Geometry-controlled Onset of Inertial Drag in Granular Impact

TL;DR

The paper addresses how intruder geometry, especially cone half-angle φ, controls both the onset of inertial drag and the peak force during granular impact. Using controlled drops of conical projectiles with φ = 20°–70° into a glass-bead bed and high-bandwidth accelerometry, it identifies a geometry-dependent crossover speed $v_c$ above which inertial drag dominates. In the inertial regime, the peak force scales as $F_{\text{peak}} \propto v^2$ and collapses across geometries when normalized by $ \tan\phi$, i.e., $F_{\text{peak}}/\tan\phi \propto v^2$. The findings show that cone geometry governs both the timing and strength of inertial momentum transfer into the grains, providing a geometry-aware extension of existing granular impact models.

Abstract

The impact of solid intruders into granular media is commonly described by a combination of quasi-static resistance and an inertial drag force proportional to the square of the impact speed. While intruder geometry is known to influence force magnitudes, its role in controlling the onset of inertial drag has remained largely unexplored. Here we present systematic impact experiments using conical intruders spanning a wide range of apex angles. By measuring the peak acceleration during impact, we show that the emergence of a well-defined inertial response depends sensitively on cone geometry. Blunt cones exhibit quadratic scaling with impact speed over the full range of velocities studied, whereas sharper cones display a delayed transition to inertial behavior at higher speeds. We define a geometry-dependent crossover speed marking the onset of the inertial regime and find that it scales approximately linearly with the cone angle through $\tanφ$. Once the inertial regime is established, the peak force collapses when rescaled by $\tanφ$, indicating that cone geometry controls the effective momentum transfer to the grains. These results demonstrate that intruder geometry governs not only the magnitude of inertial drag, but also the impact speed at which it becomes dominant.

Figures (6)

• Figure 1: Peak acceleration $a_{\text{peak}}$ as a function of impact speed $v$ into glass beads for a steel sphere with diameter $D = 2.6$ cm. Markers show mean values and error bars indicate standard deviation. The dashed red line shows the known $a_{\text{peak}} \propto v^2$ scaling for sphere impacts (valid for $v\gtrsim$ 1.5 ms$^{-1}$) and is plotted as a reference to validate the accelerometer response.
• Figure 2: Representative acceleration traces for conical intruders impacting a granular bed. Left panel: blunt $70^{\circ}$ cone with impact speed $v = 1.1$ ms$^{-1}$ showing a smooth, reproducible inertial peak. Right panel: sharp $30^{\circ}$ cone with impact speed $v = 1.8$ ms$^{-1}$ with an inertial peak. Inset: same sharp cone with impact speed $v = 1.2$ ms$^{-1}$ exhibiting an intermittent acceleration signal associated with localized force chain events and failure to form a collective inertial response.
• Figure 3: Peak acceleration $a_{\text{peak}}$ as a function of impact speed squared $v^2$ for conical intruders with varying cone half-angle. Blunt cones exhibit quadratic scaling over the full speed range, while sharper cones deviate at low speeds and only approach quadratic scaling above a geometry-dependent crossover speed. This trend indicates that the onset of the inertial regime is delayed as cone angle is decreased.
• Figure 4: Geometry-dependent characteristic crossover speed $v_c$ associated with the emergence of inertial drag, plotted as a function of cone angle $\phi$. For blunt and very sharp cones, the transition to inertial behavior is sharp and $v_c$ is well defined, varying approximately linearly with $\tan \phi$ (shown by the black dashed line). For intermediate cone angles, the crossover occurs gradually over a finite range of impact speeds, and the plotted value should be interpreted as an effective onset rather than a unique threshold.
• Figure 5: Geometry-dependent characteristic crossover speed $v_c$ associated with the emergence of inertial drag, plotted as a function of $\sin \phi$. It can be seen that there is no collapse as with $\tan \phi$, hence the scaling of the onset to inertial drag cannot be explained by the change in projected area of the conical intruder.