We report on the penetration of cylindrical projectiles dropped from rest into a dry, noncohesive granular medium. The cylinder length, diameter, density, and tip shape are all explicitly varied. For deep penetrations, as compared to the cylinder diameter, the data collapse onto a single scaling law that varies as the 1/3 power of the total drop distance, the 1/2 power of cylinder length, and the 1/6 power of cylinder diameter. For shallow penetrations, the projectile shape plays a crucial role with sharper objects penetrating deeper

Newhall, K. A

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University of PennsylvaniaScholarlyCommonsDepartment of Physics Papers Department of Physics12-24-2003Projectile-Shape Dependence of Impact Craters inLoose Granular MediaK. A. NewhallUniversity of California, Los AngelesFollow this and additional works at: https://repository.upenn.edu/physics_papersPart of the Physics CommonsAt the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles. Currently, he is a faculty member at thePhysics Department at the University of Pennsylvania.This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/614For more information, please contact repository@pobox.upenn.edu.Recommended CitationNewhall, K. A. (2003). Projectile-Shape Dependence of Impact Craters in Loose Granular Media. Physical Review E, 68 (6),060301-1-060301-3. http://dx.doi.org/10.1103/PhysRevE.68.060301Projectile-Shape Dependence of Impact Craters in Loose Granular MediaAbstractWe report on the penetration of cylindrical projectiles dropped from rest into a dry, noncohesive granularmedium. The cylinder length, diameter, density, and tip shape are all explicitly varied. For deep penetrations,as compared to the cylinder diameter, the data collapse onto a single scaling law that varies as the 1/3 power ofthe total drop distance, the 1/2 power of cylinder length, and the 1/6 power of cylinder diameter. For shallowpenetrations, the projectile shape plays a crucial role with sharper objects penetrating deeper.DisciplinesPhysical Sciences and Mathematics | PhysicsCommentsAt the time of publication, author Douglas J. Durian was affiliated with University of California, Los Angeles.Currently, he is a faculty member at the Physics Department at the University of Pennsylvania.This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/614Projectile-shape dependence of impact craters in loose granular mediaK. A. Newhall and D. J. DurianDepartment of Physics & Astronomy, UCLA, Los Angeles, California 90095-1547, USA~Received 9 September 2003; published 24 December 2003!We report on the penetration of cylindrical projectiles dropped from rest into a dry, noncohesive granularmedium. The cylinder length, diameter, density, and tip shape are all explicitly varied. For deep penetrations,as compared to the cylinder diameter, the data collapse onto a single scaling law that varies as the 1/3 powerof the total drop distance, the 1/2 power of cylinder length, and the 1/6 power of cylinder diameter. For shallowpenetrations, the projectile shape plays a crucial role with sharper objects penetrating deeper.DOI: 10.1103/PhysRevE.68.060301 PACS number~s!: 45.70.Cc, 83.80.FgAnyone who has walked across a beach realizes that eventhough the sand moves under the pressure of a step, at somepoint their foot is supported. Someone who is running leavesa deeper and larger footprint, or crater, than someone who iswalking. This phenomenon applies to anything dropped intosand; no matter how energetic the impact, the sand will even-tually stop the projectile. For an object of mass m that isdropped from rest and that moves downwards a total distanceH ~including motion throughout the impact!, the gravita-tional potential energy mgH is transfered from the projectileto the sand. Some of this energy is dissipated and some of itis converted back to gravitational potential energy associatedwith the size and shape of the crater. Fundamentally, theenergy is expended by an average stopping force ^F& appliedby the sand as the projectile moves a total distance d frominitial impact to rest:^F&d5mgH . ~1!Thus, a straightforward measurement of d vs H provides in-formation about granular mechanics, a topic of widespreadinterest @1,2#. A full understanding of the stopping forces andthe transfer of energy may enable a fundamental explanationfor such impact-related phenomena as the complex sequenceof crater morphologies @3–6# and the spectactular granularjets formed by a collapsing hole @7,8#.The penetration of spheres into loose granular media hasbeen reported previously. In Ref. @9#, the total drop distanceH was varied by up to a factor of 1000; the projectile densityrp was varied by a factor of 60; the projectile diameter Dpwas varied by a factor of 4; the grain density rg was variedby a factor of 2; and the tangent of the draining repose anglem was varied by a factor of 2. The resulting penetrationdepths are reproduced in Fig. 1, along with the observedscaling lawd50.14F S rprgm2D 3/2Dp2HG 1/3. ~2!From the combined dependence on projectile density anddrop height, (rpH2/3)1/2, we note that the penetration depth isnot a function of either the sphere’s kinetic energy, ;rpH ,or its momentum, ;rpH1/2, at impact. In Ref. @6# a differentdefinition of crater depth, suitable when the spheres com-pletely submerge, gave a different scaling.The behavior in Fig. 1 cannot be explained by currenttheories. Granular hydrodynamics and a rate-independent‘‘plowing’’ force both predict different scaling than Eq. ~2!~see discussion in Ref. @9#!. But how general is the observa-tion? In particular, how does the crater depth depend uponthe shape of the projectile? One would expect a pointed pro-jectile to penetrate deeper than a blunt projectile. Perhapsthis may be accounted for by the value of the numericalcoefficient in Eq. ~2!. Or perhaps the functional form ofdepth vs drop distance may be entirely different, with d;H1/3 arising as an accident of the projectile’s sphericalshape.To explore the role of projectile shape on impact crater-ing, we perform a series of experiments with the same basicprotocol as in Ref. @9#. Namely, we fill a 1000 ml beaker~diameter 4 in.! with dry, noncohesive glass beads ~diameter0.2 mm, draining respose angle 24°). The depth of filling is2 in. for most trials, but is increased to 4 in. for deep impacts.The density of the granular medium is rg51.51 g/cc, whichcorresponds to random-close packing with a volume fractionof about 63%. As in Ref. @9#, this state is achieved by swirl-ing the sample horizontally, at first rapidly and then evermore slowly, until the surface is level and at rest. Instead ofdropping spheres, we now drop a wide variety of cylindersand measure their depth of penetration. Cylinder materialsinclude wood ~poplar, rp50.5 g/cc), aluminum (Al, rp52.7 g/cc), and high-density tungsten carbide (WC, rp517.1 g/cc). We use four different cylinder diameters, DpP$0.25 in., 0.50 in., 1.0 in., 2.0 in.%, and four different cyl-inder lengths, LP$0.5 in., 1 in., 2 in., 4 in.%. To vary theprojectile shape, progressively sharper conical tips are ma-chined at one end of these cylinders; we use four differentFIG. 1. Crater depth vs scaled total drop distance. All data arefrom Ref. @9# for spherical projectiles. The solid line is the power-law scaling given by Eq. ~2!.RAPID COMMUNICATIONSPHYSICAL REVIEW E 68, 060301~R! ~2003!1063-651X/2003/68~6!/060301~3!/$20.00 ©2003 The American Physical Society68 060301-1cone angles, $180° (i.e., flat),135°,90°,45°%. To achieve avertical orientation at impact, the cylinders are suspended bya thin loop of thread, tied through an accurately machinedhole. For each cylinder, the drop distances are varied aswidely as possible within certain limitations. The minimumis set either by the penetration depth for a cylinder placedinfinitesimally above the surface of the medium or by thetendency of ~long or light! cylinders to tip over after a shal-low penetration. The maximum drop distance is set either bythe depth of the sand or by the tendency of cylinders to beginto tumble during free fall. No trial is accepted unless thecylinder remains vertical after impact.Four sets of penetration depth vs drop distance data aredisplayed in Fig. 2, all for cylindrical projectiles of length 2in. and diameter 0.25 in. This includes WC and Al rods withflat ends, as well as Al rods with 135° and 90° tips. Evi-dently WC penetrates deeper than Al, in accord with theirdensities. The three Al rods all penetrate to approximatelythe same depth, in spite of their different tip shapes. Further-more, all data are adequately described by a 1/3 power law,d;H1/3, just as for spherical projectiles. Therefore, the pro-jectile shape seems to play no crucial role and the prior scal-ing law is not special to spheres.The results in Fig. 2 severely constrain the generalizationof Eq. ~2! to aspherical projectiles. A natural hypothesis isthat the crucial parameter is the mass per unit cross sectionalarea, (2/3)rpDp for spheres and rpL for cylinders. For cyl-inders, equal variation of density and length should lead toequal variation of penetration depth. Thus, we must haved50.14F S rpLrgm2D 3/2Dp1/2HG 1/3 ~3!for consistency with Eq. ~2!.To test this expectation, we display depth data vs scaleddrop height, @(rpL)/(rgm2)#3/2Dp1/2H , in Figs. 3~a!–3~d!.This allows us to plot Eq. ~3! as a single curve, onto whichwe look for all data to collapse. Each subplot is for a differ-ent rod diameter; symbol types distinguish rods according totheir length, density, and tip shape. Evidently, the degree ofcollapse is better for deeper penetrations and is worse forshallower penetrations. In Fig. 3~a!, for the 0.25 in. diam-eters, the quality of collapse is comparable to that foundearlier for spheres. Comparison of the four subplots showsthat the onset of collapse appears correlated with rod diam-eter. In particular, collapse by Eq. ~3! seems to be attainedwhen the penetration is larger than the rod diameter. Forshallower penetration, Eq. ~3! fails and the actual depth de-pends on tip shape. In this regime, sharper tips cause deeperpenetration. The tendency of the data to deviate from d;H1/3 for small H, concomitant with the onset of shape de-pendence, can also be seen in close reinspection of Fig. 2.In conclusion, the crater depth vs drop height scaling ob-served previously for spheres is robust and can be simplygeneralized to projectiles of different shapes. The d;H1/3power law is not an accident of projectile shape. The onlyaccident is that it holds for spheres even when the penetra-tion is not greater than the sphere diameter. The indepen-dence of deep penetrations on projectile shape, as observedhere, should heighten the importance of finding a full theo-retical understanding. It also offers a clue: The dissipation ofthe projectile’s energy in the granular medium must occurthroughout a volume that is much larger than the crater. Pre-sumably, this involves a cooperative effect of many grains,such as long force chains successively created then brokenby the projectile impact. The finite size of the granularsample should then play a role, especially for deeper penetra-tions, and this could be tested experimentally. So far, wehave only ensured that the sample is sufficiently deep as tonot affect the penetration depths.To bring this manuscript full circle, we return to a walk onthe beach. In particular, we generalize the depth scaling lawto projectiles with noncircular cross section and we predictFIG. 2. Crater depth vs drop distance for four different cylindri-cal projectiles, all with diameter Dp50.25 in. and length L52 in.The solid lines represent fits to a 1/3 power law, d;H1/3.FIG. 3. Crater depth vs scaled total drop distance. Data are forcylindrical objects, divided into graphs by cylinder diameter, aslabeled. Values of L varied from 0.5 in., indicated by the smallestsymbols, to 1.0 in., to 2.0 in., to 4 in. by larger symbols. Tungstencarbide is denoted by 3 , flat-ended aluminum by h , 135°-tip alu-minum by s , 90°-tip aluminum by , , 45°-tip aluminum by Land flat-ended wood by h . The line in each subplot is Eq. ~3!.RAPID COMMUNICATIONSK. A. NEWHALL AND D. J. DURIAN PHYSICAL REVIEW E 68, 060301~R! ~2003!060301-2the depth of footprints in dry noncohesive sand @rg51.59 g/cc, m5tan(38°) @9##. Making three substitutions,rpL with the mass per cross sectional area m/A , Dp with2AA/p , and H with v2/(2g), we arrive atd50.11S m/A3/2rgm2 D 1/2S Av2g D 1/3. ~4!The area factors have been distributed to make more appar-ent that this expresssion is dimensionally correct; at constantmass, the depth decreases with area as d;A25/12. For impactby a foot, we take A58330 cm2, m570,000 g, and v5100 cm/s ~corresponding to a drop height of 5 cm!. Forthese numbers Eq. ~4! predicts a footprint depth of 6.7 cm, inreasonable accord with experience.We thank Franc¸oise Queval for coordinating the ‘‘Re-search Experience for Undergraduates’’ program at UCLA;we thank John de Bruyn and Lev Tsimring for useful discus-sions. This material is based upon work supported by theNational Science Foundation under Grant Nos. DMR-0305106 and PHY-0243625.@1# H.M. Jaeger, S.R. Nagel, and R.P. Behringer, Rev. Mod. Phys.68, 1259 ~1996!.@2# J. Duran, Sands, Powders, and Grains: An Introduction to thePhysics of Granular Materials ~Springer, New York, 2000!.@3# Impact and Explosion Cratering, edited by D.J. Roddy, R.O.Pepin, and R.B. Merril ~Pergamon Press, New York, 1978!.@4# H.J. Melosh, Impact Cratering: A Geologic Process ~OxfordUniversity Press, New York, 1989!.@5# K. Holsapple, Annu. Rev. Earth Planet Sci. 21, 333 ~1993!.@6# A.M. Walsh, K.E. Holloway, P. Habdas, and J.R. de Bruyn,Phys. Rev. Lett. 91, 104301 ~2003!.@7# S. Thoroddsen and A. Shen, Phys. Fluids 13, 4 ~2001!.@8# R. Mikkelsen, M. Versluis, E. Koene, G.-W. Bruggert, D. vander Meer, L. van der Weele, and D. Lohse, Phys. Fluids 14,S14 ~2002!.@9# J.S. Uehara, M.A. Ambroso, R.P. Ojha, and D.J. Durian, Phys.Rev. Lett. 90, 194301 ~2003!.RAPID COMMUNICATIONSPROJECTILE-SHAPE DEPENDENCE OF IMPACT . . . PHYSICAL REVIEW E 68, 060301~R! ~2003!060301-3

Projectile-Shape Dependence of Impact Craters in Loose Granular Media

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Projectile-Shape Dependence of Impact Craters in Loose Granular Media

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