Crack propagation and coalescence(11)

RFPA

C.A.Tang,S.Q.Kou/EngineeringFractureMechanics61(1998)311±324321

wing-cracks(stagee-finFig.4).Immediatelyafterthatandwithoutanincreaseintheaxialload,theentirerowofthesmaller¯awssuddenlyandspontaneouslyproducedwing-cracks(stageginFig.4),whichcoalescedwitheachotherleadingtoamacroscopicshearfailureuponfurtherloading(stepiinFig.4).Comparedwiththeaxialloadrequiredtoinitiatetheaxialsplittinginuncon®nedorinlateraltensionconditions,theloadrequiredtoinitiatemacro-shearfailureinthecon®nementconditionwasmuchhigher(51.6MPa).Thesesimulationsclearlyshowthedramaticin¯uenceofcon®nementconditionsonthefailuremodesofsamples.ThenumericalresultssurprisinglymimictheexperimentalobservedphenomenareportedbyHoriiandNemat-Nasser[4])inthesampleswithasimilarpre-existingsetupof¯awgeometry(seeFig.9ofHoriiandNemat-Nasser[4]).Inparticular,thenumericalresultspresentedhereappeartosupporttheconclusiondrawnbyHallbaueretal.[26]thattheformationofthemacroscopicfractureplaneismorelikelytheproductofatensileratherthanashearprocess.Itcanbeconcludedthatbothsplittingandshearfailuresoftenobservedinaxialcompressionexperimentsofrocksandotherbrittlematerialsmaybethe®nalmanifestationofearliertensilecrackgrowthinducedunderoverallcompression.Inrealgeomaterials,aspointedoutbyHoriiandNemat-Nasser[4],isolatedpre-existingGri th-typecracksareseldomseentobethemajorsourceofmicrocracking,andmanysources(orstressconcentrators),otherthanthepre-existingcrack-like¯aws,havebeenidenti®ed.Setsofgrainboundaries,sti orsoftinclusions,andlow-aspectratiocavities,aswellassuitablyorientedinterfacesoftwodi erentminerals,willinitiatemostofthemicrocrackswhenloaded[27,28].Therefore,itisessentialthatthenumericalmodelsimulatingthefailureprocessofgeomaterialsshouldbeabletoconsiderthesetypesofinhomogeneities[Fig.1(b)].Figs.5and6illustratetheuseofRFPA2Dtosimulateamorerealisticmaterialcontaininginhomogeneitiesonthegrainscale.Thepurposeofthissimulationwasnotto®ndaquantitativematchwiththeexperimentalresults,butshowthatthequalitativetrendsweresimilartowhathappensinarealgeomaterial.Theuniaxialcompressionwassimulated.ThesimulationshowninFigs.5and6indicatesthatthefailuremodestronglydependsonthemechanicalandgeometricpropertiesofthegrainsandinclusions.Theultimatefailuremodewas,asexpected,highlya ectedbythegrainsorinclusions.AdetailedexaminationofFigs.5and6showsthat,atthebeginningofthedeformationprocess,afewlocalfractureswereinitiatedaroundsomeofthegrainsorlocatedintheweakzones(asshowninstagesa±cinFigs.5and6).Thedamagewasdistributedthroughthesampleinanuncorrelatedway.Asexpected,whenmicrocracksoccurred,theygeneratedtensilestressesintheadjacentelementsnearthecracktips.Thesetensilestressesandtheheterogeneityinthemechanicalandgeometricalpropertiesofgrainsorinclusionsinitiatedmicrocracksanddrovethepropagationofthesemicrocracksinthesample.Asfracturepropagated,morecracksdevelopedwiththeirinclinationsparalleltotheverticaldirectionofthecompressivestress(seestagedandstagee).Itwasfoundthatsomeofthecracksdevelopedalongthegrainboundariesandotherstransectedthegrains(stagee).Furtherincreasedaxialdisplacementresultedinthesplittingfailurebycoalescenceofcracksinanunstablemanner(stagese±g).Fig.7showsthestress±strainrelationofthesimulatedsample.Themarkers,a,b,FFFi,inthecurvecorrespondtothefailurestagesindicatedinFigs.5and6.Thesimulationdoesnotonlyreproducephenomenainthepre-failureprocess,butalsothephenomenainthepost-failure