Crack propagation and coalescence(4)

RFPA

314C.A.Tang,S.Q.Kou/EngineeringFractureMechanics61(1998)311±324

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