Calibration of the modified Mohr-Coulomb fracture model by use of localization analyses for three tempers of an AA6016 aluminium alloy

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ContentslistsavailableatScienceDirect

International Journal of Mechanical Sciences

journalhomepage:www.elsevier.com/locate/ijmecsci

Calibration of the modiﬁed Mohr-Coulomb fracture model by use of localization analyses for three tempers of an AA6016 aluminium alloy

Henrik Granum

^a^,^∗

, David Morin

^a^,^b

, Tore Børvik

^a^,^b

, Odd Sture Hopperstad

^a^,^b

aStructural Impact Laboratory (SIMLab), Department of Structural Engineering, NTNU – Norwegian University of Science and Technology, Trondheim, Norway

bCentre for Advanced Structural Analysis (CASA), NTNU, Trondheim, Norway

a r t i c le i n f o

Keywords:

Ductile fracture AA6016 Uncoupled damage Crack propagation Numerical simulations

a b s t r a ct

ThispaperpresentsanovelcalibrationprocedureofthemodifiedMohr-Coulomb(MMC)fracturemodelbyuse oflocalizationanalysesandappliesitforthreetempersofanAA6016aluminiumalloy.Thelocalizationanalyses employtheimperfectionbandapproach,wheremetalplasticityisassignedoutsidethebandandporousplasticity isassignedinsidetheband.Ductilefailureisthusassumedtooccurwhenthedeformationlocalizesintoanarrow band.Themetalplasticitymodeliscalibratedfromnotchtensiontestsusinginversefiniteelementmodelling.The porousplasticitymodeliscalibratedbyuseoflocalizationanalyseswherethedeformationhistoriesfromfinite elementsimulationsofnotchandplane-straintensiontestsareprescribedasboundaryconditions.Subsequently, localizationanalysesareusedtoestablishthefailurelocusinstressspaceforproportionalloadingconditionsand thustodeterminetheparametersoftheMMCfracturemodel.Finiteelementsimulationsofnotchtensionand in-planesimplesheartestsaswellastwoloadcasesofthemodifiedArcantestareusedtovalidatethecalibrated fracturemodel.Thepredictionsbythesimulationsareingoodagreementwiththeexperiments,eventhough somedeviationsareseenforeachtemper.Theresultsdemonstratethatlocalizationanalysesareacost-effective andreliabletoolforpredictingductilefailure,reducingthenumberofmechanicaltestsrequiredtocalibratethe MMCfracturemodelcomparedtothehybridexperimental-numericalapproachusuallyapplied.

1. Introduction

Modellingandsimulationofductilefractureinmetallicmaterialsis anactiveresearchﬁeldwheresigniﬁcantprogresshasbeenmadeover thelastdecades.Thisresearchisimportantsinceindustriesliketheau- tomotiveindustrywanttoutilizethematerialstothebrinkoffailure.

Thus,thedemandforaccuratepredictionsoffracturebynumericalsim- ulationsisincreasing.Reliabledesignofstructuralcomponentsagainst ductilefracturerequiresarobustnumericalframeworkabletoaccu- ratelydescribethedamageandfracturepropertiesofthematerial.In manylightweightmetals,whichhavereceivedspecialattentionbythe automotiveindustryinrecentyears,strengthandductilityareinversely proportionalproperties.Asstrengthisoftenfavouredinthiscase,the ductilityimposesagreatchallengein designofsafetycomponentsof suchmaterials.

Nucleation,growthandcoalescenceofmicroscopicvoidsatvarious lengthscalesisknowntobethephysicalmechanismgoverningductile failure.Studieshaveagreedthatthestressstateaﬀectstheductilityofa metallicmaterial[1–3].Theinﬂuenceofthehydrostaticstressstatewas discoveredearlyandhassincebeenincludedinseveralfracturemodels.

∗Correspondingauthor.URL:https://www.ntnu.edu/kt/fractal. E-mailaddress:henrik.granum@ntnu.no(H.Granum).

URL:http://www.ntnu.edu/kt/fractal(H.Granum)

Morerecently,theinfluenceofthedeviatoricstressstateonductilityhas beenproventhroughexperiments,see[4],forexample.Thisledtopro- posalsofbothnewandmodifiedversionsofexistingfracturemodelsto incorporatethisdependence.Avarietyofapproachestomodelductile fracturearecurrentlyavailable.Notablementionsareporousplasticity, continuumdamagemodels,forminglimitcurvesanduncoupleddam- agemodels.Thelatterapproachispopularduetoitssimplicity,where thedamageevolutionis uncoupledfromtheconstitutiveequationin contrasttoporousplasticityandcontinuumdamagemodels.Material degradation isthusnotaccountedforandthedamageismerelyrep- resentedbyascalarvariable.Thiscomeswiththeadvantagethatthe fracturemodelmaybecalibratedindependentlyoftheplasticitymodel, simplifyingtheidentificationofmodelparameterssignificantly.Theun- coupledfracturemodelsareusuallypresentedonlocusformwherethe failurestrainisdefinedbythestressstate.Bythisapproach,thevalid- ityofthefailurestrainisconfinedtoproportionalloadingpaths.Dam- ageisoftenaccumulatedbyanintegral-basedapproachwheredamage evolveswithincrementsoftheequivalentplasticstrainovertheplastic strainpath.Byemployingsuchadamageaccumulationapproach,the modelisjustifiedintheliteraturetobevalidinsimulationsinvolving

https://doi.org/10.1016/j.ijmecsci.2020.106122

Received29June2020;Receivedinrevisedform15September2020;Accepted27September2020 Availableonline2October2020

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curvesfromthesimulationstothoseoftheexperiments.Theaccuracy ofthisapproachreliesonexperimentsthatcoverarangeofstressstates andexhibitclosetoproportionalloadingpathsallthewaytofracture.

Thelatterrequirementisdifficulttofulfilformoststressstates.Asanal- ternativetothisapproach,localizationanalysesmaybeusedtopredict ductilefailure.Anunderlyingassumption hereisthatstrainlocaliza- tionisaprecursortofailure,andthusmayberegardedastheonsetof fracture.Mechanicaltestsarethenonlyneededtocalibratetheconstitu- tiveequationsusedinthelocalizationanalysesandthereisnorequire- mentforproportionalloadingpaths.Morinetal.[7]combinedunitcell simulationsandlocalizationanalysestopredictfailureofasteelunder non-proportionalloading.Thenumericalresultswerevalidatedagainst experimentalresultsreportedbyBasuandBenzerga[8]andfoundtobe ingoodagreement.Theversatilityandeffectivenessofthelocalization analysesweredemonstratedbyMorinetal.[9]wherefailureloci of metalsweregeneratedfromlocalizationanalysesandappliedtoanad- vancedhigh-strengthsteelsubjectedtoproportionalloadingpaths.The resultswereevaluatedagainst3DunitcellanalysesbyDunandandMohr [10]andproventogivecomparableresultsinafractionofthecompu- tationaltime.Grubenetal.[11]appliedanexperimental-numericalap- proachtodeterminethestrainlocalizationandductilefractureoftwo dual-phasesteels.Fourteststhatcoveredstressstatesfromsimpleshear toequi-biaxial tensionwereconducted.Numericalsimulationsofthe testswereperformed,andthefailurestrainswereestimatedbycom- parisontotheexperimentaldata.Localization analysesbyuse ofthe imperfectionbandapproachwereconductedtopredicttheonsetoflo- calization.Theresultsindicatedthatthelocalizationanalysesprovided conservativevalues ofthefailurestrainsandthatthestressstatein- sidethebandtendstomovetowardsageneralizedshearstatepriorto localization.Bergoetal.[12]usedunitcellsimulationsandlocaliza- tionanalysestocalibratefailurelociforthreedifferentsteels.Thestudy wasconfinedtogeneralizedtensionstressstatesandthusonlythede- pendenceonstresstriaxialitywasincluded.Theuncoupledplasticity andfracturemodelswerecalibratedbasedonasingleuniaxialtension testandmicromechanicalsimulationsusingunitcells,metalandporous plasticityandlocalizationtheory.Thepredictedductilitywassomewhat conservativeforWeldox460Eandnon-conservativeforWeldox900E, butaccurateforWeldox700E.Itwasemphasizedthattheaccuracyof thelocalizationanalysesreliesheavilyonanaccuratecalibrationofthe porousplasticitymodel.However,itwasnotedthatthisapproachis wellsuitedtoreducetheexperimentalprogrammerequiredtocalibrate fracturemodels.

Wierzbickietal.[13],Lietal.[14],Grubenetal.[15]andBaietal.

[16]haveallpresentedstudiescomparingthepredictivecapabilitiesof diﬀerentuncoupledfracturecriteriaforvarioussteelsandaluminium alloys.Severalofthecriteriaevaluatedareheuristicextensionsofwell- knowncriterialike theMohr-Coulomb(MC),Cockcroft-Latham (CL), Rice-Tracey(RT)andWilkinscriteriatonameafew.AsWierzbickietal.

[13]pointedout,thequalityofafracturecriterionintendedforindus- trialapplicationmayberoughlymeasuredbytheperformanceandthe cost.Here,performanceisdeﬁnedbytheaccuracyofsimulationscom- paredtoexperimentaltests,whilethecostisrelatedtothenumberof mechanicaltestsneededtocalibratethemodelandthecomplexityre- latedtothis.Amongthecriteriawithonlyonemechanicaltestrequired forcalibrationistheCLcriterion.Ithasbeenusedwithsuccessinmany

T351aluminiumalloyandaTRIP690steel,andvalidateditagainstvar- iousmechanicaltests.Accuratepredictionsoffractureinitiationwere obtained,butitwasnotedthatthepredictionswerelessaccurateingen- eralizedtension.DunandandMohr[20]investigatedthepredictiveca- pabilityoftheMMCfracturemodelbycomparingpredictionstofracture experimentsonTRIP780steel.Ninedifferentexperimentswereusedin thecomparisoncoveringwiderangesofstresstriaxialityandLodepa- rameter.Fractureinitiationwascorrectlypredictedinallsimulationsof thetests.Itwassuggestedthattheunderlyingphysicsofthefracture modelisoflessimportancethanitsmathematicalflexibility,implying thateventhoughphenomenologicalfracturemodelsaremotivatedby micromechanicalobservations,theirabilitytofitexperimentaldatais asuperiorcharacteristic.However,afracturemodelwithhighflexibil- ityallowserroneouscalibrationandrequiresdetailedknowledgebythe user.

A modiﬁcationof theMMC fracturemodeldenotedthe Hosford- Coulomb (HC) fracturemodelwas proposed byMohr andMarcadet [5], wherethevonMisesequivalentstresswas replacedbytheHos- fordequivalentstressincombinationwiththenormalstressactingon theplaneofmaximumshear.TheHCfracturemodelisbasedonthe extensivestudyon3DunitcellsbyDunandandMohr[10],thusithas amicromechanicalfoundationincontrasttotheMMCfracturemodel.

TheHCfracturemodelwaspresentedonlocusform,anddamageac- cumulationwastakencareofbyanintegral-basedapproach.Fracture experimentsonthreediﬀerentsteelswereconductedandthreeexper- iments wereused tocalibratethefracturecriterion.Whencompared againsttheexperiments,thesimulationsshowedgood agreementfor thethreematerialsandfracturewasaccuratelypredictedinallcases.

GorjiandMohr[21]andZhangetal.[22]investigatedductilefracture inthealuminiumalloyAA6016.InGorjiandMohr[21],theHCfrac- turemodelwasemployedincombinationwithananisotropicplasticity modeltopredictshearfractureindeepdrawingtests.Eightcupdraw- ingexperimentswereusedinthecalibrationprocesstoincreasethero- bustnessofthefracturemodel.Theresultsshowthattheplasticityand fracturemodelscanpredictthelocationandtheonsetoffracturewith goodaccuracy.InZhangetal.[22],ananisotropicDruckeryieldfunc- tionandafracturecriterionproposedbyLouetal.[23]wereemployed, whereasimplesheartestandtwonotchtensiontestswithdiﬀerentradii wereusedinthecalibrationofthefracturecriterion.Bycomparingthe experimentstothesimulationsitwasfoundthattheonsetoffracture wasaccuratelypredictedintestsrangingfromsimplesheartouniaxial tension.

Inthepresentstudy,weapplyanisotropicplasticityandfracture modeltopredictductilefractureinvariousexperiments,wherespeci- mensaretakenfromAA6016aluminiumalloysheetsinthreediﬀerent tempers.TheMMCfracturemodelwasselectedandcalibratedbyuseof localizationanalysesbasedontwomechanicaltests.Thestudyisanat- uralextensiontotheworkbyBergoetal.[12]whereastresstriaxiality dependantfracturemodelwasinvestigatedtogetherwithanisotropic plasticitymodel.Theaimofthestudyistoassesstheaccuracyofthe calibratedMMCfracturemodelbycomparisonagainstmechanicaltests, wherethemodel’sabilitytopredictfractureinitiationandcrackpropa- gationisevaluatedforarangeofstressstates.Theresultsdemonstrate thattheuseoflocalizationanalysestocalibrateafracturemodelhas

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Table1

ThechemicalcompositionofAA6016inwt%.

Si Mg Fe Cu Mn Cr Zn Ti Al

1.3160 0.3490 0.1617 0.0081 0.0702 0.0025 0.0084 0.0175 Balance

thepotentialtobeacost-eﬀectiveandaccuratewayofpredictingduc- tilefractureandcrackpropagation.

2. Materialsandmechanicaltests 2.1. Materials

Experimentswereconductedonthreedifferenttempersofthealu- miniumalloyAA6016.Thematerialsweredeliveredas1.5mmthick plateswith in-plane dimensions625mm ×625 mm in tempers T4, T6andT7byHydroAluminiumRolledProductsinBonn.Thisalloy ismainlyusedintheautomotiveindustryasouterbodypanelsdueto itsexcellentsurfacequality,goodformability,andhighstrength.Toob- tainthevarioustempers,allplateswerefirstsolutionheat-treatedat 530°CbeforebeingairquenchedtoreachtemperT4.TempersT6and T7werethenobtainedforsomeoftheplatesbyartificialageingfor5h at185°Candfor24hat205°C,respectively.Thechemicalcomposi- tionofthealloyisgiveninTable1.Theyieldstrengthofthetempers rangesfrom about135MPaforT4to245MPafor T6,andtheulti- matetensilestrengthrangesfromroughly200MPaforT7tojustbelow 300MPaforT6.Allmechanicaltestswerecarriedoutwiththelongi- tudinalaxisalongtherollingdirection,unlessspecifiedotherwise.The initialthicknessofallspecimenswasmeasuredandfoundtobesimilar tothenominalplatethicknessof1.5mm.AnInstron5985seriesuniver- saltestingmachinewasusedinalltests,wheretheforcewasmeasured bya30kNloadcellattachedtotheactuator.AProsilicaGC2450camera orientatedperpendiculartothein-planeaxesofthespecimencaptured imagesfromalltests.Allspecimenswerespray-paintedwithaspeckle patterntoenable2Ddigitalimagecorrelation(2D-DIC)byuseofthe in-housesoftwareeCorr[24].

2.2. Uniaxialtensiontests

InGranumetal.[25],uniaxialtensiontestsinthreediﬀerentdirec- tionswithrespecttotherollingdirection(0°,45° and90°)oftheplates wereconducted.Additionaluniaxialtensiontestsintherollingdirec- tionoftheplatewereconductedonlyfortemperT4inconjunctionwith thematerialtestprogrammepresentedinthisstudy.Thiswasdoneto monitorthenaturalageingthatoccursintemperT4underprolonged roomtemperaturestorage,resultinginsoluteclustering.Thiseﬀectis knowntoslightlystrengthenthealloy.Thespecimenhadagaugelength of70mmandawidthof12.5mm,andisdepictedinFig.1a).Thetests wereconductedwithacrossheadvelocityof2.1mm/min,resultingin aninitialstrainrateof5 ×10⁻⁴s⁻¹inthegaugeregion.Avirtualex- tensometerwithaninitiallengthof60mmwasusedtoextractdisplace- mentsbyuseof2D-DIC.

2.3. Notchtensiontests

Notchtensiontestsintherollingdirectionwithtwodiﬀerentnotch radiiwereconducted,withgeometryinspiredbythespecimensusedin Ericeetal.[26].ThetwospecimensaredenotedNT10andNT3,and thegeometriesaredepictedinFig.1b)andFig.1c),respectively.The NT10specimenhadanotchradiusof10mm,whiletheNT3specimen hadanotchradiusof3.35mm.Theminimumwidthofthenotchwas 5mmforbothgeometries.Thespecimensweretestedwithacrosshead velocityof0.6mm/min.Forcemeasurementfromtheloadcellandim- agesfromthecamera wererecordedat2Hz.Twosetsofvirtualex- tensometersavailableineCorrwereusedinthepost-processingofthe

experiments,oneglobalandonelocal.Theinitiallengthoftheglobal andlocalvirtualextensometerswas15mmand2mm,respectively,for bothtestgeometriesandthevirtualextensometerswereplacedcentric tothenotchradius.

2.4. Plane-straintensiontests

Thegeometryoftheplane-straintension(PST)specimenisdepicted inFig.1d).Ithadalengthof100mmandawidthof40mm.Theopening ofthenotchwas10mmandthewidthatthenarrowestpointinsidethe notchwasmeasuredto17.33mm.Theforcewasmeasuredbytheload cellandimagesweretakenbythecameraat2Hz.Thetestswerecon- ductedwithmechanicalclamps,wheretheclampedregiononeachside ofthegaugewasapproximately40mm×35mm.Priortotesting,the clampedregionsofthespecimenweresandeddowntoensuregoodgrip betweentheclampsandthespecimen.Thetestswereconductedwith acrossheadvelocityof0.4mm/min.Twoglobalvirtualextensometers withaninitiallengthof10mmpositionedapproximately16mmfrom thecentreofthenotchwereusedtoextractthedisplacementsintheDIC calculations.ThedisplacementsfromtwovirtualextensometersineCorr wereusedtoensurethatnorotationswereenforcedduringloading.A localvirtualextensometerwithagaugelengthof2mmplacedcentric acrossthenotchwasusedtoobtainlocalmeasurementsfromthetests.

2.5. In-planesimplesheartests

Thein-planesimpleshear(ISS)specimenhadagaugelengthof5mm andthegeometryisdepictedinFig.1e).Thetestswereconductedwith acrossheadvelocityof0.15mm/mininanattempttoobtainaninitial strainrateinthegaugeregionof 5 ×10⁻⁴s^-1.Avirtualextensome- terineCorrspanningacrossthegaugeregionwithaninitiallengthof 10.05mmwasusedtoextractdisplacements.Acameraaimedperpen- diculartothein-planesurfacecapturedimagesandforcemeasurements wererecordedbytheloadcell,bothat1Hz.

2.6. ModiﬁedArcantests

SixmodiﬁedArcanspecimenswerecutfromaplateofeachtem- per.ThegeometryofthespecimenisgiveninFig.1f).Thespecimen wasclampedbyfourloadingbracketsusing12M6-boltsasshownin Fig.2.Twodiﬀerentloadcaseswereapplied byalteringtheloading direction𝛽;three with𝛽 = 90° andthree with𝛽 = 45° asshown in Fig. 2a)andFig. 2b), respectively. All tests wereconducted witha crossheadvelocityof1mm/min,similartothosecarriedoutin[27]. Thetestsareabbreviated“Arcan𝛽” todistinguishbetweenthetwoload cases.IntheArcan90tests,thespecimenisloadedinatensionmode wheretheloadaxiscoincideswiththespecimen’slongitudinalaxis.The pinconnectingthemountingbracketstothetestmachineallowsthe mountingbracketsandspecimentorotatewhenloaded,enablingthe specimentotearopen.IntheArcan45tests,thespecimenissubjected toamixed-modeloading.Thewidthatthenarrowestpointinthegauge sectionwasmeasuredto32mmwithnegligiblevariationsbetweenthe specimens.Avirtualextensometerofinitiallength10.5mmspanning acrossthenotchalongthelongitudinalaxisofthespecimenineCorr wasusedtoextractdisplacementsbyuseof2D-DIC.

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Fig.1. Geometryoftestspecimenswithmeasuresinmm:a)uniaxialtension,b)andc)notchtension,d)plane-straintension,e)in-planesimpleshearandf)modiﬁed Arcan.

Fig.2. TestsetupofamodiﬁedArcanspecimenwitha)𝛽=90° andb)𝛽=45°

3. Experimentalresults 3.1. Uniaxialtensiontests

Engineeringstress-straincurvesofrepresentativetestsfromGranum etal.[25]areplottedinFig.3a).Aslightvariationinelongationat fracturebetweenthedifferenttensiledirectionsisseenforeachofthe tempers,whiletheflowstressshowedamaximumdeviationof3%.The LankfordcoefficientsR_𝛼 werecalculatedfor alltensiontests andare listedinTable2,where𝛼denotestheanglewithrespecttotherolling direction.Allcoefficientsarebelowunityandsomewhathigherinthe rollingdirection.Thesimilarityinflowstressbetweenthethreedirec- tionssuggeststhatthealloyexhibitsisotropicpropertieswithrespect tostrengthandstrainhardening.However,theLankford coefficients suggestthatthematerialisprone tothinningandexhibitsmoderate

Table2

LankfordcoeﬃcientsR_𝜶forrepre- sentativetensiontests.

Temper R 0 R 45 R 90

T4 0.58 0.45 0.44 T6 0.69 0.48 0.55 T7 0.77 0.57 0.62

anisotropyinplasticﬂow.Theequivalentstrainattheonsetoffracture wasfoundbyuseofDICwhereacharacteristicelementsizeof0.57mm wasused.MultipleDICsimulationswithaslightlyshiftedpositionof themeshwereconductedtoavoidresultsbiasedbytheDICmesh.For thetestsintherollingdirection,theaveragelogarithmicfracturestrain cameoutas0.70,0.33and0.73fortempersT4,T6andT7,respectively.

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Fig.3. Engineeringstress-straincurvesfrom:a)representativetestspresentedinGranumetal.[25]andb)representativetestsintemperT4conductedinconjunction withthemechanicalteststomonitortheeﬀectofnaturalageing.

Thesevaluesrepresentthestrainwherefractureinitiatesatthesurface ofthespecimen.

Theengineeringstress-straincurvesforthreesetsoftemperT4tests arepresentedinFig.3b).TheT4–1curveisarepresentativetestinthe rollingdirectionfromFig.3a).T4–2isatestconductedinconjunction withthemodifiedArcantestsandT4–3isatestconductedinconjunc- tionwiththeplane-straintensionandin-planesimplesheartests.The numberingofthetestsindicatestheordertheywereperformedin,i.e., theT4–1testwasconductedatanearlierpointintimethantheT4–2 test,whichwasperformedpriortotheT4–3test.Itisseenthatnatural ageingincreasesthestrengthastheT4–2andT4–3curvesareshiftedup- wardscomparedtotheT4–1curve.However,thenaturalageingseems tohavereachedsaturationwhentheArcantestsweredone,asthedif- ferencebetweentheT4–2andT4–3curvesisnegligible.Thiseffectwas alsoinvestigatedbyEngleretal.[28]forthesamematerialandthe readerisreferredtothisworkforathoroughdiscussionofthis phe- nomenon.DuetothenegligibledifferencesbetweentheT4–2andT4–3

curves,theneedformultiplecalibrationsoftheplasticitymodelforthis temperwasdeemedunnecessary.

3.2. Notchtensiontests

TheexperimentalresultsfromtheNT10andNT3testsareshownin Fig.4.Therepeatabilityofthenotchtensiontestswasexcellent,and thusonlyonetestperconﬁgurationisplotted.Thedisplacementwas extractedfrom theglobalvirtualextensometerwhilethelogarithmic strainiscalculatedfromthedisplacementmeasuredbythelocalvir- tualextensometer.Theforcelevelsbetweenthetwotestgeometriesare similarwhilethedisplacementsareslightlylargerforthelargestnotch radius.Thelocalstrainisalsoseentobehigherinthetestswiththe largestnotchradius.Thisisexpectedasthesharpernotchradiuscon- ﬁnesthegaugesectionmorethanthelargernotchradius,resultingin higherstresstriaxialitieswithinthisregion.

Fig.4. Experimentalresultsfromthea)NT10andb)NT3testsintermsofforce-displacementandlogarithmicstrain-displacementcurves.

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Fig.5. Fracturesurfacesofa)NT10andb)NT3testsforthethreetempers.

Fig.6. Force-displacementandlogarithmicstrain-displacementcurvesfrom plane-straintension(PST)tests.

Fracturedspecimensfromthesixdifferentnotchtensiontestswere examinedbyvisualinspectionandareshowninFig.5.Onlyonetest perconfigurationisshownasnegligibledifferenceswereseenbetween fracturesurfacesofrepeatedtests.Aslantfracturesurfacewasfound foralltests,eventhoughsomeNT3testsdisplayed roughshearlips.

Ingeneral,thefracturesurfaceswererougherfortemperT7thanfor tempersT4andT6, andshear lipsweremoreprominentin theNT3 teststhanintheNT10tests.

3.3. Plane-straintensiontests

Theforce-displacementandlogarithmicstrain-displacementcurves fromrepresentativeplane-straintensiontestsareshowninFig.6.Du- plicatetestsarenotshownduetotheexcellentrepeatability.Theresults areinaccordancewiththetrendsseenforthenotchtensiontests,where themostprominentdiﬀerencebeingthesimilarityinelongationatfrac- turebetweentempersT6andT7.Thefractureinitiatedin thecentre ofthespecimenforalltestsandpropagatedinastraightlinetowards thefreeedges,perpendiculartotheloadingdirectionasseeninFig.7. ForthetestsintemperT6,thecrackpropagatedinstantlyresultingin asuddenlossofload-carryingcapacity,whilethetestsin temperT4 exhibitedslightlyslowercrackpropagation.OnlytheT6–3testexperi- encedcompletefracture,wherethespecimenwaspulledapart.Inthe restofthetests,theforceleveldroppedbelowathresholdlimitatwhich thetestwasstoppedautomatically.ForthetestsintemperT7,thecrack propagatedslowlyandittookapproximately40sfrominitiationtocom- pletion.ByinspectionofthefracturedT6–3specimenshowninFig.7,

aslantfracturesurfacewasobserved,wherethecrackwasseentoﬂip totheotheradmissibleshearband.

3.4. In-planesimplesheartests

Theforce-displacementcurvesfromthesheartestsareshowntothe leftinFig.8wherethenumberedmarkerscoincidewiththenumbered imagesontheright-handside.Owingtoslightscatter,resultsfromall repeattestsarepresented.Thestrainfieldsobtainedby2D-DICreveal thatstrainslocalizedinathinbandacrossthegaugesectioninalltests (notshownforbrevity).Byinspectionoftheimages,fractureseemingly occurredsimultaneouslywithinthisband,astheoriginoffractureiniti- ationwasdifficulttopinpointexactly.In-planerotationswereobserved inalltests,resultinginanangleofthelocalizeddeformationbandcom- paredtotheloadingdirection,asevidentfrom theimagesin Fig.8. Theangleofthebandwithrespecttotheloadingdirectionwassimilar between repetitionsandtempers.Thedropin theforce-displacement curves,particularlyfortemperT7,ispresumedtooccurduetothecom- binedeffectofmaterialsofteningandareareductioninthelocalized deformationband.ThesheartestsindicatedthattemperT7hassupe- rior ductilitycomparedtotempersT4andT6, andthehighductility makes itdifficulttopinpointtheonsetof fracturein thetemper T7 tests.Byinspectionoftheimages,theonsetoffractureisanticipated tooccursomewherebetweenpoint2and3intherepresentativeforce- displacementcurvefortemperT7inFig.8.Allspecimensdisplayeda smoothandflatfracturesurfacethroughthethickness,andtherewere negligibledifferencesamongrepetitionsandtempers.

Theforce-displacementcurvesfromthemodiﬁedArcan45andAr- can90tests areshownin Fig.9a)andFig.9b),respectively.Overall, thetrendsareinaccordancewiththeothermechanicaltestspresented showingthattemperT6givesthehighestpeakforcefollowedinturn bytempersT4andT7.Thepeakforcesareconsistentlyhigherinthe Arcan90teststhanintheArcan45tests,andtheratiobetweenthepeak forcesinthetwoloadcasesisalmostidenticalamongstthetempers.The displacementatpeakforceissmallerintheArcan45teststhaninthe Arcan90tests,buttheﬁnaldisplacementislargerintheformer.This islinkedtothecrackpathsbeinglongerintheArcan45teststhanin theArcan90tests,especiallyfortempersT4andT6.AsFig.9b)indi- cates,theArcan90-T6testexperiencedasuddenlossofload-carrying capacityasthecrackpropagatedinstantlyacrossthespecimenbetween twoimagesofthetest.EventhoughtemperT7isthemostductileal- loycondition,thetestsintemperT4exhibitsthelargestdisplacements.

Thisislinkedtothecombinationofadequatestrength,work-hardening andductilityintemper T4,whichseemstobemorefavourable than thehighductilityandlowwork-hardeningseenfortemperT7inthese tests.

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Fig.7.Fracturedplane-straintensionspecimensandfracturesurfaceoftestspecimenT6–3.Theredlinesonthepicturesindicatethecrackpath.(Forinterpretation ofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.8.Force-displacementcurvesfromthein-planesimpleshear(ISS)testswherethenumberedmarkerscoincidewiththeimagesontheright-handside.

Fig.9. Force-displacementcurvesfroma)theArcan45testsandb)theArcan90tests.

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Fig.10. FracturedspecimensshowingthediﬀerentfracturemodesofthemodiﬁedArcantestswithcorrespondingfracturesurfaces.

Fracturedspecimens fromthe modifiedArcan tests areshownin Fig.10.Intheupperpartofthefigure,thethreedifferentfracturemodes observedinthetestsaredisplayed.Acurvedcrackpathwasobservedin boththeArcan45-T4andtheArcan45-T6tests.IntheArcan45-T6tests, thecrackwasarrestedapproximately10mmfromtheedgeandthetests werestoppedastheforceleveldroppedbelowalowerthreshold.Inthe Arcan45-T4tests,thecrackwasarrestedapproximately5mmfromthe edgewhenstopped,butduringthedismantlingofthespecimensfrom theloadingbrackets,thespecimenswerepulledapart.Byinspectionof thefracturesurfacesfromtheArcan45-T4tests,allthreetestsexhib- itedthealternatingslantfracturephenomenon.Thecrackinitiatedand propagatedinaslantfracturemodeuntilarrested.Byinspectionofthe Arcan45-T6specimen,thefractureinitiatedinaV-modeandpropagated inthismodeforapproximately3mmbeforeittransitionedtoaslant fracturemode.Thealternatingslantfracturephenomenonwasnotob- servedinanyofthesetests.TheV-modewasalsoseeninGrubenetal.

[29]forArcan45specimensmadeofDocol600DLsteel.IntheArcan45- T7tests,fractureinitiatedinaV-modewithinthesameareaasforthe Arcan45-T4andArcan45-T6tests,andthecrackinitiallyfollowedasim- ilarcurvedpath.However,thecrackpathdeﬂectedabruptlyafterafew millimetresandpropagatedperpendicularlytothelongitudinalaxisof thespecimenasshowninFig.10.Thefracturesurfaceswereroughwith evidentshearlips.AlltheArcan90testsexhibitedasimilarcrackpath forallthreetempers,butdiﬀerencesinthefracturesurfaceswereseen.

IntheArcan90-T4tests,thefractureinitiatedandpropagatedinaslant fracturemodewherethecrackwasseentoﬂipabruptlythroughoutthe deformationinalltests.OneoftheArcan90-T6testsinitiatedinaV- modebeforeatransitiontoslantfracturewasseen,whilethetwoothers initiatedinaslantfracturemode.BoththeArcan90-T4andArcan90-T6 testsshowedasmoothfracturesurface,whiletheArcan90-T7testshad arougherfracturesurface,similartowhatwasseenintheArcan45-T7 tests.

4. Modellingandsimulation 4.1. Stressstateparameters

Anyadmissiblestressstatecanbe expressedbythethreeordered principalstresses𝜎1≥𝜎2≥𝜎3givenby

𝜎1= 2

3𝜎vMcos(𝜃)+𝜎h (1a)

𝜎2= 2 3𝜎vMcos

(𝜃−2𝜋 3

)

+𝜎h (1b)

𝜎3= 2 3𝜎vMcos

(𝜃+2𝜋 3

)

+𝜎h (1c)

where0≤𝜃≤^π₆ isthedeviatoricangle,𝜎vM=√

3𝐽2 isthevonMises equivalentstress,and𝜎h=I₁/3isthehydrostaticstress.Here,J₂andI₁ arethesecondprincipaldeviatoricstressinvariantandtheﬁrstprinci- palstressinvariant,respectively.Thestressstatemaybeconveniently describedbythestresstriaxialityTandtheLodeparameterL.Thestress triaxialityisdeﬁnedastheratiobetweenthehydrostaticstress𝜎hand thevonMisesequivalentstress𝜎vM,viz.

𝑇= 𝜎h

𝜎vM

(2)

TheLodeparameterdescribesthedeviatoricstressstate,andisde- ﬁnedeitherintermsofthedeviatoricangle𝜃ortheorderedprincipal stresses(𝜎1,𝜎2,𝜎3)as

𝐿=√ 3tan

(𝜃−𝜋 6 )

= 2𝜎2−𝜎1−𝜎3

𝜎1−𝜎3

(3)

TherangeoftheLodeparameterisL∈[−1,+1],whereL=−1,0 and+1representstatesofgeneralizedtension,generalizedshearand generalizedcompression,respectively.

(9)

4.2. Plasticitymodel

Theconstitutiverelationisgivenbythehigh-exponentyieldsurface proposedbyHershey[30]andHosford[31],theassociatedﬂowrule andanextendedVocehardeningrule.Eventhoughthematerialexhibits moderateplasticanisotropyaccordingtotheLankford coeﬃcient,an isotropicplasticitymodelisapplied.Theequivalentstressisgivenby theprincipalstressesas

𝜎eq=(1

2(||𝜎1−𝜎2||^𝑎+||𝜎2−𝜎3||^𝑎+||𝜎3−𝜎1||^𝑎))¹𝑎

(4) whereaisaparametercontrollingthecurvatureoftheyieldsurface.

Thisparameteris settoa= 8in thisstudyassuggestedbyHosford [32]basedonpolycrystalplasticitycalculations.Theyieldfunctionis expressedas

𝜙= 𝜎eq− 𝜎M= 𝜎eq−(

𝜎0+𝑅(𝑝))

≤0 (5)

where𝜎Misthematrixmaterialﬂowstress,𝜎0istheyieldstress,Risthe hardeningvariableandpistheequivalentplasticstrain.Thehardening variableisdeﬁnedbyanextendedVocehardeningruleontheform 𝑅(𝑝)=

∑3 𝑖=1

𝑅_𝑖(𝑝)=

∑3 𝑖=1

𝑄_𝑖( 1−exp(

−𝐶_𝑖𝑝))

(6)

whereR_iarehardeningtermsthatsaturateatdiﬀerentlevelsofplastic strain.ThehardeningparametersQ_iandC_iworkinpaircontrollingthe strainhardeningofthematerial.

4.3. MMCfracturemodel

FractureinthesimulationsisgovernedbyamodiﬁedMohr-Coulomb fracturemodel.TheversionoftheMohr-Coulombmodelusedin this studywasinspiredbytheoneproposed byBaiandWierzbicki[19]. Theytransformedtheoriginalmodelintolocusformwherethefailure strain̄𝜀f,i.e.,theequivalentplasticstrainatfailure,wasdeﬁnedinterms ofthestresstriaxialityTandtheLodeangleparameter ̄𝜃 .Thelatter isanormalizedparameteroftheLodeangleandiswithin therange

̄𝜃∈ [−1,+1].Inthisstudy,theLodeparameterLisusedinsteadofthe Lodeangleparameter ̄𝜃,andtheexpressionforthefailurestrain ̄𝜀f is thengivenas[19]

𝜀f(𝐿,𝑇)=

⎧⎪

⎨⎪

⎩ 𝐾̂𝐶2

[

̂𝐶3+

√3 2−√

3 (̂𝐶₄^∗− ̂𝐶3

)(sec(−Lπ 6

)

−1)]

×

⎡⎢

⎢⎣

√ 1+ ̂𝐶₁²

3 cos(−𝐿𝜋 6

) + ̂𝐶1

(𝑇+1 3sin(−𝐿𝜋

6 ))⎤⎥

⎥⎦

⎫⎪

⎬⎪

⎭

−¹ 𝑛

(7)

where

̂𝐶₄^∗=

{1 for −1≤𝐿≤0

̂𝐶4 for 0<𝐿≤1 (8)

Themodelhassixparametersthatmustbecalibrated: ̂𝐶1 governs thepressuredependence; ̂𝐶2andKcontroltheoverallductility; ̂𝐶3de- terminestheLodeparameterdependence; ̂𝐶4governstheasymmetryof thefailurelocusbetweenstatesofgeneralizedtensionandcompression;

andincreasingvaluesofnshifttheductilityupwardsanddecreasethe stresstriaxialityandLodeparameterdependence[19].

Damageis introduced by the damage variableD which is setto evolvewithincrementsoftheequivalentplasticstrainp,givenas 𝐷(𝑝)=

∫

𝑝 0

d𝑝

𝜀f(𝐿,𝑇) (9)

Thematerialisundamagedinitsinitialconﬁguration,i.e.,D=0, andfractureinitiateswhenD=1.Whereasthefailurelocusisvalidfor proportionalloadingonly,thedamageaccumulationruleisassumedto accountfornon-proportionalloadpathsinanapproximateway.

4.4. Finiteelementmodelling

The finiteelement (FE) simulations of themechanical tests used in the calibration process were conducted with the implicit solver of Abaqus[33]withdisplacement-controlledloading.Allsimulations withtheMMCfracturemodelwereconductedusingtheexplicitsolver of Abaquswith velocity-controlledloading. Thespecimensweredis- cretizedbyuseof8-nodelinearbrickelementswithselectivereduced integration,denotedC3D8inAbaqus.Fractureismodelledbyelement erosion,whereelementsareremovedwhenDinEq.(9)reachesunity.In thesimulationsoftheNT3,NT10andPSTtests,threesymmetryplanes wereutilized,andthereducedmodelswereoptimizedwithrespectto thenumberofelements.Thevalidityofthereducedmodelswithop- timizedmeshdesignwas verifiedagainstselected simulationsof the fullspecimenwithadense,uniformmesh.Thedifferencesinthepre- dictedcrackinitiationandpropagationbetweensimulationswiththe optimizedanduniform meshdesignswerenegligible.Themaximum timestepintheimplicitsimulationswasselectedsothateachsimula- tionhadaround200increments.Allsimulationswereconductedwith 5elementsoverthehalf-thicknessandanin-planediscretizationinthe gaugeregionwithacharacteristicelementsizeof0.15mm.Thisresulted ininitiallycubic-shapedelementsinthegaugeregion.

Anin-planeviewoftheFEmodelswiththemeshdepictedonthe initialgeometryofthespecimensisshowninFig.11.Inallsimulations, exceptforthesimulationsofthePSTtests,theloadwasassignedtoa referencenodepositionedinthecentreofthepinhole.AnMPCbeam constraintwasusedtoconnectthereferencenodetotheelementset ontheboundaryofthespecimen,tomimictheeffectofapinpulling thespecimen.Thisisvisualizedbytheredlinesandthebluereference nodesinFig.11.Thisapproachlimitsthenumberofelementsinthe FEmodelssignificantlyandpresumesthattheomittedpartmovesasa rigidbody.Also,therotationsinducedbythepinnedlinkisrecognized inthismodellingapproachbyallowingthereferencenodetorotatein- plane.InthesimulationofthePSTtest,theclampedareawasassumed tobehaveasarigidbodyandthusomittedinthemodel.Theboundary conditionswerethusassignedtotheedgesborderingtotheclamped regionsofthemodel.TheISSandmodifiedArcantestspecimenswere modelledaccordingtoFig.11d)-e),whereonlythrough-thicknesssym- metrywasutilized.IntheISSmodel,tworeferencenodesconnectedto theedgesofthespecimenwereappliedtoallowforin-planerotations.

InthemodifiedArcanmodel,thepartofthespecimengrippedbythe clampingframewaspresumedtomoveasarigidbodyandwasthus omittedfromthemodel.In-planerotationswereaccountedforbyin- sertingreferencenodescoincidingwiththecentreofthetwoloading pinsshowninFig.2.Thereferencenodeswereconnectedtotheedges ofthespecimenbyanMPCbeamconstraint.Thisapproachsimplifies themodelsubstantiallyandenablesfeasiblecomputationaltimeswith thedesireddiscretization.Inallexplicitsimulations,thevelocitywas rampedupoverthefirst10%ofthesimulationtime,andtheenergy balancewascarefullycheckedtoensurequasi-staticloadingconditions.

4.5. Localizationanalyses

The localizationanalyses were conductedusing the imperfection bandapproach,followingtheworkbyRice[34].Adetaileddescription oftheapproachusedinthisstudycanbefoundinMorinetal.[9],and onlyabriefoverviewisgivenherein.Asolid,homogenousbodythatis homogeneouslydeformedisconsidered.Withinthisbody,athinplanar bandisassumedtoexistwherestressandstrainratesareallowedtobe diﬀerentfromtheirvaluesoutsidetheband.However,continuingequi- libriumandcompatibilityacrosstheimperfectionband areenforced.

Thenormaltothebandisdenotednandasketchofasolidbodywith aplanarbandisdepictedinFig.13.Localizationissettooccurwhen thestrainrateinsidethebandbecomesinﬁnite.Thecriticalorientation ofthebandisnotknownonbeforehandandlocalizationanalysescov- eringarangeofbandorientationsmustbeconductedtoﬁndtheone

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Fig.11. Finiteelementmeshesoftestspecimen:a)NT3,b)NT10,c)PST,d)ISSande)modiﬁedArcan.

Fig.12. Experimental(crosses)andnumericalforce-displacementandlogarithmicstrain-displacementcurvesofthethreetempersfora):NT10andb):NT3.

exhibitingthelowestductility.Thefailurestrainistakenastheequiva- lentplasticstrainoutsideoftheimperfectionbandatlocalizationinside theband.

The solid body is governed by theplasticity model described in Section4.2.Insidetheband,aporousplasticitymodelisusedtorepre- sentthematerialbehaviour,whichenablesasimpleapproachtointro- duceanimperfectionbynucleationandgrowthofvoids.Theheuristic extensionoftheGurson-Tvergaardmodel[35,36]proposedbyDæhli etal.[37]isadopted,wheretheyieldconditionisgivenby

Φ = (𝜎eq

𝜎M

)2

+2𝑞1𝑓cosh (3𝑞2

2 𝜎h

𝜎M

)

−( 1+𝑞3𝑓²)

=0 (10)

where𝜎eqistheHershey-HosfordequivalentstressdefinedinEq.(4), 𝜎MistheflowstressofthematrixmaterialaccordingtoEq.(5),q₁,q₂,q₃ aretheTvergaardparameters,𝜎histhehydrostaticstress,andfisthe voidvolumefraction.Theevolutionofvoidvolumefractionisdefined as

̇𝑓= ̇𝑓g+ ̇𝑓n+ ̇𝑓s=(1−𝑓)tr𝐃^p+𝐴ṅ𝑝+𝑘s𝑓𝜅(

𝝈^′)𝝈^′∶𝐃^p 𝜎eq

(11)

where ̇𝑓gisthevoidgrowthrate, ̇𝑓nisthevoidnucleationrateand ̇𝑓s

representsthecontributionfromvoidsofteninginsheartotheporosity evolution[38].TheparametersA_nandk_sgovernvoidnucleationand

voidssofteninginshear,respectively.Furthermore,𝝈′isthedeviatoric stresstensor,D^pistheplasticrate-of-deformationtensordeﬁnedbythe associatedﬂowruleand𝜅(𝝈′)isafunctionofthesecondandthirdin- variantofthedeviatoricstresstensor,J₂andJ₃,respectively,viz.

𝜅( 𝝈^′)

=1−27 4

𝐽₃²

𝐽₂³ (12)

Byincludingthetermforvoidsofteninginshearintheevolution equation,thephysicalmeaningofthevoidvolumefractionfislostandit shouldbeinterpretedasadamageparameter,assuggestedbyNahshon andHutchinson[38].ThereaderisreferredtoDæhlietal.[37]fora detailedaccountoftheporousplasticitymodelusedinthelocalization analyses.

5. Calibration

5.1. Calibrationofhardeningparameters

Thecalibrationprocedurefollowsasimilarapproachasemployed ine.g.MohrandMarcadet[5].Theuniaxialtensiontestswereusedto obtainaninitialestimateofthehardeningparameters.Aspreadsheet wasusedtoﬁttwoofthethreehardeningtermstothetruestress-strain curveuptonecking.InversemodellingoftheNT10testsbyuseofthe

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Fig.13. CalibrationapproachfortheporousplasticitymodelparametersA_nandk_s.TheplotshowsdataforthetemperT6calibration.

Fig.14. Comparisonbetweenresultsfromthelocalizationanalyses(SLM)andthecalibratedMMCfracturemodel.

Table3

MaterialparametersoftheextendedVocehardeningrule.

Temper 𝜎₀(MPa) Q ₁(MPa) C ₁ Q ₂(MPa) C ₂ Q ₃(MPa) C ₃ T4 135.0 19.04 87.05 142.22 10.06 75.00 3.08

T6 245.1 6.45 438.98 109.39 11.13 2.58 9.05

T7 152.8 3.76 2316.10 57.11 38.34 25.81 4.59

optimizationtoolLS-OPT[39]wasthenemployedtocalibratethehard- eningparameters.Theinitialestimatewasusedasastartingpointin theoptimization,wherethefirsthardeningtermwaskeptfixedandthe secondandthirdhardeningtermscouldchange.SequentialFEsimula- tionswerethenconductedwithdifferenthardeningparameterswhere LS-OPTemployedageneticoptimizationalgorithmtofindtheoptimal setofparameters.Theforce-displacementcurvesfromtheNT10tests wereusedastargetsintheoptimizations.Thefiniteelementmodelem- ployedispresentedinSection4.4andthecalibratedhardeningparam- etersaredisplayedinTable3.Theforce-displacementandlogarithmic strain-displacementcurvesfromthesimulationsareplottedassolidlines togetherwiththeexperimentsascrossesforthetwonotchtensiontests inFig.12.Thegoodagreementbetweenthesimulationsandexperi- mentsforbothtestsillustratesthevalidityofthecalibratedplasticity model.

5.2. CalibrationoftheMMCfracturemodel

Thepredictivecapabilityof thelocalizationanalysesrelies onan accuratedescriptionofthematerialbehaviour insideandoutsidethe imperfectionband.Theplasticitymodelusedoutsidethebandwascal- ibratedasdescribedintheprevioussection.Fortheporousplasticity

modelusedinsidetheband,theparametersintroducedbyTvergaard [36]weregivenstandardvalues,i.e.,q₁=1.5,q₂=1.0,𝑞3=𝑞²₁=2.25. ThenucleationrateA_nandthevoidshearingparameterk_swerecal- ibratedbasedonlocalizationanalysesfollowingtheprocessdepictedin Fig.13,whereastheinitialvoidvolumefractionf₀wassettozero.The deformationgradientF(t)wasextractedfromthecriticalelementinthe through-thicknesscentreofanNT10andaPSTsimulationandassigned asboundaryconditionsinlocalizationanalyses.Aseriesoflocalization analyses withvaryingnucleationrateA_n andvoid shearingparame- terk_swasconducted.AccordingtoNahshonandHutchinson[38],the voidshearingparameterissuggestedtobeintherange1≤ k_s≤3for structuralalloys,thusthreevalueswithinthisrangewereinvestigated:

k_s={1.0, 2.0, 3.0}.TheoptimalvalueofA_nwasfoundwhenlocaliza- tionoccurredforastrainoutsidethebandsimilartotheexperimental failurestrainforagivenk_s,asdepictedintheplotinFig.13.Theoptimal valueofk_swasnotsearchedforandthecalibratedvalueofk_swascho- senfromthethreevaluesinvestigated.Theexperimentalfailurestrain wastakenfromthecriticalelementinasimulationbasedonbothglobal andlocalmeasurementsfromDICresults.Ingeneral,failurewasfound toinitiatejustbeforetheﬁnaldipintheforcelevels,asitwasassumed thatstrainlocalizationinitiatedatthispoint.TheplotinFig.13shows theresultsfromthelocalizationanalysesfortemperT6wheretheopti-

(12)

Fig.15. FailurelocioftheMMCfracturemodelforthethreediﬀerenttempersofAA6016.

Table4

Calibratedvaluesofthepa- rametersintheporousplas- ticitymodel.

Temper A n k s

T4 0.006 2.0 T6 0.0125 2.0 T7 0.005 2.0

malvalueofthenucleationrateA_nisdepicted.Thecalibratedvaluesof theparametersintheporousplasticitymodelaregiveninTable4.

Withthemetalandporousplasticitymodelscalibrated,localization analyseswithproportionalloadingconditionswereconducted,i.e.,as- signingadeformationwherethestresstriaxialityTandtheLodepa- rameterLareconstantduringtheentiresimulation.Fromtheseanal- yses,failurestrainscoveringaconsiderableregionofthestressspace wereobtained,eveniflocalizationwasnotachievedforalltheapplied

stressstates.TheparametersoftheMMCfracturemodelwerefinally optimizedagainstthiscloudofpointsinaPythonscript.Approximately 100pointswereusedineachoptimizationtoensureasolidbasisfor theidentification.Thebasin-hopping algorithm[40]availableinthe SciPypackage[41]wasemployedtodeterminetheoptimalsetofmodel parameters.Thealgorithmaimstofindtheglobalminimumofacost function,heredefinedasthedifferencebetweenthefailurestrainscal- culatedwithMMCfracturemodelandlocalizationanalysesforawide rangeofstressstates.Withintheglobalsteppingalgorithm,alocalmin- imizationiscarriedoutusingasequentialleastsquaresprogramming (SLSQP)method[42].Approximately60basin-hoppingiterationswere performedtofindtheoptimalsetofparametersusingthedefaultpa- rametersofthealgorithm(thereaderisreferredto[41]formoredetails uponthesenumericalaspects).Thecalibratedparametersaregivenin Table5.

ThecalibratedMMCfracturemodelandthetargetpointscalculated bythelocalizationanalyses(giventheabbreviationSLM)areshownin Fig.14forselectedvaluesofthestresstriaxiality.Fromtheﬁgureitisev- identthatthemaintrendsarecapturedinthecalibrationofthefracture

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Fig.16. Comparisonbetweenexperimental(crosses)andnumerical(solidlines)force-displacementandlocalstrain-displacementcurvesoftemperT4:a)NT10,b) NT3,c)PSTandd)ISS.Allcurvesareplottedtofracture.

Table5

CalibratedparametersofthemodiﬁedMohr-Coulombfracturemodelbased onlocalizationanalyses.

Temper K ̂𝐶 1 ̂𝐶 2 ̂𝐶 3 ̂𝐶 4 n

T4 0.9969 0.01000 0.5075 0.8820 1.0056 0.01122 T6 0.9988 0.01135 0.5081 0.8847 1.0066 0.01000 T7 0.9611 0.00100 0.4815 0.8672 1.0005 0.00138

model,eventhoughthefitaroundgeneralizedshear(L=0)isnotalways good.ThedependenceonthestresstriaxialityaroundL=0issmallac- cordingtothelocalizationanalyses,resultinginthepointsforL=±0.2 insomecasesoverlappingeachother.Thisbehaviourisnotaccurately capturedbythecalibratedfracturemodel,whereanevidentdependence onthestresstriaxialityisseenaroundL=0.Thefitisaccurateforgen- eralizedtensionandcompressionforallthreetempers.Thesomewhat flatfailurelocuspredictedbythelocalizationanalysesisamongstother linkedtotheuseofaHersheyyieldsurfacewithexponenta=8.Thein- fluenceoftheyieldsurfacecurvatureonductilefailurewasinvestigated byDæhlietal.[37],whereHersheyyieldsurfaceswithexponenta=2 (i.e.,equaltothevonMisesyieldsurface)anda=8wereinvestigated byuseoflocalizationanalyses.Theresultssuggestthatayieldsurface

witha=8displaysaﬂatterfailurelocuscomparedtotheyieldsurface witha=2.ThereaderisreferredtoDæhlietal.[37]forfurtherdetails ontheinﬂuenceoftheyieldsurfacecurvatureonductilefailure.

The resultsfrom thecalibration of theMMC fracturemodelsare showninFig.15.Theoverallshapeofthethreefracturesurfacesinthe leftcolumnofthefigureissimilar,wherebothanevidentstresstriaxial- ityandLodeparameterdependenceisseen.Themonotonicdecreasein ductilityforincreasingstresstriaxialityisshowninthemiddlecolumn ofthefigureforgeneralizedcompression(L=1),generalizedtension (L=−1)andgeneralizedshear(L=0).Therateofdecreaseinductil- ityforincreasingstresstriaxialityissimilarforgeneralizedcompression andgeneralizedtensionforalltempers.However,therateofdecrease inductilityforincreasingstresstriaxialityismuchlowerforgeneralized shear,especiallyfortempersT4andT6.Generalizedshearexhibitsthe lowestductilityfollowedbygeneralizedtensionandgeneralizedcom- pression,wherethedifferencebetweenthetwolatterissmall.Temper T7exhibitsamuchstrongerdependenceonthestresstriaxialityforgen- eralizedshearcomparedtotempersT4andT6.Thisisclearlyvisualized intherightcolumnwherethefailurelociareplottedasafunctionof theLodeparameterforselectedvaluesofthestresstriaxiality.Itshould benotedthatthisstrongstresstriaxialitydependencewasnotpredicted bythelocalizationanalysesandisaresultofthecalibrationoftheMMC

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fracturemodel.Theductilityinsimpleshearisconsiderablyhigherfor temperT7thanfortemperT6,whiletemperT4issomewherein between.Theasymmetryofthefailurelociisevidentforalltemperswhere thelowestductilityforselectedvaluesofthestresstriaxialityisfound forslightlynegativevaluesoftheLodeparameter.Thehighductilityfor T=0andL=±1fortemperT6,higherthanforbothtempersT4andT7, issomewhatpeculiar.Thelocalizationanalysesdidnotprovideresults forthesestressstatesandthispartofthefailurelocusisobtainedby extrapolation.However,forthestressstatesachievablebyexperiments, thefailurelocusfortemperT6exhibitslowerductilitythanthefailure locifortempersT4andT7.

Thehighest andlowestductility ontheplane stressfailurelocus is found forthe stress statesrepresenting uniaxialtension (L = −1, T=0.33)andplane-straintension(𝐿=0,𝑇=1∕√

3),respectively.Itis worthnotingthattheductilityishigherinuniaxialtensionthaninequi- biaxialtension(L=1, T=0.67)foralltempers.Thisdiﬀerencewould notbepossible toexpresswiththeHosford-Coulombfracturemodel, wheretheductility isforcedtobeequalinuniaxialandequi-biaxial tension.Thecuspontheplane-stressfailurelocusforuniaxialtension andthevalleytowardssimpleshear(L=0,T=0)foralltemperscatego- rizesthesematerial’sfracturebehaviourasLodeparameterdominated.

Astresstriaxialitydominatedfracturebehaviourwouldexhibithigher

ductilityinsimpleshearcomparedtouniaxialtension,andthusnocusp wouldappearintheplane-strainfailurelocusforuniaxialtension.

6. Numericalresultsanddiscussion 6.1. Materialtests

Theresultsfromthesimulationsofthematerialtestsonthenotch tension(NT),plane-straintension(PST)andin-planeshear(ISS)speci- menswiththeMMCfracturemodelareshowninFigs.16–19.Theex- perimentalresultsarepresentedasdiscretecrosses,whilethesolidlines representthesimulations.Bycomparisonoftheresponsecurvesfortem- perT4showninFig.16,thepredictionsbytheMMCfracturemodelare ingeneralfoundtobegood.Whencomparingtheforce-displacement curvesforthefourtests,theagreementforthenotchtensiontestsisex- cellent,whiletherearesomedeviationsforthePSTandISStests.These deviationsareexpectedwhenmodellingamoderatelyanisotropicma- terialwithanisotropicyieldsurface.Theonsetoffractureisaccurately predictedfortheNT10test,whileitisslightlyconservativefortheNT3 andPSTtests.FortheISStest,theresponsecurvesdeviatealreadyat yieldingandtheonsetoffractureispredictedforalargerdisplacement thanintheexperiment.Thereasonforthisdeviationislinkedtothe

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textureofthealloy,requiringananisotropicyieldcriteriontocapture thebehaviourasdiscussedinAchanietal.[43].Engleretal.[28]in- vestigatedthemicrostructureandtextureofanAA6016sheetintemper T4whereacharacteristiccuberecrystallizationtexturewasfound.This textureleadstoarelativelyhighyieldstressinshearcomparedtouni- axialtension[43].Thus,theyieldstressinasheartestisnotexpected tobeaccuratelypredictedwithanisotropicyieldsurface.Considering thatthetextureisnotalteredbytheheat-treatment,thisbehaviourisex- pectedforalltempers.TheconflictingfracturepredictionfromthePST andISStestsillustratesthedifficultiesoffindingasetofparametersthat accuratelydescribestheonsetoffractureinbothtests.

TheresultsfromsimulationswiththeMMCfracturemodelfortem- perT6areshowninFig.17.ThepredictionoffractureinthePSTtestis slightlyconservative,whilethefracturepredictionsfortheNT10,NT3 andISStestsareslightlynon-conservative.Theimpressiveaccuracyin thepredictionsofthetemperT6testsisevidentastheleastaccuratepre- dictionisobtainedfortheNT10test,whichwasusedinthecalibration.

Asexpected,thedeviationsbetweentheforce-displacementcurvesfor theISStestinitiatedalreadyatyield.Despitethis,accurateprediction ontheonsetoffractureisalsoobtainedforthistest.

TheresultsfromthesimulationswiththeMMCfracturemodelfor temperT7areshowninFig.18.Theagreementbetweentheexperimen- talandnumericalresponsecurvesandthepredictedonsetoffractureis

excellentfortheNT10,NT3andPSTtests.Theonlynotabledeviation amongstthesetestsistheshiftinthelocalstrainforthePSTtests,re- sultinginaslightlyhigherstrainattheonsetoffractureinthesimula- tioncomparedtotheexperiment.Asmentionedearlier,theexactonset offractureintheISStestisdiﬃculttodeterminebasedontheforce- displacementcurves.Thedisplacementatwhichtheonsetoffractureis predictedinthesimulationappearstobeareasonableestimatewhen inspectingimagesfromthetestatasimilardisplacement.

The predictive capability of the MMC fracture model has been demonstratedintermsofglobalandlocalresponseparametersforfour diﬀerent materialtests.InFig.19,thestressstatehistoriesextracted fromsimulationsofthesameexperimentsarepresented.Thesolidlines aretakenfromsimulationswherethecriticaldamageparameterisset artiﬁciallyhighandtheendofthelinesindicatetheonsetoffracturein theexperiments.Fractureintheexperimentswasdeterminedbasedon bothlocalandglobalmeasurementsforNT10,NT3andPSTtests,while globalmeasurementswereusedforISS.Thedotsindicatetheonsetof fracturepredictedbytheMMCfracturemodel.Thestressstatescovered bytheexperimentsincludemainlynegativevaluesoftheLodeparame- terandpositivevaluesofthestresstriaxiality.Thestressstatehistories aretakenfromthethrough-thicknesscentreelementfortheNT10,NT3 andPSTtests,whichcorrespondstotheelementsubjectedtothelargest equivalentplasticstrain.InthesimulationsoftheISStests,theelement

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Fig.19. Evolutionofthestressstate(i.e.,Lodeparameterandstresstriaxiality)asfunctionoftheequivalentplasticstrainextractedfromthecriticalelement.The predictedfracturebytheMMCfracturemodelisindicatedbythedotswhiletheendofthesolidlinesgivestheonsetoffractureinthetests.

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Fig.20. Experimentalandnumericalforce-displacementcurvesfortheArcan45testsina),c)ande)andcorrespondingcrackpathsontheundeformedconﬁguration inb),d)andf).

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Fig.21. Experimentalandnumericalforce-displacementcurvesforthea)Arcan90-T4,b)Arcan90-T6andc)Arcan90-T7tests.

subjectedtothelargestvalueofthedamageparameterDatthedisplace- mentoffractureintheexperimentistakenasthecriticalelement.The damageparameterwasusedtodeterminethecriticalelementsincethe largestvalueoftheequivalentplasticstrainwasfoundonthethrough- thicknesssurfacewithinthenotch.Thisregionis heavilyaffectedby thein-planerotationsthatoccurinthetests,whichmakestheelement subjectedtothelargestvalueofequivalentplasticstrainanunsuitable choicefortheISStests.ThechosencriticalelementinallISStests is locatedonthein-planesurfacewithinthegaugeregionwherestrains localize.Amongtheeightintegrationpointswithinthecriticalelement, theonesubjectedtothelargestvalueofequivalentplasticstrainischo- seninalltests.WhencomparingthepredictionsbytheMMCfracture modelwiththeexperimentalvaluesinFig.19,i.e.,comparingthedots totheendpointofthesolidlines,thetrendsaresimilartotheonesin Figs.16–18.ByinspectionofFig.19,itisevidentthatthestressstate historiesarequitesimilaramongthedifferenttempersapartfrominthe ISStest.Thereasonforthisisthevaryingpositionofthecriticalelement amongthesimulationsoftheISStest.Thedifferenceinstrength,work-

hardeningandductilitybetweenthethreetempersresultsindifferent deformationprocesseswhichaffectthepositionofthecriticalelement, emphasizing thedifficultiesfacedwithanin-planesimplesheartest.

RothandMohr[44]investigatedthechallengesrelatedtodetermining thestraintofailureforsimpleshearforawiderangeofsheetmetals.

Amongstthestudy’sconclusions,itwasstatedthattheshapeofthespec- imenplaysasignificantroleandthatdifferentmaterialpropertiessuch asstrength,work-hardening andductility requiredifferent shapesof thespecimen.ByinspectionofthestressstatehistoriesfortheISStests, temperT6isclosesttoexhibitaproportionalloadingpath,suggesting thatthegeometryoftheISStestspecimenissuitableforthematerial propertiesofthistemper.Ideally,boththestresstriaxialityandtheLode parametershouldbeequaltozeroallthewaytofractureinasheartest.

EspeciallytheISStestfortemperT7exhibitsaloadingpathwhereTand Lvarymarkedlythroughoutthedeformation,makingitlesssuitableto useastargetinacalibrationprocess.Thisisoneofthereasonswhythe PSTtestwaschoseninordertocalibratethevoidshearingparameter k_s intheporousplasticitymodelandnottheISStest.Consideringthe

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Fig.22. ThestrainﬁeldsofArcan45-T6andArcan90-T7fromDICandFEsimulationstakenwhenthecrackhadpropagatedapproximatelyhalfwaythroughthe specimen.

rathersimpleplasticitymodelchosenandthatonlytwomaterialtests wereusedinthecalibration,thepredictionsbytheMMCfracturemodel aredeemedsatisfactory.

Tofurtherassess thepredictive capabilities of theMMC fracture model,themodiﬁedArcantestswith𝛽=45° and𝛽=90° weresimulated.

Here,themodel’sabilitytopredictbothcrackinitiationandpropagation istested.InFig.20theforce-displacementcurvesfromexperimentsand simulationsoftheArcan45testsareshowntotheleft,withcorrespond- ingcrackpathsontheundeformedconﬁgurationtotheright.Despite thesmallinaccuraciesseeninthepredictionsforthematerialtests,ex- cellentagreementbetweentheexperimentalandnumericalresultsis seenfortheArcan45tests,bothintermsofforce-displacementcurves andcrackpaths.Theonsetoffractureisinitiatedatthecorrectdisplace- mentandpositiononthespecimenforallthreetempers.Additionally, thesimulatedcrackpropagationoccursmostlyalongthecorrectpaths atsimilarvelocitiesastheexperimentalones.Eventhesomewhatsur- prisingstraightcrackpathseenintheArcan45-T7testwaspredicted accurately.Theslanted fracturesurfaceobservedin theexperiments wasnotpredictedinanyofthesimulations.Topredictslantedfracture, thethrough-thicknesssymmetrymustbeabandonedandamuchdenser meshaccompaniedbyacoupleddamagemodelismostlikelyrequired [45].However,thecrack inthetemperT7simulationpropagatedin atunnellingmodefrominitiationtocompletefracture.Thestresstri- axialityinsidethenotchwherefractureinitiatedwasbetween0.3and 0.4,whiletheLodeparameterwasapproximatelyequalto−1forall tempers.Justinfrontofthepropagatingcrack,aregionwithstresstri- axialitybetween0.6and0.7andaLodeparameterclosetozerowas presentforalltempers.

Fig.21 showstheexperimentalandnumericalforce-displacement curvesfortheArcan90tests.Theonsetoffracturewasaccuratelypre- dictedfortempersT4andT7,whilefortemperT6fractureoccurred slightlylater inthesimulation thaninthe experiment.Additionally,

therewasaslightdeviationintheforcelevelinpartsoftheresponse curvebeforetheonsetoffracturefortemperT6ofunknownreasons.

ThecrackpropagationwasaccuratelypredictedfortempersT4andT7, wheretheagreementbetweentheexperimentalandnumericalresponse curveswasgoodthroughoutthewholedeformationprocess.Theveloc- ityofthepropagatingcrackfortemperT6wasnotaccuratelycaptured in thesimulations, wherealowervelocitythaninthetestswaspre- dicted.

Theonsetof fracturein thesimulationwas foundtooccurafew millimetreswithinthenotchandnotatthefreesurface.Thisoccurred eventhoughthelargestvalueoftheequivalentplasticstrainwasfound on thefree surface.However,byinspection ofthestressstateatthe onsetoffracture,theregioninsidethenotchwasfoundtobesubjected toahigherstresstriaxialityandaLodeparameterclosertozerothan onthefreesurface.Fractureinitiatedinthisregionbeforepropagating perpendicularly totheloadingdirection.Theagreementbetweenthe crack patternin theexperimentandsimulationwasexcellent forall tempersandisnotshownforbrevity.

ThestrainﬁeldsofanArcan45-T6andArcan90-T7testareshown in Fig.22 fromboth DICandFEsimulations. Thestrainﬁelds were takenwhenthecrackhadpropagatedapproximatelyhalfwaythrough thespecimeninbothtests.Themagnitudeofthestrainsisconsistently higherintheFEsimulationsthanintheDICsimulationsowingtothe densermeshusedintheformer.However,thequalitativetrendsare similar between thetwosetsof simulationsforbothtests.Anarrow zonewithlocalizedstrainsinfrontofthepropagatingcrackiscorrectly predictedinbothcases. IntheArcan45-T6test,thereisaband with slightlyhigherstrainsacrossthespecimenwhichisnotfullydeveloped intheFEsimulationsandthusonlypartiallypredicted.

7. Conclusions

Thispaperhaspresentedanovelcalibrationprocedureofthemod- iﬁedMohr-Coulomb(MMC)fracturemodelbyuseoflocalizationanal- ysesoftheimperfectionbandtypeandapplieditforthreetempersof