Infinite Penetration of a Projectile into a Granular Medium

An object falling in a fluid reaches a terminal velocity when the drag force and its weight are balanced. Contrastingly, an object impacting into a granular medium rapidly dissipates all its energy and comes to rest always at a shallow depth. Here we study, experimentally and theoretically, the penetration dynamics of a projectile in a very long silo filled with expanded polystyrene particles. We discovered that, above a critical mass, the projectile reaches a terminal velocity and, therefore, an endless penetration.

Figure 1

(a) Experimental $z$ vs $t$. Two types of trajectories are clearly observed; those that end in the bulk of the material ($m=35$ to 84 g), and those that reach the bottom of the silo ($m=92$ to 182 g). (b) $v$ vs $z$ for the above trajectories, where the previous distinctive behavior is even clearer. The speeds were obtained by numerical differentiation of the data in (a). Each point in the plots shown in (a) and (b) represents the average of five repetitions and the error bars correspond to the standard deviation. (c) Theoretical $z$ vs $t$ and (d) $v$ vs $z$, respectively, obtained by numerically solving Eq. (1), see text. The colored shadows are error bars arising from the propagation of the parameters uncertainties (for the sake of clarity, some shadows were omitted). The dashed lines are the separatrices, corresponding to $m_c=85.94\mathrm{g}$. Inset: theoretical trajectories for some masses above $m_c$ for depths as large as 20 m.

Figure 2

(a) $V_T^2$ vs $m$. The crossing point with the horizontal axis, where $V_T^2=0$, defines a critical mass, see text. (b) $Z_F$ vs $m$. The values of $Z_F$ (black points) were obtained from Fig. 1b. The red squares represent the experimental minimum depths ($z_{\min }$) where intruders remain at rest inside the silo. The solid line is the solution of Eq. (3). The curve diverges at a critical mass $m_c=85.94\mathrm{g}$ (dotted vertical line). Inset: $z_{\min }$ vs $-\ln (1-mg/\kappa \lambda )$. The slope of the line is $\lambda$.

Figure 3

MDS in a 2D silo. The net force acting on a falling intruder (mass 17 g). The oscillatory broken line ($a/g$ vs $z$) is for one experiment (i. e. considering only one bed) and the green line is a smoothed average of five trajectories obtained using five different beds. Snapshots of the intruder and its surroundings corresponding to large fluctuations are depicted. The color intensity represents the forces acting on the particles. On the left side the intruder strongly decelerates due to the built up of force chains, and on the right side the force chains weaken and the intruder accelerates. At around 90 cm of depth, the average net acceleration is close to zero and the intruder falls with a terminal speed (see also Supplementary Fig. 5 [22]).