Mesoscale models for concretehomogenisation and damage behaviour.docx

Mesoscale models for concretehomogenisation and damage behaviour.docx

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Mesoscalemodelsforconcretehomogenisationanddamagebehaviour

FiniteElementsinAnalysisandDesign42（2006）623–636

Mesoscalemodelsforconcrete:

Homogenisationanddamagebehaviour

P.Wriggers∗,S.O.Moftah

InstituteforMechanicsandComputationalMechanics,UniversityofHannover,Hannover,Germany

Received27October2005;receivedinrevisedform27October2005;accepted20November2005

Availableonline9January2006

Abstract

Inthispaperthree-dimensionalgeometricalmodelsforconcretearegeneratedtakingtherandomstructureofaggregatesatthemesoscopiclevelintoconsideration.ThegenerationprocessisbaseduponMonteCarlo’ssimulationmethodwhereintheaggregateparticlesaregeneratedfromacertainaggregatesizedistributionandthenplacedintotheconcretespecimeninsuchawaythatthereisnointersectionbetweentheparticles.Forhighvolumefractionsofaggregates,newalgorithmsforgeneratingrealisticconcretemodelsareproposed.

Thegeneratedgeometricalmodelsarethenmeshedusingthealignedapproachinwhichthefiniteelementboundariesarecoincidentwithmaterialsinterfacesandthereforetherearenomaterialdiscontinuitieswithintheelements.

Thefiniteelementmethod（FEM）isusedinthedirectcomputationoftheeffectivepropertiesofconcrete.Theresultsobtainedfromthenumericalsimulationsandthesubsequenthomogenisationarethencomparedwithexperimentaldata.Furthermorenumericalsimulationsofthedamageandfractureprocessofconcreteareperformedusinganisotropicdamagemodeltomodeltheprogressivedegradationofconcrete.Finally,aconcreteblockisinvestigatedwherenumericalandexperimentalresultsarediscussed.

©2005ElsevierB.V.Allrightsreserved.

Keywords:

Finiteelement;Homogenisation;Sievecurves;Concrete;Damage

1.Introduction

Concreteisthemostwidelyusedconstructionmaterialintheworldduetoitsgoodstrengthanddurabilityrelativetoitscost.Ithasagreatvarietyofapplicationsinthefieldofcivilengineering.Inbuildingconstruction,concreteisusedforfootings,foundations,columns,beams,girders,wallslabs,andallimportantbuildingelements.

Concretehasahighlyheterogeneousmicrostructureanditscompositebehaviourisexceedinglycomplex.Therefore,reli-ablepredictionsofthebehaviourofthematerialbasedexclu-sivelyonexperimentalstudieshavebecomelimited.Forob-tainingadeeperunderstanding,theoreticalstudiesbasedonmicromechanicsanalysisoftheinteractionbetweenvariouscomponentsofconcretehavebeendevelopedfordeducingthemacroscopicconstitutivebehaviourofconcrete.However,themicrostructureandpropertiesoftheindividualcomponents

∗Correspondingauthor.Tel.:

+495117622220.

E-mailaddress:

wriggers@ibnm.uni-hannover.de（P.Wriggers）.

0168-874X/$-seefrontmatter©2005ElsevierB.V.Allrightsreserved.doi:

10.1016/j.finel.2005.11.008

ofconcreteandtheireffectsonthemacroscopicmaterialbe-haviourhavenotbeentakenintoaccount.Forsuchdetailstobeincludedintothecomputationalanalysis,concreteneedstobeanalysedasamulti-scalecompositematerialwherethemi-crostructureisrealisticallysimulated.Numericalsimulations,coupledwiththeoryandexperiment,areconsideredtobeanextremelyimportanttoolforsuccessfullyexaminingmaterialpropertiesbymeansofcomputationalmaterialsscience.

Inthenumericalsimulationofconcreteatamesoscopiclevelitisevidentthatseveralparameterssuchastheshape,sizeanddistributionofcoarseaggregateswithinthemortarmatrixsig-nificantlyinfluencethemechanicalbehaviourofconcrete.Theaimofthisworkistogeneraterandommesostructuremodelsoftheconcretematerialatthemesoscopiclevelbasedupongivenparametersandprobabilitydistributionofaggregatepar-ticlescoordinatesandsizes.Then,thegeneratedmesostructuremodelsaretobemeshedusinganautomaticmeshingprepro-cessorforcomputationaltestingofthematerialwiththefiniteelementmethod（FEM）inordertoanalysetheeffectsofthemesostructureconstituentsonthemacroscopicresponseoftheconcretematerial.

Severalmesoscopicmodelsofconcretestructurehavebeendevelopedforstudyingtheinfluenceofmaterialcompositionontheoverallbehaviour.Bazantetal.[1]developedatrussmodeltosimulaterealisticallythespreadofcrackinganditslocalisation.SchornandRode[2]studieddamageprocessesofconcreteusingaframeworkmodel.AlatticemodelpresentedbySchlangenandvanMier[3]seemedtobeapromisingtoolforthesimulationoftypicalfailuremechanismandcrackfacebridginginconcrete.Anotherapproachforsimulatingthestruc-tureofconcretebyafiniteelementmeshhasbeendevelopedbyWittmannandhisco-workers[4–6].Inthisapproach,me-chanicalandnonmechanicalpropertiescanbemorerealisti-callysimulatedforconcretewithdifferentcompositions.Wangetal.[7]proposedaprocedureforgeneratingrandomaggregatestructuresbasedontheMonteCarlorandomsamplingprincipleandthendevelopedamethodofmeshgenerationforstudyingthenonlinearbehaviourofconcrete.

Inthegenerationofrandomconcretestructurestheshapeofaggregateparticleshastobetakenintoaccountinordertostudytheeffectofaggregateshapesuponthemechanicalbe-haviourofconcrete.Aggregateshapeshaveasignificantinflu-enceonthestressdistributionwithintheconcretematerialand,therefore,onthecracksinitiationanddamageaccumulationuptothemacroscopicfailure.Two-dimensionalmodelsweregen-eratedbyZaitsevandWittmann[8]tostudycrackpatternsinconcreteusingpolygonalandcircularshapesforaggregates.Wangetal.[7]andWittmanetal.[5]havegeneratedroundedaggregateparticlesfollowingthemorphologicallawdevelopedbyBeddowandMeloy[9].Forangularshapesofaggregateparticles,Wangetal.[7]adjustedshapesofrandomlygener-atedpolygonstohaveprescribedelongationratios.However,forthree-dimensionalmodels,Bazantetal.[1],GuidoumandNavi[10],SchlangenandvanMier[3]andSchornandRode[2]justassumedthataggregateparticleshavesphericalshapes.Morerecently,Häfneretal.[11],Leiteetal.[12]andZohdi[13]haveusedellipsoidfunctionswithvaryingparametersforobtainingvariousaggregateshapes.Garboczi[14]describedamathematicalprocedureusingsphericalharmonicfunctionstocompletelycharacteriseconcreteaggregateparticlesbasedonthree-dimensionalimagesacquiredviaX-raytomography.

Aggregatesizedistributionplaysanessentialroleinconcretemixdesignandoptimisation.Aproperselectionofaggregatesizedistributionaffectsthemainpropertiesofconcretesuchasworkabilityofconcretemix,mechanicalstrength,permeabilityanddurability.Thesizedistributionofaggregateparticlescanbeeitherdescribedbymeansofgradingcurvesorobtainedfromsieveanalysis.Fullercurveisoneofthemostacceptablegradingcurveswhichprovideoptimalaggregatepackingandthebestpropertiesofconcrete.Consequently,mostresearchersincludingSchlangen[15],SchlangenandvanMier[3],vanMieretal.[16]andWittmannetal.[6]appliedtheFullercurvetogeometricalmodellingofconcrete.

Inregardtothesimulationofaggregatespatialdistribution,differenttechniqueshavebeendevelopedtogenerateameso-scopicstructurethatresemblestherealconcrete.Bazantetal.[1],SchlangenandvanMier[3],Wittmannetal.[5]andWangetal.[7]adoptedthetake-and-placemethodforgeneratingran-

domparticlemodelsoflowparticlevolumefractions.DeSchut-terandTaerwe[17]usedanotherapproachforthegenerationofrandomconcretestructuresbasedonthedivide-and-fillmethod.Astochastic-heuristicalgorithmwasdevelopedbyLeiteetal.[12]forgeneratingamorerealisticthree-dimensionalstructureofconcrete.Forachievingveryhighaggregatevolumefrac-tion,vanMierandvanVliet[18]usedanalternativealgorithmcalledtherandomparticledropmethodforarrangementofag-gregateparticles.Adifferentapproachcalledthedistinctele-mentmethodwasdevelopedandextensivelyappliedbyCun-dallandStrack[19]forsimulatingthebehaviourofgranularsolidssuchassand.

NumericalanalysisofheterogeneousmaterialsusingtheFEMrequiresthediscretisationofthecreatedmesoscopicmodels.Differentmeshingtechniqueshavebeenappliedforthediscretisationofcomplexmicrostructures.Alignedmesh-ingapproacheshavetheadvantageofexplicitlyrepresentingtheboundariesbetweenparticlesandmatrix.However,themeshgenerationofrandomheterogeneousmaterialsisratheratediousprocessinthreedimensions.Thealignedapproachhasbeenusedbyseveralauthors,e.g.[17,20–22,7,6]forgenerat-ingfiniteelementmeshesintwo-dimensionalcases.Inrecentyears,ithasbeenrecognisedthattheinterfacialtransitionzone（ITZ）betweenaggregatesandmortarhasagreatinfluenceontheinitiationofmicrocracksinconcrete.Wangetal.[7]developedameshgenerationmethodbasedontheadvancingfrontapproachinwhichtheITZdomainsaremodelledusingGoodmantypeelements.However,Eckardtetal.[20]disre-gardedtheITZandassumedthatthebondbetweenaggregatesandmortarisrigid.

SchlangenandvanMier[3]andSchornandRode[2]usedaprojectionmethodofaregularmeshontotherandomaggregatestructure.InthemodelsofvanMierandvanVliet[18],vanMieretal.[16],Schlangen[15]andSchlangenandvanMier[3]theaggregates,matrixandITZpropertiesareassignedtoalatticeofbeamelements.AsimilartechniquehasbeenappliedbyLeiteetal.[12]inwhichthemodelsareidealisedaspla-nartrussesandframesfortwo-dime