Pe r gamon Int. ZImpact Engng, Vol. 19, Nos. 5--6, pp. 531-570,1997 1997 Elsevier Science Ltd Printed inGreat Britain. All fights reserved PII:S0734-743X(97)00016-X0734-743X/97 $17.00+0.00 DYNAMICUNIAXIALCRUSHINGOFWO O DS. R. REI D*a ndC. P ENG Department ofMechanicalEngineering, UMIST,POBox88,SackvilleStreet,Manchester,M601QD,U.K. (Received30October1996;inrevised f orm30January1997) Summar y- - Exper i ment alresults are provi ded from aseries oftests on the uniaxial dynamic crushing ofcylindrical speci mens offivespecies of woodselected forthedensityrangetheycoverandtesteduptoi mpact velocities ofapproximately 300 ms -1.An account ofthe macro-deformation and micro-deformation modes resulting from quasi- staticand dynamic uniaxialcompression is given. Measurements ofthe force pulses generated by the i mpactofthe wood specimens onthe end ofaHopkinson bar load cell show thatsignificant enhancement s ofthe initial crushing strengthsof thespecimensoccurunderdynamicloadingconditions.Thedeformationmechanismsof woodare localised underquasi-staticcompression andunderdynamicloadingconditionstheybecomeevenmorelocalised and propagate through the material as crushing wave fronts whi ch havesome ofthe characteristics ofshock waves. Asimpleshockmodelbaseduponarate-independent, rigid,perfectly-plastic, locking(r-p-p-l)idealisationofthe st ress-st rai n curves for wood is proposed to provide af as torder understanding ofthe dynamic response. This modelisparticularlysuccessfulinpredictingthedynamicenhancement of thecrushingstrengthof speci mensloaded acrossthegrainasconfi rmedbycomparisonsbet weentheexperimentaldataandtheoreticalresults.Itisless successful forthosecompressed alongthegrain.Thesourceofthediscrepancy isdiscussed andexplanationsare provided for the fairlyconstant crushingstress enhancementfactor observed atlowtomoderateimpact velocities, forthehi ghimpactvelocityatwhi chshock-type responseisinitiatedandfortheexistence ofclearlydelineated crushfrontswhi chcharacterise t hesespecimens. 1997ElsevierScienceLt d.Keywords:cellularmaterials,wood,impact energyabsorption, crushingstrength,microinertia,shockwaves. NOTATION Ao CL Cs hcell L Lo M At* mw Sr t u v Vc Vcrit vl Vo Vs x Xo 7 OL El /91 cross- sect i onal areao f t hecyl i ndri cal speci men l ongi t udi nal el ast i cwavespeedi n0 speci mens shockspeedrelat ivet ot heproject i l e axiall engt ho f wo o dcel ll engt ho f t heuncrushedport i ono f woodspeci men initiall engt ho f t hewoodspeci men masso f t hebacki ngdi sc combi nedmasso f woodspeci menandbacki ngdi sc masso f t hewo o dspeci men dynami ccrushi ngstressenhancement factor,Sr = g r D / O ' c rt i me di spl acement o f t heendmass(backi ngdi sc)M vel oci t yatt i met l owes t i mpact vel oci t yatwhi chcrushi ngi sinit iat edi n0 speci mens criticalimpact vel oci t yforfullspeci menl ocki ng 0 speci menvel oci t yatwhi chcrushi ngceases i mpact vel oci t y i mpact vel oci t yatwhi chshockwavesaregenerat edin0 speci mens currentl engt ho f t hecrushedmaterialstrainedt oel init iall engt ho f t hecrushedport i on = P o/ pt= 1-ell ocki ngstrainparameter l ocki ngstrain densi t yo f t hecrushedmaterial *Corresponding author. 531 532S RReidandC.Peng Do P~ 17 I-Ilk O" O"A O-comp O'cr(O) crD(0) O"R O"T Crys initialdensity cellwallmaterialdensity,po/p~=relativedensity specificenergy(energyperunitmass)ofspeci men mi ni mumvalueof 17toproducefulllockingof thespeci men stressbehindcrushingwavefrontattimet axialstaticcrushingstress appliedstressinaxialdirectioncorrespondingtouniaxialyieldinthecellwalls crushingstressforgrainorientationat0tothedirectionof loading initialdynami ccrushstressforgrainorientationat0tothedirectionof loading radialstaticcrushingstress tangentialstaticcrushingstress yieldstressof cellwallmaterial 1.I NTRODUCTI ON Therehavebeennumerousstudiesof cellularstructuresandcellularmaterialssuchasmet al ring systems,honeycombs, pol ymeri cfoamsandwood,someofwhichhavebeenrevi ewedinthecontextofdynami c compressi on by Reide t al . [ 1].Manyaspects ofthebehavi our ofcellularsolidsaresummari sed wellinthebook by Gi bsonandAshby[2].Onereasonfor thisinterestis thecurrentandpotentialuseofthesematerialsfor packagi ng, asi mpact energyabsorbersandtheiruseascorematerialsinlightweight sandwichstructures.Gi bsonandAshbydevot eachapterof theirbook[2]totheselectionof materials forlowspeedi mpact applications.Woodinparticular,whilstbeingoneof theoldestconstructional materials,hasalsobeenusedasaprotectivematerialforhighvelocityi mpact eventsforcenturies(see Johnson[3,4])andisoftenusedasani mpact energyabsorbingmaterialincont ai nment structures surroundingsyst emswhichmaydisintegrateor,forexampl e, aspackagi ngmaterialintransportation flasks for nuclearfuel.Theresponseofwoodathighratesofloadingisthereforeofsome interest.There havebeenfewsystematicstudiesofthebehavi our ofwoodunderi mpact loadingconditionsasnotedby Johnson[3].Thispaperpresentstheresultsof onesuchstudyinwhichspeci mensof woodwere subjectedtouniaxialstrainunderquasi-staticandi mpact loadingconditions. Anintroductiontothestudy ofwoodunderdynami c uniaxialcrushingconditionsandwhenpenetrated byflat-endedcylindricalprojectilewasgiven byReide t al . [5],whoprovi dedexperi ment al datafroma preliminarystudyof YellowPineandAmeri canOak.Thesensitivityof thecrushingstressesof these woodstoloadingratewasnoted.Forexampl eatani mpact vel oci t yof around100ms -1thecrushing strength(i.e.thestressatwhichinelasticdeformat i onisinitiated)ofthetwowoodsincreasedbyfactors ofbet ween2and4dependinguponthedensityofthewoodandtheorientationofthegrainwithrespecttothedirectionof loading. Inthepresentpaper,moreextensivei mpact testdatathanthosegivenin[1,5]areprovi dedonthe significantchangesinthecrushingstressesof fivespeciesof wood,Balsa,YellowPine,Redwood,Ameri canOakandEkkiasthei mpact velocityisincreased.Thewoodstestedhaddensitiesranging approxi mat el y from 260- 1200kgm3 andweresubjectedtouniaxialcompressi on underi mpactloading conditionsati mpact velocitiesintherange30- 300ms1 approximately. Loadi ngbot halongthegrain andacrossthegrainisconsideredandthedeformat i onmechani smscorrespondingtoeachare contrasted. Themai nemphasi sinthepaperisplacedonpredictingtheinitialcrushingstrengthsof thevarious specimenstestedintermsof thei mpact velocity,thedensityof thewoodanditsmaterialproperties. Simpletheoreticalmodel saredevel opedtodescribetheeffectsof phenomenawhicharegovernedby inertialeffectsactiveatthescaleof thecellularstructure,i.e.microinertialeffects.Fr omamaterials scienceperspective, themodel sarebasedonthepremi sethatwoodisrate-insensitive,theexplanation for theenhancement ofthecrushingstressesbeingsoughtintheinfluenceoftheseinertialeffects rather thaninviscoelasticorviscoplasticeffects.Inthissenseasimilarapproachisfol l owedtothatproposed byJahsman[6]inaveryinteresting,earlierstudyof thedynami cbehavi ourof syntacticfoams.Thepredictionsof asi mpl eshockmodelarepresentedinwhichtheenhancement of thecrushing stressisattributedprimarilytothepropagat i onof aplanesurfaceof stressdiscontinuitythroughthe cellular material.Despiteitssimplicity,it isshownthatthemodelagreeswellwithexperi ment al datafor Dynamic uniaxial crushing of wood533 crushingacrossthegrainbutthat,forarangeof velocitiesuptoacertainvaluewhichdependsonthe densityofthewood,it underestimatestheenhancement oftheinitialcrushingstressalongthegrain.The reasonsfor thisarediscussedandanot hermicroinertialeffectissuggestedasthesourceofthecrushing stressenhancement forwoodloadedinthisdirection. To illustratetherangeofphenomenai nvol ved whenwoodiscrushed,thequasi-staticbehavi our ofthe uniaxialcrushingofwoodsisbri efl ydescribedinthenext section.Thisalsoprovi desareferencepoint forthediscussionof thedynami ctestdatawhichfollows. 2.QUASI-STATICCOMPRESSI ON Typicalst ress-st rai ncurvesfor uniaxialcompressi ontestsonavarietyofwoodsaregi venbyGi bson andAshby[2]summari si ngtheirearlierworkwithEasterlingandHarryson[7]andwithMaiti[8].A detailedaccount ispresentedof thedeformat i onmechani smsi nvol vedwhenwoodiscompressedinits threeprincipaldirectionsnamel yaxial(alongthegrain),radial(acrossthegraintransverse tothegrowth rings)andtangential(acrossthegrainparalleltothegrowthrings).Elasticdeformat i onoccursby distortionsof thecellstructurewhicharedistributedthroughoutthevol umeof thespecimen. Inelastic deformat i onoccursastheresultof thecol l apseandgrossdeformat i onof thecellsinamanner qualitativelysi mi l ar to thatobserved inmacroscopi c cellularsyst ems as discussedbyseveralauthorsand summari sedin[1].Inelasticdeformat i oninwoodtendstobealocalisedphenomenon, especi al l yfor speci mens l oadedal ong thegrain.Followingtheattainmentoftheinitialcrushingstress,thedeformat i on progressesthroughthegrowthandmultiplicationofcrushingzonesatanappliedloadwhichiscloseto theinitialcrushingload. Fig.1showstypicall oad-compressi oncurvesforcylindricalspeci mensof thewoodstestedinthis study.Thespeci mens werenomi nal l y 75mmindi amet er and75mm highandwerecontainedinathick- wal l edcylindricalsteelvesselof 75mminternaldi amet ertoi mposeuniaxialstrainconditions.The moi st urecontentof thespeci mensweremeasuredintheconventionalmanner usingamoi st uremeter. For theBalsa,Pine,RedwoodandOak speci mens themoi st ure contentwaswithintherangeof9to12%. The Ekki speci mens hadahighmoi st urecontentintherange15-24%. Ekki isnormal l y usedinamari ne environment. (Notethat,inthispaper,theonl ydetaileddatapresentedforspeci mensof Ekki relateto thoseloadedacrossthegrain.)Freespeci menstendtosplitandfracture(seeFig.2),althoughtheir behavi our uptoand j ust beyondinitialcrushingisvery closetothatshowninFig.1 sincethePoi sson' s ratiosfor woodtendtoberelativelysmall[2,9].Forspeci mensloadedalongthegrain,thereisadistinct loaddropaftertheinitiationofcrushingfol l owedbyaloadplateauforallofthewoods.Forspeci mens loadedacrossthegrain,crushingcontinuesessentiallyatorslightlyabovethecrush-initiationload, increasingatanincreasingrate(i.e.thecurveisconvextowardsthedi spl acement axis)withdisplace- ment.Thereis nostrain-softening.Therateofincreaseoftheloadissignificantly higherfor woodswith higherinitialdensitiessuchasOakandEkki.Inallcases,thedeformat i onreachesastagewherethe crushingoftheindividualcellsbecomesexhaustedandfurtherdeformat i onistheresultof solidphase compressi onof thecellwallmaterial.Thisgivesamuchstifferresponseasthematerial' l ocksup' .Theessentialdifferencesbet weenloadingalongthegrainandtransversetothegrainlieinthe differencesinthecol l apsemechani smsinducedinthecellularstructureof thewood.Theresponseto compressi onalongthegrainisext remel ycompl exandtheinitiationof crushingcaninvolvefailureofthepyrami dal endcapsof thecellsforlowdensitywoodssuchasBal sa[7]ortheonsetof various buckl i ngfailuressuchasconcert i naprogressi veaxialcrushingof thecells(fol d-mi crobuckl i ng, see [2,9])or Eul er-t ype buckl i ngwhichcanleadtoki nk-bandformat i on especi al l yinthemoredensewoods [9].Fig.2showsanexampl eof thet ypeof deformat i onthatoccursinfree(i.e.withoutanylateral constraint)speci mensofhighdensity(inthiscaseEkki )l oadedalongthegrain.Compressi on acrossthe grain(despitetheinfluenceof theraysinenhancingthecrushstressbyapproxi mat el y40%whenthe wood iscrushedintheradialasopposed tothetangentialdirection)issomewhatmorestraightforward.Ithasbeendescribedint ermsof plasticbendi ngcol l apseof at wo-di mensi onal hexagonal cellulararray whichprovi desagoodapproxi mat i ontothestructureof manywoods[7].Underquasi-staticloading conditionsthedeformat i on alsotendstobemoredispersedthroughthevol ume ofthespeci menthanthat producedbyloadingalongthegrain. Theconsequencesof theseobservationsonthestructuralf or mof thecellstructureandthe mechani smsof failurearethatsi mpl erelationshipscanbeformul at edbet weentheinitialcrushing 5 3 4 S. R. Rei dandC. Peng 5 0 04 5 0400 : }50 z:~00 ~. oO o200 1.50 100 5O 0 5 0 0(a) (~~0ak / / / [ / / / , - . 0~Re d wo o d . - " "O' P i n e --~_--. . . . . . . . . . . . . . . ~-" ~/"-.. l tittiitI iiiI I I I I J [ I ii~ I E I : ii~ I ii; I 4iI I iI I I I I010~30:504 0 5 0 6 0Di s p l a c e me n t ( n a m)45 O 4 0 03 5 0z3 0 025 0s20 01 o( 1 O0 50 0 t0E k k ~9 ( ' ( a k~'' -. /"i / / i9 0 P i n et t/t, / t~J / / - 9 0 ~B a l s a01 0 ' ~ 0:301 0 5 0 6 0Di s p l a c e n l e n t ( r a m)Fig.1.Quasi-static uniaxialload-displacementcurvesforavariety ofwoods: (a)along(0); (b)across(90 )thegrain s t r e s s e s a ndt he r e l a t i v e d e n s i t y o f t he wo o d , i . e. t he a v e r a g e d e n s i t y o f t he w o o d s a mp l e , P0,d i v i d e d b yt he d e n s i t y o f t he c e l l wa l l ma t e r i a l , Ps.T h e s e aree x p r e s s e d as f o l l o w s [ 2] :era150 (Po/Ps) MN m ~2(1) err= 1.4crT= 7 0 ( po/p.~,) 2 MN m - 2 , ( 2 )Dynamic uniaxial crushing of wood535 IFig. 2.Compressedfree Ekki specimens: (a) kink bands in 0 specimen; (b) shear failure in 90 specimen. wherethesufficesA,RandTrefertothethreeprincipaldirectionsof thewoodstructure,viz.axial, radialandtangential.Psisvirtuallythesameforallwoodsandhasavalueof 1500kgm -3[2].The numeri cal factorsarederivedempi ri cal l ywhilsttheindicesarisenaturallyf r omtheparticularcollapse mechani smsconsideredinthestructuralmodels. Thedifferencesincompressi vestrengthintheradial andtangentialdirectionsarerelativelysmallwhencomparedwithdifferencesbet weenthestrength alongthegrainandthatacrossthegrain.Inthetestsreported, thespeci mensl oadedacrossthegrain werecutwiththegrainperpendi cul artotheiraxis(i.e.thedirectionof loading)butweregenerally neitherfullyradialnorfullytangentialspecimens. Therefore, nodistinctionismadehereinbet weenthe tangentialandradialdirections.Asinglecrushingstress t ransverse tothegraindefi ned bycrcr(90)equal t oO"Rhasbeenused.Speci menscutinthiswayarereferredtoas90 speci mensherein.Theaxial crushingstress is referred toastrot(0),theangl esi mpl y indicatingthedirectionofl oadi ngrelative tothe graindirectionandsuchspeci mensarereferredtoas0 specimens. Compari sonsbet weenEqns(1)and (2)andthemeasuredinitialcrushingstressesareshowninTable1.Thelowvaluesforthe90 Ekkispeci mensareduetotheveryhighmoi st urecont ent ( ~20%) of thespeci menstested.The' f r ee'speci menhadasi mi l armoi st urecontenttothespeci menstesteddynamically.Asnotedabove, theinelasticdeformat i ontendstobelocalisedinwood.Itdevel ops bythegrowthoflocalcrushzonesandcrushbands,ofteninitiatedattheendsof thespeci menadjacent totheloading platens.Overal l deformat i onof thelaterallyconstrainedspeci menoccursbythepropagat i onof crush 536S.R.ReidandC.Peng Table1.Uniaxialquasi-staticparameters for wood specimens testedunder laterallyconstrained conditions (uniaxialcompression); 75mmdiameter75mmhigh WoodandgrainInitialdensity,Initialcrushstress,Theoreticalcrushstress,LockingSpecificlocking orientationP0(kg m3)a~r(N mm2)Eqns(1)or(2)(N mm -2)strain,c Ienergy(kJkgi) BalsaO'27727.027.70.6863.4 Balsa90:2641.62.20.6510 Y.PineO38343.338.30.6462.2 Y.Pine9 0 3965.l4.90.6015.2 Redwood0 '36743.036.70.6559.0 Redwood90'40910.55.20.5816.3 A.Oak0:72575.072.50.3330.4 A.Oak9 0 69512.315.00.3716.0 Ekki90'*10512434.4- -Ekki90"t 18318.243.50.133.7 * Freespecimen. fronts,thesebeingmorediffusefor90 specimensbutvery distinctfor 0 specimens.For0 specimens thedeformationoccursin'compressiblekinkbands'asshowninFig.3.Thesearesimilartothekink bandsproducedinthecompressivefailureofunidirectional,fibre-reinforcedpolymercomposites [10,11]buttheconsiderablecompressibilityarisingfromcellularcollapsemakesthemdistinctive.The anglebetweenthenormaltothekinkbandandthedirectionof loadingincreaseswiththedensityof the wood, theangle beingalmost zero for Balsa(giving theappearanceof a' pl ane' crush front),35 for Oak andapproximately45 forEkki.ThislatterfeaturehasbeenbrieflydiscussedbyReide t al . [1]. Aconsequenceof thelocalisednatureof thedeformationinwoodisthat,whilstitisconventionalto convertload-displacementcurvesintostress-straincurvesbydefiningstressasloaddividedbycross- sectionalareaandstrainas change inspecimenheight divided bytheundeformed height,itmustalways beborneinmindthatintheinelasticrangethestrainfieldishighlynon-uniformthroughoutthe specimen.However,therapidincreaseintheslopeoftheload-displacementcurvesoccurswhenthe crushed zoneshavegrown to encompass most of theinitialvolume ofthematerial.At thispointonecan logicallydefineacriticalstrain,thelockingstrain,e~,whichreflectsthetransitionfromthecellwall collapsemechanismtosolidphasecompressionofthemajorityofthecellwallmaterial.Usinganyof theconventionaldefinitionsof strainbeforelockingoccursproducesastrainmeasurewhichvaries accordingtothesizeof thespecimenandisthereforenotrepresentativeof thematerial.Adoptingthe procedure describedabove allowstheload-displacementcurvesto beconverted intostress-straincurves togiveanapproximatecontinuumrepresentationfortheuniaxialdeformationmodesof thewood specimens.Fig.4gives thestress-straincurves corresponding totheload-displacement curvesshownin Fig.1.Whendiscussingtheuniaxialcrushingofcellularmaterials,thekeypropertiesaretheinitial crushingstress,Crcr, andthelockingstrain,el,[2].Thesearelistedfor thefive woodsin TableItogether withthevaluesgiven by Eqns(1)and(2).Thegraphicalconstructionusedtodefinethesecharacteristic parametersfromthestress-straincurvesisalsoshowninFig.4.Itisnotedthatthecrushstressforthe EkkisampleissignificantlylessthanthevaluegivenbyEqn(2).Thisisbecauseofthehighmoisture content. Maitie t al . [8]discussedtheshapesofthestress-straincurvesforavarietyofcellularsolids includingwood.InparticulartheyproducedthefollowingequationtoestimateelforawoodofdensityPo, Cl - 1(~ (PO/P~ ).( 3)In Ref.[8]a valuea=2issuggested.Resultsfrom more recenttests[9]indicatethatthisissatisfactory for low densitywoods such asBalsa but it cannot beused for moderateand high densitywoods.For Pine andRedwood,c~= t.35ismoreappropriateandfor Oakc~ isapproximately equaltot .3.For thesamples of Ekkitested,thetransitiontolockingismoregradual.However,defininglockingasshowninFig.4 resultsinavaluefore~of0.13andavalueforc~of1.1.Theseresultssimplydemonstratethat characterisingtheinelasticpropertiesofcellularmaterialsintermsof thetwoparameters~rcr andel, whilstconvenient,becomesimprecisewhentherelativedensityofthematerialincreases.Jahsman[6] andothershaveusedmoreaccuraterepresentationsforthestress-straincurvesoftheirmaterials. Dynamicuniaxialcrushingof wood537 Fig.3.Split0 constrainedspecimens:(a)progressive' pl ane' crushfrontinBalsa;(b)multiplekinkbandsinOak. 5 3 8 S. R. Re i dandC. Pe ng " I (o)// '' L , : 9 0 / *i 8 0 ,0- i; : 6 H o" * . /4 0 " ' , . . . . . . . - "~ , / .. . . . . . o: ~ . t . . , ~1' 3 o i /10i~ 0 . 0 0, 10 . 2 0. 3 0B a l s a . . . . .i i i i i l i l l l J l l J l l [ i t ~ l ~ l l t t [ l l l l l , ~ l l , ~ l t l l ~0 . 4 0 . 5 0 , 6 0 . 7 0 . 8Ax i a l s t r a i nt 201 1 01 0 0%2 +k *9 0g oz7 0~'6 0% - - 5 0. ~4O 3 0:~201 0C] ( b )0E k k i i -.r !:! I //:~ 9 0 P i n et / 9 0 Oa ki/ i/" ~