Prediction of welding distortion and residual stress in a th

发布时间：2024-11-06

国外关于焊接变形和参与应力分析的研究文献

Available online at http://

ComputationalMaterialsScience43(2008)

353–365

http:///locate/commatsci

Predictionofweldingdistortionandresidualstressina

thinplatebutt-weldedjoint

DeanDenga,*,HidekazuMurakawab

ResearchCenterofComputationalMechanicsInc.,TogoshiNI-BLDG7-1,Togoshi,Shinagawa-ku,Tokyo142-0041,Japan

JoiningandWeldingResearchInstitute,OsakaUniversity,11-1,Mihogaoka,Ibaraki,Osaka567-0047,Japan

Received1October2007;receivedinrevisedform22November2007;accepted4December2007

Availableonline29January2008

Abstract

Inautomotiveindustry,thinplatepartsarecommonlyused.Duringassemblingprocess,weldingtechnologyisusuallyemployedbecauseofhighproductivity.Weldingdistortionoftenoccursinthinplateweldedstructuresduetorelativelylowsti ness.Thedistortioncausesproblemsnotonlyintheassemblingprocessbutalsointhe nalproductquality.Therefore,predictionandreductionofweldingdeformationhavebecomeofcriticalimportance.Inthisstudy,three-dimensional,thermo-elastic–plastic,largedeformation niteele-mentmethod(FEM)isusedtosimulateweldingdistortioninalowcarbonsteelbutt-weldedjointwith1mmthickness.Tocomparewiththelargedeformationtheory,thesmalldeformationtheoryisalsousedtosimulatetheweldingdeformationandweldingresidualsstress.Meanwhile,thecharacteristicsofweldingtemperature eld,plasticstraindistributionandweldingresidualstressinthinweldedplatesarealsoexaminednumerically.Experimentsarealsocarriedouttomeasuretheweldingdistortioninthethinplatebutt-weldedjoint.Bycomparingthesimulationresultswiththemeasurements,itisfoundthattheresultspredictedbythethermo-elastic–plastic,largedeformationFEMmatchtheexperimentalvalueswell.Moreover,usingtheinherentstrainsobtainedbythethermo-elastic–plasticFEM,http://paringtheresultssimulatedbytheelasticFEMwiththosepredictedbythethermo-elastic–plasticFEM,itisveri edthattheinherentstrainmethodcane ectivelypredicttheweldingdeformationinthethinplatebutt-weldedjointwith1mmthickness.Ó2007ElsevierB.V.Allrightsreserved.

PACS:07.05.Tp;47.11.Fg;65.40.De;81.20.Vj

Keywords:Weldingdistortion;Numericalsimulation;Thinplate;Plasticstrain;Finiteelement;Inherentstrain;Nonlinearanalysis

1.Introduction

Distortioninaweldedstructureistheresultofthenon-uniformexpansionandcontractionoftheweldandsur-roundingbasematerial,causedbytheheatingandcoolingcycleduringweldingprocess.Weldingdistortionhasnega-tivee ectsontheaccuracyofassembly,externalappear-ance,andvariousstrengthsoftheweldedstructures.Inmanycases,additionalcostsandscheduledelaysareincurredfromstraighteningweldingdistortion.Onthe

Correspondingauthor.Tel.:+81337853033;fax:+81337856066.E-mailaddress:deng@rccm.co.jp(D.Deng).

otherhand,increasingly,thedesignofengineeringcompo-nentsandstructuresreliesontheachievementofsmalltol-erance.Forthesereasons,predictionandcontrolofweldingdeformationhavebecomeofcriticalimportance.Inthepastdecades,alotofexperimentsandnumericalanalyseshavebeenconductedforpredictingweldingdis-tortion,andalotoffundamentalknowledgehasbeenalsoestablished[1–7].However,thereisverylimitedliteraturedescribingthepredictionandmeasurementofweldingdeformationinthethinplateweldedstructuresespeciallyfortheseweldedstructurewhoseplateorwallthicknessislessthan3.0mm.Recently,Liangetal.[8–10]haveestablishedanumberofdatabasesofweldinginherentdeformationsfortypicalthinplateweldedjointusing

0927-0256/$-seefrontmatterÓ2007ElsevierB.V.Allrightsreserved.doi:10.1016/http://matsci.2007.12.006

国外关于焊接变形和参与应力分析的研究文献

354D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

experimentalmethodandinverseanalysismethod,andsomemeaningfulachievementshavebeenobtained.

Inautomotiveindustrythethinplatepartsarecom-monlyemployed,andalotofthinplatepartsareassembledbyarcweldingprocessbecauseofhighproductivity.Duetorelativelysmallsti ness,signi cantweldingdistortionoftenoccurs.Tocomprehensivelyunderstandthecharacteristicsofweldingdeformationinthinplateweldedstructure,itisnecessarytodofurtherfundamentalresearchesbymeansofbothexperimentandnumericalsimulation.

Inthisstudy,three-dimensional,thermo-elastic–plastic,largedeformation niteelementmethod(FEM)isemployedtosimulateweldingdistortionandweldingresid-ualstressinalowcarbonsteelbutt-weldedjointwith1mmthickness.Tocomparewiththelargedeformationtheory,thesmalldeformationtheoryisalsousedtosimulatetheweldingdeformationandweldingresidualsstress.Mean-while,thecharacteristicsofweldingtemperature eld,plas-ticstraindistributionandweldingresidualstressinthinweldedplatesarealsoexaminednumerically.Experimentsarealsocarriedouttoverifythenumericalsimulationmethod.Moreover,usingtheinherentstrains(plasticstrains)computedbythethermo-elastic–plasticFEM,anelasticFEM,inwhichthelargedeformationistakenintoaccount,isalsoutilizedtoestimatetheweldingdeforma-tioninthesamebutt-weldedjoint.2.Experimentalprocedure

Inthisstudy,asimpleexperimentiscarriedouttomea-sureweldingdeformationinthethinplatebutt-weldedjoint.Thebutt-weldedjointconsistsoftwothinmildsteelsheets.Thedimensionofeachsheetis100mmÂ100mmÂ1mm.Theweldingmethodisgasmetalarcwelding(GMAW).Theshieldinggaswas80%Ar+20%CO2.TheweldingwireisYGW16[11].ThedetailedweldingconditionsareshowninTable1.

Toimitatetheweldingconditionsusedinautomobileindustry,aninitialgapbetweenthetwoplatesissettobe0.4mmandapartiallyweldingisperformedinthejoint.Thelengthofweldinglineisabout60mm.Thespecimensareweldedwithoutexternalconstraints.Intheexperi-ments,threeidenticalbuttjointsareperformed.Fig.1showsthepictureofacompletedjoint.

Afterwelding,thede ectionatthecenteroftheweldinglineismeasuredusingasimplemethodasshowninFig.2.Intheexperiments,aVerniercaliperisusedtomeasurethevaluesofh0andh1.Inthis gure,wisthedi erencebetweenh0andh1.Themeasuredvalues(w)ofthesethreejointsare2.20mm,1.90mmand2.10mm.Approximately,

Table1

WeldingconditionsParameterValue

Current(A)65.0

Voltage(V)17.0

Weldingspeed(mm/min)780.0

Fig.1.Thinplatebutt-weldedjoint.

w/2canberegardasthede ectionatthecenteroftheweld-ingline.Theaveragevalue(w)ofthesethreebuttjointsis2.067mm.Thismeanstheaveragevalueofthede ection(w/2)attheweldinglineis1.03mm.

3.Predictionoftemperature eld,residualstressanddeformationusingthermo-elastic–plasticFEM

Inthissection,basedonABAQUScode[12]asequentiallycoupledthermo-elastic–plastic niteelementcomputationalprocedureisdevelopedtocalculatetemperature eld,weldingresidualstressandweldingdeformations.The3D niteele-mentanalysesareperformedtonumericallystudytheweldingdistortionandweldingresidualstressinbutt-weldedthinplates.Inordertocapturethenonlineargeometricalbehaviorsinathinplatestructure,thelargedeformationtheoryisincor-poratedintothermo-elastic–plasticFEM.

Fig.3showsthe niteelementmeshmodelusedinthesimulation.Thedimensionsofthe niteelementmodel

Shieldinggas owrate(L/min)15.0

Tip-to-workdistance(mm)25.0

Wirediameter(mm)0.9

Travelangle(°)45.0

国外关于焊接变形和参与应力分析的研究文献

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365355

Fig.3.Finiteelementmodel,gapandbeadshape.

arethesameasthoseoftheexperimentalspecimen.Ele-mentmeshesaredenserinthevicinityoftheweldcenter-line,whilethemeshesbecomegraduallycoarseraway

fromtheweldzone.Thelengthofeachelementintheweld-ingdirectionis2.5mm.Thenumberofdivisioninthethicknessdirectionisfour.Thetotalnumberof8-nodebrickelementis4000.

Inthepresentstudy,toconsiderthebeadshapeintheFEmodel,threedimensionsoftheweldbeadobtainedintheexperimentsaremeasured.Oneistheheightofweldreinforcement;andtheothertwoarethebreadthsbetweenthetwotoesontheuppersurfaceandthebottomsurface,respectively.IntheFEmodel,thebeadshapeisroughlydeterminedbasedonthesethreeparameters.ThebeadshapeisshowninFig.3.

Inthethermalanalysis,theweldingconditionsareassumedtobethesameasthoseusedintheexperiment.3.1.Heatsourceandthermalanalysis

Weldingheattransferanalysiswithgivenweldingcondi-tionsisperformedinthe3Dthinplatemodel.Inthisstep,temperaturehistoriesateachelementnodesarecomputedduringtheweldingprocess.3D,8-nodes,linearbrickandheatelements(DC3D8)[12]areselectedforthethermalanalysis.TemperaturedependentphysicalpropertiesofthemildcarbonsteelasshowninFig.4[13]areemployedinheattransferanalysis.Inthisstudy,solid-statephasetransformationisneglectedbecausethein uenceofphasetransformationontheweldingdeformationandweldingresidualstressisinsigni cantinthelowercarbonsteel[14].Duringthewelding,thegoverningequationfortransientheattransferanalysisisgivenby:qc

ðx;y;z;tÞ¼ÀrÁqðx;y;z;tÞþQðx;y;z;tÞð1Þ

whereqisthedensityofthematerials[g/mm3],cisthespe-ci cheatcapacity[J/(g°C)],Tisthecurrenttemperature[°C],qistheheat uxvector[W/mm2],Qistheinternalheatgenerationrate[W/mm3],x,yandzarethecoordi-natesinthereferencesystem[mm],tisthetime[s],and$isthespatialgradientoperator.

Thenon-linearisotropicFourierheat uxconstitutiveequationisemployed:q¼ÀkrT

ð2Þ

wherekisthetemperature-dependentthermalconductivity[J/(mms°C)].

Inthisstudy,theheatfromthemovingweldingarcisappliedasavolumetricheatsourcewithadoubleellipsoi-daldistributionproposedbyGoldak[15],whichisexpressedbythefollowingequations:Forthefrontheatsource:

Qðx0;y0;z0;tÞ¼6p 3 ffQwÀ3x02=a2À3y02=b2À3z02=caee2

1eð3Þ

1bcppFortherearheatsource:

Qðx0;y0;z0;tÞ¼

6p 3 frQwÀ3x02=a2À3y02=b2a2bcpp

eeÀ3z02=c2

2eð4Þ

wherex0,y0andz0arethelocalcoordinatesofthedoubleellipsoidmodelalignedwiththeweldedpipe;ffandfrareparameterswhichgivethefractionoftheheatdepositedinthefrontandtherearparts,respectively.Becausethetemperaturegradientinthefrontleadingpartissteeperthaninthetailingedge,ffandfrareassumedtobe1.33and0.67,respectively.Qwisthepoweroftheweldingheatsource.Itcanbecalculatedaccordingtotheweldingcur-rent,thearcvoltageandthearce ciency.Thearce -ciencyg,isassumedtobe80%fortheGMAWweldingprocess.Theparametersa1,a2,bandcarerelatedtothecharacteristicsoftheweldingheatsource.The

parameters

国外关于焊接变形和参与应力分析的研究文献

356

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

Table2

ParametersoftheheatsourceParameterValue(mm)a12.0a24.0b1.2c

1.0

oftheheatsourcecanbeadjustedtocreateadesiredmeltedzoneaccordingtotheweldingconditions.Theval-uesoftheseparametersusedinthepresentsimulationaresummarizedinTable2.

WhenstructuralanalysisFEMcodessuchasABAQUSandMARCareusedtosimulatethetemperaturedistribu-tioninweldingprocess,the uid owandsolidi cationofmaterialintheweldpoolcannotbedirectlyconsideredbecausethecoupledproblembetweensolidandliquidisnotinvolvedinthesesoftwareatpresent.However,thee ectofthe uid owhassigni cante ectsonthetemper-aturedistributionandtheshapeofweldpool.Ifthee ectofthe uid owisneglected,thehighesttemperatureinweldpoolwillbeveryhigh.Accordingtotheauthor’sexperience,thepeaktemperatureinweldingpoolishigherthan3000°Cinsomecaseswhenthe uid owe ectisneglected.Thisphenomenonismuchdi erentfromtherealisticsituation.Okagaitoetal.[16]measuredthesur-facetemperaturedistributiononTIGweldpoolinSUS304steel.Theirresearchsuggeststhatthehighesttem-peratureonthemoltenpoolsurfaceisapproximately1750°C.Inthisstudy,toconsiderthe uid owanarti -ciallyincreasedthermalconductivityintheweldpoolisused.Thethermalconductivityisassumedtobetwiceaslargeasthevalueofroomtemperaturefortemperatureabovethemeltingpoint.

Thethermale ectsduetosolidi cationoftheweldpoolaremodeledbytakingintoaccountthelatentheatforfusion.Thevalueofthelatentheatis270J/g[17].Theliq-uidustemperatureTLandthesolidustemperatureTSareassumedtobe1500°Cand1450°C,respectively.

Heatlosses(qc)duetoconvectionareconsideredforallthesurfacesusingNewton’slaw:qc¼ÀhfðTsurÀT0Þ

ð5Þ

wherehfis lmcoe cientforconvection[W/(mm2°C)],Tsuristhesurfacetemperature[°C],andT0isambienttem-perature[°C].

Inthisstudy,atemperature-dependent lmcoe cient[18]isused,andtheambienttemperatureisassumedtobe20°C.

Radiationheatlosses(qr)areaccountedforallthesur-facesbyusingStefan–Boltzmanlaw:

qr¼ÀerðT4ÀT4

sur0Þ

ð6Þ

whereeisemissivity,risStefan–Boltzmanconstantforradiation.

Inthisstudy,theemissivityisassumedtobe0.2[19,20].

Theuser-de nedsubroutinestoABAQUScodeareuti-lizedintheheattransferanalysistomodelheat uxes,con-vectionandradiationboundaryconditions.3.2.Mechanicalanalysis

Thesame niteelementmodelsusedinthethermalanal-ysesareemployedinmechanicalanalyses,exceptfortheelementtypeandboundaryconditions.Therestraintcon-ditionsareshowninFig.3bythearrows.TheC3D8Iele-menttype[12]isusedtosimulatethestress–strain eld.Theanalysesareconductedusingthetemperaturehistorycalculatedbythethermalanalysesastheinputinformation.

Forthemildsteel,becausephasetransformationhasaninsigni cante ectontheweldingresidualstressandthedeformation,thetotalstraincanthereforebedecomposedintothreecomponentsasfollows:etotal¼eeþepþeth

ð7Þ

Thecomponentsontheright-handsideofEq.(7)corre-spondtoelastic,plasticandthermalstrain,respectively.

TheelasticstrainismodeledusingtheisotropicHooke’slawwithtemperature-dependentYoung’smodulusandPoisson’sratio.Fortheplasticstraincomponent,aplasticmodelisemployedwiththefollowingfeatures:theVonMisesyieldsurfaceandtemperature-dependentmaterialproperties.Fig.5showsthetemperaturedependentmechanicalproperties[13].Becausethee ectofworkhard-eningisnotsigni cantinmildsteel,itisneglectedinthisstudy.

Inthisstudy,thethicknessofthespecimenisonly1mm,soitcanbeexpectedthatthegeometricallynonlinearphe-nomenonprobablyoccurduringwelding.Toexaminethedi erencebetweenthenumericalresultscomputedbythe

国外关于焊接变形和参与应力分析的研究文献

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365357

largedeformationtheoryandthatcalculatedbythesmalldeformationtheory,boththetwomethodsareusedtopre-dictweldingdistortionandweldingresidualstress.Thus,inthemechanicalanalysisstage,twosimulationcases(caseAandcaseB)areperformed.ThelargedeformationtheoryisconsideredincaseA,whilethesmalldeformationtheoryisusedincaseB.3.3.Simulationresults

3.3.1.Characteristicsofweldingtemperature eld

Fig.6showsthetemperaturehistoriesatthecenteroftheweldingline.Fromthis gure,itcanbeseenthatthepeaktemperatureatthetopsurfaceoftheweldpoolisabout1800°C.Inthesame gure,itcanalsobeobservedthatthedi erencebetweenthepeaktemperatureatthetopsurfaceandthatatthebottomsurfaceisnotsigni -cant.Thethermale ectsduetosolidi cationoftheweldpoolareconsideredintheFEmodel,sothesetwocoolingcurvesre ectthesolidi cationphenomenaoftheweldpool.Exceptforthepeaktemperature,thetemperaturehis-torycurvesofthetopsurfaceandthebottomsurfaceinthefusionzonehavenodi erence.Fig.7showsthetempera-turehistoriesofthetopsurfaceandthebottomsurfaceintheheat-a ectedzone(HAZ)ofthemid-section.Thedistancebetweenthispositionandtheweldcenterlineis3.2mm.This gureindicatesthatboththetopsurfaceandthebottomsurfaceintheHAZhavealmostthesamepeaktemperatureandtheidenticalcoolingrate.IntheFEmodel,theplatethicknessofthefusionzoneislessthan2mm,andtheplatethicknessisonly1mminotherparts.Itisthesmallplatethicknessthatresultedinaneventem-peraturedistributionthroughthicknessduringwelding.3.3.2.Weldingresidualstress

Inthemechanicsanalysis,boththelargedeformationtheoryandthesmalldeformationtheoryareusedtosimu-

lateweldingresidualstressanddeformationinthebutt-weldedjoint.Fig.8showsthelongitudinalresidualstressdistributionsofthemiddlesection,whicharecomputedbythelargedeformationtheory(caseA).Fig.9showsthelongitudinalstressdistributionsofthemiddlesection,whichpredictedbythesmalldeformationtheory(caseB).Duetoarelativelylargeout-of-deformationaregener-atedafterweldingincaseA,thereisadi erencebetweenthelongitudinalstressofthetopsurfaceandthatofthebottomsurface.Onthecontrary,thereisalmostnodi er-encebetweenthelongitudinalstressofthetopsurfaceandthatofthebottomsurfaceincaseB.

Fig.10showsthetransverseresidualstressdistributionsofthemiddlesectionpredictedbycaseA.Thereisalsoasigni cantdi erencebetweenthetransverseresidualstressofthetopsurfaceandthatofthebottomsurface.Itisvery

国外关于焊接变形和参与应力分析的研究文献

358D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

clearthatinthefusionzoneandtheHAZthetransverseresidualstressofthetopsurfaceismuchlargerthanthatofthebottomsurface.Thedi erenceresultsfromthetrans-versebendingdeformation.Fig.11showsthetransverseresidualstressdistributionsofthemiddlesectioncomputedbyecaseB.This gureindicatesthatthereisalmostnodif-ferencebetweenthetopsurfaceandthebottomsurface.3.3.3.Weldingdeformation

Fig.12showsthecontoursofthede ectiondistributioncomputedbythelargedeformationtheory.Fromthis g-ure,itcanbeobservedthatalargelongitudinalbendingandatransversebendingareproducedafterwelding.Atthetwoendsofthegaps,themaximumde ectionisabout1.7mm.Fig.13showsthecontoursofthede ectiondistri-butionpredictedbythesmalldeformationtheory(caseB).

This gurealsore ectsthatbothlongitudinalbendingandtransversebendingaregeneratedinthebutt-weldedjoint.ThedeformationmodeissimilartocaseA,howeverthemagnitudesofde ectionaremuchsmallerthancaseA.Fig.14showsthede ectiondistributionsofcaseAandcaseBalongthemiddleline(lineAB),whichisde nedinFig.12.Inthis gure,pointAshowninFig.12isassumedtobetheorigin.Theexperimentalmeasurement(de ection)atthecenteroftheweldinglineisalsoplottedinthesame gure.Itisclearthatthede ectionatthecenteroftheweldinglinepredictedbycaseAismuchclosetotheexperimentalvalue.Fromthesame gure,itcanbealsoknownthattheexperimentalvalueissigni cantlylargerthanthenumericalresultcomputedbycaseB.Thisinfor-mationsuggeststhatwhenweldingdeformationofathinplateweldedjointorstructureissimulatednumericallyitisnecessarytoconsidergeometricallynonlinearphenome-non.Otherwise,http://paringthetransverseshrinkageofthetopsurfacewiththatofthebottomsurface,itcanbeobservedthattheformerissmallerthanthelatter.How-ever,thedi erenceisverysmall.Fig.16showstheY-direc-tionaldisplacementdistributionofthemiddlesectioncomputedbythesmalldeformationtheory.This http://paringFig.15withFig.16,itcanbeconcludethattheY-directionaldisplacementspredictedbycaseAareclosetothosecomputedbycaseBonthewhole.

3.3.4.Plasticstraindistribution

Fig.17showstheplasticstraindistributionsinthelon-gitudinaldirection(weldingdirection)ofthemiddlesec-tion.Theplasticstrainvaluesaretheaverageonesthroughthicknesspredictedbythelargedeformationthe-

国外关于焊接变形和参与应力分析的研究文献

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365359

Fig.12.De ectiondistributionofCase

Fig.13.De ectiondistributionofCaseB.

oryandthesmalldeformationtheory.Fromthis gure,itcanbeobservedthattheplasticstraindistributionspre-dictedbythetwocaseshavenosigni cantdi erence.Boththerangeofthelongitudinalplasticstrainvaluesandthedistributionshapesofthesetwocasesarefairlyclose.Fig.18showstheplasticstraindistributionsinthetrans-versedirectionofthemiddlesectioncomputedbyCaseAandCaseB.This gureindicatesthatthetransverseplasticstraindistributionsofthetwocasesarequitesimilar.

FromFigs.17and18,itisknownthattherangeofthelongitudinalplasticstraindistributionislargerthanthatofthetransverseplasticstraindistribution.ThetransverseplasticstrainrangeconcentratesalmostonlyinthefusionzoneandtheHAZ,whilethelongitudinalplasticstrainrangedistributesinarelativelylargerange.Generally,therangeandmagnitudeoftheplasticstraincomponentaremainlygovernedbythepeaktemperatureandtherestraintconditions[21].Duringwelding,becausetherestraintintensityinthelongitudinaldirection(weldingdirection)islargerthanthatinthetransversedirection,therangeofthelongitudinalplasticstrainisrelatively

large.

国外关于焊接变形和参与应力分析的研究文献

360D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

3.3.5.Discussions

Basedonthesimulationresults,itisknownthatboththelongitudinalplasticstrainandthetransverseplasticstraindistributenarrowlyinthefusionzoneanditsvicin-ity.Formildsteel,theplasticstrainsaremainlygovernedbythethermo-mechanicalbehavioroftheweldmetalandthebasemetalnearthefusionzoneduringwelding,sothemagnitudesanddistributionspredictedbythelargedeformationtheoryandthesmalldeformationtheoryarefairlysimilar.Onthecontrary,the naldeformationofthethinplatebutt-weldedjointshowsasigni cantlyglobalcharacteristic.Itisveryclearthateventhoughthelargedeformationtheoryandthesmalldeformationtheorypre-dictsimilarplasticstrainsinthebutt-weldedjoint,how-ever,the nalweldingdeformationsespeciallytheout-of-

deformation(de ection)aresigni cantdi erent.Therea-sonisthatbeyondtheplasticstrainzonetheweldingdefor-mationismainlygovernedbytheelasticstrainandthestrain–displacementrelationship.Bycomparingwiththeexperiment,itisclearthatthede ectionatthecenteroftheweldinglinepredictedbythelargedeformationtheoryisveryclosetotheexperimentalmeasurement.Theresultsimulatedbythesmalldeformationtheoryismuchsmallerthanthemeasurement.Thisindicatesthatwhenthethermo-elastic–plasticFEMisusedtopredicttheweldingdeformationinathinplatestructurethegeometricallynon-linearphenomenonshouldbecarefullyconsidered.Itisveryinterestingtonotethateventhoughthemagnitudesoftheweldingdeformationpredictedbythetwotheoriesaremuchdi erent,however,thedeformationmodesaresimilar.BothFigs.12and13showthatthede ectiondis-

国外关于焊接变形和参与应力分析的研究文献

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365361

tributionofthebutt-weldedjointhasasaddle-shapedmode,whichisgenerallyoppositetothatofafullweldingbutt-weldedjoint[13].Becauseapartiallyweldingisper-formedinthebuttjoint,thegapsremainedatthetwoendscausedthesaddle-shapedmodeafterwelding.

Becauserelativelylargelongitudinalbendingandtrans-versebendingispredictedbythelargedeformationtheory,boththelongitudinalstressandthetransversestressofcaseAhavesigni cantgradientdistributionsthroughthickness.Onthecontrary,asmallout-of-deformationissimulatedbythesmalldeformationtheory,soeitherthelongitudinalstressorthetransversestressofcaseBhasalmostnogra-dientthroughthicknessinthebutt-weldedjoint.4.Predictionofweldingdeformationusinginherentstrainmethod

4.1.Inherentstrain

Althoughthethermo-elastic–plasticFEMcanbeusedtosimulateweldingtemperature eld,weldingresidualstressandweldingdeformation,averylongcomputationaltimeisneededbecausetheweldingmechanicalbehaviorishighlynonlinearproblemincludingmaterialnonlinearity,geometricalnonlinearityandsometimescontactnonlinear-ity.Besidesthethermo-elastic–plasticFEM,elasticFEMbasedoninherentstraintheory[21–26]http://paringwiththethermo-elastic–plasticFEM,onlyaveryshortcomputa-tionaltimeisneededtocompletethesimulationevenforalargeandcomplexstructure.Moreover,onlytheelasticmodulusandthePossion’sratioatroomtemperatureareusedintheelasticFEM,andthetemperaturedependentmaterialpropertiesarenotneeded.Inthisstudy,theinher-entstrainmethodisemployedtosimulatetheweldingdeformationinthethinplatebutt-weldedjoint.

Basedonexperimentalobservationsandtheoreticalanalysis,itisfoundthatthetotalweldingdistortionofaweldjointismainlyproducedbyfourcomponents,namelylongitudinalshrinkage(dx),transverseshrinkage(dy),lon-gitudinalbending(hx)andtransversebending(angulardis-tortionhy).Thefourfundamentaldeformationcomponentsarealsocalledinherentdeformations[10].Accordingtothenumericalresultsobtainedbythethermo-elastic–plasticFEM,thefourinherentdeformationcomponentsinacross-sectionofthebutt-weldedjointcanbecalculatedusingthefollowingequations[8–11]:

Zx¼epdydzð8ÞdZx1y¼hep

ydydzð9Þh12

Zx¼h3epxðzÀh=2Þdydzð10Þh12

Zy¼h3epyðzÀh=2Þdydzð11Þwhereepxistheplasticstrainintheweldingdirection(lon-gitudinaldirection);epdirection;histhethicknessyistheplasticstraininthetransverseoftheplate,andzisthecoor-dinateinthethicknessdirection.

Thesourceinducedweldingdeformationandweldingresidualstressiscalledinherentstrain[22].Whentheinher-entdeformationsofaweldjointareknown,theycanbetransferredintothecorrespondinginherentstraincompo-nents[26].WhentheinherentstrainsareintroducedintoanelasticFEM,itisusuallyassumedthateachcomponenthasauniformdistributionalongtheweldingline.Therefore,theaveragevaluesofaweldjointareoftenusedtoapprox-imatelyrepresenttheinherentstrainsforthewholejoint.Theaveragevaluesofthefourinherentdeformationsinthebutt-weldedjointcanbecalculatedusingthefollowingformulae:

d1ZLx¼Ldxdxð12Þw d1Z0

Ly¼Ldydxð13Þw h1Z0

Lx¼Lhxdxð14Þw0

h1ZLy¼Lhydxð15Þw0whereLwisthelengthoftheweldingline,Listhelengthofthespecimen.

4.2.Elastic niteelementbasedoninherentstrain

IntheelasticFEM,the4-nodeplateisused.Forthinplatedeformation,de ectionw(x,y)isassumedtobeequaltothede ectionofmid-plane,w0(x,y).Whenlargedeformation(geometricallynonlinearphenomenon)iscon-

国外关于焊接变形和参与应力分析的研究文献

362D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

sidered,thestrain–displacementrelationscanbede nedasfollows:

"eibou01 ow 2# o2w x¼exþex¼oxþ00

2oxþÀzox2ð16Þ"e¼eiov01 o 2# 2 yyþeby¼oyþw02oyþÀzow0

oy2ð17Þc¼ci ou0ov0ow0ow0 o2w0

xyxyþcbxy¼oyþoxþoxoyþÀ2zoxoyð18Þwhereu0,v0andw0arethedisplacementatmid-plane,ex,ey

andcxyaretotalstrains,andei;ex;eiyandcixyarein-plane

strains,andebThecurvaturexbyandcb

(jxybendingstrains.

x)inaplaneparalleltothex–zplaneandthecurvature(jy)inaplaneparalleltothey–zplaneandthetwistingcurvature(jxy),whichrepresentsthewarpingofthex–yplane,canbede nedasfollows:o2jw0

x¼Àox2

ð19Þj¼Ào2w0

yoy2

ð20Þ

jÀ

o2w0

xy¼oxoy

ð21Þ

Fig.19showsschematicallyabutt-weldedjoint,theaveragelongitudinalshrinkage(d joint.Inthis

x)canbetrans-formedintothein-planestraincomponent(eÃdinaldirection;theaveragetransverseshrinkagex)inlongitu-(d y)canbechangedintothein-planestraincomponentðeÃtransversedirection;theaverageangulardistortionyÞðh

yÞcanbeconvertedintothecurvatureðjÃandtheaveragelongitudinalbendingyÞðh alongthey-axis;

xÞcanbetrans-formedintothecurvatureðjÃxÞalongthex-axis.TheseinherentstraincomponentsareintroducedintotheelasticFEMasinitialstrains,andthetotalweldingdistortioncanbeestimatedthroughelastic niteelementanalysispro-cedure[26]asshowninFig.20.

InFig.19,{e*},{e},{ee},{f},{u}and{r}arevectorsofinherentstrain,totalstrain,elasticstrain,equivalentnodalload,nodaldisplacement,andresidualstress,and[B],[D],and[K]arestrain–displacementmatrix,elasticstress–strainmatrixandsti nessmatrix,respectively.

4.3.Elastic niteelementmodel

Theelastic niteelementmodel,theareaintroduced

inherentstrains,andtheinitialgapsareshowninFig.21.Accordingtothesimulationresultscomputedbythethermoelastic–plasticFEMtherangeofthe

longitudinal

Fig.21.TheFEmodel,theareaintroducedinherentstrainsandtheinitialgap.

国外关于焊接变形和参与应力分析的研究文献

D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365363

plasticstrainislargethanthatofthetransverseplasticstrain.Intheelasticanalysis,itisassumedthatallthefourinherentstraincomponentsareintroducedintothesamearea.Thefouraverageinherentstraincomponentsaredeterminedbythefollowingequations:

eÃx¼dx=Bis

dy=BiseÃ¼

ð22Þð23Þð24Þð25Þ

hÃx¼hx=Bis

hÃy¼hy=Bis

whereBisisthebreadthoftheelementsinwhichtheinher-entstrainsareintroduced.

4.4.Simulationresultsanddiscussions

Fig.22showsthecontourof nalde ectiondistribu-tion.BycomparingwithFig.11,itcanbeconcludedthatthede ectiondistributionpredictedbytheelasticFEMmatchesthatcomputedbythethermo-elastic–plasticFEMwell.Fig.23showsthede ectiondistributionalongthelineABasde nedinFig.20.Thecorrespondingde ec-tiondistributionpredictedbythethermo-elastic–plasticisalsoplottedinthesame gure.Itisclearthatthedi erencebetweenthetwocurvesisverysmall.

Fig.24showstheY-directionaldisplacementdistribu-tionsalonglineABpredictedbytheelasticFEMandthethermo-elastic–plasticFEM.Theaveragevaluesofthetopsurfaceandthebottomsurfacesimulatedbythethermo-elastic–plasticFEMareplottedinthis gure.This guremeansthatthetwodistributedcurvesarequitesim-ilar.Throughcarefullyobservation,itcanbefoundthat

thetransverseshrinkagepredictedbythethermo-elastic–plasticFEMisslightlylargerthanthatestimatedbytheelasticFEM.IntheelasticFEmodel,theaveragetrans-verseinherentstrainofthewholejointisused.Inthethermo-elastic–plasticFEmodel,thetransverseshrinkageshavenon-uniformdistributionalongtheweldinglineandthemaximumvalueisatthecomparedlocation(lineAB).Thesetwofactorsaremainreasonswhichresultedinaslightlydi erencebetweenthetwocurvesshowninFig.24.

WhentheelasticFEMisusedtopredictweldingdefor-mation,theattentionismainlypaidtothetotal

deforma-

Fig.22.De ectiondistributionpredictedbyelasticFEM.

国外关于焊接变形和参与应力分析的研究文献

364D.Deng,H.Murakawa/ComputationalMaterialsScience43(2008)353–365

tionratherthanthelocaldeformation.Fromthisview-point,itcanbeconcludedthatthetransverseshrinkagepredictedbythelargedeformationelasticFEMmatchesthatcomputedbythethermo-elastic–plasticFEMwellonthewhole.

FromFigs.12,22,23and24,itcanbeconcludedthattheelasticFEmodelhasaccuratelyreproducedtheweldingdeformation,whichissimulatedbythethermo-elastic–plasticFEmodel.Basedonthepresentstudy,itcanbeinferredthatwhentheinherentstrainsofeachjointinvolvedinathinplatestructureareknown,theproposedelasticFEMcane http://paringthecomputationaltime,itisfoundthatthecomputationaltimetocompletetheelasticFEanalysisisfarshorterthanthatusedbythethermo-elastic–plasticFEanalysis.Inthepresentstudy,thetotalcomputationaltimeofboththethermalanalysisandthethermo-mechan-icalanalysisisapproximately12hoursforthethermo-elas-tic–plasticFEmodel,whereasthecomputationaltimeoftheelasticFEmodelisshorterthan1min.Ontheaspectofcomputationaltime,theelasticFEMhasasigni cantadvantageoverthethermo-elastic–plasticFEM.Thus,thismethodisapromisingalternativetopredictweldingdefor-mationforpracticallargethin-plateweldedstructures.

5.Conclusions

Inthisstudy,thethermo-elastic–plasticFEMisusedtosimulatetheweldingtemperature eld,residualstressanddistortioninathinplatebuttjoint.Meanwhile,experi-mentsarecarriedouttomeasuretheweldingdeformation.Moreover,theelasticFEMbasedoninherentstraintheoryisalsoutilizedtosimulateweldingdeformationinthesamebuttjoint.Accordingtothenumericalandexperimentalresults,thefollowingconclusionscanbedrawn.

(1)Basedonthesimulationresults,itisknownthatthere

almostisnotemperaturegradientthroughthicknessinthethinplatebuttjointduringwelding.

(2)Althoughthelargedeformationtheoryandthesmall

deformationtheorypredictthesimilarplasticstrains,however,the nalweldingdeformationsespeciallytheout-of-deformationcomputedbythetwometh-odsaremuchdi erent.Bycomparingwithexperi-ment,itisclearthatthenumericalresultcalculatedbythethermo-elastic–plastic,largedeformationFEMisinagoodagreementwiththemeasurement.Therefore,topreciselypredictweldingdeformationinathinplatestructure,itisnecessarytoconsiderthegeometricallynonlinearproblem.

(3)Becauseofrelativelylargelongitudinalbendingand

transversebendingdeformations,boththelongitudi-nalandthetransverseresidualstressespredictedbythethermo-elastic–plastic,largedeformationFEMhaveagradientthroughthickness.

(4)TheelasticFEMwithconsideringlargedeformation

canbeusedtopredictpreciselyweldingdeformationinthethinplatebuttjoint.Moreover,thecomputa-tionaltimeismuchshorterthanthatusedinthethermo-elastic–plasticFEM.Fortheautomotiveindustryapplication,theelasticFEMbasedontheinherentstraintheoryisapromisingmethodtopre-dictweldingdeformation.

References

[1]G.Verhaeghe,PredictiveFormulateForWeldDistortion–ACriticalReview,AbingtoPublishing,2000.

[2]D.Radaj,WeldingResidualStressandDistortionCalculationandMeasurement,WoodheadPublishingLtd.,DVSVerlag,2003.

[3]D.Deng,H.Murakawa,Y.Ueda,InternationalJournalofO shoreandPolarEngineering14(2)(2004)138–144.

[4]G.HJung,C.L.Tsai,WeldingJournal83(6)(2004)177s–187s.

[5]ZhiLiFeng,ProcessesandMechanismsofWeldingResidualStressandDistortion,WoodheadPublishingLtd.,Cambridge,UK,2005.[6]L.-E.Lindgren,ComputerMethodsinAppliedMechanicsandEngineering195(48–49)(2006)6710–6736.

[7]D.Deng,W.Liang,H.Murakawa,JournalofMaterialsProcessingTechnology183(2-3)(2007)219–225.

[8]W.Liang,S.Sone,M.Tejima,H.Serizawa,H.Murakawa,TransactionsofJWRI33(2004)45–51.

[9]W.Liang,D.Deng,H.Murakawa,TransactionsofJWRI34(1)(2005)113–123.

[10]W.Liang,D.Deng,S.Sone,H.Murakawa,WeldingintheWorld49

(11/12)(2005)30–39.

[11]R.Suzuki,T.Nakano,KobeSteelEngineeringReports52(3)(2002)

74–78.

[12]ABAQUS/Standard,vols.1,2and3,Version6.4,Hibbitt,Karlsson

&SorensenInc.,2003.

[13]D.Deng,Theoreticalpredictionofweldingdistortioninthincurved

structureduringassemblyconsideringgapandmisalignment,Doc-toralThesis,OsakaUniversity,2002.

[14]D.Deng,Y.Luo,H.Serizawa,M.Shibahara,H.Murakawa,