Crack propagation and coalescence

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PERGAMONEngineeringFractureMechanics61(1998)311±324

Crackpropagationandcoalescenceinbrittlematerialsundercompression

C.A.Tanga,1,S.Q.Koub,*

aLNM,InstituteofMechanics,ChineseAcademyofSciences,Beijing,100080,People'sRepublicofChinabDivisionofMiningEngineering,LuleaÊUniversityofTechnology,S-97187,LuleaÊ,Sweden

Received25January1998;receivedinrevisedform17July1998;accepted9August1998

Abstract

Twoparticularcasesconcerningcrackpropagationandcoalescenceinbrittlematerialshavebeenmodeledbyusingtherockfailureprocessanalysiscode,RFPA2D,andtheresultshavebeenvalidatedbyreportedexperimentalobservations.Firstly,axialcompressionofnumericalsamplescontaininganumberoflarge,pre-existing¯awsandarowofsuitablyorientedsmaller¯awsaresimulated.Ithasbeencon®rmedthatunderaxialcompression,wing-cracksnucleateatthetipsofthepre-existing¯aws,growwithincreasingcompression,andbecomeparalleltothedirectionofthemaximumfar-®eldcompression.However,coalescenceofthewing-cracksmaybeineithertensilemodeorshearmode,oracombinationofbothmodes.Thenumericalresultsshowqualitativelyareasonablygoodagreementwithreportedexperimentalobservationsforsampleswithsimilar¯awarrangements.Thenumericalresultsdemonstratethat,withacon®ningpressure,thecrackgrowthisstableandstopsatsome®nitecracklength;whereasalateraltensilestressevenwithasmallvaluewillresultinanunstablecrackgrowthafteracertaincracklengthisattained.Secondly,failuremodeinasamplecontaininginhomogeneitiesongrainscalehasalsobeensimulated.Theresultsshowthatthefailuremodestronglydependsonthemechanicalandgeometricpropertiesofthegrainsandinclusions.#1998ElsevierScienceLtd.Allrightsreserved.

Keywords:Numericalsimulation;Cracks;Coalescence;Brittlematerial;Heterogeneity

*Authortowhomcorrespondenceshouldbeaddressed.1CurrentlyatCentreforRockburstsandInducedSeismicityResearch,NortheasternUniversity,Shenyang,110006,People'sRepublicofChina.

0013-7944/98/$-seefrontmatter#1998ElsevierScienceLtd.Allrightsreserved.PII:

S0013-7944(98)00067-8

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1.Introduction

Numerousexperimental[1±8]andtheoretical[3,6,9±11]e ortshavebeendevotedtotheunderstandingofthecrackinitiation,propagation,andcoalescenceofpre-existing¯awsinbrittlematerials.Uniaxialcompressionexperimentswitharock-likemodelmaterialcontainingdouble¯awsconductedbyReyesandEinstein[6]showedthatpre-existingnon-persistent¯awscoalesceintwodi erentmodes:(1)ifthepre-existing¯awsoverlap,thecoalescenceoccursthroughinterconnectionofthedevelopingwingcracks;(2)ifthepre-existing¯awsdonotoverlap,coalescenceoccursthroughsecondarycracks.FurtherexperimentalworkonthistopichasbeencarriedoutbyBobetandEinstein[7],andShenetal.[8].Experimentswithsamplescontainingmultiplepre-existing¯awswerecarriedoutbyNemat-NasserandHorii[3].Itwasshownthatforacertainoverallorientationofthe¯awsthegrowthoftheout-of-planecracksmaybecomeunstable,leadingtopossiblemacroscopicfaulting.Althoughfracturemechanicsprovidesafundamentalbasisinunderstandingcrackbehavior,theuseoffracturemechanicstodescribethepropagationandcoalescenceofmultiplecracksingeomaterialsisarduous.Asamatteroffact,muchofthetheoreticalstudyoffractureisfocusedonanindividualcrack.Althoughthisapproachismostsuccessfulwhenfractureoccursbythepropagationofasinglecrack,forrockmechanicsorcivilengineeringandgeophysicalproblems,theanalysisofmaterialfailuremodescharacterizedbymultiplecrackingeventscanbetreatedthroughdi erentapproaches.Severalnumericalmodelshaveemergedasusefultoolstosimulatefailurebymultiplecrackingeventsandtostudythegeneralbehaviorsofbrittlefractures[3,6,9±18].Forexample,bycombiningasmearedcrack/damagemechanicsapproachwithastrain-basedfailurecriterion,coalescencethroughsecondarycrackswasanalyzedbyReyesandEinstein[6].SimilarworkwasperformedbyShenandStephansson[10]byusingtheDDM(discontinuousdeformationmethod)modelwithamodi®edG-criterion.IntheworkofNemat-NasserandHorri[3],aclosedformsolutionwasobtainedforaregularsetof¯aws,assumingthatthepre-existingstraight¯terontheyimprovedtheirmodeltoconsiderthepossiblecurvedbranchingpath[4].ZaitsevandWittmann[16]havepublishedapaperoncrackpropagationinthespecimenwithacompressiveloadbutwithoutconsiderationoftheinteractionamongthedistributedcracks.WithintheframeworkofBEMandDDM,crackpropagation,interactionandcoalescence,andthesizee ectsonstrengthforbrittlespecimenhavebeenstudiedbyCarpinterietal.[17].AdamagemechanicsmodelhasbeenusedbyHanandSwoboda[18]tosimulateasetof¯awsdistributedregularlywiththesameorientation.Inthepresentpaper,twoparticularcasesconcerningcrackpropagation,interactionandcoalescenceinbrittlematerialsarestudiedwithanumericaltoolcalledtherockfailureprocessanalysiscode,RFPA2D,developedrecentlybyTang[19].Thepurposeofthisapproachistoaccentuatethefundamentallydi erentmechanismswhichmaybeinvolvedinfailureeitherintensionorshearmode,dependingonthecon®nementconditions.MotivatedbytheobservationsofthemodelexperimentsconductedbyHoriiandNemat-Nasser[4],numericalmodelsamplescontainingpre-existingmultiple¯awssimilartothesetupintheirexperimentshavebeenadopted.The¯awsmaybepre-existingcracks,orsomeirregularinhomogeneitiessuchasweakinclusions.Fromthemathematicalpointofview,thisrepresentsaratherdi cultprobleminnumericalmodeling,whichhasnotbeendealtwithbefore.Thesecondsimulated

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casedealswithfailuremodeinasamplecontaininginhomogeneitiesonagrainscale.ThesimulationsshowninthispaperseemtodemonstratethatRFPA2Disapowerfulnumericaltechniqueforsolvingthiskindofproblem,whichinvolvescrackinitiation,propagation,interactionandcoalescenceinbrittlematerialswithpre-existing¯awsorotherinhomogeneities.However,moreworkhastobedoneforageneralconclusion.

2.BriefdescriptionofRFPA2Dandthemodelsetup

RFPA2Dcode[19],developedattheCenterforRockburstsandInducedSeismicityResearch(CRISR),NortheasternUniversity,People'sRepublicofChina,canbeusedtomodeltheobservedevolutionofdamageorcrackinitiation,propagationandcoalescenceinbrittlematerialsbyallowingthelinearelasticelementstofailinabrittlemanner.Themethodhasbeenusedformodelingprogressivefailureandassociatedseismicitiesinbrittlerock[20,21]andpillarsinundergroundmining[22].Insteadofusingafracturemechanicsapproachwherefracturepropagationiscontrolledbythefracturetoughnessandisrelatedtoastressintensityfactorattheadvancingcracktip,afailureapproachisadoptedinthecode,RFPA2D,wheremicrofracturingoccurswhenthestresslevelinanelementsatis®esacertainstrengthcriterion[19].Inthepresentinvestigation,aCoulombcriterionenvelopewithatensilecut-o [23]isusedsothattheelementsmayfaileitherinshearorintension.Thesimulationproceedsasfollows.Anexternaldisplacementinonedirectionofthesample(stresscon®nementmaybeappliedintheperpendiculardirection)isappliedandthestressanddeformationineachelementarethencomputed.Theexternaldisplacementintheloadingdirectionisthengraduallyincreasedstepbystep.Whenstressesinsomeelementsatisfythestrengthcriterion,theelementisdamagedeitherinshearorintensionandbecomesaweakelement.Althoughthefailedelementisgivenaverylowelasticmodulus,itisnotremovedfromthemesh,whileinmanylatticemodelsthespringorbeamelementsthatexceedcertainthresholdstrengthareremovedfromthemesh[12].Thestressanddeformationdistributionthroughoutthesamplearethenadjustedinstantaneouslyaftereachelementrupturetoreachanequilibriumstate.Atpositionswithincreasedstressduetostressredistribution,thestressmayexceedthecriticalvalueandfurtherrupturesarecaused.Theprocessisrepeateduntilnofailureelementsarepresent.Moreexternaldisplacementisthenapplied.Inthispaperwe®rstconsideranumericalsamplecontainingpre-existingcrack-like¯awsforsimulatingthefractureprocess,whichmaybehelpfulinunderstandingmacroscopicshearfailureduetothecoalescenceofthetensilecracks.Themodel,asshowninFig.1(a),containselevensmall¯aws(8.6mmforeach¯awinlength)andeightlarge¯aws(28.8mminlengthforeach),withthesmall¯awsarrangedinarow.Toensureclarityinthefollowingdiscussions,wemarktheeightlarge¯awswiththenumbersfrom1to8.Themeshforthemodelconsistsof240Â120=28,800elementswithageometryof200Â100mminsize.Alltheelementshavethesamesizeinscale(squareinshape).Anaxialcompressionisappliedtothesamplewithdi erentcon®ningconditions(compression,tensionandnocon®nement).Theresultsarevalidatedbycomparingthemwithsomepublishedexperimentalresultsobtainedforasimilar¯awgeometrysetup.

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Fig.1.Thenumericalmodelsforthesimulations:(a)Modelcontainingarowofsmall¯awsandseverallarger¯aws;and(b)modelcontainingrandomlydistributedinhomogeneities.

Then,anumericalsample[asshowninFig.1(b)]representingamorerealisticbrittlematerialcontaininginhomogeneitiesingrainscaleisusedtosimulatethecrackinitiation,propagationandcoalescence.Twotypesofheterogeneity(mesoscopicandmicroscopicheterogeneity)ofmaterialpropertiesareconsideredinthesample.Themesoscopicheterogeneityrepresentsthevariationofthesize,shape,mechanicalproperties(suchasstrength,Young'smodulusandPoisson'sratio)andthevolumepercentageofthegrainsorinclusions.Themicroscopicheterogeneityrepresentsthevariationofthemechanicalpropertiesfortheelementscomposingthegrainsortheinclusions.Inthismodel,thesizeandtheshapeofthegrainsandinclusionsaredesignedarbitrarilytosimulatethephysicalmicrostructureofgeomaterials.Everygrainorinclusionhasitsownphysicalpropertiesandgeometricproperties.ThedistributionofthestrengthandelasticityparametersoftheelementsforeachgrainorinclusionfollowsaWeibull'sdistributionlawwitharandomspatialdistribution.Thedistributionisde®nedbytwoparameters,i.e.onematerialparameterandonehomogeneityindex.Theformerrepresentseitherstrength,orYoung'smodulus,orPoisson'sratio,whicharerelatedtotheexpectionvaluesoftheindividualparameteroftherockelementsinthemodel.Thelattercontrolstheshapeofthedistributionfunctionrelevanttothedegreeofthematerialheterogeneity.ThemeshforthemodelinFig.1(b)is140Â70=9800elements(200Â100mm).Thedi erentgreyscalesrepresentvaluesoftheYoung'smodulusofelementsordi erentvaluesofthenormalizedshearstressoftheelements,wheretheshearstressisnormalized

by

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thestrengthparameterofthecorrespondingelement.Thebrighterthecoloris,thehigherwillbethevalueoftheelasticmodulusorthenormalizedshearstress.

3.Numericalresultsanddiscussions

ThesimulationresultsrelatedtothemodelinFig.1(a)foraxialcompressionwithoutcon®ningstressareshowninFig.2.Itisshownthat,underuniaxialcompression,thelarger¯awsnucleatedcracks®rst.AsshowninstagebinFig.2,wing-cracksinitiatedfromthetipofthe2nd¯aw,thenfrom3rd,6thand7th¯awsinstagec.Thesecracksgrewoutoftheirownplanesinastablemanner(staged),buttherateofgrowthincreaseddramaticallyafterasuitablelengthwasattained(stagee).Inthisstage,thecrackinteractionandcoalescenceoccurredbetweenthe3rdand4th¯awsandbetweenthe5th,6thandthe8th¯aws.Thecoalescencebetweenthe3rdand4th¯awsandbetween5thand6th¯awswascausedbythetensionfailure,whereasthecoalescencebetweenthe6thand8th¯awswascausedbyshearfailure.Thecrackpropagationandcoalescence®nallyledtoanunstablefailurebyaxialsplittingasshowninstageg-1tog-3,whilemanyofthesmall¯awsdidnotevennucleateanywing-cracks(stageg-1).Itisworthnotingthatthefractureprocessfromg-1tog-3inFig.2waswithinonestep,whichmeansthatnoextraexternalenergywasneededforthisprocess.Althoughtheoveralldirectionofthecrackpropagationtendedtobeparalleltotheloadingdirection,somedetailsofthecrackpropagationpathsareinterestingtonote.ItcanbeseenfromstageeinFig.2that,afterthecoalescenceofthe5thand6th¯aws,thewing-crackfromthetipof5th¯awcontinuedtopropagate,curvedtowardsanearbysmall¯aw(stagef)andwentthroughthe¯aw,stillbykeepingitsoriginalpropagationdirection(stageg-1tog-3).Asimilarprocesswasobservedforthecrackinitiatedfromthetipofthe4th¯aw.Itisfoundthatgenerallythecracksinitiatedfromthe¯awtipspropagatedina®naldirectionparalleltotheaxialload,butsometimesthepathsofsomecracksdeviatedduetoanearby¯aw(seethe1st,4thandthe5th¯awinstageg).Infact,thegrowthofthewing-crackbecameunboundedwhenweappliedasmalltensilestressnormaltotheaxialcompressiondirection.Whentheaxialloadreached8.1MPa,thecracksinitiatedatthetipsofthe3rd¯awsuddenlycoalescedwiththe1stand4th¯awsandpropagatedinanunstablemanner,joiningthesmall¯awsandleadingtoaxialsplitting(seestagee-1toe-5inFig.3).Theresultsrevealthattheaxialcompressionrequiredtoinduceunstablegrowthofcrackextensionsdecreasesdramaticallyasaslightlateraltensionisapplied.InuniaxialloadingconditionasshowninFig.2,thecorrespondingloadtoinitiateaxialsplittingfailureis16.3MPa,whichistwicetheloadinthecasewiththelateraltension(Fig.3).Theseresultsalsosupportmanyresearchers'explanationstotheexperimentallyobservedphenomenonofaxialsplittingundercompressionofrocksamplesthatisbelievedtobetheresultoftensioncracksdevelopedatthetipsofthepre-existing¯aws[24,25].Thesituationchangedcompletelywhenlateralcompressivestresswaspresent.Forthenumericalsamplewithexactlythesamegeometryofthepre-existing¯awsasinFigs.2and3,anaxialcompressionwithalateralpressureof6Mpawasapplied.AsshowninFig.4,atthebeginning,asinthecaseoftheuncon®nedsample,larger¯awsnucleatedwing-cracks(stagedinFig.4).However,thesecracksweresoonarrested,andsomeofthesmaller¯awsnucleated

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Fig.2.Crackpropagationandcoalescenceinbrittlerockunderaxialcompressionwithoutcon®nement(simulatedwithRFPA2D).

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Fig.3.Crackpropagationandcoalescenceinbrittlerockunderaxialcompressionwithalateraltensionof1MPa(simulatedwithRFPA2D).

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Fig.4.Crackpropagationandcoalescenceinbrittlerockunderaxialcompressionwithacon®nementof6MPa(simulatedwithRFPA2D).

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Fig.5.Crackpropagationandcoalescenceinbrittlerockcontaininginhomogeneitiesunderuniaxialcompression(simulatedwithRFPA2D).(Di erentgreyscalesrepresentdi erentvaluesofYoung'smodulus).

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Fig.6.Crackpropagationandcoalescenceinbrittlerockcontaininginhomogeneitiesunderuniaxialcompression(simulatedwithRFPA2D).(Di erentgreyscalesrepresentdi erentvaluesofnormalisedshearstress).

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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

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Fig.7.Stress±strainrelationforbrittlerockcontaininginhomogeneitiesunderuniaxialcompression(simulatedwithRFPA2D).

process.Thefailuremodeseemstoindicatethatthefailureinaxialcompressionofbrittlematerialsisthe®nalmanifestationofearliertensilecrackgrowthinducedunderoverallcompression.

4.Conclusions

Twoparticularcasesconcerningcrackpropagationandcoalescenceinbrittlematerialsaremodeledbyusingarockfailureprocessanalysiscode,RFPA2D.Numericalsimulationsreproducedqualitativelythegeneralobservationsmadeinlaboratoryexperiments.Themostsigni®cantresultthatemergedfromthenumericalsimulationisthee ectoflateralstressonthefailuremechanisms.Thenumericalsimulationsofsampleswithsuitablyoriented,pre-existing¯awsunderaxialcompressionshowthatmicrocracksmaynucleateattipsofthe¯awsandgrowinthedirectionoftheaxialload.However,thepresenceofslightlateraltensionresultsinout-of-plane,curvedcrackgrowthinanunstablemannerwhenasuitablecriticallengthwasattained.Inthatcase,thecracksgrowalmostspontaneouslyandwithoutanincreaseintheaxialcompression,leadingtothecollapseofthesampleinthemannerofaxialsplitting.Ontheotherhand,nounstablegrowthoccursifalateralcon®ningpressureisappliedtothesample.Asamatteroffact,thecracksinitiatedfromthe¯awtipsstoppropagationafterreachinga®nitelength.Itiscon®rmedthat,underaxialcompression,thenucleation,growth,interactionandcoalescenceofmicrocracksarethedominantcontrollingsourcesthatleadstomacroscopicfailureofrocks.However,thecoalescenceofthewing-cracksmaybeineithertensilemodeorshearmode,oracombinationofbothmodes.Thenumericalresultsshowqualitativelyareasonablygoodagreementwithreportedexperimentalobservationsforthesampleswithsimilar¯awarrangements,andsupporttheexperimentallyinferredconclusionthatbothsplittingandshearfailuresoftenobservedinexperimentsinaxialcompressionofrocksandotherbrittlematerialsarethe®nalmanifestationofearliertensilecrackgrowthinducedunderoverallcompression.TheuseofRFPA2Dtosimulateamorerealisticmaterialcontaininginhomogeneitiesongrainscaleshowsthatthefailuremodestronglydependsonthemechanicalandgeometricpropertiesofthegrainsandinclusions.Theresultagainshowsthatthefailureof

brittle

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materialsinaxialcompressionisthe®nalmanifestationofearliertensilecrackgrowthinducedunderoverallcompression.

Acknowledgements

TheworkreportedinthispaperwasaccomplishedwhileProfessorTangoftheNortheasternUniversityofTechnology,University,thePeople'sRepublicofChina,wasaguestofLuleaSweden,anditformspartofaresearchcollaboration.ThiscollaborationwasmadepossiblebypartialfundingfromNationalNaturalScienceFoundation,People'sRepublicofChina(no.UniversityofTechnology,Sweden,andtheSwedishNuclearFuel59472018)andfromLuleaandWasteManagementCo.(SKB).

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