Dynamics of Drag and Force Distributions for Projectile Impact in a Granular Medium

Our experiments and molecular dynamics simulations on a projectile penetrating a two-dimensional granular medium reveal that the mean deceleration of the projectile is constant and proportional to the impact velocity. Thus, the time taken for a projectile to decelerate to a stop is independent of its impact velocity. The simulations show that the probability distribution function of forces on grains is time independent during a projectile’s deceleration in the medium. At all times the force distribution function decreases exponentially for large forces.

Figure 1

Snapshots of a projectile in the three distinct regimes of its motion in a bidisperse mixture of particles (cylinders). Experiment: the larger cylinders of the bidisperse mixture are colored black for visualization and are 40% larger in diameter than the grey cylinders. The dashed line shows the location of the surface after the collapse is complete. Simulation: the shading of each particle is proportional to the sum of the magnitudes of all the normal forces acting on that particle; this renders visible the instantaneous force chains. The projectile is 9.8 times as large in diameter and 657 times as massive as the smallest particles.

Figure 2

(a) Position $y(t)$ and (b) velocity $v_y(t)$ of the projectile as a function of time for different impact velocities, from both experiment (○) and simulation (solid lines). The two vertical dot-dashed lines give approximate boundaries between three regions: impact, where the projectile rapidly decelerates (cf. Fig. 3); penetration, where the mean acceleration is constant, as illustrated by a dashed line fit in (a) of a parabola to the results from experiment and simulation for each $v_0$; and collapse, where the projectile has almost stopped and the particles above it are collapsing to fill the transient crater left by the penetration. The ordinate for (b) for each successive impact velocity $v_0<363\mathrm{cm}/\mathrm{s}$ is shifted by $30\mathrm{cm}/\mathrm{s}$ for clarity. Inset: normalized acceleration of the projectile versus impact velocity from experiment (●) and simulation (◇).

Figure 3

The time series of the force on the projectile obtained from the simulation. The three regimes of motion are separated by dot-dashed lines. Inset: the power spectrum of the projectile acceleration during the penetration regime (0.02–0.15 s) for a projectile with initial velocity $v_0=238\mathrm{cm}/\mathrm{s}$ is described by $f^{-\alpha }$ with $\alpha =2.1\pm 0.2$.

Figure 4

Vertical component of the force computed for every particle in contact with the projectile during part of the penetration regime ($0.100<t<0.108\mathrm{s}$ in Figs. 2, 3). Each force grows, reaches a maximum (representing the inclusion of a particle in a particular force chain), and then decreases. Each type of line represents a particular particle; thus, the two arrows correspond to the same particle that appeared first at 12.344 cm and then reappeared at 12.501 cm. The projectile impact velocity was $v_0=238\mathrm{cm}/\mathrm{s}$. The average of the $y$ component of the force on the projectile during this interval was 0.57 N.

Figure 5

Probability distribution of normal contact forces between grains for a projectile with $v_0=112\mathrm{cm}/\mathrm{s}$ at the following times: during impact ($t_1=0.02\mathrm{s}$, ●), early in the penetration regime ($t_2=0.05\mathrm{s}$, ■), late in the penetration regime ($t_3=0.12\mathrm{s}$, $+$), and during collapse ($t_4=0.20\mathrm{s}$, ▲). The distribution decays exponentially for $F>F^{*}$. The dependence of $F^{*}$ on the impact velocity is shown in the inset; the slope is $0.047\pm 0.004\mathrm{mN}\mathrm{s}/\mathrm{cm}$. Each curve was obtained by averaging over 50 runs.