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A Numerical Investigation of the Continuous Bending Under Tension Test

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A Numerical Investigation of the Continuous Bending Under Tension Test
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   JournalofMaterialsProcessingTechnology 211 (2011) 1948–1956 ContentslistsavailableatScienceDirect  Journal   of    Materials   Processing   Technology  journalhome   page:www.elsevier.com/locate/jmatprotec A   numerical   investigation   of    the   continuous   bending   under   tension   test A.   Hadoush a , 1 , A.H.   van   den   Boogaard b , ∗ , W.C.   Emmens c a Materialsinnovationinstitute,P.O.Box5008,2600GADelft,TheNetherlands b UniversityofTwente,P.O.Box217,7500AEEnschede,TheNetherlands c TATASteelRD&T,P.O.Box10000,1970CAIJmuiden,TheNetherlands a   r   t   i   c   l   e   i   n   f   o  Articlehistory: Received8May   2010Receivedinrevisedform14June2011Accepted18June2011 Available online 25 June 2011 Keywords: IncrementalsheetformingBendingundertensionCyclicbendingForce–displacementInstabilityFEM a   b   s   t   r   a   c   t In   this   paper   the   continuous   bending   under   tension   test   is   analyzed   by   numerical   simulation.   The   abilityof    achieving   high   strains   bycombined   stretching   and   bending   isconsidered.   This   deformation   mode   hassimilarities   with   the   deformation   that   takes   place   in   incremental   sheet   forming   (ISF)   and   may—at   leastpartly—explain   the   high   strains   that   are   observed   there.   The   sensitivity   of    the   numerical   model   to   meshdiscretization   is   studied   as   well   asthe   influence   of    different   material   models.   An   isotropic   hardeningmaterial   model   and   two   mixed   isotropic/kinematic   hardening   material   models   are   used.   The   results   forthethree   models   are   very   similar,   for   the   shape   of    the   load   curves,   but   not   for   the   point   of    necking.   Anumerical   analysis   of    the   cyclic   force–displacement   curve   of    the   CBT   test   is   presented.   This   analysis   isfocused   onthe   pattern   of    the   cycle   and   the   evolution   of    the   cycle   during   the   test.   The   loss   of    stabilityfor   inhomogeneous   stress   distributions   is   analyzed   and   the   importance   of    bending   in   stabilizing   thedeformation   under   tension   is   demonstrated.   Stability   is   lost   if    the   complete   cross   section   is   in   astate   of tensile   stress. © 2011 Elsevier B.V. All rights reserved. 1.Introduction Thecontinuousbendingundertension(CBT)testcanbeseenasatensiletestonstripmaterial,withadditionallocalbendingbyasetofrollsthatistravellingoverthelengthofthestrip.Themaineffectofadditionalbendingisthattherequiredtensileforceforthesameamountofelongationisreduced(MarciniakandDuncan,1992).Hadoushetal.(2007)presentedasimplified2-dimensionalfiniteelementmodelfortheCBTtesttostudythecontributionofbendinginstabilizingthedeformationofastriptohighstrain.TheCBTtestwasproposedbyBenedyketal.(1971)toinvestigatematerialprop- ertiesathighlevelsofstraining.EmmensandvandenBoogaard(2009a)identifiedtheCBTtestasanincrementalformingprocessandshowedexperimentallythathighlevelsofstrainareobtainedforvariousmaterials.IntheCBTtest,sheetmaterialdeformsincrementallyratherthancontinuouslyasinastandardtensiletest.ThedeformationaroundrollsintheCBTtestbearsresemblancewiththedefor-mationaroundthesphericaltoolinincrementalsheetforming(ISF)andthisresemblancemotivatedtheresearchdescribedinthispaper. ∗ Correspondingauthor. E-mailaddress: a.h.vandenboogaard@utwente.nl(A.H.vandenBoogaard). 1 Presentaddress:FacultyofEngineering,TheHashemiteUniversity,P.O.Box330127,Zarqa13115,Jordan. IncrementalsheetformingisadisplacementcontrolledprocessperformedonaCNCmachine.Aclampedblankisdeformedbythemovementofthetoolthatfollowsaprescribedtoolpathasintro-ducedbyMatsubara(1994).   Anextensiveoverviewoftheprocessisgivenby Jeswietetal.(2005).   InISF,strainsareobservedthatareoftenfarabovetheforminglimitcurveforthematerialundercon-sideration.Inclassicalsheetformingoperationsthedeformationwouldbecomeunstable,leadingtotheinceptionofnecking,butinISFthedeformationappearstobestabilized.Severalmechanismshavebeenproposedinliteraturetoexplaintheincreasedforma-bility:shear,contactstress,bending,andcyclicstraining.ThesemechanismsarediscussedindetailinarecentreviewpaperbyEmmensandvandenBoogaard(2009b).Toseparatetheeffectof bendingundertensionfromotherstabilizingeffectsinISF,theCBTtestisinvestigated.Inthispaper,a3DfiniteelementmodeloftheCBTtestispre-sented.Themodelisusedtoobtainprocessknowledgeandtovalidateassumptionsabouttheobservedextendedformability.TherelationbetweentheCBTtestandISFoperationsisnotdiscussed,butwillbethesubjectofaseparatepaper(EmmensandvandenBoogaard,2011).Thebasiccomponentsofthenumericalmodeltopredicttheforce–displacementcurveoftheCBTtestarepresentedinSection2.Then,explanationsontheshapeoftheforce–displacementcurvearepresentedinSection3.Theseexplanationsarebasedonpro- cessmechanicsandprocesscharacteristicsoftheCBTtest.Finally,apreviouslyclaimedstabilitycriterionforbendingundertensionisinvestigatedbynumericalanalysis. 0924-0136/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2011.06.013   A.Hadoushetal./JournalofMaterialsProcessingTechnology  211 (2011) 1948–1956 1949 Fig.1. CBTsetup(EmmensandvandenBoogaard,2009a). 2.Processdescriptionandnumericalmodel Inthissection,theCBTexperimentwillbeintroducedbrieflyandaFEMmodelwillbedevelopedandvalidated.Inthenextsectionresultsfromthemodelwillbecomparedwithexperimentalresults.TheexperimentalsetupandresultsoftheconsideredCBTtesthavebeenpresentedinmoredetailinEmmensandvandenBoogaard(2009a).  2.1.TheCBTexperimentandFEMmodel AphotographoftheCBTsetupisshowninFig.1.Intheright- handpartofthepictureaverticaltensilestripisvisiblethatisclampedinatensilemachine.Asetof3rollsbendsthestrip,of which1rollisvisibleinthepicture.Therollsarefreelyrotat-inginbearingsandthecompleterollsetismovingupanddownrepeatedlywhilethestripisstretched.IntheFEMmodel,thetensilestripisplacedhorizontallyandtherollsetismovingrepeatedlyfromlefttoright,aspresentedinthe2-dimensionalschematicinFig.2.Therollsetismodeledby3 analytical,frictionless,cylindersof15mmdiameter.Inlongitudinaldirection,therollsareseparatedfromeachotherby17.5mm.   Therollsetcantravelinthelongitudinaldirectiononly.First,thecentralrollisplacedsuchastofitthespecimeninbetweentherollswith-outdeformingit.Thenthebendingofthespecimenisintroducedbythemovementofthecentralrolldownwardsover2.3mm   plustheinitialsheetthickness.Thecyclicmovementoftherollsetinlongitudinaldirectionintroducesbendingatcontinuouslychang- Fig.3. Schematicofthespecimen(dimensionsinmm,   thedrawingisnottoscale)(EmmensandvandenBoogaard,2009a). ingpositions.Therollsetmoveswithastrokeof100mm   andavelocityof66mm/s,whilethecrossbarmoveswith2.5mm/s.ThespecimenusedintheCBTexperimentisschematicallyshowninFigure3.Ithasauniformthicknessof1mmandpiecewise uniformwidth.Themiddlepartofthespecimenhasthesmallestwidth(20mm).   Thecyclicbendingisperformedonlyinthemiddlepartofthespecimen.Experimentally,itisobservedthatthepartthatexperiencescombinedtensionandbendingdeformsplasticallyasshowninFig.4.Becauseofthegeometryandappliedloadingcon- ditions,plasticdeformationofthewiderpartscanbeneglectedformildsteel.Becauseofsymmetryalongthelongitudinalaxis,onlyhalfofthemiddlepartofthespecimenismodeled.Themodeledpartofthespecimenis200mminlengthand10mm   inwidth.ItcanbeobservedinFig.4thatthereisstilllateralcontractionand thereforea2DplanestrainstatecannotbeassumedintheCBTsimulation.Inthenexttwo   sectionstheinfluenceofthemeshdiscretizationandthematerialmodelonthepredictedforce–displacementcurveisstudied.First,threedifferentFEmeshsizesareusedtoinvestigatethemeshdependency,thenastudyontheeffectoftheappliedhardeningmodelispresented.  2.2.Meshsizedependency Aregularmeshisusedwith8triangularshellelementsusedtodiscretizethe10mm   half-widthofthespecimen.Theshellele-mentisacombinationofadiscreteKirchhofftriangularelementforbendingandaconstantstrainmembraneelementasproposedbyBatozetal.(1980).   A3-pointintegrationschemeintheplane,com-binedwith7integrationpointsinthicknessisapplied,resultingin21integrationpointsperelement.Thetrianglesarelargeatthelongitudinalsymmetrylineandsmallatthefreeedgeofthestripwithanelementsizeratioof4to1.Thisdistributionrepresentsthenon-uniformstressdistributionalongthespecimenwidththatisaresultofthestress-freeedge.Becausethestressdistributionnearthesymmetrylineisquiteuniforminwidth-directionahighaspectratio(width/length)isnotdetrimental.Force–displacementcurveswerevirtuallyequalforuniformlyandnonuniformlydistributedmeshes.Inlongitudinaldirectionauniformelementlengthisusedthatshouldbesmallenoughtorepresentthebendingbytherolls,even Fig.2. Continuousbendingundertensiontestdescription.  1950  A.Hadoushetal./JournalofMaterialsProcessingTechnology  211 (2011) 1948–1956 Fig.4. Unusedandusedspecimens:untested(top),tensiletested(middle)andCBTtested(bottom).Auniformdeformationisobservedinthewhiterectangle. Fig.5. Samplesofmeshdensities:coarse(left),intermediate(middle)andfine(right).Thesamplesrepresent2mmstriplengthand10mmhalf-width,withthefreeedgeon   thebottomandthesymmetrylineonthetopofeachsample. afterconsiderableelongation.Threelongitudinalelementlengthsareusedandtheyareclassifiedas:coarse(1mm),intermediate(0.5mm)   andfine(0.25mm).   Themodeledstripdoesnotrequireageometricalimperfection—asoftenusedinthesimulationofatensiletest—becauseanonuniformdeformationisinducedanyhowbythelocalbendingaction.SamplesofthedifferentmeshdensitiesareshowninFig.5.ThepredictedlongitudinalforceattheclampededgeversusthecrossbardisplacementisshowninFig.6forthethreedif-ferentmeshes.Thedifferentmeshespredictthesamepatternof theforce–displacementcurveandanalmostequalforcelevel.Thedifferenceinthepredictedforcesisaresultofthespatialdis-cretization.Asexpected,ahigherlevelofoscillationisobservedinthecoarsemesh.Thefinemeshsimulationfinishes15completecyclesandfailsduringcycle16.Morestablecyclesarepredictedwiththecoarseandtheintermediatemeshes.Alargerelementisexpectedtosmooththeachievedstrainandthatresultsindelayingthelocalizationofdeformation.Inthenextsection,theintermedi-atemeshisusedfortimeefficiency.Forthedetaileddiscussionof theforce–displacementcurveinSection3thefinemeshisapplied.  2.3.Materialmodel Inthiswork,materialmodelsareusedwithparametersthatarerepresentativeformildsteelDC06,takenfromVanRiel(2009).TheanisotropicplasticbehaviorismodeledwiththequadraticHill’48yieldfunction(Hill,1948)withparametersbasedonthe R -values.Hardeningismodeledwithacombinationofisotropichardeninggovernedbyapowerlawrelationandkinematichard-eningrepresentedbytheArmstrong–FrederickmodelaspresentedbyChaboche(1991).   Threeparametersetsareusedforthehard-eningmodeltoseetheeffectonthesimulationresults.Thefirstmodel(isotropic)onlyusestheisotropicpartofthemodel.Becausebendingandunbendingofthestripresultsincyclictensileandcom-pressivestresses—atleastintheouterfibres—itcanbeassumedthattheBauschingereffectplaysaroleintheanalysis.Toinves-tigatetheinfluenceoftheBauschingereffectonthepredictedforce–displacementcurvetwo   differentsetsofparameterswereused(iso/kin1andiso/kin2).Bothsetswerefittedtothesamecyclicsheartests,butthesecondsetusedahigherweighingfac-torforthetransientzoneafterloadreversal.Theimplementation 050100150200010002000300040005000Cross bar displacement (mm)    P  r  e   d   i  c   t  e   d   l  o  n  g   i   t  u   d   i  n  a   l   f  o  r  c  e   (   N   ) FineCoarseIntermediate 9091929394952800300032003400360038004000Cross bar displacement (mm)    P  r  e   d   i  c   t  e   d   l  o  n  g   i   t  u   d   i  n  a   l   f  o  r  c  e   (   N   ) CoarseIntermediateFine Fig.6. Thepredictedforce–displacementdiagramsfortheentireCBTtest(left)andazoomedinpartoftheprocess(right).   A.Hadoushetal./JournalofMaterialsProcessingTechnology  211 (2011) 1948–1956 1951 00.20.40.6−600−400−2000200400600 ε x      σ   x IsotropicIso / kin 1Iso / kin 2 Fig.7. Stress–straincurvesforthecomponentsin  x -direction. ofthemodelsfollowstheprocedureasgiveninZienkiewiczandTaylor(2005).Thecyclicstress–strainresponseforthe3parametersetsispre-sentedinFig.7.Duetothedifferentfittingprocedures,asmall differenceinthestress–straincurveisobservedevenforthefirstpartofthetestthatrepresentsmonotonicloading.Afteraloadreversal,thefirstisotropic/kinematichardeningmodelshowsanon-sharpelastic/plastictransitioncomparedtothesharpelas-tic/plastictransitionthatisobservedintheisotropicmaterialmodel.Thesecondisotropic/kinematicmodelclearlyshowsearlyre-yieldingandanincreasedhardeningrateafterloadreversal.Becausethehardeningrateisanimportantparameterinthesta-bilityofdeformation,itcouldbeexpectedthatthelastparametersetshowsextendedstability.ThepredictedforceattheclampededgeforeachparametersetisplottedversusthecrossbardisplacementinFig.8,togetherwith experimentalresults.Forthisanalysis,themodelwithintermediatemeshsizewasusedbecauseoftimeefficiency.Ingeneral,thedif-ferentparametersetspredicttheexperimentalforce–displacementcurvesuccessfully.Duetoafinitetime,neededforreversingthemovementoftherollsetintheexperimentandtheinstantaneousreversalinthesimulation,theexperimentalandsimulationresultsareshiftedslightly.InFig.8thestartandendofacycleareindicated with‘S’and‘E’respectively.TheBauschingereffectthatisincludedintheisotropic/kinematicmaterialmodelsshowsnosignificantdifferenceinthepatternofthepredictedforcecomparedtothepredictedforcebytheisotropicmaterialmodel,butasmalleffectontheforcelevelcanbeseen.Thecombinedisotropic/kinematicmaterialmodelsshowsearlierlocalizationthantheisotropicmate-rialmodel.Thiscontradictswiththeexpectationthatkinematichardeningmodelscanstabilizetheprocessbytheincreasedhard-eningrateafterstressreversal.Itisconcludedthatastandardshellelementdescriptionandanisotropichardeningmaterialmodelissufficienttopredicttheexperimentalforce–displacementcurveoftheCBTtestformanycycleswithgoodagreementforstablecycles.ThenumericalmodeldoesnotpredicttheformabilitylimitofCBTtestcorrectlybecauseofmaterialandmeshdependency.Mostsurprisingly,mod-elingtheBauschingereffectwithkinematichardeninghasnosignificantinfluenceonthepredictedforce–displacementrelation,althoughitdoesinfluencethenumericalstability.Therelativelylowinfluenceofthematerialmodelshowsthatthepatternoftheforce–displacementcurveismainlydeterminedbythemechan-icsoftheprocessratherthanthematerialbehavior.Theprocessmechanicswillbediscussedinthefollowingsection. 3.Force–displacementcurve Thepurposeofthissectionistoanalyzetheshapeoftheforce–displacementcurveandvalidateanhypothesisabouttheforcepeaksthatwaspresentedintheexperimentalworkof EmmensandvandenBoogaard(2009a).Theanalysesareper-formedwiththefinemeshaspresentedinSection2.2.Theforce–displacementcurveincludesaparticularnumberof cyclesdependingontheexperimentalsettingsandthetestedmate-rial.Threerepresentativecyclesoftheforce–displacementcurveareplottedinFig.9.Acycleconsistsmainlyoftwo   peaksandtworelativelysteadyparts.Thepeaksareobservedafterthemomentthattherollsetchangesitstravelingdirection.Thesteadypartsrepresentthemainpartofthecycle.Asexplainedearlier,ignoringthedecelerationandaccelerationoftherollsetinthesimulationresultsinpredictingasmallertimeintervalofacycleinthesim-ulationcomparedtotheexperimentaltimeinterval.Thisexplainstheincreasinglagofthemeasuredforceprofilecomparedtothepredictedforceprofile.  3.1.Semi-steadypart  IntheCBTtest,onlymaterialregionsthatarecurrentlybeingbendeddeformplastically.Theforce–displacementcurveofacom-pletetestwas   presentedinFig.8.Becauseofthebendingaction,a lowertensileforceisrequiredtostretchthestripcomparedtotheforcerequiredtostretchthesamecross-sectionundertensiononly.Althoughthetensileforceisdecreasedbybendingaction,materialhardeningwillincreasethetensileforceaftersomedeformation, justlikeinanordinarytensiletest.Becausethecross-sectionareareducesuponstretching,thegrowthofthetensileforcereducesandfinallythetensileforcewilldecreasebecausethecross-sectionareareducesfasterthanthematerialhardeningincreasestheforce.Thisis,again,justlikeinanordinarytensiletest,butatalowerlevelbecauseoftheadditionalbendingaction.Athighelongationthematerialgetsthinnerandtherelativeproportionofbendingcomparedtostretchingreduces.Then,alsothereductionoftensileforcebybendingbecomessmaller.Withinonecycleitcanbeobservedthateveryfirsthalfofacycleshowsahigheraverageforcethanthesecondhalfofthesamecycle.Thiscanbeexplainedbytheadditionalpositive/negativecontribu-tionoftheleft/right(down/up)movementoftherollset.Becausematerialispulledoutofthedeformationzonefromoneside,thepartbetweenthepeaksisnottrulyinasteadystate.Duetothedimensionoftherollset,thestripisbentandunbentuptodif-ferentlevels.Threezonesareidentifiedbasedonthefrequencyofbending/unbendingasshowninFig.10.   Inzone1,themate-rialisbentandunbentonceforeachpassoftherollset.Similarly,thematerialexperiences2setsofbending/unbendingand3setsofbending/unbendingdeformationinzone2andzone3,respec-tively.Thecyclicforward/backwardmovementoftherollsetshiftstheactivelocationoftheplasticdeformationthroughthesezonesassumingthatonlythebentmaterialdeformsplastically.Thenumericallyachievedresultsofthelongitudinalforceforthedifferentcyclicpatternsareplottedversusthecumulativedis-placementoftherollsetinFig.11.   At0mmdisplacement,therollsetisattherightnearthefixedclampandtheforcetransducerandat100mmtherollsetisattheleftnearthemovingcrossbar.Acom-pletecycleconsistsofmovingfromright(0mm)   toleft(100mm)andbacktorightagain.Thematerialmigratesfromonezonetoanotherinthedirec-tionofthecrossbar.ThecalculateddisplacementsofseveralnodesthroughthelengthofthestripisshowninFig.12.   Thecrossbarpullsthematerialtotheleft,whilethethreerollsaremovingbetweenthepositionsindicatedby[1–3]intheinitialpositionnearthefixedclamp,wherethetensileforceismeasured,andinitialposition

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