InternationalJournalofImpactEngineering28(2003)161182In-planedynamiccrushingofhoneycombsaniteelementstudyD.Ruana,G.Lua,*,B.Wangb,T.X.YucaSchoolofEngineeringandScience,SwinburneUniversityofTechnology,Hawthorn,Victoria3122,AustraliabDepartmentofMechanicalEngineering,BrunelUniversity,Uxbridge,MiddlesexUB83PH,UKcDepartment of Mechanical Engineering, Hong Kong University of Science & Technology, Clear Water Bay, Hong KongReceived25October2001;receivedinrevisedform24June2002AbstractThe in-plane dynamic behaviour of hexagonal aluminium honeycombs has been studied in this paper bymeansofniteelementsimulationusingABAQUS.Theinuencesofhoneycombcellwallthicknessandimpact velocity on the mode of localised deformation and the plateau stress are investigated. Deformationmodes inthe X1directionchange signicantlywithdifferent values of cell wall thickness andimpactvelocity. Threedeformationmodesareobserved. Theyaresummarisedinamodeclassicationmapandtwoempirical formulae are obtainedfor critical velocities at whichdeformationmode changes. Twodeformationmodes are observedinthe X2direction. The plateaustresses are relatedtothe cell wallthickness by a power law for a given velocity and they are proportional to the square of velocity for a highvelocity. An empirical formula for plateau stress at high impact velocities is derived in terms of the cell wallthicknessandvelocity.r 2002ElsevierScienceLtd.Allrightsreserved.1. IntroductionHoneycombis atype of cellular material withatwo-dimensional arrayof hexagonal cells(Fig. 1a). It is used in a variety of applications: frompolymer and metal honeycombs foradvancedaerospacecomponentstoceramichoneycombsforhigh-temperatureprocessing[1].Muchtheoretical andexperimental workhas beencarriedout onhoneycombs under staticloading [1,2].One particularfeatureof honeycombsis softening withineach celland localisationofdeformation(Fig. 1b).Inrecentyears,theniteelementmethodhasbeenusedtoanalysethestaticcrushingbehaviourofhoneycombs.NumericalsimulationsbyPapkaandKyriakides[3,4]*Correspondingauthor.Fax:+61-3-9214-8264.E-mailaddress:glu@swin.edu.au(G.Lu).0734-743X/02/$ - seefrontmatter r 2002ElsevierScienceLtd.Allrightsreserved.PII:S 0 7 3 4 - 7 4 3 X( 0 2 ) 0 0 0 5 6 - 8provide a good insight into the crushing process of honeycombs. Silva and Gibson [5] studied theeffect of randomly removed individual cell edges on the Youngs modulus and the plateau stress inregular hexagonal honeycombs. GuoandGibson[6] investigatedthebehaviour of intact anddamagedhoneycombsusingABAQUS. Theyanalysedtheeffect of isolateddefectsof varioussizes and that of the separation distance between two defects on the elastic and plastic behaviourof honeycomb. It was impossible to conduct such investigations basedmainly ona micro-mechanicalanalysisofasinglecell.Honeycombsbehavedifferentlyunderdynamicloading, mainlyintheirdeformationmodesand stress levels. Stronge et al. [79] proposed one-dimensional models to account forthe strain-softening within each cell and strain rate effect. Dynamic tests were performed by Baker et al. [10],andWuandJiang[11] byusingagasguntostudytheout-of-planepropertiesof aluminiumhoneycombs. ZhaoandGary[12] studiedthein-planeandout-of-planecrushingbehaviourofaluminiumhoneycombs using the split Hopkinsonpressure bar (SHPB). However, in thesestudies,onlyafter-testspecimenswereobservedandthewholecrushingprocessofthecellsandspecimens was not studied. No theoretical model has been proposed for two-dimensionalspecimens.NomenclatureA aparameter(Eq. (11))B aparameter(Eq. (11))h honeycombcellwallthickness(Fig. 1)L0originallengthofhoneycombsampleintheX1directionl lengthofhoneycombcelledge(Fig. 1)p exponent(Eq. (11))s honeycombcellsize(Fig. 1)t timeX1horizontalloadingdirection(Fig. 2)X2verticalloadingdirection(Fig. 2)Greeklettersd displacementedlocking(ordensication)strainofhoneycombv impactvelocityvc1criticalvelocityatwhichdeformationmodechangesfromXmodetoVmodevc2criticalvelocityatwhichdeformationmodechangesfromVmodetoImoder densityofhoneycombrsdensityofhoneycombcellwallmaterials plateaustressofhoneycombs0staticplateaustressofhoneycombsysyieldstressofhoneycombcellwallmaterialD.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 162Inthispaper,byusingtheniteelementmethod,westudythein-plane(X1andX2direction)deformationofhexagonal aluminiumhoneycombsunderimpactloading. Theinuencesofcellwall thickness and impact velocity on the deformation mode and plateau stress are discussed. Theresultsaresummarisedwithamodeclassicationmapandempirical formulaefortheplateaustress.2. FiniteelementmodelTheFEmodel usedisshowninFig. 2. It has16cellsintheX1directionand15intheX2direction. The cell size is s 4:7 mm and l 2:7 mm (corresponding to b 1201) for all cases andthe cell wall thickness h varies from 0.08 to 0.5 mm. ABAQUS/EXPLICIT [14] was employed forthedynamicanalysis. Thecell wall material wasassumedtobeelastic, perfectlyplasticwithaYoungs modulus of 69 GPa and a yield stress of 76 MPa. Each edge of the cell wall was modelledwith threeshell elements (type S4R). The model consisted of a total of2280 shell elements.Eachhexagonalcellwasdenedasasingleself-contactsurface.Self-contactwasalsodenedbetweenthe outside faces of the cell which might contact other cells during crushing. Through aFig. 1. Aluminiumhoneycomb:(a)Undeformedhoneycomb;(b)Deformedhoneycombinaquasi-staticcompressiontest.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 163convergencestudy, it wasdeterminedthat veintegrationpointsthroughthethicknessof theelementsprovidedsufcientlyaccurateresults.A constant velocity was applied to a rigidplate, which crushed the honeycomb. The velocityvwas variedfrom3.5to280 m/sinorder tostudy theeffectofthe loadingrate.When crushinginthe X1 direction, all degrees of freedom of the left edge of the specimen were xed and the top andbottom edges were free. A horizontal constant velocity (X1 direction) was applied to the right faceof theplate. Similarly, forimpact inthevertical (X2) direction, all degreesof freedomof thebottomedgewerexedandtheleftandrightedgeswerefree. Avertical constantvelocitywasappliedtothetopfaceofthestrikingrigidplate.3. Resultsandanalysis3.1. Deformationmodes3.1.1. DeformationmodesintheX1directionFigs. 35 show deformation modes in theX1direction under various impact velocities for caseh 0:2 mm. For Fig. 3, v 3:5 m/s. Initial localisation occurs when the displacement of the rightedge is small (Fig. 3a) and this produces an X shaped band starting from the stricken end. Withtheincreaseofthedisplacement, asecondlocalisedX bandisdevelopedfromthexededgeFig. 2. FEmodelandarigidimpactplateusedinthesimulation.Thevelocityisconstantduringcrushing.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 164anditintersectswiththerstlocalisationbandtoformarhombusatthecentreofthespecimen(Fig. 3b and c). As the crushing proceeds, more localised bands occur with one more layer of cellscrushedalongtheX bands (Fig. 3d). After that, localisationtakes placewithinthecentralrhombus (Fig. 3e and f). Finally, when deformation within the rhombus exhausts, more localisedFig. 3. Crushingof ahoneycombinthe X1direction, h 0:2 mm, v=3.5 m/s. Deformations arelocalisedinitiallywithin an X shaped band. (a) d 5:6 mm (t 1:6 ms); (b) d 11:9 mm (t 3:4 ms); (c) d 18:2 mm (t 5:2 ms); (d)d 24:5 mm (t 7:0 ms); (e) d 34:0 mm (t 9:7 ms); (f) d 40:3 mm (t 11:5 ms); (g) d 46:6 mm (t 13:3 ms); (h)d 56:0 mm(t 16:0 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 165bands occur near the loading edge (Fig. 3g) until the honeycomb is completely crushed(Fig. 3h). The X shaped localisation band was also observed in our quasi-static experiments, seeFig. 1(b).At a higher impact velocity of 14 m/s (Fig. 4), a number of cells near the right edge of the modelisslightlycrushedwithinaV shapedblockandnoobviouslocaliseddeformationbandareobservedat thebeginningof deformation(Fig. 4a). Afterwards, alocaliseddeformationbandFig. 4. Same conditions as Fig. 3, except v=14m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t 0:85 ms); (c)d 18:2 mm(t 1:3 ms);(d) d 24:5 mm(t 1:75 ms);(e) d 34:0 mm(t 2:43 ms);(f) d 40:3 mm(t 2:88 ms);(g) d 46:6 mm(t 3:33 ms);(h) d 56:0 mm(t 4:0 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 166occursneartheloadingedge(Fig. 4b). ThebandisoftheshapeofX butisslightlyslimmerthanthat underalowerspeedimpact (asinFig. 3). Withfurtherdeformation, morelocalisedbands develop progressively (Figs. 4ce). When the displacement increases to about 40 mm, a newobliquelocalisedbandoccurs,fromthexededgeandgrowstowardsalongtheexistingonesFig. 5. SameconditionsasFig. 3, except v=70m/s: (a)d 5:6 mm(t 0:08 ms); (b)d 11:9 mm(t 0:17 ms); (c)d 18:2 mm (t 0:26 ms); (d) d 24:5 mm (t 0:35 ms); (e) d 34:0 mm (t 0:49 ms); (f) d 40:3 mm (t 0:58 ms);(g) d 46:6 mm(t 0:67 ms);(h) d 56:0 mm(t 0:8 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 167which seem to have stopped developing (Figs. 4fh). Then the bands interact with each other andthesampleistotallycrushed.NoobviouslocalisedX orV shapedbandhasbeenfoundthroughthewholecrushingprocess whenthe impact velocityis higher, for example, at 70 m/s (Fig. 5). Onlyalocalisedtransversebandperpendiculartotheimpactisobservedattheloadingedgeanditcontinuestopropagate, layer bylayer, tothexededge. This deformationis inamanner of plane wavepropagation[13].Figs. 6 and7 showthe deformationmodes for cell wall thicknesses of 0.5 and0.08 mm,respectively. Impact velocitiesarethesameasinFig. 4(14 m/s). ThelocalisedbandshowninFig. 6(a) is almost the same as that obtained in our experiment (Fig. 1b) and by Guo and Gibson[6] for the static case. Fig. 6(a) shows a localised band starting fromthe impact edge forh 0:5 mm.However,forthesamevalueofdeformation,noobviouslocalisedbandisobservedforh 0:2 mminFig. 4(a). InFig. 6(b), thecellsinthelocalisedbandarecrushedmorethanthoseinFig. 4(b). Fig. 6(e)showsthatthelocalisedbandfromthexededgeoccurswhenthecompresseddisplacementoftherightedgeis34 mm,whileforh 0:2 mm,thebandoccursatadisplacement of about 40 mm, as inFig. 4(f). All these indicate that increasingthe cell wallthicknessof thehoneycombhasthesameeffect asdecreasingtheimpact velocityintermsofdeformationmode. Further evidence for this is that for thinwall cells nolocalisedbandisdevelopedfromthexededge(Fig. 7), withthedeformationmodebeingsimilartothecaseofthickerwallcells(Fig. 5).3.1.2. ModeclassicationmapThe above deformation characteristics may be summarised as follows. Oblique localised bandsare present when the impact velocity is low (3.5 and 14 m/s). The lower the velocity, the wider theX shaped band (Figs. 3 and 4). With the increase of the impact velocity, localised bands tend tobe transverse (Fig. 5); also a localised band starting from the xed edge occurs later in the process.For example, for an impact velocity of v 3:5 m/s, the localised band which starts from the xededge is observed when crush deformation reaches d 11:9 mm (t 3:4 ms) (Fig. 3b). However forv 14 m/s, it canonlybeobservedat d 40:3 mm(t 2:88 ms) (Fig. 4f). Nolocalisedbandappearsfromthexededgewhenthevelocityishighenough(Fig. 5).Atthisstage,wemaybroadlyclassifyalltheobserveddeformationmodesintheX1directioninto three types. Type one is of X shaped deformation mode, as shown in Fig. 3(a). The featureof this modeis that X shapedlocalisedbandscanbeobservedclearlywhenhoneycombiscrushedbyadeformationassmallas5.6 mm(d=L0 7%).Type three is I mode, as shown in Fig. 5. In this mode, there is no obvious oblique localisedband through the whole crushing process and only vertical bands normal to the loading directionarefound. Typetwoisatransitional modebetweentheX andI modes, suchasshowninFig. 4(a),andisnamedtheVmode.Thelocalisedbandsareoblique,buttheydonotformacompleteX shapewhenahoneycombiscrushedbyadisplacementof5.6 mm(d=L0 7%).Fig. 8 is a sketch of these three modes. The I and X modes are distinctive and the V modeislessso.Basedonthisconvention,weplot,logarithmically,thedeformationmodesintheX1directionof honeycombswithdifferent cell wall thicknessinFig. 9. Deformationmodesareall of Xmode at low velocities and I mode at high velocities. The critical velocity at which deformationD.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 168mode switches from one type to another depends on the cell wall thickness. For cell wall thicknessbetween 0.08 and 0.4 mm, the deformation is of X mode when the velocity is lower than 7 m/s.However, for cell wall thickness of 0.5 mm, X mode occurs when velocities are as high as 14 m/s.Fig. 6. Same conditions as Fig. 3, except h 0:5 mm, v 14 m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t 0:85 ms); (c) d 18:2 mm (t 1:3 ms); (d) d 24:5 mm (t 1:75 ms); (e) d 34:0 mm (t 2:43 ms); (f)d 40:3 mm(t 2:88 ms);(g) d 46:6 mm(t 3:33 ms);(h) d 56:0 mm(t 4:0 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 169The lowest velocities for an I mode are 70 m/s for honeycombs with a cell wall thickness of 0.1and 0.2 mm, 100 m/s for a cell wall thickness of 0.3 mm, and 140 m/s for a cell wall thickness of 0.4and0.5 mm,respectively.Fig. 7. Same conditions as Fig. 3, except h 0:08 mm, v 14 m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t 0:85 ms); (c) d 18:2 mm (t 1:3 ms); (d) d 24:5 mm (t 1:75 ms); (e) d 34:0 mm (t 2:43 ms); (f)d 40:3 mm(t 2:88 ms);(g) d 46:6 m(t 3:33 ms);(h) d 56:0 mm(t 4:0 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 170From a dimensional analysis, a nondimensional critical velocity v=s=r1=2is often adopted fordynamic response of solids. We follow a similar argument for the whole honeycomb specimen andhere that this dimensional group is only dependent upon another group, h=l; assuming the size ofa honeycombspecimenis immaterial. But since sph=l2andrph=l [1], we wouldexpectvph=lp; which corresponds to a straight line in Fig. 9. From Fig. 9 vc1is almost independent ofh=l;butvc2ph=lp:Theempiricalequationsforthetwocriticalvelocitiesare,respectively,vc1 14 m=s 1andvc2 277h=lpm=s: 2Fig. 8. Sketchofthethreetypesofthedeformationmodes.(m/s)lh

c220.03"I" mode"V" mode(transitional mode)"X" mode

c1 0.2

(m/s)h/l0.110100= 277= 14 (m/s)Fig. 9. Deformation mode map in terms ofh=land v (() denotes deformation mode type oneX mode, with Xshaped localised bands, (D denotes deformation mode type threeI mode, with the localised bands beingperpendicular totheloadingdirection, (J) denotes deformationmodetypetwoV mode, whichis in-betweenmodesoneandthree).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 1713.1.3. DeformationmodesintheX2directionThe deformation modes in the X2 direction for honeycombs with a cell wall thickness of 0.2 mmare shown in Figs. 10 and 11. The impact velocities are 7 and 70 m/s, respectively. When v 7 m/s,lightlycrushedcellsformanarch at thelowerpart of themodel (Figs. 10aandb). Thisissimilar to the V mode for the X1 direction (Because the cells that form this V shaped band areless severely crushed than those in the X1direction, the V mode observed in the X2direction isless visible than that in the X1direction.). Localised bands are composed of heavily crushed cells,perpendicular to the loading direction, at the upper middle position of the model (Fig. 10b). ThisissimilartotheI modeintheX1direction. However, whentheimpactvelocityisrelativelyhigh, say 70 m/s, the upper loading edge is crushed rst and it forms a localised band horizontally(Fig. 11). When v 70 m/s, there is no V mode and only I mode is observed. As the crushingcontinues,morelocalisedbandsareformedlayerbylayer,asshowninbothFigs. 10(c)(f)andFigs. 11(b)(f).Fig. 10. Crushing of a honeycomb in the X2 direction, h 0:2 mm, v 7 m/s: d 5:6 mm (t 0:8 ms); (b) d 11:9 mm(t 1:7 ms);(c) d 18:2 mm(t 2:6 ms);(d) d 34:0 mm(t 4:9 ms);(e) d 40:3 mm(t 5:8 ms);(f) d 46:6 mm(t 6:7 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 172Fig. 12 shows the deformation of a honeycomb with a cell wall thickness of 0.5 mm. The impactvelocity is 7 m/s. The V mode can be observed rst (Fig. 12a). Then heavily crushed cells formhorizontallylocatedbands(Fig. 12bf).However,therstnoticeablelocalisedbandoccursataposition that is one cell-layer lower than that in Fig. 10. This indicates that the position of the rstlocalised band occurs further from the loading edge with the increasing of the cell wall thicknessearlierlayersarestrongerbecauseoftheirlateralinertia,whichisprobablythesameastheeffectoflateralinertiainstrutproblem.3.2. Plateaustress3.2.1. X1directionIn the simulations, the force between the honeycomb and the striking rigid plate are calculated.A typical force-displacement curve is shown in Fig. 13 for v 14 m/s and h 0:2 mm, and Fig. 14demonstrates the effect of the impact velocity. In general, there are initial, and for high velocities,subsequent peaks. However, because our main interest is in the plateau stress, which is importantFig. 11. SameconditionsasFig. 10, exceptv 70 m/s: (a) d 5:6 mm(t 0:8 ms); (b) d 11:9 mm(t 1:7 ms); (c)d 18:2 mm(t 2:6 ms);(d) d 34:0 mm(t 4:9 ms);(e) d 40:3 mm(t 0:58 ms);(f) d 46:6 mm(t 0:67 ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 173forenergyabsorption, andthevaluesof thepeakpointsareheavilyinuencedbythecut-offfrequencyofawavelterintheFEcalculation, thesepeakforcesarenotdiscussedhere. Therelationshipbetweentheforceandthedisplacementforothercell wall thicknessissimilar, butwithdifferentvaluesforpeakandplateauforces.The values of the dynamic plateau stresses, which are calculated as the ratio of the plateau forcetoloadingsectional area, arelistedinTable1. Theoretical staticplateaustressesarecalculatedFig. 12. Same conditions as Fig. 10, except h 0:5 mm, v 7 m/s: (a) d 5:6 mm(t 0:8 ms); (b) d 11:9 mm(t 1:7 ms);(c) d 18:2 mm(t 2:6 ms);(d) d 34:0 mm(t 4:9 ms).0 10 20 30 40 50 60050100150200Initial peak force Plateau forceForce (N)Displacement (mm)Fig. 13. Dynamicforce-displacementcurveofahoneycombintheX1direction,h 0:2 mm,v 14 m/s.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 1740 10 20 30 40 50 600255075100=35m/s=14m/s=7m/s=3.5m/sTheoretical static forceForce (N)Displacement (mm)Fig. 14. Dynamicforce-displacement curveintheX1directionof honeycombswiththesamecell wall thicknessof0.2 mmatdifferentimpactvelocities.Table1Plateaustressforhoneycombsunderimpactloadingv(m/s) h(mm)0.08 0.2 0.3 0.4 0.5s0a(MPa) Static 0.0506 0.317 0.712 1.266 1.978Calculateddynamicplateaustress s(MPa)X1direction 7 0.0513 0.332 0.770 1.349 2.25214 0.061 0.365 0.829 1.478 2.38535 0.134 0.550 1.106 1.899 3.13470 0.556 1.600 2.491 3.777 5.772100 1.061 2.966 4.677 6.802 10.268140 1.959 5.378 8.784 12.500 16.983200 3.731 10.286 16.700 23.700 31.698280 7.546 20.492 33.500 47.009 60.403X2direction 14 0.069 0.453 1.009 1.775 2.58635 0.193 0.771 1.636 70 0.501 1.549 2.960 100 0.953 3.014 4.734 140 1.815 5.260 8.575 ValuesofparameterAinEq. (3) 96.7 255.3 405.5 560.5 736.5aObtainedfromEq. (4.26b)inRef.[1]multipliedbythefactorof1.15.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 175followingGibsonandAshby[1], whichiss0 23sysh=l2: (Wemultiplytheir formulafor theplateaustressbyafactorof1.15toaccountfortheplanestrainconditionofthecellwalls.)Forthe same cell wall thickness, dynamic plateau stresses are higher than the theoretical static values.When the impact velocity is sufciently high, the deformation mode is a propagation of a planeplasticwave.Globally,thestaticstressstraincurvesofhoneycombsexhibitahardeningwithalocking strain, though at a micro-level for each cell, there is a softening. This suggests that a shockwave theorymaybeapplicableinthiscase.Thesimplestformforthedynamicstressisthen[13]s s0rv2ed s0Av2; 3where s is the dynamic plateau stress, s0is the static plateau stress as listed in Table 1 for varioushoneycombs, r is the density of the honeycomb, v is the impact velocity, edis the locking strain ofhoneycombsunderstaticloading, andAisaparameterwhichequalstor=ed: Thedensitiesofthesehoneycombscanbecalculatedbyusingthefollowingequation[1]:r rs23phl: 4Thecalculatedvaluesofdensityforhoneycombswithcellwallthicknessof0.08,0.2,0.3,0.4and 0.5 mm are 91.9, 229.8, 344.7, 459.6 and 574.5 kg/m3, respectively. The static locking strain ofhoneycombsdecreaseswithcell wall thickness. However, itsactual valueisnot available. Weassume that the static locking strain for honeycombs with wall thickness of 0.08, 0.2, 0.3, 0.4 and0.5 mmare0.95,0.9,0.85,0.82and0.78,sothatEq. (3)besttstheFEresultsintheregionofhigh velocity. The values of parameter A ( r=ed) in Eq. (3) for h 0:08; 0.2, 0.3, 0.4 and 0.5 mmarethencalculatedandlistedinTable1.Figs. 15(a)(e) show the change of plateau stress s s0 with impact velocity, where calculateddynamicplateaustressesfromFEareplottedasdiamondsymbols.Forallthevaluesofthecellwall thickness, theplateaustressincreaseswithimpactvelocity. Whenthevelocitiesarehigherthanacertainvalue,plateaustresses s s0showagoodcorrelationtovelocitiesbyasquarelaw, corresponding to a slope of 2 in the double logarithmic plot. The slope is lower than 2, whenhisrelativesmall, say0.08 mm, andlargerthan2, whenhisrelativelarge, 0.30.5 mminourcalculation.ThevalueofparameterAinEq. (3)isrelatedtothecellwallthicknessorh=l;fromtheleast-squarettingasA 4742hl 23115hl 0:75 5andisplottedinFig. 16.Thestaticplateaustressforvarioush=l canbedescribedby[1]s0 1:15 23syshl 2: 6ThusEq. (3)canbere-writtenass 0:8syshl 24742hl 23115hl 0:75" #v27D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 1761 10 1001E-30.010.1110 FE data-0=96.72-0 (MPa) (m/s)1 10 1000.010.1110FE data -0=255.32-0 (MPa) (m/s)1 10 1000.010.1110FE data-0=405.52-0 (MPa) (m/s)1 10 1000.1110FE data (m/s)-0=560.52(a)(b)(d) (c)-0 (MPa)1 10 1000.1110FE-0=736.52 -0 (MPa) (m/s) (e)Fig. 15. ChangeofplateaustressesintheX1directionwiththeimpactvelocity:(a)h 0:08 mm;(b)h 0:2 mm;(c)h 0:3 mm;(d)h 0:4 mm;(e)h 0:5 mm.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 177or,notingthat sys 76 MPainourFEcalculationssys 0:8hl 262hl 241hl 0:01" #106v28ThisistheequationforthedynamicplateaustresswhenthevelocityishighforanImode.Forexample,whenv 100 m/sssys 1:39hl 20:41hl 0:0001: 9Forh=l oftherangeof0.030.18,Eq. (9)canalsobeapproximatedbyapowerlawssys 0:9hl 1:2: 10Equations for some higher velocities are listed in Table 2, and the power laws have an exponentsmallerthan2.Fig. 17showstherelationshipbetweentheplateaustressandthecellwallthicknesswhentheimpact velocityis kept constant. Withinthe impact velocityrange of 7280 m/s, the plateaustressesshowageneralgoodcorrelationtothecellwallthicknessortheratioofh=l byapowerlaw,whichisssys Bhl p; 11Here, B is a constant and p is an exponent. Both B and p depend on the impact velocity, as shownin Fig. 17. p is equals to 2 at low velocities, which is the same as static cases. At high velocities, p isless than 2. Both Band p obtained here fromcurve tting equal or approximate to the0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200100200300400500600700800 values of parameter A A=4742(h/l)2+3115(h/l)+0.75Ah/lFig. 16. ParameterAinEq. (3)versush=l withthebestttedcurve.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 178correspondingvaluesinTable2. Inthiscase, Eq. (8) canbeusedasanempirical formulafordynamicplateaustressathighimpactvelocity.3.2.2. X2directionThe force-displacement curve in the X2 direction is similar to that in the X1 direction, but differsin peak forces and plateau forces. The plateau stresses in the X2direction are given in Table 1 and0.01 0.1 11E-41E-30.010.1=280m/s /ysh/l/ys=5.4(h/l)1.1/ys=2.9(h/l)1.2/ys=1.5(h/l)1.2/ys=0.9(h/l)1.2/ys=0.6(h/l)1.2/ys=0.7(h/l)1.7/ys=0.9(h/l)2.0/ys=0.9(h/l)2.1=200m/s=140m/s=100m/s=70m/s=35m/s=14m/s=7m/sFig. 17. Plotof s=sysintheX1directionversush=l:Table2ExpressionsofplateaustressderivedfromEq. (8)forhighvelocitiesVelocity(n)(m/s) sintheformofEq. (8) sintheformofpowerlaw140ssys 2:0hl 20:8hl 0:0002ssys 1:5hl 1:2200ssys 3:28hl 21:64hl 0:0004ssys 2:7hl 1:1280ssys 5:66hl 23:21hl 0:0008ssys 5:4hl 1:2D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 179Figs. 18(a)(c). Eq. (3) is assumed to be able to describe the plateau stress in the X2 direction whenimpact velocity is high enough. The values of density r and locking strain edare the same for thehoneycomb in both X1and X2directions when the cell wall thickness is the same. Forh 0:08 mm (h=l 0:03) shown in Fig. 18(a), FE results are on a straight line when the velocity isbetween14and140 m/s. Inthiscase, itcanbesaidthattheplateaustressesarerelatedtotheimpactvelocitybyasquarepowerlaw.InFigs. 18(b)and(c),circlesymbolsareonthestraightlineswhenvelocitiesarelargerthan70 m/s, forhoneycombwithcell wall thicknessof 0.2 mm(h=l 0:07) and 0.3 mm(h=l 0:11), respectively. This indicates againthat plateau stressesincreasewiththeimpact velocityfollowingasquarelawwhenthevelocitiesarelargerthanacriticalvelocity,andatarateslowerthanasquarelawwhenthevelocitiesaresmallerthanthiscriticalvelocity.Similarly, when the impact velocity is constant, plateau stresses in the X2 direction also increasewith the cell wall thickness by power laws, as shown in Fig. 19. The exponent is 2 at low velocities,whichisthesameasthestaticcases.Itislessthan2athighvelocities.Bandparethesamefor10 1000.010.11FE data (m/s)10 1000.1110FE data (m/s)10 1000.1110FE data (m/s)(a) (b)(c)_0=96.7v2-0 (MPa)_0=255.32 -0 (MPa)_0=405.52-0 (MPa)Fig. 18. ChangeofplateaustressesintheX2directionwiththeimpactvelocity:(a)h 0:08 mm;(b)h 0:2 mm;(c)h 0:3 mm.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 180loadinginbothX1andX2directionsatv 100and140 m/s.Eq. (8)isalsoapplicableforhigh-speedimpactintheX2direction.4. ConclusionsBothdeformationmodesandplateaustressesofhoneycombsunderin-planedynamicloadingarerelatedtothecellwallthicknessandimpactvelocity.Three different types of deformation modes are observed when the honeycomb is loaded in theX1direction. LocaliseddeformationcausesobliqueX shapedbands(X mode) at thelowimpactvelocity, butproducesvertical localisedbands(I mode)perpendiculartotheloadingdirection at high impact velocity. Atransitional V mode is present when the velocity ismoderate. The localised bands occur closer to the loading edge when the impact velocity increases.Reducingthethickness of thecell wall has thesameeffect as increasingtheimpact velocity,leadingtoformationoflocalisedbandsclosertotheloadingedge.V modeisalsoobservedintheX2directionwhencrushingatlowvelocity. Thecellsthatcomposethe Vshaped bandsare lessheavilycrushedthanthoseintheX1direction.OnlyImodeisobservedathighimpactvelocity.ThisImodelocalisedbandcontinuestogrowintheadjacentcelllayers.ThepositionoftherstlocalisedbandintheImodedifferswiththeimpactvelocityandcell wall thickness. Thehighertheimpactvelocityandthethinnerthewallthickness,thenearertherstlocalisedbandoftheImodetotheloadingedge.Arelationshiphasbeenestablishedbetweentheplateaustresses, thecell wall thicknessandvelocity (Eq. (8)). Plateau stresses show a good correlation to the cell wall thickness or the ratio of1E-30.010.1/ys0.01 0.1h/l/ys=1.5(h/l)1.2=140m/s/ys=0.9(h/l)1.2/ys=0.7(h/l)1.3/ys=0.7(h/l)1.6/ys=1.3(h/l)2.0=100m/s=70m/s=35m/s=14m/sFig. 19. Plotof s=sysintheX2directionversush=l:D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182 181h=l by a power law for a given impact velocity. The exponent equals to 2 at low velocities, which isthesameasthestaticcase.However,athighvelocities,theexponentislessthan2.The plateau stresses in both X1and X2directions increase with the impact velocity by a squarelawwhenthevelocityissufcientlyhigh,whichisinagreementwiththeReidPengequation.AcknowledgementsThe authors wish to thank the Australia Research Council for the nancial support and WorleyFEAfor an academic license of ABAQUS. The fourth author (T.X. Yu) would like toacknowledge the support fromthe HongKongResearchGrant Council (RGC) under grantHKUST6079/00E.References[1] Gibson LJ, Ashby MF. Cellular solids: structures and properties, 2nd ed. Cambridge: Cambridge University Press,1997.[2] Klintworth JW, Stronge WJ. Elasto-plastic yield limits and deformation laws for transversely crushedhoneycombs.IntJMechSci1988;31:27392.[3] PapkaSD, KyriakidesS. In-planecompressiveresponseandcrushingof honeycombs. IntJMechPhysSolids1994;42:1499532.[4] Papka SD, Kyriakides S. 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