Collisional model of energy dissipation in three-dimensional granular impact

Drag force regime in dry and immersed granular media

The drag force acting on an intruder colliding with granular media is typically influenced by the impact velocity and the penetrating depth. In this paper, the investigation was extended to the dry and immersed scenarios through coupled simulations at different penetrating velocities. The drag force regime was clarified to exhibit velocity dependence in the initial contact stage, followed by the inertial transit stage with a F∼z^{2} (force-depth) relationship. Subsequently, it transitioned into the depth-dependent regime in both dry and immersed cases. The underlying rheological mechanism was explored, revealing that, in both dry and immersed scenarios, the granular bulk underwent a state relaxation process, as indicated by the granular inertial number. Additionally, the presence of the ambient fluid restricted the flow dynamics of the perturbed granular material, exhibiting a similar rheology as observed in the dry case.

Imperfect bodies sink imperfectly when settling in granular matter

From Mars rovers to buildings, objects eventually sink and tilt into a fluidized granular bed due to gravity. Despite the irregular shape of realistic granular intruders, most research focus on the settling of "perfect" objects like spheres and cylinders. Here, we systematically explore the penetration of "imperfect" solids-from stones to bodies with carefully controlled asymmetries-into granular beds. A cylinder with two halves of different roughnesses rotates toward the granular region next to the smoother surface and deviates from the vertical direction. We demonstrate that even small irregularities in the surface of an object may produce substantial changes in the penetration process. Using computer simulations, we show that defects concentrate granular force chains, thus producing decisive forces on the intruder. Furthermore, we demonstrate that tilting and migration of sinking bodies can be correctly predicted by a simple mechanical model based on a unified force law.

Contacts, motion, and chain breaking in a two-dimensional granular system displaced by an intruder

We investigate numerically how the motion of an intruder within a two-dimensional granular system affects its structure and produces drag on the intruder. We made use of discrete numerical simulations in which a larger disk (intruder) is driven at constant speed amid smaller disks confined in a rectangular cell. By varying the intruder's velocity and the basal friction, we obtained the resultant force on the intruder and the instantaneous network of contact forces, which we analyze at both the cell and grain scales. We found that there is a bearing network that percolates forces from the intruder toward the walls, being responsible for jammed regions and high values of the drag force, and a dissipative network that percolates small forces within the grains, in agreement with previous experiments on compressed granular systems. In addition, we found the anisotropy levels of the contact network for different force magnitudes and regions, that the force network can reach regions far downstream of the intruder by the end of the intruder's motion, that the extent of the force network decreases with decreasing the basal friction, and that the void region (cavity) that appears downstream of the intruder tends to disappear for lower values of the basal friction. Interestingly, our results show that grains within the bearing chains creep while the chains break, revealing the mechanism by which bearing chains collapse.

Virtual Energy Management for Physical Energy Savings in a Legged Robot Hopping on Granular Media

We discuss an active damping controller to reduce the energetic cost of a single step or jump of dynamic locomotion without changing the morphology of the robot. The active damping controller adds virtual damping to a virtual leg spring created by direct-drive motors through the robot’s leg linkage. The virtual damping added is proportional to the intrusion velocity of the robot’s foot, slowing the foot’s intrusion, and thus the rate at which energy is transferred to and dissipated by the ground. In this work, we use a combination of simulations and physical experiments in a controlled granular media bed with a single-leg robot to show that the active damping controller reduces the cost of transport compared with a naive compression-extension controller under various conditions.

The Bernoulli effect in horizontal granular flows

The Bernoulli effect that commonly occurs in continuous fluids is simultaneously increasing a fluid's velocity and decreasing static pressure or the fluid's gravitational potential energy. Although, the Bernoulli effect has already been extensively explored, there is a lack of research on the relationship between flow velocity and pressure in a discrete medium. In the present study, this relationship in horizontal granular flows excited by vertical vibration is experimentally studied. It was found that the random motion and horizontal directed motion of the granules restrict each other so that the total pressure remains almost constant with respect to time and height. In fact, it implies that the Bernoulli effect occurs in the granular flows. It was also found that the pressure constant of the Bernoulli effect depends on the vibrating intensity and frequency, which reflects the energy transfer in the granular flows. Our results show a dynamic property of the granular flows, which is different from continuous fluids, even though it is similar to some extent.

Effect of two parallel intruders on total work during granular penetrations

The intrusion of single passive intruders into granular particles has been studied in detail. However, the intrusion 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 less explored. Here, we used 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 total work (W) of the two rods over the depth of intrusion was measured, and the instantaneous velocities of particles over the duration of intrusion were calculated by simulations. We found that the total work done by the intruders changes with distance between them. We observed a peak in W at a gap spacing of ∼3 particle diameters, which was up to 25% greater than W at large separation (>11 particle diameters), beyond which the total work plateaued. This peak was likely due to reduced particle flow between intruders as we found a larger number of strong forces-identified as force chains-in the particle domain at gaps surrounding the peak force. Although higher friction caused greater force generation during intrusion, the gap spacing between the intruders at which the peak total work was generated remained unchanged. Larger intruder sizes resulted in greater total work with the peak in W occurring at slightly larger intruder separations. Taken together, our results show that peak total work done by two parallel intruders remained within a narrow range, remaining robust to most other tested parameters.

Efficacy of simple continuum models for diverse granular intrusions

Granular intrusion is commonly observed in natural and human-made settings. Unlike typical solids and fluids, granular media can simultaneously display fluid-like and solid-like characteristics in a variety of intrusion scenarios. This multi-phase behavior increases the difficulty of accurately modeling these and other yielding (or flowable) materials. Micro-scale modeling methods, such as DEM (Discrete Element Method), capture this behavior by modeling the media at the grain scale, but there is often interest in the macro-scale characterizations of such systems. We examine the efficacy of a macro-scale continuum approach in modeling and understanding the physics of various macroscopic phenomena in a variety of granular intrusion cases using two basic frictional yielding constitutive models. We compare predicted granular force response and material flow to experimental data in four quasi-2D intrusion cases: (1) depth-dependent force response in horizontal submerged-intruder motion; (2) separation-dependent drag variation in parallel-plate vertical-intrusion; (3) initial-density-dependent drag fluctuations in free surface plowing, and (4) flow zone development during vertical plate intrusions in under-compacted granular media. Our continuum modeling approach captures the flow process and drag forces while providing key meso- and macro-scopic insights. The modeling results are then compared to experimental data. Our study highlights how continuum modeling approaches provide an alternative for efficient modeling as well as a conceptual understanding of various granular intrusion phenomena.

Intrusion into Granular Media Beyond the Quasistatic Regime

The drag force exerted on an object intruding into granular media is typically assumed to arise from additive velocity and depth dependent contributions. We test this with intrusion experiments and molecular dynamics simulations at constant speed over four orders of magnitude, well beyond the quasistatic regime. For a vertical cylindrical rod we find velocity dependence only right after impact, followed by a crossover to a common, purely depth-dependent behavior for all intrusion speeds. The crossover is set by the timescale for material, forced to well up at impact, to subsequently settle under gravity. These results challenge current models of granular drag.

Surprising simplicity in the modeling of dynamic granular intrusion

Granular intrusions, such as dynamic impact or wheel locomotion, are complex multiphase phenomena where the grains exhibit solid-like and fluid-like characteristics together with an ejected gas-like phase. Despite decades of modeling efforts, a unified description of the physics in such intrusions is as yet unknown. Here, we show that a continuum model based on the simple notions of frictional flow and tension-free separation describes complex granular intrusions near free surfaces. This model captures dynamics in a variety of experiments including wheel locomotion, plate intrusions, and running legged robots. The model reveals that one static and two dynamic effects primarily give rise to intrusion forces in such scenarios. We merge these effects into a further reduced-order technique (dynamic resistive force theory) for rapid modeling of granular locomotion of arbitrarily shaped intruders. The continuum-motivated strategy we propose for identifying physical mechanisms and corresponding reduced-order relations has potential use for a variety of other materials.

Impact in granular matter: Force at the base of a container made with one movable wall

In geotechnics as well as in planetary science, it is important to find a means by which to protect a base from impacts of micrometeoroids. In the moon, for example, covering a moon base with regolith, and housing such regolith by movable bounding walls, could work as a stress-leaking shield. Using a numerical model, by performing impacts on a granular material housed in a rectangular container made with one movable sidewall, it is found that such wall mobility serves as a good means for controlling the maximum force exerted at the container's base. We show that the force exerted at the container's base decreases as the movable wall decreases in mass, and it follows a Janssen-like trend. Moreover, by making use of a dynamically defined redirecting coefficient K(X), proposed by Windows-Yule et al. [Phys. Rev. E 100, 022902 (2019)2470-004510.1103/PhysRevE.100.022902], which depends on the container's width X, we propose a model for predicting the maxima measured at the container's base. The model depends on the projectile and granulate properties, and the container's geometry.

Rolling away from the Wall into Granular Matter

The sedimentation of solid objects into granular matter near boundaries is an almost virgin field of research. Here we describe in detail the penetration dynamics of a cylindrical object into a quasi-2D granular medium. By tracking the trajectory of the cylinder as it penetrates the granular bed, we characterize two distinct kinds of motion: its center of mass moves horizontally away from the lateral wall, and it rotates around its symmetry axis. While the repulsion is caused by the loading of force chains between the intruder and the wall, the rotation can be associated to the frictional forces between the grains and the intruder. Finally, we show the analogies between the sedimentation of twin intruders released far from any boundaries, and that of one intruder released near a vertical wall.

Power-Law Scaling of Early-Stage Forces during Granular Impact

We experimentally and computationally study the early-stage forces during intruder impacts with granular beds in the regime where the impact velocity approaches the granular force propagation speed. Experiments use 2D assemblies of photoelastic disks of varying stiffness, and complimentary discrete-element simulations are performed in 2D and 3D. The peak force during the initial stages of impact and the time at which it occurs depend only on the impact speed, the intruder diameter, the stiffness of the grains, and the mass density of the grains according to power-law scaling forms that are not consistent with Poncelet models, granular shock theory, or added-mass models. The insensitivity of our results to many system details suggests that they may also apply to impacts into similar materials like foams and emulsions.

The role of initial speed in projectile impacts into light granular media

Projectile impact into a light granular material composed of expanded polypropylene (EPP) particles is investigated systematically with various impact velocities. Experimentally, the trajectory of an intruder moving inside the granular material is monitored with a recently developed non-invasive microwave radar system. Numerically, discrete element simulations together with coarse-graining techniques are employed to address both dynamics of the intruder and response of the granular bed. Our experimental and numerical results of the intruder dynamics agree with each other quantitatively and are in congruent with existing phenomenological model on granular drag. Stepping further, we explore the ‘microscopic’ origin of granular drag through characterizing the response of granular bed, including density, velocity and kinetic stress fields at the mean-field level. In addition, we find that the dynamics of cavity collapse behind the intruder changes significantly when increasing the initial speed . Moreover, the kinetic pressure ahead of the intruder decays exponentially in the co-moving system of the intruder. Its scaling gives rise to a characteristic length scale, which is in the order of intruder size. This finding is in perfect agreement with the long-scale inertial dissipation type that we find in all cases.

Impact-Induced Energy Transfer and Dissipation in Granular Clusters under Microgravity Conditions

The impact-induced energy transfer and dissipation in granular targets without any confining walls are studied by microgravity experiments. A solid projectile impacts into a granular target at low impact speed (0.045≤v_{p}≤1.6  m s^{-1}) in a laboratory drop tower. Granular clusters consisting of soft or hard particles are used as targets. Porous dust agglomerates and glass beads are used for soft and hard particles, respectively. The expansion of the granular target cluster is recorded by a high-speed camera. Using the experimental data, we find that (i) a simple energy scaling can explain the energy transfer in both soft-particle and hard-particle granular targets, (ii) the kinetic impact energy is isotropically transferred to the target from the impact point, and (iii) the transferred kinetic energy is 2%-7% of the projectile's initial kinetic energy. The dissipative-diffusion model of energy transfer can quantitatively explain these behaviors.

Collision-based understanding of the force law in granular impact dynamics

We study the stopping force felt by an intruder impacting onto a granular medium. Variations in the shape of the intruder can influence the penetration depth by changing the inertial drag. We find this observed correlation can be explained by associating the velocity-dependent inertial drag to the energy dissipation that occurs through intermittent collisions of force-chain-like clusters, the mean behavior of which can be statistically described. In consequence, the stopping force can be captured through a proposed collisional model with good accuracy, and the observed impact dynamics data can be reproduced quantitatively.

Constitutive relation for the system-spanning dynamically jammed region in response to impact of cornstarch and water suspensions

We experimentally characterize the impact response of concentrated suspensions consisting of cornstarch and water. We observe that the suspensions support a large normal stress-on the order of MPa-with a delay after the impactor hits the suspension surface. We show that neither the delay nor the magnitude of the stress can yet be explained by either standard rheological models of shear thickening in terms of steady-state viscosities, or impact models based on added mass or other inertial effects. The stress increase occurs when a dynamically jammed region of the suspension in front of the impactor propagates to the opposite boundary of the container, which can support large stresses when it spans between solid boundaries. We present a constitutive relation for impact rheology to relate the force on the impactor to its displacement. This can be described in terms of an effective modulus but only after the delay required for the dynamically jammed region to span between solid boundaries. Both the modulus and the delay are reported as a function of impact velocity, fluid height, and weight fraction. We report in a companion paper the structure of the dynamically jammed region when it spans between the impactor and the opposite boundary [Allen et al., Phys. Rev. E 97, 052603 (2018)10.1103/PhysRevE.97.052603]. In a direct follow-up paper, we show that this constitutive model can be used to quantitatively predict, for example, the trajectory and penetration depth of the foot of a person walking or running on cornstarch and water [Mukhopadhyay et al., Phys. Rev. E 97, 052604 (2018)10.1103/PhysRevE.97.052604].

Archimedes' law explains penetration of solids into granular media

Understanding the response of granular matter to intrusion of solid objects is key to modelling many aspects of behaviour of granular matter, including plastic flow. Here we report a general model for such a quasistatic process. Using a range of experiments, we first show that the relation between the penetration depth and the force resisting it, transiently nonlinear and then linear, is scalable to a universal form. We show that the gradient of the steady-state part, K ϕ , depends only on the medium's internal friction angle, ϕ, and that it is nonlinear in μ = tan ϕ, in contrast to an existing conjecture. We further show that the intrusion of any convex solid shape satisfies a modified Archimedes' law and use this to: relate the zero-depth intercept of the linear part to K ϕ and the intruder's cross-section; explain the curve's nonlinear part in terms of the stagnant zone's development.

Asteroid surface impact sampling: dependence of the cavity morphology and collected mass on projectile shape

In-situ exploration and remote thermal infrared observation revealed that a large fraction of Solar System small bodies should be covered with granular regolith. The complex and varied geology of the regolith layer may preserve the historical records of the surface modification and topographic evolution experienced by asteroids, especially cratering processes, in which the projectile shape plays a crucial role. Regarding the impact sampling scheme, the projectile-shape dependence of both the cavity morphology and the collected mass remains to be explored. This paper studies the process of the low-speed impact sampling on granular regolith using projectiles of different shapes. The results demonstrate that the projectile shape significantly influences the excavation stage, forming cavities with different morphologies, i.e., cone-shaped, bowl-shaped and U-shaped. We further indicate that the different velocity distributions of the ejecta curtains due to the various projectile shapes result in various amounts of collected mass in sampler canister, regarding which the 60° conical projectile exhibits preferable performance for impact sampling scheme. The results presented in this article are expected to reveal the dependence of the excavation process on projectile shape under micro gravity and provide further information on the optimal designs of impact sampling devices for future sample-return space missions.