On the effect of plastic anisotropy, strength and work hardening on the tensile ductility of aluminium alloys

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International Journal of Solids and Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

On the effect of plastic anisotropy, strength and work hardening on the tensile ductility of aluminium alloys

Bjørn Håkon Frodal

^a^,^b

, David Morin

^a^,^b

, Tore Børvik

^a^,^b

, Odd Sture Hopperstad

^a^,^b

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

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

a rt i c l e i n f o

Article history:

Received 3 June 2019 Revised 28 August 2019 Accepted 3 October 2019 Available online xxx Keywords:

Ductile failure Strain localization Anisotropy Work hardening Porous plasticity

a b s t r a c t

Theinfluenceofplastic anisotropy,yield strengthandwork hardeningonductile failureisstudiedby nonlinearfiniteelementsimulationsandstrainlocalizationanalysesoftensiletestsindifferentmaterial orientations.Threealuminiumalloys withdifferentgrainstructuresandcrystallographictextures,heat- treatedtothree conditions giving riseto differentyield strengthand work-hardening behaviours,are considered.Theanisotropicyieldsurfacesofthealloys,obtainedbythecrystalplasticityfiniteelement method,areusedinthenumericalsimulationsofductilefailureinthetensiletests.Inaddition,ayield surfaceforanisotropicmaterialisincludedforcomparison.Theseyieldsurfacesarecombinedwiththree stress-straincurvesrepresentativeofthedifferentheat-treatments,resultinginarangeofrelevantmodel materialswithdifferentplasticanisotropy,yieldstrengthandworkhardeningusedinthenumericalin- vestigations.Finiteelementsimulationsoftensiletestsinsevenin-planedirectionsarecarriedout,i.e., 0°,15°,30°,45°,60°,75°and90°tothereferencedirection,andthenon-proportionalloadinghistories areusedinthesubsequentstrainlocalizationanalyses.Plasticanisotropyisfoundtohaveamarkedin- fluenceonthetensileductilityandtoinducefailureanisotropy,i.e.,avariationinthefailurestrainwith loadingdirection.Theshapeand extensionofthe regionsofconcentratedplasticflowinthefiniteel- ementsimulationsvarywithtensiledirectionfortheanisotropicmaterials.Inagreementwithprevious experimentalevidence,thestrainlocalizationanalysespredictavariationofthefailurestrainwithtensile directionthatappearstocorrelatewiththevariationoftheLankfordcoefficient,indicatingthatthefail- ureanisotropyiscloselylinkedtotheplasticanisotropy.Thestrainlocalizationanalysespredictahigher ductilityformaterialswithloweryieldstrengthandhigherworkhardening,asthesefeaturesleadtoa moredistributedplastic deformationandastressstatewithalowerstresstriaxialityintheneck.This redistributionoftheplasticdeformationmakesthetensilespecimenlesspronetostrainlocalizationand subsequentductilefailure.Theinfluenceofyieldstrengthandworkhardeningisfurtherfoundtodepend ontheplasticanisotropy.

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

1. Introduction

The thermo-mechanical processing of metals influences microstructural characteristics such as the grain structure and the crystallographic texture, and determines the plastic behaviour of these materials. As a result, extruded profiles, rolled plates and other formed structural components typically exhibit plastic anisotropy. The strength of the plastic anisotropy varies, and is mostlygoverned bythe crystallographictexture(EnglerandRan- dle,2009).Using crystalplasticitytheory,whichaccountsforthe crystallographictextureofmaterials,the yieldingandplasticflow

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

ofmetals are well described(Zhang et al., 2015; 2016). Numeri- cal simulations of materials withcrystal plasticityare in general computationallyexpensive, andphenomenologicalplasticitymod- elsarethuspreferredwhenrelativelylargestructuralcomponents are considered. These models may include an anisotropic yield function,typically incorporatingone orseverallineartransforma- tionsof thestress tensor(Barlat etal., 2005),which iscalibrated fromeitheralargenumberofexperimentaltests(Fourmeauetal., 2011) or crystal plasticity simulations (Zhang et al., 2015; 2016;

Frodaletal.,2019).

Theprocessofductilefractureincludesnucleation,growthand coalescenceof microscopic voidsat second-phaseparticles orin- clusions, and depends markedly on the local stress state and

https://doi.org/10.1016/j.ijsolstr.2019.10.003

0020-7683/© 2019 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/ ) Pleasecitethisarticleas:B.H.Frodal,D.MorinandT.Børviketal.,Ontheeffectofplasticanisotropy,strengthandworkhardeningon

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Nomenclature Symbols

σ

^Cauchy^stress^tensor

λ

˙ ^Plastic^multiplier

˙

q Non-uniformityratevector

˙

p Equivalentplasticstrainrate C^t Materialtangentstiffnesstensor D Rate-of-deformationtensor F Deformationgradienttensor I Second-orderidentitytensor L Velocitygradienttensor N Nominalstresstensor

n Unitnormalvectortoimperfectionband

R Rotationtensor

^Yield^function

φ

^,

θ

Localizationbandangles

φ

0,

θ

0 Initiallocalizationbandangles

σ

0 Initialyieldstress

σ

^t ^True^stress

σ

I,

σ

II,

σ

III Orderedprincipalstresses

σ

M Matrixﬂowstress

σ

h Hydrostaticstress

σ

vm vonMisesequivalentstress

ε

f Macroscopicfailurestrain

ε

l Logarithmicstrain

ϕ

^Equivalent^stress

ξ

^Strain^rate^ratio

A Cross-sectionarea a Yieldsurfaceexponent A₀ Initialcross-sectionarea D Cross-sectiondiameter E,

ν

^Elastic^coeﬃcients

F Measuredforce

f Voidvolumefraction f₀ Initialvoidvolumefraction

L Lodeparameter

p Equivalentplasticstrain p_f Localequivalentfailurestrain Q_i,

θ

i Isotropichardeningparameters q_i Tvergaardparameters

S_k,S_l Principalvaluesoftransformedstresstensors T Stresstriaxialityratio

Abbreviations

ED Extrusion/referencedirection ND Normal/thicknessdirection TD Transversedirection

microstructural characteristics in a complex way (Pineau et al., 2016).Inturn,thelocalstressstateisgovernedbytheyieldingand plasticflow of the material,and itfollows that the strength and workhardeningofamaterialcaninfluencetheductilitymeasured inan experimental test.Ifthematerialexhibitsplasticanisotropy becauseofthethermo-mechanical processing, themeasuredduc- tilitycouldalsodependonthedirectionofloading.Foraluminium alloys,experimentsshow that thetensileductilitydecreases with increasingyieldstress(Lloyd,2003;Westermannetal.,2014;Ped- ersen et al., 2015; Hannard et al., 2016) and is markedly influ- encedalsoby plastic anisotropy(Fourmeau etal., 2013;Khadyko etal.,2019).Numerical simulations indicatethat thisvariation in tensileductilityis partiallydue todifferencesin thedeformation andlocalstress state within theneck region ofthe tensilespecimen, as a higher yield strength is typically associated with re- ducedwork hardening (Dæhli et al., 2016). A higher stress level

Fig. 1. Normalized failure strain versus tensile direction obtained from tensile tests on a recrystallized AA6063 alloy ( Khadyko et al., 2019 ) and a non-recrystallized AA7075 alloy ( Fourmeau et al., 2013 ) with different crystallographic textures.

may also accelerate void nucleation at second-phase particles or inclusions(Pineau etal.,2016).Ductile failure canalso becaused byplasticanisotropy,e.g.,triggeringshearbandsinductilemateri- als(Benzergaetal.,2019).

There are three main sources of anisotropic failure in metals:

plasticanisotropy,whichprimarily stemsfromthecrystallographic texture,morphological anisotropy,whichoriginatesfromtheshape and preferred orientation of particles and voids, and topological anisotropy, which is a result of the spatial distribution of parti- clesandvoids.Albeit,thesetypesofanisotropyoriginatefromthe microscale, their effect is usually observed at the macroscale as a variation in the failure strain with loading direction, i.e., failureanisotropy.Experimentalevidencefromtensiletestsonsmooth axisymmetric specimens (Fourmeau et al., 2013) andflat rectangular specimens (Khadyko etal., 2019) indicates that the failure anisotropyobservedforsomealuminiumalloyscorrelateswiththe plasticflowanisotropyasexpressedbytheLankfordcoefficients.In Fig.1,failureanisotropy(i.e.,thatthefailurestrainvarieswithten- siledirection)isillustrated forarecrystallized AA6063alloywith recrystallizationtexture(Khadykoetal.,2019)anda fibrous,non- recrystallized AA7075 alloy with deformation texture (Fourmeau etal.,2013).Whereasthefailure anisotropyissignificantforboth materials, thevariationof thefailure strain withtensiledirection isoppositeforthetwoalloys.

Basedonunitcellsimulations,Keralavarmaetal.(2011)inves- tigatedtheeffectsofinitialporosity,initialvoidaspectratio,stress triaxialityandanisotropyparameters,andshowedthatthevoidas- pectratio,inadditiontotheplasticanisotropyparameters,cansig- niﬁcantlyaffecttheoverallductilityofanisotropicsolids.Morere- cently,LegarthandTvergaard(2018)performedthree-dimensional unitcell simulationsinvestigatingthethreesources ofanisotropic failure. Theyfoundthat the presenceofplastic anisotropy ampli- ﬁes thepredictionsobtained fordifferentinitialvoid shapes,and that there was a clear interaction betweenthe effects of plastic anisotropy,void shape andvoid spacing.Also experimentally the arrangementofsecond-phaseparticleshasbeenobservedtohave an effect on the failure process aswell asthe failure anisotropy (Hannard etal., 2018). Agarwaletal. (2002)studiedthe cracking ofsecond-phaseparticlesinanextrudedaluminiumalloy.Theyob- servedthatforagivenstrainlevel,thenumberfractionofcracked particlesvarieddependingontheloadingdirection.Thusvoidnu- cleationduetoparticlecrackingcanleadtofailureanisotropy.

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Useofunitcellsimulationsisanattractivewayofstudyingthe mechanismsofductilefailure,asinformationofthelocaldeforma- tion ﬁeldscan be employed to geta more profoundunderstand- ing ofthegrowthandcoalescenceofvoids.Intheunit cellmod- ellingframework,ductilefailureisusuallyassumedtocorrespond totheonsetofvoidcoalescence.However,strainlocalizationisof- ten a strongindicator forimminent ductilefailure, asplastic de- formationanddamageevolutionlocalizeinanarrowregionprior to failure initiation.Based onunit cellsimulations, Teko˘gluet al.

(2015) showed that dependingon the stress triaxiality, strain lo- calizationoccurssimultaneouslyorpriortovoidcoalescence.Thus, thestrain localizationphenomenoncan beconsideredasan indi- catorforincipientductilefailure.

The imperfection band approach to localization analysis, first proposed by Marciniak and Kuczyński (1967) for plane stress states, andlater extended by Rice (1976)to a generaland rigor- ousformulation,canbeappliedtostudyandpredicttheinitiation of ductile failure. A material with an imperfection is considered wherethepropertiesareslightlydifferentinsidetheimperfection compared to the restof the material.When the material is sub- jectedtoloading,deformationtendstoconcentrateinsidetheim- perfectionandthistendencypromoteslocalizationofdeformation in the material. The imperfection is taken in the form of a pla- narband,andthestressandstrainfieldsinsideandoutsideofthe band arehomogeneousbutdifferent.Localizationbylossofellip- ticity occurswhenthestrain ratebecomesinfiniteinsidethe imperfectionband.Totriggerlossofellipticity,theimperfectionband mustincorporateasofteningmechanism(RudnickiandRice,1975) inthecaseofassociatedplasticflow, andthisisusually achieved byuseofaporousplasticitymodeldescribingtheconstitutivebe- haviourinsidetheband.Thematerialoutsidethebandisdescribed either by metal plasticity or porous plasticity. The imperfection bandapproachhasrecentlybeenusedinseveralstudies,andgood quantitativeagreementisobservedbothwithunitcellsimulations (Morinetal., 2018a;Morinet al., 2019; Reddi etal., 2019;Vish- wakarma andKeralavarma, 2019) andexperimental tests(Gruben et al., 2017; Morin et al., 2018b; 2019). Whereas several studies haveusedunit cellsimulationstoinvestigatevoidgrowthandco- alescenceinanisotropic materials(Keralavarmaetal., 2011;Dæhli etal.,2017a;LegarthandTvergaard,2018;Frodaletal.,2019),localization analyses with finite element-based unit cells have so far only been performed for isotropic materials (Barsoum and Faleskog, 2007; Barsoum and Faleskog, 2011; Dunand and Mohr, 2014; Dæhli et al., 2017b; Guo and Wong, 2018; Vishwakarma andKeralavarma,2019).Usingthesecomputationallyexpensivefi- nite element models to perform strain localization analyses for anisotropic solidsisstill difficultevenwithmodern computers.A

large numberof localization band orientationshas to be investi- gatedwithinathree-dimensionalsetup foreachloadcaseandre- sultsinprohibitivecomputationaltimes.

Inthis study,the influence ofplastic anisotropy,strength and workhardeningontheinitiationofductilefailureintensionisin- vestigatednumericallywiththeuseofthestrain localizationthe- ory.Thus,incipientductilefailurewillinthefollowingbeconsid- eredtooccur atthe instancewhen strain localizationis firsten- countered in the material. Experimental data from tension tests on three extruded aluminium alloys obtained in previous studies(Khadykoetal., 2014;Frodaletal., 2019)is usedasbackdrop for the numerical study. These alloys have different grain struc- turesandcrystallographictextures,andweresolutionheat-treated andartificially aged to three conditions giving differentstrength and work-hardening behaviour. The anisotropic yield surfaces of the alloys were obtained by crystal plasticity simulations. Based ontheseexperimentalresults,asetoffictitious,butrelevant,alu- miniummaterialsaredesignedthatexhibitdifferentcombinations ofstrength,workhardening andplasticanisotropy.Finiteelement simulationsoftensiletestsonsmoothaxisymmetricspecimensare performedforeachofthesematerialsinsevenin-planedirections, i.e., 0°, 15°,30°, 45°, 60°, 75°and 90° to thereference direction.

Subsequently, the non-proportionalloading histories fromthe ﬁ- nite element simulations are used in strain localization analyses to predict incipient ductile failure of the tensile specimens, and thus to investigate the effect of plastic anisotropy, strength and work hardening on thetensile ductility. It is importantto investigate these effects together in order to disclose any interaction effectson thetensileductility. Inorder toincorporatethe plastic anisotropyof thematerials,the porousplasticitymodelproposed byDæhlietal.(2017a),incorporatingtheanisotropicyieldcriterion Yld2004-18p(Barlatetal.,2005),isappliedinallsimulations.

2. Experimentalbackground

The tensile ductility of the aluminium alloys AA6060, AA6082.25 and AA6082.50 has been examined experimentally in previous studies (Khadyko et al., 2014; Frodal et al., 2019).

These alloys were provided by Hydro Aluminium as extruded rectangular proﬁles, with a thickness of 10 mm and a width of 83 mm, from which axisymmetric tensile specimens were ma- chined. The specimens were solution heat-treatedand artiﬁcially aged to three different tempers, namely temper O (annealed), temperT7(overaged)andtemperT6(peakstrength).

Thethreealuminiumalloyshavedifferentgrain structuresand crystallographic textures (Frodal et al., 2017) leading to differ- entplastic anisotropy (Frodal etal., 2019). TheAA6060 alloyhas

M10x1.0

Ø6

R7.25

16 5 40 5 16

Through-thickness Cross-section

Fig. 2. Axisymmetric tensile specimen with the ﬁnite element mesh. The through-thickness and cross-section mesh is shown from the centre of the specimen. Dimensions are in mm.

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a recrystallized grain structure comprising equi-axed grains, and exhibits a cube texture with a minor Goss component. A typical ﬁbrous, non-recrystallized grain structure is observed forthe AA6082.25alloy,whichhasacubetexturewithorientationsalong the

β

^-fibre. ^The ÂA6082.50 âlloy ^has recrystallized grain struc- turewithlargeelongatedgrainsandarotatedcubetexture(Frodal etal., 2017). Forfurtherdetails aboutthematerials, thereaderis referredtoKhadykoetal.(2014)andFrodaletal.(2017,2019),and forfurtherinformationonthetexturecomponentsinFCCmateri- als,see,e.g.,EnglerandRandle(2009).

Axisymmetrictensilespecimens,seeFig.2,wereusedtodeter- mine the work-hardeningresponse andductile failure properties ofthe materials (Khadyko etal., 2014; Frodalet al., 2019). All of thespecimenswereorientedalongthetransversedirection(TD)of theextrudedproﬁle.Adisplacement-controlled testmachinewith aconstantcross-headvelocityof1.2mm/minwasusedtoperform thetests. Duringtesting, theforceanddiametersalongtheextru- siondirection(ED)andthicknessdirection(ND) ofthe minimum cross-section of the specimen were continuously measured until fracture usinga loadcell and an in-houselaser-based measuring system(Frodaletal.,2017),respectively.

Thecurrentareaofthespecimencanbeestimatedby A=

π

4D₁D₃ (1)

whereD₁ andD₃ are the measureddiameters inED andND, respectively.The truestress overtheminimumcross-sectionarea is

σ

^t⁼ ^F_A ⁽²⁾

whereF is the measured force. Assuming plastic incompressibil- ityandnegligibleelasticstrains,thelogarithmic(ortrue)strain is givenby

ε

l =ln

_A

0

A

(3)

whereA₀ istheinitialcross-sectionarea ofthespecimen,and

σ

^t

and

ε

l represent average valuesover the minimum cross-section areaofthespecimen.

Fig. 3 presents the true stress-strain curves from the tensile testsinTD plottedup to the point offailure, wheremarked dif- ferences between the behaviour of the different alloy and temper combinations can be observed. Note that failure is here de- ﬁnedasthepointofmaximumtruestress,andanabruptdecrease inthe stress level is observed after thispoint. The strength and workhardeningofthedifferenttempersofthesamealloyaredis- tinct. In general, the O tempers have the highest work hardening,buttheloweststrength.TheT6tempershavethelowestwork hardeningandthehigheststrength,whiletheT7tempersarebe- tweentheOandT6temperswhenitcomestostrengthandwork hardening,seealsoSection3.2.Comparingthealloys,thestrength clearlyvariesbetweenthem,andalso thework hardeningis different. Typically, the strength of the two AA6082 alloys for the sametemperissimilarandhigherthan thatoftheAA6060alloy.

Theonlyexception isforthe Otemper,wheretheAA6082.25 al- loyhashigherstrengththan thetwo other alloys.The reasonfor thisisprimarilythatforthetwoAA6082alloysintempersT6and T7the precipitate numberdensities are higherthan forthe lean AA6060alloy,whereasfortheOtemperthemaincontributionsto theyield strengthcome fromelementsinsolid solution,the dis- persoidnumber densityandthe grain structure(sub-grain structure).

Comparing the point of failure for the various alloy-temper combinations,makes it apparent that the AA6060alloy isby far themostductilealloyandthe Otemperisthemostductiletem- perforeachalloy.EventheleastductiletemperoftheAA6060al- loy,i.e,theT6temper,hasamuchhigherfailurestrainthanallof

Fig. 3. True stress-strain curves from tension tests of the aluminium alloys (a) AA6060, (b) AA6082.25, and (c) AA6082.50 in tempers T6, T7 and O. All tests were performed with tensile direction along TD of the extruded proﬁle. The data is taken from Khadyko et al. (2014) and Frodal et al. (2017, 2019) .

thetempersofthetwoAA6082alloys.ComparingthetwoAA6082 alloys, it is observed that the AA6082.25 alloy has,in general, a higherfailure strain thantheAA6082.50 alloy,forthesametem- per.The lower ductility observedfor theAA6082.50 alloycan be linkedtothegrainstructureofthisalloy,seeSection5.

Fig.4showstheaveragefailurestrain fromthetensiletestsin TDversustheinitial yieldstressat0.2%plasticstrain. Itisclearly visiblethatthemagnitudeofthefailurestrain,andthustheduc- tilityofthematerials,varywithyieldstrength,andalsothediffer- encein ductility betweenthe alloysisevident. In previous stud- iesonvariousaluminiumalloys,ithasbeenfoundthatthefailure straintendstodecreaselinearlywithincreasingyieldstrengthfor similarmicrostructure(Lloyd,2003;Westermannetal.,2014;Ped- ersenetal.,2015;Hannardetal.,2016),andthereaderisreferred tothesestudiesfordetaileddiscussionsonthephysicalinterpreta-

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Box 1. Overview of the strain localization analyses ( Morin et al., 2018a ) ¹.

Fig. 4. Failure strain in tension versus initial yield stress at 0.2% plastic strain for the three alloys in different tempers.

tion of theseexperimental trends.In short, theyield strength of a material is closely linked to its work hardening, and typically asthestrengthincreases,thework hardeningdecreases,whichis negativeforthe ductility,seeSection 4.Ahigherstress levelmay also accelerate voidnucleation at second-phaseparticles (Pineau etal.,2016).

3. Numericalmethods 3.1. Strainlocalizationtheory

The strain localizationtheoryis usedherein toinvestigatethe influenceofstrength,workhardeningandplasticanisotropyonthe tensilefailureofductilematerials. Atmoderatestresstriaxialities, localization hasbeenfound to occur simultaneously asvoid coa- lescence (Teko˘gluetal.,2015).Thisistrueunderarandomdistri- butionofvoids(Reddietal.,2019;VishwakarmaandKeralavarma, 2019),andstrain localizationcan thus beusefulin predictingin- cipientductilefailure.Asalreadymentioned,wewilldefineincip- ientductilefailureastheinstancewhenstrain localizationisfirst encounteredinthematerial,andthelocationoffailureinitiationis withinthefiniteelementwherestrainlocalizationoccursfirst,i.e., acriticalelement,seeSection3.3.

The imperfection band approach proposed by Rice (1976) is usedinthisstudy.Thismethodconsidersamaterialconsistingof two homogeneous regions, which are separated by a thinplanar imperfectionband, andsubjected toan overall uniformdeforma- tion.The constitutiveequations inside and outsideof the imper- fectionbandareallowedtobedifferent,withtherequirementthat equilibriumand compatibility conditions are enforced across the band.Abriefoverviewofthegoverningequationsoftheimperfec- tionbandapproachisgiveninBox1.ThereaderisreferredtoRice (1976),NeedlemanandRice(1978),andMorinetal.(2018a,b)for furtherdetails.

While this method does not impose any restrictions on the constitutive equations of the material inside or outside the imperfection band, the sameapproach asin Nahshonand Hutchin- son (2008),Gruben etal. (2017),andMorin etal.(2018a, 2018b, 2019) is used in the current work. A porous plasticitymodel is used to represent the material inside and outside of the imperfectionband.Itisassumedthatanydamagemechanismoccurring outsideofthebandisnegligibleandthattheporosityhereiszero (f=0),whereas insidetheband animperfectionisintroduced by pre-existingvoids(f₀>0).Formoderatestresstriaxialities,encoun- tered inphysical tension tests, this is usually an appropriate as- sumption(Xue etal., 2010;2013;Westermann etal., 2014). Note thatalsoother typesofimperfectionscanbeincludedbothinside and outside the band, e.g., void nucleation (Morin et al., 2018a;

2018b; 2019) andvoidsoftening in shear(Nahshon andHutchin- son,2008;Morinetal., 2018a) canbeintroducedintheconstitu- tiveequations.

Anoverviewoftheporousplasticitymodelusedhereinisgiven inBox 2.The heuristic modiﬁcationof the Gurson (1977) model proposed by Dæhli et al. (2017a) and applied by Morin et al.

(2018b) is used in the current study. This extension introduces the equivalent stress of the Yld2004-18p yield function (Barlat et al., 2005) into the constitutive equations, in order to include anisotropic yielding andplastic ﬂow. Forzero porosity, the yield criterionreducestotheoriginalYld2004-18pyieldfunction(Barlat etal.,2005).Theporousplasticitymodelintroducesmaterialsoft- eninginsidetheimperfectionband,whichtriggerslossofelliptic- ityofthegoverningequations,i.e.,strainlocalization.Whenanas- sociatedﬂowruleisadopted,materialsofteningisrequiredforloss ofellipticitytooccur(RudnickiandRice,1975)andthusforthelo- calizationconditions(Box1)tobemetforreasonablestresslevels.

NotethattheGurson-Tvergaardyieldfunction(Gurson,1977;Tver- gaard, 1981), Equation(15), isderived using von Misesplasticity,

1Note that in Morin et al. (2019) , there is a typo in Equations (26) and (27). The correct expressions are here given in Equations (5) and (6) of Box 1 .

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Box 2. Overview of the porous plasticity model ( Dæhli et al., 2017a ).

andtheonlymodiﬁcationintroducedhereisthatthemacroscopic vonMisesequivalentstressisreplacedbyEquation(16).Alsoother porousanisotropic plasticitymodelshavebeen developedby,e.g, BenzergaandBesson(2001),andSteglichetal.(2010).Theporous plasticitymodelusedherehasbeenvalidatedagainstunitcellsim- ulationsby Dæhliet al.(2017a),andforfurther details regarding theaccuracy ofsuch heuristically extendedmodels, thereader is referredtoDæhlietal.(2017a,2019).

The strain localization theory by the imperfection band approach has been implemented in a stand-alone Fortran pro- gramme,asdescribedindetailbyMorinetal.(2018a).Theporous plasticitymodelhasbeen implementedinto ausermaterial sub- routine(UMAT)forAbaqus/Standard(Abaqus,2014).

3.2.Finiteelementanalyses

The imperfectionbandanalysesare drivenbyloadinghistories extractedfromﬁniteelementanalysesoftensiletests.Theaxisym- metrictensilespecimen ismodelled inAbaqus/Standard, andthe ﬁniteelementmeshispresentedinFig.2.Lineareight-nodesolid elementswithselectivereducedintegration (C3D8)are used.The dimensionsoftheelementslocatedinthecentreofthespecimen are0.10×0.15×0.15mm³,withtheshortestelementlengthalong thetensiledirection.

The material behaviour of the tensilespecimen is defined by theporousplasticitymodeldescribedinBox2.Whenrunningthe finiteelement simulationsof thetensiletests, theinitial porosity issettozero(f0=0), thusreducingthemodeltotheanisotropic Yld2004-18p plasticity model (Barlat et al., 2005) with isochoric plastic flow. The effect of porosity in the finite element simulations is thus neglected as fwill be zerothroughout thesesimu- lations.Isotropicelasticityis assumedwithaYoung’s modulus of E=70000MPaandaPoisson’sratioof

ν

=0.3,whicharerelevant valuesforaluminiumalloys.

Inordertostudytheeffectofplasticanisotropyonthetensile ductility, typical yield surfaces fortextured aluminium alloysare employed in the simulations. The AA6060and AA6082.25 alloys havetypicalrecrystallizationanddeformationtexture,respectively, whereastheAA6082.50alloyhasatypicaltextureofanalloywith largerecrystallizedgrains,seeSection2.Inaddition,theyieldsur- faceofanisotropicalloywithrandomtextureisincludedforcom- parison.Toemphasise thattheyield surfacesfromthesealloysin thefollowingwillbecombinedwithﬂowstresscurvesthatdonot

belongtotherespectivealloy,wewillconsiderthealloysasmodel materials andrenamethem accordingly. Themodel materials are thus denotedalloyA, B,C andDwithyield surfacebelongingto alloyAA6060,AA6082.25AA6082.50andanisotropicmaterial,respectively.Theyieldsurfacesofthemodelmaterialsarepresented inFig.5depictedintheED-TDplane.Itisapparentthattheyield surfacesforthealloysaredistinctduetothecrystallographictex- ture. Theseanisotropic yield surfaceshavepreviously been found byFrodal etal.(2019) usingcrystalplasticityfiniteelement analyses.The yieldsurfaceofthe isotropicmaterial (alloyD)isgiven bythe Yld2004-18pyieldfunction withall anisotropycoefficients equaltoone, thusreducingit toanisotropichigh-exponent yield function.The exponents oftheselected anisotropic yieldsurfaces areapproximately12.Thus,tolimittheinfluenceoftheyieldsur- face curvature on the imperfection band analyses, as studied by Dæhlietal.(2017b),theexponent oftheisotropicmaterial isset toa=12.Thelistofanisotropy parametersisomittedinthispa- per, and the readeris referred to Frodal et al.(2019) forfurther details.

Fig.6 presentsthenormalizedyield stressesandLankford co- efficients as function of the tensiledirection in the ED-TD plane obtainedwiththeselectedyieldsurfaces.TheLankfordcoefficient isdefinedas

R_α= d

ε

⊥

d

ε

^ND ⁽²²⁾

whered

ε

⊥istheincrementalstraininthedirectionperpendicular totheloadingdirectionlyingintheED-TDplane andd

ε

ND isthe incrementalstraininthethicknessdirection(ND).Thus,theLank- fordcoefficient R_α givesthe evolutionofthe cross-section ofthe specimen.The0°directionisalong ED,andistakenasthe refer- encedirectioninthisstudy,whilethe90°directionisalongTDof theextrudedprofile.Boththevaluesandthevariationofthenor- malizedyield stressesandLankfordcoefficientsare markedlydif- ferentfortheanisotropicyieldsurfaces.Thetwocurvesappearto exhibittheoppositetrendforagivenanisotropicyieldsurface,i.e., when thenormalized yieldstress has a maximum/minimum,the Lankford coefficient tends to have a minimum/maximum. These extremaarecausedbythecrystallographictextureofthealloys.

In orderto study theinﬂuence of strength andwork hardening on ductile failure, selected work-hardening curvestypical for aluminium alloys will be used. The selected work-hardening be- haviouris takenfromtheAA6082.25alloyartiﬁciallyaged tothe three conditionsgivenin Section 2.The work-hardeningrule de-

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Fig. 5. Yield surfaces depicted in the ED-TD plane for (a) alloy A (AA6060), (b) alloy B (AA6082.25), (c) alloy C (AA6082.50), and (d) alloy D (isotropic). Contours of increasing normalized shear stress are plotted in 0.1 increments, with the maximum value in the centre.

Fig. 6. (a) Normalized yield stress and (b) Lankford coeﬃcient versus tensile direction for uniaxial tension in the ED-TD plane obtained with the four selected yield surfaces.

ﬁned inBox2 wascalibratedinFrodaletal.(2019) usinganop- timization procedure. Fig. 7 presents the normalized ﬂow stress curves and displays the large difference in work hardening be- tweenthethreetempers.Thecorrespondinginitialyieldstressand work-hardeningparametersaregiveninTable1.

In the following, the simulation procedure consists of ﬁ- nite element analyses of the tensile tests in seven in-plane di-

rections, i.e., 0°, 15°, 30°, 45°, 60°, 75° and 90° with respect to the reference direction (ED). Strain localization theory will be used to predict the logarithmic failure strain in each simulation of a tensile test, see Section 3.3. Note that only initiation of failure is predicted by using the strain localization theory in the post-processing of the ﬁnite element simulation results.

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Table 1

Initial yield stress and work-hardening parameters ( Frodal et al., 2019 ).

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

O 57.6 2661.3 44.6 382.2 32.6 120.8 91.0

T7 163.6 1300.1 28.9 1301.2 40.7 52.3 232.9

T6 299.5 470.5 28.5 485.0 29.8 50.0 279.4

Fig. 7. Normalized ﬂow stress curves representing the work-hardening behaviours used in the numerical study.

3.3.Localizationanalyses

Whenperforming theimperfectionbandanalyses,thematerial insidetheimperfectionbandisdescribedbytheporousplasticity modelinBox 2withan initialporosity of f₀=0.005,whileout- side,theporosityissettozeroasintheﬁniteelementsimulations.

Thisvalueoff₀ givesfailurestrainsusingthelocalizationanalyses thatarereasonableforaluminiumalloys,andischosenforall the materialsinordertoisolate theeffectsofstrength,work harden- ingandplasticanisotropyonthefailurestrain.Theporousplastic- ityparametersbyTvergaard(1981)areheregivenstandardvalues ofq₁=1.5,q₂=1.0and q₃=q²₁, butcould alternatively be cali- bratedforeachcombinationofyieldsurfaceandﬂowstresscurve, asdonebyDæhlietal.(2017a).

The location of ductile failure initiation is not known a pri- ori,butitisreasonabletoassumethatfailure isﬁrstencountered withintheneckofthetensilespecimen.Accordingly, allelements within this region are examined forstrain localization usingthe imperfectionbandapproachasdescribedinBox 1.The numerical procedureisasfollows(Morinetal.,2018b):

1.ThedeformationgradientF(t)ofeachelementwithintheneck regioniscalculatedbasedonthe nodaldisplacements andthe isoparametricshapefunctions.

2. An imperfection analysis is run for each of these elements based on the extracted deformation gradient F(t) for a large numberof band orientations(approximately 700unique band orientations for each element) deﬁned by

φ

0∈[0,

π

^] ^and

θ

0∈[0,2

π

^], ûsingâ ^domain^reduction^methodâs^describedⁱⁿ

Morinetal.(2018a).

3. For each element, a localfailure strain p_f iscalculated as the minimumoverallimperfectionbandorientationsoftheequiv- alentplasticstrainoutsidetheimperfectionband atlossofel- lipticityinsidetheband.

4. Using the relationship between the local equivalent plastic strainpoftheelementsandthemacroscopiclogarithmicstrain

ε

l from the ﬁnite element simulation of the specimen, the macroscopic failure strain

ε

f corresponding to strain localiza- tionwithintheactualelementisfound.

5. The actual logarithmicfailure strain corresponds to theminimum value of

ε

f over the neck region and its position is as- sumedtobethelocationoffailureinitiation.

Forfurtherdetailsonthenumericalprocedurethereaderisre- ferredtoMorinetal.(2018b,2019).Inthesubsequentsections,the localizationband willreferto theband forwhichlossofelliptic- ityoccursﬁrstinthecriticalelement,thustheimperfectionband givingthelowestmacroscopicfailurestrain

ε

f.

4. Numericalresults 4.1. Macroscopicbehaviour

Fig. 8 presents the true stress-strain curves from the ﬁnite element analyses of the tensile tests along the reference direction (ED). All of the materials represented by the yield surfaces in Fig. 5 are shown with the three work-hardening behaviours, and are plotted until failure predicted by the imperfection band approach. It is evident that both the plastic anisotropy, and the strength and work hardening have a pronounced effect on the failure strain, whereas the stress-strain curves are almost iden- tical betweenthe differentyield surfaces in the reference direction.Intheother tensiledirections,variationsareobservedasthe yield stress varies withtensile direction according to the plastic anisotropy deﬁnedbytheyield surface,seeFig. 6a.Notethat,al-

Fig. 8. True stress-strain curves from the ﬁnite element analyses of the tensile tests along the reference direction (ED), where the curves are plotted until failure as predicted by the strain localization theory for each material. The location of predicted failure, is shown with the corresponding symbols used in Fig. 9 : ( ) alloy A, ( ) alloy B, ( ✦ ) alloy C, ( ) alloy D. The uppermost curve is for temper T6, the inter- mediate for temper T7 and the lower for temper O.

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Fig. 9. (a) Failure strain versus initial yield stress for loading along the reference direction (ED), and (b) failure strain versus tensile direction in the ED-TD plane for the alloys with the temper T6 work hardening.

though the macroscopic stress-strain curvesare indistinguishable in the reference direction, the local stress state varies with the plasticanisotropy,seeSection4.2.

InFig.9a,thepredictedfailurestrainisplottedagainsttheini- tial yieldstress forall materials. The effectofstrength andwork hardeningisseentovarywiththeplasticanisotropy.Forthelow- eststrengthandhighestworkhardening(temper O),alloysAand D have approximately the same failure strain, which is signiﬁ- cantly higher than the failure strains of alloys B and C. In this temper, alloy B has somewhat lower ductility than alloy C. For higher strength and lower work hardening (tempers T6 andT7), however, alloys B and C switch position, and the failure strain becomes clearly lower for alloy C than for alloy B. The difference in ductility between alloysA and Dremains small in tem- persT6andT7,butalloyAhassomewhathigherductilitythanal- loyD.Thus,thedecreaseinfailurestrain withincreasingstrength anddecreasingworkhardeningdependsmarkedlyupontheplastic anisotropy.

Fig.9bpresentsthefailurestrainversusthetensiledirectionin the ED-TDplane forthealloyswiththetemper T6workharden- ing,i.e.,thehigheststrengthandlowestworkhardening.Thefail- urestrainoftheanisotropic materialsvariessignificantlywiththe tensiledirection, whileforthe isotropicalloy(alloyD) thefailure strain isconstant.Thefailure strain,andthusthetensileductility ofalloyA,isthegreatestinED(0°direction),whereasalloysBand Chavethehighestductilitywhenloadedinthe45°direction.The lowest failure strain isobservedin the60°,15°and0°directions foralloysA,B,andC,respectively.Comparingthevariationofthe failure strain in Fig.9b withthevariation of theLankfordcoeffi- cient in Fig.6b, it isfound that thesetwo characteristics exhibit tosome extentthesametrends,andexhibit anapproximatelyin- verse correlation with the normalized yield stress in Fig.6a, see Section 4.2forfurtherdetails. Thepredictedvariation ofthefail- ure strain with tensiledirectionalso resemblesexperimental ob- servationsfromtheliteraturereproducedinFig.1.Thepredictions foralloyAexhibitthesametrendastheexperimentaldataforthe AA6063alloyinKhadykoetal.(2019),andtheexperimental find- ings in Fourmeau et al.(2013) fora AA7075 alloy are similar to the predictions foralloyB. Thesealloys havesimilar grain struc- tureandcrystallographictexturetothosepresentedinFig.1.

Plotsofthedeformedconﬁgurationofthe tensilespecimenat failure predictedby theimperfectionband analyses areshownin Fig. 10,as obtainedin the ﬁniteelement simulations of the ten-

siletests indifferent directionswith respectto the referencedirection (ED). Regions of concentrated plastic flow is observed in thecentreofthespecimen.The shapeoftheseregionsisdefined bytheplasticanisotropyasdescribedbytheyieldsurfaceandthe associatedflowrule.Betweenthematerials, differentdeformation modesare seen,and thelevel ofequivalent plasticstrain at failure varies between the materials andwith the tensile direction.

Forthetestsalong ED andTD,theregion ofconcentrated plastic ﬂowissymmetric aboutthematerialaxes duetotheorthotropic samplesymmetry, whereas in the other directions the region of concentratedplasticﬂowdevelopsatanangletotheloadingaxis.

Areasonableconjunctureisthatthesedeformationmodesleadto differentshapesoffracturedspecimens,varyingbetweencup-and- cone to slant shear fracture modes due to the anisotropic plastic ﬂow. However, with the current approach, based on a poste- riorilocalizationanalyses,onlyincipientductilefailurecanbede- scribed.ThedeformedshapesobtainedhereforalloyBaresimilar tothefracturemodesobservedexperimentallyby Fourmeauetal.

(2013).Inadditiontoplasticanisotropy,materialinhomogeneities, e.g.,the arrangementofsecond-phaseparticles,can contribute to the ductile failure process andaffect the fracture path (Hannard etal.,2018).Failure initiation,aspredictedby thestrain localiza- tiontheory,isobservedtooccurintheregionofthehighestequiv- alentplasticstrain,andforalloysB,CandD,failureinitiatesinthe centreforallloadingdirections.ForalloyA,failureinitiatesinthe centrefor all loadingdirections withthe 45°direction asan ex- ception. Inthis loading direction, failure initiation occurs further towardsthespecimenperiphery.Notethat theimperfectionband ofthestrainlocalizationanalysesshouldnotbeconfusedwiththe regionsofconcentrated plasticﬂowwithintheneckofthetensile specimensintheﬁniteelementsimulations.

Fig.11depictscontourplotsoftheequivalentplasticstrain on theminimum cross-section ofthetensile testsinthe 0°and90° directionsforalloyBintemperT6.Theequivalentplasticstrainis observedtobemoreconcentratedinthecentreofthespecimenin the0°direction,whichislinkedtotheverylowvalueoftheLank- fordcoefficient inthisdirection, seeFig.6b. Inthe90°direction, theLankfordcoefficientismarkedlyhigherandthiscontributesto a more uniform plastic strain distribution across the specimen’s cross-section, which is positive forthe tensileductility. As a result,alloyBhasaslightlyhigherfailurestraininthe90°direction thanin the0°direction, seeFig.9b,even though thestress level ishigherinthe90°direction,seeFig.6a.Theeffectofplasticflow Pleasecitethisarticleas:B.H.Frodal,D.MorinandT.Børviketal.,Ontheeffectofplasticanisotropy,strengthandworkhardeningon

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Fig. 10. Deformed conﬁguration of the tensile specimen at failure (as predicted by the imperfection band analysis) depicted in the ED-TD plane as obtained from the simulations of tensile tests in different directions with respect to the reference direction (ED): (top) alloy A, (middle) alloy B, and (bottom) alloy C, with work hardening according to temper T6. Contours of the equivalent plastic strain are shown on the deformed meshes. The Lankford coeﬃcient R αof the corresponding test is depicted on the top of each mesh.

onductilitywillbefurtherdiscussedinthenextsection—andpar- ticularlyinconnectionwithFig.14.

4.2.Microscopicbehaviour

In thissection, welookmore closelyintothe microscopicbehaviour of the critical element in the neckregion of the tensile specimen,i.e.,thelocationoffailureinitiation.Relevantquantities bothinside andoutsideoftheimperfection band fromthestrain localizationanalysesare investigatedinorder tofurther interpret theeffects ofstrength,work hardening andplastic anisotropyon strainlocalization.Itisthereforeusefultodeﬁnecertainstressin- variants,such asthestresstriaxialityratioandLodeparameterto beusedinthefollowing.Thestresstriaxialityratioisdeﬁnedas T=

σ

h

σ

vm

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where

σ

h=¹3tr(

σ

) ^is ^the hydrostaticstress and

σ

vm=

3 2

σ

^:

σ

isthevonMisesequivalentstress,

σ

^being^the^stress^deviator.^The

Lodeparameterisdeﬁnedas L=2

σ

II−

σ

I−

σ

III

σ

I−

σ

III

(24)

where

σ

I≥

σ

II≥

σ

III are the ordered principal stresses. Notethat theLodeparameterisL=−1forgeneralizedaxisymmetrictension, L=0forgeneralizedshear,andL=+1forgeneralizedaxisymmet- riccompression.

Fig. 12 presents quantities fromthe imperfection band analyses obtainedinside andoutside ofthe critical imperfectionband inthecriticalelement fromthesimulationsofthetensiletestsin thereferencedirection(ED)foralloysA,B,CandDintemperT6.

ThenormalizedvonMisesstress,insideandoutsidetheband,to- getherwiththenormalizedvoidvolumefractioninside theband, areplottedagainstthelogarithmicstraininFig.12a.ThevonMises

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Fig. 11. Minimum cross-section at failure (as predicted by the imperfection band analysis) of alloy B with the temper T6 work hardening showing contours of the equivalent plastic strain for simulations of tensile tests in the: (a) 0 °direction, and (b) 90 °direction.

Fig. 12. Local behaviour in the regions inside (dashed lines) and outside (solid lines) of the imperfection band in the critical element for the materials with work hardening according to temper T6 loaded along ED: (a) normalized von Mises stress and normalized void volume fraction, (b) stress triaxiality ratio, (c) equivalent plastic strain, and (d) Lode parameter versus logarithmic strain over the neck.

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stressissimilar outsideoftheband forall alloys, whereasinside theband thematerialexperiences porosity inducedsoftening beforelocalization occurs. Initially, the evolution of the porosity is similarforallmaterials,butwithstrainingtheporosity insidethe band grows differently depending on the material and thus the plasticanisotropy.AsshowninFig.9,alloysAandChaverespec- tively thehighest andlowest failure strain forthis conﬁguration.

SofteningisseentooccurearlierforalloyCthanfortheotheral- loys,owing totherapid increaseoftheporosity atalower value ofthelogarithmicstrainintheneck.Thestresstriaxialityisplot- ted against the logarithmic strain in Fig. 12b. After necking, the stress triaxiality increases with straining, but witha higher rate insidetheimperfectionbandthanoutside,andstrainsofteningin- sidethe band coincides witha rapidincrease ofthe stress triaxiality. Compared with the other alloys, alloyC has higher stress triaxiality both outside and inside the imperfection band for all strains,whichexplainsthelowerductility.Anearlierrapidincrease ofthestress triaxialityinsidetheband isseenforthealloyswith thelowest ductility, namely alloysB and C. The equivalentplasticstrain pinsideandoutsideofthecriticalimperfectionband is plottedinFig. 12casa function ofthe logarithmicstrain

ε

l over the neck.Outside of the imperfection band, the equivalentplasticstrain evolves similarly forall alloys, whereas insidethe evo- lutiondiffers. The material inside the band experiences a higher equivalentplastic strain rate than outside to compensate forthe porosity-inducedsoftening, whichoccursatdifferentstrain levels forthefourmaterials. Fig.12ddisplaysthe Lodeparameter Lin- sideandoutsideoftheimperfectionbandasafunctionofthelog- arithmicstrain

ε

lovertheneck.Itisapparentthatthestressstate driftsfromgeneralizedtension(L=−1)towardsgeneralizedshear (L=0) inside the imperfection band, as also observed by Morin etal.(2018a,b).Thestress state isalso observedto differslightly fromgeneralizedtensionoutsideoftheimperfectionband dueto the plastic anisotropy. For the materials with the lowest ductility,e.g., alloyC,thestressstate insideofthe bandisobserved to changerapidlyfromgeneralizedtensiontowardsgeneralizedshear atalower logarithmicstrain thanforthe moreductilematerials, e.g.,alloyA.

InFig.13thesamequantitiesasinFig.12areshown,buthere thesimulationsofthetensiontestsinthereferencedirection(ED) foralloyAintempersO,T7andT6areaddressed.Thetrendsseen inFig.13arerepresentativefortheotheralloysaswell.Theporos- ityis seentogrow thefastestfortheT6temperandtheslowest forthe Otemper, i.e., a higherwork hardening leadsto a lower growthrateoftheporosityasafunctionofthelogarithmicstrain overtheneckofthetensilespecimen,seeFig.13a.Inthesimula- tionsfortheOtemper,thestresstriaxialityincreaseswithaclearly lower rate than for the other two tempers withhigher strength andlowerworkhardening,seeFig.13b.Thus,foragivenlogarith- micstrainafternecking,thestresstriaxialityisdeﬁnitelythelow- estfortheOtemper,whichispartlythereasonforthehigherten- sileductility.Thestresstriaxialityratioisfoundtobehigherclose to strain localization forthe T6 temper than for the T7temper, both inside and outside of the imperfection band. From Fig. 13c it isevident that the lower strength andhigher work hardening ofthe Otemper lead to a more gradual increase in the equiva- lentplasticstrainasthelogarithmicstrainovertheneckincreases comparedwiththeT6andT7tempers.Thehigherworkhardening contributesto amoreuniformplasticstrain distributionthrough- outthecross-sectionofthetensilespecimen,whichdelaysthefor- mationofaneckandthusstrainlocalization.Albeitnotasappar- ent,theequivalentplasticstrain insideandoutsideoftheimper- fectionbandevolvesfasterfortheT6temperthanfortheT7tem- per.As seen in Fig. 13d, the stress state appears to drift earlier fromgeneralizedtensiontowardsgeneralizedshearfortempersT6 andT7than for temper Odueto the higher strength andlower

work hardening. It seems asifthe plastic anisotropy mighthave astrongerimpactonthestress statewithintheneckregionfora materialwithlowerworkhardening.Alowerwork-hardeningrate givesa sharper andfaster evolvingneckregion which willaffect thestressstateinthenecksothatitdriftsearlierfromitsoriginal stateofuniaxialtension.

Fig.14presentsthemicroscopicbehaviourofalloyBintemper T6fromsimulationsofthetensiontestsinthe0°,45°,and90°directions. Inagreement with Fig.6a, the von Misesstress level is clearlydifferentinthethreedirections,seeFig.14a.Theevolution oftheporosityisalsodifferentinthethreedirections,wherethe voidgrowthistheslowestinthe45°direction,whichhasthelow- estvonMisesstress.Initially,thevoidgrowthislowerinthe0°di- rectionthaninthe90°direction,butthischangesintheﬁnalstage beforelocalization.FromFig.14b,itcanbeobservedthatthestress triaxialitylevelinsidetheimperfectionbandatagivenlogarithmic strain over the neck is higher in the 0° direction with the low- estductility. Outsidetheimperfectionband, the90°directionhas aslightlylower stress triaxialitylevelthan the45°direction,and the0°directionis alsoherethedirectionwiththe higheststress triaxialitylevel.

Inthe 45°direction, theequivalentplasticstrain rateis lower than in the other directions, seeFig. 14c. This indicates that the plastic deformation ismore dispersed over the specimen’scross- section, which is favourable to prevent strain localization. Again, thestress state isseen todrift fromgeneralizedtension(L=−1) towardsgeneralizedshear(L=0) insideoftheimperfectionband, andthedriftoccursfirstinthe0°directionexhibitingthelowest ductility.Duetotheplasticanisotropydefinedbytheyieldsurface andtheassociatedplasticflowrule,theLodeparameterisseento evolve differently for different tensiledirections also outside the imperfectionband.

5. Discussion

ThefailurepredictionspresentedinFig.9obtainedinthestrain localizationanalysesarefoundtocapturethetrendsobservedex- perimentally. In agreement withthe experimental data inFig. 4, thelocalizationanalysesinFig.9agivealowerfailurestrain fora material witha higher strengthand lower work hardening. Also, the failure strain is observed to varywith plasticanisotropy and theinfluenceofstrengthandworkhardeningisdifferentdepend- ing on thealloy. Albeitgood agreement isachieved betweenthe numericalandexperimental trends,therearecertain mechanisms notincludedinthenumericalstudythatcanhaveasubstantialef- fect onthefailure strain.Forinstance,intheexperimentsforthe AA6060andAA6082.25 alloys, thedifference inthe failurestrain betweentheOandT7tempersandtheT7andT6tempersissimi- lar,butnumericallya smallerdifference isobserved betweenthe T7 and T6 tempers. A reasonable explanation for this finding is thatthestresslevelfortheT6temperissufficientlyhightomake void nucleation a moreimportant mechanismfor damage evolution(Pineauetal.,2016),whichisnotcapturedbytheporousplas- ticity modelused herein, consideringonly growthof pre-existing voids.FortheAA6082.50alloy,alargedecreaseinthefailurestrain between the O and T7tempers is observed experimentally. This findingandtheloweroverallductilityobservedfortheAA6082.50 alloy can be linked to the grain structure, asthe large grainsof thisalloyincrease the amountofintercrystalline fracture andre- ducethetensileductility(Frodaletal.,2017).

Althoughplasticanisotropy relatedto thecrystallographictex- turecanleadtovariationsinthefailurestrainbothbetweenalloys and withtensile direction asseen in the numerical study,other sources of anisotropy can inﬂuence the ductile failure process andaffectthefailure anisotropy.Evenmaterials exhibitingnearly isotropic yielding and plastic ﬂow can exhibit failure anisotropy

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Fig. 13. Local behaviour in the regions inside (dashed lines) and outside (solid lines) of the imperfection band in the critical element for alloy A with the three different ﬂow stress curves loaded along ED: (a) normalized von Mises stress and normalized void volume fraction, (b) stress triaxiality ratio, (c) equivalent plastic strain, and (d) Lode parameter versus logarithmic strain over the neck.

becauseofmorphological ortopologicalanisotropycaused bythe shape, orientation and spatial distribution of voids and particles (Hannard et al., 2018). As investigated by Agarwal et al. (2002), the loading direction can affect the particle cracking process, as thenumberfractionofcrackedparticlescandependonthedirec- tion ofloading.Thus,voidnucleation byparticle crackingcanin- troduceanisotropiceffects.Alsothespatialdistributionofparticles andclusterscaninﬂuencethevoidcoalescenceprocessandleadto failureanisotropy(Hannardetal.,2018).Ithasalsobeenshownby unit cell analyses that the void aspect ratio can signiﬁcantly af- fecttheoverallductility(Keralavarmaetal.,2011)andthatplastic anisotropycanamplifythiseffect(LegarthandTvergaard,2018).

It was quite apparent fromFig. 10 that the plastic anisotropy has an inﬂuence on the deformation mode at incipient ductile failure of the specimen, as the shape and extension of the regions ofconcentrated plastic ﬂow vary with tensiledirection for the anisotropic materials. It is a reasonable conjuncture that the variation in shape of the regions can change the failure mode fromcup-and-conefailure toslantshearfailure dependingonthe plastic anisotropy and loading direction. These failure modes are typically observed experimentally foranisotropic materials (Chen et al., 2011; Fourmeau etal., 2013). Thiswas also recentlystud- iedbyBenzergaetal.(2019),whoshowedthatanisotropicplastic- itycaneffectivelytriggershearbandsandcausefailureofductile materials.

From the strain localization analyses, it was evident that the stress states in the regions inside and outside of the imperfec-

tionbandinthecriticalelementdependontheplasticanisotropy of alloys A, B, and C, see Fig. 12. The stress state inside of the imperfection band was seen to be strongly affected by the plastic anisotropy,and that thishas a great inﬂuence on the failure strain predicted by the strain localization theory. Higher work- hardeningrateisfavourablefordelayingfailureasitdelaysneck- inganddistributestheplasticdeformationoverawiderareaofthe specimen’s cross-section. This will in turn affect the local stress state insidethe specimenso that,e.g., thestress triaxialityis re- ducedforagivenvalueofthelogarithmicstrainovertheneck,cf.

Fig.13.

The failure strain varies withtensile direction for anisotropic materialsandtheplasticanisotropyofalloyBwasfoundtoaffect thestressstate bothinside andoutsideoftheimperfectionband, seeFig.14.A lowerstress levelwasobservedto give anincrease inthe ductility, butalso thevalue of theLankford coeﬃcient af- fectsthe ductility, since it governs the distributionof the plastic deformationacross thespecimencross-section.A Lankfordcoeﬃ- cient closeto unitywill distributethe plasticdeformations more uniformlyandthusbepositivefortheductility,cf.Fig.11.Albeita lowerstresslevel,causedbytheanisotropyinyieldstress,appears toincreasetheductility,theeffectofthestresstriaxialityandLode parameterseemsto beevengreater. Asan example,inthe simulations ofthe tensiletests ofalloy B,the stress level is lower in the0°directionthaninthe90°direction,butevensotheductility islowerintheformerdirection.Thereasonforthelowerductility inthe 0° directionis that the stress triaxialityand the Lode pa- Pleasecitethisarticleas:B.H.Frodal,D.MorinandT.Børviketal.,Ontheeffectofplasticanisotropy,strengthandworkhardeningon

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Fig. 14. Local behaviour in the regions inside (dashed lines) and outside (solid lines) of the imperfection band in the critical element for alloy B with temper T6 work hardening loaded in the ED-TD plane: (a) normalized von Mises stress and normalized void volume fraction, (b) stress triaxiality ratio, (c) equivalent plastic strain, and (d) Lode parameter versus logarithmic strain.

rameterevolveinafavourablewayforlocalization,whichismore importantthanthelowerstresslevel.

6. Concludingremarks

Threealuminiumalloyswithdifferentgrainstructureandcrys- tallographictexture, solutionheat-treated and artiﬁciallyaged to threeconditionsgivingdifferentstrengthandwork-hardeningbe- haviours, were considered in the study. Previous experimentson thesematerials hadshownthat thetensileductility ofthe alloys decreasedwithhigherstrengthandlowerworkhardening,andthe ductilitywasdifferentdependingonthealloy.

The influenceofplasticanisotropy,strengthandwork harden- ingonductilefailurewasstudiedbynonlinearfiniteelementsim- ulationsandstrainlocalizationanalyses oftensiletestsinvarious directions.The anisotropicyield surfacesofthealuminiumalloys, previouslyobtainedbythecrystalplasticityfiniteelementmethod, wereusedto constructaset ofmodelmaterials.Theseyield sur- facesandanisotropicyieldsurfacewerecombinedwiththreeflow stresscurvesrepresentativeforthedifferentheat-treatments.Thus, a total of 12 model materials, with different plastic anisotropy, strengthandworkhardeningwereconstructedandusedinthenu- merical investigations. Finite element simulations of tensile tests onsmooth axisymmetric specimens were conductedin seven in- plane directions, and the deformation gradient history extracted fromthenumericalsimulations were used todrive thestrain localizationanalyses.

Plastic anisotropy was found to have a marked influence on the tensile ductility and to induce failure anisotropy. The shape and extension of the regions of concentrated plastic flow in the finite element simulations varied with tensile direction for the anisotropic materials.The highlydeformedregions werefound to vary in shape so that the deformation mode at incipient ductile failure changes from a flat to a slant shear mode depending on theloadingdirectionandplasticanisotropy.Inagreementwithex- perimental evidence from the literature (Fourmeau et al., 2013;

Khadyko etal., 2019),the strain localization analyses predicteda variation of the failure strain with tensile directionthat appears tocorrelatewiththevariationoftheLankfordcoeﬃcient,thusin- dicating thatthefailure anisotropy iscloselylinked totheplastic anisotropyforthesealuminiumalloys.

The strain localization analyses predict a higher ductility for materialswithlowerstrengthandhigherworkhardening,asthese features lead to a moreuniform plastic strain distribution and a stress state witha lower stress triaxialityinthe neck.The redis- tribution of plastic deformation due to the high work hardening makes thetensilespecimenlessprone tolocalizationandductile failure.The inﬂuenceofstrengthandwork hardeningontheten- sileductilityisalsofoundtodependontheplasticanisotropy.

Acknowledgements

The ﬁnancial support of this work from the Centre for Ad- vanced Structural Analysis (CASA),Project No. 237885,Centre for