Inclined Fiber Pullout from a Cementitious Matrix: A Numerical Study

24 Pages·2016·3.91 MB·English

materials
Article
Inclined Fiber Pullout from a Cementitious Matrix:
A Numerical Study
HuiZhang1andRenaC.Yu2,*
1 CollegeofCivilEngineering&Architecture,ZhejiangUniversity,Hangzhou310058,China;
[email protected]
2 ETSIdeCaminos,C.yP.,UniversityofCastilla-LaMancha,CiudadReal13071,Spain
* Correspondence:[email protected];Tel.:[phone](ext.6313)
AcademicEditors:TimonRabczukandPattabhiBudarapu
Received:30August2016;Accepted:20September2016;Published:26September2016
Abstract: Itiswellknownthatfibersimprovetheperformanceofcementitiouscompositesbyacting
asbridgingligamentsincracks. Suchbridgingbehaviorisoftenstudiedthroughfiberpullouttests.
Therelationbetweenthepulloutforcevs. slipenddisplacementischaracteristicofthefiber-matrix
interface. However,sucharelationvariessignificantlywiththefiberinclinationangle. Inthecurrent
work,weestablishanumericalmodeltosimulatetheentirepulloutprocessbyexplicitlyrepresenting
the fiber, matrix and the interface for arbitrary fiber orientations. Cohesive elements endorsed
with mixed-mode fracture capacities are implemented to represent the bond-slip behavior at the
interface. ContactelementswithCoulomb’sfrictionareplacedattheinterfacetosimulatefrictional
contact. Thebond-slipbehaviorisfirstcalibratedthroughpull-outcurvesforfibersalignedwiththe
loadingdirection,thenvalidatedagainstexperimentalresultsforsteelfibersorientedat30◦ and60◦.
Parametricstudiesarethenperformedtoexploretheinfluencesofbothmaterialproperties(fiberyield
strength,matrixtensilestrength,interfacialbond)andgeometricfactors(fiberdiameter,embedment
lengthandinclinationangle)ontheoverallpulloutbehavior,inparticularonthemaximumpullout
load.Theproposedmethodologyprovidesthenecessarypull-outcurvesforafiberorientedatagiven
angleformulti-scalemodelstostudyfractureinfiber-reinforcedcementitiousmaterials. Thenovelty
liesinitscapacitytocapturetheentirepulloutprocessforafiberwithanarbitraryinclinationangle.
Keywords: fiber-reinforcedconcrete;pulloutresponse;internalfrictionresistance
1. Introduction
Sinceconventionalconcretetendstofailinabrittlemannerunderexcessiveloading,fibersare
often added to improve its ductility and durability. Typical examples of high-performance
fiber-reinforced cement-based composites are slurry infiltrated fiber concrete (SIFCON) [1–3],
engineered cement composites (ECC) [4–6], steel fiber-reinforced self-compacting concrete [7,8]
or high performance concrete [9], as well as ultra-high toughness cementitious composites [10].
The effectiveness of the given type of fibers, steel fibers in particular, is often assessed through a
pullout test, in which the force required to pull a fiber out of the hardened concrete is measured.
Thetransmissionofthisforceisachievedthroughtheinterfacialbond,definedastheshearstressat
theinterfacebetweenthefiberandthesurroundingmatrix[7,11–13]. Inordertocapturetheinterface
failure, the stress criterion [14,15] or the energy criterion [16–18], as well as cohesive approaches
wherebondstressisdeterminedbytherelativeslipbetweenthefiberandmatrix[12,16,19–21]have
beenadopted.
Naamanetal.[12]pointedoutthatalignedfibers(i.e.,loadisappliedatthefiberlongitudinal
direction)exhibittwotypesofshearbondsattheinterface:elasticandfrictional.Whentheelasticshear
stressattheinterfaceexceedsthebondstrengthoftheinterface,thebondbecomesfrictionalinnature.
Materials2016,9,800;doi:10.3390/ma9100800 www.mdpi.com/journal/materials
Materials2016,9,800 2of24
Thismeansthatthebond-brokenprocesswasconsideredabrupt,notgradual.Basedonthishypothesis,
Naamanetal.[12]developedafundamentalstudyofthebondinfiber-reinforcedcementcomposites,
as well as the relationship between pullout curves and bond shear stress vs. slip curves. In their
analysis,acompletepulloutcurvewasanalyticallyderivedfromanassumedbond-sliprelationship,
andthedualprobleminwhichthebond-sliprelationshipwasobtainedfromanexperimentalpullout
curvewassolved. Furthermore,theyrelatedthefrictionalbehavioroftheinterfacebyshrink-fitand
fiber-matrix misfit theory. Other examples of aligned fibers can be found in [22,23]. For instance,
byfocusingontheinterfacialproperties,Chanvillard[22]developedamicro-mechanicalmodelto
accountforthedifferentphenomenaobservedinanon-straightfiber. Ellisetal.[23]conductedthe
simulationofasinglefibertakingintoaccountthefibermorphology. Eventhoughgoodagreement
wasobtained,onlyalignedfiberswereconsideredinthesetwomodels.
Whenfibersarerandomlydistributed,however,besidesbondstrengthandfrictionalongthe
interface, additional phenomena, such as fiber bending, matrix spalling and local friction effects,
areinvolved[24–27]. Furthermore,thecontributionsofthesemicro-mechanismsaredependenton
thefiberinclinationangleandthefibermaterialproperties[13,18,24–26,28,29]. Thecontributionof
fiberbendingincreaseswithinclinationangle. Thefibercurvaturealsoplaysanimportantroleinthe
distributionofpressureagainstthesurroundingmatrix. Duetothelocalcurvatureofthefiberand
residualstressattheinterface,thematrixtendstocrackandspall[24,26,29]. Thishasadirectimpact
ontheeffectiveembedmentlengthanddeformationwithinthefiber. Moreover, snubbingfriction
nearthefiberexitpointcanincreasethepull-outresistance[24]. Considerableeffortshavebeenput
intomodelingthementionedphenomenainvolvedinthepulloutprocessofinclinedfibers. Some
modelsconsideredthebendingmechanismofinclinedfiberswhileignoringthematrixspalling. For
example,MortonsandGroves[30]employedanelementarybeamtheorytocalculatetheforceneeded
toproduceaplastichingeinthefiberasitiswithdrawnfromthematrix. Inthisway,themagnitudeof
theadditionalpull-outforceduetofiberinclinationwasaccountedfor. Themodelreproducedwellthe
experimentalobservationsforlowerinclinationangles,butfailedtodosoforsteeperones. Lietal.[24]
assumedanexponentialincreaseofthepulloutloadwithfiberorientationtocapturethemaximum
pulloutload;however,detailedinformationoftheentirepulloutprocesswasomitted. Bytreating
the fiber as a beam on an elastic foundation with variable stiffness and the possibility of spalling,
LeungandLi[18]developedamicro-mechanicalmodeltoanalyzethecoupledfiberbending-matrix
spallingmechanismandtodeterminecrackbridgingstressinrandomfiber-reinforcedbrittlematrix
composites. Afterwards,themodelwasextendedtoductilefibers[31]. Nevertheless,frictionwasleft
outofthepicture. Morecomprehensivemodels,likethatofCailleuxetal.[29]andFantillietal.[20],
involveconsiderableparameters,whichcanonlybeestimatedthroughpullouttestsofaligned,aswell
asinclinedfibers. Inaddition,tediousnumericaliterativeproceduresarerequired.
In spite of the above models, detailed simulations to consider all of the aforementioned
phenomena have been seldom conducted to explicitly model the entire pullout process for fibers
atanarbitraryangleandconsideringbothfiberbendingandmatrixspalling. Inthecurrentwork,
weendeavortodoso. Cohesivemodelscapableofrepresentingmixed-modefractureandCoulomb’s
frictionareemployedtosimulatetheinterfacebetweenfiberandmatrix: boththede-bondingand
frictionalphases.Inaddition,fiberbendingandmatrixspallingaretakenintoaccountnaturallythanks
to the explicit representation of the fiber, the matrix and the interface in between. The rest of the
paperisorganizedasfollows. Theinterfacebondcharacteristicsandmatrixspallingarepresentedin
Section2. ModelcalibrationandvalidationaregiveninSection3. Numericalresultsandparametric
studiesaredepictedinSections4and5. Finally,relevantconclusionsaredrawninSection6.
2. BondCharacterizationandMatrixSpalling
Inthissection,twoessentialfeaturesthatdeterminethepulloutresponseofafiberwitharbitrary
orientation are explained in detail; namely bond characterization and the quantification of matrix
spalling. Theformerdealswiththeidentificationofthedistinctmechanismsbehindinterfacebond
Materials2016,9,800 3of24
deteriorationandfrictionfrompullouttestsofbothalignedandinclinedfibers. Thelattertacklesthe
matrixfailurenearthefiberexitpointwhenaninclinedfiberispulledout.
2.1. InterfaceBondCharacterization
Theshearstressvs. sliprelationshipisconsideredtobeaconstitutivepropertyoftheinterface.
Itscharacterizationisofprimaryimportanceforpredictingboththemechanicalandfractureproperties
offiber-reinforcedcomposites. Naamanetal.[11]attributedthepresenceandcombinationofdistinct
componentstothecomplexnatureofthebond: physicalandchemicaladhesion,themechanicaleffect
ofdeformedorhookedfibers,theentanglementoffibersandfriction,whichisgreatlyinfluencedby
confinement. Inthecurrentwork,weconcentrateonthecaseofasinglesmoothstraightfiberbeing
pulledoutfromacementitiousmatrix. Consequently,onlytheinterfaceadhesionandfrictionaredealt
with. ByfittingwiththeexperimentaldataofLeungandShapiro[26],thecontributionofinternal
frictiontointerfacestrengthwillbeidentifiedandquantified. Itneedstobeemphasizedthatthesame
methodologycanbeappliedforhookedordeformedsteelfiberswithadditionaleffortstorepresent
thedetailedfibergeometry.
Either for an aligned fiber or an inclined one, there exists certain frictional resistance after
debonding when being pulled out. This friction is mainly dominated by the surface roughness,
anditisoftenassumedtobeconstantregardlessofthefiberorientation. Thiscomponentisdefinedas
internalfriction,τ (s),whichisafunctionoftheslipdisplacement,s. Asforinclinedfibers,however,
f
the mechanisms are quite different. Pullout load can be decomposed into a parallel force and a
perpendicularone. Theformerpullsthefiberout, whilethelatterbendsthefiberandchangesits
directionduringthepulloutprocess. Inaddition,duetothenon-uniformcompressionattheinterface,
thebrittlematrixmayspall,andthefiberembedmentlengthisreduced.
Inordertotakeintoconsiderationtheabovephenomena,weproposeagenericconstitutivelaw
thathasthreeconstituentstogoverntheinterfacedeteriorationprocessasfollows:
τ(θ,s) = τ (s)+τ (s)+µp(θ) (1)
b f
whereµisCoulomb’sfrictioncoefficient,θisthefiberinclinationanglewithrespecttotheexternalload
direction,whereas p(θ)isthecompressivestressasafunctionofθ. Thefirstterm,τ (s),illustratesthe
b
interfacialbondrelationduetointernalphysicalandchemicalcohesion;itisoftenadecayingfunction
oftheslipdisplacement,s. Inaddition,τ (s)iscohesiveinnature;thus,itcanbemodeledthrough
b
cohesiveelements[32,33]withMode-IIfracturecapacity. Thesecondterm, τ (s),isacontribution
f
frominternalfriction,whichresiststhemotionbetweenthefiberandthematrix;itcanbeconstantor
decreasing;seeFigure1. Thethirdtermgivestheshearstressduetodryfriction,whichplaysarole
onlywhenthefiberisorientedatanon-zeroanglewithrespecttotheloaddirection.
Inthecurrentwork,thebondstressisassumedtobelinear-decreasingwithrespecttos,whereas
internalfrictionisassumedtobeconstant;namely:
(cid:18) (cid:19)
s
τ (s) = (τ −τ ) 1− , 0≤ s ≤ s (2)
b max fc s bc
bc
τ (s) = τ (3)
f fc
whereτ isthemaximumshearstressresistedattheinterface;itcoverstheeffectofboththeinternal
max
bondandinternalfriction. s isthecriticalslipdisplacement,whenthebondiscompletelybroken.
bc
τ istheinternalfrictionalresistance,whichisapropertyoftheinterface.
fc
TheconstitutivelawencompassedinEquations(2)and(3)highlightsagradualfailureatthe
interface; see Figure 1. This, on the one hand, differs from the abrupt bond failure assumed by
Naanmanetal.[12];ontheotherhand,itappearsparticularlyimportant,sinceinmostpracticalfiber
applications,theslipdisplacementislessthan1mm;seeYuetal.[34]. Whenworkinginacorrosive
environment, the maximum allowed crack opening in steel fiber-reinforced composites is around
Materials2016,9,800 4of24
0.3mm. Thismakesitespeciallyimportanttobeabletocapturethedetailedfailureprocessatsmall
slipvalues.

τ
max

sser τ f Constant friction
ts
rae Decaying friction
h
s
d
n
o
B
Slip displacement
Figure1.Bondstressvs.relativeslipattheinterface:thecontributionoffriction.
2.2. MatrixSpalling
Duringthepulloutprocessofinclinedfibers,duetothelocalcurvatureandstretchingofthefiber
segment at the free end, matrixspalling is inevitable [24,26,29]. Detailed modeling of the spalling
process is not an easy task. Since the deterioration process occurs within a narrow band near the
interface, excessive mesh distortion often leads to convergence problems and, thus, impedes the
furthermodelingoftheentirepulloutprocess.Consequently,simplificationsareoftenassumed,sothat
thespallingpartistakenofffromthenumericalrepresentation,oncethematrixtensilestrengthis
reached[27,29,35,36]. Thelengthoftheerodedmatrixalongthefiberdirectionistheso-calledspalling
length, denoted as L by Laranjeira et al. [35]. A simplified failure criterion was adopted in [35],
sp
i.e.,if thespallingforceimposedbyfibercurvatureislargerthantheresistanceprovidedbythematrix
atthecrackedsurface,thematrixwillspall. Accordingly,anestimatedlengthofthespalledmatrixcan
beobtainedasfollows:
aL2 +bL +c =0 (4)
sp sp
where: √
a = 2 + cosθ , b = df , c = −Pmax sinθ. (5)
sinθ sin2θ sinθ ft
Intheaboveexpressions,d standsforthefiberdiameter,θ istheinclinationangle,P isthe
f max
peakpulloutloadofanalignedfiber,whereas f isthetensilestrengthofthematrix. Theestimated
t
values from Equation (4) closely match the ones measured through scanning electron microscopy
(SEM)byLeungandShapiro[26].
Inthecurrentwork,Equation(4)isadoptedasafirstapproximationtorepresentthematrixwedge,
whichistobespalledofflateron. Alineofcohesiveelementsconnectsthewedgetothemainpartof
thematrix. Trialrunsareperformedtodeterminethemomentatwhichthematrixwedgeshouldbe
deactivated. Fromthenon,theelementswithinthematrixwedgeceasetocontributetotheoverall
stiffness. Itbearsemphasisthatsuchatreatmentisonlytosavecomputationaltimewithoutdetriment
tothemodeledphenomenon,sincethefirstprincipalstresseswithinthematrixarechecked,andthe
spalledlengthisadjustedifnecessary,toensurethatthematrixwouldnotpresenthyper-strength.
3. FiniteElementMethodology: ModelCalibrationandValidation
Asmentionedbefore,theobjectiveistoexplicitlymodelthematrix,thefiberandtheinterfacein
between.Inthissection,a2Dsetting,asshowninFigure2a,ischosen.Boththefiberandthematrixare
representedwithfour-nodeiso-parametricelements(K-L-M-NandG-H-I-J),whereastheinterfaceis
VersionAugust17,2016submittedtoMaterials 6of29
VersionAugust17,2016submittedtoMaterials 6of30
bythematrixatthecrackedsurface,thematrixwillspall. Anestimatedlengthofthespalledmatrix
canbeboybtthaeinmeadtriaxccaottrhdeincrgactoketdhesuwrfaocrek,tohfeLmaaratrnixjewiriallestpaalll..[A34n]eastsimfoaltleodwlesn:gthofthespalledmatrix
canbeobtainedaccordingtotheworkofLaranjeiraetal.[34]asfollows:
aL2 +bL +c = 0 (5)
aLsp2sp+bLsspp+c=0 VersionAugust19,2016submitted(to5)Materials 6of30
VerswionhAeruegust19,2016submittedtoMaterials 6of30
bythemawtrhixeVraeertstiohneAcuraguckste1d9,s2u01r6fasacueb=,matiht=stepeindms2ptiqona2tM+qriax+tescrwiiosnaciilsosnl2lsq2qsqqp,,ablbl.==Anssiiddnenffsqqti,m,cca=t=ed��lPemnPgamxtbcfhaatsyxnifonttsfhbqitneehemoqbsapttaarilinlxeedadtmathcacteroicrxrdaicnkgedto(s6tuh)6refoawfc3(e6o0,)rkthoefmLaartrainxjewiriallestpaall.l.[3A4n]aesstfiomlloatweds:lengthofthespalledmatrix
canbeobtainedaccordingtotheworkofLaranjeiraetal.[34]asfollows:
129 in1w29hiicnhw,bdhyifctihhs,etdmhfeaistfirtihbxereafitdbtrhieaemdciraeamtceker,teedqr,isqsuitrshfatehceein,intchlcielniVnmearastaitioitonorAnnixuaagwunnstgig1ll9lle,es,20,pP1Pa6mslmaulx.bamxiAsititntsehdetethosptMeiemaptaerkaeiatlapsekdulplloeunulgtlotlohuaotdfloothfatedhseopafaltlihlgenedeadmliagtnriexd aL2 +bL6o+f30c =0 (5)
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itshegon1vu3e5m1r3n9etteoirmdict3heab.elyw1i4oFan0ivtishc1net3oora3iahubtleetillwmstsiehitAtvhalyieefest,ifcrdmnw11lihm33eaxe01htewiserwsnsiinfit,cm.thetbthoaIediretsmoebocgnn,nsneaewetheeenstt11sediho33pohsd01mebwardotsbeeehlnfitedeaalthceeossabfoeiauodrcnnfbeurlorttueo,oeeiniFsdvwsfm,egeipfatsthgdhywhloetmeaueerr:irebotneidertemcueaaeyt.rnaagsse2uFsliohnodfmibrrrtlndeseeeooi.cda.sessamuTtrTtutntlorh-thhnehlzectrttniehoeeehsant11guex,33nregailt23nongsphnitetsihb,ilrnlioswtoieslisrcfielccauncaihaittesaplthrcil,ttlcnortehrtiyaIhhundnehnomsniiaeniemnscmngtsahsgtetfitpeitorheocrnaarelelirbedslxocnradoaeeumt.efsecctidscrTpmtntlepprhorehermogavpeetneeninlhyrnlnmattpeeefesmc(sltdwonaaStiiwiitriprmEotodcrmsms11tfrirMaif33xakooaati45ldl.eh,ss)tttansecTridewtbiladoteoihnryyrxrpetiteinvemoo,Loyvttaasehnoedelds(stteuu.eShtheodsdeiTnEepmseametshgMftevawaefrofiaarErt)tenoonemvitqbbettmdmrcd.ryehrlioiiie(vxSraniL5Emmnmalah)eeqwralaluuiuap.tltnapnhsene(nhuis5cdsadgresae)thtoftaigaacmrifimitfr[olntnenhfe2orimdnas6gooseepcte]tenSmee.rElarhisyaiapqnfxsaeoslm.ptp.nre(ermi5aIotrrt)ttoxefaocciadnht[mlc2cotew6oeasoee]tnhd.ildioyetninsrctmeihttbroamoutthcritbenheepeertephseoenmitnottmheeednmtoaauttrwtixhthwicahedtt,hgseeuchatreatmentisonlytosavecomputational
weeiτncrlhe,rteomiiscrr111111fh.e333344aa678901,nCtc11111tot44444eoas01234nih3bniisln(.esysyIctepe-bfsFuwFhJdfoqeeeinireAergau;nug1111itcd-4444ecnww.ieksmsr1234tnett-nd2eierhAr.enmo111111edbte333333bfsFeeenlKco456789nnso.eyIealrenioadngetg,nu-gentmls.Lmmwtt.etamrttih3d.ahhifsoi-2)s.opyomenee.Ieeibtcs2ntpeTnrstrnhonnFh.huehcDzeneoedethtite1111111eesedpes.wran3333333-imecess2345678s-.ropsoeikcpsioieIbite2neovrattocTensae-twmttihhrxpetDsenreoioodtdmlhehymfoiileharaocda-,ontnnspaeeoathpiauhlempstncredrsmggesdele2niaIhetwercllexabocmtn,t-Doeerodlo,dtkaiihssttieweas1111111havttttlttnptetge3333333.hwrnheutotmfrte2345678siidaeesesonhheoefoneriggdentttfedacaudnsebeetriintggytgciwmttihlnhtttvtheastleoiseeumeteu:syahmir,edithhsoamstct,enrnhpitipsisimtltrper.trhsihceetwmasieseicrtgeeeiahafaweevadbiteIhm.wfsrnomsdtax,eellcnnthn-egerlolteIakisieddleifeetsweaasfeotvostetettedhowoempsahcirjbedhrstnehmddneunsnetiaonrocaolerioaeim.dntleoeossevalsauefrbhletvIcaisgegcfFttleehnxaketltsw.melnaruheoealtlndiotnd,idpntosa,hsoeesolAdrgtvhiotggtwtempintl.sdrsslwatcieewe.uitbheeihuftd,yai.toodreccnatfdnhdfen2alerornFol:iioehwdgulhtttsaaedten,nlreshaintimnltedrliodwottpbeiohsscheeyr.tebonisumdootsiareifcnooTooFeoiinm.axopobllsvemcefr111111lmhrfInuiddFpittffpett333333eaknectaaspeahge456789homhlditjnet,oedEpunsaalonaaessugeh.rtiealrwjteqdoeinlsdnldrtu.ctesmttih3eneitetrlri..hrtts2coi.oicres.nontdyiieyhmdyith2ihts(caateFcteaa,onihgttnple5t,sea,aepFthlhenellerdehivnl,t)rdedetlydoitbidtmerhefitibelttw.orornabriuaexcmchohaei-ioieeonTboooiinitsflksfnrpibrrenpswreecntmsvisttuhturentutieatsfeftnuhradenaeeulhththtedeteeaeametaoetlelemdEripemeodansentctfireilccracaulmrhsjqtocetauneortgtmfegeuredlttna,fhesatrhaiselei.afsnremthnraspsonido,stletiethtaesasn(ceeictihad,iancretnt5eseinesn.nassslattcealeiseritres,)nihdttf,lcaplfixdudyrf,etnsnfthgtdeietorufiats,nehrep,ceihibspgewocedhaorsfmtuysnthesaedlri0sbomehtruepotntseo,vcssiureac,deeeahtesvdtrteiaextmtieavhtehlltim.mwentoctlfisheeitivahfih1111ddirirlahsmIrmseesm3333enseraeotlreane,eeaanahts6789iepcusdsirtanfipdasnaosstnetteetdtrnmadrtnefiiht,tltaepneobnlrdcneeddrpimaatesatsntereroilrete.rrctsaodepatih3nuryftoxrdcusttsnieemeshoeFiiadt,,bhigptigci.sttensytemnwrrrratrxhottthhtayesrtycaercomtmoethtpmhaiptiooeimo:,notdiobrhFhxemasteietprnexrteeseihmnmahrfeibeiyesieenhtiaifidwtxAdbtenwiurionmertennndnolschcahnrrhpacea-emnowtusmsddiepocmokassaattooerittttealtttttnr,iehjpetotettyenrourhnhieuhiaeedtppitmhrslreidniesmscdoaerhtdsltoentxieeefierstcarcpwgodceonoteemahttintnmosteisaen,anleenseesbxroeaaulacs,itahlidnltarndchbetontsigseminvatctoitrnwrmrtbttirxhelrmtneiiv-owtempheerpteeteotyfesrrdshteartoieasehsiitteayunaxeenaennhiaosueouearitelrltettctn.ppoeeasoertnmavhrwfilleidoglesmnorttctntehosdgrafirrnltwintthspehmeeetmyxeaanoernheedscas,eetahnsstwedddttr,seesdsteaptcstwreettwaseoemlrwrgi,oh,seaioonnmvsaunnaiwixtestesrxpeetnnoihtsttocfstrdebeupwyehnpossaachtwemiiw-tia,tuncrrcmeetttvwvmnwshtcrtetih,ilehhehaoaltowfbeahhooelesoeiddeisotuleaiutteaneetoceomnrseidnhhdgmhinlodridtoanntsfitnddewnxieeetireetlmnaspaan-otretnowountl,tshnsttnatuilepctretttrotwlhhotiheiooioewunpmentorxtpentienaohtrdmfthnpdgegetamrhtiwmeitawnrrhcnetetiyaeiaicaheochatsbtowes:atidtnetienutteplnrrrsihhgimniinnitaaiiamxsnoxeeeextiitllutlnwhnnoasemmltddyertaedotjetesrugosilrexesesicsacctwxaevewapaoeldlisuiltcerbihloicedtifmrnoisnatnupctlothoeyuiltenctotpastmemnrrt,eisiabosssoatuenirannnditaxertlcdeyel,vtttoahhleeeidnfimasrutsaitroteprnirtxhin,acttihtpheaelfismbtrraeetsrsaixenswdwotiuhthledininntotheterpfmarecaseterniinxt
As mentioned before, we aim to explicitly model the matrix, the fibre and the interface in
142 four-node iso-paraAmsemtriecnetil1oe39mnee3dn.tFbsi,enfwioterheee,lreemwaesenttahmiemetihntotodeoreflaoxgcpyel:icitmsitolddyieslmccraeoltidibserealdtitohinneatonmdbpaevattariwlriixsde,aoettinfoh.ntewIfionb-nraeod2aeDnldisnetethteinign,tearsfascheoiwnn in Fig. 2, both the fibre and the matrix are represented with
N 114434 sFeigg.m2ebn.ts111444.012ThbfoeeutcwrNo-1enn39eosntdi3b.te.NeutIwtFinisivAenoeeMisa-nt11pe44.mb012aeeDeIrlhnenabfomatemasiutvoeewre2nin-tAoDentettreidousmnindscr.emgbeeteto,eteIhifinslfnonaoeortgdstam-ieh,opo,2snaaelewDohresnadgioesnmtyshswebt:ae,oeteimttnwmrfrwionifoncargidhtenco,eie,enlebfslaeoreewFescmeuFgxitiaewsamirspgelg-hsilnen.ag.ebioecttoin2ort2swmnicdtaht,,,vs.ltneyew.bietboIoioirhTnsmnointnonehhetao-areFehtxpendtedip2haacgderDlrtsoe.lifahbcnvat2mfitisyheasth,cebtlleeeeiytbirfitattdeuoirimninmbaittccsahtitgraveioon,eeotdredthdlrfnaheiiaefsbaesxmesotlcees,nhcesheufigtierhdthvnaibhemrosetvmere-tswteidin,omehaslfiniwuaotaneseabrncrwiddihttrnrxdrsmeoeee,i.rxftiFateainaa,hristiangehsTesttetsd.rodehorhsimt2eex-ihhti,epnppnhaefiobtratteaaecebrooiwirrnisrortrixfeenpehtaasnnentacamrearttesiorfhierrenaetnfsfdeeipdciastretoctfirewurgwetfeipbhiosticstrrtievioweehnedve-seniioraenensnl-ntcntenoeeredbtmoeddedrtedfdtieabhsheewceyewneladliiimaitvnnttisihnhenicea,oottrohwuixeprshaaivoierreersfelrotaeafhwpstre,wetasoihsen-nenstthoeeioddrnweftwanelicinrtiehnfeaicsegiosvdeirsnceredtibseydaincotohepsaiviresloafwt,waos-nshoodwenlinine
N M M 114434 KsFeigg.m2ebn.tfssoe.ugmrT-nehonNe111Ldt444s234e.coiTssFnoehi-ggsMep.tmaic2troebuaMnn.mttssitev.itteurTitchibveeeelchebomaenvhesnatiiotvstsuiuc,otrwiuvrhoeFoefibrfgeet.thhah2saebev.tihiinoneutteirernfrotaeffcaretfchaieceseiignsiostvegdreofirsanvccereeedFrtiinsibsgeeygd.doa2vinebcborty.onheepadsaivcibreoyshloaaefwcsto,iwshva0oeses-nsilvhoaeodwwela,nwlian,isneasshshoowwnniinn s0
n Fnig.2b.s w �sc �s0
KJ n KJ n IL IL J KJ KJ�s0 I sL IsLI ww �tc �ss0cscsc �scsc s�s0 �scsc scs
c
N G N H �sct �tc �tc �
M � c
M
G G H H N (Ga) G H H N M (b) �tc �tc
N M N
K n L Figure2M. (a)A2DKinterfaceelemsLentI-J-Kw-LfoMrmedbetweentwofour-nodesolidelementsG-H-I-J
K L
J I andK-L-M-NF,KiwguhnreerFJei2gs.uar(nead)2.wAL(aa2)rDeAtih2nIeDtetairnnfatgecerefnatecielaelemalenemdnetnnIot-rJIm--JK-aK-lL-dLsifsfooprrlmmaceeewddmbbeenetttwwjueeemennptwst,woanofodufonr-unisro-tdnheoedsnoeolirdsmoelaliedlmeelnetmsGen-Hts-IG-J-H-I-J
directionofthaenJdtwKo-alLnin-dMeKs--eNLg-,mMIwe-NhnKte,srw;e(hbse)raetnhsdeabnwodnawdre-LasrtlehJipethlteaawtnagn(seghneontiwtaialnIlaafnondrdannoloirnrmmeaaarl-ldddiesipcsrlpealacaescmineemgncteanjustem)ju.pms.ps.
K n L s w
G HJ Th14e5abI3o.v11e4.5cCooGn3n.fi1sg.tiuCGtruoanttisivtoHietnurtieisHlvaietmiroeplnJalsteimofonersnmfoteradtmriainxtr,tihxfiG,Iebfircbeor,ema,namHndedtrhtcheieailinnsttoeerfrtffawacciaiaarleltrtAraanNnsiSstYiiotSino1nzo2nz.0oe,nuesingitsAPDL
In order to explicitly model the physical phenomena involved in the pullout process, the
programming14l6anguage. Regardingtheconstitutivelaws,linearelasticityforthematrixandbilinear
In order to explicitly model the physical phenomena involved in the pullout process, the
146 commercial software ANSYS 12.0 is employed. Both the matrix and the fibre are represented
plasticitywith14i7sotropichardeningforsteelfiberareadopted,respectively.
G 3.1. Exp11H44e78rimcaeosnFa11mtni44a2g89dlmDuKSraceee-osL4tr2unc--.M2npsiDa(to-aiolNt)dfu4,AsLet-woienv2ufhsoeDtnetdwrrlgeieaunawasctsrentatesrudrnfuGaadrSAccraeewhtlNuaerarplseSraeiosemrYlpoltFaiSehs,enidHnogec1dutlt19tiKiraIedv92e--nlLJe9.2e-g0-le.KMymel(n-e-aiLNet)mesim,Aanfweoletp2nrhasmDmlentosrdiee(pyndps(tnelepaolodrbnaflryaeadnmcftneeowweadereaell.ree1td1emhni88tseeh22tpBnew))tmlt,oa,aIoc-ntaJweh-wgftKmoerhn-uihLexittrilnhia-felonalteeranojmeunddldmmeeaedlnstpaasohtbsoirsetec.mltrtiwiidafiacxelbendenriladaseetpnmwn.lbadodecTienlfmhoitbntuseehiren-Glaneitino-nrjuHdtefimeek-arpbIisf-nrsoJar.leeicdkmiaeiaallnertmteeircmeannrptasesltGpiaits-icrHotei-ncspI-eJlnastetidc
constitutive laws are respectively employed for the matrix and the fibre. The interfacial transition
1I4n514o93r.d1.eCrotnostaitsusteivsesrtehlaetioenfsfefocrt1mo45aftr3fii.1xb.,eCfirobnryseti,ietaulndtidvestrhtereleaitnniogtnestrhffoarcomianalttrrriexa,innfisbfirtoei,roacnnedzmothneeenintterefafficiacliternancsyitioinnzotneerms of
zone(ITZ)betFwigeueren2t.h(ea)cAem2DenintitteirofaucsemeleamtreinxtaI-nJ-dK-tLhfeorfimbeedrbiestwreepenrotdwuocfeoudr-ansodceohsoelsidiveeleemleenmtseGn-tHs-cIa-Jpable
150 In order to explicitly model the physical phenomena involved in the pullout process, the
maxi1m46um crIanckorbdreidrgtoingeaxnfpdolriKcc-ieLtl-yaMn1m4-d6No,tdwoethlaeltrheeesnaepnrhgdyywsiaacbarelstophrehpettaninoognme,neLtniaealuainnngdvonalonvredmdaSlihndaisptphilreaocep[mu2le6lno]tucjutamrprprieos.dcesosu,tthe
commercial software ANSYS 12.0 is employed. Both the matrix and the fibre are represented
pullo14u7t tceosmtsmfeorrcisatleesolffitwbearres oAfNdSiY1f4f7Ser1e2n.0t yisieeldmpstloreynedg.thsBaonthdtihneclminaattriioxnanadngtlhees fiatbr0e◦,ar3e0◦reoprre6se0n◦.ted
All of the fibers are 03..51.mComnsitnitudtiivaemreeltaetrionasndfor2m2amtrimx,fiinbrlee,nangdtht.heTinhteerpfaucilalolturtansspiteicoinmzeonnes are blocks
145
of25.4mm×12.7mm×9.5mminsize,withaneffectivefiberlength(equaltotheembedmentlength
In order to explicitly model the physical phenomena involved in the pullout process, the
146
in this case), L , of 10 mm. The material parameters for the matrix, the fiber and the interface are
f commercial software ANSYS 12.0 is employed. Both the matrix and the fibre are represented
147
giveninTable1,whereastheyieldandtensilestrengthsofthefourtypesoffibersarelistedinTable2.
AdditionallylistedinTable2isthecriticalfiberlength,i.e.,themaximumembeddedlengthforafiber
tobepulledoutfromamatrixwithoutrupture[24]. Itisrelatedtothemaximumshearstress,τ ,
max
fibertensilestrength, f ,fibercross-section, A ,andperimeter, p ,asfollows
r f f
Af fr (πd2f/4) fr df fr
L = = = . (6)
c
p τ πd τ 4τ
f max f max max
Itneedstobepointedoutthatthisestimationisforalignedfibersonly;inthecaseofinclined
ones,thislengthisconsiderablysmallerduetothecontributionoffiberbending.
Materials2016,9,800 6of24
Table1.Materialparametersforthematrixandthefibergivenin[26].
ρ E ν fc ft
(kg/m3) (GPa) - (MPa) (MPa)
Matrix 2100 30 0.20 36.5±2.5 fc/12(estimated)
Steelfiber 7800 200 0.33 - -
Table2. Yieldandtensilestrengthofthefourtypesoffiberstestedin[26];thecorrespondingLc is
listedforadiameterof0.5mm,where(1)and(2)arecalculatedusingthetensilestrength, fr,andthe
yieldstrength, fy,respectively.
FiberType 1 2 3 4
fy(MPa) 275 469 635 954
fr(MPa) - 783 847 1023
(1)Lc(mm) 12.7 21.7 29.4 44.2
(2)Lc(mm) - 36.3 39.2 47.4
3.2. IdentificationoftheFiber-MatrixInterfaceProperties
Inordertoextracttheinterfacepropertiesfromexperimentalpulloutcurves,wefirstintroduce
theconceptofapparentinterfacialshearstress,whichisdefinedas:
τ∗ = 1 (cid:90) Leτ(s)ds (7)
Le 0
whereτ(s)istheactualbondstressdistributionattheinterfaceandL istheinitialembedmentlength,
e
thevalueofwhichisequaltoL .
f
Assumingthefiber-matrixinterfacepropertyisuniform,themaximumpulloutloadandpeak
frictionalloadarerespectivelycalculatedas:
P = πd L τ , P = πd L τ . (8)
max f e max f f e fc
ThevaluesforP andP aredeterminedfromthepulloutresponseofalignedfibers,asshown
max f
inFigure3. Notethatthevaluesfor P and P areaveragedforthefourtypesoffiberslistedin
max f
Table2toobtainthoseofτ andτ ,aswellastheirstandarddeviationsinTable3. Thecriticalslip
max fc
forinterfacialbond,s ,0.3mm,isdeterminedthroughtrialanderror,sothatthefirstdecayingbranch
bc
ofthenumericalpulloutresponses,asdemonstratedinFigure3,shouldfallwithintheexperimental
range. AsregardsthefrictioncoefficientgiveninTable3,itisestimatedaccordingtotheexperimental
resultsofChanvillard[22],whichwasalsoadoptedbyLaranjeiraetal.[27].
60
Experiment
50
Pmax Numerical
N) 40 simulation
d(
a
o
ut l30 Lower and upper experimental range
o
ull20
P Numerical approximation
Pf
10
0
0 2 4 6 8 10
Displacement(mm)
Figure3.Experimentalrange(lightbluelines)forpulloutcurvesforalignedfiberType2(yieldstrength
469MPa)[26]anditsbi-linearnumericalapproximation(reddottedline). Pmax andPf arethepeak
andthetransitionalpulloutforces.
Materials2016,9,800 7of24
Table3.Extractedparametersofthefiber-matrixinterfacefromtheexperimentaldataofLeungand
Shapiro[26].
τmax τfc sbc µ
(MPa) (MPa) (mm) -
2.7±0.1 0.5±0.1 0.3 0.60
3.3. NumericalModel
Thein-planedimensionsandboundaryconditionstosimulatethepullouttestsperformedby
LeungandShapiro[26]areillustratedinFigure3. Notethatwithinatwo-dimensionalplainstress
framework,thefiberthickness,T ,iscalculatedthroughEquation(9)sothatthecontactareaatthe
f
interfaceisthesameasthatoftheoriginalone. Inthesameway,thefiberheight,H ,isdetermined
f
viaEquation(10)sothatthemomentofinertiaoftheoriginalfiberisthesame. Forthecaseofafiber
diameterof0.5mm,T andH arecomputedas0.785and0.36mm,respectively.
f f
πd L
f f
T = (9)
f 2L
f
(cid:32)πd4/64(cid:33)1/3
f
H = (10)
f T /12
f
AdditionaldescribedinFigure4aretheboundaryconditionsimposed. Verticaldisplacements
arepreventedonthetopandbottomsides,whereashorizontalmovementsareimpededontheleft.
Therightendofthefiberisfixedintheverticaldirectionsothatonlyhorizontalmovementispermitted.
Notethatthesameboundaryconditionsareimposedforinclinedfibers. Thepullingprocessiscarried
outwithintervalsof0.001mminthehorizontaldirectionuntil0.3mm,followedwithincrementsof
0.01mmuntiltheend.

25.4mm
0.36mm
30º 60º
10mm
12.7mm
Figure4.In-planedimensionsandboundaryconditions(thesameforallcases,shownonlyforaligned
fibers)forthepullouttestsperformedbyLeungandShapiro[26],withfiberinclinationanglesof0◦,
30◦and60◦.
3.4. MeshDescription
Atypicalmeshanddetailedelementdistributionaroundthefiberfortheinclinationangleof
30◦ aredemonstratedinFigure5. Notethattherightendofthefiberleansonamatrixwedge,which
willspalllateron. Forthisparticularcase,thematrixandthefiberconsistof2198and154four-node
solidelements,respectively,whereas289two-nodecontactpairsareplacedattheinterface. Themesh
worth noting that this numerical model is mesh sensitive. To specify, contact is a
Materials2016,9,800 highly nonlinear problem so the mesh density will influence the efficiency of 8of24
calculation. Moreover, a coarse mesh will lead to higher stiffness of elements thus
sensitivityanalysiscpauesrinfgo rlamrgeedr ptuolliancgh fioercvees. aAbftaerl atrniacle anbde tewrroere (ndistchuesscedo mlatperu),t tahtiiso mneaslh eifsfi thcei encyandaccuracy
isgoingtobepreseonvetreadll bienstS oence ttioo cnom4p.romise between the computational efficiency and accuracy.

Figure5. Typicalmesh(left),zoomedinaroundthefiber(topright)anddiscretisationofthefiber
Figure 3. Meshing
(bottomright).
On the other hand, the effect of matrix spalling should be investigated during the
4. NumericalResultsandDiscussion
simulation of inclined fibre pullout. As crack grows, increasing bending of the
In this section, mesh sensitivity analysis is first conducted along the fiber transverse and
debonded segments of the fibre occurs and the most superficial pieces of matrix
longitudinaldirectionsinordertodeterminetheparticularmeshtobeemployedforfurtherstudies.
wedge tend to spall off. Some literatures provided methods to calculate the spalled
Second, the entire pullout load vs. slip displacement curves are extracted to compare with those
obtainedexperimenletnaglthly (BbryanLdte (u1n98g5)a, nLdaraSnhjeairpa i(r2o01[02)6. ]A.fTtehr igredom,tehtrey vanodn dMimeisnesisonsst roef ssspaallnedd thefirstprincipal
stressevolutionsarweedegxep isl odreteedrmbinoetdh, thfoisr wtehdegefi cbane brea enstdabtlihsheedm aas tar cixom.pFoinneanlt lbye,fothree caplcuulllaotiuont. workisobtained.
Furthermore, ANSYS can deactivate specific elements. After being killed, these
4.1. MeshSensitivityAnalysis
elements contribute near-zero stiffnesses to the overall stiffness matrix and the
Themeshsensitivityanalysisiscarriedoutfortheinclinationangleof30◦ andtheyieldstrength
solution-dependent state variables (such as stress, plastic strain, creep strain, etc.) are
of 635MPa (Type 3in Table 2). Two kinds ofmesh sensitivities are studied: therefinementin the
10
transversedirectio nandalongtheaxialdirectionofthefiber. Theformeristocheckthecapacityof
themeshinthefibertobearbendingmoments,whereasthelatteristoassessifthediscretisationis
fineenoughtoresolvethesliplength. Inthetransversedirection,thefiberissplitintoone,twoor
fourdivisions(seeFigure6a);thecorrespondingload-displacementcurvesareplottedinFigure6b.
Ontheonehand,convergenceisachievedasthemeshisrefined. Ontheotherhand,slightdifferences
among the three curves are perceivable. This means that the mesh sensitivity in the transversal
directionisrathermodest. Meshesoftwodivisionsacrossthetransversedirectionareemployedfor
furtherstudies.
Alongthelongitudinaldirectionofthefiber,fourdifferentelementsizesareconsidered,0.303mm,
0.222 mm, 0.135 mm and 0.068 mm, which lead to 33, 45, 74 and 148 divisions along the 10-mm
length(seeFigure7a);thecorrespondingpulloutloadvs. slipenddisplacementcurvesaredepictedin
Figure7b. Notethatasthemeshgetsfiner,theresponseissmoother,andconvergenceisachieved.
Specifically, the same peak load is obtained for the two finer meshes. In order to keep a balance
betweenthecomputationalefficiencyandaccuracyoftheresultssought,themeshsizeof0.135mmis
selectedforfurtherstudies. Itneedstobeemphasizedthatthemeshsensitivityinthelongitudinal
directionismorepronouncedthanthatinthetransversedirection. Inaddition,fromFigure7b,itis
observed that the maximum pullout load was achieved at a slip displacement of 0.07 mm, which
agreeswiththestatementofMortonandGroves[30],whoclaimedthatthisvalueshouldbeofthe
orderof,butlessthanhalf,afiberdiameter.
Materials2016,9,800 9of24
80
1 division

2 divisions
60 4 divisions
)N
( d
a
ol tu40
o
llu
P
20
0
0 2 4 6 8 10
Displacement (mm)

(a) (b)
Figure6.(a)Differentnumbersofdivisionsinthefibertransversedirection(one,twoandfour);and
(b)thecorrespondingpulloutloadvs.displacementresponses.

Coarse Fine Finer Finest
(a)
80

)N 70 60
( d
ao 40
l tu 50
o
llu 20
P
30 0
0 0.1 0.2 0.3 0 2 4 6 8
Displacement (mm)
Coarse Fine Finer Finest
(b)
Figure7.(a)Differentelementsizes(numberofdivisions)alongthefiber;and(b)thecorresponding
pulloutload-displacementresponses.
Materials2016,9,800 10of24
4.2. ValidationagainstExperimentalPulloutLoadvs. DisplacementResponse
Inordertoverifythepreviouslydevelopedmethodology,wecomparetheentirepulloutcurves
withtheirexperimentalcounterpartsgivenbyLeungandShapiro[26].Thiscomparisonisdisplayedin
Figure8forfibersinclinedat30◦ and60◦ withfourdifferentyieldstrengthsgiveninTable2. Notethat
boththepeakloadsandthegeneraltendencyarewellcaptured;thenumericalcurvesfallwithinthe
experimentalrange;inparticular,therisingtailattheendofeachpulloutprocessisalsoreproduced.
70 70
60 60
50 50
40 40
30 30
20 20
10 10
0 0
0 2 4 6 8 0 2 4 6 8
275MPa-30° 275MPa-60°
70 70
60 60
50 50
40 40
30 30
20 20
10 10
)
N 0 0
( d 0 2 4 6 8 0 2 4 6 8
a 469MPa-30° 469MPa-60°
o
l tu 70 70
o 60 60
llu 50 50
P
40 40
30 30
20 20
10 10
0 0
0 2 4 6 8 0 2 4 6 8
635MPa-30° 635MPa-60°
70 70
60 60
50 50
40 40
30 30
20 20
10 10
0 0
0 2 4 6 8 0 2 4 6 8
954MPa-30° 954MPa-60°
Displacement (mm)

Experiment Numerical simulations

Figure8.Numerical-experimental[26]comparison:completepulloutcurvesforthefouryieldstrengths
giveninTable2;thefiberisinclinedeitherat30◦(leftcolumn)or60◦(rightcolumn).
Itneedstobepointedoutthatasthefiberyieldstrengthincreases,thenumericalpulloutcurves
exhibitmoreoscillations. Thismeansthattherelativeslipbetweenfiberandmatrixismoreunstable;
thus,finermeshesareneededtoreducethenoiseobservedinthosecurvesforfibersofhigherstrength.
4.3. StressEvolutionwithintheFiber
Taking Type 2 fiber (yield strength 469 MPa) with an inclination angle of 30◦ as an example,
thevonMisesstressevolutionsforseveralcharacteristicpointswithinthefiberareexamined. These