Strain heterogeneities in the rolling direction of steel sheets submitted to the skin pass: A finite element analysis

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ContentslistsavailableatScienceDirect

Journal

of

Materials

Processing

Technology

j ou rn a l h o m epa ge :w w w . e l s e v i e r . c o m / l o c a t e / j m a t p r o t e c

Strain

heterogeneities

in

the

rolling

direction

of

steel

sheets

submitted

to

the

skin

pass:

A

ﬁnite

element

analysis

A.M.

Giarola

a

_,

_P.H.R.

_Pereira

b

_,

_P.A.

_Stemler

c

_,

_A.E.M.

_Pertence

b

_,

_H.B.

_Campos

b

_,

M.T.P.

Aguilar

d

_,

_P.R.

_Cetlin

b,∗

a_Instituto_Federal_de_Educac¸_ão,_Ciência_e_Tecnologia,_Department_of_Exact_Sciences,_Faz._Varginha,_Rodovia_{Bambuí-Medeiros}_km_05,_Caixa_Postal_05,

Bambui,MinasGerais,Brazil

b_Universidade_Federal_de_Minas_Gerais,_Department_of_Mechanical_Engineering,_Av._Antonio_Carlos_6627,_31270-901_Belo_Horizonte,_Minas_Gerais,_Brazil c_Universidade_Federal_de_Minas_Gerais,_Department_of_{Metallurgical}_and_Materials_Engineering,_Av._Antonio_Carlos_6627,_31270-901_Belo_Horizonte,

MinasGerais,Brazil

d_Universidade_Federal_de_Minas_Gerais,_Department_of_Materials_and_{Construction,}_Av._Antonio_Carlos_6627,_31270-901_Belo_Horizonte,_Minas_Gerais,_Brazil

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received30January2014 Receivedinrevisedform 12September2014 Accepted19September2014 Availableonline28September2014

Keywords: Skinpass

Strainheterogeneities Lüdersbands Finiteelementanalysis

a

b

s

t

r

a

c

t

Hotrolled,annealedorgalvanizedlowcarbonsteelsheetsdisplayyieldpointsintensiletesting.These areassociatedwithdeformationheterogeneitiesalongthetestspecimen,knownasLüdersbands,which causeanunacceptablesurfaceappearanceinsteelsheetformedproducts.Thisproblemisusually elim-inatedthrough alightcoldrollingpass(“skin-pass”)inthematerialasalaststepinitsindustrial manufacturing,imposingareductionofthicknessofabout1–2.0%onthesheet.Existingexperimental resultsforthesestrainheterogeneitiesalongtherollingdirectionareanalyzed,aswellascorresponding ﬁniteelementanalyses.Theconstitutivebehaviorofthematerialisdescribedbyacurvedisplayingan initialstresspeakfollowedbytheworkhardeningofthematerial.TheFEAsledtodistributionsand shapesofLüdersbandsinthesheetssimilartotheexperimentalones.Thebandsnucleateintermitently andpropagatefromthesurfaceofthesheets,attheinitialcontactregionbetweenthematerialandthe rolls.Forlowandhighfrictionbetweenthematerialandtherollandforlowsheetthicknessreductions (0.5%and1.0%),alimitedinﬂuenceoftherolldiameterontheLüdersbandsdistributionwasfound.For sheetthicknessreductionsof2.0%andhigher,highfrictionbetweentherollandthematerialandlarge rolldiameters,thedistributionofLBsinthesheetswasdifferentfromthatobtainedforsmallroll diame-tersandlowfrictionbetweentherollsandthematerial.Increasingthicknessreductionsintheskin-pass ledtochangesintheLüdersbandsdistribution,involvinganincreasinghomogeneityinthedeformation. Itissuggestedthatthecriticalthicknessreductionintheskin-passisassociatedtoaminimumvolume fractionofmaterialdeformedaboveacertainstrainlevel.

1. Introduction

Skinpassrolling(ortemper rolling)of steel sheetsinvolves athickness reductionof ≈1–2%,usually followsthehotrolling, annealingorgalvanizingofthematerialandaimsatLüdersbands (henceforth called LBs) prevention, sheet ﬂatness and surface topographycontrol(Green,2002;Kijima,2013,2014).TheLBs cor-respondtolocalizeddeformationbandsand areassociatedwith pronouncedstrainheterogeneitiesinthematerial,causingan unac-ceptablesurfaceappearanceinsteelformedparts(Dieter,1976). Theobjectiveoftheskin-passinDualphase,TRIP,IFandaustenitic

∗ Correspondingauthor.Tel.:+553134093504;fax:+553134433783. E-mailaddress:[email protected](P.R.Cetlin).

stainlesssteelsisonlytheﬂatnessandsurfacetopographycontrol, sincetheydonotdisplaytheLBsproblem(Green,2002).Thetensile testingofthesematerialsbeforetheskin-passleadstostress–strain curvesthatdo notdisplayanyieldpoint(henceforthcalledYP) involvinganinitialstresspeakfollowedbyayieldplateau,caused bythenucleationandpropagationofLBs.Ontheotherhand,the skinpassofcarbonsteelandferriticstainlesssteelssheets,which displayYPsintensiletestingbeforetheskin-pass,mustalso elim-inatetheLBsprobleminadditiontocontrollingthesheetﬂatness andsurfacetopography(Green,2002).Henceforththeterms“LBs” and“deformationbands”willbeusedinterchangeably.

Inoneoftheearlieststudiesontheskinpassofsteels,Hundy (1955)coulddetectnostrainheterogeneitiesintherolling direc-tionofskin-passprocessedCarbonsteelsheetswhoseYPhadbeen eliminatedbypreviousheatingto700◦Cfollowedbyquenching. http://dx.doi.org/10.1016/j.jmatprotec.2014.09.015

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Fig.1.LBsintheskin-passofsheetswiththicknessreductionsof(a)0.9%and(b)2.4%viewedthroughchemicaletchingonthelongitudinaledgeofthematerial;initial thicknessofthesheet:19.05mm,diameterofrolls:254mm(Hundy,1955).

Kijima(2013)presentedarecentandcomprehensiveanalysisof theskin-passforsteelsnotdisplayingYPsintensiletestingbefore theskin-pass.Twodifferentrollradius(250mmand50mm),an initialsheetthicknessof0.69mmandthreethicknessreductions (≈0.5%,≈1.0%and≈2.0%)wereconsidered.Aﬁniteelement anal-ysis(FEA)indicatedthattheeffectivestrainatthesheetcenterline andsurfaceincreased(notmonotonically) alongthecontactarc forbothrolldiameters,butthestrainheterogeneityinthe mate-rialprocessed withthe250mmrolls washigherthan withthe 50mmrolls,especially forthetwo higherreductions.Nostrain heterogeneityorLBswerereportedalongthelongitudinal direc-tionoftherolledsheet.Kijima(2013)alsoconcludedthattheFEA oftheskin-passleadstoadequateresultsiftheelasticrollsare replacedbyrigidoneswithaﬂatteneddiameterpassingthrough theentryand exit pointsin rollingand thepoint of minimum thickness.

Hundy(1955)observedthatformaterialsdisplayingYPsin ten-siletesting before theskin-pass,a light longitudinalstretching aftertheskin-passledtosurfacemarkingsperpendiculartothe rollingdirection,which,correspondedtosuccessiveyielded(LBs) andunyieldedregionsintherolledsheet.Healsofoundthatfor thicknessreductionsinexcessofabout3%,nodeformation pat-ternoftheabovementioned typewasobserved,leading tothe

conclusionthatforincreasinglevelsofdeformationintheskinpass thedeformationtendedtobemoreuniform.

Hundy (1955) also analyzed the LBs in the through thick-nessofanunstretchedmaterial19.05mmthick;theblacketched regions inFig.1 correspondtoyielded regions(LBs), indicating thatforthicknessreductionsintheskinpassof0.9%and2.4%,the nucleationofa largenumberofLBsinthematerialisobserved. In addition, the increase in the reduction also led to a more homogeneousdeformation.Theapproximatespacingbetweenthe centerlinesofsuccessiveLBs(measuredgraphicallyasindicatedin Fig.1forthe0.9%reduction)isofabout2.5mm.Attentioniscalled tothe“Y”shapeofsomeoftheLBsemanatingfromthematerial surfaceinFig.1(oneofthemindicatedbytherectangle)andtothe curvatureoftheLBs.Inaddition,morethanonesetofLBs,crossing oneanother,areobserved.

TheanalysismethodemployedbyHundydoesnotallowan eval-uationofthestrainlevelseitherintheLBsorintheregionsbetween them;ontheotherhand,theetchedregionssuggestthatthetop andbottomregionsofthematerial(close tothesurfaceswhich underwentcontactwiththerolls)havebeenfullydeformed.Hundy alsoremarksthattheeliminationoftheLBs problemthrougha skinpassisreachedforreductionsinthicknessstillassociatedwith highlyinhomogeneousdeformationinthematerial.

Fig.2.Surfacemarkingsontheedgeofa1.27mmthicksteelsheetofa0.040%C,afterskin-passrollingwiththicknessreductions(a)0.2%,(b)0.4%,(c)0.7%,(d)1.3%,(e)2.2% and(f)3.2%,rolldiameter30mm;thelowerhalfofeachpicturecorrespondstoapolishedsurface(ButlerandWilson,1963).

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Fig.3. LüdersbandsontherollingplanerevealedbyetchingwithFry’sreagent (Lake,1985).

ButlerandWilson(1963)performedextensiveexperimentson theskin-passrollingof1.27and6.35mmthicklowCarbonsteel sheets(0.038–0.053%C)withvariousgrainsizes,utilizinga two-highrollingmillwith30mmand355mmdiameterrolls,running at15m/minand withparafﬁnlubrication. Theyconcluded that thesurfaceoftherolledsheetwascomposedbyyielded(LBs)and unyieldedregions,andthatincreasingthicknessreductionsinthe skinpassledtoanincreaseinthethicknessoftheLBsanda conse-quentdecreaseintheregionsbetweenthem.

ButlerandWilson(1963)alsoexaminedthestrain heterogene-ityintheskin-passrolledsheetsalongtheirlongitudinaledgeand centralthickness.Fig.2showssomeoftheirresults,indicatinga highlevelofstrainheterogeneity.Theyshowedthatchanges in therolldiameterandthesheetthicknessledtosome modiﬁca-tionsintheobserveddeformationbandswhich,however,followed thesamegeneraltrendanddifferedonlyinsomedetails(basically asmallercurvatureoftheLBsasthesheetthicknessdecreased). Typicalapproximatebandspacing,measuredvisuallybetweenthe centersoftwosuccessiveLBs,asshowninFig.2d,is≈0.45mm.

AccordingtoButlerandWilson(1963),thestressappliedduring therollingofthematerialundergoesacyclicvariation,involving aninitialhighvalueleadingtothenucleationofasmallyielded regionnearthesurface,whichpropagatesrapidlytowardthe cen-terofthesheetonaplaneofmaximumshearstress,followedby arelaxationofthestressestoavaluenolongerabletopropagate theyielding.Theappliedstressthenrisesagain,leadingtoa repeti-tionofthebandnucleation.Accordingtotheproposedmodel,LBs shouldnucleatebasicallyattherollingentranceandwouldthusbe relativelyindependentoftherolldiameter.

Lake(1985)submittedvarioussteelsheets(capped,Aluminum killed and HSLA, with initial thickness varying from 0.58 to 3.15mm)tolaboratoryskin-passrollingwith150mm(dry and lubricated)and38mm(dry)diameterrolls;healsoanalyzedsteel sheets submitted to the skin-pass under industrial conditions. Rollingspeedsrangedfrom0.6m/sto2.2m/s.Chemicaletching oftheskin-passrolledsamplesrevealedagainLBssimilartothose alreadydetectedbyHundy(1955)andButlerandWilson(1963). Thestrainheterogeneityintheskin-passrolledmaterialwas evalu-atedwithtransmissionelectronmicroscopy,anditwasfoundthat, aspredictedbyHundy andButlerandWilson, avery heteroge-neousstraindistributionstillprevailsintherolledsheetforthe eliminationoftheyieldpointofthematerialaftertheskin-pass.

Lake(1985)reportsaseriesofvaluesfortheLüdersbands spac-ingaftertheskinpass.Thespecimenswerechemicallyetchedto theirmidplanes,agedat300◦Cfor30mintosensitizedeformed areasandetchedwithFry’sreagent.Fig.3displaysaresultreported

Fig.4. Stress–straincurvesfortheworkhardeningandfortheunlockingand/or multiplicationofdislocationsandforthejointactionofthetwomechanisms(Green, 2002).

bythisauthor,whostatesthatthistechniquedoesnotdelineate preciselywhichareashavebeendeformedandthattheaccuracy ofthemeasuredaveragespacingisapproximately±8%.The anal-ysisofthefigureshowsthattheLBsarenotparalleltoeachother anddisplaya“wavy”aspect,involvinglargevariationsinthe dis-tancebetweenthebands.Lakedoesnotdefineclearlywhatisa “bandspacing”,anditisgatheredthatitwouldbethe“average” dis-tancebetweentwodarkregions,thuscorrespondingtothelighter regionspossiblydisplayingadeformationlowerthanthoseinthe banditself.ThemainprobleminLake’stechniqueisthatitis dif-ficulttoevaluatethelevelofstrainatwhichregionsaremoreor lessetched,makingitimpossibletoestablishaclearcriterionfor themeasurementofthebandspacing.Themeasurementofthis spacingemployingtheindicationsonthelateralfaceoftherolled sheets,utilizingtheapproximatecenter–centerdistancebetween thebands,asshowninFig.2,willthusbehigherfromthosereported byLakeandbasedonthetechniqueillustratedthroughFig.3.

Lake(1985)reportsanincreaseintheLBsspacingcausedbythe skin-passwiththeincreaseininitialthicknessofthematerial, ran-gingfromaspacingof≈0.18mmforaninitialthicknessof0.5mm toalmost0.6mmforasheet3.0mmthick;thesevalueswere col-lectedforlaboratoryandindustrialskin-passes.Theseareofthe sameorderofmagnitudeasthosemeasuredforFig.2(≈0.45mm), butlowerthanforthe19.05mmthicksheetinFig.1(≈2.50mm). NosigniﬁcanteffectoftherollingspeedwasdetectedontheLBs spacing.

Lake(1985)alsoreportedthattheLBsspacingafterskin-pass rollingwiththe150mmdiameterrolls,dryandlubricated,witha thicknessreductionof0.8%indicatedasmallincreaseinthespacing whenthelubricationisapplied.Thedryskin-passrollingofthe samematerial,butwiththe38mmdiameterrolls,ledtoadecrease ofabout15%intheLBsspacing.

IthasbeenwidelyacceptedthattheYPsinlowCarbonsteels, withtheircharacteristicinitialpeak stressfollowed bya stress plateauassociatedwiththenucleationandpropagationofLBs,is causedbydislocationlockingbyinterstitialNitrogenandCarbon atoms(CottrelandBilby,1949),inthesocalled“strainaging”of thematerial.Theinitialmobiledislocationdensityofaged mate-rialsislow,leading tohighstressesfor theunlockingand high speedinitialmovementofthedislocations.Amassive multiplica-tionofmobiledislocationsensues,causingasuddenloweringof theiraveragespeedandconsequentlyalsooftheappliedstress.On theotherhand,theincreaseindislocationdensityworkhardensthe material,leadingtoanoverallsituationschematicallydescribedby Fig.4(Green,2002),whichshowsthateachpointinthematerial

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Fig.5. Deformationheterogeneitiesinthelongitudinalsectionofasheetaftertheskinpassconsideringtwoconstitutivebehaviorsforthematerial(AorB)andreductions ofthicknessof0.5,1.0and2%(Yoshidaetal.,2008).

undergoesaninitialhighappliedstresswhichfallsprecipitously andthenrisesagainatamuchslowerrateduetowork harden-ing.Itisinterestingtoremarkthatthisistheapproachadoptedby CorusIndustriesintheiranalysisofskin-passphenomena(Green, 2002).TheabovedescribedeventswereﬁrstproposedbyJohnston andGilman(1959)asamodelforthediscontinuousyieldinginLiF andprovedtobeofwideapplicationtovariousmaterials.

Aconsequenceoftheabovesituationisthatdeformation com-monlystartsinregionsdisplayingstressconcentrationsintensile samples,asdemonstratedbySunetal.(2003),andthenspreadsto therestofthespecimen,whichthusdisplaysdeformationbands (LBs)thatpropagateunderanapproximatelyconstantstress(the LowerYieldStressintensiletesting).Itisexperimentallydifﬁcult(if notimpossible)toaccuratelymeasuretheinitialpeakstress illus-tratedinFig.4,sincetheexperimentalset-upsactuallyreﬂectthe averageappliedexternalstresses,andnotthelocallydeveloped concentratedstress.Hemerichetal.(2011)havefoundaninitial stresspeakinthestress–straincurveofsteelwhichishigherin theirspecialbendingtestthanundercommontensiletestingand concludedthattensiletestingisunabletoadequatelycapturethe upperyieldstress.Yoshidaetal.(2008)alsofoundmuchhigher initialstresspeaksintensiletestsofstrainagedmaterial,when thestressconcentrationstypicaloftensiletestingaresubstantially decreased.

Schwaband Ruff(2013) presenteda detailedanalysisofthe yieldpointphenomenaintensiletesting,withspecialemphasison theestablishmentofacriterionforthedeterminationofthelower yieldstress.Theanalysisisbasedonconsiderationsofstress triax-ialitiesjustaheadandbehindthepropagatingLBs.Theirmodelis basedontwohypotheses:(a)thetrueupperyieldstressismuch higherthantheupperyieldstrengthcommonlymeasuredin ten-siletestsand(b)thetruematerialbehaviorafterfirstyieldingis not theobservedone (involving ayield plateauduring theLBs propagation),butinsteadwithstrainhardeningstartingatthetrue loweryieldstrengthofthematerial;inotherwords,thematerial behaviorshouldbedescribedbyacurvesimilartothatinFig.4. The authors present extensive experimental resultsconfirming theirassumptions, and alsoshowthat consideringhundredsof performedtensiletestsandcorrespondingFEAs,aconsistent simu-lationoftheobservedbehaviorscanonlybereachedifasufficiently highvalueoftheinitialpeakstressisconsidered.Adetailedanalysis ofasituationwherethispeakstressistakenas500MPaandthetrue loweryieldstressas150MPaforastrainof0.01ispresented, lead-ingtoanaccuratedescriptionoftheexperimentalresults,including thepropagationofLBs.Theauthorsalsodemonstratedthatslight misalignmentsinthegripsoftensiletestsleadstothesuppression oftheupperyieldstress.

It seemsthat theonlyFEAof theskin-passintheliterature consideringastress–strainbehaviorofsteelsgivenbyacurve sim-ilartothatinFig.4isthatbyYoshidaetal.(2008),whocovered aspectsconnectedto(a)theconstitutivedescriptionoftheinitial material(beforetheskin-passrolling),(b)theanalysisoftheLBs formationcausedbytheskin-passand(c)theeliminationofthe yieldpointontensiletestingfollowingtheskinpass.The constitu-tivebehaviorofthematerialemployedfortheFEAwasderivedfrom dislocationmultiplicationandworkhardeningconsiderations(as discussedabove),andledtoarangeofstress–straincurves.Twoof thesecurveswereemployedintheFEAanalysesoftheskin-pass: onewithaninitialstresspeakof≈620MPa(MaterialA)andthe otherwithaninitialstresspeakof≈380MPa(MaterialB).Forboth materials,theinitialstresspeakwasfollowedbysofteningdown toaflowstressof≈280MPaatastrainof≈0.02,andasubsequent stagewherethematerialunderwentworkhardening.Theinitial thicknessofthematerialwas1.2mm,therolldiameter250mm andthecoefficientoffrictionbetweentherollsandthematerial 0.2;therollswererigid(noelasticdeformation).Theresulting dis-tributionofdeformationinthelongitudinalsectionoftherolled sheets,aftertheskinpass,isdisplayedinFig.5.Itisimportantto remarkthatYoshidaetal.(2008)statethatstrainlocalization char-acteristicsareaffectedbythemeshsizeandthatfinermesheslead tohigherstrainlocalizations.Inaddition,itisalsostatedthatthe aimoftheirpaperisnottodiscussthedetailsofnumericalaspects ofsimulation,buttodemonstratehowitstronglydependsonthe upperyieldpointphenomena.Theseauthorsgivenodetaileither concerningthemeshdensityorthetimestepsintheirskinpass simulations.

Forthesamereductioninthicknessintheskinpass,Material AleadstothepresenceofregularlyspacedLBsalongtherolling directionandtohigherlevelsofstrainintheLBsthanfor mate-rial B, where deformationis much less heterogeneousthan for MaterialA.ItisinterestingtonoticethatYaritaandItoh(2008) employedintheirFEAoftheskin-passastressstraincurve display-inganinitialsmallstresspeakfollowedbywhatseemstobeayield plateau;hisresultsaresimilartothosereportedformaterialBby Yoshidaetal.(2008)inthesensethatnodeformationbands(LBs) arereportedinthelongitudinaldirectionoftheprocessedsheet. Yoshidaetal.(2008)remarkedthatthedetectionofLBs associ-atedwiththeadoptedstress–strainbehaviordependsonthemesh densityinthematerial.Itisnoteworthythatthetruestress–strain curveutilizedbySchwabandRuff(2013)intheiranalysisis sim-ilartotheabovementionedMaterialA.Yoshidaetal.(2008)also performedFEAsemployingcorrugatedrolls,whichledtothe for-mationofLBsinthematerialintheregionswherethecorrugations causedlocalstressconcentrationsinthematerial,indicatingthe

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Fig.6.Effectivestress–effectivestraincurvesadoptedfortheFEAsinthepresent paper.

specialimportanceofthesestressconcentrationsinthenucleation ofLBs.

AcomparisonofFigs.2and5(MaterialA)indicates similari-tiesbetweentheexperimentalandsimulatedresults.Ontheother hand,theFEAresultsinFig.5donotindicatepronounced differ-encesindeformationheterogeneitybetweenthesurfaceandthe mid-planeofthematerial,inoppositiontotheexperimentalresults reportedbyHundy(1955)andLake(1985).

Asfarasthepresentauthorsareawareof,theliteraturepresents noFEAsfortheskinpassrollingofsteelsheetsdisplayingYPsin ten-siletestingbeforetheskin-pass,coveringthefollowingaspects:(a) theeffectofrolldiameteronthedistributionofLBsinthematerial, (b)theeffectofvaryingfrictionconditionsontheLBsdistribution, (c)thenucleationofLBsand(d)thestraindistributionintheLBs forincreasingreductionsofthicknessintheskinpassandits rela-tionshiptotheeliminationoftheYieldPointElongation(YPE)by theskin-pass.ThepresentpaperpresentsFEAscoveringarange ofthesevariables,foramaterialdisplayingaconstitutive behav-iorsimilartothatofmaterialAinYoshidaetal.(2008)andtothe materialinSchwabandRuff(2013).Inadditiontotheabovepoints, FEAsareperformedforvariousvaluesoftheinitialstresspeak.Itis shownthat(a)asfoundbyLake(1985),rolldiameterhasalimited influenceontheLBsdistributionintherolledsheets,(b)friction conditionsalsohavealimitedinfluenceontheLBsdistributionin thematerial(c)LBsnucleatewherethefirstcontactbetweenthe materialandtherollsoccurs,aspredictedbyButlerandWilson (1963)(d)anincreaseintheskin-passreductionleadstochangesin thestraindistributionintheLBs.Itissuggestedthattheelimination oftheYPEbytheskinpassmaybeconnectedtothevolume frac-tionofthematerialstrainedabovethedeformationcorresponding totheminimumstressinthestress–straincurve.

2. Materialandmodeling

2.1. Material

Theconstitutivebehaviorofthematerialutilizedinmostofthe presentFEAsfollowscloselythesuggestionsbyYoshidaetal.(2008) andSchwabandRuff(2013)inordertoadequatelydescribetheLBs phenomenabothintensiletestingandintheskin-pass;thereisan initialpeakof600MPa,theflowstressthendropsto220MPafora strainof0.005,dropsagainto200MPaatastrainof0.02andthen presentsalinearworkhardeninguptoastrainof0.1andaflow stressof700MPa;theresultingstress–straincurvecorrespondsto theuppercurveinFig.6.Thiscurvewasnotobtainedbydirect experimentaltests, which involve many difficulties,as already

discussed.Thefinalstressof700MPaissubstantiallyhigherthan theoneusuallyobservedforlowcarbonsteels(about400MPafor astrainof0.1,asindicatedbyKijima(2013)),andwasadoptedin ordertodecreasethestrainheterogeneities,which,formaterials withfinalstresslevelsof400MPawillbemorepronouncedthanfor thepresentsimulations.Thisisduetothefactthattheinitial post-peaksofteningofthematerialcausesastrainconcentrationwhere thisstrainwastriggered,wherefurtherlocalizeddeformationwill proceedtilltheworkhardeningofthematerialanddecreasesin theappliedstressesinducethetransferofdeformationtoanother region.Highworkhardeningratesaftertheinitialdecreaseinflow stress(leadingtohighstressessuchas700MPa)thusmakemore difficultthestrainconcentrations.Simulationswerealsorunwith thefollowinginitialstresspeakvalues:350,500and600MPa.The materialwasconsideredasrigid-plastic,strainrateinsensitiveand displayingisotropicsofteningandhardening.Theconsiderationof astrainratesensitivityandaparabolicworkhardeningdidnotalter significantlythesimulationresults,leadinghowevertolonger com-putationaltimes.Itisrecognizedthatthechoiceoftheinitialstress strainbehaviorofthematerialcannotbe,atpresent, experimen-tallyjustified,butallexistingevidencepointtoabehaviorsimilar totheadoptedone,forsteelssubjecttostrainagingassociatedwith dislocationlockingbyinterstitialatoms(CandN).Thedevelopment ofexperimentaltechniquesallowingtheaccuratemeasurementof theinitialstress–strainbehaviorofagedsteelswouldbeofgreat interest.

2.2. Finiteelementanalyses

Simulationswere performedusing theimplicit DEFORM–2D software,version10.0(ScientiﬁcFormingTechnologies Corpora-tion,Columbus,OH).Duetothesymmetryintherollingprocess, onlyhalfofthesheetthicknesswasconsidered,asdisplayedin Fig.7.

AccordingtoKijima(2013),theFEAoftheskin-passfor mate-rialsnotdisplayingYPsbeforetheskinpasscanemployrigidrolls withadiameterincreasedbytherollﬂattening.Consideringthis alsovalidformaterialsdisplayingYPs,theevaluationoftheeffect oftherolldiameterwasperformedforrigidrollswithdiameters of150,500,750,1000,1500and3000mm,rotatingatanangular speedsuchthattheperipheralvelocityinallcaseswassimilarto thatforthe150mmrollsrotatingat6.24rd/s,andthereductionin thicknesswastakenas1%(typicalofmanypracticalsituations).In mostcasesthesheetwas1.0mmthickand40mmlong;the reduc-tionsinthicknesswereof0.5,1.0,2.0,3.0,3.5,and4.0%oftheinitial thickness,andtherollshadadiameterof150mm,leadingtomuch shortercomputationaltimeinrelationtothesimulationsforlarger rolls.Asimulationisalsopresentedforaninitialthicknessofthe materialof19.05mmandarigidrollwithadiameterof254mm, inordertocomparetheresultswiththoseinFig.1.Allresultsare reportedreplicatingthesimulationsinrelationtothehorizontal symmetryaxis,sothatthefullthicknessofthematerialisviewed. Ashearfrictionfactor(m)betweenthematerialandtherolls wastakenas0.12formostsimulations(Dieter,1976).Simulations werealsoperformedform=0.1,0.2,0.4,0.6,0.8and1.0,aswellas forCoulombfrictioncoefﬁcients=0.1,0.2,0.3,0.5and0.6.This lastvalue(=0.6)alreadycorrespondstothestickingbetweenthe materialandthecylinder,consideringthevonMisesyield crite-rion.Themeshhad5300elements/mm2_in_the_longitudinal_section

ofthesheetand thetimestep was5×10–6slong.The simula-tionresultsweresensitivetothenumberofelements, asfound byYoshidaetal.(2008);for784elements/mm2 _in_the

longitudi-nalsectionofthesheet,noLBswereobservedforsheetthickness reductionof2.5%ormoreintheskinpass.Forthehigherdensity meshutilized(5300elements/mm2_),_this _did_not_occur_even_for

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Fig.7. SchematicillustrationoftherollingprocessintheFEAoftheskinpass.

thicknessreductionofabout3%,nostrainheterogeneitiescould beobservedonthesheetplaneaftertheskin-passofthematerial; sincehisvisualtechniquesprobablycouldnotdetectstrain het-erogeneitiesatthislevelofthicknessreduction,itwasconsidered thatthemeshdensityof5300elements/mm2_is_adequate_for_the

presentFEAs.Forthismeshdensity,theconsiderationofan elastic-plasticbehavior(EP)ofthematerialdemandedstep-timesmuch lowerthanthosementionedaboveandledtoextremelylongand unpracticalcomputationaltimesforthesimulations,especiallyfor highinitialstresspeakvaluesandlargerolldiameters.Theobtained strainheterogeneitiesandLBsdistributionsforanEPmaterialwere similartothoseobtainedforarigid-plastic(RP)material,butthe formerledtoasomewhatmoreheterogeneousstraindistribution andlowerstrainsclosetothesheetsurfacethanthelatter.Froma qualitativepointofview,regardingthestrainheterogeneitiesand thedistributionofLBsinthesheet,simulationswithaRP mate-rialledtoconclusionssimilartothosebasedonsimulationswith EPmaterial,butwithmuchshortercomputationaltimes.Itisthus consideredthatforthecurrentavailablecomputationalresources andthelevelofanalysispresentlydiscussed,aRPplasticsimulation leadstotechnicallyreasonableresultsandfeasiblecomputational times,providedasufﬁcientlyﬁnemeshisutilized. Adiscussion coveringacomparisonbetweenthesimulationswithEPandRP materialiscoveredintheAppendixofthepresentpaper.

Theeffectivestraindistributioninthematerialwasanalyzed semi-quantitativelythroughcolorshadesinthelongitudinal sec-tionof therolledsheets, and quantitativelythrough graphsfor theeffectivestrainalong arolledsheetlengthof1.4mm,taken afterthesheethadalreadyreachedasteadystaterollingcondition. Suchdatawerecollectedalongthesheetsurfaceanditscenterline. Thedistanceof1.4mmwassufﬁcienttocoveraboutfoureffective straincycles(foursuccessiveLBs);theoriginofthisdistancewas alwaystakenasapointwhereaminimumofeffectivestrainatthe centerlineoftherolledsheetwasobserved.

3. Resultsanddiscussion

3.1. TheeffectoftherolldiameterandthesimilarityoftheFEA resultswiththeexperimentalones

Fig.8displaystheresultsoftheFEAsfortherolldiametersof150, 500,7501000,1500and3000mm(coveringthewiderangeof skin-passrollinginlaboratoriesandindustries)andshearfrictionfactor m=0.12.TheﬁgurealsoincludesvaluesoftheparameterL/Hforthe varioussituations,whereListhelengthoftheroll/materialcontact andHisthethicknessofthematerial;this parameteris widely usedinplasticityanalysesforthecharacterizationofthe geomet-riccharacteristicsofformingprocesses.Thestrainheterogeneityis

Fig.8. Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling;sheetthickness1.0mm,thicknessreduction1%,shearfrictionfactor0.12androll diameters150,500,750,1000,1500and3000mm.

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Fig.9.Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling; sheetthickness19.05mm,thicknessreduction0.9%,shearfrictionfactorm=0.12 androlldiameter254mm.

qualitativelysimilarforallcases,exceptfortheLBsspacing,which risesfrom≈0.27mmalongthesheetcenterline,forthe150mm rolldiameter,to≈0.38mmfortheotherrolldiameters,inatrend similartothatreportedbyLake(1985).Itisdifficulttocomparethe presentspacingvalues,whichwerevisuallymeasuredbetweenthe centersofsuccessiveLBs,andthosereportedbyLake(1985),who employedadifferentmeasurementcriterion,asalreadydiscussed. Itisthusconsideredthatthepresentresultsarewithinthe uncer-taintiesinvolvedintheexistingdata,andoneconcludesthatexcept fortheLBsspacing,therigidrolldiameterhasalimitedinfluence ontheLBsdistributionintheskinpassform=0.12,which corre-spondstoa relativelywell-lubricatedconditionintheskin-pass rolling.Undersuchfrictionconditions,theinitialandthe elasti-callyflattenedrolldiametersdonotleadtoappreciabledifferences inthepredictedLBsdistribution.Fromthepointofviewof com-putingtime,theuseofthe150mmrolldiameterledtomuchfaster calculations,withoutanymajorlossofinformationwithregardto theotherlargerrolldiameters.

Fig.9showsthedistributionoftheeffectivestraininthesheet thicknessaftertheskin-passrollingforasituationsimilartothatin Fig.1,indicatingagoodrelationshipbetweentheLBsdistributionin bothﬁgures.Thebandcurvaturesand“Y”shapedbandsemanating fromthesheetsurfacesaresimilarinthetwosituations,butthe measuredbandspacingissomewhatdifferent(≈2.50mmforthe experimentalvaluesand≈3.30mmforthesimulations).Itshould benotedthattheFEAsinFig.8showarecurringpresenceofthe “Y”shapedbandsemanatingfromthesheetsurfaces,whichseems tobeafeatureoftheLBsformationintheskinpass,forthepresent FEAs.AcomplexsetofLBsisobservedinFig.9,withmainLBssimilar tothoseinFig.1,butalsootherLBsintersectingthemainones, whichcanalsobeobserveduponacloserexaminationofFig.1.

Finally,anFEAwasperformedforathinsheet(0.2mmthick, 150mmdiameterroll),whichledtoastraindistributionsimilarto thoseinFig.8,butwithabandspacingof0.10–0.12mm,matching anextrapolationdowntothe0.2mmthicknessofLake’sresults forvaryingsheetthickness.ItisconcludedthatthepresentFEAs, performedwithashearfrictionfactorm=0.12,describewellthe existingexperimentalresultsforthestrainheterogeneitiesinthe skinpassofsteelsheets.

3.2. Theeffectofthefrictionbetweentherollandthematerial Figs.10and11displaythedistributionofLBsalongthe longitu-dinaldirectionoftherolledsheetsforvariousvaluesoftheshear frictionfactor(m)andCoulombfrictioncoefﬁcient(),respectively

Fig.10.Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling; sheetthickness1.0mm,thicknessreduction1.0%,rolldiameter150mmandvarious shearfrictionfactors(m).

Fig.11.Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling; sheetthickness1.0mm,thicknessreduction1.0%,rolldiameter150mmandvarious Coulombfrictioncoefﬁcients().

andarigidrolldiameterof150mm,whichcorrespondstoa typ-icalskin-passrollingconditioninlaboratories.Itcanbeobserved thatthefrictionconditionsintheinterfacebetweenthematerial andtherollshavelittleinfluenceontheLBsdistribution,in accor-dancewithLake’s(1985)findingsforlubricatedandunlubricated skin-passrolling.FEAssimulationsoftheeffectivestrainevolution onthesheetsurfacealongtherollinggapforthevariousfriction conditionsindicate thatmostof thedeformationoccursalmost immediatelyattherollinggapentry;afterwards,aregionwhere verylittleornostrainincrementisobserved(possiblyassociated withstickingbetweentherolland thematerial) followedby a smallerregionwithanincreasingstrainrisingtoitsfinalvalue. Thiscorrespondstoasituationwherethefrictionconditionshave littleinfluenceontheLBsdistribution,sincetheseareestablished basicallyattherollgapentrance,aswillbedemonstratedinthe nextsection.

3.3. Theeffectofchangesintheeffectivestress–effectivestrain curve

Fig.12displaysthedistributionofeffectivestrainintherolled sheetsforaﬁnalstressof400and700MPaatastrainof0.1and ini-tialpeakof600MPa.Theresultsweresimilarforbothstressvalues, intermsoftheLüdersbandsdistributioninthesheet.Asexpected, strainsinthedeformedregionswerelowerforthesimulationswith 700MPathanforthesimulationswith400MPa,asindicatedbythe lightercolorsinthe700MParesults,inrelationtothe400MPaone. Fig. 13 displays the distribution of effective strain in the rolled sheets for initial stress peaks in the adopted effective stress–effective strain curves of 350, 500 and 600MPa and a stress of 700MPa at a strain of 0.1. The distribution of LBs is

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Fig.12.Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling; sheetthickness1.0mm,thicknessreduction1.0%,shearfrictionfactorm=0.12,roll diameter150mm,initialstresspeakof600MPaandﬁnalstressof400MPaand 700MPaforastrainof0.1intheadoptedeffectivestress–effectivestraincurves.

Fig.13.Effectivestraindistributioninthesheetthicknessaftertheskin-passrolling; sheetthickness1.0mm,thicknessreduction1.0%,shearfrictionfactorm=0.12, rolldiameter150mmandvariousinitialstresspeaksintheadoptedeffective stress–effectivestraincurves.

qualitativelysimilarforthethreesituations,butdisplayingmore diffuse,lowerstrainsandadecreasingbandspacingasthestress peakislowered.The“Y”patternclosetothesurfaceofthesheets forthe600MPastresspeak,forexample,cannotbeobserved any-moreforthe350MPastresspeak.Itisinterestingtoobservethat Yoshida’sresultsfollowthesametrendasthepresentones:fora 380MPastresspeak(Fig.5,MaterialB)alowstrainheterogeneity isobserved,probablyalsoduetoalowermeshdensityemployed byYoshida,whoseaimwasjusttodemonstratethepresenceofLBs inthematerial.

3.4. ThenucleationoftheLBsintheskinpass

Fig.14showssuccessivestepsinaFEAsimulationfortheevents connectedtothenucleationof LBsintheskin-pass.Thedashed verticallinesinFig.14correspondtotheposition,ineachstep,of theinitialcontactpointinFig.14a.

Fig.14adisplaysthemomentwherethereisstillappreciable straininginthelast LBinthesheet,involvinga highstrainrate inthisband.AnewbandhasalreadynucleatedinFig.14b(still notseenintheeffectivestraindiagram,butalreadydetectedin thestrainratediagram),propagatingtowardthesheetcenterline, aspredictedbyButlerandWilson(1963).Duetotheshorttime elapsedbetweenFig.14aandb,stillnoappreciablestraincanbe observed.ThesituationissimilarinFig.14c,butthestraininghas stoppedforthebandalreadynucleated(strainrateofzero)and predominatesforthenewband,whichﬁnallycanbeseeninthe effectivestraindiagraminFig.14d.Itisinterestingtonoticethat twonewbandsarealsoemanatingoutwardlyfromthesheet cen-terline,inFigs.14candd;thiscanbeseeninitiallyonlyinthestrain ratediagrams,asusual.

InFig.14dthenewdeformationbandsemanatingfromthesheet centerlinestartgrowingbackwards(seethestrainratediagram), towardtheinitialband,inaprocesscompletedinFig.14e,where onecanalreadyseetheindicationsofthisphenomenoninthestrain diagram.Fig.14fshowsthenucleationofanewbandat approxi-matelythesameregionasthepreviousone,whosestrainratestarts decreasing.Thenewbandjoinsthepreviousband,establishingthe alreadymentioned“Y”patternfortheLBs.InFig.14g,thestrain ratehasfallentozerointhepreviousband,andpredominatesonly inthenewlynucleatedband,andthe“Y”patterncanalreadybe seeninthestraindiagram.Theprocessisstartedalloveragainfor anewsetof“Y”shapedbandsinFig.14h,whichisquitesimilarto Fig.14a.

InordertoanalyzeingreaterdetailthenucleationoftheLBs, Fig.15aexhibitsanexpandedviewofthestrainratedistribution duringtheearlystagesofaLB;thestrainrateinthevicinityof theregionoftheﬁrstcontactwiththerollishigherthanalongthe

Fig.14.EffectivestrainandeffectivestrainratedistributionsinthesheetthicknessduringthenucleationoftheLBs,rolling;sheetthickness1.0mm,thicknessreduction 1.0%,shearfrictionfactorm=0.12,androlldiameter150mm.

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Fig.15.EffectivestrainratedistributioninthesheetthicknessduringthenucleationoftheLBs,rolling;sheetthickness1.0mm,shearfrictionfactorm=0.12,thickness reduction1.0%androlldiameter150mm.

bandandsomestrainingoccursinthesurfaceregionofthematerial beforetheactualcontactwiththeroll.

Fig.15billustratesthefactthatanewLBisalsonucleatedin thesamehighstrainrateregion(formingthe“Y”pattern)afterthe formationoftheimmediatelypreviousband,andthenemanates towardthecenterofthesheet.Therepeatednucleationofbands atthesheetsurface,attheinitialroll/materialcontactregion justi-ﬁestherelativelylimitedinﬂuenceoftherolldiameterandofthe frictionconditionsbetweentherollsandthematerialontheLBs distributioninthesheet.

Itis consideredthat thestress concentrationattherollgap entranceisofspecialimportancefortheLBsnucleation.Thiscan beappreciatedinYoshidaetal.(2008)FEAsresultsforcorrugated rolls,whereeachcorrugationimposesastressconcentrationonthe sheetandleadstothenucleationofanLB.Itisthusbelievedthat Kijima’s(2014)analysisofthetransferofthecylindertopography tothesheetwouldalsoprobablydisplaythetriggeringofLBsfor

eachoftheadoptedhemisphericalcorrugationsontherollsurface, iftheadoptedstress–straincurveforthematerialdisplayedan ini-tialstresspeak,similarlytothesituationpredictedbyYoshidaetal. (2008).

3.5. Effectofincreasingthicknessreductions

Fig.16showsthroughcolorshadedisplaystheeffectivestrain distributioninthelongitudinalsectionofthesheetsafterthe skin-passforthicknessreductionsof0.5,1.0,2.0,3.0,3.5and4.0%,initial thicknessof1.0mm,shearfrictionfactorm=0.12,androll diam-eter150mm.Fig.17allowsamorequantitativeevaluationofthe variationofstrainalongthesurfaceandthecenterlineoftherolled sheets,forthesameconditionsshowninFig.16.

Lake(1985)showedthatthecriticalstrainintheskin-pass lead-ingtoasmoothstress-straincurveofthematerialintensiletesting aftertheskin-pass,for materials0.8–1.2mm thick,rangesfrom

Fig.16.Distributionofeffectivestraininthelongitudinalsectionofthesheetaftertheskinpass,forareductionof0.5,1.0,1.5,2.0,3.0,3.5,and4.0%insheetthickness,roll diameter150mm,frictionshearfactorm=0.12andsheetthickness1.0mm.

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Fig.17.Variationintheeffectivestrainalongtherolledsheet,forreductionsofthicknessof(a)0.5%,(b)1.0%,(c)2.0%,(d)3.0%,(e)3.5%,and(f)4.0%atthematerialsurface, quarterthicknessandcenterline,sheetthickness1.0mm,frictionshearfactorm=0.12,rolldiameter150mm.

≈1.3%to≈2.0%.Thelowerthicknessreductionscorrespondtograin sizes≈0.015mmandthehighesttograinsizes≈0.007mm.Itis interestingtoremarktheagreementoftheseﬁndingswiththose intheFEAsbyYoshidaetal.(2008),whoobtainedvaluesforsuch acriticalthicknessreductionrangingfrom≈1%to≈2%.

TheexaminationoftheLBsdistributionforthicknessreductions of1%and2%inFig.16conﬁrmsthepresenceofahighly hetero-geneousstraindistributionofstraininthematerial,asindicated byHundy(1955),ButlerandWilson(1963)andLake(1985).The curvesinFig.17indicatethatevenforlowreductionsinthe skin-pass(0.5%inFig.15a)thesurfaceofthesheetsisfullydeformed, withstrainsrising from≈0.05to≈0.10asthicknessreductions increasefrom0.5%to4.0%.TheresultsintheFEAsbyYoshidaetal.

(2008)displayedinFig.5(MaterialA),displaylightgrayregions indicatingthepresenceofverylowstrainsalongthesheetsurface, especiallyforthelowthicknessreductions.Thisdiffersfromthe presentresults,probablybecauseYoshida’ssimulationsinvolved anEPmaterialbehavior.

AccordingtoKijima(2013),a Coulombcoefﬁcientof friction =0.3shouldbeadoptedforahighfrictionindustrialsituation. SimulationssimilartothosewhoseresultsarereportedinFig.16 wereperformedforalargediameterroll(D=3000mm)and=0.3, inordertoevaluatesituationsinvolvinglargeﬂattenedroll diame-ters/highfrictions.ThecorrespondingresultscanbeseeninFig.18, whereitshouldbepointedthattheeffectivestrainscaleisdifferent fromthatinFig.16.

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Fig.18.Distributionofeffectivestraininthelongitudinalsectionofthesheetaftertheskinpass,forareductionof0.5,1.0,1.5,2.0,3.0,3.5,and4.0%insheetthickness,roll diameter3000mm,Coulombfrictioncoefﬁcient0.30andsheetthickness1.0mm.

ThecomparisonoftheresultsinFigs.16and18indicatesthat thedistributionofdeformationinthesheetforlowthickness reduc-tions(0.5%and1%)issimilarforthetwoextremerollingsituations (lowfrictionandrolldiameterinFig.16,highfrictionandroll diam-eterinFig.18),butthebandspacingishigherforthelatter.On theotherhand,forareductionofthicknessof2%,the deforma-tionclosertothesurfaceofthesheet,thebandthicknessandtheir spacingare differentforthetwo situationsbeingconsidered; a largerolldiameterandhighfrictionleadstoalargeranddeeper strainclosetothesheetsurfacesandtoalargerbandspacing.For evenhigherthicknessreductions(3.0,3.5and4.0%)the deforma-tionalongthesheetthicknessismorehomogeneousinthehighroll diameter/highfrictionsituationthanforsmallrolldiameter/low frictionone.Intheformer,thepresenceofbandsalongthesheet thicknessismuchlesspronouncedthaninthelatter.

Materialsnotexhibitinginitialstresspeaksintheirtensile test-ingdonotdisplayLBseitherinthetensiletestorintheskin-pass (Yaritaand Itoh,2008;Kijima,2013).Consequently,theregions intheheterogeneouslydeformedmaterialinFigs.16–18,strained

beyondthestraincorrespondingtotheminimuminthestressin thestress–straincurveshowninFig.6(astrainof≈0.02),would notdisplayYieldPointElongations(YPEs)uponfurthertensile test-ing,sincetheirbehaviorwouldbeidenticaltothatofamaterialnot displayinganinitialstresspeak.Thissituationsuggeststhatfora minimumaveragestraininthematerialof≈0.02,theYPEwould beeliminatedin tensiletestingofthesheetaftertheskinpass. Inaddition,thedeformationpeaksalong thesheetcenterlinein Fig.17notonlyrisebutalsobecomebroaderasthickness reduc-tionsintheskin-passareraised,increasingthevolumefractionof thedeformedmaterial.Itisagainsuggestedthatthecritical thick-nessreductionintheskinpasscorrespondstoaminimumvolume fractionofmaterialinthesheetdeformedtoaminimumstrain cor-respondingtotheminimumstressinthestress–straincurve(≈0.02 inthepresentcase).

Thepresent FEAsallowthecalculationoftheaveragestrain intherolledsheetasthethicknessreductionintheskin-passis increased;theresultisdisplayedinFig.19,forasmallroll diam-eterandlowfriction(Fig.19a)andforalargediameterandhigh

Fig.19.Variationinareafractiononthelongitudinalsectionoftherolledsheet,displayingstrainsabove0.02andintheaveragestraininrolledsheetforincreasingthickness reductionsintheskin-pass.

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friction(Fig.19b).Fig.19aindicatesthataminimumaveragestrain inthematerialof≈0.02isobtainedthroughathicknessreduction of≈1.3%,whereasFig.19bindicatesathicknessreductionof≈1.0%, whicharesimilartothelowervalueforthecriticalreduction rec-ommendedbyLake(1985).Fig.19alsodisplaystheareafractionof materialdeformedbyatleast0.02inthelongitudinalsectionofthe rolledsheet,forincreasingthicknessreductionsintheskin-pass. Underthepresent2Dsimulations,thisareafractioncorresponds directlytothevolumefractionofthematerialdeformedbyatleast 0.02.ForFig.19a,athicknessreductionof≈1.3%leadstoabout ≈40%ofthevolumeofthematerialdeformedbyatleast0.02;the samevolumepercentageisobtainedforathicknessreductionof 1.0%inFig.19b.Itisthussuggestedthatthecriticalthickness reduc-tionforthecompleteeliminationoftheYPEinthetensiletesting ofmaterialaftertheskin-passshouldimposeaminimum aver-agestraininthematerialsimilartothestrainfortheminimum stressaftertheinitialstresspeakinthestress–strainbehaviorof thematerial,correspondingtoaminimumof≈40%ofthevolumeof thematerialdeformedatleastuptothisstrain.Fromtheindustrial pointofview,asafetymarginshouldbeadoptedforboththe min-imumaveragestrainandthevolumefractiondeformedinorderto guaranteethecompleteeliminationoftheLBsproblem.

4. Conclusions

AFEAanalysisoftheskinpassrollingperformedfor materi-alsdisplayingyieldpointsbeforetheskin-passandarigid-plastic behaviorindicatedthat:

(1)ThereareprofuseLüdersBandsintherolledmaterial,with dis-tributionsandshapesalongtherollingdirectionsimilartothose experimentallyobservedbothinlaboratoryexperimentsandin industrialpractice.

(2)ThedistributionofLüdersBandsinthesheetssubmittedtothe skin-passwitharolldiameterof150mmisrelatively indepen-dentofthefrictionconditionsbetweenthematerialandthe rolls.

(3)Forlow frictionbetweenthematerialandtheroll, alimited butcertaininﬂuenceoftherolldiameterontheLüdersbands distributionwasfound;asomewhatincreasedstraincloseto thesheetsurfacewasobservedastherolldiameterisdecreased. (4)Forlowsheetthicknessreductions(0.5%and1.0%)simulations fortheextremesituations:(a)lowfrictionbetweenthematerial andtherollandrolldiameterof150mmand(b)highfriction betweenthematerialandtherollandrolldiameterof3000mm, againalimitedbutcertaininﬂuenceoftherolldiameteronthe Lüdersbandsdistributionwasfound.

(5)Forsheetthicknessreductionsof2.0%andhigher,highfriction betweentherollandthematerialandlargeﬂattenedroll diam-etersleadtoadistributionofLBsandofeffectivestraininthe sheetsdifferentfromthatobtainedforsmallrolldiametersand lowfrictionbetweentherollsandthematerial.

(6)TheLüdersbandsnucleateintermittentlyandgrowatthe ini-tialcontactpointbetweentherollsandthematerialattheroll gapentrance,inaccordancewiththetheoryproposedby But-lerandWilson.Itissuggestedthatthisresultsfromthestress concentrationinthisregionofthematerial.

(7)Thesurfaceofthe1.0mmthickmaterialdisplayedhigh lev-elsofeffectivestrainaftertheskin-passevenforthesmallest simulatedreduction ofthickness (0.5%);for low roll diame-tersandfrictionbetweenthematerialandtherolls,unstrained regionscanbeobservedevenforthicknessreductionsofabout 3.0–4.0%.

(8)Thethicknessreductionintheskinpassnecessaryforthe elim-inationofLüdersbandsproblemsuponfurtherprocessingof

thesheetisassociatedwithahighlyheterogeneous distribu-tionofstraininthesheet,involvingarelativelylargefraction ofunstrainedmaterial.

(9)It is suggested that the critical thickness reduction for the elimination of the Lüders Bands problems by the skin-pass corresponds to a minimum average strain in the material corresponding to the strain at the minimum stress in the stress–strain inthematerial,aftertheinitialstresspeak;in addition,itissuggestedthataminimumof_≈40%ofthematerial volumeshouldbedeformedatleastuptothisstrain.Undersuch conditions,largerolldiametersandhighfrictionbetweenthe materialandtherollswouldleadtoalowerthicknessreduction necessaryfortheeliminationoftheLBsproblemintherolled sheet.

Acknowledgments

The authorsare gratefulfor thefinancial supportof CAPES-MinistryofEducation ofBrazil,CNPq–MinistryofScienceand TechnologyofBrazil(CNPq301.034/2013-3),FAPEMIG– Founda-tionfortheSupportofResearchofMinasGeraisState,Graduate Programin Metallurgicaland MiningEngineeringofUFMGand Graduate Program in Mechanical Engineeringof UFMGfor the financialsupportofthepresentactivities.ThesupportofDr.Chris Fisher (Scientific Forming Technology Corporation, SFC) in the solutionof problems duringthe present simulationsis warmly acknowledged.

AppendixA. Commentsontheelastic-plastic(EP)andrigid plastic(RP)materialsimulationsoftheskinpass

Fig. A1 shows the results of a simulation of the skin-pass with a 1% thickness reduction, material with a 350MPa initial stresspeak,meshwith≈20,000elements(5300elements/mm2_),

Coulomb coefﬁcient of friction =0.3, rigid cylinder diameter 150mm,forelastic-plastic(EP)andrigid-plastic(RP)materials.It canbeseenthattheLBsdistributionisqualitativelysimilarinboth simulations,withtheEPonedisplayingalowerbandspacingthan theRPone.Inaddition,thedeformationclosetothesheetsurfaceis deeperandmorehomogeneousfortheRPsimulationthanforthe EPone,wherethepresenceofapparentlyundeformedregionsis suggested.TheEPsimulationalsoindicatesthepresenceofthe“Y” shapeddeformationregionsclosetothesheetsurface,which can-notbeobservedintheRPsimulation.Thisdifferencebetweenthe EPandRPsimulationsisassociatedwiththefactthatanystrainin theRPsimulationisalreadyplastic,whichisnotthecasewiththeEP simulation,wheretheelasticstrainsarerecovereduponunloading. TheRPresultscorrespondtosomewhatlowerstrain hetero-geneitiesthan forthe EPones.Thismeans thattheﬁndings in thepresentinvestigationactuallyinvolvesmallunderestimatesin thestrainheterogeneities,lendingfurthercredencetothevalidity

Fig.A1.Resultsofthesimulationoftheskin-passwitha1%thickness reduc-tion,materialwitha350MPainitialstresspeak,meshwith≈20,000elements (5300elements/mm2_),_Coulomb_coefﬁcient_of_friction₌_0.3,_rigid_cylinder

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Fig.A2.Resultsofthesimulationoftheskin-passwitha1%thicknessreduction, EPandRPmaterialwitha350MPainitialstresspeak,meshwith≈1000,≈2000 and≈4000elements,Coulombcoefﬁcientoffriction=0.3,rigidcylinderdiameter 150mm.

ofthepresenceofstrainheterogeneitiesinthepresentRPﬁnite elementanalyses.

TheconsiderationofthematerialasEPandtheﬁnemesh uti-lizedinthepresentinvestigationrequireaverysmalltime-step inordertoreachconvergence,demandingalongcomputational time.Thistimestepmustbefurtherdecreasedasthepeakstressis raised,leadingtolongercomputationaltimes.Astherolldiameter isincreased,longersheets(andthushighernumberoftotalsteps) mustbesimulatedinordertoreachanadequatesteadystatetobe reported,leadingtoevenlongercomputationaltimes.For exam-ple,forarolldiameterof150mmandastresspeakof350MPain Fig.A1above,theEPsimulationtimeisabout100h,incontrast toabout15hforthecorrespondingRPsimulation.Theincreasein thestresspeakandtherolldiameterleadtoexponentialincreases inthecomputationaltimes,easilymakingitunpracticalwiththe presentcomputationalresourcesavailable.

Onewayofshorteningthecomputationaltimesistodecrease themeshsize,withapronouncedlossofdetailsinthestrain dis-tribution.Fig.A2 showsresultsfor simulationswithEP andRP materialoftheskin-passwitha1%thicknessreduction,material witha350MPainitialstresspeak,meshwith≈1000,≈2000and ≈4000elements,Coulombcoefﬁcientoffriction=0.3,rigid cylin-derdiameter150mm.Thereisaclearlossofdetailsanddecreasein thestrainheterogeneityofthesheetasthenumberofelementsin themeshisdecreased,whencomparedwiththesituationdescribed inFig.A1,butcomputationaltimesaredowntoreasonabletimes. Itcanalsoagainbeobservedthat simulationswithRPmaterial leadstoalowerstrainheterogeneitythanthesimulationswithEP material,buttheoverallpatternofLBsissimilarforbothsituations. AccordingtoYoshidaetal.(2008),thestrainlocalization char-acteristicsareaffectedbythemeshsize,andﬁnermeshesleadto higherstrainlocalizations.Inaddition,itisalsostatedthattheaim oftheirpaperisnottodiscussthedetailsofnumericalaspectsof simulation,but todemonstrate howit stronglydependsonthe upperyieldpointphenomena.Theseauthorsgivenodetail con-cerningtheirmeshdensity or thetimestepsin theirskinpass simulations.TheirresultsforanEPanalysisand materialwitha stresspeakof380MPa(Fig.5,MaterialBinthemainbodyofthe paper)almostdonotindicatethepresenceofLBs,whereasfora materialwithastresspeakofabout600MPa,diffusebandsare indi-cated.TheresultsinFig.A2,forEPandRPsimulationswith_≈1000 elementsalsodonotindicatethepresenceofLBsinthematerial, indicatingthatYoshidaetal.probablyperformedtheirsimulations

Fig.A3.Resultsofthesimulationoftheskin-passwitha1%thicknessreduction, materialwitha450MPainitialstresspeak,meshwith≈1000elements,Coulomb coefﬁcientoffriction=0.3,rigidcylinderdiameter500mm,EPandRPmaterial.

Fig.A4.Resultsofthesimulationoftheskin-passwitha1%thicknessreduction, materialwitha450MPainitialstresspeak,meshwith≈1000elements,Coulomb coefﬁcientoffriction=0.3,rigidcylinderdiameter800mm,EPandRPmaterial.

withsimilarmeshes,thatallowshortercomputationaltimesthan ﬁnemeshes.

Figs.A3andA4presentcomparisonsoftheresultsofEPandRP simulationsemployingameshof≈1000elements,astresspeakof 450MPaandcylinderdiametersof500and800mm,respectively. ThesituationintheseﬁguresissimilartothatshowninFig.A1, butobviouslywithfarlessdetailsintheLBsandstrain distribu-tions,andpointtothesameconclusionsalreadyreachedforthe ﬁnemeshes(seeparagraphimmediatelyaboveFig.A1inthemain bodyofthepaper).Itisinterestingtoobservethattheincreasein rolldiameterfrom500mmto800mmleadstoqualitativelysimilar resultsespeciallyforthesimulationswithEPmaterial.

Evenforthe≈1000elementmesh,itisestimatedthatthe com-putationaltimefora150mmrolldiameterandastresspeakof 600MPawouldbeofaround45daysfortheavailable computa-tionalresources;fora3000mmrolldiameteritisestimatedthatit wouldtake3–6months,withoutanyeventualintermediate adjust-mentsnecessaryforsolvinglocalconvergenceproblems.

Itisthusconsideredthatforthecurrentavailablecomputational resourcesandthelevelofanalysispresentlydiscussed,aRP plas-ticsimulationleadstotechnicallyreasonableresultsandfeasible computationaltimes,providedasufﬁcientlyﬁnemeshisutilized.

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