Particle Scale Dynamics in Granular Impact

We perform an experimental study of granular impact, where intruders strike 2D beds of photoelastic disks from above. High-speed video captures the intruder dynamics and the local granular force response, allowing investigation of grain-scale mechanisms in this process. We observe rich acoustic behavior at the leading edge of the intruder, strongly fluctuating in space and time, and we show that this acoustic activity controls the intruder deceleration, including large force fluctuations at short time scales. The average intruder dynamics match previous studies using empirical force laws, suggesting a new microscopic picture, where acoustic energy is carried away and dissipated.

Figure 1

(a) Six selected frames, starting at 2.75 ms after impact and spanning $475\mu \mathrm{s}$, showing the end of a typical compressional event which generates an acoustic pulse, which disconnects from the intruder. (b) A space-time plot of $G^2$ in an angular region under the bottom half of the intruder (half-annulus) over time. The $x$ axis is time, and the $y$ axis is radial distance from the bottom of the intruder, where the top of the plot corresponds to the bottom edge of the intruder. The slope of the disturbances gives a consistent acoustic speed of $\sim 325\mathrm{m}/\mathrm{s}$. (c) The sum of the response in the space-time plot above, after subtracting background inhomogeneities. Calibrating this will yield our measurement of instantaneous force, as shown later, where the range shown above [0–1.1 arbitrary units (AU)] maps to an intruder acceleration range of 0–27 g.

Figure 2

(a) Comparing the intruder trajectory to the photoelastic response from Fig. 1c shows that the intruder acceleration is very well correlated to the photoelastic or acoustic fluctuations in high-speed videos. We time-average the photoelastic response (thick, blue line) to match the time scale of the acceleration measurement $a_{\mathrm{int}}$ (black, dashed line), which has limited time resolution. Rescaling the photoelastic measurement gives extremely close agreement with the measured deceleration (both the mean and fluctuations). The calibrated photoelastic force measurement without time filtering (thin, red line) shows much larger fluctuations at a much shorter time scale. (b) The inset shows the calibration of photoelastic response versus 2D pressure (force per width of intruder or piston) from experiment (black dashed line) and from a static test (blue circles), with good agreement.

Figure 3

Photoelastic pulses decay as they propagate away from the intruder. We observe a thin angular slice of opening angle of $\pi /8\mathrm{rad}$, extending $25d$ below, and centered directly beneath the intruder. We use 40 different pulses from different impacts of a single intruder ($D=21.17d$), where the intruder velocity at the pulse emission varies between 2 and $6\mathrm{m}/\mathrm{s}$. We then plot the natural logarithm of $G^2$ per area as a function of depth for each pulse, normalized by the total intensity in the pulse (wave intensity will decrease as $1/r$ moving away from a point source in 2D, and this effect has already been accounted for in this plot). The imposed fit (thick, red line) is $\exp (-r/L)$, where $L$ is the decay length, roughly 10 particle diameters.

Figure 4

(a) The fluctuating term $\eta (t)$ for a single impact (typical for all impacts), where $\eta (t)\sim G^2(t)/G_{\mathrm{avg}}^2(t)$, as discussed in the text. (b) The autocorrelation and (c) the PDF of the combined fluctuating signals for all impacts for each intruder ($\sim 20$ runs per intruder). The semilog PDF plot shows $P(\eta )\sim \exp (-\eta )$. We see an autocorrelation time of $\sim 1\mathrm{ms}$, which gives a typical event time, which agrees with video frames in Fig. 1a. The lack of dependence on intruder size suggests a collective mechanism and not a simple combination of uncorrelated, random force chains.