Crack propagation and coalescence(3)

RFPA

C.A.Tang,S.Q.Kou/EngineeringFractureMechanics61(1998)311±324313

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.