The effect of strain rate sensitivity evolution on deformation stability during superplastic forming

Theeffectofstrainratesensitivityevolutionondeformation

stabilityduringsuperplasticforming

M.A.Nazzal,M.K.Khraisheh∗,F.K.Abu-Farha

CenterforManufacturing&DepartmentofMechanicalEngineering,UniversityofKentucky,Lexington,KY40506-0108,UnitedStates

Abstract

Themostimportantcharacteristicofsuperplasticmaterialsisthehighsensitivityofflowstresstodeformationrate.Ingeneral,aconstantstrainratesensitivityindexvalueisusuallyusedforcalibratingmodelsdescribingsuperplasticdeformation.However,experimentalresultsindicatethatthestrainratesensitivityindexdependsonstrainrate,strainanddoesnotremainconstantduringdeformation.Inthiswork,theeffectsofstrainratesensitivityvariationonthestabilityofdeformationduringsuperplasticformingareexaminedusingfiniteelementsimulationsinconjunctionwithamicrostructure-basedconstitutivemodel.ThemodelisexperimentallycalibratedandvalidatedfortheAZ31magnesiumalloy.Theresultsclearlyshowtheimportanceofaccountingforthevariationofstrainratesensitivityinmodelingandsimulatingsuperplasticforming.© 2007 Elsevier B.V. All rights reserved.

Keywords:Superplasticforming(SPF);Strainratesensitivity;Finiteelementanalysis;Deformationstability

1.Introduction

Superplasticmaterialsareauniqueclassofpolycrystallinesolidsthatexhibitextraordinaryuniformtensileductilitywhendeformedundercertainconditionsoftemperatureandstrainrate.SuperplasticmaterialsincludesomeAlalloys,Tialloys,Cualloys,Mgalloys,ceramicsandcomposites.Thesemateri-alsarecharacterizedbyextremelylowflowstressesandmarkedstrainratesensitivityoftheflowstressduringlargedeformationatelevatedtemperatures.Recently,theAZ31magnesiumalloyhasreceivedsignificantattentionduetotheincreasingdemandforlightweightstructuralcomponentsespeciallyfortheauto-motiveindustry.Unfortunately,theinferiorductilityofthisalloyatroomtemperaturehindersitsusageinsomemanufacturingprocesses,suchassheetmetalforming.Warmforminghasbeencarriedouttoenhancetheformabilityofthisalloy.Yet,amoreattractiveattributeofthisalloyisitssuperplasticbehaviorathighertemperatures[1,2].Superplasticitystretchesthelimitsofformabilityofmagnesiumalloysbeyondconventional,offer-ingmoreopportunitiesformagnesiumusageintheautomotivesector.

Correspondingauthor.Tel.:+18592576262x219;fax:+18593231035.E-mailaddresses:manazz0@engr.uky.edu(M.A.Nazzal),khraisheh@engr.uky.edu(M.K.Khraisheh),rurouni@engr.uky.edu(F.K.Abu-Farha).

0924-0136/$–seefrontmatter© 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jmatprotec.2007.03.097

∗

Forsuperplasticmaterials,thestrainratesensitivityindexshouldbegreaterthanorequalto0.3andforthemajorityofsuperplasticmaterialsitliesintherangeof0.4–0.8[3].Strainratesensitivityrepresentsthecapacityofthematerialtoresistneckingandinfluencestheoveralldeformationandstabilityduringsuperplasticdeformation[3].Therefore,inordertocap-turethedeformationcharacteristicsofsuperplasticmaterials,thestrainratesensitivityhastobedeterminedaccurately.Mostconstitutivemodelsandfiniteelement(FE)simulationsofsuper-plasticforming(SPF)applyconstantstrainratesensitivityindexvalueduringdeformation.However,experimentalstudieshaveshownthatthestrainratesensitivityvarieswithstrainrate,strain,andtemperature[4,5].Hence,theuseofconstitutivemodelsthatdonottakeintoaccountstrainratesensitivityindexvari-ationmightleadtomisleadingresults,especiallywhenlargedeformationisachieved.

ThecriticalissuesthatcurrentlyhamperthewidespreaduseofSPFaremostlyrelatedtomaterialbehaviorduringformingandincludelowproductionrate,becauseoflowdeforma-tionratesused,andlimitedpredictivecapabilitiesofmaterialdeformationandfailure.Severalinvestigators[6–8]tackledtheproblemoflowproductionratebyconstructingvariablestrainrateformingpaths,derivedfromstabilityanalysis,thathelpinreducingtheformingtimewhilemaintainingtheintegrityoftheformedpart.Variablestrainrateformingpathssuggestthatintheinitialstagesofdeformation,highstrainratesareusedtodeformtheworkpieceandasmoredeformationtakesplace

190M.A.Nazzaletal./JournalofMaterialsProcessingTechnology 191 (2007) 1–192

andthedeformationstabilityiscompromised,thestrainrateisadjustedtoalowervalue.TheeffectivenessofthesespathswasvalidatedexperimentallybyKhraishehetal.[9].However,thevariationinstrainratesensitivityindexthatplaysaveryimpor-tantroleindeterminingthedeformationstabilitywasignoredinthesestudies,andaconstantvaluewasused.

Inthiswork,amulti-scalestabilityanalysisthatincorpo-ratesbothgeometrical(necking)andmicrostructuralaspectsalongwithFEsimulationsareemployedtostudytheeffectsofstrainratesensitivityvariationontheSPFprocess.Itisimpor-tanttonotethatthestabilitycriterionandFEsimulationsarebasedonexperimentallyvalidatedandcalibratedconstitutiverelationswithmicrostructuralevolution[5].TheeffectsofstrainratesensitivityvariationonvariousSPFparametersareinves-tigatedincludingvariablestrainrateformingpaths,thicknessdistributionoftheformedpart,andthepressure–timeschedulesgeneratedusingFEsimulations.

2.Constitutivemodel,stabilitycriterion,andFEanalysis

Inordertodescribethedeformationbehaviorduringsuper-plasticdeformation,constitutiveequationsthatcorrelateflowstresswithstrain,strain-rate,temperatureandothermicrostruc-turalquantitieswhichincludegrainsizeandcavitationhavetobeestablished.Theconstitutivemodelusedinthisworkisgivenby[10–12]:

󰀁󰀂1/m

¯kσ

˙=¯ε(1)

dp1−fa¯d=d0+cε¯)fa=fa0exp(ψε

(2)(3)

(fa)areused.Asimplelineargraingrowthmodelsimilarto

theoneusedbyCaceresandWilkinson[10]isusedhereasgivenbyEq.(2).Sincecavitationisprimarilycontrolledbytheplasticflowofthesurroundingmatrix[3],plasticity-controlledgrowthisonlyconsideredandthecavitationevolutionmodelisdescribedbytheexponentialrelationgiveninEq.(3).

Incorporatingthemodifiedconstitutiveequation(1)alongwiththeevolutionequations(2)and(3)intotheframeworkofHart’sstabilityanalysis[13],astabilitycriterionaccountingforgeometricalinstabilities,microstructureaspects,andstressstatewasdevelopedbyThuramallaandKhraisheh[8];whichhasthefollowingformattheonsetofinstability:aγ∗+m+aζ∗=1

(4)

˙aretheeffectivestrainandstrainrate,respectively,¯andε¯whereε

¯theeffectiveflowstress,mthestrainratesensitivityindex,ptheσ

graingrowthexponent,dtheaveragegrainsize,d0theinitialgrainsize,fa0andfatheinitialareafractionofvoidsandthecurrentareafractionofvoids,respectively,ψthevoidgrowthparameterandkandcarethematerialparameters.Inordertoaccountforthechangeinmicrostructureduringdeformation,evolutionequationsforgrainsize(d)andareafractionofvoids

Thefirsttermintheaboveequationcorrespondstostrainhardeningduetograincoarsening,thesecondtermcorrespondstostrainratesensitivityandthethirdtermrepresentstheinflu-enceofcavitation.Theparameter(a)isafunctionoftheratiobetweentheprincipalplasticstrainsintheplaneofthesheet.Thestabilitycriterioncanbesolvedforagiveneffectivestrainratetoyieldacriticaleffectivestrainattheonsetofinstability.Theprocessisthenrepeatedfordifferenteffectivestrainratestoyieldanoptimumvariablestrainrateformingpath.

ThefiniteelementsimulationshavebeenperformedusingthecommercialfiniteelementsolverABAQUS.UserdefinedsubroutinesarecompiledtoimplementtheconstitutivemodelintotheFEcode[12].Adeeprectangularboxthatis60cmlongby40cmwideby20cmdeepwitha2cmflangearounditisused.Theinitialthicknessofthesheetis2mm.Duetosymmetry,aquarterofthesheetismodeled.Sevenhundredandfourfullyintegratedbilinearmembraneelementsareusedtomodelthesheet.Ontheotherhand,thedieismodeledusing231triangularrigidelements.Abuiltinpressurecontrolalgorithmaimedatobtainingapracticalloadcurveatlowcomputationalcostisusedintheanalysis.Formoredetailsaboutthepressurecontrolalgorithmused,seeNazzaletal.[12].3.Discussionandresults

TocalibratetheconstitutiveequationspresentedinSection2,asetofconstantstrainrateuniaxialtensiletestsandstrainrate

Fig.1.(a)Stress/strainratecurve[5];(b)strainratesensitivityindex/strainratecurve[5].

M.A.Nazzaletal./JournalofMaterialsProcessingTechnology 191 (2007) 1–192191

Fig.2.(a)Optimumvariablestrainrateformingpaths.(b)Pressure–timeschedule.(c)Sheetthicknessdistributionforthefirstrun(constantm)andthesecondrun(variablem).

jumptestsfortheMgAZ31alloyhastobeconducted.Abu-FarhaandKhraisheh[5]haveperformedsuchtestsforawiderangeoftemperaturesandtheyconcludedthattheoptimumsuperplas-ticbehaviorforthisalloyisattainedat400◦C;therefore,thistemperatureisonlyconsideredinthiswork.Fig.1(a)presentsthestress/strainratecurvewhichshowsasigmoidalvariationofflowstresswithstrainrate.Superplasticityoccursinregion2,wherethestrainratesensitivityindexhashighvaluesatmoder-atestrainrates,accompaniedbyverylargeelongation.Themostcommontechniqueusedtodeterminethestrainratesensitivityindexvalue,m,isfromtheslopeofthecurveatregion2.Thisslopeevaluationresultsinthewell-knownbell-shapedm/strainratecurveshowninFig.1(b).However,onecaneasilynoticethehighvaluesofm(upto0.83)obtainedfromslopeevalua-tion,whichdonotseemtoberealistic.Ontheotherhand,itiswellknownthatthestrainratejumptestisthemostaccuratemethodofdeterminingthestrainratesensitivityindex.Resultsobtainedfromthismethodareonlyusedinthisanalysisandthebellshapedm/strainratecurveisignored.Fig.1(b)showsthem/strainratecurvewhenmismeasuredusingstrainratejumptests.

Fig.1(b)isfrequentlypresentedinmanystudies,yethardlyadoptedinmodelingandsimulationefforts.Usually,asinglevalueform,whichisthemaximumone,isused.However,inthiswork,theimportanceofaccountingforthevariationofstrainratesensitivityindexonthestabilityofSPFishighlightedandFEsimulationsareperformedtovalidatethestabilityanalysisresults.

Inordertostudytheeffectsofstrainratesensitivityvaria-tiononthestabilityofsuperplasticdeformation,optimumstrainrateformingpathsarederived,basedonthestabilityanalysisdescribedinSection2,fortwoparticularcases.Forthefirstcase,thestrainratesensitivityindexistakentobeconstantandequaltothemaximumvaluedepictedfromstrainratejumptestsresults;whichis0.66.However,forthesecondcase,thestrainratesensitivityisconsideredvariableaccordingtothejumptestsshowninFig.1(b).Forbothcases,thestressstateisassumedtobebalancedbiaxial.Fig.2(a)showstheoptimumstrainrateformingpathsforbothcases.Itisseenfromthisfigurethatathighstrainrates,thecriticalstrainachievedforaspecificstrainrateismuchhigherwhenmisconsideredconstant.However,aswemovetowardslowerstrainrates,theoptimumvariablestrainratepathsbecomeclosertoeachotheraswellasthecriticalstrains.Thisobservationcanbeexplainedasfollows:athighstrainrates,thestrainratesensitivityindexislowwhencom-paredtoitsmaximumvalue,whichreducestheamountofstabledeformationforthecaseofvariablemcomparedtothecaseofconstantm;ontheotherhand,atlowstrainrates,mapproachesitsmaximumvalueandtheoptimumvariablestrainratepathforthecaseofvariablemmovestowardtheoneforthecaseofconstantm.

ThreeFEsimulationsfortheblowformingofaboxarecon-ductedtoexaminetheeffectsofstrainratesensitivityvariationontheSPFprocess.Inthefirstrun,aFEmodelthatusesasinglevalueform(maximum)isused,andtheloadiscontrolledaccord-ingtothevariablestrainratepathderivedforaconstantmvalue,showninFig.2(a).Inthesecondrun,theFEmodelaccountsforthevariationofm,andtheloadiscontrolledaccordingtothepathderivedforthevariablemcase.TheresultsofthetworunsareshowninFig.2(b)and(c).FromFig.2(b),itisobservedthattheformingtimewhenmisconsideredconstantis15%lessthanthatforthevariablemcase.However,thesheetthicknessdistri-butioniscomparableforbothcases.Thisisexpectedsincethestabilityanalysiswillguaranteestabledeformationduringform-

192M.A.Nazzaletal./JournalofMaterialsProcessingTechnology 191 (2007) 1–192

Fig.3.(a)Sheetthicknessdistributionforthesecondrun.(b)Sheetthicknessdistributionforthethirdrun.

ing;whichwasattainedbyreducingtheformingpressureandincreasingtheformingtimeforthevariablemcase.Thisresultshowstheadvantageofusingtheoptimumvariablestrainratewhichactsasasmartcontrol.Theoptimizationtoolrecognizesthefactthatwhenmislower,theamountofstabledeformationachievedislowerandhencetheformingtimeisprolongedandtheformingpressureisreducedtoallowmorestabledeforma-tionatlowerstrainrates.Inthethirdrun,thepressureprofilegeneratedinthefirstsimulationrun(constantm)isusedtoformasheetwithaFEmodelthataccountsforstrainratesensitivityindexvariation.Thissimulationcaserepresentstherealscenariofollowedbytheindustry.Thestrainratesensitivityindexvariesduringdeformationwhilethemodelsusedarebasedoncon-stantm;hencethepressureprofilesfollowedaredesignedformaterialthatdoesnotundergostrainratesensitivityvariation.Becauseofthat,prematurethinningandfailuretakeplace.Fig.3showsacomparisonbetweenthesheetthicknessesgeneratedforthesecondrunandthethirdrun.Itisclearlydemonstratedthatstrainratesensitivityindexvariationhasadrasticeffectonthethicknessdistributionoftheformedpart;theformedpartinthethirdrunundergoesseverelocalizedthinningwhencomparedtotheoneobtainedfromthesecondrun.4.Conclusions

Inthiswork,optimumvariablestrainratepathsfordiffer-entscenarioswerederivedusingmultiscalestabilityanalysis.FEsimulationswerethenconductedtogenerateoptimumpressure–timeprofilesandtovalidatetheresultsobtainedfromstabilityanalysis.Theresultsclearlyhighlightedtheimportanceofaccountingforstrainratesensitivityvariationondeformationstabilityduringsuperplasticforming.Thepresentanalysiscanbefurtherimprovedbyconsideringstrainratesensitivityasafunctionofstrainrateandstrain;insteadofstrainrateonly.Thisissueiscurrentlyunderinvestigation.

Acknowledgement

ThesupportoftheNationalScienceFoundation,CAREERAward#DMI-0238712,isacknowledged.References

[1]L.Tsao,C.Wu,T.Chuang,Evaluationofsuperplasticformabilityofthe

AZ31magnesiumalloy,Mater.Res.Adv.Techniques92(2001)572–577.[2]F.Abu-Farha,M.K.Khraisheh,DeformationcharacteristicsofAZ31mag-nesiumalloyundervariousformingtemperaturesandstrainrates,in:ProceedingsoftheEighthESAFORMConferenceonMaterialForming,Cluj-Napoca,Romania,2005,pp.627–630.

[3]J.Pilling,N.Ridley,SuperplasticityinCrystallinesolids,TheInstituteof

Metals,London,UK,19.

[4]M.A.Khaleel,K.Johnson,C.Hamilton,M.Smith,Deformationmodeling

ofsuperplasticAA-5083,Int.J.Plast.14(1998)1113–1154.

[5]F.Abu-Farha,M.K.Khraisheh,Mechanicalcharacteristicsofsuperplastic

deformationofAZ31magnesiumalloy,J.Mater.Eng.Perform.,2007,inpress.

[6]C.H.Johnson,C.H.Hamilton,H.M.Zbib,S.Richter,Designingoptimized

deformationpathsforsuperplasticTi6Al4V,in:N.Chandra,etal.(Eds.),AdvancesinSuperplasticityandSuperplasticForming,TheMinerals,Met-alsandMaterialsSociety,1993,pp.3–15.

[7]X.D.Ding,H.M.Zbib,C.H.Hamilton,A.E.Bayoumi,Ontheoptimization

ofsuperplasticblow-formingprocesses,J.Mater.Eng.Perform.4(1995)474–485.

[8]N.V.Thuramalla,M.K.Khraisheh,Multiscale-basedoptimizationofsuper-plasticforming,Trans.NAMRI/SME32(2004)637–3.

[9]M.K.Khraisheh,F.Abu-Farha,M.Nazzal,K.Weinmann,Combined

mechanics-materialsbasedoptimizationofsuperplasticformingofmagne-siumAZ31alloy:modeldevelopmentandexperimentalvalidation,CIRPAnn.55(2006)233–236.

[10]C.H.Caceres,D.S.Wilkinson,Largestrainbehaviorofasuperplastic

copperalloy-deformation,ActaMetall.32(1984)415–422.

[11]M.J.Stowell,Cavitygrowthandfailureinsuperplasticalloys,Met.Sci.17

(1983)92–98.

[12]M.A.Nazzal,M.K.Khraisheh,B.M.Darras,Finiteelementmodelingand

optimizationofsuperplasticformingusingvariablestrainrateapproach,J.Mater.Eng.Perform.13(2004)691–699.

[13]E.W.Hart,Theoryofthetensiletest,ActaMetall.15(1967)351–355.