A SPHERICAL CAVITY-EXPANSION PENETRATION

1、~)PergamonInt..LSolidsStructuresVol.34,Nos31-32,pp.4127-4146,1997©1997ElsevierScienceLtdAllrightsreserved.PrintedinGreatBritain0020-7683/97$17.00+.00PII:S0020-7~3(97)00017-6ASPHERICALCAVITY-EXPANSIONPENETRATIONMODELFORCONCRETETARGETSM.J.FORRESTALSandiaNationalLaboratories,Albuquerque,NewMexico87185-0303,U.S.A.andD.Y.TZOUDepartmentofMechanicalandAerospaceEngineering,UniversityofMissouri,Columbia,Missouri65211,U.S.A.(Received1March1996;inrevisedform5December1996)Abstract--Wedevelopedaspherical,ca。

2、vity-expansionpenetrationmodelforconcretetargets.Targetconstitutivedescriptionsidealizedpressure-volumetricstrainasincompressibleorlinearcompressibleandidealizedshearstrength-pressureasMohr-Coulombwithatensioncutoff.Weshowedtheseidealizedmaterialdescriptionswerereasonableapproximationstotriaxialmaterialdata.Predictionsfromthecompressiblepenetrationmodelareingoodagreementwithdepthofpenetrationdatafora30.5-mm-diameter,1.60kg,ogive-nose,rodprojectileforstrikingvelocitiesto1100m/s.©1997ElsevierScien。

3、ceLtd.INTRODUCTIONAnalyticalmethodsforpenetrationmechanicsbeganwiththeworkofBishopetal.(1945).Theydevelopedequationsforthequasi-staticexpansionsofcylindricalandsphericalcavitiesandusedtheseequationstoestimateforcesonconicalnosepunchespushedslowlyintometaltargets.Later,Goodier(1965)developedamodeltopredictthepenetrationdepthofrigidsphereslaunchedintometaltargets.Thatpenetrationmodelincludedtargetinertialeffects,soGoodier(1965)approximatedthetargetresponsebyresultsfromthedynamic,sphericallysymmetr。

4、ic,cavity-expansionequationsforanincompressibletargetmaterialderivedbyHill(1948)anddiscussedbyHill(1950)andHopkins(1960).Recently,Forrestaletal.(1995)developedsphericalcavity-expansionpenetrationmodelsforspheri-cal-nose,rigidrodsthatpenetrateductilemetaltargets.Modelpredictionsareinreasonablygoodagreementwithdepthofpenetrationdata.Cavity-expansionpenetrationmodelshavealsobeenusedtostudypolycarbonatetargets.Wrightetal.(1992)presentmodelingandpunchdata,andFlecketal.(1990)presentmaterialdatathatsho。

5、whighductilityathighstrainratesisthemainreasonforthegoodballisticperformanceofpolycarbonatetargets.RadinandGoldsmith(1988)andWrightetal.(1993)presenttheseballisticimpactdata.ForrestalandLuk(1992)developedasphericalcavity-expansionmodelforsoiltargetsandcomparedmodelpredictionswithfieldtests.Predictionswereinreasonablygoodagreementwithprojectiledecelerationanddepthofpenetrationdata.InadditionForrestal(1986)developedacylindricalcavity-expansionmodelforpenetrationintodry,porousrockandcomparedpredict。

6、ionswithprojectiledecelerationdata.Weobservedthatthecylindricalcavity-expansionapproximationoverpredictstheearlytimedecelerationresponseandunderpredictsthelaterdecelerationresponse.Bycontrast,thesphericalcavity-expansionapproximationisinreasonablygoodagreementfortheentiredecelerationresponse(ForrestalandLuk,1992).Inthisstudy,wepresentasphericalcavity-expansionpenetrationmodelforconcretetargetsandcomparedpredictionswithdepthofpenetrationdatafora30.5-mm-diameter,1.60kg,ogive-nosesteelrod(Forrestal。

7、etal.,1996).Sphericalcavity-expansionpenetrationmodelsapproximatethetwo-dimensionaltargetresponsewithequationsderivedfrom41274128M.J.ForrestalandD.Y.Tzousphericallysymmetric,cavity-expansionanalyses(Forrestaletal.,1995).Thus,thespheri-callysymmetric,cavity-expansionanalysesareusedasinputtothepenetrationequations.Targetconstitutivedescriptionsidealizepressure-volumetricstrainasincompressibleorlinearcompressibleandidealizeshearstrength-pressureasMohr-Coulombwithatensioncutoff.Weshowlaterthatthesei。

8、dealizedmaterialdescriptionsarereasonableapproxi-mationstotriaxialmaterialdata(JoyandEhrgott,1993).Inthefollowingsections,wederiveequationsforthesphericallysymmetriccavity-expansionproblems,presentourpenetrationequations,andcomparemodelpredictionswithdepthofpenetrationdata.SPHERICALLYSYMMETRICCAVITY-EXPANSIONPROBLEMFORMULATIONAsphericallysymmetriccavityisexpandedfromzeroinitialradiusatconstantvelocityV.AsshowninFigsla,b,thisexpansionproducesplasticandelasticresponseregions.ForslowenoughV,therear。

9、ethreeregionsofresponse(Fig.1b):anelasticregion,aregionwithradialcracks(thematerialreachesitstensilestrength),andaplasticregion(thematerialreachesitsshearstrength).AsVincreases,theradiallycrackedregiondiminishesandiseliminatedeventually.Thatis,forlargeenoughcavity-expansionvelocity1,1,theresponseiselastic-plastic(Fig.la).InFig.lb,theplasticregionisboundedbyr=litandr=ct,thecrackedregionisboundedbyr=ctandr=Clt,andtheelasticregionisboundedbyr=Cltandr=catwhereristheradialEuleriancoordinate,tistimes,。

10、candclareinterfacevelocities,andCdistheelastic,dilatationvelocity.Thematerialintheplasticregion(ForrestalandLongcope,1990)isdescribedbyalinearpressure-volumetricstrainrelationandaMohr-Coulombyieldcriterion.Thus,VtctCdtr(a)VtCtCltCdt(b)Fig.1.Responseregions:(a)theelastic-plasticproblemand(b)theelastic-cracked-plasticproblem.Cavity-expansionpenetrationmodel4129p=I~(1-po/p)=/~(la)p=(ar+ao+a4,)/3;ao=a~(lb)ar-aO=2p+z;r=[(3-2)/3]Y(lc)wherepisthepressure;P0,Paredensitiesoftheundeformedanddeformedmateria。