Local jamming via penetration of a granular medium

We present a series of measurements examining the penetration force required to push a flat plate vertically through a dense granular medium, focusing in particular on the effects of the bottom boundary of the vessel containing the medium. Our data demonstrate that the penetration force near the bottom is strongly affected by the surface properties of the bottom boundary, even many grain diameters above the bottom. Furthermore, the data indicate an intrinsic length scale for the interaction of the penetrating plate with the vessel bottom via the medium. This length scale, which corresponds to the extent of local jamming induced by the penetrating plate, has a square root dependence both upon the plate radius and the ambient granular stress near the bottom boundary, but it is independent of penetration velocity and grain diameter.

Figure 1

(Color online) Granular penetrometer apparatus used for penetration measurements. (a) Actuator consists of a high-torque (0.44-Nm) computer-controlled stepper motor with a 200-mm range of linear motion and a distance-dependent voltage divider serving as a measure of the depth of penetration. The force cell has a capacity of 110 N acting in either tension or compression. Measurements are controlled and monitored via computer using a commercial data acquisition board. The entire actuator and penetration assembly is rigidly attached to a wall. $z=0$ represents the bottom of the vessel containing the grains. (b) Preparation method of the granular sample as described in the text. The descending support is lowered via a mechanical lift such that grains were never in free-fall motion. This filling procedure was performed prior to every measurement.

Figure 2

(Color online) Height dependence of penetration force $F\left(z\right)$ for $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ and a $r=12.7\mathrm{mm}$ penetrating plate. The fill height of the vessel, $z_{\max }$, increases from left to right for the data shown. The three different indicated regimes of the penetration curves (linear/hydrostatic, wall supported, and bottom dominated) are discussed in the text.

Figure 3

(Color online) Penetration force $F\left(z\right)$ and differential penetration force $\Delta F\left(z\right)$ as a function of height above the bottom of the vessel, $z$, for two different fill heights using $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ diameter glass beads and a 9.5 mm radius penetrating plate. (a) $F\left(z\right)$ for $z_{\max }=112\mathrm{mm}$ (open squares) and $F\left(z\right)$ for $z_{\max }=230\mathrm{mm}$ (open triangles). $F_{\mathit{\text{bulk}}}\left(z\right)$ for $z_{\max }=112\mathrm{mm}$ (solid line) is obtained by translating the $F\left(z\right)$ data for $z_{\max }=230\mathrm{mm}$ to be aligned with $F\left(z\right)$ for $z_{\max }=112\mathrm{mm}$ near the top grain surface. The location of $F_0=F_{\mathit{\text{bulk}}}(z=0)$, as described in the text, is also indicated. Data are plotted for every third data point for the $z_{\max }=230\mathrm{mm}$ measurement. (b) and (c) $\Delta F\left(z\right)$ on linear and semilogarithmic scales, respectively. Solid lines are fits to the equations described in the text. Note that the minimum in $\Delta F\left(z\right)$ results from the smooth vessel bottom used for these data.

Figure 4

(Color online) $\Delta F\left(z\right)$ for a series of different vessel bottom boundaries ($d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ and $r=12.7\mathrm{mm}$). Vessel bottoms and penetrating plates were machined from aluminum (the beaded bottom was coated with 3-mm-diam glass beads, and the Teflon bottom was covered with a thin Teflon sheet). Circularly grooved boundaries are composed of concentric grooves with a depth and width as indicated. The radial grooved bottom consisted of 16 radial grooves, 0.79 mm wide and 0.79 mm deep. Error bars have been suppressed for presentation and markers are shown for every tenth data point.

Figure 5

(Color online) Vessel diameter dependence of penetration forces using $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm},r=12.7\mathrm{mm}$, and a smooth-bottomed vessel. (a) $F\left(z\right)$ for a series of different vessel diameters for $z_{\max }=112\mathrm{mm}$ (solid symbols) and $z_{\max }=230\mathrm{mm}$ (open symbols) fillings of the medium. Points are shown for every 20th data point and error bars have been suppressed for clarity of figure. (b) $\Delta F$ vs $z$ plotted using a semilogarithmic scale with color coding identical to panel (a). Arrow indicates trend of results for increasing vessel diameter. Data for 54-mm-diam vessel are not shown due to lack of significant exponential behavior. Error bars and points at $z>\zeta$ have been suppressed for clarity of figure. Color coding is identical to panel (a). (c) Parameters $\lambda$ and $\zeta$ as a function of vessel diameter. The right vertical axis for $\zeta$ is plotted using a scale six times larger than the left vertical scale for $\lambda$.

Figure 6

(Color online) Velocity dependence of $F\left(z\right)$ and $\Delta F\left(z\right)$ for $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm},r=12.7\mathrm{mm}$, and a smooth-bottomed vessel. (a) $F\left(z\right)$ for a series of velocities for both $z_{\max }=112\mathrm{mm}$ (solid symbols) and $z_{\max }=230\mathrm{mm}$ (open symbols) measurements. Points are shown for every 20th data point, and error bars have been suppressed for clarity of figure. (b) $\Delta F\left(z\right)$ with points shown for every fifth data point, and error bars suppressed for clarity. Color coding is identical to panel (a). (c) Parameters $\lambda$ and $\zeta$ as a function of the penetrating plate velocity plotted on a semilogarithmic scale. The right vertical axis for $\zeta$ is plotted using a scale six times larger than the left vertical scale for $\lambda$.

Figure 7

(Color online) Thickness dependence of the penetrating plate for $F\left(z\right)$ and $\Delta F\left(z\right)$ as measured for $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm},r=12.7\mathrm{mm}$, and a smooth-bottomed vessel. (a) $F\left(z\right)$ for both $z_{\max }=112\mathrm{mm}$ (solid symbols) and $z_{\max }=230\mathrm{mm}$ (open symbols) measurements. (b) $\Delta F\left(z\right)$ with color coding identical to panel (a). (c) Fitting parameters $\lambda$ (left axis) and $\zeta$ (right axis) as a function of the penetrating plate thickness.

Figure 8

(Color online) Grain diameter dependence of the penetration forces employing spherical glass beads (Jaygo Inc.), $r=12.7\mathrm{mm}$, and a smooth-bottomed vessel. (a) $F\left(z\right)$ for a series of different diameter glass beads and “playsand” media for shallow and deep fillings. Points are shown for every 20th data point, and error bars have been suppressed for clarity of figure. (b) and (c) $\lambda$ as a function of grain diameter for smooth (open symbols) and circularly grooved (solid symbols) bottom container measurements respectively for a series of different penetrating plate radii. For smooth-bottomed vessels, the “playsand” data had no local minima and a value of $\lambda =7.15\pm 0.07\mathrm{mm}$ for the data shown.

Figure 9

(Color online) $\Delta F\left(z\right)$ for a series of different filling depths, $z_{\max }$, for $r=12.7\mathrm{mm}$ and $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ (left column) and $d_{\mathit{\text{grain}}}=0.56\pm 0.04\mathrm{mm}$ (right column). For each panel, $z_{\max }$ increase from left to right as $z_{\max }=62\mathrm{mm},82\mathrm{mm},101\mathrm{mm},116\mathrm{mm},135\mathrm{mm},153\mathrm{mm},174\mathrm{mm},$ and $193\mathrm{mm}$ (arrows indicate trend of results for increasing filling depth). Panels (b) and (d) are results obtained using a circularly grooved bottom with a groove size of 1.6 mm. Note the persistence of the length scale behavior to larger values of $z$ in the grooved bottom data compared to the smooth bottom data.

Figure 10

(Color online) Length scale $\lambda$ as a function of the stress at the bottom of the vessel, as measured by $F_0$, for $d_{\mathit{\text{grain}}}=0.56\pm 0.04\mathrm{mm}$ (triangles) and $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ (squares) using both grooved (solid symbols) and smooth (open symbols) bottom vessels. Solid lines are individual fits to the power law $\lambda \propto {\left(F_0\right)}^{1∕2}$.

Figure 11

(Color online) Penetrating plate radius $r$ dependence of the penetration force $F\left(z\right)$ using $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ glass beads with $z_{\max }=112\mathrm{mm}$ and plotted on a semilogarithmic scale. Points are shown for every tenth data point, and error bars have been suppressed for clarity.

Figure 12

(Color online) $F_0$ measured for $z_{\max }=112\mathrm{mm}$ vs area of penetrating plate for different grain diameters. Solid line is proportional to area of penetrating plate. Square cross-section penetrating plates are included in the $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ measurements. Deviations for the force being proportional to the area for the smallest penetrating plates $(r=4.4\mathrm{mm})$ used is likely due to the penetration force being comparable to the drag force of the insertion rod $(r=3.2\mathrm{mm})$.

Figure 13

(Color online) $\Delta F\left(z\right)$ for a series of different radii penetrating plates using both $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ (left column) and $d_{\mathit{\text{grain}}}=0.56\pm 0.04\mathrm{mm}$ (right column). For each panel, the plate radii increase from left to right as $r=6.4\mathrm{mm},$ $9.5\mathrm{mm},$ $12.7\mathrm{mm},$ $15.9\mathrm{mm},$ $19.1\mathrm{mm},$ and $25.4\mathrm{mm}$ (arrows indicate trend of results for increasing penetrating plate radii). All data are plotted on a semilogarithmic scale to demonstrate the exponential behavior of $\Delta F\left(z\right)$. Panels (b) and (d) are results obtained using a circularly grooved bottom with a groove size of 1.6 mm. Note the persistence of the length scale behavior to larger values of $z$ in the grooved bottomed data. For comparison, error bars are only shown for the $r=25.4\mathrm{mm}$ penetrating plate.

Figure 14

(Color online) Length scales $\lambda$ (lower points) and $\zeta$ (upper points) as a function of penetrating plate area for different grain diameters and a smooth bottomed vessel. Data are included for both circular and square cross-section penetrating plates. Solid lines are individual comparisons to the power law $\lambda \propto A^{1∕4}$ and $\zeta \propto A^{1∕4}$.

Figure 15

(Color online) Length scale $\lambda$ as a function of the quantity $\sqrt{F_0∕r}$ for circularly grooved (solid symbols) and smooth (open symbols) bottomed vessels for (a) $0.92\pm 0.04\mathrm{mm}$ diameter grains and (b) $0.56\pm 0.04\mathrm{mm}$ diameter grains. Data points include those derived from $z_{\max }$ dependence and plate size dependence measurements. Solid lines are linear fits with an intercept at the origin as described in the text.

Figure 16

(Color online) Scaled differential penetration force $(\Delta F\to \Delta F∕F_0)$ vs scaled height $(z\to z∕{(F_0∕r)}^{1∕2})$. Data in each panel are shown for both height and disk radius dependence. (a) and (b) $d_{\mathit{\text{grain}}}=0.92\pm 0.04\mathrm{mm}$ using smooth and circularly grooved bottom vessels, respectively. (c) and (d) $d_{\mathit{\text{grain}}}=0.56\pm 0.04$ using smooth and circularly grooved bottom vessels, respectively. Color coding is identical to the data shown in Figs. 9, 13. Dotted lines are used for data where the plate size was varied and solid lines are used for data where $z_{\max }$ was varied. Error bars are suppressed for clarity.