Abstract
Experiments on the low-speed impact of solid objects into granular media have been used both to mimic geophysical events1,2,3,4,5 and to probe the unusual nature of the granular state of matter6,7,8,9,10. Observations have been interpreted in terms of conflicting stopping forces: product of powers of projectile depth and speed6; linear in speed7; constant, proportional to the initial impact speed8; and proportional to depth9,10. This is reminiscent of high-speed ballistics impact in the nineteenth and twentieth centuries, when a plethora of empirical rules were proposed11,12. To make progress, we developed a means to measure projectile dynamics with 100 nm and 20 μs precision. For a 1-inch-diameter steel sphere dropped from a wide range of heights into non-cohesive glass beads, we reproduce previous observations6,7,8,9,10 either as reasonable approximations or as limiting behaviours. Furthermore, we demonstrate that the interaction between the projectile and the medium can be decomposed into the sum of velocity-dependent inertial drag plus depth-dependent friction. Thus, we achieve a unified description of low-speed impact phenomena and show that the complex response of granular materials to impact, although fundamentally different from that of liquids and solids, can be simply understood.
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Main
To measure dynamics, we use a line-scan digital CCD (charge-coupled device) camera to image a finely striped transparent rod attached vertically to the top of the projectile (see the Methods section). The instantaneous speed is the key quantity, deduced from the displacement of the striped pattern between successive frames. The temporal precision is 20 μs, set by the 50 kHz frame rate of the line-scan camera. The position resolution is 100 nm, set by the 3.8 μm per pixel magnification divided by the square root of the number of pixels. These combine to give a velocity resolution of 0.5 cm s−1. Besides measurement fidelity, another advantage of our method is that it applies even to very deep impacts—as long as the striped rod does not submerge.
Our complete dynamics data set is shown in Fig. 1, which shows position z, velocity v, and acceleration a, versus time t, for initial impact speeds, v0, ranging from 0 to −400 cm s−1. Time is measured from initial impact; position is measured upwards from the granular surface, opposite to gravity. A striking feature is that, although the final position is approached smoothly, the velocity vanishes abruptly with a discontinuity in acceleration. Similar behaviour is evident in data from an embedded accelerometer (P. B. Umbanhowar, private communication, 2004; J. C. Amato, private communication, 2005). This is counter to the viscous approach to a stable equilibrium, where acceleration vanishes continuously, but it permits the stopping time, tstop, to be easily gauged from the velocity versus time data. Note that tstop actually decreases with increasing impact speed; surprisingly, deeper penetration requires less time. Evidently, granular matter is very different from ordinary solids and liquids in its resistance to penetration.
We begin the analysis by comparing our impact data with the seemingly contradictory trends reported in refs 6,7,8,9,10. First, the stop time is plotted versus v0 in the inset of Fig. 1b. As in ref. 8, for a 4.46-cm-diameter steel cylinder dropped sideways onto a bed of 0.46–0.64 cm diameter rods, the stopping time, tstop, seems constant for fast impacts; however, this is only the limiting behaviour as tstop increases for slower impacts. This trend for stopping time with impact speed may also be seen in the raw data of ref. 10. Second, the instantaneous acceleration is plotted versus depth in the inset of Fig. 1c for the case v0=0. As in ref. 9, for 4-cm-diameter spheres of varying mass released from rest at the surface of a bed of 40 μm sand grains, a seems equal to −g plus a Coulomb friction term proportional to depth; however, this is only the limiting behaviour as we find in the main plot that the acceleration at z=0 increases with impact speed. Third, the absolute final penetration depth, d, is plotted in Fig. 2 versus both the total drop distance, H=h+d, and versus the impact speed, v0=−(2g h)1/2, where h is the free-fall distance and g=980 cm s−2. As in refs 6,13, for a wide variety of projectiles and grains, d is well approximated by (d02H)1/3, where d0 is the minimum penetration depth for h=0. However, as in ref. 7, for steel spheres and glass beads ranging from 45 to 425 μm, the penetration depth, d, is also well approximated by d0+α|v0|. Though the experimental systems of refs 6,7,8,9,10 are all different, and were interpreted by four different force laws, they are actually all consistent with the data reported here. This suggests that a single, unified, force law underlies all of the observations.
Tsimring and Volfson14 proposed that the total force on the projectile is
equal to the sum of gravity plus Coulomb friction plus inertial drag. This is similar to the Poncelet force law, F0+c v2, used in high-speed ballistics11,12, but accounts for the depth-dependence of Coulomb friction in granular media. Tsimring and Volfson argued that the form of F(z) should vary from quadratic to constant owing to the shapes of the projectile and of the growing crater excavated by its motion. They also showed how such a force law can approximately account for a depth scaling of d=(d02H)1/3.
The entire dynamics data set of Fig. 1 can be used to test the form of equation (1). Combining it with Newton’s second law, the acceleration at a given fixed depth zi should be quadratic in speed: a+g=F(zi)/m+v2/d1. For each drop height, we therefore examine the acceleration and the speed when the projectile passes through five different fixed depths, zi={0,−1,−2,−3,−4}±0.1 cm. The acceleration values are shown versus speed in Fig. 3a, where each point represents a different drop height and where the five colours represent the five fixed depths. The results are quadratic in speed and furthermore, crucially, have the same proportionality factor d1=8.7±0.7 cm for all five depths. This is demonstrated by the dotted curves in the main plot, which become parallel lines when plotted versus speed squared in the inset. The good agreement shows that the projectile experiences a force m v2/d1 that is independent of depth. This result can be expressed as 0.8ρgDb2v2, and hence can be interpreted as an inertial force required for the projectile to mobilize a volume Db3 of granular media with density ρg.
The form of the Coulomb friction-like term may now be examined using the value of d1 deduced above. As equation (1) gives F(z)/m=a+g−v2/d1, we evaluate the right-hand side and plot the results versus depth in Fig. 3b. There the curves represent data for different drop heights; the open symbols represent extrapolation to v=0 of a+g data at fixed depth for many drop heights, from Fig. 3a. This produces a good collapse of our entire data set, for all drop heights and for all times. It shows that the projectile experiences a force F(z) that is independent of speed and that increases with depth. The form of this force is similar to F(z)/(m g)=1+[3(z/d0)2−1]exp(−2|z|/d1), shown in Fig. 3b as a grey dashed curve, which is constructed to give d=(d02H)1/3 exactly15. This varies from quadratic to linear, and saturates at great depths, just as argued by Tsimring and Volfson. However, the data are better fitted over most of the range to a simpler, linear form F(z)=k|z| with k/m=1,040±10 s−2, shown by the dotted line. This result can be expressed as k=20μ ρgg Db2, which is larger than expected for ordinary Coulomb friction and observed previously for objects pushed horizontally in a granular medium16.
Altogether we have shown that a force law, ΣF=−m g+k|z|+m v2/d1, can account for the salient features of existing granular impact data. Here, the high quality and range of the dynamics data are sufficient to allow the individual depth- and velocity-dependent terms to be isolated and demonstrated. The order of magnitude of the latter force is ρgDb2v2, set by grain density and ball area. The order of magnitude of the former force is larger than μ ρgg Db2|z|, set by Coulomb friction. Although the detailed scaling has yet to be checked, the order-of-magnitude consistency of these expressions reveal that the characteristic length scale is given by the diameter of the projectile, Lc=Db. This explains why the penetration depth is within an order of magnitude of the ball diameter for a wide range of impact speeds. As gravity is the only system parameter whose units contain time, the characteristic time and velocity scales must therefore be Tc=(Db/g)1/2 and Vc=(Dbg)1/2. Indeed, these explain the typical stopping time and the velocity beyond which the stopping time is constant, seen in the inset of Fig. 1b. In addition, finally, the force law explains the acceleration discontinuity at stoppage, seen in Fig. 1c, as Δa=−g+(k/m)d. The only puzzles that remain are the linear form of the Coulomb friction term plus the precise values of k and d1, as well as their scaling with system properties. In addition to accounting for the full behaviour of the projectile, the fundamental forces demonstrated here to act between the projectile and the medium must also govern the injection of energy needed to explain spectacular features in the medium such as crater morphologies4,5 and jet splash heights17,18,19,20.
Methods
Spherical glass beads (diameter range 250–350 μm) are used as a dry non-cohesive granular medium with density ρg=1.52 g cm−3, draining angle of repose θr=24∘ and friction coefficient μ=tan(θr)=0.45. A clear plexiglass tube (outer diameter: 8 inches; inner diameter: 7.5 inches; height: 12 inches) is set on a sieve (US sieve size 45–60). The same windbox used in previous experiments13 is attached under this container. The glass beads are poured into the container to a depth of about 20 cm. The medium is fluidized, and gradually de-fluidized, by a uniform upflow of N2 gas before each impact to ensure a homogeneous medium with a flat surface. The volume fraction occupied by the beads is 0.590±0.004 after fluidization21. A steel sphere of diameter Db=2.54 cm is used as a projectile. An acrylic transparent square rod (length: 8 inches; square cross-section: 1/8×1/8 inches) is glued vertically on top of the sphere. A horizontally striped transparent sheet is affixed to one side of the rod. The width and space of the stripes are 0.2 mm. A small metal tip is glued to the top of the rod. The total mass of the projectile plus rod is m=69.2 g, giving the ball an effective density of ρb=8.07 g cm−3. The projectile is held by an electromagnet, which is turned off to commence free-fall of the projectile. A line-scan CCD camera (1,024 pixels, 8 bits deep, 50 kHz frame rate) is placed to capture light transmitted through the striped pattern. Images are also acquired before and after each impact for calibration of length scale.
To analyse video data, the velocity is first deduced from the frame rate and the location of the peak in the cross correlation of two successive images of the striped pattern. Next, the impact time and speed are identified from fits of raw v(τ) data to [v0−g(τ−τ0)]H(τ−τ0)+ [v0+a0(τ−τ0)+j0(τ−τ0)2/2]H(τ0−τ), where the fitting parameters are {v0,a0,j0,τ0} and H(x) is the Heaviside function. The stop time and acceleration discontinuity are identified by fits to v(t)=Δa(t−tstop) over a small window before tstop, where t=τ−τ0. Position is then deduced from . The results are consistent with, but more accurate than, the values of h=v02/(2g) and measured directly by a microtelescope mounted to a height gauge. Finally, acceleration is deduced from fits of the velocity data to line segments, where the fitting windows are adjusted so that the acceleration uncertainty is smaller than the larger of 0.5% or 0.005 g. An example of raw video data and analysis is given in the Supplementary Information, along with a discussion of possible effects due to atmospheric air pressure.
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Acknowledgements
This work was supported by the National Science Foundation (D.J.D.) and the Japan Society for the Promotion of Science Postdoctoral Fellowships for Research Abroad (H.K.).
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Katsuragi, H., Durian, D. Unified force law for granular impact cratering. Nature Phys 3, 420–423 (2007). https://doi.org/10.1038/nphys583
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DOI: https://doi.org/10.1038/nphys583