Low-Speed Impact Craters in Loose Granular Media

We report on craters formed by balls dropped into dry, noncohesive, granular media. By explicit variation of ball density ${\rho }_b$, diameter $D_b$, and drop height $H$, the crater diameter is confirmed to scale as the $1/4$ power of the energy of the ball at impact: $D_c\sim ({\rho }_b{D}_b^{3}H{)}^{1/4}$. Against expectation, a different scaling law is discovered for the crater depth: $d\sim ({\rho }_b^{3/2}{D}_b^{2}H{)}^{1/3}$. The scaling with properties of the medium is also established. The crater depth has significance for granular mechanics in that it relates to the stopping force on the ball.

Figure 1

The diameter and depth of craters formed by balls dropped into 0.2 mm diameter glass beads, as a function of total energy loss, $E=mgH$. Each data point represents the experimental result for a single drop height $H$. Plotting vs $mgH$ collapses the diameter data, as seen earlier in Ref. 6, onto a $1/4$ power law. By contrast, the depth data does not collapse, but follows a $1/3$ power law. Symbol, ball type, ball density, and ball diameter are as follows: ○ for hollow plastic, $0.26\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $2.54\mathrm{c}\mathrm{m}$; □ for wood, $0.83\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.59\mathrm{c}\mathrm{m}$; ◇ for nylon, $1.10\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.59\mathrm{c}\mathrm{m}$; × for nylon, $1.10\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $2.54\mathrm{c}\mathrm{m}$; + for silicon rubber, $1.10\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.52\mathrm{c}\mathrm{m}$; △ for acrylic, $1.20\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.59\mathrm{c}\mathrm{m}$; ● for live ball, $1.20\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $3.82\mathrm{c}\mathrm{m}$; ■ for dead ball, $1.30\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $3.82\mathrm{c}\mathrm{m}$; ◆ for delrin, $1.40\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.59\mathrm{c}\mathrm{m}$; ▲ for Teflon, $2.20\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.59\mathrm{c}\mathrm{m}$; ▼ for ceramic, $3.90\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.27\mathrm{c}\mathrm{m}$. NB: Data for three even denser balls are omitted: stainless steel, $7.90\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $2.54\mathrm{c}\mathrm{m}$; lead, $11.3\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.13\mathrm{c}\mathrm{m}$; and tungsten carbide, $16.4\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$, $1.91\mathrm{c}\mathrm{m}$. These produce craters that also scale as $D_c\sim H^{1/4}$ and $d\sim H^{1/3}$; however, they do not exhibit the same $D_c\sim E^{1/4}$ collapse as the less dense balls.

Figure 2

Scaling of crater size with ball density and diameter. Each point represents the result of a $D_c\propto H^{1/4}$ or $d\propto H^{1/3}$ power-law fit to the size vs $H$ data sets of Fig. 1, divided by the labeled combination of ball diameter or density and drop height; error bars are comparable to symbol size. All data are for 0.2 mm diameter glass beads. Note that the three densest balls do not obey the $D_c\sim {\rho }_b^{1/4}$ scaling; data for these balls are boxed in both (a) and (b).

Figure 3

Scaling of crater size with grain-grain friction coefficient, $\mu =\tan {\theta }_r$, for 1-inch diameter nylon ball dropped into the media specified in Table . Each point represents the result of a $D_c\propto H^{1/4}$ or $d\propto H^{1/3}$ power-law fit to the size vs $H$ data sets, divided by the labeled combination of ball and grain densities, ball diameter, and drop height; error bars are comparable to symbol size.