is defined by the relation between the discharge flow rate…
Granular flow in a silo demonstrates multiple nonlocal rheological phenomena due to the finite size of grains. We solve the Nonlocal Granular Fluidity (NGF) continuum model in quasi-2D silo geometries and evaluate its ability to predict these nonlocal effects, including flow spreading and, importantly, clogging (arrest) when the opening is small enough. The model is augmented to include a free-separation criterion and is implemented numerically with an extension of the trans-phase granular flow solver described in arXiv:1411.5447, to produce full-field solutions. The implementation is validate…
We investigate the mixing characteristics of dry granular material while draining down a silo with multiple exit orifices. The mixing in the silo, which otherwise consists of noninteracting stagnant and flow regions, is observed to improve significantly when the flow through specific orifices is stopped intermittently. This momentary stoppage of flow through the orifice is either controlled manually or is chosen by the system itself when the orifice width is small enough to cause spontaneous jamming and unjamming. We observe that the overall mixing behavior shows a systematic dependence on the…
Granular material is often handled using silos with inserts in industrial processes to prevent "rat-holing" and similar behaviour, optimise mixing behaviour, and prevent segregation. We study the mass flow rate and the static zones of granular silos using the $μ(I)$ rheology in the incompressible Navier-Stokes equations. We also extend the $μ(I)$ model to incorporate pseudo-dilatancy and nonlocal fluidity. We find that insert shape has a significant effect on flow rate and static material. We also find that the effect of inserts on flow rate is not strongly affected by the extensions, while th…
For granular materials falling through a circular exit at the bottom of a silo, no continuous flow can be sustained when the diameter D of the exit is less than 5 times the characteristic size of the grains. If the bottom of the silo rotates horizontally with respect to the wall of the silo, finite flow rate can be sustained even at small D. We investigate the effect of bottom rotation to the flow rate of a cylindrical silo filled with mono-disperse plastic beads of d = 6 mm diameter. We find that the flow rate W follows Beverloo Law down to D = 1.2d and that W increases with the rotation rate…
We investigate, at a laboratory scale, the collapse of cylindrical shells of radius $R$ and thickness $t$ induced by a granular discharge. We measure the critical filling height for which the structure fails upon discharge. We observe that the silos sustain filling heights significantly above an estimation obtained by coupling standard shell-buckling and granular stress distribution theories. Two effects contribute to stabilize the structure: (i) below the critical filling height, a dynamical stabilization due to granular wall friction prevents the localized shell-buckling modes to grow irreve…
The granular silo is one of the many interesting illustrations of the thixotropic property of granular matter: a rapid flow develops at the outlet, propagating upwards through a dense shear flow while material at the bottom corners of the container remains static. For large enough outlets, the discharge flow is continuous; however, by contrast with the clepsydra for which the flow velocity depends on the height of fluid left in the container, the discharge rate of granular silos is constant. Implementing a plastic rheology in a 2D Navier-Stokes solver (following the mu(I)-rheology or a constan…
We present an experimental study of segregation of granular matter in a quasi-two dimensional silo emptying out of an orifice. Size separation is observed when multi-sized particles are used with the larger particles found in the center of the silo in the region of fastest flow. We use imaging to study the flow inside the silo and quantitatively measure the concentration profiles of bi-disperse beads as a function of position and time. The angle of the surface is given by the angle of repose of the particles, and the flow occurs in a few layers only near the top of this inclined surface. The f…
Using both a continuum Navier-Stokes solver, with the mu(I)-flow-law implemented to model the viscous behavior, and the discrete Contact Dynamics algorithm, the discharge of granular silos is simulated in two dimensions from the early stages of the discharge until complete release of the material. In both cases, the Beverloo scaling is recovered. We first do not attempt quantitative comparison, but focus on the qualitative behavior of velocity and pressure at different locations in the flow. A good agreement is obtained in the regions of rapid flows, while areas of slow creep are not entirely …
We conducted simple experiment and numerical simulation of two-dimensional granular discharge flow driven by gravity under the influence of an obstacle. According to the previous work (Zuriguel {\it et al.,\,Phys.\,Rev.\,Lett.}\,{\bf 107}: 278001, 2011), the clogging of granular discharge flow can be suppressed by putting a circular obstacle at a proper position. In order to investigate the details of obstacle effect in granular flow, we focused on particle dynamics in this study. From the experimental and numerical data, we found that the obstacle remarkably affects the horizontal-velocity di…
"Granular elasticity," useful for calculating static stress distributions in granular media, is generalized by including the effects of slowly moving, deformed grains. The result is a hydrodynamic theory for granular solids that agrees well with models from soil mechanics.
Although fully elastic when static, granular media become transiently elastic when being slowly sheared -- during which both the elastic energy and stress relax. Starting from this observation, we cogently derive the framework for granular hydrodynamics, a set of differential equations consistent with general principles of physics, especially reversible and irreversible thermodynamics. In addition, an expression for the granular elastic energy is reviewed and further discussed.
We measure the flow of granular materials inside a quasi-two dimensional silo as it drains and compare the data with some existing models. The particles inside the silo are imaged and tracked with unprecedented resolution in both space and time to obtain their velocity and diffusion properties. The data obtained by varying the orifice width and the hopper angle allows us to thoroughly test models of gravity driven flows inside these geometries. All of our measured velocity profiles are smooth and free of the shock-like discontinuities ("rupture zones") predicted by critical state soil mechanic…
An self-contained elastic theory is derived which accounts both for mechanical yield and shear-induced volume dilatancy. Its two essential ingredients are thermodynamic instability and the dependence of the elastic moduli on compression.
We present a differential equation for the flow rate of granular materials during the discharge of a silo. This is based in the energy balance of the variable mass system in contrast with the traditional derivations based on heuristic postulates such as the free fall arch. We show that this new equation is consistent with the well known Beverloo rule, providing an independent estimate for the universal Beverloo prefactor. We also find an analytic expression for the pressure under discharging conditions.
Granular material flowing through bottlenecks like the openings of silos tend to clog and to inhibit further flow. We study this phenomenon in a three-dimensional hopper for spherical and shape-anisotropic particles by means of X-ray tomography. The X-ray tomograms provide information on the bulk of the granular filling, and allows to determine the particle positions and orientations inside the silo. In addition, it allows to calculate local packing densities in different parts of the container. We find that in the flowing zone of the silo particles show a preferred orientation and thereby a h…
Experiments on isotropic compression of a granular assembly of spheres show that the shear and bulk moduli vary with the confining pressure faster than the 1/3 power law predicted by Hertz-Mindlin effective medium theories (EMT) of contact elasticity. Moreover, the ratio between the moduli is found to be larger than the prediction of the elastic theory by a constant value. The understanding of these discrepancies has been a longstanding question in the field of granular matter. Here we perform a test of the applicability of elasticity theory to granular materials. We perform sound propagation …
In spite of many attempts to model dense granular flow, there is still no general theory capable of describing different types of flows, such as gravity-driven drainage in silos and wall-driven shear flows in Couette cells. Here, we summarize our recent proposal of the Stochastic Flow Rule (SFR), which is able to describe these cases in good agreement with experiments, and we focus on testing the theory in more detail against brute-force simulations with the discrete-element method (DEM). The SFR is a general rate-independent constitutive law for plastic flow, based on diffusing spots of fluid…
We study the formation of shocks on the surface of a granular material draining through an orifice at the bottom of a quasi-two dimensional silo. At high flow rates, the surface is observed to deviate strongly from a smooth linear inclined profile giving way to a sharp discontinuity in the height of the surface near the bottom of the incline, the typical response of a choking flow such as encountered in a hydraulic jump in a Newtonian fluid like water. We present experimental results that characterize the conditions for the existence of such a jump, describe its structure and give an explanati…
Composite of two kinds of grains is modeled in two-dimension and the elasticity is calculated using molecular dynamics method implementing Gear predictor-corrector of fifth order. It has been observed that same composite concentration can be represented by several configurations of the two kinds of grains. Simulation results show a peak or maximum value of a parameter proportional to Young's modulus, which quantitatively agrees with reported experiment results. Ratio of used in simulation has influenced average value of the mentioned parameter monotonically, it is increasing as the ratio incre…
We propose the description of the granular matter which is based on distribution of dry friction coefficients. Using such a concept and a simple one-dimensional packing of grains we solve the silo problem. The friction coefficients at contacts are determined both by geometry of packing configuration and the stress distribution in a medium. Within such an approach the Janssen coefficient is determined and its dependence on the particle-particle and boundary-particle friction coefficients is obtained. Also we investigate the conditions for the appearance of the maximum in the pressure distributi…
Granular gravity driven flows of glass beads have been observed in a silo with a flat bottom. A DC high electric field has been applied perpendicularly to the silo to tune the cohesion. The outlet mass flow has been measured. An image subtraction technique has been applied to visualize the flow geometry and a spatiotemporal analysis of the flow dynamics has been performed. The outlet mass flow is independent of voltage, but a transition from funnel flow to rathole flow is observed. This transition is of probabilistic nature and an intermediate situation exists between the funnel and the rathol…
The flow rate of water through an orifice at the bottom of a container depends on the hydrostatic pressure whereas for a dry granular material is nearly constant. But what happens during the simultaneous discharge of grains and liquid from a silo? By measuring the flow rate as a function of time, we found that: (i) different regimes appear, going from constant flow rate dominated by the effective fluid viscosity to a hydrostatic-like discharge depending on the aperture and grain size, (ii) the mixed material is always discharged faster than dry grains but slower than liquid, (iii) for the mixt…
Electrostatic theory preserves charges, but allows dipolar excitations. Elasticity theory preserves dipoles, but allows quadrupolar (Eshelby like) plastic events. Charged amorphous granular systems are interesting in their own right; here we focus on their plastic instabilities and examine their mechanical response to external strain and to external electric field, to expose the competition between elasticity and electrostatics. In this paper a generic model is offered, its mechanical instabilities are examined and a theoretical analysis is presented. Plastic instabilities are discussed as sad…
Transmission of forces in static granular materials are studied within the framework of the force network ensemble, by numerically evaluating the mechanical response of hexagonal packings of frictionless grains and rectangular packings of frictional grains. In both cases, close to the point of application of the overload, the response is non-linear and displays two peaks, while at larger length-scales it is linear and elastic-like. The cross-over between these two behaviors occurs at a depth that increases with the magnitude of the overload, and decreases with increasing friction.
A general approach is presented for understanding the stress response function in anisotropic granular layers in two dimensions. The formalism accommodates both classical anisotropic elasticity theory and linear theories of anisotropic directed force chain networks. Perhaps surprisingly, two-peak response functions can occur even for classical, anisotropic elastic materials, such as triangular networks of springs with different stiffnesses. In such cases, the peak widths grow linearly with the height of the layer, contrary to the diffusive spreading found in `stress-only' hyperbolic models. In…
Granular elasticity, an elasticity theory useful for calculating static stress distribution in granular media, is generalized to the dynamic case by including the plastic contribution of the strain. A complete hydrodynamic theory is derived based on the hypothesis that granular medium turns transiently elastic when deformed. This theory includes both the true and the granular temperatures, and employs a free energy expression that encapsulates a full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. For the special case of stationary granu…
The dense flow of air bubbles in a two-dimensional silo (through an aperture D) filled with a liquid is studied experimentally. A particle tracking technique has been used to bring out the main properties of the flow: displacements of the bubbles, transverse and axial velocities. The behavior of the air bubbles is observed to present similarities with non-deformable solid grains in a granular flow. Nevertheless, a correlation between the bubble velocities and their deformations has been evidenced. Moreover, a new discharge law (Beverloo-like) must be considered for such a system, where the flo…
In this paper, we have designed and employed a suspended-wall silo to remove Janssen effect in order to explore directly the local pressure dependence of Granular Orifice Flow (GOF) systematically. We find that as Janssen effect is removed, the flow rate Q changes linearly with the external pressure. The slope α of the linear change decays exponentially with the ratio of the silo size and the size of the orifice Φ/D, which suggests the existence of a characteristic ratio λ (~2.4). When Φ/D > λ, α gradually decays to zero, and the effect of external pressure on the GOF becomes negligible, where…
By mixing glass beads with a curable polymer we create a well-defined cohesive granular medium, held together by solidified, and hence elastic, capillary bridges. This material has a geometry similar to a wet packing of beads, but with an additional control over the elasticity of the bonds holding the particles together. We show that its mechanical response can be varied over several orders of magnitude by adjusting the size and stiffness of the bridges, and the size of the particles. We also investigate its mechanism of failure under unconfined uniaxial compression in combination with in situ…
We investigate experimentally the mechanical response to shear of a 2D packing of grains across the jamming transition. First, we develop a dedicated experimental setup, together with tracking and photoelastic techniques in order to prepare the packing in a controlled fashion and to quantify the stress and strain tensors at the grain scale. Second, we install a inflating probe (a 2D "balloon"), which shears the packing with a cylindrical symmetry. We probe experimentally stresses and strains for strain amplitudes as low as $10^{-3}$ and for a range of packing fractions within $2\%$ variation a…
Granular jamming has been identified as a fundamental mechanism for the operation of robotic grippers. In this work, we show, that soft particles like expanded polystyrene beads lead to significantly larger gripping forces in comparison to rigid particles. In contradiction to naive expectation, the combination of jamming and elasticity gives rise to very different properties of the jammed phase, compared to hard-particle systems. This may be of interest also beyond the application in robotic grippers.
Due to the non-linearity of Hertzian contacts, the speed of sound in granular matter increases with pressure. Under gravity, the non-linear elastic description predicts that acoustic propagation is only possible through surface modes, called Rayleigh-Hertz modes and guided by the index gradient. Here we directly evidence these modes in a controlled laboratory experiment and use them to probe the elastic properties of a granular packing under vanishing confining pressure. The shape and the dispersion relation of both transverse and sagittal modes are compared to the prediction of non-linear ela…
We study experimentally the distribution of local stresses in a granular material confined inside a vertical cylinder. We use an image correlation technique to measure the displacement field of the container induced by the forces exerted by the grains on the inner wall. We describe an optimization procedure based on the linear theory of elastic shells to deduce the distribution of these forces from the measured displacement field. They correspond to the stress field of the granular material close to the container's inner wall. We first confirm the validity of Janssen's description for various …
The mechanical response of naturally abundant amorphous solids such as gels, jammed grains, and biological tissues are not described by the conventional paradigm of broken symmetry that defines crystalline elasticity. In contrast, the response of such athermal solids are governed by local conditions of mechanical equilibrium, i.e., force and torque balance of its constituents. Here we show that these constraints have the mathematical structure of a generalized electromagnetism, where the electrostatic limit successfully captures the anisotropic elasticity of amorphous solids. The emergence of …
We study the effect of fluids on the angle of repose and the segregation of granular matter poured into a silo. The experiments are conducted in two regimes where: (i) the volume fraction of the fluid is small and it forms liquid bridges between particles, and (ii) the particles are completely immersed in the fluid. The data is obtained by imaging the pile formed inside a quasi-two dimensional silo through the transparent glass side walls. In the first series of experiments, the angle of repose is observed to increase sharply with the volume fraction of the fluid and then saturates at a value …
In a recent paper [Phys. Rev. X 12, 031021], we reported experimental observations of ``ultrastable'' states in a shear-jammed granular system subjected to small-amplitude cyclic shear. In such states, all the particle positions and contact forces are reproduced after each shear cycle so that a strobed image of the stresses and particle positions appears static. In the present work, we report further analyses of data from those experiments to characterize both global and local responses of ultrastable states within a shear cycle, not just the strobed dynamics. We find that ultrastable states f…
We study the transport properties of particles draining from a silo using imaging and direct particle tracking. The particle displacements show a universal transition from super-diffusion to normal diffusion, as a function of the distance fallen, independent of the flow speed. In the super-diffusive (but sub-ballistic) regime, which occurs before a particle falls through its diameter, the displacements have fat-tailed and anisotropic distributions. In the diffusive regime, we observe very slow cage breaking and Peclet numbers of order 100, contrary to the only previous microscopic model (based…
We present experimental results on the shape of arches that block the outlet of a two dimensional silo. For a range of outlet sizes, we measure some properties of the arches such as the number of particles involved, the span, the aspect ratio, and the angles between mutually stabilizing particles. These measurements shed light on the role of frictional tangential forces in arching. In addition, we find that arches tend to adopt an aspect ratio (the quotient between height and half the span) close to one, suggesting an isotropic load. The comparison of the experimental results with data from nu…
The present work reports a novel investigation into the collisional dynamics of particles in the vicinity of the outlet of a 2D silo using molecular dynamics simulations. Most studies on this granular system focus in the bulk of the medium. In this region contacts are permanent or long-lived, so continuous approximations are able to yield results for velocity distributions or mass flow. Close to the exit, however, the density of the medium decreases and contacts are instantaneous. Thus, the collisional nature of the dynamics becomes significant, warranting a dedicated investigation as carried …
The transmission rules of interparticle forces between granular particles were analyzed, and a discrete packing model was proposed to calculate the static pressure at the bottom of granular material confined in a vertical column. The mechanical analysis and numerical simulation results indicate that the silo effect is caused by the frictional contacts between border particles and inner walls, the static pressure at the bottom depends on the external load initially, and then tends to a saturation pressure (Pn) in an exponential form. The saturation pressure (Pn) is positive linear related to th…
Segregation and mixing of granular mixtures during heap formation have important consequences in industry and agriculture. This research investigates three different final particle configurations of bi-disperse granular mixtures -- stratified, segregated and mixed -- during filling of quasi-two dimensional silos. We consider a larger number and relatively wider range of control parameters than previous studies, including particle size ratio, flow rate, system size and heap rise velocity. The boundary between stratified and unstratified states is primarily controlled by the two-dimensional flow…
The response to a localized force provides a sensitive test for different models of stress transmission in granular solids. The elasto-plastic models traditionally used by engineers have been challenged by theoretical and experimental results which suggest a wave-like (hyperbolic) propagation of the stress, as opposed to the elliptic equations of static elasticity. Numerical simulations of two-dimensional granular systems subject to a localized external force are employed to examine the nature of stress transmission in these systems as a function of the magnitude of the applied force, the fric…
Very recently, To et al.~have experimentally explored granular flow in a cylindrical silo, with a bottom wall that rotates horizontally with respect to the lateral wall \cite{Kiwing2019}. Here, we numerically reproduce their experimental findings, in particular, the peculiar behavior of the mass flow rate $Q$ as a function of the frequency of rotation $f$. Namely, we find that for small outlet diameters $D$ the flow rate increased with $f$, while for larger $D$ a non-monotonic behavior is confirmed. Furthermore, using a coarse-graining technique, we compute the macroscopic density, momentum, a…
We investigate the Green function of two-dimensional dense random packings of grains in order to discriminate between the different theories of stress transmission in granular materials. Our computer simulations allow for a detailed quantitative investigation of the dynamics which is difficult to obtain experimentally. We show that both hyperbolic and parabolic models of stress transmission fail to predict the correct stress distribution in the studied region of the parameters space. We demonstrate that the compressional and shear components of the stress compare very well with the predictions…
