D. D. RadfordG. J. McShaneV. S. DeshpandeN. A. Fleck1e-mail: naf1@eng.cam.ac.ukDepartment of Engineering,University of Cambridge,Trumpington Street,Cambridge, CB2 1PZ,United KingdomDynamic Compressive Responseof Stainless-Steel SquareHoneycombsThe dynamic out-of-plane compressive response of stainless-steel square honeycombs hasbeeninvestigatedfor impact velocities rangingfromquasi-staticvalues to300 ms1.Square-honeycomb specimens of relative density 0.10 were manufactured using a slottingtechnique, andthestressesonthefrontandbackfacesofthedynamicallycompressedsquare honeycombs were measured using a direct impact Kolsky bar. Three-dimensionalnite element simulations of the experiments were performed to model the response andtohelpinterprettheexperimentalresults.Thestudyhasidentiedthreedistinctfactorsgoverningthedynamicresponseof thesquarehoneycombs: material ratesensitivity,inertial stabilization of the webs against buckling, and plastic wave propagation. Mate-rial rate sensitivity and inertial stabilization of the webs against buckling cause the frontand back face stresses to increase by about a factor of two over their quasi-static valuewhen the impact speed is increased from 0 to 50 ms1. At higher impact velocities, plasticwave effects cause the front face stress to increase linearly with velocity whereas the backfacestressisalmost independent of velocity. Theniteelement predictionsareinrea-sonable agreement with the measurements. DOI: 10.1115/1.2424717Keywords: honeycombcores,impacttesting,dynamicloads,materialratedependence,dynamic buckling1 IntroductionSeveralrecentinvestigationshaverevealedthatmetallicsand-wichpanelshaveanadvantageover monolithicplatesof equalmass in blast resistant structural applications 14. The dynamicperformance of sandwich panels is strongly dependent upon theircore topology and the square-honeycomb core is a promising can-didate1,2. Thepresent studyisacombinedexperimental andnumerical investigation of the dynamic out-of-plane compressiveresponse of square honeycombs in a sandwich conguration.The out-of-plane compression of aluminium hexagonal honey-combsunderlowspeedimpact isfairlywell understood. Intheautomotive industry, the focus of attention has been on the energyabsorption capacity of the hexagonal honeycomb of relative den-sity 0.01 0.04 subject to impact speeds o below 30 ms1. Atthesespeeds, thedynamicstrengthenhancement of thehoney-combs is primarily due to inertial stabilization of the honeycombwebsagainstelasticbuckling:thehoneycombsareinaxialequi-libriumand, consequently, theforcesontheimpactedanddistalends are approximately equal; see for example Zhao and Gary 5and Wu and Jiang 6. There exists little experimental data on thedynamic compression of honeycombs at high speeds o100 ms1 apart from the investigation by Harrigan et al. 7 onthe dynamic crushingof aluminiumhexagonal honeycombs ofrelativedensity 1%. Theymeasuredthestressesontheim-pactedendof thesespecimensandconcludedthat plasticwaveeffects lead to an elevation in the dynamic plateau strength.An optimization study by Xue and Hutchinson 2 has revealedthat squarehoneycombswithrelativelydensitiesofabout 10%,and made from stainless steel which displays a strong strain hard-eningresponse arepromisingfor applicationsinblast resistantsandwich plates. The honeycombs of interest in such blast appli-cations are different from those considered by Harrigan et al. 7:theyhavehigherrelativedensitiesandaremadefromamaterialwithahighstrainhardeningcapacity. XueandHutchinson8haverecentlyreportedniteelementsimulationsofthedynamicresponseofsuchstainlesssteelsquarehoneycombssubjectedtocompressivevelocitiesintherange20200 ms1. Theirsimula-tions highlighted three factors contributing to the dynamicstrengthenhancement of thestainlesssteel squarehoneycombs:materialratesensitivity,inertialstabilizationofthewebsagainstbuckling, andplasticwavepropagation. Fortheloadingratesofinterest in blast applications, they argued that plastic wave propa-gation and plastic buckling occur over comparable timescales, andsubstantialaxialplasticstrainingcanoccurpriortotheonsetofbuckling.The current study is an experimental validation of the ndingsof XueandHutchinson8, andhas thefollowingscope. Thedynamic out-of-plane compressive response of square-honeycomblatticematerialisinvestigatedforappliedcompressivevelocitiesranging from quasi-static values to 300 ms1. The stresses on theimpacted and distal ends of =0.10 stainless-steel square honey-combs are measured using a direct impact Kolsky bar, and high-speed photography is employed to observe the dynamic deforma-tion modes. The effect of specimen height is explored byperformingtestsonspecimensof height H=6 mmand30 mm.The experimental measurements are compared with three-dimensional nite element simulations to gauge the delity of thesimulations and to help interpret the experimental ndings.2 Experimental Investigation2.1 Specimen Conguration and Manufacture. Squarehoneycombshavebeenmanufacturedfrom AISItype304stain-less steel sheets of thickness b=0.30 mm using the technique de-veloped by Ct et al. 9. The sheets were cropped into rectanglesof height H equal to 6 mm and 30 mm, and length 21 mm. Cross-slots Fig. 1 of width b=0.31 mm, and spacing L=6 mm, werecut by electro-discharge machining EDM and the square honey-comb was assembled as sketched in Fig. 1. A clearance of 10mbetween sheet and slot allowed for easy assembly while providing1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in theJOURNALOF APPLIED MECHANICS. Manuscript received January 20, 2006; nal manu-script received May 9, 2006. Review conducted by Robert M. McMeeking.658 / Vol. 74, JULY 2007 Copyright 2007 by ASME Transactions of the ASMEDownloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termsasufcientlytight t toassurestability. BrazingwasconductedwithNi-Cr 25-P10wt % at a temperature of 1120Cinanatmosphereof dryargonat 0.030.1 mbar, andthebrazewasapplied uniformly over the sheets. Capillarity draws the braze intothejoints, andresultsinanexcellent bond. All specimenscom-prised33cellsandhaddimensions21 mm21 mmH. Therelativedensityofthesquare-honeycombistorstorderinb/ Lgiven by = 2bL1giving a relative density of 10%. This was conrmed by weighingthe specimens.2.2 Propertiesof theConstituent Materials. Theuniaxialtensileresponseofthe AISI304stainlesssteelusedtomanufac-ture the square honeycomb specimens was measured at a nominalstrainrateof103s1,andthetruetensilestressversuslogarith-mic strain curve is plotted in Fig. 2a. This material was tested intheas-brazedconditiontomatchthat of theas-manufacturedspecimens, andthemeasured0.2%offset yieldstrengthYandultimatetensilestrengthUTSwere300 MPaand700 MPa, re-spectively. Post-yield, the stainless steel exhibits a linear harden-ing response with a tangent hardening modulus Et1.4 GPa.Stout and Follansbee 10 have investigated the strain-rate sen-sitivity of the AISI 304 stainless steel for strain rates in the range104s1 104s1. Their data are replotted in Fig. 2b, wherethedynamicstrengthenhancement ratioRisplottedagainst theplastic strain rate pfor 103s1p104s1. Here, R is denedas the ratio of the stress dp=0.1 at an applied strain rate ptothestress 0p=0.1 at thequasi-staticrate p=103s1. Themeasured stress versus strain histories presented by Stout and Fol-lansbee 10 indicate that R is reasonably independent of the levelofplasticstrainpat whichRiscalculated. Thus, thedynamicstrengthd versus plastic strainphistory can be estimated fromthe relationdp = Rp0p 2where Rp is given in Fig. 2b. In the nite element simulationsoftheexperimentspresentedbelow,weemploythisprescriptionfor the strain-rate sensitivity of the 304 stainless steel, with 0pgiven by the measured quasi-static =103s1 stress versusstrain history Fig. 2a. As an example, the estimated true tensilestress versus logarithmic strain histories of the AISI 304 stainlesssteelatthreeselectedadditionalvaluesofappliedstrainrateareincluded in Fig. 2a.2.3 Quasi-static Compressive Response of the SquareHoneycombs. The quasi-static compressive response of thesquare honeycombs was measured in a screw-driven test machineat an applied nominal strain rate of 103s1. A laser extensometerwas employed to measure the average nominal compressive strain while the nominal stress was inferred from the measurementsfrom the load cell of the test machine.Themeasuredout-of-planequasi-staticcompressiveresponsesof thesquare-honeycombspecimens areplottedinFig. 3. Thespecimensdisplayapeakstrengthfollowedbysofteningand,-nally, rapid hardening upon densication at a strain D0.60.7. Notethat theH=30 mmspecimenhasalowerpeakstrengthanddisplaysmoreabruptsoftening.Ctetal. 9havealreadynotedthat the quasi-static collapse mode involves tor-Fig.1 SketchofthemanufacturingtechniqueforthesquarehoneycombFig. 2 a Themeasuredquasi-statictensilestressversusstrainresponseof theAISI 304stainlesssteel andtheesti-matedhighstrainrateresponseatthreeadditional valuesoftheappliedstrainrateusingthedataofStoutandFollansbee10. b The dynamic strength enhancement ratio R as a func-tion of plastic strain rate pfor the AISI 304 stainless steel at aplastic strainp=0.1 10.Fig. 3 Quasi-static compressive stress versus strain responseof the =0.10 square honeycombs, of cell height H=6 mm andH=30 mmJournal of Applied Mechanics JULY 2007, Vol. 74 / 659Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termssional plastic buckling of the cells, with the peak static strength saccurately predicted by a plastic bifurcation analysis 9.2.4 Dynamic Test Protocol. The dynamic out-of-plane com-pressive response of the square honeycombs was measured from aseries of direct impact tests in which the forces on the faces of thehoneycombweremeasuredviaastrain-gaugedKolskybar 11.Twotypesoftestswereconductedtomeasuretheforcesontheimpactedanddistal ends of the specimens, referredtosubse-quently as the front and back faces, respectively.In the front face conguration Fig. 4a, the test specimen isattachedtooneendofthestrikerbarsometimesknownasthebacking mass, 7 and the combined striker bar and specimen arered from a gas gun so that the specimen impacts the Kolsky barnormally and centrally. In the back face conguration Fig. 4b,the specimen is placed centrally on the stationary Kolsky bar andthe striker bar is red from the gas gun and impacts the specimen.These two independent tests allow for a measurement of the tran-sient force on both the impacted and distal faces of the specimen.The kinetic energy of the projected striker governs the level ofcompressionattainedandtheimposedtransient velocityat oneendof thespecimen. Wewishedtocompressthespecimensatapproximatelyconstantvelocityandchosethestrikermassesac-cordingly. In the experiments conducted at lowvelocity o50 ms1 and at intermediate velocity 50 ms1o200 ms1strikers of massM=2.4 kg and 0.5 kg, respectively,wereemployed.AstrikerofmassM=0.090 kgsufcedforthehighvelocityo200 ms1experiments. Themeasurementsandnite element simulations presented below show that these strikermasses are sufcient to provide almost constant velocity compres-sionof thesquare-honeycombspecimensfor nominal compres-sive strains of up to 40%.Thestrikerwasgiventherequiredvelocitybyringitfromagasgunofbarrellength4.5 manddiameter28.5 mm.Nosabotwasemployedasthecylindrical strikerhadadiameterequal to28.0 mm. Theburstingof copper shimdiaphragms formedthebreech mechanism of the gun. The impact experiments were per-formed at velocities ranging fromapproximately 10 ms1to300 ms1. The velocity of the projectile was measured at the exitof the barrel using laser-velocity gates and the impacted end of theKolskybar wasplaced100 mmfromtheopenendof thegunbarrel.The setup of the Kolsky pressure bar 11 is standard. A circularcylindricalbaroflength2.2 manddiameter28.5 mm wasmadefromthemaragingsteel M-300yieldstrength1900 MPa. Thepressure history on the impacted end of the bar was measured bydiametricallyoppositeaxial straingaugesplacedapproximately10diametersfromtheimpact endof thebar. Theelasticstrainhistoriesinthebarsweremonitoredusingthetwo120TMLfoil gaugesof gaugelength1 mminahalf-Wheatstonebridgeconguration. Astrain bridge amplier of cutoff frequency500 kHz was used to provide the bridge input voltage and a digitalstorageoscilloscopewas usedtorecordtheoutput signal. Thebridge systemwas calibrated dynamically over the range ofstrains measured during the experiments and was accurate towithin 3%. The longitudinal elastic wave speed was measured at4860 ms1, giving a time window of 800s before elastic reec-tions from the distal end of the bar complicated the measurementof stress.Theresponsetimeandaccuracyof themeasurement systemweregaugedfromaseriesofcalibrationtests. Wereportthere-sults of one such representative test as follows. A maraging steelstriker bar of diameter 28.5 mm and length 0.5 m was red at theKolsky bar at a velocityo=9.0 ms1. The stress versus time re-sponse measured by the strain gauges on the Kolsky bar is plottedinFig. 5.Withtimet =0correspondingtotheinstantofimpact,thestresspulsearrivesatthegaugelocationatt =66s.Elasticwavetheorypredicts that theaxial stress inthebar is co/ 2=175 MPa, where and c are the density and longitudinal elasticwave speed of steel, respectively. The measured peak value of thestress is within 1% of this prediction. However, the measurementsystemhasaniteresponsetime, withthestressrisingtothispeak value in approximately 10s see the inset in Fig. 5. Thisrisetimeplacesanoperational limit onmeasuringthedynamicresponse of the square honeycombs. It becomes signicant at thehigher velocities because signicant compression of the specimenis achievedwithinthe rst 10s. The measuredstress inthecalibration test drops back to zero at t =270s; this is the time forpropagationof anaxial stresswavedownthebar, followedbyreection of the elastic wave from the distal end of the striker barback to the strain gauge.3 Experimental Results for the Dynamic Compressionof Square HoneycombsWerst present thedynamiccompressionresponseof theH=30 mmsquare-honeycombspecimens andthencontrast thesemeasurements with those for the H=6 mm specimens.3.1 The H=30 mmSquare Honeycombs. The measuredfront and back face axial stress versus normalized time tot / HhistoriesfortheH=30 mmspecimensarepresentedinFigs. 6a, 7a, and 8a for impact velocities o=20 ms1,50 ms1, and 240 ms1, respectively. The time t is measured fromthe instant of impact and thus the normalized time tis a measureof the nominal compressive strain of the square-honeycomb speci-Fig. 4 Sketches of the direct impact Kolsky bar setup for mea-suring the stress versus time histories in a front face and bback face congurations. All dimensions are in mm.Fig. 5 Stress versus time history measured in the Kolsky barduring a calibration test. A 0.5 m long steel striker is red at theKolsky bar ato=9.0 ms1.660 / Vol. 74, JULY 2007 Transactions of the ASMEDownloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termsmens, assumingcompressionat auniformvelocityoover thedeformation history. In these gures, the front and back facestressesaredenedfromthemeasuredfront faceforceFfandback face force Fb asf FfAo4aandb FbAo4brespectively, where Ao=2121 mm2is the cross-sectional area ofthesquare-honeycombspecimens. High-speedphotographicse-quences of thedeformationof thespecimens inthefront facecongurationaregiveninFigs. 6b, 7b, and8b. Theinter-frame times in Figs. 6b and 7b are 100s while the sequencesinFig.8bweretakenat40sintervals. Theexposuretimeofeachphotographis20%oftheinterframetimeinallcases.Thedust clouds inthe high-speedphotographs are associatedwithtearingof thesquarehoneycombsalongthebrazedjoints. Thiswas conrmed by post-test visual inspections of the dynamicallytestedspecimens. Tearingofthejointswasalsoobservedinthequasi-static tests. The dynamic measurements show two qualita-tivelydistinct behaviors: 1 theresponsefor o=20 ms1and50 ms1, and 2 the response foro=240 ms1.1. o=20 ms1and 50 ms1. The measured front and back facestresses equalize within t 10s or ot / H=0.010.02.Recall that the response time of the measurement system is10s. Thus, the earlydifferences betweenthe front andback face stresses are likely to be associated with the mea-surement system and we conclude that the specimens are inaxial equilibrium over almost the entire deformation history.Similar to the quasi-static case, the square-honeycombspecimenshaveadistinct stresspeak. Thesepeakstressesincreasewithimpact velocityandat o=50 ms1theyareapproximatelytwicethequasi-staticvalue. Material strainratesensitivityalonecannotaccountfortheincreaseinthepeakstressanddynamicstabilizationof thewebsagainstbuckling is expected to play an important role.2. o=240 ms1. The measuredfront face stress is approxi-mately constant over the deformation history to withinringing of the measurements up to the specimen densi-cation strain ofot / H0.9. Moreover, the front face stressexceedsthebackfacestressover thedeformationhistory.Thisindicatesthatthespecimenisnotinequilibrium,withwavepropagationeffectsplayingadominant role. Thisissubstantiated by the high-speed photographs Fig. 8b:shortening of the specimen is concentrated near the im-pacted end, with the distal end of the specimen undergoingonly small plastic deformations.The measured peak front face stress fpand back face stress bpare normalized by the peak quasi-static value s and are plotted inFig. 9asafunctionoftheimpactvelocityo. Overtherange0o50 ms1, the front and back face stresses remain equal andattain double their quasi-static values. Foro50 ms1, the backface stress remains approximatelyconstant at its value for o=50 ms1while the front face stress increases approximately lin-early with velocity. These observations suggest that the followingthree mechanisms provide the dynamic strength enhancements inthese experiments.1. Material strainrate sensitivity. The dynamic strengthen-Fig. 6 a Measured front and back face stress versus norma-lised time histories in the H=30 mm honeycomb specimens im-pactedat o=20 ms1;and bthecorrespondinghighspeedphotographicsequenceof thedeformationinthefront facecongurationat aninterframetimeof 100 s. Theniteele-ment predictions constant velocity boundary condition are in-cluded in a.Fig. 7 a Measured front and back face stress versus norma-lised time histories in the H=30 mm honeycomb specimens im-pactedat o=50 ms1;and bthecorrespondinghighspeedphotographicsequenceof thedeformationinthefront facecongurationat aninterframetimeof 100 s. Theniteele-ment predictions constant velocity boundary condition are in-cluded in a.Journal of Applied Mechanics JULY 2007, Vol. 74 / 661Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termshancement of thesquarehoneycombs duetothematerialstrain rate is estimated from the measurements of Stout andFollansbee10 byassuminguniformcompressionof thesquare honeycombs at a rate p=o/ H. The strengthen-hancement ratioRp isplottedversusimpact velocityinFig. 9. Comparisons withthemeasureddynamicstrengthenhancement ratiosbp/ sandfp/ ssuggest that the stressenhancements foro20 ms1are largely due to the mate-rial strainratesensitivity. Material ratesensitivitycannot,however, account for the strength enhancements at thehigher velocities.2. Dynamicbuckling. As discussedinSection2.3, thepeakquasi-static strength of the square honeycombs is set by tor-sional plastic buckling of the cells of the square honeycomb.Under dynamicloading, inertial stabilizationresults inanenhancedbucklingstrengthduetotheactivationofhigherorder bucklingmodes. AbrahmsonandGoodier 12 andCalladineandEnglish13 haveelaboratedonthisinthecasefor rods whichhaveattainedaxial equilibrium. Thishigher order buckling is clearly seen in the high-speed pho-tographs inFig. 8b. Inthe velocityrange 20 ms1o50 ms1, the front andbackface stresses are approxi-matelyequal but material ratesensitivityalonecannot ac-count for the dynamic strength enhancement. The additionalstrengthening is attributed to the enhanced dynamic bucklingstrength of the square honeycombs due to inertial stabiliza-tion.3. Plastic wave propagation. At high impact velocities, a plasticwave moves alongthe prismatic axis of the honeycomb.Prior totheonset of bucklingof thewebs of thesquarehoneycombs, one-dimensional elasticplastic wave theorycan be used to estimate dynamic stress on the front face ofthehoneycombasthewavepropagatesthroughthethick-ness of the honeycomb. In the small strain, rate-independentlimit, the front face stress is given byf =FfAo= Y + ccplo Ysce Y + ccplo5whereceandcplaretheelasticandplasticwavespeeds,respectively, of the honeycomb parent material of yieldstrengthY. Thedensityof thehoneycombis c sinterms of the density s of the parent material. During propa-gationoftheplasticwavetothebackface, thebackfacestress has no inertial contribution and is given by b=Y . Itremainstospecifyanappropriatevaluefor Yinordertodeterminefandb.HerewearbitrarilytakeYtobetheyield strength of the solid material at a strain rate p=104s1the nominal strain rate for a honeycomb of heightH=30 mmcompressedat o=300 ms1 inorder toillus-trate the predictions of the one-dimensional wave model,and plot Eq. 5 with cplEt/ s410 ms1in Fig. 9. Thegoodagreementofthefrontfacestresspredictionwiththeexperimental measurementsfor o50 ms1suggeststhatone-dimensional plastic wave propagation sets the peakfront face stresses at these higher velocities.3.2 The Effect of Specimen Height. The measured front andbackfacestress versus timehistories of theH=30 mmandH=6 mmsquare-honeycombspecimensforanimpact velocityo=100 ms1are plotted in Figs. 10a and 10b, respectively. Plas-tic wave effects play a signicant role in the H=30 mm specimenwiththefront andbackfaceforces equalizingonlyfor ot / H0.1. Recall that the plastic wave speed in 304 stainless steel iscpl410 ms1. Thus, we expect the plastic wave to reach the rearfaceatthelongertimeot / H0.25.Theniteelementcalcula-tions reported subsequently show that material rate sensitivity re-sults in the specimen attaining axial equilibrium sooner than esti-matedfroma rate independent plastic wave theorydue toanincreased plastic wave speed resulting from smearing out of theplasticshock. Thewidthoftheshockfrontinamaterialwithalinearviscousratesensitivityscalesas/ scplwheresisthedensity of the material andthe viscosity; see Kaliski and Wlo-darczyk 14. In contrast, the front and back face stresses in theH=6 mm specimen are approximately equal over the entire defor-mationhistory. Thisispartlyduetothesmearingoftheplasticshockwaveandpartlyaresultofthe10sresponsetimeofthemeasurement system.The measured peak front and back face stresses normalized bythe correspondingquasi-static peakstrength for the H=6 mmFig. 8 a Measured front and back face stress versus norma-lised time histories in the H=30 mm honeycomb specimens im-pacted ato=240 ms1; and bthe corresponding high speedphotographicsequenceof thedeformationinthefront faceconguration at an inter-frame time of 40 s. The nite elementpredictionsconstant velocity boundary condition are in-cluded in a.Fig. 9 Measuredpeak front face stress fpandback facestressbpversusimpact velocityointheH=30 mmhoney-comb specimens. The measured dynamic stresses are normal-ized by peak quasi-static stress s from Fig. 3. The predictionsof the dynamic stresses based on material strain-rate sensitiv-ity and one-dimensional plastic wave propagation are included.662 / Vol. 74, JULY 2007 Transactions of the ASMEDownloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termssquare honeycombare plottedinFig. 11as a functionof theimpact velocityo. IncludedinFig. 11areestimatesofthedy-namicfront andbackfacestressesbasedonthematerial strain-ratesensitivityandthefront facestresses duetoplasticwavepropagation effects, as discussed in Sec. 3.1. Again, Y in Eq. 5is interpreted as the yield strength of the solid material at a strainratep=104s1. The basic mechanisms of dynamic strength en-hancements in these specimens are similar to those in the H=30 mmspecimens.However,intheseshorterspecimens,mate-rial rate sensitivity smears the plastic shock over the entire heightH=6 mm of the specimenandthus the front andbackfacestressesareapproximatelyequalovertherangeofvelocitiesin-vestigated here. Note that the front face stresses for the H=6 mmandH=30 mmspecimens areapproximatelyequal forvelocities o50 ms1Figs. 9 and 11. This conrms that veloc-ity rather than applied nominal strain rate is the governing param-eter at these higher rates of compression.4 Finite Element InvestigationA limited nite element FE investigation of the dynamic com-pression of the square-honeycomb specimens has been performed.The aims of this investigation are:1. Todeterminetheaccuracyof three-dimensional niteele-ment calculations in predicting the dynamic compressive re-sponse of the square honeycombs;2. Tousetheniteelementcalculationstoinvestigatetheef-fect of striker decelerationonthe measureddynamic re-sponse; and3. Todemonstratetheeffectoftheresponsetimeofthemea-surement systemonthemeasuredearlytimedynamicre-sponse of the square honeycombs.4.1 Three-Dimensional FESimulations of the DynamicCompressionof theSquareHoneycombs. All theFEsimula-tions were conducted using the explicit version of the commercialniteelement packageABAQUS. Thegeometryof thehoney-comb specimens was identical to that employed in the experimen-talinvestigationandthehoneycombsweremodeledusinglinearshell elements S4R in the ABAQUS notation with b as the thick-nessof thehoneycombwalls. Ameshsensitivitystudyshowedthat an element size of b/ 2 sufced to give a converged solution.All computations reportedhere employedsucha mesh. Rigid,massless plates discretized using four-noded rigid elements,R3D4inthe ABAQUSnotationweretiedtoboththefrontandbackfaces of thespecimens andthegeneral contact optioninABAQUS was employed to provide hard, frictionless contact be-tween all surfaces in the model. The tie constraint is appropriate ifnegligible sliding occurs at the interface between the honeycombandstrikermassandKolskybar.High-speedphotographsoftheexperiments suggest that this is anappropriate assumptionformodeling purposes.Mostofthecomputationswereconductedbycompressingthespecimen at a constant applied velocity as follows. The front rigidplate was constrained to move only in the axial direction i.e., inthe direction of the height H of the specimen while the back facewasfullyclamped. A constant velocityointheaxial directionwasimposedonthefrontrigidplateandtheaxialforcesonthefront andbackfaces weremonitoredas afunctionof timetodetermine the front and back face stress versus time history.In a few simulations, the experimentally applied loadings in thefront andbackfacecongurations werealsomimicked. Inthefront face conguration, a point mass Mwas attached to the backface. Thespecimen, backfaceandmassMwerethengivenaninitialvelocityointheaxialdirectionwiththefrontrigidplatefully clamped and the back face and point mass restricted to moveonlyintheaxialdirection.Similarly,inthebackfacecongura-tion, thebackfacewasfullyclampedandthepoint massnowattachedtothefrontfacewasgivenaninitialvelocityoalongwiththerigidfront plate. Thepoint massandfront platewereconstrained to move only in the axial direction.4.2 Material Properties. It was assumed that the square-honeycomb specimens comprised AISI 304 stainless-steel sheets.Unless otherwise specied, the stainless steel was modeledasJ2-owtheoryratedependent solidofdensityf=8060 kg m3,Youngs modulus E=210 GPa and Poisson ratio =0.3. Theuniaxialtensiletruestressversusequivalentplasticstraincurvesat plastic strain rates 103s1p104s1were tabulated inABAQUSusingtheprescriptiondescribedinSec. 2.2andem-Fig. 10 The measured front and back face stress versus nor-malisedtimehistoriesofthe aH=30 mm;and bH=6 mmsquare-honeycomb specimens for an impact velocity o=100 ms1. Theniteelement predictions constant velocityboundary condition are included in the gures.Fig. 11 Measured dynamic peak front fp and back bp facestressesintheH=6 mmhoneycombspecimensasafunctionof the impact velocityo. The measured dynamic stresses arenormalized by peak quasi-static stress s from Fig. 3. The pre-dictions of the dynamic stresses based on material strain-ratesensitivityandone-dimensional plasticwavepropagationareincluded.Journal of Applied Mechanics JULY 2007, Vol. 74 / 663Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termsploying the data of Fig. 2.A comparison between the quasi-static FE predictions and mea-surements is included in Fig. 3; see Zok et al. 15 for details ofsuchcalculationsandreasonsforthediscrepanciesbetweenthemeasurements and the FE predictions.4.3 Effect of Initial Imperfections.Theeffectofinitialim-perfections on the nite element predictions of the dynamic com-pressive response was rst investigated. Dynamic compressiontypically results in the activation of higher order buckling modesas seen in the high-speed photographs presented above. The buck-lingwavelengthsareastrongfunctionofthematerialpropertiesand compression velocity as discussed by Abrahamson andGoodier 12 for the dynamic buckling of rods. Consequently, weinvestigated the effect of the magnitude and mode of imperfectionon the dynamic compressive response of the square honeycombs.Fourmodesofinitialimperfectionswereconsidered.ModesIandII arethetwolowest staticeigenmodes withtheModeIIwavelength half that of Mode I. In order to investigate the effectof a distribution of imperfection wavelengths we also consideredmodes which are the sum of the rst 20 and 40 static eigenmodesequal maximum amplitude for each mode. These will be subse-quentlyreferredtoas Modes III andIV, respectively. All fourmodes are shown in Fig. 12 for the H=30 mm honeycomb.TheFEpredictionsof thestressversustimehistoriesof theH=30 mmhoneycomb for applied velocities o=20 ms1and240 ms1areplottedinFigs. 13aand13b, respectively. Thefront and back face stresses are almost equal for o=20 ms1andthus for the sake of clarity only the front face stresses are showninFig.13a.Ineachcase,resultsareplottedforthefourinitialimperfectionmodeswithamaximumimperfectionamplitudeof0.05b. We observe that the choice of the initial imperfection modehas a negligible effect on the predicted stress versus time histories.Infact, fortherangeofcompressivevelocitiesconsideredhere,theFEcalculationspredict aresponsethat isinsensitivetothemode of the imperfections for imperfection amplitudes in therange 0.02b0.1b. This holds for both the H=30 mmand H=6 mmhoneycombs.All thecalculationsreportedsubsequentlyemploytheModeIimperfectionwithamaximumimperfectionamplitude of 0.05b.4.4 Sensitivity of Response to Choice of LoadingCondition. The striker mass Min the experiments was chosen togive an approximately uniform compression velocity over a nomi-nal compressive strain of about 40%. Here we employ FE calcu-lations to verify this and also to compare the FE predictions of thestriker velocityover the deformationhistoryof the specimenswith those inferred from measurements.FEpredictions of thefront andbackfacestresses of theH=30 mmsquarehoneycombwitho=20 ms1areshowninFig.14for constant velocitycompressionandfor projectileimpactwithagiveninitial velocity. Intheimpact simulations, apointmassM=2.4 kg was attached to the square honeycomb to repre-sent thestriker employedintheexperiments. Bothsets of FEpredictions are nearly identical for ot / H0.2 with small discrep-ancies between the two sets of simulations forot / H0.2.We proceed to estimate the velocity reduction of the striker inthe experiments and to compare these estimates with the FE pre-Fig. 12 The four modes of initial imperfections introduced intothe FE model of the H=30 mm square honeycomb: a Mode I;b Mode II; c Mode III; and d Mode IV. A section through themidplane of the honeycomb is shown in each case.Fig. 13 FE predictions of the front and back face stress versustimehistoriesfortheH=30 mmcompressedatavelocity ao=20 ms1frontface;and b o=240 ms1frontandbackface.Inbothcases,resultsareshownforthefourmodesofinitial imperfections with an imperfection amplitude 0.05b.Fig. 14 A comparisonbetweentheFEpredictionsforacon-stantappliedvelocityandforimpactboundaryconditions H=30 mm honeycomb specimen witho=20 ms1664 / Vol. 74, JULY 2007 Transactions of the ASMEDownloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termsdictions. Thisreductionismostsevereinthebackfacecongu-rationsincethestrikerisdeceleratedbyaforceassociatedwithdrivingalongtheplasticshockwave. Thestrikervelocityversustime relation in the back face conguration is given bybt = o 1M0tFf dt 6where Ffis the measured force exerted by the square-honeycombspecimens in the corresponding front face conguration. The nor-malized striker velocitybt / o inferred from measurements inthe back face conguration of the H=30 mm specimens are plot-ted in Fig. 15 as a function of the normalized timeot / Hfor theo=20 ms1and 50 ms1cases. These calculations were per-formed with M=2.4 kg to match the striker mass employed in theexperiments.ItisclearfromFig.15that,whilethereisanegli-giblereductioninvelocityfortheo=50 ms1case,thevelocityofthestrikerreducesbyabout20%fortheo=20 ms1caseatot / H0.35. The corresponding FE predictions of the striker ve-locity in the back face conguration are included in Fig. 15 mea-sureddirectlyfromthevelocityofthepointmassintheFEcal-culations. The FE calculations predict slightly less velocityreductionsthanthoseestimatedfromtheexperimental measure-ments. Thisislikelytoberelatedtotheringingintheexperi-mental measurementswhichresultsinanoverestimationof thefront face force. Similar estimates for the other impact velocitiesconrmedthat negligiblestrikervelocityreductionsoccurredinallexperimentswitho50 ms1.TheseexperimentalestimatesandFEpredictionsconrmthat theimpact experimentscanberegardedasconstant velocityexperimentsoverthepractical de-formation regime 0ot / H0.4.4.5 Accuracy of Predictions for the H=30 mmSquareHoneycombs.FEpredictionsofthefrontandbackfacestresseson the H=30 mm specimens are included in Figs. 6a, 7a, and8a for a constant velocity of o=20 ms1, 50 ms1, and240 ms1, respectively. Thesimulationswereterminatedwithinthe FE code when large rotations of the shell elements resulted ina loss of accuracy of the simulations. The FE simulations capturethe measured stresses in the o=20 ms1and 50 ms1experimentstoreasonableaccuracy. TheFEpredictions of thedeformationmodefortheo=50 ms1caseareplottedinFig.16att =55sand 155s. It is noted that the observed deformation Fig. 7bandpredicteddeformationmodes Fig. 16arequitedifferent;asimilar discrepancyindeformationmode was observedfor o=20 ms1althoughthecomparisonisnotshownexplicitly.Thisdiscrepancy is likely to be associated with the tearing of thebrazed joints in the experiments and not accounted for in the FEcalculations. Important differences between the measured and FEpredictions of the compressive response at o=20 ms1and50 ms1are as follows.1. The FE simulations predict a sharp rise of the stresses on thefront and back faces due to the presence of the elastic stress wave.Such a sharp rise is not observed in the experiments. This is likelyto be related to the fact that the response time of the measurementapparatus is about 10s and thus the experimental measurementsareunabletocapturetheinstantaneousriseinstressduetothearrival of the elastic wave.2. At ot / H0.20, the FEsimulations predict a suddenin-crease in the stresses on the back face due to contact between thecell wallsasseeninFig. 16fortheo=50 ms1simulation. Inreality, amuchgentlerincreaseinthemeasuredstressesoccurs.We attribute this discrepancy to tearing of the brazed joints of thesquare honeycombs as evidenced from the dust clouds in the pho-tograph at t =255s in Fig. 8b. This effect is not included in theFEsimulationsandislikelytoresultinanoverpredictionofthemeasured stress.The comparisons betweenthe FEsimulations andmeasure-mentsfortheo=240 ms1case Fig.8bshowlargerdiscrep-Fig. 15 Experimental estimates and FE predictions of the nor-malized striker velocity bt / o as a function of the normalizedtimeot / Hfor thebackfacecongurationwithinitial strikervelocities ofo=20 ms1and 50 ms1Fig. 16 The FE predictions of the deformation mode of the H=30 mm honeycomb specimen, subjected to a constant appliedvelocityo=50 ms1at: at =55 s;and bt =155 s. Thesetimes correspond to the times of the high speed photographsin Fig. 7b. A section through the midplane of the honeycombis shown.Journal of Applied Mechanics JULY 2007, Vol. 74 / 665Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 06/17/2015 Terms of Use: http://asme.org/termsancies. The differences between the measurements and predictionsofthestressesaremainlyduetotheslowresponsetimeofthemeasurement apparatus, with the predictions and measurements inreasonable agreement for ot / H0.2. The oscillations in the mea-sured stress history are due to exural waves induced in the Kol-sky bar by slight misalignment of the impact and are neglected inthe FE simulations. Again, the FE calculations predict a rise in thebackfacestressat ot / H0.4. Thisriseisnot observedintheexperiments due to tearing of the brazed joints of the specimens.4.6 FE Simulations to Elucidate the Role of Material RateSensitivity. The measurements indicate that the plastic wavereachesthedistal endofthespecimenearlierthanpredictedbyrate independent, small strain plasticity theory. For example, withaplasticwavespeedofcpl410 ms1andaninputspeedofo=100 ms1,itispredictedthattheplasticwavewillincreasethestress on the back face to 100 MPa at ot / H0.25. However, themeasurements inFig. 10suggest that the plastic wave arrivesmuch earlier, at ot / H0.1. The FE predictions are in reasonableagreement with the experimental measurements for t 10s, thatis forot / H0.01 and 0.03 for theH=30 mm and 6 mm speci-mens, respectively. This conrms that thehighstress measure-ments over 0.1ot / H0.25 are not an artefact of the measure-ment system but are probably related to an increased plastic wavespeed associated with the rate sensitivity of the 304 stainless steel.Weproceedtochecktheeffect ofmaterial ratesensitivitybyperforming FE simulations with the 304 stainless steel modeled asarateindependent J2owtheorysolidwithauniaxial tensilestress versus plastic strain response given by the p=103s1datain Fig. 2a. A comparison between the rate independent and ratedependent FE predictions constant applied velocity of the frontandbackfacestressesonH=30 mmand6 mmsquarehoney-combs is showninFigs. 17a and17b, respectively, for o=100 ms1. The rate dependent and rate independent FE calcula-tions predict that the plastic wave arrives at the backface atot / H0.15 andot / H0.2, respectively. Thus, the FE calcula-tions demonstrate that material strain rate sensitivity can accountfor theincreasedplasticwavespeed. Thedynamicstrengthen-hancement due to material rate effects are also clearly seen in Fig.17: the rate dependent FE calculations predict that both the frontand back face stresses are about 25% higher than those in the rateindependent case.5 Concluding RemarksSquare-honeycombspecimensof relativedensity =0.10andheightsH=30 mmand6 mmweremanufacturedbyslottingto-gether 304 stainless-steel sheets and then brazing together the as-sembly. Theout-of-planecompressiveresponseof thesespeci-mens was measured for velocities ranging from quasi-static valuestoo=300 ms1. The stresses on both the front and back faces ofthe square honeycombs were measured in the dynamic tests usinga direct impact Kolsky bar.Torsional plasticbucklingof thewebs is thecollapsemodeunder quasi-static loading. Three distinct mechanisms govern thedynamic response of the square honeycombs: i material rate sen-sitivity; ii inertial stabilization of the webs against buckling; andiii plasticwavepropagation. IntheH=30 mmspecimens, ef-fects i and ii are dominant foro50 ms1with the front andbackfacestressesincreasingbyaboutafactoroftwoovertheirquasi-staticvalues. At highervelocities, plasticwaveeffectsbe-comeincreasinglyimportantwiththebackfacestressremainingapproximately constant at its value foro=50 ms1and the frontfacestressincreasingapproximatelylinearlywithvelocity. Thepeak front face stresses in the H=6 mm and H=30 mm cases areapproximately equal at the higher velocities indicating that veloc-ity rather than strain rate governs the response of these specimensunder high rates of compression.Three-dimensional niteelement simulations capturetheex-perimental measurements toreasonableaccuracy. Discrepanciesbetween the measurements and predictions are attributed to: i aresponsetimeof 10sbythemeasurement apparatus; andiitearingof thesquarehoneycombsalongthebrazedjoints. Thistearingleadstoasignicantdropinthetransmittedloadandinthe energy absorbed by the square honeycomb.AcknowledgmentThe authors are grateful toONRfor their nancial supportthrough US-ONR IFO Grant No. N00014-03-1-0283 on The Sci-ence and Design of Blast Resistant Sandwich Structures and to theIsaac Newton Trust, Trinity College Cambridge.References1 Fleck, N. A., and Deshpande, V. S., 2004, The Resistance of Clamped Sand-wich Beams to Shock Loading, J. Appl. Mech., 71, pp. 386401.2 Xue, Z., and Hutchinson, J. W., 2004, AComparative Study of Blast-Resistant Metal Sandwich Plates, Int. J. Impact Eng., 30, pp. 12831305.3 Radford, D. D., Fleck, N. A., and Deshpande, V. S., 2005, The Response ofClamped Sandwich Beams Subjected to Shock Loading, Int. J. Impact Eng.,32, pp. 968987.4 McShane, G. J., Radford, D. D., Deshpande, V. S., andFleck, N. A., 2006,The Response of Clamped Sandwich Plates With Lattice Cores Subjected toShock Loading, Eur. J. Mech. 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