Importancia de la deformación por fluencia lenta y la difusión en gel de la reacción de álcali-sílice (RAS) en la predicción de la expansión provocada por la RAS

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Availableonlineat

ScienceDirect

www.sciencedirect.com

www.e-ache.com HormigónyAcero2018;69(S1):3–14 www.elsevierciencia.com/hya

Importance

of

creep

and

ASR

gel

diffusion

in

predicting

ASR

induced

expansion

夽

Importancia

de

la

deformación

por

ﬂuencia

lenta

y

la

difusión

en

gel

de

la

reacción

de

álcali-sílice

(RAS)

en

la

predicción

de

la

expansión

provocada

por

la

RAS

Saeed

Rahimi-Aghdam

a

,

Zdenˇek

P. Baˇzant

b,∗

a_Ph.D._candidate,_Department_of_Civil_and_{Environmental}_Engineering,_Northwestern_University,₂₁₄₅_Sheridan_Road,_CEE/A135,_Evanston,_IL_60208,_USA b_{Distinguished}_McCormick_Institute_Professor_and_W.P._Murphy_Professor,_Departments_of_Civil,_Mechanical_and_Materials_Science_Engineering,_Northwestern

University,2145SheridanRoad,CEE/A135,Evanston,IL60208,USA

Received27November2017;accepted23May2018

Availableonline30July2018

Abstract

Thepaperreviewsdevelopmentofadiffusion-basedandcreep-basedmodelforcalculatingtheevolutionofexpansionanddamageinducedby alkali-silicareaction(ASR).First,themodelofBaˇzantandSteffens(2000)isadoptedtocalculatetherateofproductionoftheASRgelwithinthe aggregate.Next,anon-lineardiffusionaccordingtoRahimi-AghdamandBaˇzant(2017)ispresentedtomodelthepenetrationofASRgelintothe micropores,nanoporesandmicrocracks.Thegeldiffusionintoporesoftheaggregatecausesexpansionanddamagetothesurroundingconcrete, andisfoundtobeanimportantmodelingaspect.ThedamageisassessedbymicroplanemodelM7,intowhichthecreepisincorporated.The creepisfoundtohaveasignificantinfluenceonthelong-termevolutionofASR-induceddamage.Thepredictionsareingoodagreementswith thelaboratoryexperimentsandthemodelappearstobereadytopredicttheASReffectsinrealstructures.

Keywords:Alkali-silicareaction;Swelling;Damage;Microplanemodel;Finiteelement

Resumen

Elartículoanalizaeldesarrollodeunmodelobasadoenladifusiónyenladeformaciónporfluencialentaparacalcularlaevolucióndelaexpansión yeldañoprovocadoporlareaccióndeálcali-sílice(RAS).Primero,seadoptaelmodelodeBaˇzantySteffens(2000)paracalcularlatasade produccióndegelparalaRASdentrodelconglomerado.Acontinuación,sepresentaunadifusiónnolinealsegúnRahimi-AghdamyBaˇzant(2017) paramodelarlapenetracióndelgelparalaRASenlosmicroporos,nanoporosymicrofisuras.Ladifusióndelgelenlosporosdelconglomerado provocaexpansiónydañoalhormigóncercano,yseencuentraqueesunaspectoimportantedemodelado.Eldañosevaloramedianteelmodelo demicroplanosM7,alcualseincorporaladeformaciónporfluencialenta.Secreequeestatieneunainfluenciaconsiderablesobrelaevolucióna largoplazodeldañoprovocadoporlaRAS.Lasprediccionessoncoherentesconlosensayosdelaboratorioyparecequeelmodeloestápreparado parapredecirlosefectosdelaRASenestructurasreales.

Palabrasclave: Reaccióndeálcali-sílice;Hinchamiento;Da˜no;Modelodemicroplanos;Elementofinito

夽 _Written_in_honor_of_Dr._Carmen_Andrade,_retiring_Director_of_CSIC,_Toroja

Institute,andPastPresident,RILEM.

∗_{Corresponding}_author.

E-mailaddress:[email protected](Z.P.Baˇzant).

https://doi.org/10.1016/j.hya.2018.05.006

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1. Introduction

The alkali-silicareaction(ASR; akaalkali-aggregate reac-tion,AAR)attacks mineralaggregatesinconcretewhen they contain imperfectly crystalline silica. The reaction produces a gel that can imbibe enormous amount of water and cause swelling.Theinducedswelling,progressingformanymonths, yearsorevendecades,oftencausessignificantstrength degra-dation and damage in concrete structures. Because drying diminishestherate ofASR,the worstdamageusually occurs in massive structures such as large bridges and dams, and nuclearpowerplantstructuresinwhichthecross-sectioncore remainsundriedforyears,evendecades.Asaresult,preventing theASR-induceddamageisoneimportantgoalofsustainable design.

TheASR-induceddamagewasfirstidentifiedbyStantonin 1942 [1].Since that timemany researchers have worked on thisproblemandavastbodyof literature hasbeengathered. Acomprehensiveliteraturereviewhasrecentlybeenpublished bySaoumaandXi[2,3].

Althoughseveralmodelshavebeenproposedtopredictthe ASR-induced damage, they considerasimplistic constitutive lawforconcrete.AnexceptionisarecentmodelAlnaggarand Cusatis[4]whichusesCusatis’latticediscreteparticlemodel, andintheparticlecontactsanexcellentmaterialmodelfor con-crete.However,allthesemodelsdonotconsiderthediffusion ofASRgelintoporesandcracks,andallexceptAlnaggarand Cusatis’neglect the effectof creep.Both phenomena signifi-cantlymitigatetheASR-induceddamage.

The currentmodels, except[4],assume thatthe only phe-nomenonthatcontrolstherateofASR-inducedswellingisthe productionof ASRgel anditswater imbibition.In modeling the diminishing rate andstoppage of ASR-induced swelling, which happen after a few months in accelerated tests, these models consider the ASR gel to reach the maximum capac-ityof waterimbibition relativelyfastandwaterimbibition to stop after that. This assumption contradicts the experimental resultsshowingthattheASRgelcanimbibeavastamountof water,even100-timesits initialvolume.The onlyreasonable justificationforthediminishingandstoppageofASR-induced swellingisthediffusionofASRgelintothesurroundingpores, andalsointothemicrocracksproducedbythediffusioninthe aggregate pieces and surrounding cement mortar. After sig-nificant damage the diffusion rate exceeds the rate of water supplywhichcausestheswellingtoslowdownandeventuallyto stop.

This study reviews the recent development of a compre-hensive model for ASR [7,31] and provides some improved explanations and justifications of its main features, which include:

1. The delay in ASR due toproduction of ASR gel andits diffusionintotheporesandexpandingcracks.

2. Fracturingofthesolidframeworkofconcreteasatwo-phase medium, caused by diffusion of the ASR into pores and cracksintheaggregatesandsurroundingmortar.

3. Thedifferencesincrackingpatternsofthedamageinduced bytheASRunderdifferentconfinementconditions,applied stressstatesandboundaryconditions.

4. TheeffectofcreeponmitigatingtheASRdamage.

5. TheeffectofalkalicontentontheASR-induceddamageand swelling.

6. The effect of temperature on ASR-induced damage and swelling.

Analysis of theseeffects must be coupled witha realistic model for ASR gel production within the reactive aggregate pieces.SuchamodelwasdevelopedbyBaˇzantandSteffens[8] andwasalsousedandimprovedbyAlnaggarandCusatis[4]. Herethismodeliscoupledwiththeanalysisofthe aforemen-tionedphenomena.

Analysis oftheASRgeldiffusionisanessentialaspect of prediction.Anonlineardiffusionmodelhasbeendevelopedand calibratedforthispurpose.TopredicttheASR-induceddamage inlargerstructures,amacroscopiccontinuummodel,endowed withalocalizationlimiterpreventingspuriousinstabilityof soft-eningdamage,isnecessary.Themicroplaneconstitutivedamage model[5,6]ismostrealisticforthispurpose.Itcandistinguish differentdamagepatternsundervariousstressstates.Itslatest version,M7,isadopted.Sincecreepofthesolidframework sig-nificantlymitigatestheASRdamage,aconcretecreepmodelis incorporatedintoM7.

2. ReviewofsimpliﬁedmodelofkineticsofASRgel production

First, weneed tointroduceamodelfor the kineticsof gel production in mineralaggregates. To thisend, the simplified modelof BaˇzantandSteffens[7],asimprovedbyBaˇzantand Rahimi-Aghdam[8,9]andused,inimprovedform,byAlnaggar andCusatis[4],isadopted.

Since the shape of aggregate grains is not too important, sphericalgrainsofdiameterDareconsidered(Fig.1a).TheASR reaction occurs at various randomly located discrete sources insidethegrain.Thosenearthegrainsurfacewillbeactivated firstandthetimetoactivatethedeepersourceswillgrowwith the depth,z.Thus,as anaveragebehaviorofmanyaggregate grains,weintroduceasmearedcontinuummodelinwhichwater diffuses radially into a spherical aggregate grain [8], with a sphericalfrontofradiuszatwhichtheASRreactiontakesplace. TheASRreactionisassumedtooccurinstantlyatthefrontof waterpenetration(Fig.1a).Thereactionissloweddownbythe diffusionofwaterthroughtheaggregatepiece,andparticularly throughthelayerofASRgelalreadyformed.

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Waterdiffusionfront

ITZ

PotentialASRgelsource ActivatedASRgelsource

x z

dz ITZ

D

(i) (ii) (iii)

a ASRgelformationduetowaterdiffusionintoreactiveaggregate:(i)earlystageofdiffusion;(ii)late stageofdiffusion;(iii)idealizationwithsphericaldiffusion

K+

Na+ OH

-OH

-SiO2 SiO2

SiO2 SiO2

SwellingASRgel Non-SwellingASRgel

(i) (ii) (iii)

b ASRreactionprocessandschematicillustrationofASRinduced damage:(i)alkali-silicareaction; (ii)formationofswellingandnon-swellingASRgel;(iii)ASR-induceddamageandcracking

p Non-SwellingASRgel

SwellingASRgel Gelinfiltration Crackproduced

(i)

_(ii)

x

Pressure profile

c ASRgeltransport:(a)ASRgelinfiltrationandproducedglobaldamage;(b)oneidealgel infiltrationpath

Figure1.FormationoftheASRgelanditsdiffusion.

Inthisstudy,thesamesetofequationsasBaˇzantand Rahimi-Aghdam(2017)isusedtocalculatew(t)andthemassofimbibed water, wi(t).It shouldbe noted that we assumethe ASR gel

toimbibewaterforyears,andevendecades.Themainreason forthedecelerationoftheASR-inducedswellingisthe diffu-sionofASRgelintothecracksproducedbyASR(and,often, diminishingofwatersupply).Severalotherstudiesassumedthe ASRgeltoimbibeonlyan assumedlimited amountofwater foronlyanassumed limitedtime, consideringthese phenom-enatodiminishandstopaccordingtoachosenschedule.This simplification disagrees with the experimental results, which showASRgelcanimbibeanalmostunlimitedamountwithout anyparticular timelimitation.Theseoversimplificationswere

necessarybecausethediffusionofASRgelintotheporeswasnot considered.

3. DiffusionofASRGelintosurroundingpores

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EffectiveASRgelvolumefraction EffectiveDarcycoefficient

ASRgelvolumefraction

Pressure profile effectongelpermeabilitymaybeintroducedas

Forsimplicitythepressuregradientcanbereplacedbyaveragepressuregradient

b SincetheASRGelcaninfiltrateinrandomdirections,weconsideredaveragevolumetrictransport instead:

c Theinelasticstrain,

d Azeropressuregradientisconsideredfortheeasilyaccessiblevolume, ν0 .Fortheremainingpart,

ν_0,aparabolicprofileseemsappropriate.

e ASRgelpressureevolution(Exponentialalgorithm)

a

Figure2.CalculatingthediffusionofASRgelandInducedpressureinsidepores.

acontinuumpoint.However,consideringalltheuncertainties and simplifications, it appears sufficient to replace the flow velocity, z˙, with the velocity of the diffusion front, dx/dt, and the pressure gradient withthe average pressure gradient ∇p = p/x. Using these reasonable simplifications, we can writethesimplifiedDarcydiffusionequationasgiveninFig.2a. TheASRgelfirstpenetratesatnear-zeropressurepintothe emptyandeasilyaccessibleporesof volumefraction vo.The easilyaccessibleporesareofthreekinds:(1) poresinsidethe aggregate,thedistributionandsizeofwhichdependsstrongly onthetypeofaggregate;(2)thecementpastepores(sincethe hardenedcementpastealwayscontainsemptypores,dueto self-desiccation);and(3)thebiggerporesintheinterfacetransition zone(ITZ)surroundingtheaggregate,moreeasilyfillabledue toalowermassdensityandhigherporosityinthatzone.

Notethat,inrealstructures,thefillingofemptyporesof vol-umefractionvocantakemanyyearsanddecades.Thisisthe reasonforthegreatdelayintheinitiationoftheASR-induced expansion.Onlyaftertheemptyporesgetfilled,further imbi-bition of watercan produce asignificant pressurein the gel, producecracksandpenetratethem(Fig.1bandc).

Weassumethatoneorseveralbasicgelsourcesformineach reactiveaggregate(whenitcontainsreactivesilica).Fromeach gelsource,thegeldiffusesinmultiplerandomdirections(Fig.1a andc)intonearbypores,microcracksandITZ.Notethatthegel that penetratesinto cementpaste(fartherfromthe aggregate) will calcifyandthus no longer imbibewaterand swell, thus becomingharmless.ThedirectionofASRgeldiffusionis ran-domandthedistancesfromthedifferentgelsourcestothefronts ofdiffusionpathsalsovaryrandomly(Fig.1c).Therefore,onthe

continuumscale,itisreasonabletoconsideravolumetric aver-agediffusioninsteadofadirectionaldiffusion.Thevolumetric diffusionequationcanbewrittenasshownFig.2b,wherepis the averagegelpressure;b isaneffectiveDarcy permeability (dimensionm2/Nsorms/kg),whichisproportionaltotheactual Darcypermeability,bD;andvefisaneffectivevolumefraction. Itisclearthatpermeabilitymustincreaseduetocracking.If wedenotethecrackopeningasδandusethecrackbandmodelas alocalizationlimiter[10,11],thenwehaveδc ≈ lo∈

,where loisamaterialcharacteristiclength,and ∈istheinelasticpart ofaveragetensilestrainacrossthecrackband.

Sincetheinducedcrackscanrunrandomlyinanydirection, expansioninalldirectionsisexpected,i.e.,theinelasticstrain mustbevolumetric,∈_vconsideredtobeafunctionofthe princi-palinelasticstrains.Amongthem,theincreaseofpermeability canbe causedonlybythosetensilestrainsthat exceeda cer-tain empirical finitethreshold, ∈₀ (which ishereassumedto be0.001%).Thus,theeffectofinelasticstrain(damage)ongel permeabilitymaybeintroducedasseenFig.2c.

4. EvolutionofASRgelpressureatconstanttotalgel mass

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pressurepuptovolumev0.Beyondthat,aparabolicpressure

drop(assketchedinFig.2d)ismostrealistic.

Now that we have assumed the self-similar profile in the pores, we can use an explicit computational algorithm to calculatethepressureevolution.However,veryshorttimesteps wouldbeneededtoensurenumericalstabilityandconvergence. Thislimitationbecomesaseriousprobleminsimulating long-termASR.Thesameproblemwasfacedinmodelinglong-term creepofconcretestructures.Thecreepproceedsrapidlyatfirst, whichneedstimestepsintheorderofsecondsatthebeginning. Buttoruncalculationsupto,e.g.,50years,thetimestepsfor creep integrationneed tobe extendedto monthsin duration. Thesame increaseinthe timestep mustpossible heresince, afterseveralyears,thepressureevolutionbecomesveryslow.

Therefore,ananalogoftheunconditionallystable exponen-tial algorithm for creep [12,13] has been developed for gel diffusion.Similartopiece-wisestrainconstancyinthe uncon-ditionally stable exponentialalgorithm for creep [12,13], we assumethemassofgeltobeheldconstantduringeachtimestep Δtandallowthepressuretorelaxduetodiffusion.Attheend ofeach timestep,the pressureandgelmassare correctedby anabruptchange.Fig.2edescribestherelaxationofpressureat theconstantgelmass,withtpplayingtheroleofcharacteristic time.Itshouldbenotedthatthepressurerelaxationequationis nonlinearsincetpdependsonvariablessuchasvolumefractions andpermeability,whichthemselvesdependonpressure. How-ever,sincethesevariablesdonotchangesignificantlyduringthe timestep,tpmaybeconsideredasaconstantduringeachtime stepandcalculatedusingthe valuesfromprevious timestep. Thisallowsthepressurerelaxationequationtobeintegratedby separationofvariables.

5. Constitutivelawforthesolidpartofconcrete andincorporationofcreep

GenerallyinstructuresundergoingASR,theconcrete experi-encescomplexmultiaxialstressstates.Thus,anisotropicdamage shouldbeexpected.Theanisotropicdamageofconcretecanbe describedeffectivelybythemicroplaneconstitutivemodelM7 [5,6].Thebasicideaof microplanemodelistoformulatethe constitutivelawintermsofthevectorsofstressandstrainacting onagenericplaneofanyorientationinthematerial microstruc-ture,calledmicroplane.Theuseofvectors,insteadoftensors, issimilartotheTaylormodels,usedforplasticityof polycrys-talline metals,butthereare majordifferences; especially,the staticconstraintinsteadofakinematiconeisconsidered.Inthis study,thelatestversionofmicroplanemodel,M7,isused.Model M7hasbeendemonstratedtogiverathergoodpredictionsofthe behaviorofquasibrittlematerialsoverabroadrangeofloading conditions.

TheASR-induceddamageinstructuresevolvesoveryears, evenmanydecades.Therefore,creepandshrinkageplayamajor role.Creepandshrinkagecanevenhaveasignificanteffecton relativelyshortlaboratoryexperiments.Todeterminethecreep andshrinkageinundamagedconcrete,RILEMmodelB4[14]is usedinthisstudy.However,theASRdamageinthepresenceof externalloadscausesfracturing.Thus,thecreepandshrinkage

cannotbecalculatedsimplyusingaplaincreepmodel,suchas B4.Itneedstobecalculatedusingacombinationofthecreep modelwiththemicroplanemodelM7.

In undamaged material between the expanding pores and cracks,thecreepisagingviscoelasticandlinearinstress,while thefracturingdamageleadingtocracksdependsonstress non-linearly. Asaresult,the creepstrain canbe consideredtobe additivetofracturing strain.Since the ASR-induced pressure anddamageevolveintimeandinfluencetheconstitutivelaw, therate-typecreepmodel[15]mustbeused,ratherthan hered-itaryintegral-type.Theratetypecreepmodelcanbestructured accordingtotheKelvinorMaxwellchain.Theformerismore convenientsincemodelB4specifiesthecompliance,ratherthan relaxation,function(Fig.3a).

The springanddashpotsmoduli of theKelvinunits inthe chain canbeobtained by discretizingthecontinuous retarda-tionspectrum[16].Duetoaging,thespectrumissignificantly different foreverytimestep andeveryintegrationpoint. The retardationtimesareselectedtomakeageometricprogression withquotient10andmustcoverthewholetimerangeof inter-est(Fig.3a).Timeintegrationallowinganarbitraryincreaseof thetimestepsascreepslowsdownneedstousetheexponential algorithm[12],whichisunconditionallystable.Onthetensorial level,theexponentialalgorithmleadstotheincremental stress-strainrelationasshowninFig.3b,inwhich, ∈cr_ij isthecreep strainincrementtensorthatiscalculatedfromBazant’s expo-nentialalgorithmforKelvinchainmodel; ¯Eistheincremental moduluswhichisobtainedfromtheexponentialalgorithm;εsh istheshrinkagestrainincrement,andαΔT isthethermalstrain. To combine M7 with creep, all the stress predictions in the microplanemodelarewritteninaformanalogoustotheequation inFig.3b.

Itshouldbementionedthatinthisstudy,forsimplicity,model B4,whichgivesaveragecreepforcrosssection,isused. How-ever,duetodamageandnon-uniformityofstressitisbeneficial tousepointwisecreeplawssuchasXMPSmodel[17].

6. Two-phasemediumforloadingofconcrete bypressureinporesandcracks

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Aging creep: (a) Kelvin chain; (b) continuous retardation spectrum; (c) discrete retardation spectrum

On the tensorial level, exponential algorithm (Bažant 1971) leads to the incremental stress-strain relation

Effective young modulus from exponential algorithm

Creep strain

Shrinkage strain Temperature strain Two- Phase Medium for loading of concrete by pressure in pores and cracks

Considering aggregate size (small and big), mass of imbibed water,wi

Total stress in the two phase medium stress in solid phase

Volumetric stress in the two phase medium Volumetric stress in solid phase

Effective porosity

size of small and big aggregate size.

= masses of imbibed water for aggregate sizes D_s and D_b

Weights

a)

b)

c)

d)

Figure3.Incorporationofcreepandtwo-phasemedium.

ofevaporablewateriscontainedinthenanopores(onlyafew atomswide).Thiswaterbehavespartlyasaload-bearingsolid. Therefore, the Biot medium relations need to be adapted forconcrete.BaˇzantandRahimi-Aghdam(2017)modifiedBiot relationsbyusingeffectiveporosityφinsteadofBiotcoefficient. Thesamemodificationisusedhere.Usingeffectiveporosityφ insteadofBiotcoefficientα,theincrementalvolumetricand ten-sorialtwo-phaseequilibriumrelationscanbewrittenasFig.3c.

7. ModelingASRforvariousaggregatesizes

Forsimplicity,alltheprecedingequationsinthisstudyare formulatedforoneequivalentaggregatesize.Butinreality,the

sizes ofreactiveaggregatesare distributedstatisticallyovera broadrange.

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-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

0 100 200 300 400 500

AxialSt

rain(%)

Time(day)

Unconfined,0MPa Multon etal.(2006)

-0.02 0 0.02 0.04 0.06 0.08

0 100 200 300 400 500

RadialSt

rain(%)

Time(day)

-0.04 0 0.04 0.08 0.12 0.16 0.2 0.24

0 100 200 300 400 500

Re

la

ti

ve

V

olum

e

chan

ge

(%)

Time(day)

a)CalibrationofMoltonetal.(2006)experimentsforfreeexpansioncondition

b)StressStatesforMulton etal.(2006)

Confinement:

1. Unconfined.

2. Confinedwith3mmsteelring. 3. Confinedwith5mmsteelring.

i)Unconfined ii)Confined

Loading:

1. Noexternalloading.

2. 10MPacompressiveaxialload. 3. 20MPacompressiveaxialload.

Steelring

Figure4.CalibrationofMultonetal.experimentandstressstates.

thetotalmassof imbibed waterwi iscalculatedas shownin Fig.3d.

8. EffectofvariousstressstatesonASR-induced expansion

Severalnumerical andexperimental studiesshowed a sig-nificant effect of applied stress state and confinement on ASR-induced expansion and damage [4,19–22]. Particularly, experiments showed that the effect of loading on the ASR-inducedvolumeexpansionisminimalandusuallytheapplied stress changes onlythe direction of expansion. For instance, largecompressivestressinonedirectiontransferstheexpansion tootherdirectionswithsmallerornocompression.Toanalyze theabilityofthe proposedmodel topredict theASR-induced expansionanddeteriorationatdifferentstressstates,weconsider somepublishedexperimentaldata.Firstweanalyzethe accel-eratedlaboratorytestsofMultonandToutlemonde(2006).The durationofthesetestswas450daysandtheaccelerated ASR-inducedexpansionwascomparabletothatobtainedduring5to 50yearsinactualstructures(inwhichthereactionisnot accel-erated).Inthesetests,concretecylindersof diameter130mm andheightof120mm,withthewater-cementratioof0.45,was used.ToacceleratetherateofASR,potassiumhydroxidewas dissolvedinthemixingwatertoincreaseNa2Oeqto1.25%of

themassofcement.

Themodelwasfirstcalibratedforthecaseoffreeexpansion. Two aggregate sizes were considered: D=9mm for 85% of aggregates, and D=4.2mm for 15%. Fig. 4a shows the calibrated results for the free expansion case. As the figure

shows, considering two aggregate size is enough, and the predictedinitialexpansionhasnodelay.

Multon and Toutlemonde (2006) analyzed the effect of various stress states on the ASR-induced expansion. Fig. 4b shows the confinement used and the loading conditions. In total,theytestedspecimensatninedifferentstressstates.Here we use one of them (the free expansion) to calibrate our model and then analyze the calibrated model ability to pre-dictASR-inducedswellinginotherstressstates.Fig.5ashows the plot of predicted vs. experimental results for the axial deformationsandFig.5b showsthesamefortheradial defor-mations. Thesefigures demonstrate that the model is ableto predict the ASR-induced expansion in good agreement with the experimental results. In particular, they confirm that the present model canpredict the so-called ‘expansion transfer’, i.e., the load-induced transfer of ASR expansion to another direction.

Itshouldbenotedthatinthecylindersconfinedbytubular steel envelopes,the concrete was considered toslide against thesteel.Forsurethisslidewasnotfrictionlessandthefriction couldbesignificant.However,thefrictioncoefficientwasnot reported.Insimulations,itwasassumedtobe0.15.

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0

0.1 0.16 0.012

0.008 0.004 0.000 -0.004 0.04 0.03 0.02 0.01 -0.01 0.12 0.08 0.04 0.08 0.05 0.04 0.02 -0.02 -0.02 -0.02 -0.04 -0.04 -0.06 -0.06 -0.08 -0.01 _-0.01 -0.02 -0.02 -0.02 -0.03 -0.03 -0.04 -0.04 -0.04 -0.05 -0.06 -0.08 -0.1 -0.08 -0.2 -0.02 0.00 0.02 0.04 0.05 0.08 0.10 0.12 0.14 -0.16 -0.12 -0.08 -0.04 0 0.20 0.15 0.12 0.08 0.04 0.00 -0.04 0 0 0 100 100 100 100 200 200 200 200 300 300 300 300 400 400 400 0 0

100 200 300 400

400 500

500

0 100 200 300 400 500

0 0

100 200 300 400 500

0 0 0 100 100 200 200 300 300 400 400 500 500 0 0 0.01 0.01 -0.01 0.00 0.02 0.02 0.03 0.03 0.04 0.05 0.06 0.05 0.04 0.04 0.02 0.02 0 0 100 100 200 200 300 300 400 400 500 500

0 100 200 300 400 500

0

100 200 300 400 500

500

0 100 200 300 400 500

500 500

Axial Strain (%)

Radial Strain (%)

Time (day)

Time (day) Time (day)

Time (day)

Time (day) Time (day)

Time (day) Time (day) Time (day) Time (day) Time (day) (g) (g) (h) (h) (a) (b) (e) (e) (f) (f) (d) (d) (a) (a) (b) (b) (c) (c) Unconfined 10 MPa Multon and Toutlemonde (2006) Unconfined 20 MPa Multon and Toutlemonde (2006)

3 mm confinement 0MPa Multon and Toutlemonde (2006)

5 mm confinement 10 MPa Multon and Toutlemonde (2006) 3 mm confinement

10 MPa Multon and Toutlemonde (2006)

3 mm confinement 20 MPa Multon and Toutlemonde (2006) Unconfined 10 MPa Multon and Toutlemonde (2006) Unconfined 20 MPa Multon and Toutlemonde (2006) 5 mm confinement 20 MPa Multon and Toutlemonde (2006)

3 mm confinement 0 MPa Multon and Toutlemonde (2006)

3 mm confinement 10 MPa Multon and Toutlemonde (2006)

3 mm confinement 20 MPa Multon and Toutlemonde (2006) 5mm confinement 0 MPa Multon and Toutlemonde (2006) 5mm confinement 20 MPa Multon and Toutlemonde (2006) 5mm confinement 10 MPa Multon and Toutlemonde (2006) 5 mm confinement

0 MPa Multon and Toutlemonde (2006)

Predicted Axial strains for different stress and confinements conditions (experimental results by Melton et al.

(2006)

Axial Strain (%)

Radial Strain (%)

Predicted Axial strains for different stress and confinements conditions (experimental results by

Melton et al.(2006)

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(i) (ii) (iii) Unconfined

NoExternalload

Unconfinedwith

axialload _Noload _axialload

(iv)

a) Crackingpatternsfordifferentstressstates:(i)unconfined-unloaded;(ii)radialconfined-unloaded; (iii)unconfinedaxiallyloaded;(iv)confinedaxiallyloaded

0 0.01 0.02 0.03 0.04 0.05 0.06

0 20 40 60 80 100 120

Expansion

(%)

Time(day) Mortar

T=20°C,C_alk=1.2% BenHaha(2006)

(ii)

0 0.02 0.04 0.06 0.08

0 100 200 300 400 500

Expansion

(%)

Time(day)

Concrete

T=20°C,Calk=1.2%

BenHaha(2006) (i)

b)

0.6 0.7 0.8 0.9 1 1.1

0 100 200 300 400 500

Re

la

ti

ve

c

omp

re

ssi

ve

st

re

n

g

th

Time(Day)

Concrete

T=20°C ,C_alk=1.2% BenHaha(2006)

0.6 0.7 0.8 0.9 1 1.1

0 100 200 300 400 500

Re

la

ti

ve

Y

oung

M

odulus

Time(Day)

Concrete

T=20°C ,Calk=1.2% BenHaha(2006) 0.6

0.7 0.8 0.9 1 1.1

0 100 200 300 400 500

Re

la

ti

ve

t

e

nsile

st

re

n

g

th

Time(Day)

Concrete

T=20°C ,C_alk=1.2% BenHaha(2006)

c) Mechanical properties change due to ASR reaction: (a) compressive strength of affected concrete versus unaffected one; (b) tensile strength of affected concrete versus unaffecte done; (c) Young’s modulus of affected concrete versus unaffected one

Figure6.CrackingpatternforASR-induceddamageatdifferentstressstatesandthedegradationofmechanicalpropertiesduetoASR-induceddamage.

9. TheeffectofASRonmechanicalproperties

Asmanystudiesshowed,theASR-inducedexpansion gener-atesmicrocracksandcracksthatweakentheconcrete[23–32]. Here we analyze the influence of ASR on: (1) the com-pressive strength; (2) the tensile strength; and (3) Young’s modulus.

Themodelpredictionsarecomparedwiththeexperimental dataofBenHaha[25,26].Hisexperimentsincludeaccelerated testsofconcreteprismsofdimensions70 ×70 × 280 mm, submerged inwater.Since no measurements arereported for autogenousshrinkageandswelling,theireffectsareneglected,

althoughtheymighthavehadconsiderableeffectsonthe defor-mations.

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force, increased toreach the strength limit, is applied tothe specimenatdifferentreaction times.Althoughit seems ratio-nalthatthecompressivestrengthdeclinesduetoASR,because it generates micro- and macro-cracking, there is a disagree-mentamongvariousexperimenters.Forinstance,ClarkandOno [27,29]foundthatASRcandecreasethecompressivestrength significantly(upto40%),whileMonette[28]didnotseeany sig-nificantchange.However,inthisregarditshouldbenotedthat theASRandtheagingduetocementhydrationhaveopposite effects,theformerdecreasingthestrengthandthelatter increas-ingit.Anothersourceofincreaseincompressivestrengthmay betheextraC–S–Hthatisproducedoutsidetheaggregatewhen

theASRgelgetscalcified.Thesecompetingeffectsmustbethe mainreasonforthisdiscrepancy.

ToisolatetheeffectofASR,weconsidertherelative com-pressive strengthsof specimenswithandwithoutthereactive aggregates.Forcalculatingtheagingduetohydration,themodel byRahimi-Aghdametal.[33]isused.Fig.6cshowsthe com-pressive strengthratio forconcretes affectedorunaffectedby ASR, atvariousreaction times, andshows that the compres-sive strengthdecreasesbyabout 5%duetoASR,whichisin agreementwiththeexperimentalresults.

ThesameprocedureisusedtodeterminetheeffectofASR onthetensilestrengthandYoung’smodulus.AsFig.6cshows

WeightofASRbasegel WeightofASRbasegel

forcompletereaction

Alkalicontentatwhich noreactionoccurs

0

0.02 0.04 0.06 0.08

0 100 200 300 400 500

Expansion(%)

Time(day)

Calk=0.4 % Calk=0.8% Calk=1.2% Concrete,T=20°C

BenHaha(2006)

0

0.02 0.04 0.06

0 20 40 60 80 100 120

Expansion(%)

Time(day) C

Calk=0.8%

Mortar,T=20°C Alkalicontentatwhichcompletereactionoccurs

a)Effectof alkalicontentalkalicontentonASRinducedexpansioncanbeconsideredusingfollowing empirically provenrelation:

c)Effectof temperatureonASRinducedexpansioncanbeconsideredusingArrheniustyperelation:

Gelpermeability

Referencegel

permeability Temperature

Activation Energy

0 0.02 0.04 0.06 0.08

0 20 40 60 80 100 120

Expansion(%)

Time(day) T=2 °C T=40 °C T=60 °C

Mortar,Calt=1.2%

BenHaha(2006)

0 0.04 0.08 0.12 0.16

0 100 200 300 400 500

Expansion(%)

Time(day)

Concrete,Calk=1.2%

T=40 °C

T=20 °C (i)

(ii)

) i i ( )

i (

b)ASRexpansionforconcreteandmortarspecimenswithdifferent alkalicontents.

d)ASRexpansionforconcreteandmortarspecimenswithdifferent temperatures.

=1.2%

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thattheeffectsofASRonthetensilestrengthandonYoung’s modulusaremorepronounced.Themodelisabletopredictthese effectswell.TheASRexpansionisherefoundtodecreasethe tensilestrengthandYoung’smodulusbynearly15%.

10. EffectofalkalicontentonASRinduced-expansion

The alkali content can have a considerable influence on the ASR reaction[34,35]. The availabilityof alkali ions and hydroxyl ions is what controls the ASR kinetics. Often, we shouldassesstheASRfor the concreteinwhichreactive sil-ica content is insufficient to complete the ASR reaction. To modelthis,wemustrelatetheamountoftheASRgelproduced tothealkalicontent.Fig.7ashowstheempiricalrelationthat isconsideredinthisstudy, inwhichCalk isthealkalicontent (ratioofthemassofalkalitocementmass),C_alk0 isthealkali contentatwhichASRstopsordoesnotbegin,andC∗_alk isthe alkali contentatwhich alkaliions are adequatefor complete reaction.In this study, we assumeC∗_alk = 0.1%, andC_alk∗ is setequalto1.25%forconcreteand1%formortar.Thevalue of C_alk∗ is smaller for mortar sincec/a is higher inamortar and we calculate the alkali content as a function of cement (c/a = cement-to-aggregateratio,bymass).Itwouldbe bet-tertofindanempiricalequationforC_alk∗ asafunctionofc/a,but therearenotenoughexperimentaldatatoverifythatequation.

Weconsidertwosetsofexperimentstoassesstheeffectof alkalicontent.Thefirstisthesameasthatalreadyconsidered inanalyzingtheeffectofASRonthedegradationof mechan-icalproperties. ThesecondisBenHaha’s(2006,2007)setof testsofsaturatedmortarprismsofsize40×40×160mm.The experimentalandpredictedresultsfordifferentalkalicontents arecomparedinFig.7a.Thepredictionsareingoodagreement withtheexperimentalresults.

11. EffectoftemperatureonASR-inducedexpansion

Likeinallchemicalreactions,thetemperaturechangealters the ASR kinetics [31,36–41], andthis causes changes inthe ASR-inducedexpansionanddamage.Thetemperatureeffectis consideredtofollowtheArrheniusequation(Fig.7b)for per-meabilityofwaterthroughtheASRgel,forinitialdiffusivityof cementmortararoundtheaggregate,andforpermeabilityofthe ASRgel; here T=current absolute temperature, R=universal gasconstant.

For the experimental comparisons and calibration, Ben Haha’s tests [25,26]were used again toassess the ability of themodelinpredictingtheeffectoftemperature.Fig.7bshows theexperimentalvs.predictedresultsforconcreteandmortar prisms.Thefitsareseentobequiteclose.

12. Conclusion

Themechanicalanalysisaswellastheattainmentofagood agreementofnumericalpredictionswiththeexperimental obser-vationsconfirmthatboththelong-termcreepandthelong-term diffusion,whichcausestheASRgeltopenetrateintoporesand newcracksinthemineralaggregatesandcementmortarnear

aggregatepieces,areimportantmechanismsinASRdamageto structures.Theymitigatethedamagesubstantially.

Acknowledgment

Partial financial supportsfrom the NEUP Program of the U.SDepartmentofEnergyundergrantDE-AC07-05/D14517, and from the U.S. National Science Foundation under grant CMMI-1153494,bothtoNorthwesternUniversity,aregratefully acknowledged

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