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

Abstract

We experimentally and computationally study the early-stage forces during high-speed impacts into granular beds. Experiments consist of impacts into 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 intruder velocity, the elastic modulus of the grains, the mass density of the grains, and the intruder size according to power-law scaling forms that are not consistent with Poncelet models or granular shock theory. We find that the early-stage forces are inherently related to wave propagation, but the current theory of shocks in granular media cannot capture them, suggesting that a new theory is needed. The insensitivity of our results to many system details suggest that they may also apply to impacts into similar materials like foams and emulsions.High-speed impact by an intruder into a granular bed is a ubiquitous process with broad relevance in many disciplines, including ballistics [1-4], robotics [5,6], astrophysics [7], and earth science [8]. The forces exerted by the grains on the intruder are often described using Poncelet drag, which is dominated by a velocity-squared force with a nearly constant drag coefficient [9][10][11][12]. However, as noted by Pica Ciamarra et al.[13] and many others [3,12,[14][15][16][17][18], the initial impact forces are consistently larger than expected from Poncelet drag or similar drag models, which is sometimes attributed [12] to a shock wave at impact [19]. These large forces are short lived but may be most important for determining damage to the grains and projectile [3,20] and for capturing the dynamics of a highly transient process like a jumping animal or robot [5,6]. However, there is currently no theory that captures these initial forces. In this Letter, we use experiments and simulations to demonstrate that peak forces during the initial stages of granular impact obey simple, power-law scaling forms that depend only on the impact speed, the mass density and stiffness of the grains, and the intruder size. These scaling laws do not fit within the framework of any existing theory related to impact, including Poncelet models and the theory of shocks in granular media [19,[22][23][24][25][26][27], implying that a new theory is needed for this process.We uncover these scaling laws through experiments and simulations of impacts into granular beds with varying grain and intruder properties. Experiments involve circular intruders striking a collection of more than 10,000 photoelastic disks (3 mm thick) confined between two Plexiglas sheets (0.91 m × 1.22 m × 1.25 cm). These experiments have been used previously to study the microscopic origins of the Poncelet drag laws during the penetration regime [15,16,21] as well as the speed and spatial structure of the shocks propagating away from the point of impact [19]. Here, we focus on the intruder dynamics during the initial stages. Intruders are machi...

Viscous-like forces control the impact response of shear-thickening dense suspensions

“…2003; Goldman & Umbanhowar 2008; Clark, Petersen & Behringer 2014; Zhao et al. 2015; Krizou & Clark 2020). For all experiments and theoretical models, we find that these scalings can be well approximated by The prefactors and can vary with intruder properties, and we will examine how they depend on , and, for conical intruders, cone angle .…”

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“…[37], where the concepts of shear jamming and the DST are unified. Observation of a universal scaling law for impacts on dry granular media was recently reported [38]. It would be interesting to investigate whether such a scaling law also exists in the case of suspensions.…”

Finite amplitude waves in jammed matter

“…1(b)), which can lead to the formation of shocks and craters. [30][31][32] In such cases, the initial impulse also induces the formation of a strongly non-linear supersonic wave. However, due to the lack of continuous deformation, these waves are nonuniform shocks because there is no well-defined compressed state.…”