We present data for the time-dependence of wooden spheres penetrating into a
loose non-cohesive packing of glass beads. The stopping time is a factor of
three longer than the time $d/v_\circ$ needed to travel the total penetration
distance $d$ at the impact speed $v_\circ$. The acceleration decreases
monotonically throughout the impact. These kinematics are modelled by a
position- and velocity-dependent stopping force that is constrained to
reproduce prior observations for the scaling of the penetration depth with the
total drop distance.Comment: 4 pages, experimen

D. J. Durian

J. Duran

M. A. Ambroso

R. D. Kamien

R. L. Brown

English

arXiv

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Dynamics of shallow impact cratering

We present data for the time dependence of wooden spheres penetrating into a loose noncohesive packing of glass beads. The stopping time is a factor of 3 longer than the time d ∕ v0 needed to travel the total penetration distance d at the impact speed v0. The acceleration decreases monotonically throughout the impact. These kinematics are modeled by a position- and velocity-dependent stopping force that is constrained to reproduce prior observations for the scaling of the penetration depth with the total drop distance

Ambroso, M. A

Kamien, Randall D

Durian, Douglas J

ScholarlyCommons@Penn

University of PennsylvaniaScholarlyCommonsDepartment of Physics Papers Department of Physics10-2005Dynamics of Shallow Impact CrateringM. A. AmbrosoUniversity of California, Los AngelesRandall D. KamienUniversity of Pennsylvania, kamien@physics.upenn.eduDouglas J. DurianUniversity of Pennsylvania, djdurian@physics.upenn.eduFollow this and additional works at: https://repository.upenn.edu/physics_papersPart of the Physics CommonsThis paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/609For more information, please contact repository@pobox.upenn.edu.Recommended CitationAmbroso, M. A., Kamien, R. D., & Durian, D. J. (2005). Dynamics of Shallow Impact Cratering. Physical Review E, 72 (4),041305-1-041305-4. http://dx.doi.org/10.1103/PhysRevE.72.041305Dynamics of Shallow Impact CrateringAbstractWe present data for the time dependence of wooden spheres penetrating into a loose noncohesive packing ofglass beads. The stopping time is a factor of 3 longer than the time d ∕ v0 needed to travel the total penetrationdistance d at the impact speed v0. The acceleration decreases monotonically throughout the impact. Thesekinematics are modeled by a position- and velocity-dependent stopping force that is constrained to reproduceprior observations for the scaling of the penetration depth with the total drop distance.DisciplinesPhysical Sciences and Mathematics | PhysicsThis journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/609Dynamics of shallow impact crateringM. A. Ambroso,1 R. D. Kamien,2 and D. J. Durian1,21Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USAReceived 21 March 2005; published 20 October 2005We present data for the time dependence of wooden spheres penetrating into a loose noncohesive packing ofglass beads. The stopping time is a factor of 3 longer than the time d /v0 needed to travel the total penetrationdistance d at the impact speed v0. The acceleration decreases monotonically throughout the impact. Thesekinematics are modeled by a position- and velocity-dependent stopping force that is constrained to reproduceprior observations for the scaling of the penetration depth with the total drop distance.DOI: 10.1103/PhysRevE.72.041305 PACS numbers: 45.70.Ht, 45.70.Cc, 83.80.Fg, 89.75.DaGranular impact is a phenomenon of natural interest. Onefocus of recent work is the size and morphology of the crater,and the analogy with planetary cratering 1–3. Another isthe dramatic splash produced by the collapsing void 4,5.Still another is the final depth of penetration d, because itprobes granular mechanics via the depth-averaged stoppingforce Fs=mgH /d, where m is the projectile mass, g=980 cm/s2, h is the free-fall distance, and H=h+d is thetotal drop distance 6–10. For shallow impact by spheres6,10, and for deeper impacts by cylinders with various tipshapes 7, the penetration depth scales asd/d0 = H/d01/3, 1where d0 is the minimum penetration for h=0. The inset ofFig. 1 shows Eq. 1 agreeing with data over nearly threedecades in H. While this constrains the stopping force, itdoes not reveal a unique form. For example, Eq. 1 isequally consistent with Fsz2 and Fsv4/3, where z and v,respectively, are the instantaneous depth and speed of theprojectile 6. Which, if either, of these possible stoppingforces is correct? How does the nature of the stopping force,and the resulting transfer of energy from the projectile to themedium, conspire to produce the subsequent granular splashand the final crater morphology? The unintuitive response ofgranular media to external forcing is a topic of widespreadinterest beyond the specific example of impact 11–13.Recently impact dynamics have been measured by high-speed video 14–16 and by an embedded accelerometer17. In Ref. 14 the total upward force is found to be F=−mg+ mg+kd. The stopping force, in parentheses, is in-dependent of time but has a value that depends on the impactspeed v0. Solution of F=ma gives a penetration depth ofd=mv02 / 2k. In Ref. 15, the total force is found to beF=−mg+k	z	. The stopping force is Coulomb friction andincreases with time. Solution of F=ma gives a penetrationdepth of similar form to Eq. 1: d /d0= H /d01/2 with d0=2mg /k. In Refs. 16,17, the acceleration decreases withtime. The various reported force laws thus appear contradic-tory, both in terms of their time dependencies and in terms oftheir predicted penetration depths. Furthermore, none of thereported force laws is consistent with the dH1/3 observa-tion of Eq. 1.In this paper we measure cratering dynamics in the unex-plored regime of shallow impact, where the projectile neversubmerges. Our approach is to measure position vs time withan optical method, both faster and more precise than imag-ing. As in Refs. 16,17, we find that the acceleration de-creases throughout impact. Our theoretical approach is toconsider possible instantaneous force laws whose depth av-erages reproduce the observed scaling of Eq. 1. The bestcandidate depends on both position and speed, and suggeststhat the seemingly disparate results of Refs. 6,14–17 maynot be contradictory but instead may represent limiting casesof a common force law that holds for both shallow and deepimpacts.Our materials and penetration depth measurements areidentical to those of Ref. 10. The medium consists of glassbeads, diameter Dg=0.30±0.05 mm, prepared at 59% pack-ing fraction by slowly turning off a fluidizing upflow of air.The projectiles are wooden spheres of diameter Db=1.49 orFIG. 1. Color online The vertical position z of the bottom of asphere vs time t, for typical conditions as labeled. The impact be-gins at z=0, t=0, and gravity points in the −z direction. Onlyevery 1/100 datum is shown. The inset shows the final penetrationdepth vs total drop distance, both scaled by the minimum penetra-tion depth d0. The small symbols represent data for four woodenspheres from Ref. 10. The large symbols represent additional runs,for which in this paper we report on impact dynamics  is for theexample in the main plot. The values of d0 are 0.63 and 0.44 cmfor the 1.99 and 1.49 in. diameter spheres, respectively.PHYSICAL REVIEW E 72, 041305 20051539-3755/2005/724/0413054/$23.00 ©2005 The American Physical Society041305-1=1.99 in., and of density b=0.7 g/mm3. These are held anddropped from rest via a suction mechanism. The drop dis-tance and final penetration depth are measured with a tele-scope mounted to a height gauge. Data for the woodenspheres of Ref. 10, and for the dynamics runs reportedhere, are shown in the inset of Fig. 1 to obey Eq. 1.The time-dependent vertical position of the projectile ismeasured optically. A laser =532 nm is placed about 1.5m from the sample, along with a cylindrical lens that fans thebeam into a thin sheet. An aperture is used to select thecentral portion of the beam, where the intensity is nearlyconstant, and to set its size to be slightly greater than Db. Onthe other side of the projectile, we align a second apertureequal to Db. Behind this we place a large planoconvex lens tofocus the light onto a photodiode. As such, the collected lightintensity varies nearly linearly with projectile position. Ineach run the photocurrent starts at a maximum, decreases asthe projectile freely falls onto the beam, goes to zero whenthe beam is fully blocked, and then increases as the projectilefalls further and light passes over its top; impact occurs dur-ing this last phase. To calibrate we hold the ball in severalknown positions, with light passing above and below, and wefit to the particular cubic polynomial expected for an alignedGaussian beam.Typical depth vs time data are displayed in Fig. 1. Theimpact occurs at t=0, and the projectile comes to rest inabout 0.03 s. The solid curve through the t0 free-fall datais not a fit, but rather −v0t−gt2 /2 with v0=2gh. We digi-tized the photocurrent using a 12-bit analog-to-digital con-verter operating at 105 points per second. Finer digitizationand faster capture could be achieved at marginally greaterexpense. The fidelity with which impact dynamics can becaptured by our method exceeds high-speed video. Notwith-standing, there exist certain limitations. One is that for largeenough drop heights the grains splash into the laser beam. Alesser limitation is that the minimum drop height was suchthat none of the beam was blocked by the projectile prior toits release. Overall we achieved over a factor of two varia-tion in total drop distance for each sphere. To differentiateposition vs time data, we fitted to a cubic polynomial withGaussian weighting that nearly vanishes at the edges. Forfitting windows that are too small, the velocity and accelera-tion results are noisy; for fitting windows that are not toolarge, the depth-averaged acceleration equals gh /d as re-quired by energy conservation. This check gives confidencein both our data and our differentiation procedures.Data for position-velocity-acceleration vs time, for bothspheres and all drop heights, are displayed in Fig. 2. Alllengths are scaled by the final penetration depth d, and alltimes are scaled by the time d /v0 required to move a distanced at the impact speed v0. To within measurement error, thedata all collapse according to this scaling. By construction,the scaled position data must decay from 0 to −1 with aninitial slope of −1; the scaled velocity data must decay from−1 to 0; and the depth average of the scaled accelerationmust equal 1 /2. We find that the spheres all come to rest atabout 3v0t /d, roughly three times longer than if they movedthe same distance at constant speed. Since v0h1/2 and dH1/3 have similar scaling, the impact duration is nearlyconstant as in Refs. 14,17. We also find that the accelera-tion decreases with time, in accord with Refs. 16,17 but incontrast to Refs. 14,15. Note that the scaled value of gravi-tational acceleration is −gd /v02=−d / 2h; this ranges from−0.05 to −0.10 for our runs shaded gray region in Fig. 2,bottom panel and is generally small compared to the projec-tile acceleration.Our impact dynamics data can now be compared withexpectations for various candidate force laws. For examplethe simple ad hoc form F=−mg+k	z		v	 best agrees withEq. 1 if the exponents are related by = 4−2 /3. Theagreement becomes exact for F=−mg+k	z	2. Predictionsfor this special case =2,=0, and also for =0,FIG. 2. Color online Vertical components of position, velocity,and acceleration vs time, for all runs; red orange dots are for Db=1.99 in. 1.49 in.. To within measurement errors, the data col-lapse when lengths are scaled by d and times are scaled by d /v0,where d is the final penetration depth and v0 is the impact speed. Byconstruction, the scaled position data all decay from 0 to −1 with aninitial slope of −1 gray lines. Note that the acceleration is notconstant gray dashed line ad /v02=1/2, but rather decreases withtime. The data rule out the power-law forces consistent with Eq. 1,but are consistent with a modified Poncelet model.AMBROSO, KAMIEN, AND DURIAN PHYSICAL REVIEW E 72, 041305 2005041305-2=4/3, are displayed with the scaled data in Fig. 2. Evi-dently, the decay of zt is too fast for the former and tooslow for the latter. The actual behavior lies between theseextremes. A marginally acceptable fit, also shown, is attainedfor =1/2 ,=1.Better fits can be achieved if the stopping force equals aconstant plus a term that grows with speed. If the drag isviscous, then the force law is given by the Bingham model,F=−mg+ F0+b	v	. If the drag is inertial, then the forcelaw is given by the Poncelet model, F=−mg+ F0+cv2.The Bingham model has recently been advocated for granu-lar impact 8, while the Poncelet model has long been usedfor ballistics applications 18. For both, position vs speedcan be found by writing a=v dv /dz, separating variables,and integrating. All our dynamics data are shown again inphase space plots of scaled velocity and acceleration vsdepth, as well as acceleration vs speed, in Fig. 3. The bestone-parameter fits to these models give F0−mg / bv0=0.065 and F0−mg / cv02=0.20, respectively; 2 issmaller for the Poncelet model by a factor of 2. These fitsnot shown are both acceptable, but not quite as nice as theone displayed. Still, neither model predicts the observed pen-etration depth scaling of Eq. 1. The Bingham model givesd= mv0 /b1− ln1+1/ where = F0−mg / bv0, andthe Poncelet model gives d= m / 2cln1+1/ where = F0−mg / cv02. Furthermore, the predicted initial accel-erations, and the final penetration depths, are unphysical inthe limit v0→0.We now modify the Poncelet model along the lines sug-gested by Tsimring and Volfson 19. Since the constant termrepresents friction, it should depend on depth according tohydrostatic pressure 15,20 and the geometry of the granularmedium near the projectile: F=−mg+ Fz+cv2. For anyFz this can be recast as an ordinary differential equation forkinetic energy vs position and solved by use of an integratingfactor, exp−2cz /m. To exactly recover the observed pen-etration depth scaling, Eq. 1, and hence be consistent withthe observed depth-averaged stopping force, we find that thefriction term must be Fz /mg= 3z /d02−1exp−2	z	 /d1+1 where d1=m /c. This vanishes for shallow penetrationand approaches a constant for deep penetration, consistentwith the special limits advocated in Ref. 19. Altogether theprojectile acceleration in our model isa/g = 3z/d02 − 1exp− 2	z	/d1 + v2/gd1 . 2For high speeds and deep impacts the velocity term domi-nates. For shallow impacts, the leading behavior is a /g−1+2	z	 /d1+ 3/d02−2/d12z2+v2 / gd1. For zero impactspeed, the initial acceleration is −g, as expected.To compare our force model with data, it is convenient towork in phase space since velocity vs depth can be computeddirectly as vv02 = 1 − 	z/d	3 − 	z/d	d0/d21 − d0/d2 e−2	z	/d1. 3For v0=0, the relation is v2=2gd0	z /d0	− 	z /d0	3exp−2	z	 /d1. Since the predicted dynamics depend on thevalue of d /d0, the apparent collapse of data in Fig. 2 wouldthen represent clustering about an average to within measure-ment error. Setting d /d0=2.77, the average value for all ourruns, the best one-parameter fit to Eq. 3 is for d /d1=1.2.This gives an excellent description of the data, as shown bythe solid purple curves in Figs. 2 and 3.As the ball comes to rest, in all acceptable fits, the timebecomes about 3d /v0 and the upward acceleration becomes1–2g. After stopping the acceleration abruptly vanishes, asseen directly in measurement by accelerometer 17. The dis-continuity is more apparent here in plots of acceleration vsposition or speed Fig. 3 than in plots of acceleration vstime Fig. 2.In conclusion, we have measured the dynamics of impactto further constrain the form of the force law responsible forprior observations of final penetration depth vs total dropdistance, dH1/3 6,7,10. For two sphere sizes, and a factorof 2 in drop distance, the position vs time data appear tocollapse when scaled by impact speed and final depth. Sev-eral possible force laws can now be ruled out altogether. Twovelocity-dependent force laws are consistent with dynamicsbut not with penetration depth data. The best candidate is amodification of the Poncelet model along the lines suggestedin Ref. 19. It features inertial drag plus a particular depth-dependent friction term designed to exactly recover the pen-etration depth scaling of Eq. 1. The predicted kinematicsEqs. 2 and 3 agree very well with our present data. Themodel has two important length scales, the minimum pen-etration depth d0 and an inertial drag length d1. The former isFIG. 3. Color online Velocity and acceleration vs position, andacceleration vs velocity inset, for all runs, scaled as in Fig. 2; redorange dots are for Db=1.99 in. 1.49 in.. Excellent fits to themodified Poncelet model are shown as solid purple curves. Thevarious dashed curves represent the same power-law forces as inFig. 2.DYNAMICS OF SHALLOW IMPACT CRATERING PHYSICAL REVIEW E 72, 041305 2005041305-3given by d0= 0.14/	3/2b /g3/4Db where 	 is the tangentof the repose angle, g is the grain density, b is the balldensity, and Db is the ball diameter 10. For the systemexamined here, d1 is about twice as large as d0 but we do notyet know the dependence of d1 on system properties. Thiscould be deduced, and the model could be further tested, bymeasurement of impact dynamics over a broad range of con-ditions. At one extreme, the “dry quick sand” examined inRef. 15 for v0=0 is reproduced very well by our modelwith d1=3d0 21.We thank D. Lohse, L. S. Tsimring, and P. B. Umban-howar for sharing their results prior to publication. We thankT. C. Lubensky and P. T. Korda for helpful discussions. Thismaterial is based upon work supported by the National Sci-ence Foundation under Grant No. DMR-0305106.1 J. C. Amato and R. E. Williams, Am. J. Phys. 66, 141 1998.2 Y. Grasselli and H. J. Herrmann, Granular Matter 3, 2012001.3 A. M. Walsh, K. E. Holloway, P. Habdas, and J. R. de Bruyn,Phys. Rev. Lett. 91, 104301 2003.4 S. T. Thoroddsen and A. Q. Shen, Phys. Fluids 13, 4 2001.5 D. Lohse, R. Bergmann, R. Mikkelsen, C. Zeilstra, D. van derMeer, M. Versluis, K. van der Weele, M. van der Hoef, and H.Kuipers, Phys. Rev. Lett. 93, 198003 2004.6 J. S. Uehara, M. A. Ambroso, R. P. Ojha, and D. J. Durian,Phys. Rev. Lett. 90, 194301 2003; 91, 149902E 2003.7 K. A. Newhall and D. J. Durian, Phys. Rev. E 68, 060301R2003.8 J. R. de Bruyn and A. M. Walsh, Can. J. Phys. 82, 439 2004.9 X.-J. Zheng, Z.-T. Wang, and Z.-G. Qiu, Eur. Phys. J. E 13,321 2004.10 M. A. Ambroso, C. R. Santore, A. R. Abate, and D. J. DurianPhys. Rev. E 71, 051305 2005.11 R. L. Brown and J. C. Richards, Principles of Powder Me-chanics Pergamon Press, Oxford, 1970.12 H. M. Jaeger, S. R. Nagel, and R. P. Behringer, Rev. Mod.Phys. 68, 1259 1996.13 J. Duran, Sands, Powders, and Grains: An Introduction to thePhysics of Granular Materials Springer, New York, 2000.14 M. P. Ciamarra, A. H. Lara, A. T. Lee, D. I. Goldman, I.Vishik, and H. L. Swinney, Phys. Rev. Lett. 92, 1943012004.15 D. Lohse, R. Rauhe, R. Bergmann, and D. van der Meer, Na-ture London 432, 689 2004.16 K. E. Daniels, J. E. Coppock, and R. P. Behringer, Chaos 14,S4 2004.17 P. B. Umbanhowar private communication.18 M. E. Backman and W. Goldsmith, Int. J. Eng. Sci. 16, 11978.19 L. S. Tsimring and D. Volfson unpublished.20 I. Albert, J. G. Sample, A. J. Morss, S. Rajagopalan, A. L.Barabasi, and P. Schiffer, Phys. Rev. E 64, 061303 2001, andreferences therein.21 See EPAPS Document No. E-PLEEE8-72-135510 for com-parison of the modified Poncelet model with the impact dy-namics data of Refs. 14, 15, 17. This document can bereached via a direct link in the online article’s HMTL referencesection or via the EPAPS homepage http://www.aip.org/pubservs/epaps.html.AMBROSO, KAMIEN, AND DURIAN PHYSICAL REVIEW E 72, 041305 2005041305-4

Dynamics of Shallow Impact Cratering

Dynamics of shallow impact cratering

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