The penetration by a gravity driven impact of a solid sphere into a granular medium is studied by two-dimensional simulations. The scaling laws observed experimentally for both the final penetration depth and the stopping time with the relevant physical parameters are here recovered numerically without the consideration of any solid friction. Collisional processes are thus found as essential in explaining the physics of the qualitatively observed phenomena whereas frictional processes can only be considered as secondary effects in the granular penetration by impact.
Understanding the penetration dynamics of intruders in granular beds is relevant not only for fundamental physics, but also for geophysical processes and construction on sediments or granular soils in areas potentially affected by earthquakes. In this work, we use Lock-in accelerometry to study the penetration of intruders into quasi-2D granular matter fluidized by lateral shaking. We observed that there are two well-defined stages in the penetration dynamics as the intruder sinks into the granular material.
DEM simulations were used to study the sensitivity of a granular system to the change of the mechanical properties of the grains, applied to the penetration of a cylindrical intruder near a vertical wall. The simulations reproduce a real experimental setup, changing the friction, the Young modulus and the restitution coefficient of the particles.
We study experimentally the influence of confinement on the penetration depth of impacting spheres into a granular medium contained in a finite cylindrical vessel. The presence of close lateral walls reduces the penetration depth, and the characteristic distance for these lateral wall effects is found to be of the order of one sphere diameter. The influence of the bottom wall is found to have a much shorter range.
Impact dynamics is measured for spherical and cylindrical projectiles of many different densities dropped onto a variety non-cohesive granular media. The results are analyzed in terms of the material-dependent scaling of the inertial and frictional drag contributions to the total stopping force. The inertial drag force scales similar to that in fluids, except that it depends on the internal friction coefficient. The frictional drag force scales as the square-root of the density of granular medium and projectile, and hence cannot be explained by the combination of granular hydrostatic pressure …
We study the dynamic process occurring when a granular assembly is displaced by a solid impactor. The momentum transfer from the impactor to the target is shown to occur through sporadic, normal collisions of high force carrying grains at the intruder surface. We therefore describe the stopping force of the impact through a collisional based model. To verify the model in impact experiments, we determine the forces acting on an intruder decelerating through a dense granular medium using high-speed imaging of its trajectory. By varying the intruder shape and granular target, intruder-grain inter…
When an object penetrates into granular matter near a boundary, it experiences a horizontal repulsion due to the intruder-grain-wall interaction. Here we show experimentally that a square cuboid intruder, released from rest with no initial velocity, neara vertical wall to its left, goes through three distinct kinds of motion: it first tilts clockwise, then 'slides' away from the wall,and finally tilts counterclockwise. This dynamic highly favors both repulsion and penetration of the cuboid intruder as compared to that observed from its release farther away from the wall
We present a heuristic explanation for the anomalous (density-gradient-dependent) heat current in collisional granular fluids. Inelastic grain collisions lead to highly non-equilibrium states which are characterized by large spatial gradients and/or temporal variations in the granular temperature. It is argued that the heat current in such non-equilibrium states is driven by the temperature gradient averaged over the typical collision time and/or mean free path. Due to the density dependence of the inelastic energy loss, the nonlocal averaging of the temperature gradient leads to an effective …
Predicting the force exerted on an object as it penetrates a granular medium is of interest in engineering, locomotive, and geotechnical applications. Current models of granular drag, however, vary widely in applicability and parameterization, and the physical origin of the granular resistive force itself is a subject of debate. Here we perform constant speed penetration experiments, combined with calibrated, large-scale molecular dynamics simulations, at velocities up to 2 m/s to test the effect of impact velocity on the depth dependent `hydrostatic' drag force. We discover that the evolution…
We present a series of measurements examining the penetration force required to push a flat plate vertically through a dense granular medium, focusing in particular on the effects of the bottom boundary of the vessel containing the medium. Our data demonstrate that the penetration force near the bottom is strongly affected by the surface properties of the bottom boundary, even many grain diameters above the bottom. Furthermore, the data indicate an intrinsic length scale for the interaction of the penetrating plate with the vessel bottom via the medium. This length scale, which corresponds to …
We present experimental measurements of penetration depths for the impact of spheres into wetted granular media. We observe that the penetration depth in the liquid saturated case scales with projectile density, size, and drop height in a fashion consistent with the scaling observed in the dry case, but that penetration depths into saturated packings tend to be smaller. This result suggests that, for the range of impact energies observed, the stopping force is set by static contact forces between grains within the bed, and that the presence of liquid serves, primarily, to enhance these contact…
We study the quasi-static penetration of a flexible beam in a two-dimensional dense granular medium lying on an horizontal plate. Rather than a buckling-like behavior we observe a transition between a regime of crack-like penetration in which the fiber only shows small fluctuations around a stable straight geometry and a bending regime in which the fiber fully bends and advances through series of loading/unloading steps. We show that the shape reconfiguration of the fiber is controlled by a single non dimensional parameter: L/Lc, the ratio of the length of the flexible beam L to Lc, a bending …
We measure the quasi-static friction force acting on intruders moving downwards into a granular medium. By utilizing different intruder geometries, we demonstrate that the force acts locally normal to the intruder surface. By altering the hydrostatic loading of grain contacts by a sub-fluidizing airflow through the bed, we demonstrate that the relevant frictional contacts are loaded by gravity rather than by the motion of the intruder itself. Lastly, by measuring the final penetration depth versus airspeed and using an earlier result for inertial drag, we demonstrate that the same quasi-static…
We present evidence for the interactions between a ball and the container boundaries, as well as between two balls, that are mediated by the granular medium during impact cratering. The presence of the bottom boundary affects the final penetration depth only for low drop heights with shallow filling, in which case, surprisingly, the penetration becomes deeper. By contrast the presence of the side wall causes less penetration and also an effective repulsion. Repulsion is also found for two balls dropped side-by-side.
We study the behavior of cylindrical objects as they sink into a dry granular bed fluidized due to lateral oscillations, in order to shed light on human constructions and other objects. Somewhat unexpectedly, we have found that, within a large range of lateral shaking powers, cylinders with flat bottoms sink vertically, while those with a "foundation" consisting in a shallow ring attached to their bottom, tilt besides sinking. The latter scenario seems to dominate independently from the nature of the foundation when strong enough lateral vibrations are applied. We are able to reproduce the obs…
We report on the penetration of cylindrical projectiles dropped from rest into a dry, noncohesive granular medium. The cylinder length, diameter, density, and tip shape are all explicitly varied. For deep penetrations, as compared to the cylinder diameter, the data collapse onto a single scaling law that varies as the 1/3 power of the total drop distance, the 1/2 power of cylinder length, and the 1/6 power of cylinder diameter. For shallow penetrations, the projectile shape plays a crucial role with sharper objects penetrating deeper.
The non-Newtonian character of granular flow in a vertical pipe is analyzed. The time evolution of the flow velocity and the velocity fluctuations (or granular temperature) is derived. The steady state velocity has a power law dependence on the pipe width with an exponent between 3/4 and 3/2. The flow becomes faster the more efficient the collisional cooling is, provided the density remains low enough. The dependence of collisional cooling on the solid fraction, the restitution coefficient and a possible electric charge of all grains is discussed in detail.
Liquid drop impact and penetration into a granular layer are investigated with diverse liquids and granular materials. We use various size of SiC abrasives and glass beads as a target granular material. We also employ ethanol and glycerol aqueous solutions as well as distilled water to make a liquid drop. The liquid drop impacts the granular layer with a low speed (~ m/s). The drop deformation and penetration are captured by a high speed camera. From the video data, characteristic time scales are measured. Using a laser profilometry system, resultant crater morphology and its characteristic le…
When a large blunt object penetrates into a granular material, the force response exhibits an initial nonlinear relationship with depth that is widely attributed to the formation of a shear jammed stagnant zone. We present an experimental study of a model 2D granular system that preserves common aspects of slow granular penetration, such as a linear force law for large depths, while also allowing for direct and precise measurement of a stagnant zone. We also probe structural anisotropies throughout the bed and examine their characteristics inside and outside the stagnant zone. Finally, we simu…
When an intruder strikes a granular material from above, the grains exert a stopping force which decelerates and stops the intruder. Many previous studies have used a macroscopic force law, including a drag force which is quadratic in velocity, to characterize the decelerating force on the intruder. However, the microscopic origins of the force law terms are still a subject of debate. Here, drawing from previous experiments with photoelastic particles, we present a model which describes the velocity-squared force in terms of repeated collisions with clusters of grains. From our high speed phot…
We present data for the penetration of a variety of spheres, dropped from rest, into a level non-cohesive granular medium. We improve upon our earlier work [Uehara {\it et al.} Phys. Rev. Lett. {\bf 90}, 194301 (2003)] in three regards. First, we explore the behavior vs sphere diameter and density more systematically, by holding one of these parameters constant while varying the other. Second, we prepare the granular medium more reproducibly and, third, we measure the penetration depth more accurately. The new data support our previous conclusion that the penetration depth is proportional to t…
Penetration of Rigid Rods, Flexible Rods, and Granular Jets into Low-Density Granular Media
Instituto de Física, Benemérita Universidad Autónoma de Puebla（普埃布拉自治大学物理研究所）；Gulliver, CNRS, ESPCI Paris, Université PSL（高勒维尔实验室，法国国家科学研究中心，巴黎高等物理化工学院，巴黎文理研究大学）
From geotechnical applications to space exploration, auger drilling is often used as a standard tool for soil sample collection, instrument installation, and others. Focusing on granular flow associated with the rotary drilling process, we investigate the performance of auger drilling in terms of sampling efficiency, defined as the mass ratio of the soil sample collected in the coring tube to its total volume at a given penetration depth, by means of experiments, numerical simulations, as well as theoretical analysis. The ratio of rotation to penetration speed is found to play a crucial role i…
We report measurements in a 2-dimensional, gravity-driven, collisional, granular flow of the normal force delivered to the wall and of particle velocity at several points in the flow. The wall force and the flow velocity are negatively correlated. This correlation falls off only slowly with distance transverse to the flow, but dies away on the scale of a few particle diameters upstream or downstream. The data support a picture of short-lived chains of frequently colliding particles that extend transverse to the flow direction, making transient load-bearing bridges that cause bulk fluctuations …
We examine the impact of a solid sphere into a fine-grained granular bed. Using high-speed X-ray radiography we track both the motion of the sphere and local changes in the bed packing fraction. Varying the initial packing density as well as the ambient gas pressure, we find a complete reversal in the effect of interstitial gas on the impact response of the bed: The dynamic coupling between gas and grains allows for easier penetration in initially loose beds but impedes penetration in more densely packed beds. High-speed imaging of the local packing density shows that these seemingly incongruo…
The impact of single passive intruders into granular particles has been studied in detail. However, the impact force produced by multiple intruders separated at a distance from one another, and hence the effect of their presence in close proximity to one another, is largely unexplored. Here, we use numerical simulations and laboratory experiments to study the force response of two parallel rods intruding vertically into granular media while varying the gap spacing between them. We also explored the effect of variations in friction, intruder size, and particle size on the force response. The ne…
We report on measurements of self-diffusion coefficients in discrete numerical simulations of steady, homogeneous, collisional shearing flows of nearly identical, frictional, inelastic spheres. We focus on a range of relatively high solid volume fractions that are important in those terrestrial gravitational shearing flows that are dominated by collisional interactions. Diffusion over this range of solid fraction has not been well characterized in previous studies. We first compare the measured values with an empirical scaling based on shear rate previously proposed in the literature, and high…
Discrete element method simulations of confined bidisperse granular shear flows elucidate the balance between diffusion and segregation that can lead to either mixed or segregated states, depending on confining pressure. Results indicate that the collisional diffusion is essentially independent of overburden pressure. Because the rate of segregation diminishes with overburden pressure, the tendency for particles to segregate weakens relative to the re-mixing of particles due to collisional diffusion as the overburden pressure increases. Using a continuum approach that includes a pressure depen…
Flows of hard granular materials depend strongly on the interparticle friction coefficient $μ_p$ and on the inertial number ${\cal I}$, which characterizes proximity to the jamming transition where flow stops. Guided by numerical simulations, we derive the phase diagram of dense inertial flow of spherical particles, finding three regimes for $10^{-4} \lesssim {\cal I} \lesssim 10^{-1}$: \textit{ frictionless, frictional sliding, } and {\it rolling}. These are distinguished by the dominant means of energy dissipation, changing from collisional to sliding friction, and back to collisional, as $μ…
We investigate the unjamming response of disordered packings of frictional hard disks with the help of computer simulations. First, we generate jammed configurations of the disks and then force them to move again by local perturbations. We study the spatial distribution of the stress and displacement response and find long range effects of the perturbation in both cases. We record the penetration depth of the displacements and the critical force that is needed to make the system yield. These quantities are tested in two types of systems: in ideal homogeneous packings in zero gravity and in pac…
We study the effect of pre-collisional velocity correlations on granular shear flow by molecular dynamics simulations of the inelastic hard sphere system. Comparison of the simulations with the kinetic theory reveals that the theory overestimates both the energy dissipation rate and the normal stress in the dense flow region. We find that the relative normal velocity of colliding particles is smaller than that expected from random collisions, and the discrepancies in the dissipation and the normal stress can be adjusted by introducing the idea of the collisional temperature, from which we conc…
We explore the process of initiating motion through a granular medium by measuring the force required to push a flat circular plate upward from underneath the medium. In contrast to previous measurements of the drag and penetration forces, which were conducted during steady state motion, the initiation force has a robust dependence on the diameter of the grains in the medium. We attribute this dependence to the requirement for local dilation of the grains around the circumference of the plate, as evidenced by an observed linear dependence of the initiation force on the plate diameter.
Direct measurements of the acceleration of spheres and disks impacting granular media reveal simple power law scalings along with complex dynamics which bear the signatures of both fluid and solid behavior. The penetration depth scales linearly with impact velocity while the collision duration is constant for sufficiently large impact velocity. Both quantities exhibit power law dependence on sphere diameter and density, and gravitational acceleration. The acceleration during impact is characterized by two jumps: a rapid, velocity dependent increase upon initial contact and a similarly sharp, d…
We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution $α$ and $β$, respectively. The basic quantities characterizing the distribution function $f(\mathbf{v},\bmω)$ of linear ($\mathbf{v}$) and angular ($\bmω$) velocities are the second-degree moments defining the translational ($T^\text{tr}$) and rotational ($T^\text{rot}$) temperatures. The deviation of $f$ from the Maxwellian distribution parameterized by $T^\text{tr}$ and $T^\text{rot}$ can be measured by the cumulants associated with the fourth-degree velocity momen…
Understanding the penetration dynamics of intruders in granular beds is relevant not only for fundamental Physics, but also for geophysical processes and construction on sediments or granular soils in areas potentially affected by earthquakes. While the penetration of intruders in two dimensional (2D) laboratory granular beds can be followed using video recording, it is useless in three dimensional (3D) beds of non-transparent materials such as common sand. Here we propose a method to quantify the sink dynamics of an intruder into laterally shaken granular beds based on the temporal correlatio…
We study in this work the dynamics of a granular impurity immersed in a low-density granular gas of identical particles. For description of the kinetics of the granular gas and the impurity particles we use the rough hard sphere collisional model. We take into account the effects of non-conservation of energy upon particle collision. We find an (approximate) analytical solution of the pertinent kinetic equations for the single-particle velocity distribution functions that reproduces reasonably well the properties of translational/rotational energy non-equipartition. We assess the accuracy of t…
We consider the penetration dynamics of a vertical cylinder into a dry granular medium subjected to successive impacts. The depth of the impactor below the free surface $z_N$ first evolves linearly with the impact number $N$ and then follows a power-law evolution $z_N \propto N^{1/3}$. The depth reached by the cylinder after a given number of impacts is observed to increase with the impact energy but to decrease with its diameter and the density of the granular medium. We develop a model that accounts for the quasi-static and inertial granular forces applying on the cylinder to rationalize our…
In a recent paper [A. Santos, G. M. Kremer, and V. Garzó, \emph{Prog. Theor. Phys. Suppl.} \textbf{184}, 31-48 (2010)] the collisional energy production rates associated with the translational and rotational granular temperatures in a granular fluid mixture of inelastic rough hard spheres have been derived. In the present paper the energy production rates are explicitly decomposed into equipartition rates (tending to make all the temperatures equal) plus genuine cooling rates (reflecting the collisional dissipation of energy). Next the homogeneous free cooling state of a binary mixture is anal…
This work summarizes a series of studies on two-dimensional granular impact, where an intruding object strikes a granular material at high speed. Many previous studies on granular impact have used a macroscopic force law, which is dominated by an inertial drag term proportional to the intruder velocity squared. The primary focus here is on the microscopic force response of the granular material, and how the grain-scale effects give rise to this inertial drag term. We show that the inertial drag arises from intermittent collisions with force-chain-like structures. We construct a simple collisio…
We study the appearance of large-scale dynamical heterogeneities in a simplified model of a driven, dissipative granular system. Simulations of steady-state gravity-driven flows of inelastically colliding hard disks show the formation of large-scale linear structures of particles with a high collision frequency. These chains can be shown to carry much of the collisional stress in the system due to a dynamical correlation that develops between the momentum transfer and time between collisions in these "frequently-colliding" particles. The lifetime of these dynamical stress heterogeneities is se…
We carry out constant volume simulations of steady-state, shear driven flow in a simple model of athermal, bidisperse, soft-core, frictionless disks in two dimensions, using a dissipation law that gives rise to Bagnoldian rheology. Focusing on the small strain rate limit, we map out the rheological behavior as a function of particle packing fraction $ϕ$ and a parameter $Q$ that measures the elasticity of binary particle collisions. We find a $Q^*(ϕ)$ that marks the clear crossover from a region characteristic of strongly inelastic collisions, $Q<Q^*$, to a region characteristic of weakly inela…
In order to improve our understanding of landing on small bodies and of asteroid evolution, we use our novel drop tower facility to perform low-velocity (2-40 cm s^-1), shallow impact experiments of a 10 cm diameter aluminum sphere into quartz sand in low effective gravities (~0.2-1 m s^-2). Using in situ accelerometers, we measure the acceleration profile during the impacts and determine the peak accelerations, collision durations and maximum penetration depth. We find that the penetration depth scales linearly with the collision velocity but is independent of the effective gravity for the ex…
A new numerical method is presented to efficiently simulate the inelastic hard sphere (IHS) model for granular media, when fluid and frozen regions coexist in the presence of gravity. The IHS model is extended by allowing particles to change their dynamics into either a frozen state or back to the normal collisional state, while computing the dynamics only for the particles in the normal state. Careful criteria, local in time and space, are designed such that particles become frozen only at mechanically stable positions. The homogeneous deposition over a static surface and the dynamics of a ro…
We report the characteristics of the temporal fluctuations in the local force delivered to the wall of a 2D hopper by a granular medium flowing through it. The forces are predominantly impulsive at all flow rates for which the flow does not permanently jam. The average impulse delivered to the wall is much larger than the momentum acquired by a single particle under gravity between collisions, reflecting the fact that momentum is transferred to the walls from the bulk of the flow by collisions. At values larger than the average impulse, the probability distribution of impulses is broad and dec…
We present here a detailed granular flow characterization together with force measurements for the quasi-bidimensional situation of a horizontal cylinder penetrating vertically at a constant velocity in dry granular matter between two parallel glass walls. In the velocity range studied here, the drag force on the cylinder does not depend on the velocity V_0 and is mainly proportional to the cylinder diameter d. Whereas the force on the cylinder increases with its penetration depth, the granular velocity profile around the cylinder is found stationary with fluctuations around a mean value leadi…
A new segregation criterion based on the inelastic Enskog kinetic equation is derived to show the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the different parameters of the system. In contrast to previous theoretical attempts the approach is not limited to the near-elastic case, takes into account the influence of both thermal gradients and gravity and applies for moderate densities. The form of the phase-diagrams for the BNE/RBNE transition depends sensitively on the value of gravity relative to the thermal gradient, so that it is possib…
We consider a dilute gas of inelastic hard spheres enclosed in a slab under the action of gravity along the longitudinal direction. In addition, the gas is subject to a white-noise stochastic force that mimics the effect of external vibrations customarily used in experiments to compensate for the collisional cooling. The system is described by means of a kinetic model of the inelastic Boltzmann equation and its steady-state solution is derived through second order in gravity. This solution differs from the Navier-Stokes description in that the hydrostatic pressure is not uniform, normal stress…
The most widely used model for granular gases is perhaps the inelastic hard-sphere model (IHSM), where the grains are assumed to be perfectly smooth spheres colliding with a constant coefficient of normal restitution. A much more tractable model is the inelastic Maxwell model (IMM), in which the velocity-dependent collision rate is replaced by an effective mean-field constant. This simplification has been taken advantage of by many researchers to find a number of exact results within the IMM. On the other hand, both the IHSM and IMM neglect the impact of roughness -- generally present in real …
