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IFTVOCEMIXEDHARDENINGLAWOFEQ2XPARAMETER10MKSATIA,MKJAINTHINWALLEDSTRUCTURES1272018500515503GEOMETRYOFTOOLING,SHEETTHICKNESS,SHEETELASTICANDPLASTICPROPERTIES,SHEETSHEETCONTACT,ANDFRICTIONBETWEENALLTOOLANDSHEETCONTACTINGSURFACESTHEABOVEGEOMETRIC,MATERIALANDPROCESSPARAMETERSCOULDBESTUDIEDBYSYSTEMATICALLYVARYINGTHEIRVALUESANDOBSERVINGTHEMATERIALOWANDTHEEVOLUTIONOFTHESTRESSSTATEOFTHESHEETWITHTHEPROGRESSIONOFTHEPUNCHGENERALLY,SUCHDETAILSAREQUITEDICULTTOSTUDYEXPERIMENTALLYDUETOVERYSMALLCLINCHINGAREA,NOTAMENABLETOCAMERAVIEWING,ANDTHEHIGHSPEEDOFTHEPROCESSTHECOMMERCIALFECODEABAQUSEXPLICITWASUSEDTOSIMULATETHEDIELESSCLINCHINGPROCESSTHISCODEWASPREFERREDOVEROTHERAVAILABLECODESDUETOITSABILITYTOSIMULATEMULTIPLETOOLSHEETANDSHEETSHEETINTERACTIONSTHATOCCURINCLINCHINGANDITSMOREROBUSTCONTACTALGORITHMANDSOLVERALSO,THEEXPLICITSOLVEREXHIBITSSIGNICANTTIMESAVINGOVERIMPLICITSOLVERASTHEMODELSIZEINCREASESANAXISYMMETRIC2DDIELESSCLINCHINGMODELWASDEVELOPEDDUETOTHEAXISYMMETRICNATUREOFTOOLING,THECLINCHGEOMETRY,ANDTHELOADINGCONDITIONSTHETOOLING,WITHTHEEXCEPTIONOFTHEURETHANESPRING,WASCONSIDEREDASRIGIDANDTHEBOUNDARYCONDITIONSWEREDENEDBYTOOLMOTIONSANDCONSTRAINTSTWODIERENTBLANKHOLDERTYPESWERECONSIDEREDINTHEANALYSIS,DENEDASSTEPPEDEDGEANDROUNDEDEDGEBLANKHOLDERS,ASSHOWNINFIG5,WHEREGROOVERADIUSRANDDEPTHT,ASRSTANDSECONDNUMBERSRESPECTIVELY,WEREUSEDTOREPRESENTVARIOUSSTEPPEDEDGEBLANKHOLDERSSIMILARLY,FORTHEROUNDEDEDGEBLANKHOLDER,RADIUSRANDLLETRADIUSSWEREUSEDASRSTANDSECONDNUMBERSRESPECTIVELYINTABLE2THESHEETMATERIALSWEREMODELEDUSING4NODEBILINEARAXISYMMETRICQUADRILATERALELEMENTSWITHREDUCEDINTEGRATIONCAX4RELEMENTINABAQUSEXPLICITANDENHANCEDHOURGLASSCONTROLOPTIONWASUTILIZEDTOAVOIDHOURGLASSINGPROBLEMSATLARGEPLASTICSTRAINSALARGENUMBEROFELEMENTSTHROUGHTHESHEETTHICKNESS,40INTOTAL,WEREUSEDTOMESHTHESHEETMATERIALSINTHECLINCHEDAREATOACCOUNTFORLARGETHROUGHTHICKNESSSTRAINGRADIENTPRESENTINTHEDIELESSCLINCHINGPROCESSALSO,ANASPECTRATIOOF13WASUTILIZEDTOMINIMIZEELEMENTDISTORTIONWHEREASPECTRATIOWASDENEDBYTHERATIOOFTHEELEMENTSIZEINAXIALANDRADIALDIRECTIONSTHEFEMESHSIZEINTHEAREAOUTSIDETHECLINCHINGREGIONWASKEPTRELATIVELYLARGE,TOTALING30ELEMENTSTHROUGHTHETHICKNESSWITHANASPECTRATIOOF025DUETOLARGEDEFORMATIONANDHIGHLYLOCALIZEDPLASTICSTRAININTHECLINCHREGION,ADAPTIVEREMESHINGOPTIONWITHVOLUMETRICSMOOTHINGWASUSEDTOPREVENTEXCESSIVEMESHDISTORTION,ANDANADVANCEDALGORITHMBASEDONEVOLVINGGEOMETRYWASIMPLEMENTEDTOATTAINSTABLENUMERICALSOLUTIONAMASSSCALINGFACTOROF100WASUSEDTOREDUCETHECOMPUTATIONALRUNTIMETHESHEETMATERIALWASASSUMEDTOBEISOTROPICELASTICPLASTICFOLLOWINGVONMISESYIELDCRITERIONANEXTENDEDVOCEHARDENINGLAWWASUSEDTOEXTRAPOLATETHEEXPERIMENTALPLASTICSTRESSPLASTICSTRAINDATAOBTAINEDFROMTHEUNIAXIALTENSILETESTBEYONDTHETENSILESTRESSSTRAINLIMITTOLARGESTRAINSCMEXPK1EQEQPEQP01WHERE,EQ,0ANDEQPAREEQUIVALENTSTRESS,INITIALYIELDSTRESSANDEQUIVALENTPLASTICSTRAINRESPECTIVELY,KISSTRESSCOECIENT,ANDCANDMARETWOOTHERMATERIALCONSTANTSTHETHIRDTERMONTHERIGHTSIDEOFEQ1,EQP,WASADDEDTODESCRIBECONTINUEDHARDENINGBEHAVIORATLARGERSTRAINS25THECOMMONLYUSEDSWIFTANDVOCEHARDENINGLAWSHAVEBEENFIG9ACOMPARISONOFEXPERIMENTALANDFEPREDICTEDSHAPEPROLESFORASTEPPEDEDGEBLANKHOLDER5/04ANDBSTEPPEDEDGEBLANKHOLDER5/02XPARAMETER10FIG10ACOMPARISONOFEXPERIMENTALANDFEPREDICTEDINTERLOCKREGIONSFORASTEPPEDEDGEBLANKHOLDER5/04ANDBSTEPPEDEDGEBLANKHOLDER5/02XPARAMETER10MKSATIA,MKJAINTHINWALLEDSTRUCTURES1272018500515504FOUNDTOBEINADEQUATEINREPRESENTINGTRUESTRESSSTRAINBEHAVIOROFMANYALUMINUMSHEETSMATERIALSATLARGESTRAINS26THEREFORE,AMODICATIONBASEDONUSEOFAWEIGHTFUNCTIONBETWEENTHESWIFTANDVOCEHARDENINGLAWSWASUSEDINTHISSTUDYASSHOWNBELOW|1|EQEQSWIFTEQVOCE2WHEREVOCECMEXPK1EQEQP1113ANDSWIFTKEQEQPN204INTHEABOVEEQUATIONS,K1ANDK2ARESTRENGTHCOECIENTSINEQS3AND4RESPECTIVELY,NISTHESTRAINHARDENINGEXPONENT,CANDMARETWOOTHERMATERIALCONSTANTS,ISAWEIGHTOFTHEFUNCTIONWITHVALUESINTHERANGE02509ITISTOBENOTEDTHAT,FORVALUESOF0AND1,THEMIXEDHARDENINGLAWOFEQ2REVERTSSIMPLYTOVOCEANDSWIFTLAWSRESPECTIVELYTABLE3PRESENTSTHEVALUESOFHARDENINGPARAMETERSFORAA7075OSHEETFIG6SHOWSTHEEXTRAPOLATIONOFSTRESSSTRAINCURVESTOLARGEPLASTICSTRAINSBYUSINGDIERENTWEIGHTFUNCTIONSINEQUATIONASWELLASANOTHEREXTENDEDVOCELAWTHEEXPERIMENTALLYMEASUREDCLAMPINGFORCEVERSUSPUNCHDISPLACEMENTPROLEASWELLASTWOOTHERTHEORETICALCLAMPINGFORCESWEREEMPLOYEDINTHEFESTUDYSEEEARLIERFIG4THEINTERACTIONBETWEENTOOLANDSHEETANDBETWEENTHETWOSHEETMATERIALSWASDESCRIBEDBYCOULOMBFRICTIONLAWFRICTIONCOECIENTVALUESOF015AND04WERECHOSENFORFRICTIONBETWEENSHEETTOOLSANDBETWEENTHETWOSHEETSRESPECTIVELYFROMTHELITERATURE27,28FIG7ILLUSTRATESTHEAXISYMMETRICMODELOFTHEDIELESSCLINCHINGSIMULATIONTHATALSOINCLUDESINPUTTHEAPPLIEDBLANKHOLDERCLAMPINGFORCEVERSUSTIMEPROLEATTHETOPFIG11EXPERIMENTALANDCORRESPONDINGFEPREDICTEDTHROUGHTHICKNESSCLINCHEDSHAPESFORASTEPPEDEDGEBLANKHOLDER5/04ANDBSTEPPEDEDGEBLANKHOLDER5/02XPARAMETER10FIG12ACOMPARISONOFPUNCHFORMINGFORCEVERSUSDISPLACEMENTTRACESFROMEXPERIMENTANDFESIMULATIONFORSTEPPEDEDGEBLANKHOLDER5/04XPARAMETER10FIG13FLOWSTRESSCURVESOFAA7075OSHEETATCROSSHEADSPEEDOF01AND10MM/MINMKSATIA,MKJAINTHINWALLEDSTRUCTURES12720185005155054RESULTANDDISCUS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