The role of quench rate on the plastic flow and fracture of three aluminium alloys with different grain structure and texture

three aluminium alloys with different grain structure and texture

Bjørn Håkon Frodal

^a,b,∗

, Emil Christiansen

^b,c

, Ole Runar Myhr

^a,d

, Odd Sture Hopperstad

^a,b

aStructural Impact Laboratory (SIMLab), Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), Trondheim NO-7491, Norway

bCentre for Advanced Structural Analysis (CASA), NTNU, Trondheim NO-7491, Norway

cDepartment of Physics, Norwegian University of Science and Technology (NTNU), Trondheim NO-7491, Norway

dHydro Aluminium, Research and Technology Development (RTD), Sunndalsøra NO-6601, Norway

a r t i c l e i n f o

Article history:

Received 29 October 2019 Revised 4 February 2020 Accepted 9 February 2020

Keywords:

Quench sensitivity Ductile fracture Precipitate free zones Finite element simulations Anisotropic porous plasticity

a b s t r a c t

Theyielding,plastic flowandfractureof age hardenablealuminiumalloysdependon the quenchratetoroomtemperatureafterthesolutionheat-treatmentatelevatedtemperature andbeforetheartificialageing.WeinvestigatethreeAlMgSialloyswithdifferentgrainstruc- tureandcrystallographictextureexperimentallytodeterminetheeffectsofquenchrate(ei- therwater-quenchingorair-cooling)ontheprecipitatemicrostructureand themechanical properties,i.e., yieldstress,workhardening andductility. Tensile testson smooth andV- notchspecimensand Kahnteartests areperformed tostudytheinfluenceof stressstate onplasticflowandfracture.Inaddition,finiteelementsimulationsofthemechanicaltests areperformedforoneofthealloystoinvestigatethevalidityofanextensionoftheGurson model tohigh-exponentanisotropicplasticity. Transmissionelectronmicroscopy investiga- tionsshowthatthealloysandtheirprecipitationmicrostructurearedifferentlyaffectedby thequenchrate.Commonforthethreealloysisthattheprecipitatefreezonesarounddisper- soidsandgrainboundariesbecomelarger,andtheyieldstrengthofthealloysbecomeslower, afterair-coolingthanafterwater-quenching.ThenanostructuremodelNaMowasmodifiedto accountforprecipitatefreezones,andwasabletopredictboththeprecipitationparameters andthetensileyieldstrengthofalltempersandmaterialswithareasonabledegreeofac- curacy,exceptinonecase.Inthiscase,theinhomogeneousprecipitationinthematerialis toocomplextobecapturedbytheinherentprecipitationmodelinNaMo.Duetothelower yieldstrengthandhigherwork-hardeningrateafterair-cooling,thefailurestrainisincreased forthesmoothandV-notchtensiletests.Thecrackpropagationenergy,calculatedfromthe Kahnteartests,ismarkedlyaffectedbythequenchrateandtheeffectisdifferentdepend- ingonthegrainstructureandplasticanisotropy,causedbythecrystallographictexture.The anisotropicporousplasticitymodelusedinthefiniteelementsimulationsisabletoprecisely capturethefractureinitiationinallthespecimengeometriesoftheconsideredalloy,whereas thecrackpropagationenergiesoftheKahnteartestsareslightlyoverestimated.

© 2020 The Author(s). Published by Elsevier Ltd.

ThisisanopenaccessarticleundertheCCBYlicense.

(http://creativecommons.org/licenses/by/4.0/)

∗ Corresponding author at: Structural Impact Laboratory (SIMLab), Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway.

E-mail address: [email protected] (B.H. Frodal).

https://doi.org/10.1016/j.ijengsci.2020.103257

0020-7225/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license.

( http://creativecommons.org/licenses/by/4.0/ )

1. Introduction

Agehardenablealuminiumalloysarewidelyused,suchasincarbodypanels,aeroplanefuselages, andinload-bearing componentsforstructuralapplications.Highstrengthandductility aretypicallydesiredforpracticalapplications,andalu- miniumalloysareattractive duetotheir highload-bearingcapacityandlowweight. Inaddition,propertiessuchasgood formabilityandcorrosionresistance,combinedwithagreatpotentialforrecycling,makethemappealingto,e.g.,theauto- motiveandoffshoreindustry.Thethermo-mechanicalprocessingofagehardenablealuminiumalloysinfluencesmicrostruc- tural characteristics such as thegrain structure, crystallographic texture andprecipitate structure. Thus, by changing the chemical composition, heat treatment andmechanical processing, one can control the yield stress, work hardening and plastic anisotropy of the alloy. While this flexibility is desired, it can alsogive rise to processing relatedissues such as quenchsensitivity.Thequenchrateafterhomogenizationorthesolutionheat-treatmentcanaffecttheagehardeningcapa- bilityofthesealloysandslowercoolingratescanleadtoalowerstrength(Dons&Lohne,1983).Albeithighquench rates arefavourabletoattainpreferredmechanicalproperties,itisnotalwaysachievableinpracticalapplications,andinindustry itisacommonpracticetolimitthequenchratetopreventdistortion(Strobeletal.,2016).

Quench sensitivity is primarily caused by the precipitation of non-strengthening phases on dispersoids and at grain boundariesduringcooling, andthe numberdensityof thedispersoids isknown toincrease quench sensitivity (Conserva

&Fiorini,1973;Deschamps,Texier,Ringeval,&Delfaut-Durut,2009;Dons&Lohne,1983;Milkereit&Starink,2015;Strobel etal., 2019; Strobel,Easton, Sweet,Couper, &Nie, 2011; Strobel etal., 2016). Thus, the agehardenablealuminium alloys experience a solute loss due tothis precipitation,and the formationof the strengthening precipitatesduringthe subse- quentartificialageingisreduced.High strengthalloystendtobemorequenchsensitive,asthealloyingelementsprovide a greaterdriving force fornucleation ofnon-strengthening precipitatesduringquenching(Strobeletal., 2011). Whilethe dispersoidscontributesignificantlytothequenchsensitivityofthealloys,theyarewantedduetotheir abilitytodelayand hinder recrystallizationduringthe thermo-mechanicalprocessing, whichisbeneficialforthemechanicalpropertiesofthe alloy(Remøeetal.,2017).

Whereashighstrengthalloystendtobemorequenchsensitive,alsoleanalloyswithfewdispersoidshavebeenobserved toexhibitsubstantialquenchsensitivity(Milkereit,Schick,&Kessler,2010;Strobeletal.,2011;Strobeletal.,2016).Herethe quenchsensitivityhasbeenlinkedtothesupersaturationofvacancies,asthehighnumberofvacanciesafterfastquenching leadstoapossibleincreasedrateofprecipitateformationduringartificialageing(Deschampsetal.,2009;Evancho&Staley, 1974; Falahati,Lang, & Kozeschnik,2012; Seyedrezai,Grebennikov, Mascher,& Zurob, 2009; Strobelet al., 2019;Werinos etal.,2016).Thismechanismhasalsobeenfound toaffectquenchsensitivityinalloyswithahighcontentofdispersoids (Strobeletal.,2019).

Duetovacancyand/orsolutediffusion,precipitatefreezones(PFZs)formarounddispersoidsandgrainboundariesinage hardenablealuminiumalloys(Strobeletal.,2019, 2016;Unwin, Lorimer,& Nicholson,1969). SincethePFZslackstrength- eningprecipitates,thesezonesare softerthantherestofthegrain,butaretypicallystrongerthanpurealuminiumasthe PFZsretainsomesoluteinsolidsolution(Unwinetal.,1969).ThePFZslocatedadjacenttograinboundariesarethelocation andprobablythecauseofintercrystallinefracture (Lohne& Naess,1979).Plasticdeformationwilllocalizeintheseweaker zonesand crackinitiation andgrowth canoccur more easily in thePFZs (Chen,Pedersen, Clausen,& Hopperstad, 2009;

Dowling&Martin, 1976;Khadyko,Marioara,Ringdalen,Dumoulin,& Hopperstad,2016;Morgeneyer,Starink,Wang,&Sin- clair,2008).Duetothislocalizationofdeformation,thePFZscandevelopsignificantmisorientationsrelativetotheirparent grains,whichinturncould contributetothestrengtheningofthePFZs(Christiansen,Marioara,Marthinsen,Hopperstad,&

Holmestad,2018)andthusdelayfractureinitiation.

Ingeneral,acompetitionbetweenintercrystallineandtranscrystallinefractureisobservedinagehardenablealuminium alloys.Ithasbeenreportedthat thefracturemodemaychangefromtranscrystalline tointercrystalline fractureduetothe precipitation of coarse phaseson grain boundariesand the formationof PFZs, which decreasethe fracture toughness of thealloy(DeHaas&DeHosson,2002;Dumont,Deschamps,&Brechet,2003,2004a;Dumont,Deschamps,Bréchet,Sigli,&

Ehrström,2004b;Morgeneyeretal.,2008).Asaresult,fracturetoughnessmaybesignificantlymoresensitivetothequench ratethan thestrengthof thealloy(Shuey,Tiryakio˘glu,Bray,& Staley, 2006).Dumont etal.(2003, 2004a,2004b)studied two 7000 seriesaluminiumalloysusing Kahn teartestsandfound thata lower quench rateresultedin alower fracture toughness,whichwasaresultoftheincreasedintercrystallinefracturecausedbygrainboundaryprecipitationandvariations inthework-hardeningrate.Morgeneyeretal.(2008)usedsynchrotronradiationcomputedtomographytostudythefailure mechanismaheadofthecracktipinKahnteartestsfora6000seriesaluminiumalloy.Theyfoundafteraslowquenchthat fracturewasmainlyintercrystallineandthattherewasrelativelylittledamageevolutionpriortocrackinitiation.

In metallicmaterials, the mechanismfordamage evolutionandductile fracture isnucleation, growthandcoalescence ofmicroscopic voids(Pineau,Benzerga,& Pardoen,2016). Voidsmaynucleateatconstituentparticles orinclusionseither by decohesion or by particle cracking (Maire, Zhou, Adrien, & Dimichiel, 2011), or voids may pre-exist in the material (Toda etal., 2013). Based on unit cell calculationsusing the finite element method (FEM),it has been found that dam- ageevolutionandductilefailuredependonanumberoffactorsincludingthevolumefractionofvoidsandparticles,their distributionandshape,theplasticanisotropy,strengthandworkhardeningofthematrixmaterial,andonthelocalstress state (Pineau etal., 2016). Modellingof ductilematerials isoften done using micromechanicallymotivatedhomogenized materialmodelssuchasporousplasticitymodels.Inthesemodels,theevolutionofmicrostructuralvariablesisincludedto describeeffectssuchasmaterialsoftening.Thesevariablestypicallyaccountforphysicalphenomenaoccurringatthelower

t 540

185

22

min15 15

min 5 h

AC WQ 50

Fig. 1. Heat-treatment of the alloys to temper T6, with either air-cooling (AC) or water-quenching (WQ) after the solution heat-treatment.

scalesof ductilefailure. Themostrenownedmodelbased onthismicromechanicalframework, isthemodel proposedby Gurson(1977).Thismodelisattractiveduetoitssimpleformulation,incorporatingonlyasinglemicrostructuralparameter throughthevolumefractionofmicroscopicvoids.TheGursonmodelhaslaterbeenenhancedtogivebetteragreementwith unitcellcalculations(Tvergaard,1981)andextendedtoincludeeffectssuch asvoidnucleationandcoalescence(Tvergaard

&Needleman,1984).Versionsofthemodelwhichtaketheplasticanisotropyofthematrixmaterialintoaccounthavealso beenproposed(Dæhli,Faleskog,Børvik,&Hopperstad,2017;Steglich,Wafai,&Besson,2010).

Inthisstudy,theeffectofquenchrateontheplasticflowandfractureofagehardenablealuminiumalloysisinvestigated.

UsingtensiletestsonsmoothandV-notchaxisymmetric specimens,inadditiontoKahn teartests,theinfluenceofstress state on the plastic flow and fracture of three AlMgSi alloys with different grain structure and crystallographic texture isdetermined. The alloyswere solution heat-treated, cooled either inwater orairto achieve different quench ratesand artificiallyaged.It isfound that the quenchrateaffects theprecipitate structure ofthe alloysdifferently,and theslower coolingratetypicallyleadstowiderPFZsarounddispersoidsandgrainboundaries.Thesedifferencesinmicrostructureare reflectedinthe mechanicalbehaviourofthematerials. Inaddition,finiteelement simulationsofthe mechanicaltestsare performedfor one of the alloys, using the Gurson (1977) model extended by Dæhli et al.(2017) to account for plastic anisotropy.Failureinitiationisaccuratelycapturedinthefiniteelementsimulations,whereasthecrackpropagationenergy inthesimulationsoftheKahnteartestsissomewhatoverestimated.

2. Materials

Threealuminiumalloys,namely6060,6082.25and6082.50,arestudiedexperimentally.Thechemicalcompositionofthe alloysisgiveninTable1.ThealloyswereprovidedbyHydroAluminiumasextrudedrectangularprofileswithathicknessof 10mmandawidthof83mm.Threetypesofspecimensweremachinedfromtheextrudedprofilesformechanicaltesting, seeSection3.1.

All specimens were solution heat-treated and artificially aged to temper T6 (peakstrength). After the solution heat- treatment,thespecimenswerecooledtoroomtemperatureusingeitherair-cooling(AC)orwater-quenching(WQ)toattain twodifferentmicrostructuresforeachalloy,givingatotalofsixdistinctmaterials.Thesolutionheat-treatmentconsistedof keepingthespecimensinasaltbathat540^◦Cfor15min,followedbyeitherwater-quenchingtoroomtemperatureorair- coolingto50^◦Cfollowedbywater-quenchingtoroomtemperature.Inthelattercase,thetemperaturereached50^◦Cafter approximately20min withair-cooling.Subsequently, thespecimens were storedatroom temperaturefor 15min before theartificialageingtotemperT6.Toobtainthepeakstrengthcondition,thespecimenswerekeptinanoilbathat185^◦C forfivehours andthen air-cooledtoroom temperature.Fig.1illustrates thetemperaturehistoryofthespecimensduring theartificialageingwitheitherair-coolingorwater-quenchingafterthesolutionheat-treatment. Foreach combinationof

1 mm ED

ND

(a)

1 mm ED

ND

(b)

1 mm ED

ND

(c)

Fig. 2. Grain structure of the three alloys: (a) 6060, (b) 6082.25, and (c) 6082.50. Reprinted by permission from Springer Nature ( Frodal et al., 2017 ).

specimentype andheat-treatmentprocedure,thetemperatureofoneofthespecimenswasloggedbyadrilled-inthermo- couple.

Thethreealuminiumalloyshavedifferentgrainstructureandcrystallographictexture.The grainstructureofthethree alloys isshown inFig. 2. The 6060alloy hasa recrystallized grain structure comprised of equiaxedgrains of60–70 μm andexhibitsacubetexturewithaminorGosscomponent.Atypicalfibrous,non-recrystallizedgrainstructureisobserved forthe 6082.25 alloy,which hasa cube texturewithorientations along the

β

^{-fibre. Thesegrains areseveral} millimetres longinED,approximately150μminTDand10μmintheND.Inaddition,thefibrousgrainstructurecomprisessub-grains approximately2–10μm indiameter.Thelarge elongated,recrystallized grainsofthe6082.50 alloyexhibita rotatedcube texture,andareseveralmillimetreslonginED,about1–2mminTDand300–400μmintheND(Frodal,Pedersen,Børvik,

&Hopperstad,2017).

For further details about the alloys, the reader is referred to Khadyko, Dumoulin, Børvik, and Hopperstad (2014), Frodaletal.(2017)andChristiansenetal.(2018).

3. Experimentalprocedures 3.1. Mechanicaltesting

Inordertostudythedeformationandfracturemechanismsofthesematerials,differenttypesofspecimensweretested underquasi-staticloadingconditions,includingcylindricalsmoothandV-notchtensilespecimensandKahnteartestspeci- mens.Fig.3showsthegeometryofthethreetypesoftestspecimens.Thespecimensweremachinedfromthecentreofthe extrudedprofile.ThetensileaxisofthesmoothandV-notchtensilespecimenswasorientedalongthetransversedirection (TD)oftheprofile.The Kahnteartest specimensweremachinedeitherwiththetensileaxisalongtheextrusiondirection (ED)oralongTD,andalwaysperpendiculartothethicknessdirection(ND).

Adisplacement-controlledtestingmachine wasusedtoperformtheexperiments,anda constantcross-headvelocityof 1.00mm/min,0.12mm/minand1.00mm/minwereusedforthetestsonthesmoothtensilespecimens,theV-notchtensile specimensandtheKahnteartestspecimens,respectively.

Anin-housemeasuring system(Frodaletal., 2017)wasusedto measuretheminimumdiametersalong EDandND of thesmooth andV-notchtensilespecimens.Theforce andminimumdiametersofthetensilespecimenwere continuously measuredduringthetestuntilfracture.Thecurrentareaoftheminimumcross-sectionoftheaxisymmetrictensilespecimen canbeestimatedbyanellipticalareaas

A=

π

4D₁D₃ (1)

whereD₁ andD₃ arethemeasureddiametersinED andND,respectively.Thetruestressovertheminimumcross-section areaisthen

σ

t=F

A (2)

where Fis themeasured force.Assuming plastic incompressibility andnegligible elastic strains,the logarithmicstrain is givenby

ε

l=ln

A0

A

(3)

83

M10x1.0

Ø6

20 20

M10x1.0

Ø6

R0.2

(b)

36.5

11.1

(c)

Fig. 3. Geometry of the test specimens: (a) Smooth tensile specimen, (b) V-notch tensile specimen, and (c) Kahn tear test specimen with a red line indicating the virtual extensometer used. Dimensions in mm. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

whereA₀ istheinitial cross-sectionarea ofthespecimen.Itisimportanttonote that

σ

^{t and}

ε

l representaveragevalues overtheminimumcross-sectionareaofthetensilespecimen.

IntheKahnteartests,theforceFandthecross-headdisplacementuweretrackedcontinuouslyduringtesting.Inaddi- tion,thedisplacementfield onthesurfaceofthespecimenwasobtainedbydigitalimage correlation(DIC). Thedisplace- ment v across thecrack tip of the Kahn test specimenwas extractedusing a virtual extensometer (Fig. 3c).In order to quantifythe energy required toinitiate andpropagate a crack across the Kahn test specimen,the unit initiation energy (UIE)andunitpropagationenergy(UPE)weredeterminedforeachtest.TheUIEiscalculatedby

UIE= 1 bw

up 0

Fdu (4)

whereup isthedisplacementatpeakforce,bistheinitialthickness,andwistheinitialminimumwidthofthespecimen.

Similarly,theUPEiscalculatedby UPE= 1

bw _uf

up

Fdu (5)

whereu_f isthedisplacementwheretheforcehasdroppedto1% ofthemaximumforce(ASTM,2001).Theseenergymea- surescanprovideusefulinsightwhen investigatingamaterial’s resistancetocrackgrowth,asbothstrength andductility areconsidered.

Threetestswere conductedforeachcombinationofspecimentype, tensiledirectionandmaterial, i.e.,alloyandcool- ing rate (air-cooling or water-quenching) afterthe solutionheat-treatment. Hence, a total of 18 tensile testson smooth specimens,18tensiletestsonV-notchspecimens,and36Kahnteartestswereconducted.

3.2.Microstructurecharacterization

Undeformedsamples were anodized atroom temperaturefor2min usingHBF₄ (Fluoroboric acid) toreveal the grain structureunderpolarizedlight intheopticalmicroscope.FracturesurfacesofthefailedV-notchandKahn teartest speci- menswereinvestigatedinaZeissGeminiSupra55VPFESEMoperatedat20kV.

Precipitationinthedifferentalloyswascharacterizedusingtransmissionelectronmicroscopy(TEM).Thinfoilsofthema- terialswerepreparedusingconventionalthinfoilpreparationprocedures.SectionsfromtheundeformedregionsoftheKahn teartestspecimenswerecutusinganautomatedhigh-speedsaw.Thesesectionswerethenpolisheddownto ≈300μmbe- fore3mmdiameterdiskswerepunchedoutandfurtherpolisheddownto ≈100μmthicknesswithamirror-likefinish.A twin-jetStruersTenuPol3electropolishingsystemoperatedat20Vwasusedtoelectropolishthedisksandprovideelectron transparentregionsforTEMinvestigations.Theelectrolyte consistedof1/3HNO₃ (nitricacid)and2/3CH₃OH(methanol), andwaskeptat≈ −25±5^◦Cduringtheprocess. ReliableTEMinvestigationsofprecipitationrequirethethinfoilsurface

Fig. 4. TEM [001] zone axis bright field images of precipitation in (a-c) air-cooled and (d-f) water-quenched undeformed materials. The imaged areas are of comparable thickness and identical scale. Precipitates oriented along [100] and [010] matrix crystallographic directions, i.e., lying in the paper plane, are visible through the dark strain field in the surrounding matrix. Precipitates oriented along [001], i.e., out of the paper plane, appear as black dots.

Differences in contrast and precipitate appearance are largely due to different specimen tilt and diffraction conditions.

normaltobeclosetoa

⁰⁰¹

^{direction,i.e.,a[001]zoneaxisisdesiredatrelativelylowspecimentilt.Becausethethinfoil}

preparationtechniqueproducesspecimenswithaninherentvariationinqualityandthethinareasofthefinalfoilsmight not containanygrainswitha [001] zoneaxis,severalthinfoils fromeachspecimenwere prepared.Inthe6082.50 alloy, thiswasespeciallychallengingduetotheabnormallylargegrains,andelectronbackscatteringinthescanningelectronmi- croscope(SEM)wasusedtoselectsectionsfromtheKahntestspecimens.BeforeTEMinvestigations,thefinishedthinfoils werecleanedfor ≈3mininaFischione1020PlasmaCleaner.

TEMinvestigationswerecarriedoutontwodifferentinstruments.Thewater-quenched6060alloywascharacterizedona PhilipsCM30withaLaB₆filamentoperatedat150kVinapreviousstudy(Christiansenetal.,2018).Theremainingmaterials were characterized on a JEOL JEM2100 with a LaB₆ operated at200 kV. The thickness of the imaged areas in the thin foilswasmeasuredby electronenergyloss spectroscopy(EELS),usingaGatan PEELSmodel601 ontheCM30microscope anda Gatan Imaging Filtersystemon theJEM2100 microscope. Precipitation in thedifferent materials wasevaluated by measuringtheaverageneedlelength,cross-sectionalarea,numberdensity,andvolumefraction,followingtheprocedureof Andersen(1995)andMarioara,Andersen,Zandbergen,andHolmestad(2005).In6000seriesaluminiumalloysclosetopeak hardness,mostoftheprecipitateswillbeoftheneedle-shaped

β

^{phase,although}Cu-containingalloyswilllikelycontain some Q andLphasesas well (Sunde, Marioara,vanHelvoort, & Holmestad, 2018). Mostprecipitates willtherefore have identicalshapesanditispossibletoestimate theaveragesize, numberdensity,andvolume fractionofprecipitatesgiven measuredneedlelengths,cross-sections,andnumberdensities.

4. Experimentalresults 4.1. Initialmicrostructure

Fig.4showsTEMbrightfieldimagesoftheundeformedmaterials.Air-coolinghasanadverseeffectonprecipitationin all alloys, butthe precipitate microstructure ofthe alloysis differentlyaffected. The 6060alloydevelops slightlycoarser

Fig. 5. TEM images of precipitation (marked “P”) around dispersoids (marked “D”) in the dense 6082 alloys after air-cooling (a,b), and after water-quench (c,d). The image in subfigure (a) is a medium-angle annular dark field scanning TEM (STEM) image, while the other images are TEM bright field images.

The reason for applying STEM is that the air-cooled condition of 6082.25 contains many dislocations, as indicated, and their strain fields dominate the bright field contrast. Note the difference in scale between subfigure (c) and the other images.

precipitates,whicharemoresparselydistributedafterair-cooling.Inthe6082.25alloy,precipitationbecomesveryinhomo- geneousandislimitedtosmallregionsthatcontainslightlylargerprecipitatescomparedtothewater-quenchedcondition.

The6082.50alloyalsoexhibitsinhomogeneousprecipitationafterair-cooling,butnottothesamedegreeasthe6082.25al- loy.Whereprecipitateshaveformedintheair-cooled6082.50alloy,thelocalnumberdensityisgreaterandtheprecipitates aresmaller,comparedto thewater-quenchedcondition. Fig.5presentsTEMimagesofthedispersoidsresponsibleforthe inhomogeneousprecipitationinthetwo6082alloys.Air-coolingcausesdispersoidstonucleatelargeprecipitatesanddrain theirneighbouringregionofsolute, suppressingprecipitationofstrengtheningprecipitates.Becausethenumberdensityof dispersoidsinthe6082.25alloyisgreaterthaninthe6082.50alloy,thePFZsarounddispersoidsandgrainboundaries(GBs) overlap,resultinginPFZsthatspanentiregrainsandleaveonlyafewregionsofsparseprecipitation.Dislocationsobserved inthePFZsofthisalloyareprobablygeneratedduetotheconstraintthatareintroducedbythegrowthoflargeprecipitates nucleatedondispersoids.Inthe 6082.50alloy,thedispersoiddensityissubstantiallylower, sothatstrengthening precipi- tatesmayformin regionssituateda certaindistancefromeach dispersoid.Grain boundaryPFZsare alsochangedby the coolingrate.Fig.6presentsrepresentativeTEMbrightfieldimagesofgrainboundaryregionsandsurroundingPFZs.Theob- servedtrendisthattheslowcoolingrateleadstowiderPFZsaroundgrainboundaries,similartothetrendofPFZsaround dispersoids.

Table2summarisestheTEMobservations,whereaveragenumbersforneedlelength,cross-sectionalarea,numberden- sity,andvolume fractionsarepresented,alongwithapproximatePFZwidths.Precipitatestatisticswerenot performedfor theair-cooled6082.25alloy,duetoitsseverelyinhomogeneousprecipitate microstructure.Notethatthereportednumber densityfortheair-cooled6082.50 alloyisan overestimateoftheaveragenumberdensity,asitisestimatedbasedonthe denseprecipitationregions.ThePFZsaroundGBsanddispersoidswillreducetheactualnumberdensityandvolumefraction ofprecipitatesinthisalloyafterair-cooling.

ThemainresultfromtheTEMstudyisthatair-coolingreducesthetotalnumberdensityandvolumefractionofstrength- eningprecipitates in all the studiedalloys. Preciptiation is lessaffected by the cooling ratein the 6060alloy, andmost affectedinthe6082.25alloywhereitisalmostentirelysuppressed.Inthe6082.50alloy,air-coolingresultsinvery dense precipitationof fineprecipitates andlarge PFZs around dispersoidsscatteredthroughoutthe grains. This means that the precipitationmicrostructure ofthe 6082.50 alloybecomesinhomogeneous,with softregions (dispersoids withPFZs) dis- persed ina strongmatrix (preciptiate strengthened regions),where the strong matrix hasa strength comparable to the water-quenchedstate.

Fig. 6. TEM bright field images of grain boundaries (marked by “GB”) and surrounding PFZs in (a–c) air-cooled and (d–f) water-quenched materials. Black dashed lines mark the PFZ width, while the white dashed lines mark the estimated error of the PFZ width. The scale in subfigures (a–c) is the same, but for subfigures (d–f) the scale is larger for subfigure (e) than for subfigures (d) and (f).

Table 2

Average estimates of precipitation in the alloys. Precipitation in the air-cooled 6082.25 alloy is inhomogeneous, and no statistics are available.

Material Needle length Needle cross-section Number density Volume fraction PFZ width

(nm) (nm ²) (#/μm ³) (%) (nm)

6060-AC 41.8 ±2.4 23.4 ±4.2 2839 ±272 0.27 ±0.02 478 ±114 6060-WQ 40.0 ±1.0 19.1 ±0.8 5556 ±629 0.42 ±0.05 146 ±41

6082.25-AC − − − − 334 ±138 ^b

6082.25-WQ 22.9 ±0.6 7.0 ±0.3 40232 ±3563 0.64 ±0.05 61 ±16 6082.50-AC 26.5 ±0.5 8.3 ±0.3 53870 ±3826 ^a 1.18 ±0.08 211 ±33 6082.50-WQ 32.1 ±2.3 10.2 ±0.5 42183 ±7534 1.26 ±0.14 59 ±11

a The precipitation statistics for the air-cooled 6082.50 alloy are from dense precipitate regions, and the number density does not reflect the inhomogeneous precipitate microstructure.

bPFZs usually span entire grains.

4.2. Mechanicalresponse

4.2.1. TensiletestsonsmoothandV-notchspecimens

Fig. 7 presents the true stress-strain curves from the tensiletests on smooth specimens plotted up to failure. Three paralleltestswereperformedforeachmaterial.Onlytworepeattestsweresuccessfulforthewater-quenchedandair-cooled 6082.50alloy,astheotherspecimensfailedclosetothegripsection.Comparingthetwoheat-treatments,itisobviousthat thestrengthislower withair-coolingthanwithwater-quenchingforallthealloys.Thestrain tofailureisgreaterwithair- cooling than withwater-quenchingforthe 6060and6082.25 alloys. Forthe 6082.50 alloy,a largescatter is observedin thefailurestrainforthewater-quenchedmaterial,andthefailurestrainfortheair-cooledmaterialiswithinthisrange.The scatteriscausedbythelargegrainsofthisalloy,asthenumberofgrainsacrossthespecimendiameterissmall.Relatively littlescatterisobservedfortheother materials.Forthewater-quenchedmaterials,theinitialyieldstress isthelowestfor the6060alloy,whilethetwo6082alloyshaveahigherandsimilarinitialyieldstress,withthe6082.25alloyalittlehigher thanthe6082.50alloy.Withair-cooling,theinitialyieldstressofthe6082.25alloyisthelowest,andtheinitialyieldstress of the 6060 alloyis between the two 6082 alloys. Thus, the largest difference in strength due to the heat-treatment is observedforthe6082.25alloy,seeTable3.

Fig. 7. Stress-strain curves from the tensile tests on smooth specimens: (a) 6060, (b) 6082.25, and (c) 6082.50. Simulation results are shown for the 6082.25 alloy.

Table 3

Initial yield stress at 0.2% plastic strain, σ0.2, from the smooth tensile tests, with standard deviations.

Material σ0.2(MPa)

6060-AC 132.6 ±0.5

6060-WQ 176.4 ±2.7

6082.25-AC 113.4 ±3.0 6082.25-WQ 327.7 ±0.5 6082.50-AC 179.5 ±6.4 6082.50-WQ 315.0 ±1.3

Fig. 8. Stress-strain curves from the tensile tests on V-notch specimens: (a) 6060, (b) 6082.25, and (c) 6082.50. Simulation results are shown for the 6082.25 alloy.

Thestress-straincurvesuptofailure fromthe tensiletestsonV-notchspecimensare presentedinFig.8.Three repeat testsareshownforeachmaterialandthescatterissmall.ItisevidentthatthestressleveliselevatedintheV-notchspec- imenscomparedtothesmoothspecimensduetothenotchstrengtheningeffect,whichiscausedbythetriaxialstressfield inthenotch.SimilartrendsareobservedhereasinFig.7,butthefailurestrainisclearlylowerwithwater-quenchingthan withair-coolinginthetensiletestsontheV-notchspecimens—andthisholdsforthe6082.50alloyaswell.Bycomparison ofFigs.7and8,itisfound thatthefailure strain issubstantiallylower inthetensiletestsonV-notchspecimens,dueto theincreasedstresstriaxialityinducedbythesharpnotch.

LankfordcoefficientsmeasuredinthetensiletestsonsmoothandV-notchspecimensare giveninTable4.Thesecoef- ficientsgive theratiobetweentheincrementalstrainsinEDandND,andthusrepresenttheevolutionofthecross-section ofthespecimenwithplastic deformation.Smallvariationsinthe Lankfordcoefficientsare observedbetweenthemateri- als withthe different cooling rates. Forthe 6060 and6082.50 alloys, the air-cooled materials appearto be slightlyless anisotropic than thewater-quenched materials, inthe sense that theLankford coefficient iscloser to unity, whereas the oppositetrendisfoundforthe6082.25 alloy.The Lankfordcoefficientsobtainedfromthetensiletestsonthesmoothand V-notchspecimensaresimilar forthetwo6082alloys, whilethecoefficientsforthe6060alloyaremarkedlylowerinthe V-notchspecimens.The latterobservationindicates that theplastic flowis stronglyinfluenced bythe triaxialstress field imposedbythesharpnotch.

Table 4

Measured Lankford coefficients from the smooth and V- notch tensile tests, with standard deviations.

Material Smooth V-notch

6060-AC 3.67 ±0.07 1.94 ±0.15 6060-WQ 3.95 ±0.11 2.04 ±0.25 6082.25-AC 0.84 ±0.00 0.81 ±0.01 6082.25-WQ 1.10 ±0.01 0.94 ±0.01 6082.50-AC 0.32 ±0.09 0.35 ±0.04 6082.50-WQ 0.19 ±0.05 0.22 ±0.06

Fig. 9. Force-displacement curves from the Kahn tear tests showing the mean curve with the scatter in shaded colour: (a) 6060, (b) 6082.25, and (c) 6082.50. Simulation results are shown for the 6082.25 alloy.

Table 5

Unit initiation energy (UIE) and unit propagation energy (UPE) from the Kahn tear tests, with standard deviations.

Material UIE (kJ/m ²) UPE (kJ/m ²)

ED TD ED TD

6060-AC 94.0 ±0.7 102.5 ±4.0 231.0 ±2.0 250.0 ±7.1 6060-WQ 87.8 ±7.7 109.2 ±1.8 249.2 ±8.1 267.2 ±0.9 6082.25-AC 144.7 ±2.7 132.5 ±5.8 213.1 ±1.1 218.0 ±3.2 6082.25-WQ 147.6 ±9.9 135.5 ±9.5 145.1 ±9.1 118.2 ±16.0 6082.50-AC 115.9 ±3.5 81.6 ±3.9 244.8 ±24.6 153.6 ±16.7 6082.50-WQ 128.3 ±5.1 98.9 ±5.1 276.2 ±18.5 101.5 ±3.7

4.2.2. Kahnteartests

Fig.9displaystheforce-displacementcurvesfromtheKahnteartests.Thevirtualextensometersituatedacrossthenotch tip,depictedinFig.3c,isused toobtaina measureofthenotchtip openingdisplacement.Thedifferencesinstrength of theair-cooledandwater-quenchedmaterialsobservedinthetensiletestsonsmoothandV-notchspecimensarealsoseen here.Thepeakforceisconsistentlylowerfortheair-cooledthanforthewater-quenchedmaterials,andthuscrackinitiation always occursatalower forceforthe formermaterials.Variations betweenthetwoloading directionsarealsoobserved.

Forthe6060alloy,the testinthe transversedirection(TD)hasthehighestpeak force,whereas thetest intheextrusion direction(ED)hasthehighestpeakforceforthetwo6082alloys.Forthe6082.50alloy,testsalongEDgivemarkedlyhigher ductility than tests along TD, while the difference for the 6082.25 alloy is small. This large difference in ductility with loading directionisattributed tothe grain structureof the6082.50 alloy. Thelarge elongated grainsof thisalloy,which areseveralmillimetresalongED (Frodaletal.,2017),makethematerialmoresusceptibletointercrystallinefracturewhen loadedalongTD,seeSection4.3.

Table5presentstheunit initiationenergies (UIEs)andunitpropagationenergies (UPEs)calculatedbasedon theKahn tear tests, see Eqs. (4) and(5). Here, the cross-head displacement hasbeen used to calculate these energies in accord withASTM(2001).TheUIEandUPEaretheenergiesperunitarearequiredtoinitiatecrackgrowthandpropagateacrack throughthematerial,respectively. SimilarvaluesoftheUIE arefoundforthesamealloyandloadingdirection.Albeitthe force level issignificantly lower for theair-cooled materials than forthe water-quenchedmaterials, the UIEs are similar, indicatingthatthedisplacementtocrackinitiationislargerfortheair-cooledmaterials.

The UPEisgreater forthewater-quenchedmaterial thanforthe air-cooledmaterial inboth loadingdirectionsforthe 6060alloy,suggestingthatthewater-quenchedmaterialhasahigherresistancetocrackpropagation.Thistrendisreversed forthe 6082.25alloy, andtheair-cooled materialhas thegreatestpropagationenergy inboth loadingdirections.Forthe

Fig. 10. Fracture surfaces of the V-notch tensile specimens for the three alloys with air-cooling, AC (top) and water-quenching, WQ (bottom).

6082.50alloy,the loadingdirectionhasa substantial effectontheUPE, andthiseffectisgreater forthewater-quenched material than for the air-cooled material. The result is that for the 6082.50 alloy, the water-quenched material has the highestUPEinED,whereastheair-cooledmaterialhasthehighestUPEinTD.

4.3.Fractography

Togetadeeperunderstandingofthemechanismsinvolvedinthefractureprocess,afractographicanalysisofthetested specimensis performedintheSEM. Fig.10 presentsthefracture surfacesof theV-notch tensilespecimensforthe three alloyswithair-coolingandwater-quenching.Threedistinct fracturesurfaceshapesareobserved,whereeachalloyexhibits adistinctshape.The 6060alloydisplaysaflatdiamond-shaped fracturesurface.Aflatcircularcross-sectionisrecognized forthe6082.25alloy,whereasthe6082.50alloydemonstratesaslantcircularfracturesurfacewhichisaresultofthelarge grains.Thediamond-shapedfracture surfacefoundforthe6060alloyiscausedbythestrongcubetextureincombination withthesuperimposedtriaxialstressfieldoftheV-notchspecimen(Khadyko,Dumoulin,Børvik,&Hopperstad,2015).This diamond-shapedcross-sectionwillyieldinaccuraciesinthelogarithmicstrain measure,definedbyEq. (3),forlargedefor- mations,asthearea calculationusedinthestrain measureisbasedon anassumptionofan ellipticalminimumarea,see Eq. (1).

Fractureismainlytranscrystallineforthe6060and6082.25alloys,withsomeareasofintercrystallinefracture.Incon- trast,the6082.50alloyhassubstantialamountsofintercrystallinefracture,seealsoFigs.11,,and–13.Ingeneral,thesame fracturesurfaceshapesandfracturemodesareobservedfortheair-cooledandwater-quenchedmaterialsinFig.10.Theair- cooledmaterials havesmallerfracture areasthanthewater-quenchedmaterials,inagreementwithlargerfracturestrains, seeSection 4.2.1.Forthe6060 and6082.25 alloys, thefracture ismainly transcrystalline forboth theair-cooled andthe water-quenchedmaterials,whereas theamountofintercrystallinefracture isgreaterforthewater-quenchedmaterialthan fortheair-cooledmaterialforthe6082.50alloy.

Fig.11 showsthe fracture surfacesof the Kahn teartest specimens forthe 6060 alloy,including thetwo loading di- rectionsandtheair-cooledandwater-quenchedmaterials.Thefracturesurfacesareclearlydistinctandvariationsare seen betweenthetwo loadingdirectionsandthe twocoolingrates.Forthematerialsloaded alongED,transcrystallinefracture

Fig. 11. Fracture surfaces of the Kahn tear test specimens for the 6060 alloy.

is dominant.Incontrast,along TD a considerableamountof thefracture surfaceiscovered withareas ofintercrystalline fracture. Asthisalloyhas equiaxedgrains, themore probablecauseofthisdifference isthe crystallographictexture. The Gosstexturecomponentislikelytocausegreaterimbalanceandstressconcentrationsbetweengrainswhen loadedalong TD,leadingtohighlylocalizeddeformationinthe PFZsandalong grainboundaries. Thiswillinturnleadtodifferentbe- havioursalongdifferentmaterialorientationsasobservedforloadingalongEDandTD.Comparingthedifferentcoolingrates itisevidentthatthewidthofthefractureareaforthewater-quenchedmaterialissmallerthanthatfortheair-cooledma- terialloadedalongED,suggestingthatthewater-quenchedmaterialhasagreaterductility.Forthespecimensloadedalong ED, thefracture surfacesoftheair-cooled andwater-quenchedmaterials aresimilar, displayingsimilar dimplestructures.

AgreaterdifferenceisobservedbetweenthetwocoolingratesforthespecimensloadedalongTD.Theair-cooledmaterial hasalargeramountofintercrystallinefracturethanthewater-quenchedmaterial,againsuggestingthatthewater-quenched materialhasagreaterductility.

Fig.12depictsthefracturesurfacesoftheKahnspecimensforthe6082.25alloy,includingthetwocoolingratesandthe two loadingdirections.Thefracture surfacesaresimilarinall thesecases.Transcrystallinefractureis dominantwithonly minorintercrystalline fracture,i.e., someareas ofintercrystallinefracture canbe seenbetweenthe largeareasfilled with dimplesformedinthegraininterior.

ThefracturesurfacesoftheKahnspecimensforthe6082.50 alloy,includingthetwocoolingratesandthetwoloading directions,areshowninFig.13.Incontrasttothe 6082.25alloy,the6082.50alloyclearlyshowsdistinctfracturesurfaces forthedifferentcases.ForthetestsloadedalongED,onlytranscrystallinefractureisseenforthewater-quenchedmaterial.

Incontrast,large areasofintercrystallinefracture areobserved fortheair-cooledmaterial,indicating afailure mode with lowerductilitythanforthewater-quenchedmaterial.LargeramountsofintercrystallinefractureareobservedfortheKahn specimensloadedalongTDduetothegrainstructureofthisalloy.Thelargeelongatedgrains,seeFig.2,makethematerial lesssusceptibletointercrystallinefractureinEDasthefracturepathisforcedthroughthegrainsbythenotchoftheKahn specimen.The ratioofintercrystallineto transcrystallinefracture forthespecimens loadedalong TDappears tobe larger forthewater-quenchedmaterialthanfortheair-cooledmaterial.Thus,differenttrendsofintercrystallinefractureareseen forthe twoloading directions.Thisindicates that forloadingalong ED, thewater-quenchedmaterial isthemostductile, whereasitistheair-cooledmaterialthatisthemostductilewhenloadedalongTD,inagreementwithTable5.

Fig. 12. Fracture surfaces of the Kahn tear test specimens for the 6082.25 alloy.

Constituentparticlescanbeseen atthebottomofmanyofthedimplesforallmaterials, indicatingductilefracturein- volvingnucleation,growthandcoalescenceofvoids.Voidsmaybenucleatedfromconstituentparticleseitherbydecohesion orbyparticlecracking(Maireetal.,2011),ormaypre-existinthematerials(Todaetal.,2013).The6060alloycontainscon- stituentparticlesoftypeAlFeSi,whilethetwo6082alloyscontainconstituentparticlesoftypeAlFeSiMn,andtheparticles arelargerinthetwo6082alloysthaninthe6060alloy(Frodaletal., 2017).Crackedparticlescanbe foundatthebottom ofsomedimples.Theseparticlescrackedduringtheextrusionprocess,andfullyorpartiallycrackedparticlescanbefound inundeformedsamples(Frodaletal.,2017).Insomeareas,intercrystallinefracturewithahighdensityofsmallerdimples isrevealed,anditisreasonabletoassumethatthesedimplesarecausedbyvoidgrowthwithinthePFZ,possiblynucleated atgrainboundaryprecipitates(Chenetal.,2009).

5. Numericalstudy 5.1. Constitutivemodel

Inthissection,theconstitutiverelationsoftheporousplasticitymodel,whichisusedinthesubsequentfiniteelement simulations ofsome ofthe tests, are outlined. Itis assumedthat the elasticdeformations areinfinitesimal, while plastic deformationsandrotationsmaybefinite.Thephysicalmechanismsofdamageevolutionandfailurearegrowthandcoales- cenceofvoidsintheductilematrixmaterial.Acorotationalformulationoftheanisotropicporousplasticitymodelisused, andthecomponentsoftheCauchystresstensorandtherateofdeformationtensorareexpressedinaco-rotatedcoordinate systemby

σ

ˆi j=R_ki

σ

klR_{l j} ∧ Dˆ_{i j}=R_kiD_klR_{l j} (6)

whereRistherotationtensorfromthepolardecompositionofthedeformationgradientF.Therateofdeformationtensor Disadditivelydecomposedintoelasticandplasticparts

Dˆi j=Dˆ^e_{i j}+Dˆ_{i j}^p (7)

Fig. 13. Fracture surfaces of the Kahn tear test specimens for the 6082.50 alloy.

whereD^e andD^p aretheelastic andplasticrateofdeformationtensors,respectively.In theco-rotatedcoordinatesystem, therateformofthegeneralizedHooke’slawcanbeexpressedas

ˆ˙

σ

i j= E

1+

ν

^{Dˆi je+3}

(

^1−E2

ν )

^Dˆkk

δ

i j (8)

whereD^eisthedeviatoricpartoftheelasticrateofdeformationtensor,

δ

ij istheKroneckerdelta,andEand

ν

^areYoung’s

modulusandPoisson’sratio,respectively.

Plasticyieldingisgoverned bythe heuristicextension oftheGurson-Tvergaard-Needlemanmodel(Gurson,1977;Tver- gaard,1981;Tvergaard&Needleman,1984)proposedbyDæhlietal.(2017),andtheyieldfunctiontakestheform

( σ

^,p,f

)

⁼

ϕ ( σ )

σ

^y

2

+2q1f^∗cosh

q2

σ

ˆkk

2

σ

^y

−1−

(

^q1f∗

)

^{2≤0 (9)}

where

σ

yisthematrixflowstress,f^∗istheeffectivevoidvolumefractionintroducedbyTvergaardandNeedleman(1984), andq_iarethemodelparameterssuggestedbyTvergaard(1981).AsintheoriginalGurson(1977)model,thegrowingvoids are assumed to be spherical in shape. The equivalent stress

ϕ

^{is now defined by the} Yld2004-18p model proposed by Barlatetal.(2005),whichaccountsfortheplasticanisotropyofthematrixmaterial,as

ϕ ( σ )

⁼

1 4

3

k=1

3

l=1

S⁽_k¹⁾−S⁽_l²⁾

^a

¹_a

(10)

where a isan exponent determiningthe curvatureof the yieldsurface, andS_k⁽¹⁾ andS_l⁽²⁾ are the principal valuesofthe tensorss⁽¹⁾ands⁽²⁾,respectively.Thetensorss⁽¹⁾ands⁽²⁾aredeterminedbythelineartransformations

ˆ

s⁽_{i j}¹⁾=Cˆ_{i jkl}⁽¹⁾

σ

ˆkl ∧ sˆ⁽_{i j}²⁾=Cˆ⁽_{i jkl}²⁾

σ

ˆkl (11)

where

σ

^{isthedeviatoricpartoftheCauchystress tensor,andthe}fourth-ordertensorsC⁽¹⁾andC⁽²⁾ containcoefficients describingtheplasticanisotropy(Barlatetal.,2005;Dæhlietal.,2017).Foranorthotropicmaterial,Eq. (11)canbewritten

surfaceinstress space. Recently, vandenBoogaard, Havinga,Belin, andBarlat (2016) showedthat the 18anisotropy pa- rameterscanbereducedto16independentones.Thus,wewillselectcˆ⁽₁₂¹⁾=cˆ₁₃⁽¹⁾=1.Anyother choicewillgiveequivalent results,butforanisotropicmaterialtheseselectedvalueswillresultinthatallcˆ_{i j}⁽¹⁾andcˆ⁽_{i j}²⁾becomeequaltooneandthat

ϕ

⁽

σ

^{)reducestothe}Hershey–Hosfordequivalentstress.Theassociatedflowruleisused,andtheplasticrateofdeformation tensorreads

Dˆ_{i j}^p=

λ ∂

˙

∂ σ

^ˆi j

(13) where

λ

˙ ≥0istheplasticmultiplier.Thematrixequivalentplasticstrainisthenrelatedtotheplasticpowerthrough

p= t

0

˙ pdt=

t 0

σ

ˆi jDˆ^p_{i j}

(

^1−f

) σ

^{ydt (14)}

wherefisthevoidvolumefraction.

TheflowstressofthematrixmaterialisdescribedbyanextendedVocehardeningrule

σ

^y=

σ

⁰⁺³

i=1

Q_i

1−exp

−

θ

i

Q_ip

(15)

where

σ

0 istheinitialyieldstress,andQ_iand

θ

iareparameterscontrolling theworkhardening.Plasticincompressibility ofthematrixmaterialgivestheevolutionofthevoidvolumefractionas

f˙=

(

^1−f

)

^Dˆkk^p (16)

TheeffectivevoidvolumefractionisgivenbyTvergaardandNeedleman(1984) f^∗

(

^f

)

=

f if f≤fc

fc+ ^f_f^U_F^∗−_−f^f_c^c

(

^f−fc

)

^{if f}>fc (17)

wherefc isthecriticalvoidvolume fractionwhereanacceleratedvoidgrowthisinitiated, f_U^∗=1/q₁ istheultimatevalue, andf_Fisthevoidvolumefractionwherethematerialhascompletelylostitsload-carryingcapacity.Inthesubsequentfinite elementsimulations,elementdeletionisusedtodescribecrackpropagation,andtheelementisdeletedwhen f=f_F inall integrationpoints.

Finally,theloading/unloadingconditionsofplasticityaregiveninKuhn-Tuckerformas

≤0,

λ

˙ ≥0,

λ

˙

=0 (18)

whereastheconsistencycondition,usedtodeterminetheplasticmultiplier

λ

˙ ^{intheplasticdomain,isexpressedby}

λ

˙

^˙ =0 (19)

The porous plasticity model has been implemented into a user material subroutine (VUMAT) for Abaqus/Explicit (Abaqus,2014).Toensuresufficientaccuracyoftheintegrationpointvalues,sub-steppingisemployed(Dæhlietal.,2017).

5.2.Finiteelementmodellingandsimulations

Inthe following,finiteelement simulations ofthe tensiletestsandtheKahn tear testsforthe6082.25 alloyare pre- sented,whicharebasedontheporousplasticitymodeldescribedinSection 5.1.The othertwoalloysdemonstrateeffects andproperties that the porous plasticitymodel is unable to capture. The 6060 alloy developsa diamond-shaped mini- mumcross-sectionintheV-notchtensiletests,seeSection4.3,whichcanbereplicatedbycrystalplasticityfiniteelement analyses,seeKhadykoetal.(2015).Failingtosimulatethecorrectdeformationmode willyielduncertaintiesinthestrain measureandthusonthecalculatedfailure strain.Forthe6082.50alloy,thelessductilefracturemodeobservedespecially forloadingalongTD,seeSection4.3,ispoorlydescribedbytheporousplasticitymodel.Thismodelconsidersgrowthand

Fig. 14. (a) Initial yield surface of the 6082.25 alloy depicted in the ED-TD plane with contours of normalized shear stress plotted in 0.1 increments, with the maximum value in the centre, and (b) normalized yield stress and (c) Lankford coefficient versus tensile direction with respect to ED for uniaxial tension in the ED-TD plane.

Table 6

Initial yield stress and parameters of the work-hardening rule.

Material σ0(MPa) θ1(MPa) Q 1(MPa) θ2(MPa) Q 2(MPa) θ3(MPa) Q 3(MPa)

6082.25-AC 99.6 2824.3 78.8 153.1 90.8 − −

6082.25-WQ 302.1 489.1 30.1 474.5 28.7 50.4 281.8

coalescenceofvoidsinahomogeneoussolidwhichisunabletocapturethehighlylocalizeddamageevolutionandfailure modeofintercrystallinefracture.

Fortheductile6082.25alloy,damageevolutionandfailureisgovernedbynucleation,growthandcoalescenceofvoids, seeSection4.3,andtheporousplasticitymodelisusedtodescribe boththeanisotropicplasticbehaviour andtheductile failureprocessforthismaterial.IsotropicelasticityisassumedinthesimulationswithaYoung’smodulusofE=70000MPa andaPoisson’sratioof

ν

=0.3,whicharetypicalvaluesforaluminium.ThestandardTvergaard(1981)parametersq₁=1.5, q₂=1areusedtodefinethepressuresensitivityoftheyieldsurface,whichwereintroducedtogivebetteragreementwith unit cellanalyses. As analternative, theseparameterscould havebeendetermined fromunit cellcomputations(see, e.g., Dæhlietal.,2017)andmaydependon,e.g.,thework-hardeningbehaviourandplasticanisotropyofthematerial.Thus,the remainingmodelparameterstoidentifyaretheanisotropyparameterscontrollingtheshapeoftheyieldsurface(cˆ⁽_{i j}¹⁾,cˆ⁽_{i j}²⁾, a),thework-hardeningparameters(

σ

0,Q_i,

θ

i;i=1,2,3),andtheporousplasticitydamageparameters(f₀,fc,f_F).

Theanisotropyparameterscontrollingtheshapeoftheyieldsurfaceconcerningdeviatoricstressstatescanbeidentified eitherfromalargenumberofexperimentaltests(Fourmeau,Børvik,Benallal,Lademo,&Hopperstad,2011)orfrommicro- mechanicalsimulations(Frodal,Dæhli,Børvik,&Hopperstad,2019) bycombiningcrystalplasticitywiththefiniteelement method(CP-FEM).TheanisotropicyieldsurfacesofthesealloyshavepreviouslybeencalibratedbyFrodaletal.(2019)util- isingCP-FEM.Fig.14depictstheinitial yieldsurfaceofthe6082.25alloyforporosity f=0,togetherwiththenormalized initialyieldstressinuniaxialtension,andLankfordcoefficientsversustensiledirectionintheED-TDplane.

The finite element meshes of the smooth and V-notch tensile specimens andKahn teartest specimenare shownin Fig.15.Duetotheorthotropic materialsymmetry,onlyone-eighthofthesmoothandV-notchtensilespecimensandone- quarterofthe Kahn teartestspecimenare modelledto reduce thecomputational time.Linear eight-node solid elements with reducedintegration (C3D8R) are used. The dimensionsof the elementslocated in the centreof the specimens are 30×40× 40μm³,withtheshortestelementlengthalong thetensiledirection. Massscalingisusedtoreduce thecom- putationaltime,anditisensuredthattheresponseisquasi-static,i.e., thatthekinetic energyisnegligiblecomparedwith theinternal energy.Theappropriatesymmetry boundaryconditionsareenforcedandloadingisappliedto theendofthe smooth andV-notchtensilespecimens.An analytic rigidpinisused toapply theloadonto theKahn teartest specimen, whereafriction-lesssurface-to-surfacecontactformulationisusedbetweenthespecimenandtherigidpin.

Inordertodeterminethework-hardeningparametersofthematerials,finiteelementsimulationsofthetensileteston thesmoothspecimenareruninAbaqus/Explicitcombinedwiththenon-linearoptimizationsoftwareLS-OPT(Standeretal., 2015).The parameters ofEq. (15)are calibratedby minimisingthe meansquarederrorbetweenthestress-straincurves uptofailurefromthefiniteelementsimulationandtheexperimentaltests.Duringthiscalibration,thematerialisassumed to have zeroporosity (i.e., f=0). The work-hardeningparameters of the water-quenchedmaterial have previously been determined,andthereaderisreferred toFrodaletal.(2019)forfurtherdetails onthecalibrationprocess.Table6presents theresultinginitialyieldstressandthework-hardeningparameters,wheretheparametersethasbeenadjustedtoaccount fortheminorsofteningintroducedbytheinitialporositycalibratedbelow(i.e., f=f0).

(a)

TD ND

ED

(b)

TD/ED ED/TD

ND

(c)

Fig. 15. Finite element meshes of (a) the smooth tensile specimen, (b) the V-notch tensile specimen, and (c) the Kahn tear test specimen.

Table 7

Porous plasticity parameters, including initial porosity f 0, critical porosity f c, and porosity at complete material fail- ure f F.

Material f 0 f c f F

6082.25-AC 1 . 20 ·10 ⁻³ 8 . 39 ·10 ⁻³ 2 . 12 ·10 ⁻¹ 6082.25-WQ 1 . 30 ·10 ⁻³ 8 . 38 ·10 ⁻³ 6 . 08 ·10 ⁻²

The damage parameters of the porous plasticity model, including the initial porosity f₀, the critical porosity fc, and theporosity atcompletematerialfailure f_F,areherecalibratedusinga similarapproach aswiththeinitial yieldstrength andthe work-hardeningparameters.Simulations oftensiletestson thesmooth andV-notchspecimens areperformedin Abaqus/Explicit,andthedistancebetweenthepointofmaximumtruestress(i.e.,thetruestressatincipientstrainsoften- ing)isminimizedsimultaneouslyforboth testsby meansoftheLS-OPTsoftware.Thisensuresthat thepointoffailureis preciselycapturedforboththetensiletestsonthesmoothandV-notchspecimens.Theresultingporousplasticityparame- tersaregiveninTable7.

The stress-straincurvesfromthefinite elementsimulations ofthetensiletestsof thesmooth andV-notchspecimens areshowninFigs.7and8,respectively.Agoodagreementisfoundbetweenthenumericalandexperimentalresultsupto fracture forboth thesmoothandV-notchspecimens,andthepoint ofmaximumtruestress iswell captured.In thetests ofthesmoothtensilespecimens,thestressleveldropsabruptlyafterreachingthepointofmaximumtruestress.Similarly, thestressleveldropsrapidlyforthewater-quenchedmaterialinthefiniteelementsimulation,whereasthedecreaseisless swiftinthesimulationoftheair-cooledmaterial.InthetestsoftheV-notchtensilespecimens,accelerateddamageinduced softening isobserved andthestress level decreases afterthepoint ofmaximum truestress beforea sudden dropofthe stressleveloccurs.Thesametrendsareseeninthenumericalsimulations,butthesuddendropinstresslevelissomewhat delayed.Thus,moreenergyisdissipatedinthesimulations thanintheexperimentsbeforethematerial haslost allofits load-carryingcapacity,i.e.,thetensileductilityissomewhat overestimatedinthesimulations.Itshould howeverbenoted that predictions ofcrackpropagation byelement deletion are meshdependent andmoreaccurate results mightpossibly havebeenobtainedwithanevenfinermesh.

The resultingforce-displacement curvesfrom thesimulations ofthe Kahn teartestsare presentedin Fig.9.The peak force is slightly underestimated in the simulations for both the water-quenched andair-cooled materials, and the drop in theload levelis not assteep after the peak force asobserved inthe experiments. Inview of the resultsobtained in the simulations oftheV-notch specimensubjectedto tensileloading,it isreasonable toattribute thelower slope ofthe numericalforce-displacementcurvesandthehigherenergydissipationduringcrackpropagationtothemoreductilefailure modeofthefiniteelementmodel.Inaddition,errorsinthecalibratedyieldsurfacemaybecomeapparentforthisloading condition.However,thedifferencesobservedexperimentallybetweenthewater-quenchedandair-cooledmaterialsarewell capturedinthesimulations,i.e.,thepredictedloadlevelforthewater-quenchedmaterialdropsbelowthatoftheair-cooled materialwhenthedisplacementincreases.

Fig.16depictscontoursofthevonMisesequivalentstrainatfailureinitiationandduringcrackpropagationintheKahn teartestsfromthe experimentandthefinite elementsimulation oftheair-cooled 6082.25 alloyloaded alongED. Inthe experiment,thestrainfieldwascalculatedbasedonfull-fieldmeasurementsofthesurfacedisplacementfield usingdigital image correlation(DIC).Thestrainfieldsfromtheexperimentandthesimulationareinrelativelygoodagreement.Highly localizeddeformationisobservedclosetothenotchandcracktip,wheretheequivalentstrainisthehighest.Atpeakforce 1 ^{, ”butterfly} wing” shaped contours are seen close to the notch andfurther away from the notch tip the contours are morecircular.Asthecrackprogressesthroughthealloy,2 ^- 6 ^{,the”butterfly}wing” shapedcontourspersistinfrontofthe cracktip.Fig.17depictsthecorrespondingimagelocationsontheforce-displacementcurvefortheair-cooled6082.25alloy loadedalongED.

Albeit,alargesetofparametersisusedintheporousplasticitymodeltodescribetheplasticanisotropyofthematerials, only3damageparameters,f₀,fcandf_F,areincluded.Thus,thefiniteelementresultsindicatethatagooddescriptionofthe plasticdeformationandthestressstatesoccurringpriortofinalfailureiscriticalforductilefracture.Itshouldalsobenoted that all model parameters were calibrated based solelyon crystallographic texturemeasurements combinedwith crystal plasticityfiniteelementsimulations,andtwomechanicaltestssubjectingthematerialtodifferentstressstates.

6. Discussion

FromthemicrostructuralinvestigationinSection4.1,itisevidentthattheprecipitatestructureismarkedlyinfluencedby thequenchrate.Aslowerquenchrate,i.e.,air-cooling,givesareducedvolumefractionofstrengtheningprecipitateswithin thegrains,andwiderPFZsatthegrainboundariesandaroundthedispersoids.Theinfluenceofquenchrateonthenumber density,cross-sectionalarea andlength oftheprecipitatesdependsonthe alloy.The lean6060alloyhasalowernumber density of precipitates after air-cooling andthe precipitateshave a larger cross-section area andlength. The air-cooling is detrimental to the precipitate structure of the 6082.25 alloy, which has a fibrous grain structure with flat, elongated grainsandsmallsub-grains,andresultsinan inhomogeneousmicrostructurewhereentiregrainsarefreeofstrengthening

Fig. 16. Contour plot of von Mises equivalent strain during crack propagation in the Kahn tear test of the air-cooled 6082.25 alloy loaded along ED: (left) experiment and (right) finite element analysis. Extracted images from the experiment and finite element analyses at the same crack opening displacement, see Fig. 17 .

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 0

1 2 3 4 5

Displacement,v(mm)

Force,F(kN)

Fig. 17. Force-displacement curve from the finite element analysis of the Kahn tear test with the air-cooled 6082.25 alloy loaded along ED. Markers correspond to the images in Fig. 16 .