<%BANNER%>

Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.

Permanent Link: http://ufdc.ufl.edu/UFE0021240/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Soare, Stefan C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Stefan C Soare.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Cazacu, Oana.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021240:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021240/00001

Material Information

Title: Record for a UF thesis. Title & abstract won't display until thesis is accessible after 2008-02-29.
Physical Description: Book
Language: english
Creator: Soare, Stefan C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Statement of Responsibility: by Stefan C Soare.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Cazacu, Oana.
Electronic Access: INACCESSIBLE UNTIL 2008-02-29

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021240:00001


This item has the following downloads:


Full Text

PAGE 1

1

PAGE 2

2

PAGE 3

Ithankmycommitteemembersfortheirkindpatienceandunderstanding.IthankDrYoonfromAlcoaforhisconstantencouragements.ThereweremanydicultmomentsIcouldnothaveovercomewithouthissupport.IthankmycolleaguesMikeNixon,BrianPlunkettandJoelStewartfortheirfriendship,collaborationandunderstanding. 3

PAGE 4

page ACKNOWLEDGMENTS ................................. 3 LISTOFTABLES ..................................... 6 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 12 1.1YieldSurface .................................. 13 1.2IdenticationofMaterialProperties ...................... 18 1.3AReviewofSomeOrthotropicYieldFunctions ............... 21 1.3.1Hill'sQuadraticandOtherAttempts ................. 22 1.3.2Gotoh'sFourthOrderPolynomial ................... 24 1.3.3ASixthOrderPolynomial:CB2001 .................. 25 1.3.4AMoreDirectApproach:BBC2003 .................. 28 1.3.5TheUseofLinearTransformationstoModelAnisotropy ...... 28 1.4OutlineoftheDissertation ........................... 34 2FOURTH,SIXTHANDEIGHTHORDERHOMOGENEOUSPOLYNOMIALS 38 2.1Convexity .................................... 38 2.2FourthOrderPolynomialCriterionforPlaneStressStates ......... 40 2.2.1BoundsonCoecientsandData .................... 40 2.2.2IdenticationProcedureforPoly4 ................... 43 2.2.3ApplicationstotheModelingofTwoAluminumAlloysandofaMildSteel ................................ 47 2.3SixthandEighthOrderPolynomials(PlaneStress) ............. 49 2.4Extensionsto3DStressStates ......................... 54 2.5ApplicationstoDeepDrawing ......................... 57 2.5.1TheNumisheet'93SquareCupTest .................. 58 2.5.2PredictionofEaringinCylindricalCups ............... 59 2.6DiscussionandFurtherExamples ....................... 61 3AMOREGENERALAPPROACH:CRITERIAWITHTENSION-COMPRESSIONSYMMETRY/ASYMMETRY ............................ 79 3.1AGeneralFormulation ............................. 79 3.2ModelingofPlasticPropertieswithStrengthDierentialEect ...... 82 3.3ImplementationDetails ............................. 87 4

PAGE 5

103 4.1MarciniakandKuczynskiModel ........................ 105 4.2IntegrationoftheM-KEquations ....................... 109 4.2.1ComputingtheStressinZoneA .................... 109 4.2.2ComputingtheStrainsinZoneB ................... 111 4.3ApplicationstoTwoAluminumAlloys .................... 116 4.3.1AA5182-O:AFirstExperimentalDataSet .............. 116 4.3.2AA5182-O:ASecondExperimentalDataSet ............. 120 4.3.3AA3104-H19 ............................... 124 APPENDIX ASOMEALGEBRAICDETAILSREGARDINGTHECONVEXITYOFPOLY4 142 BPOLY4INITIALGUESSFORPOLY8 ....................... 145 CRETURNMAPPINGALGORITHM ........................ 147 DTHEJACOBIANFORTHEALGEBRAICSYSTEMOFZONEB ....... 159 REFERENCES ....................................... 160 BIOGRAPHICALSKETCH ................................ 165 5

PAGE 6

Table page 2-1ExperimentaldataforAA2090-T3,AA2008-T4andNUM'93mildsteel. 67 2-2Poly4coecientsforAA2090-T3,AA2008-T4andNUM'93mildsteel. 67 2-3Poly6coecientsforisotropicMises,AA2090-T3,Mat1,andMat2. 67 2-4Poly8coecientsforisotropicMisesandAA2090-T3.Forbothwehavea1=1:0000. .................................. 68 2-5Draw-inpredictionsofPoly4FEsimulationandexperimentaldata. 68 3-1Poly8coecientsforthetwomaterialsMat1andMat2. 99 3-2Poly7coecientsforthetwomaterialsMat1andMat2. 99 4-1ExperimentaldataforAA5182-O,(B), Banabicet.al ( 2005b ),AA5182-O,(W), Wuetal. ( 2003 ),andAA3104-H19.Notethatforallthreecasesonlydataat0o,45oand90oisexperimental.ThedatainbetweenwasgeneratedwithPoly4.Thebiaxialyieldstrengthandr-valuedata(whereavailable)isexperimental. .. 127 4-2Poly4coecientsforAA5182-O(B),AA5182-O(W),andAA3104-H19 127 4-3Poly6coecientsforAA5182-O(B),AA5182-O(W1),AA5182-O(W2),AA3104-H19(1)andAA3104-H19(2) ............................ 127 4-4Poly8coecientsforAA5182-O(W1),AA5182-O(W2),AA3104-H19(1)andAA3104-H19(2).Forallwehavea1=1:0000. ................... 128 6

PAGE 7

Figure page 1-1Geometricalsettingoftheuniaxialtestsforwhichdirectionalyieldstrengthandr-valuearemeasured/predicted. ........................... 36 1-2Poly4materialcharacterizationofAA2090-T3usingtheidenticationprocedurein Gotoh ( 1977 ).Directionalyieldstrengthandr-value. .............. 36 1-3ProjectiononthebiaxialplaneofthePoly4yieldsurfaceforAA2090-T3,andthePoly4biaxialyieldcurveforthesamematerial.Althoughthebiaxialyieldcurveisconvex,theyieldsurfaceisnot. ...................... 37 1-4YieldcurvesinthebiaxialplaneaccordingtoHosford'sisotropiccriterion. ... 37 2-1ProjectionsonthebiaxialplaneofthePoly4yieldsurfaceforAA2090-T3andAA2008-T4 ...................................... 69 2-2Poly4materialcharacterizationofAA2090-T3.Directionalyieldstrengthandr-value;comparisonwithYld96isalsoshown. ................... 69 2-3Poly4materialcharacterizationofAA2008-T4.Directionalyieldstrengthandr-value;comparisonwithYld96isalsoshown. ................... 70 2-4Poly4materialcharacterizationoftheNUM'93mildsteel.Directionalyieldstrengthandr-value. ...................................... 70 2-5ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA2090-T3. .......................................... 71 2-6Poly6andPoly8materialcharacterizationofAA2090-T3.Directionalyieldstrengthandr-value;comparisonwithYld2004isalsoshown. ............... 71 2-7Tooldimensionsforthesquarecupproblem. .................... 72 2-8Proleofthedrawnsquarecup,at40mmpunchstroke,anddenitionsofdraw-in(dimensionsinmm). ................................. 72 2-9ThicknessstrainvariationalongtheOBdiagonalaspredictedbyFEsimulationswithPoly4andVonMises,andasmeasuredat15mmpunchstroke. ...... 73 2-10ThicknessstrainvariationalongtheOBdiagonalaspredictedbyFEsimulationswithPoly4andVonMises,andasmeasuredat40mmpunchstroke. ...... 73 2-11Geometricalsettingandtooldimensionsforthedeepdrawingwithcylindricalpunchsimulation. ................................... 74 2-12Typicalmeshusedontheblank. .......................... 74 7

PAGE 8

............ 75 2-14ProlesofAA2090-T3drawncupssimulatedwithPoly4,2Dand3D.Yld96simulationisalsoshown,dataafter Yoonetal. ( 2006 ). .............. 75 2-15ProlesofAA2090-T3drawncupssimulatedwithPoly6-3DandPoly8-3D.TheYld2004simulationisalsoshown,withdataafter Yoonetal. ( 2006 ). ...... 76 2-16EvolutionoftheproleofthecupduringthedrawingoftheAA2090-T3blank,simulatedwithPoly6. ................................ 76 2-17ProjectionsonthebiaxialplaneofthePoly6yieldsurfacesforMat1andMat2. 77 2-18Poly6materialcharacterizationofMat1andMat2.Directionalyieldstrengthandr-value. ...................................... 77 2-19ProlesofctitiousmaterialMAT1cupaspredictedbyPoly6(FEM),andbyanalyticalformula( 2{78 )proposedby Yoonetal. ( 2006 ).Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. ................. 78 2-20ProlesofctitiousmaterialMAT2cupaspredictedbyPoly6(FEM),andbyanalyticalformula( 2{78 )proposedby Yoonetal. ( 2006 ).Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. ................. 78 3-1Poly87materialcharacterizationofMat1.Directionalyieldstrengthintensionandcompression,andr-valueintensionandcompression. ............. 100 3-2ProjectiononthebiaxialplaneofMat1yieldsurface. ............... 100 3-3Poly87materialcharacterizationofMat2.Directionalyieldstrengthintensionandcompressionandr-valueintensionandcompression. ............. 101 3-4ProjectiononthebiaxialplaneofMat2yieldsurface. ............... 101 3-5ProlesofthetwoctitiousmaterialsMAT1andMat2cupsaspredictedbythePoly87simulation.Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. ...................................... 102 3-6EvolutionoftheproleoftheMat2cupduringdrawingprocess(simulatedwithPoly87). ........................................ 102 4-1GeometricalsettingforM-Kanalysis. ........................ 128 4-2Positioningonthehardeningcurveofthetwozonesofthesheet.Deformationinstability(necking)istriggeredwhenzoneBapproachestheatportionofthehardeningcurve. ................................... 129 4-3Rotationofthegrooveduringthestretchingofthesheet. ............. 129 8

PAGE 9

Banabicet.al ( 2005b ).Directionalyieldstrengthandr-value. ............... 130 4-5ProjectionsonthebiaxialplaneofthePoly4andPoly6yieldsurfacesforAA5182-O(B).. ........................................ 130 4-6ThePoly4andPoly6biaxialyieldcurves(xy=0)forAA5182-O(B) ...... 131 4-7Theevolutionoftheratioin 4{3 duringplanestrainloading. ........... 131 4-8ForminglimitdiagramfortheAA5182-O(B)alloy. ................ 132 4-9Poly6andPoly8materialcharacterizationofAA5182-O,withdatafrom Wuetal. ( 2003 ).Directionalyieldstrengthandr-value. ................. 132 4-10ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA5182-O(W). ........................................ 133 4-11Theevolutionoftheratioin 4{3 duringplanestrainloadingforAA5182-O(W). 133 4-12Poly6andPoly8materialcharacterizationofAA5182-O,withdatafrom Wuetal. ( 2003 )andoptimizationtowardtheinscribedanisotropichexagon.Directionalyieldstrengthandr-value. .............................. 134 4-13ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA5182-O(W):optimizationtowardtheinscribedanisotropichexagon. ......... 134 4-14ThePoly6andPoly8biaxialyieldcurves(xy=0)forAA5182-O(W);withYld96biaxialpointsasinput ............................ 135 4-15ThePoly6andPoly8biaxialyieldcurvesforAA5182-O(W);thePoly6andPoly8biaxialyieldcurvesareoptimizedtowardtheinscribedanisotropichexagon. .. 135 4-16ForminglimitdiagramfortheAA5182-Oalloydescribedin Wuetal. ( 2003 );Yld96pointsinthebiaxialplaneusedasinput. .................. 136 4-17ForminglimitdiagramfortheAA5182-O(W).ThePoly6andPoly8biaxialyieldcurvesareoptimizedtowardtheinscribedanisotropichexagon. .......... 136 4-18Poly6andPoly8materialcharacterizationofAA3104-H19.Directionalyieldstrengthandr-value.Biaxialdataincludedinoptimization ............ 137 4-19ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA3104-H19. ......................................... 137 4-20ThePoly6andPoly8biaxialyieldcurves(xy=0)forAA3104-H19;biaxialpointsfrom Wuetal. ( 2003 )wereusedasinputforoptimization. ........ 138 4-21ThePoly6andPoly8biaxialyieldcurvesforAA3104-H19;optimizedtowardstheanisotropicinscribedhexagon. .......................... 138 9

PAGE 10

.. 139 4-23ForminglimitdiagramforAA3104-H19.Biaxialyieldcurvesoptimizedtowardtheanisotropicinscribedhexagon. .......................... 139 4-24Poly6andPoly8materialcharacterizationofAA3104-H19.Directionalyieldstrengthandr-value.Optimizedtowardtheinscribedhexagon. .......... 140 4-25ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA3104-H19.Optimizedtowardtheinscribedhexagon .................. 140 4-26Theevolutionoftheratioin 4{3 duringplanestrainloadingforAA3104-H19. 141 10

PAGE 11

Withintheframeworkofphenomenologicalplasticity,thisworkproposesanewlookatpolynomialanisotropicyieldfunctions.Theirrangeofapplicationshasbeenseverelylimitedinthepastbyconvexityissues.Asimpleconstrainedoptimizationschemeisproposedheretosolvethisproblem.Itisshownthathomogeneouspolynomialsoffourth,sixthandeighthordercanbepowerfultoolsformodelingalmostanytypeofvariationindirectionalpropertiesofplasticow.Theplanestressformulations,suitableforsheetformingapplications,areconsideredrst.Thenasimplemethodforgeneratingminimalist3Dextensions,basedmostlyontheplanestresscriterion,ispresented. Thepolynomialapproachisthenemployedtodevelopamoregeneralmethodofdesigningyieldsurfaces.Thenewmethodallowsforyieldsurfaceswithoutacenterofsymmetry,apropertyusefulformodelingmetalswithstrengthdierential,thatis,metalswithdierentyieldingpropertiesintensionandcompression. Severalapplicationstodeepdrawingandthepredictionofforminglimitdiagramsareconsidered.Particularattentionispaidtothepredictionofearingincylindricalcups,asamethodofvalidationoftheyieldfunction.Thepredictionoftheforminglimitcurveinbiaxialstretchingisalsoanimportanttestofthecapabilitiesofayieldfunction,themostimportantfactorbeingthecorrectdescriptionofthebiaxialyieldcurve.ItisshownthatpolynomialyieldfunctionscanbesuccessfullyusedforbothearingandFLDprediction. 11

PAGE 12

Traditionally,metalformingprocesseshavebeendevelopedbasedonexpensiveexperimentaltrials.Inrecentyears,niteelement(FE)simulationshavebeenextensivelyusedtoreducetheamountofexperimentsandtrialanderrorinvolvedintheprocessdevelopment.Keyforthesuccessofsimulationsofformingprocessesistheconstitutivemodelusedforthedescriptionoftheplasticbehavior.Withinthetheoreticalframeworkofassociativeplasticitythisreducestospecifyingtheyieldsurfaceandthehardeninglaw.Thehardeninglawdescribesthewaytheyieldsurfaceevolvesinthestressspaceduringthedeformationprocess.Thisevolutioncanbeisotropic,whichmeanstheyieldsurfaceexpandsthesameamountinalldirectionspreservingtheinitialshape,oritcanbeanisotropic,meaningthatsimultaneouslywithitsexpansiontheyieldsurfacealsochangesitsshape.Dependingonthetypeofloading,anisotropichardeningcanalsoberesponsibleforchangesinthestructuralproperties/symmetriesofthematerial.Herewefocusonlyontheeectsoftheinitialanisotropyonthedeformationprocessandthereforethehardeningoftheyieldsurfacewillbeassumedisotropic,leavingonlytheyieldsurfaceitselfasthemaintopicofthistext. Notlongago,W.F.Hosfordsynthesizedthequalitiesofagoodanisotropicyieldfunctionasfollows, Hosford ( 1993 ),pp.139: Wuetal. 2003 ),orthepredictionofearingincylindricalcups(e.g. Yoonetal. 2004 2006 ),reasonableaccuracymustleavemoreandmoreroomtothe 12

PAGE 13

Thischapterstartswithabriefreviewoftheconceptofyieldsurfacefororthotropicsheetmetalsanditsidenticationinputbasedondatafrommechanicaltests(directionalyieldstrengthandr-value).Thenpastandcurrentstateoftheartinthedevelopmentofyieldcriteriaarepresented.Itconcludeswithanoutlineoftherestofthetext. BishopandHill ( 1951 )wheretheprincipleofmaximumdissipationisproposedfortheselectionofthedirectionofplasticow:givenaplasticstrainincrementdp,theactualstateofstressistheonethatmaximizestheplasticdissipation: ()dp0;(8)2R9withf()0(1{1) Itcanbeeasilyshownthatiftheaboveinequalityissatised,thentheyieldsurfacemustbeconvexandthedirectionofplasticowmustbealongthenormaltotheyieldsurface(commonlyknownasnormalityrule,orassociativeowrule) @(1{2) 13

PAGE 14

with ();(8)2R9;(8)2R+(1{4) UsingEuler'slemmaonrstorderhomogeneousfunctionsandthenormalityrule,wethenhavethefollowingequivalencerelationfortherateofplasticwork whichpromptsforthefollowingnaturaldenitionofthehardeningparameter p=d=)d p=dWp Theyieldfunctionmustalsobepressureindependent: withpbeingthepressure,andIthesecondorderidentitytensor.Thisconditionisbasedonexperimentalfacts.Ithasbeenobservedthatsuperimposingrelativelyhigh 14

PAGE 15

1{7 withrespecttopandusing 1{2 weobtain@f @I=0()tr@f @=0()tr(dp)=0: Crystalstructure,dislocationactivity,orientationofthegrains(texture)arethemaingeneratorsofanisotropyinmetals, HullandBacon ( 2001 ), Hosford ( 1993 ), Kocksetal. ( 2000 ), Gambin ( 2000 ).Theyallcomeintoplayduringsheetfabricationbyrolling.Atmacroscopiclevel,inarstapproximation,therolledsheetcanthenbeconsideredashavingorthotropicsymmetry,withtherollingdirection(RD),thetransversedirection(TD),andthenormaldirection(ND)asthethreeorthogonalintersectionlinesofthethreeorthogonalplanesofsymmetry.Inwhatfollows,thesethreedirectionswillbetakenasthex,yandzaxes,respectively,ofanorthogonalcoordinatesystemassociatedwiththesheet(alsoreferredtoasmaterialframeoraxes).WithrespecttothiscoordinatesystemthematrixofthecomponentsoftheCauchystresstensorwillbedenotedasusual[]=266664xxyxzyxyyzzxzyz377775 15

PAGE 16

Wang ( 1970 ), Liu ( 1982 ), Boehler ( 1987 ).Whilethisapproachmightoervaluableinsightinproblemsinvolvingtheconstitutivemodelingofcomplicatedsetsofanisotropictensorialquantities,forourproblemitisbettertofollowamoredirect,andsimpler,approachbasedonthedirectrepresentationoftheyieldfunctionasafunctionofthecomponentsofthestresstensorwithrespecttothesymmetryaxes.Atarstlookthismightseemrestrictive,sinceittiestheformoftheyieldfunctiontooneparticularcoordinatesystem.However,inmostpracticalapplications,andinparticularinFEimplementations,onedealsonlywiththiscomponentformoftheyieldfunction,andthereforehavingtheaboveinvarianceconditionexpresseddirectlyincomponentformwithrespecttothematerialframeismuchmoreuseful.Thus,withrespecttothesymmetryaxeswewritetheyieldfunctionintheform Then,withrespecttothesymmetryaxes,the180orotationQ1aboutthex-axishasthematrixofcomponents [Q1]=26666641000100013777775(1{10) andfromequation 1{8 weobtain 16

PAGE 17

Thus,theyieldfunctionfinequation 1{9 isorthotropicifandonlyifitsatisesequations 1{11 and 1{12 Anothergeneralpropertyofayieldfunctionconcernsitssymmetrywithrespecttothereversaloftheloadingdirection.Ingeneral,plasticdeformationiscausedmainlybythemotionofdislocationswithinthecrystallattice.Thisdeformationmodeleavesthecrystallatticeunchangedandthereforetheyieldingofthecrystalisunaectedbythesignoftheappliedstress.Atmacroscopiclevelthismeansthattheyieldfunctionhasacenterofsymmetry:thereexists02R9suchthat Sinceinthisworkkinematichardening(meanttomodeltheBauschingereectbytranslatingtheyieldsurfaceinthestressspace)ofthematerialisnotconsideredinourconstitutivemodel,theparticularcasewhentheyieldfunctionissymmetricwithrespect 17

PAGE 18

Forexample,forauniaxialloadingtheaboveequationstatesthatthematerialyieldsatthesamestressincompressionasintension. Itshouldbenoted,however,thatplasticdeformationofacrystalcanbealsocausedbytwinning(foldingofthecrystallatticeaboutamirrorplane)adeformationmechanismfavoredespeciallyinhexagonalclose-packed(HCP)metals(magnesium,titanium),althoughitcanalsooccurinfacecentred-cubic(FCC)(aluminum,copper)andbodycentred-cubic(BCC)(iron)metals, Hosford ( 1993 ).Thisdeformationmechanismisstronglydependentonthedirectionoftheappliedstressandatmacroscopiclevelitmanifeststhroughyieldsurfacesthatdohaveacenterofsymmetry.Thiseectisreferredtointheliteratureasstrengthdierential,SD.AsymmetricyieldsurfaceswillbediscussedinChapter3ofthistext.Forthemomentwelimitourdiscussiontosymmetricyieldsurfacesonly. 1-1 .Thestressstateinthesample(parallelwiththe-directionofloading)haswithrespecttothesymmetryaxesthecomponents 18

PAGE 19

Therstorderpositivehomogeneityoftheyieldfunctionmakespossibleanexplicitformulaforitspredicteddirectionalyieldstrength: AnotherindicatorofanisotropicpropertiesmeasurableduringuniaxialtestsistheLankfordvalue,orr-value.Itisdenedastheratio, HosfordandCaddell ( 1983 ),pp.265, ln(t=t0)(1{19) wherewdenotesthestraininthewidth(TD)direction,tdenotesthestraininthethickness(ND)direction,andw,w0denotethecurrentandinitialwidth,tandt0thecurrentandinitialthickness.Fromaphysicalpointofview,thisratiomeasureshowdrawablethesheetis(itsresistancetothinning).Notethatforanisotropicmaterial(withsmoothyieldsurface)thedirectionalyieldstrengthandr-valueareconstant. Ingeneral,ther-valuedoesnotvarywithstrain, HosfordandCaddell ( 1983 ),andthereforeinsteadoftheaboveformula,theratioofthestrainincrementscanbetakenasitsdenition 19

PAGE 20

1{2 ,equation 1{20 takestheform @xysin2@f @ycos2+@f @xsin2.@f @x+@f @y(1{21) inwhichthecomponentsofthestresstensoraregivenby 1{16 Forexample,fortheuniaxialtestalongtherollingdirectionwehave=0andthen 1+1=r0(1{22) Therelevanceofthisformulaisbetterunderstoodifweconsiderthedeepdrawingprocess,seeFigure 2-11 foraschematicofit,whereablanksheetispressedwithapunchintothecavityofadie.Theangeareaoftheblank,whilestillbetweenthedieandholderismainlyinastateofuniaxialcompressionneartherim.LetusnowfocusonaninnitesimalmaterialelementoftheblankwithitsaxisparallelwiththeRDdirection,butpositionedattherimonthetransversedirectionofthesheet.Thiselementisinastateofuniaxialcompressionandassumingtension/compressionsymmetryweseefrom 1{22 thatahigherr0valueimpliesabiggery-componentoftheplasticincrement,thatisbiggerincreaseoftheheightofthenalcup.Thereasoningissimilarforanyelementalongtherim.Thusinformula 1{19 ,orin 1{20 ,thewidthdirectioninthedeepdrawingprocessisessentiallythe"radial"directionalongwhichthecupwallincreases(duetothecompressionloading,thestrainalongthewidthispositive).Thereforehighr-valuesareingeneralassociatedwithsheetmetalsthatcanbedrawnintodeepcups,whereaslowr-valuesindicatesmallerdepthsofdeep-drawing. Fromageometricalpointofview,knowledgeofthedirectionalyieldstrengthimpliesknowledgeaboutonlyonecurveontheyieldsurface.Knowingther-valueadds 20

PAGE 21

Thebiaxialyieldcurve(theintersectionoftheyieldsurfacewiththe(x;y)-plane)boundsstressstateswhichareroutinelyencounteredduringmanyoftheformingprocesses(anyprocessthatinvolvesstretching),andthereforeitrequiresamoreprecisecharacterization.Forthispurposethebiaxialvalueoftheyieldstrength,b,ismeasuredbyloadingacruciformsamplewiththestresseldx=y=b,xy=0(equi-biaxialloading).Thisindicatesonthebiaxialyieldcurvethelocationofthepointx=y.Additionaldetailontheshapeofthebiaxialyieldcurveisobtainedbymeasuringforthesameequi-biaxialstateofloadingthebiaxialr-value,rb,denedby Thisratiospeciestheinclinationoftheoutwardnormalatthecurve,attheequi-biaxialstresspoint. ThevonMises'scriterionstatesthatyieldingoccurswhenthe(Euclidean)normofthestressdeviatorreachesacertainvalue 2tr() 3I(1{25) 21

PAGE 22

2(xy)2+(yz)2+(zx)2+32xy+2xz+2yz1=2(1{26) Aslaterresearchintotheplasticityofmetalsshowed,e.g., Hershey ( 1954 ), Hosford ( 1972 ),neitherTresca'syieldfunctionnorvonMises'sholdstheabsolutetruth:FCCstructuredlatticeshaveingeneralyieldsurfaceswithshapesclosertoTresca'shexagon(enclosingit),whileBCCstructuredlatticeshaveingeneralyieldsurfacesclosertovonMises'sellipsoid(inscribedinit).Thatis,mostoftheexperimentaldatafallsbetweenthetwoyieldfunctions.Nevertheless,asweshallsee,thesetwoyieldfunctionshavebeenthestartingpointformuchofthesignicantlaterworkonanisotropicyieldfunctions. Hill ( 1948 ),isstilltodayoneofthemostpopularanisotropicyieldfunctions.ItisadirectextensionofvonMises'sisotropiccriteriontoincludeorthotropicsymmetry.Withrespecttothesymmetryaxes: Itssimpleanalyticexpressionallowsforsimpleconvexityconditions,andalsoforeasycoecientidentication(throughexplicitformulas).Theyieldfunctionisrealandconvexifandonlyifitscoecientssatisfytheinequalities, Hill ( 1990 ), Thefunctioniswellsuitedformetalslikesteel(BCC),whichhaveingeneralmildvariationsoftheirdirectionalproperties,andalsohaveellipsoidalyieldsurfaces.However,sinceonlyfourmaterialcoecientsareavailableforin-planeproperties,itcannotaccuratelypredictboththedirectionalvariationoftheyieldstrengthandofther-value.Itsidenticationusuallytakesasinputonlyoneofthetwodatasetsandhaslargeerror 22

PAGE 23

Hill ( 1950 ),pp.330,whenappliedtoearingpredictionforcirculardeep-drawncups(seethenextchapter),thequadraticcriterioncanpredictatmostfourears,thetypicalnumberformostBCCandFCCstructuredlattices.Still,someFCCalloys,likebrassandsomegradesofaluminumalloys,producecupswithsixoreveneightears(seenextchapter). Inspiteofthementionedlimitations,thesimplicityofitsformulationandtheeasewithwhichitscoecientscanbeidentied,renderedHill'squadraticastatusofuniversalitywithinthemetalformingindustry,beingadoptedforanyanalysisthatneededthedescriptionofanisotropicplasticproperties.However,experimentalevidencethatthequadraticcriterionwasnotauniversalcriterioncontinuedtogather,andaround1970'sitwasalsoobservedthatsomegradesofaluminumhadanequi-biaxialyieldstrengthratio Woodthorpeetal. ( 1970 ), MellorandParmar ( 1978 ).Sincethequadraticcriterioncouldnotmodelthistypeofanisotropy,strangelyenough,itwaslabeledatthetimeas"anomalousbehavior"(ofcourse,thereisnothinganomalousinthebehaviorofaluminum). Recognizingthelimitationsofhisquadraticcriterion,actuallyquiteearly, Hill ( 1950 ),p330,suggeststheuseofageneralhomogeneouspolynomialintheform asyieldfunctionforplanestressstates.Theintegersi;j;kareallnonnegative. Comparedwith 1{27 ,formula 1{29 mightbeconsideredatmostheuristic,but BourneandHill ( 1950 )explorethecapabilitiesofthethirdorderplanestressorthotropicpolynomial(n=3intheaboveformula): 23

PAGE 24

1{29 (thehomogeneitydegreenshouldbeaneveninteger),littleimprovementoverthequadraticcriterionisreported,thethirdorderpolynomialstillnotbeingableofareasonablesimultaneousdescriptionofyieldandr-valuevariation(althoughithasasuperiornumberofmaterialparameters,six,overtheplanestressquadraticcriterion). Todealwiththesocalled"anomalousbehavior",Hillproposedtwomorenon-quadraticcriteria.In Hill ( 1979 )itisproposedthatyieldingoccursfor withm,p,q,r,a,b,andcmaterialparameters(thefunctionissimilarwiththeoneproposedby Hosford ( 1979 )).Notable,intheaboveformulathereisnodependenceontheshearcomponentsofthestresstensor,thatis,thecriterionisapplicableonlywhentheaxesofloadingcoincidewiththesymmetryaxes.Andin Hill ( 1990 )thefollowingplanestresscriterionisproposed withm,k,a,bmaterialparameters.Althoughboth 1{31 and 1{32 improveonthepredictionofthebiaxialyieldstrength,theybringlittleornoprogressinthepredictionofthedirectionalproperties( 1{31 cannotevenbeappliedforasimpleoaxistest). ( 1977 )wasthersttoexploreformula 1{29 forn=4.Hethusconsideredthefourthorderpolynomial 24

PAGE 25

Thiswastherstyieldfunctionthatcouldsimultaneouslydescribebothyieldstrengthandr-valuedirectionalproperties.Itwasalsotherstyieldfunctioncapableofmodelingthesocalled"anomalous"behaviorofaluminum.Usingequations 1{18 and 1{21 ,Gotohobtainedaninebyninesystemoflinearequationswhichleadhimtoasetofexplicitformulasfortheidenticationofthecoecientsaiabove.Thebasicinputdataforidenticationisthen 45; 90,r0;r45;r90,and 1-2 and 1-3 .Weused=(45o+90o)=2,andascorrespondingdatatheaverages( 1-2 wenotethatthePoly4predictionsfeatureadditionalmaximaorminimaofthedirectionalproperties(around15o)overtheexperimentaldatatrend.Andalthoughthebiaxialyieldcurveisconvex,theyieldsurfaceisnotconvex,Figure 1-3 .Asthedegreeofthepolynomialincreasesinformula 1{29 ,thistypeofbehavior,withundesiredlocaloscillationsinthepredictionofthedirectionalpropertiesandlackofconvexity,becomesmoreandmoretherule,ratherthantheexception,whenthecoecientsareidentiedwithoutanyregardforthepositivityandconvexityoftheyieldfunction. 25

PAGE 26

CazacuandBarlat ( 2001 )proposedarigorousmethodtoextendanyisotropicyieldfunctionexpressedintermsofJ2andJ3toorthotropy.Theauthorsstartfromthewellknownfactthatanyisotropicyieldfunctioncanberepresentedintheform whereJi()arethethreeinvariantsofthestresstensor,J1()=tr(),J2()=tr(2),tr(3).Inparticular,forpressureindependentmaterialstheaboveformreducesto with0thedeviatoricstress.Theauthorsthenusethelistofanisotropicpolynomialinvariantsofthestresstensorfortheorthotropiccase,e.g., Liu ( 1982 ),togeneratethefollowinganisotropic,pressureindependent,polynomialgeneralizationsofJ2andJ3toorthotropicsymmetry 27(b1+b2)3x+1 27(b3+b4)3y+1 27[2(b1+b4)(b2+b3)]3z1 9(b1y+b2z)2x1 9(b3z+b4x)2y1 9[(b1b2+b4)x+(b1b3+b4)y]2z+2 9(b1+b4)xyz1 3[2b10zb5y(2b10b5)x]2xy1 3[(b6+b7)xb6yb7z]2yz1 3[2b9yb8z(2b9b8)x]2zx+2b11xyyzzx(1{37) TheorthotropiccriterionisthenobtainedbysimplysubstitutingtheaboveJo2andJo3invariantsintotheoriginalexpressionoftheisotropiccriterion 26

PAGE 27

Drucker ( 1949 )proposedthefollowingisotropiccriterion withcamaterialparameter.Note,however,thatDrucker'scriterionisstillanextensionofvonMises'squadraticanditdoesnotincludeasparticularcaseTresca'scriterionforwhicharepresentationintermsofJ2andJ3ismuchmorecomplicated,seeforexample Malvern ( 1969 ). The CazacuandBarlat ( 2001 )orthotropicextensionofthiscriterion,CB2001,isthen Thus,themethodsuggestedinthecitedpaperdoesgenerateanisotropicyieldcriteriafromisotropicones.However,itsoutcomeshouldbefurtheranalyzedonacase-by-casebasis.Forexample,forCB2001somecommentsareinorder.First,itshouldberemarkedthattheconstantcisnolongerneededintheanisotropicversionofthecriterion.Indeed,withoutlossofgeneralityitcanbemultipliedintothebkcoecientsofJo3.Second,theCB2001criterionisaquiteparticularformofasixthorderpolynomial.Thus,itsplanestressrestrictionreducestoformula 1{29 withn=6.Third,thecoecientsofthesixthorderpolynomialintheformof 1{29 dependthroughcomplicatednonlinearformulasonthecoecientsakandbkofJo2andJo3.ThisnonlinearityfurtherrestrictstherangeofapplicationsofCB2001byplacinganunnecessaryburdenontheidenticationprocessandonthepositivityandconvexityconditions(likemostoftheanisotropicpolynomialcriteria,CB2001isnotbydefaultreal-valuedandconvex).Thisisconrmedin Soareetal. ( 2007a )wherethemodelingcapabilitiesofthiscriterionareexplored.WhenappliedtothedeepdrawingofAA2090-T3theCB2001criterioncouldpredictonlyfourears.However, 27

PAGE 28

Yoonetal. ( 2006 ). ( 2000 ),andlater Banabicetal. ( 2005a ),proposedthefollowingsimplealgebraicform,closetoapolynomialform,asplanestressyieldfunction: where =Lx+My ="NxPy ="RxSy Thehomogeneitydegreemisconsideredxedand,inspiredbyHosford'stheory(seenextsection),associatedwiththecrystalclass.Thusm=8forFCCmetals,andm=6forBCCmetals.TheBBC2003yieldfunction,ascalledbyitsauthors,thenhas9materialparameters(thesameasthefourthorderpolynomialofGotoh).AnapplicationofthiscriteriontothemodelingofAA2090-T3isalsoattemptedin Banabicetal. ( 2005a ).TheresultsaresimilartotheCB2001modelingofthisalloy, Soareetal. ( 2007a ),CB2001having10planestressmaterialparameters.Thereforeitcannotpredictsixearsforthecorrespondingdrawncup.However,BBC2003enjoysoverCB2001twoessentialpropertieswhichendowitwithmuchmorepracticalvalue:itisalwaysreal-valuedandconvex(provided0a1). 28

PAGE 29

withgascalarsymmetricfunction:g(i;j;k)=g(1;2;3),foranypermutation(i;j;k)of(1;2;3).Afamoustheoremof Davis ( 1957 )thensaysthatifthefunctiongisconvex,thenthefunctionfisconvextoo(notethatgisascalarfunctionofthreearguments,whereasfhasasargumentamatrix,oratensor). Anisotropythenmeansthatbesidesthemagnitudesi,theorientationofthestresstensorwithrespecttothesymmetryaxesmustalsobeincorporatedintothefunctiong.Thesimplestmethodtoachievethisistomakealineartransformationonthestresstensor(itsdeviator,moreprecisely) Ateachmaterialparticle,therepresentationofthe(fourthorderEuclidean)tensorLisinevitablytiedtoaxedtriad(orframe)ofvectorsassociatedwiththecorrespondingparticle,inparticulartheymayrepresenttheaxesofsymmetry,andthenthetransformedtensorsincorporatesnowinformationaboutitsorientationwithrespecttothistriad.Theanisotropiccriterionisthen wherenowsiaretheprincipalstressesofthetransformedstresstensors.Thisistheessenceofthemethodbasedonlineartransformations.Oneofthebenetsofthismethodisthatitgeneratesconvexfunctions(ifalineartransformationisperformedontheargumentsofaconvexfunction,thenewfunctionisalsoconvex).Inpractice,itisused 29

PAGE 30

Hosford ( 1972 )proposedanisotropicyieldcriterionbasedontheprincipalvaluessiofthestressdeviator0intheform 2(js1s2ja+js2s3ja+js3s1ja)(1{48) Dependingonthevalueoftheexponenta2[1;1),theyieldsurfacegeneratedbythisfunctioncanspantheentiredomainbetweenTresca(m2f1;1g)andvonMises(m2f2;4g)surfaces(itcanactuallydescribealsoyieldsurfacesthatinscribevonMises'sellipsoid,butthesecaseshavenophysicalmeaninginourcase).In Hosford ( 1972 )itwasremarkedthattheexponentacanbeusedasaparametertotexperimentaldataandtheoreticalpredictionsfordierentcrystalstructures.Laterwork, Hosford ( 1979 ), LoganandHosford ( 1980 ),showedthattheexponenta=8wasinbestagreementwiththeresultspredictedbytheTaylor-Bishop-Hill(TBH)plasticitytheoryofapolycrystal, Taylor ( 1938 ), BishopandHill ( 1951 ),forFCCstructuredlattices,whereasa=6bestttedthepredictionsoftheTBHtheoryforBCCstructuredlattices,seeFigure 1-4 .Thefunction 1{48 istheisotropicfunctiongin 1{47 generallyusedasthestartingpointforanisotropictheories. Barlatetal. ( 1991 )proposedtheanisotropicyieldfunction 1{47 ,calledYld91,with[L]=2666666666666666664L11L12L13000L12L22L23000L13L23L33000000L44000000L55000000L663777777777777777775;

PAGE 31

1{48 .Notallthecomponentsareindependent.IfLin 1{46 hadasargument,insteadof0,thentosatisfythehydrostaticpressureindependenceconditionthecomponentsLikhavetosatisfytherelations:L1k+L2k+L3k=0 ThusYld91hassixmaterialparametersforgeneral3Dstressstates.ItincorporatesHill'squadraticwhentheexponenta=2,theadvantageofYld91'sformulationbeingthevariableexponentawhichallowsYld96yieldsurfacesshapesotherthanellipsoids.Thereducednumberofmaterialparameters,however,limitsthemodelingcapabilitiesofthedirectionalproperties. KaralisandBoyce ( 1993 )usedthefollowingisotropicextensionofHosford'scriterion 2[(1c)1(s1;s2;s3)+c2(s1;s2;s3)](1{49) where1isgivenby 1{48 ,and Theymotivatetheirchoicebytheneedformoregeneralisotropicyieldsurfaces,theadditionalfunction2beingcapableofmodelingyieldsurfacesoutsidethequadraticofMises.ThisisquiteirrelevantforbothFCCandBCCtypeoflattices.Theirresultinganisotropicyieldfunction,althoughincorporatesYld91,doesnotbringanysignicantimprovementoverYld91.However,theirideatocombinetwodierentyieldfunctionshasbeenasourceofinspirationforthecreationofseveralotheryieldfunctions,e.g., Barlatetal. ( 2003 ), BronandBesson ( 2004 ), Banabicetal. ( 2005a ). Barlatetal. ( 1997 )proposethefollowingextensionofYld91,knownasYld96: 2(1js2s3ja+2js3s1ja+3js1s2ja)(1{51) 31

PAGE 32

1{46 ,andx,y,ztheunitframeparallelwiththesymmetryaxes.ThefourthordertensorLin 1{46 hasthesamestructureasforYld91.Thecoecientsx,yandzarefurtherdenedby:x=x0cos221+x1sin221y=y0cos222+y1sin222z=z0cos223+z3sin223 1{46 areorderedass1s2s3,thentheabovecosinesarecomputedaccordingtotherelations:cos1=yp1;ifjs1jjs3j;oryp3;ifjs1j
PAGE 33

( 2005 ).Inordertoincreasethemodelingpowerofthelineartransformationapproach,theauthorsextendthelistofvariablesbyconsideringtwolineartransformationsC(1)andC(2),havingwithrespecttothesymmetryaxesthecomponents [C(i)]=26666666666666666640c(i)12c(i)13000c(i)210c(i)23000c(i)31c(i)320000000c(i)44000000c(i)55000000c(i)663777777777777777775;i=1;2(1{52) Thetwosetsofprincipalvaluess(i)j,i=1;2,j=1;2;3,ofthetwotransformedstresstensors arethentheargumentsofthefollowinggeneralizationofHosford'sfunction 4s(1)1s(2)1a+s(1)1s(2)2a+s(1)1s(2)3a+s(1)2s(2)1a+s(1)2s(2)2a+s(1)2s(2)3a+s(1)3s(2)1a+s(1)3s(2)2a+s(1)3s(2)3a ThelinearoperatorsCiarenolongersupposedtobesymmetric,asinthecaseofYld91,orYld96,andthenYld2004has18materialparametersfor3Dstressstates,and14forplanestressstates.ThisenablesYld2004todescribeveryaccuratelybothdirectionalpropertiesandbiaxialvaluesforyieldstrengthandr-value.In Yoonetal. ( 2006 )itisshownthatthisaccuratedescriptionofdirectionalandbiaxialpropertiesleadsalsotoaccuratepredictionoftheearingproleofdeepdrawncans.Inparticular,itisshownthatYld2004canpredictsixearsfortheAA2090-T3can. 33

PAGE 34

1{47 becomes, Barlatet.al. ( 2007 ), withgascalarsymmetricfunctionineachsetofprincipalstresses.Thus,ifg1denotestherestrictionofgwhenthesecondsetofprincipalstresseshasxedvalues,theng1((1)i;(1)j;(1)k)=gr((1)1;(1)2;(1)3),foranypermutation(i;j;k)of(1;2;3),withsimilarrelationsforg2,therestrictionofgwhentherstsetofprincipaldirectionsisxed.Thelineartransformationistheninthe(split,ordirectproduct)form IttheneasilyfollowsthatYld2004isconvex(sinceg1=aisconvex).Ontheotherhand,theJo2, 1{36 ,andJo3, 1{37 ,generalizationsproposedby CazacuandBarlat ( 2001 )arealsolineartransformationsofthecorrespondingisotropicinvariantsandthereforeCB2001canalsobeexpressedintheform 1{55 with 1{56 .However,theconvexityofDrucker'sisotropiccriteriondependsessentiallyonitsparameterc.ThisparameterisaectedbythelineartransformationonJ3.Thus,arbitrarylineartransformationsofJ3cannotbeallowedandimprovementsregardingtheconvexityofthecriterioncouldbeachievedonlybystudyingtheallowedrangeofvariationofthebkmaterialparametersofJo3. 1{29 .Incomparisonwiththelineartransformationapproach,thepolynomialapproachhasclearlytheadvantageofasimplerandmoredirectformulation.Thepricetopayforthissimplicityisthatnoteverysuchpolynomialleadstoreal-valuedandconvexyieldfunctions.InChapter2thefourthorderpolynomialisreconsideredfromthisperspective.Boundsonitscoecientsarederivedtoensureconvexityandanimprovedidenticationprogramisproposedtoavoidtheoscillatorybehaviorindirectionalpredictions.Sixthandeighthorderpolynomialsarethen 34

PAGE 35

Chapter3presentsfurthergeneralizationsandsimplicationstothepolynomialapproach.Ononehand,theseleadtosimplicationsintheoptimizationprocedureusedforcoecientidentication.WhilethepolynomialforminChapter2requirestheminimizationofaquadraticwithquadraticconstraints,thenewformrequirestheminimizationofaquadraticwithlinearconstraints.Ontheotherhand,thenewformulationallowsfortheyieldcriteriontobeasymmetricwithrespecttothereversaloftheload.Thechapterconcludeswiththesolutionofabenchmarktestforanysuchcriterion:themodelingofamaterialthatisisotropicincompressionandanisotropicintension. Finally,inChapter4thepolynomialyieldfunctionsareappliedtothepredictionoffailure(necking)duringsheetstretching,thatis,tothepredictionofforminglimitdiagrams(FLD's).Itiswellknownthatsuchpredictionsareextremelysensitivetotheshapeoftheyieldsurface.IsisshownthatthefourthorderpolynomialisnotsuitedforsuchpredictionsinthecaseofFCCaggregates,aluminumalloysinparticular.Ontheotherhand,itisshownthatthesixthandeighthorderpolynomialsarebothcapableofaccuratelimitstrainspredictionsfortheFCCclass.Thus,forpolynomialcriteriatheorderofhomogeneityhasnoconnectionwiththecrystalstructure.Rather,itisthegeneralityofthefunctionitselfthatallowsforbothellipsoidalandhexagonalshapesoftheyieldsurfaces. 35

PAGE 36

Geometricalsettingoftheuniaxialtestsforwhichdirectionalyieldstrengthandr-valuearemeasured/predicted. Figure1-2. Poly4materialcharacterizationofAA2090-T3usingtheidenticationprocedurein Gotoh ( 1977 ).Directionalyieldstrengthandr-value. 36

PAGE 37

ProjectiononthebiaxialplaneofthePoly4yieldsurfaceforAA2090-T3,andthePoly4biaxialyieldcurveforthesamematerial.Althoughthebiaxialyieldcurveisconvex,theyieldsurfaceisnot. Figure1-4. YieldcurvesinthebiaxialplaneaccordingtoHosford'sisotropiccriterion. 37

PAGE 38

Inthischapter,withintheframeworkofphenomenologicalplasticity,weinvestigatethemodelingcapabilitiesasanisotropicyieldfunctionsofhomogeneouspolynomialsoffourth,sixthandeighthorder.Planestressyieldfunctionsareconsideredrst,andthenasimplemethodtoextendthemto3Dstressstatesisprovided.Sincetheyieldsurfacemustbeconvex,wepayagreatdealofattentiontothisissue.Thechaptercloseswithseveralapplicationstothesimulationofdeepdrawingandadiscussionofearinginanisotropiccups. Groemer ( 1996 ).Letusrstnoticethatiff:Rn!R+isarstorderpositivehomogeneousfunction,thenitisuniquelydeterminedbyitsrestrictiontotheunitsphere(itssupportfunction)denotedfromnowonbyh:f(x)=fjjxjjx whereu=x=jjxjj,andjjxjjdenotestheEuclideannormofx2Rn.Whenn=2,theunitspherebecomestheunitone-dimensionalcircle.Associatingwiththiscircleapolarangle,say!2[0;2),thesupportfunctionhcanbeviewedasdenedona1Dintervalh:[0;2]!R+;h=h(!).Wethenhave Indeed,writingthehessianHofthefunctionfinpolarcoordinates(r;!)associatedwithCartesiancoordinates(x;y),andtakingintoaccounttherstorderhomogeneityoff,i.e., 38

PAGE 39

2{1 )holdstrue. Forrstorderpositivehomogeneousfunctionsdenedonspacesofhigherdimensionsthantwo,asimpleconvexityequationlike( 2{1 )isnolongeravailable.However,itcanbeeasilyshownthateverysuchfunction(andingeneral,anyfunction)isconvexifandonlyifitsrestrictiontoanytwo-dimensionalhyperplaneisconvex.Inparticular,forplanestresscaseswheretheyieldsurfaceisembeddedinR3wehave 2{1 )alonganyplanesectionthroughtheunitsphereS2. Inthecaseofarstorderhomogeneousfunctionofthegeneralform withPnahomogeneouspolynomialofdegreen,inequality( 2{1 )becomes andtheconditionsforftobeareal-valuedconvexfunctionarethen foranyplanesectionthroughtheunitsphere.Inpractice,forthepolynomialtypecriteriadiscussedinthispaperitissucienttoenforcetheseinequalitiesonlyalongadiscreteset 39

PAGE 40

Ithasbeenrstconsideredby Gotoh ( 1977 )whoalsoproposedexplicitformulasintermsofbothyieldstrengthandr-valuedataforitscoecients.However,Gotoh'sdirectapproachforcoecientidenticationcanoftenleadtoanunbalancedoverallcharacterizationofthedirectionalyieldstrengthorr-value(itcanexactlydescribefourdatapointsbutinthesametimecanhaveabigoverallerror),andoersnoprecautionswithrespecttotheconvexityoftheyieldsurface.InthissectionweproposechangestoGotoh'sidenticationprogram,changesthatstillkeeptheformulasforcoecientsexplicitwhilebalancingtheoverallerrorinpredictionandimprovingupontheconvexityoftheyieldsurface.Thesuperscript2Dineq.( 2{5 )willbedroppedfromnowonsinceinthissectiononlytheplanestresscasewillbedealtwith. 90; b,andowdatar0;r90.Thuswehave 490(2{6) 4b)(a1+a2+a4+a5)(2{8) 40

PAGE 41

90,andr0;r90mustbeaccuratelypredicted,theonlyvariableleftisthecoecienta3,orequivalently,from( 2{8 ),thebiaxialyieldstrength 2{6 )and( 2{7 ).TherestrictionofP4totheunitcircleinthisplaneis Therestrictionoftheyieldsurfacetothebiaxialplaneisthenreal-valuedif wheret:=tan(!).Using( 2{8 ),theaboveinequalityisrewrittenasaboundforthebiaxialyieldstrengthdatapoint 1= 4b>Q+a1+a2+a4+a5(2{11) Therestrictionoftheyieldsurfacetothebiaxialplaneisconvexifandonlyifthesecondinequalityin( 2{4 )issatised.Hereittakestheform where ThecoecientsA=A(!),B=B(!),andC=C(!)canbeestimatedforany!2[0;2).Foranyxed!,equation( 2{12 )hasingeneraltworealsolutionsx1(!)x2(!),andfor 41

PAGE 42

2{12 )toholdtrueforany!wemusthave AlthoughformulasforM1andM2arediculttoobtaininexplicitform,theycanbeestimatednumericallywithanydegreeofaccuracy.Then,usingequation( 2{8 ),theycanbefurtherusedtoestimatetheintervalwherethebiaxialyieldstrength (M2+a1+a2+a4+a5)1=4 Inequalities( 2{11 )and( 2{15 )arenecessaryandsucientconditionsforthebiaxialyieldcurvetobereal-valuedandconvex.Next,werepeatthesameargumentasaboveandobtainconditionsforthreemorecurvesontheyieldsurfacetobereal-valuedandconvex.ThereasoningiselementaryanditisoutlinedinAppendix A .Herewejustlisttheseconditions. Thedatapointsat=45ofromrolling, 45)4r45 45)4 where 4b(2{18) Thentheyieldcurvesintheplanesy=0andx=0arereal-valuedandconvexifandonlyifthefollowinginequalitiesaremet,respectively, 0a66p 42

PAGE 43

2{17 )toholdtrue,theyieldcurveintheplanex=yisreal-valuedandconvexifandonlyif wherewehavedenotedt:=(2= 45)4=B1=(2 45)4. wherec:=cos2,s:=sin2,and 43

PAGE 44

2{17 ),which,coupledwithequation( 2{16 )fora9,assuresexactpredictionsfor 2{21 )and( 2{22 ),weobtain (c2cs)a6+(s2cs)a8=B6B5cs(2{29) (g1g3)a6+(g2g3)a8=B7B5g3(2{30) Asmentionedbefore,ifa6anda8aredirectlysolvedforfromthesetwoequations,thereisagreatchancethat(a)althoughsomedatapointswillbematchedexactly,theoveralldatatwillbeofpoorquality,and(b)theresultingyieldsurfacewillnotbeconvex.Instead,toimprovethechancesofobtainingagooddatatandaconvexyieldsurface,wewillrequirethattheaboveequationsbesatisedinanaverageformatasmanylocationsaspossible.Thus,fori=1;2,wedenoteci=cos2i,si=sin2i,with1=15o,2=75o(or1=30o,2=60o,oranycombinationofdataonthecorrespondingintervals).Thenwenda6anda8byoptimizingthedistance(orerror)function 2Xi=1;2nw(i)1(c2icisi)a6+(s2icisi)a8(B6(i)B5cisi)2+w(i)2[(g1(i)g3(i))a6+(g2(i)g3(i))a8(B7(i)B5g3(i))]2o wheretheweightsw(i)jareasfollows:w(i)1weightsthedatapoints Thedistancefunctionisapositivedenite(assumingnotallw(i)jarezero)quadraticina6anda8,andthereforethesolutiontotheminimizationproblemexists,isunique,andcanbefoundinexplicitform.Aftersolvingfora6anda8fromthelinear 44

PAGE 45

where The'sand'sintheaboveformulasarethecoecientsappearingintheexpressionofthedistancefunction,eq.( 2{53 ): Iftheresultcomputedbyequation( 2{32 )issuchthat(a6;a8)2[0;6p 2{32 )mustbereplacedwithoneofthefollowingalternatives. Ifa6>6p 45

PAGE 46

Similarly,ifa8>6p Ifbotha6anda8areoutsidetheirboundingintervals,thenthe"corner"solutionsshouldbeaccepted.Forexample,ifa6>6p Insummary,theidenticationprogramforthecoecientsofPoly4takesasinputthedata 1. Computethecoecientsa1,a2,a4anda5accordingtoequations( 2{6 )and( 2{7 ); 2. Checkinequalities( 2{11 )and( 2{15 ).If 2{8 ); 3. Checkinequality( 2{20 ).Ifr45isoutsidetheboundinginterval,thenthisdatapointshouldbeadjustedaccordingly.Thencomputethecoecienta9usingequation( 2{16 ); 4. Computethecoecientsa6anda8accordingtoequation( 2{32 ).Ifthepair(a6;a8)satisesinequalities( 2{19 ),thenitisthesolutiontotheboundedminimizationproblem.Ifnot,dependingonthepositioningoftheabove(a6;a8)solutionwithrespecttotheboundingsquare,thenalsolutionshouldbecomputedaccordingto( 2{40 ),or( 2{41 ),ortakenasoneofthecornersofthesquare; 5. Computea7=B5(a6+a8). 46

PAGE 47

2-1 ForAA2090-T3inequalities( 2{11 )and( 2{15 )aresatisedintheform 0:8714=1= 4b>0:0435;0:6323 ItfollowsthatP4canmodeltheentirebiaxialdatasetforthismaterialwithareal-valuedconvexyieldcurve.Next,inequality( 2{20 )isalsosatisedintheform 0:5935r45=1:576920:169(2{43) Theboundingintervalsfora6anda8are 0a629:054;0a835:068(2{44) Then,duetothepeculiaroscillationinr-valuedataaround75ofromrolling,weuse1=15oand2=75oaslocationsfortheidenticationofa6anda8.Afterafewminimizationtrials(withvariousweightsstartingwithw(1)1=1:00,w(1)2=1:00,w(2)1=1:00,w(2)2=1:00),wendanacceptablematerialcharacterizationlistedinTable 2-2 .Thenalweightsusedwere:w(1)1=2:00,w(1)2=1:00,w(2)1=4:00,w(2)2=0:30(recallthatw(i)1weighs ForAA2008-T4,inequalities( 2{11 ),( 2{15 )and( 2{20 ),aresatisedintheform1:517>0:051;0:64280:90101:2230;0:13480:49155:5852,respectively,whereastheboundingintervalsfora6anda8are0a617:1410;0a820:9554.Finally,using 2-2 (thecorrespondingweightswerew(1)1=2:00,w(1)2=1:00,w(2)1=0:10,w(2)2=0:50). 47

PAGE 48

2-1 2-2 and 2-3 .ComparisonswithYld96arealsoshown.Poly4andYld96havethesamenumberofmaterialcoecientsforthedescriptionoftheinplaneplasticproperties,nine.Theirmodelingcapabilitiesareofsimilarstrength,withaslightadvantageofPoly4,duetotheexplicitidenticationformulas.BothcriteriaoerconsiderableimprovementinmodelingoverHill'squadraticcriterion,bytakingintoaccountthedirectionalvariationofboththeyieldstressandr-value.However,itshouldbenotedherethatthetwocriteriaarequitedierentatpredictingtheoverallshapeoftheyieldsurface.WhilePoly4yieldsurfacesareingeneralmuchmoresimilarinshapewiththequadraticyieldsurfaces(withtheircharacteristicovalshapenearequibiaxialstressstate),Yld96yieldsurfacesareingeneralmuchclosertotheyieldsurfaceshapespredictedbyHosfordtypecriteria, Hosford ( 1972 ).ThisaspectdierentiatesPoly4andYld96withrespecttotherangeofapplications.Forexample,forminglimitdiagrampredictionsareknowntobeextremelysensitivetotheshapeofthebiaxialyieldcurve.Forthisproblem,inparticularforaluminumalloys,Yld96shouldbetherstchoicesinceitisspecicallydesignedforbettermodelingofthebiaxialyieldcurve, Barlatetal. ( 1997 ).Ontheotherhand,Poly4canbeeasilyusedinanysituationwhereHill'squadraticissuitable,Poly4oeringanincreasedaccuracyinmaterialmodeling. Itoftenhappensthatonlythreedatapointsareavailableforyieldstrength, 45; 90,andonlythreedatapointsforr-value,r0;r45,andr90,withnoindicationaboutthevalueofthebiaxialyieldstrength.ThisisthecasewiththeNumisheet'93mildsteel, Danckert ( 1995 ),referredfromnowonasNUM93.Poly4canstillbeappliedtothemodelingofsuchdatasetsasfollows.First,thebiaxialyieldstrengthispostulatedforthismaterialtobe Pearce ( 1968 ).Then,the 48

PAGE 49

2=2 (2{45) (2{46) ForNUM93weuse1=30o,and2=60o.Thenwith1=0:3,2=0:4,1=0:2,2=0:7,andtheweightsw11=1:00,w12=0:50,w21=1:00,andw22=0:10,weobtainthecoecientslistedinTable 2-2 ,andtheresultspresentedinFig. 2-4 2{2 ),withn=6orn=8,andwiththecorrespondingorthotropicpolynomials Forn=6,thecriterionin( 2{2 )willbereferredtoasPoly6,whereasforn=8itwillbereferredasPoly8.Poly6andPoly8have16and25materialcoecients,respectively,forthedescriptionofthein-planeplasticproperties.Thus,ananalysiscomparablewith 49

PAGE 50

Schitkowski ( 1986 ).Suchsubroutinesrequiretheevaluationofthemeritfunction,constraintsandoftheirgradients.Althoughthealgebramayseemalittletedious,itisontheotherhandquitestraightforwardandweshallsketchithereratherthanintoanappendix.Thereasonisdual.First,thealgorithmisgeneralenoughtobeusedwithanyotheryieldfunction,orwithanyotherpolynomialtypestrain-ratepotential.Second,asweshallsoonseeattheendofthissectionandtheonesdedicatedtoearing,Poly6andPoly8turnouttobepowerfulyieldcriteria,capableofmodelingawiderangeofanisotropicvariationsinplasticproperties,andthereforesomedetailoftheiralgorithmicimplementationisworthwhilepursuinghere. If0ischosenasreferenceyieldstrength(forisotropichardening),then-thorderpolynomialPnwillthenpredictforauniaxialtestalongananglefromtherollingdirection Thepredictedr-valueiscomputedaccordingtotheformula (2{50) 50

PAGE 51

@x(b;b;0).@f @y(b;b;0)=@Pn (2{52) Theidenticationofthematerialcoecientsisthenperformedbyminimizingthedistancefunction 2XjnwsjQ(ai;j)( wherewsj,wrj,wsbandwrbareweightsassociatedwitheachdatapoint.Itshouldbenotedthatin( 2{53 ), 2{49 ),( 2{50 ),( 2{51 ),and( 2{52 ),respectively,denotepredictedvalues.Thegradientofthemeritfunction( 2{53 )is @ak=XjwsjQ(ai;j)( @ak+wrjR1(ai;j)rjR2(ai;j)@R1 Itisrecalledthat,duetothehomogeneityofthepolynomialfunctionsinvolved,intheaboveformulasthequantitiesQ,R1,R2,andtheirderivativesareevaluatedat(cos2;sin2;cossin),asinequation( 2{49 ),forexample. Theoptimizationprocessissubjectedtotheconstrains( 2{4 )tobeimplementedasfollows.Ontheunitsphere2x+2y+2xy=1weconsidertwofamiliesofunitcircles.The 51

PAGE 52

Abovewehave!2[0;)duetotheorthotropicsymmetryoftheyieldfunction(theyieldsurfaceinthespacex;y;xyissymmetricwithrespecttothexy=0plane).Thenimposethetwoinequalityconstraints( 2{4 )atdiscretelocationsoneach(semi)circleofthefamily.Thus,weconsiderN(v)verticalcircles,describedbyq=(q1)=N(v);q=1;:::;N(v),andoneachsuchverticalcirclewetakeM(v)points Foreachsuchpointwethenhavetheconstraints (2{57) (2{58) Thesecondfamilyisformedwithconstantlatitude(horizontal)circlesparameterizedintheform Abovewehave2[0;=2)duetotheorthotropicsymmetry,and!2[0;)duetothecentralsymmetryoftheyieldsurface.Again,weconsiderN(h)horizontalcircles,describedbyq=(q1)=(2N(h));q=1;:::;N(h),andoneachsuchhorizontalcirclewetakeM(h)points Ateachsuchpointwethenwriteagaintheconstraints( 2{57 )and( 2{58 ). 52

PAGE 53

2{57 )thisisatrivialtasksincePndependslinearlyonitscoecientsak.Thegradientsoftheconvexityconstraints( 2{58 )arealsoeasilyimplementedbynotingthat d!Pn(x;y;xy)=rPnV(2{61) where d![x;y;xy]T;A=d2 where[x;y;xy]isparameterizedby( 3{5 )or( 2{59 ).Thus,thetaskofwritingthegradientof( 2{58 )isreducedtocomputingthegradientswithrespecttoakofPn,itsrstandsecondderivatives. Inpractice,forPoly6itisenoughtotake4N(v)8,10M(v)20,1N(h)4,and10M(h)20,sothattheidentiedyieldsurfaceisreal-valuedandconvex.ForPoly8onemaytake20N(v)40,10M(v)20,2N(h)4,and10M(h)20. Thecoecienta1canbetakenas1:000fromthebeginning,theoptimizationbeingdoneontherestofthecoecients.Asinitialguesses,a(0)k,forPoly6andPoly8onecanalwaysusethecoecientsoftheisotropicvonMisescriterion.Thatis,ifIdenotestheisotropicquadraticcriterionofvonMises,thentheisotropicPoly6andPoly8arefoundfromPn=In=2.TheisotropiccoecientsforPoly6andPoly8areprovidedinTables 2-3 and 2-4 .AnalternativeinitialguessforPoly8isthecoecientvectorprovidedbyPoly4.TheconnectionbetweenthetwosetsofcoecientsislistedinAppendix B .Then,the 53

PAGE 54

TheidenticationprogramdescribedinthissectionhasbeenappliedtothemodelingofAA2090-T3.ThecoecientsofPoly6andPoly8forthismaterialarelistedinTables 2-3 and 2-4 .Thecorrespondingreal-valuedandconvexyieldsurfacesareplottedinFig. 2-5 .ThepredictionsofPoly6andPoly8fordirectionalyieldstrengthandr-valuearepresentedinFig. 2-6 ,wherecomparisonswithYld2004arealsomade.Besidestheseresults,Poly6predictionsforbiaxialyieldstrengthandbiaxialr-valueare1:033and0:668,respectively,whereasthebiaxialPoly8predictionsare1:036and0:670,respectively.FromtheseresultsweconcludethatPoly6andPoly8giveperfectsolutionstothecharacterizationofthisalloy.FurtherapplicationsofPoly6arepresentedinthenextsections. 1{11 )and( 1{12 ),itfollowsthatanyorthotropicpolynomialyieldfunctionfor3Dstressstatesmustdependontheshearcomponentsxy,xzandyzonlythroughthemonomials Then,inordertosatisfythepressureindependencecondition,onecanconsiderasyieldfunctionageneraln-thorderpolynomialofthestresscomponentssx,sy,sz,sxy,sxz, 54

PAGE 55

Thisapproachhasbeenconsideredbeforeby Arminjonetal. ( 1994 ),tobuildstrainratepotentials.Theyalsodoublecheckedtheirpolynomialform,forthecorrectnumberofcoecients,withapolynomialconstructionbasedongeneralized(anisotropic)invariants.Asimilar3Dpolynomialcriterioncanbeeasilybuiltfromasixthorderhomogeneouspolynomialandsoon.However,sincethenumberofcoecientsincreasesrapidlywiththedegreeofhomogeneity,theusefulnessofsuch3Dcriteriabecomesdoubtfulintheabsenceofasucientnumberofoutofplaneexperimentaldatapoints.Analternativeistousepolynomialcriteriaasstrainratepotentialsandtoassociatethecoecientswithtextureinformation,e.g., Arminjonetal. ( 1994 ), Zhouet.al. ( 1998 ).Yetagain,theissueofconvexitymustbedealtwithduringtheidenticationphaseevenforthisapproach.Thegeneralmethoddescribedintheprevioussectionisapplicableinthiscasetoo,withfewerconstraintsbeingneededforthistypeofpotential,duetothelargeamountofdata. Whentheidenticationoftheyieldfunctionisbasedonyieldandowdataobtainedfrommechanicaltests,theaboveapproachisclearlyunsuited,sinceoutofplanedataisscarcelyavailableforthinmetallicsheets,thereforeleavingundeterminedmostof 55

PAGE 56

Indeed,thesubspaceofdeviatoricstresses,denoteitV0,canbewrittenasadirectproductbetweenthetwodimensionalsubspacedenedbyx+y+z=0,andthesubspaceoftheshearcomponentsS:=f(xy;xz;yz)2R3g, ApressureindependentyieldfunctionfisthenuniquelydeterminedbyitsrestrictiontoV0.Equation( 2{67 )thenfollowsimmediatelyifweobservethattheplanecanbeparameterizedintheform wheres1:=xz,ands2:=yz.Asaconsequence,inequation( 2{66 )onemayalsoreplacethethedeviatoriccomponentswithsx:=s1,andy:=s2. Letusnextdenote asthatpartofthe3Dcriterionwhichcontainsthecouplingbetweentheplanestressstateandthenormalcomponent.Ontheotherhand,theplanestressrestrictionofthe3Dcriterionis 56

PAGE 57

Tocompletethe3Dformulationoftheyieldfunctionfromitsplanestressrestriction,oneonlyneedstoaddtheoutofplaneshearcouplingwiththeplanestressandnormalcomponents.Thisextensioncanbedesigneddependingonhowmanyoutofplanedatapointsareavailable.Forexample,weproposethefollowingminimalist3Dextensionofplanestressyieldfunctionsgivenbyeq.( 2{2 ): First,wenotethatthisextensioncanbereducedtoHill'48(orMises)criterion.Second,oncetheplanestressrestrictionisconvex,theabove3Dextensionisalsoconvex.Indeed,the3Dextensionisintheformf()=[f1()]2+[f2()]21=2,andthefunctionfwillbeconvexiff1andf2arepositiveconvexfunctions.Inourcase,f2isalwaysintheformf2()=2k132xz+2k232yz1=2,withk13andk23positivecoecients.Thusf2ispositiveandconvex.Ontheotherhand,f1isobtainedbyalineartransformationfromthe2Drestriction:f1()=hP(2D)n(xz;yz;xy)i1=n,andthereforef1willbepositiveandconvexif(P(2D)n)1=nispositiveandconvex. 57

PAGE 58

C ofthischapter,ortotheexcellentbookof SimoandHughes ( 1999 ),ortotheworksof Tugcuetal. ( 1999 )and Yoonetal. ( 2004 ),forexample. ThegeometricalsettingoftheproblemintheXZ-planeissimilarwiththeoneforthedeepdrawingofcylindricalcups,tobediscussedinthenextsection,andisdepictedinFig. 2-11 .IntheXZ-planetheshoulderofthepunchhasaradiusof8mm,whereastheshoulderofthediehasaradiusof5mm.ThedierenceisintheshapeofthetoolsintheXY-planewhichiscircularforthecylindricalcase,andsquareforthisproblem,depictedinFig. 2-7 .Theblankisa1501500:78mmsheet,madefromNUM'93mildsteelmodelledinSection 2.2.3 ,withPoly4coecientslistedinTable 2-2 .Theotherparametersofthesimulationweregivenasfollows, Danckert ( 1995 ): 58

PAGE 59

2-8 theproleofthenaldrawncup(at40mmpunchtravel)isshown,whereasTable 2-5 liststheamountofdraw-inat15and40mmpunchtravel.Ascanbeseen,theamountofdraw-inpredictedbythePoly4simulationagreesverywellwiththeexperimentallymeasureddata.Acomparisonanalysiswiththeblankconsideredplasticallyisotropic,andmodelledwith(ABAQUS)vonMisescriterion,wasalsoconducted.Thenalprole,andthustheamountofdraw-ininallthreedirections,predictedbytheisotropicanalysiswasvirtuallyidenticalwiththeonepredictedbythePoly4analysis.Thus,theplasticanisotropyofthesheethasverylittleoratallinuenceonthenalshapeoftheprole,andthisisduetotheinitialsquareshapeoftheblankwhichinhibitsthedevelopmentofsucheects(seealsothediscussiononearing,next).However,thestraindistributionspredictedbythetwoanalyzesdier.Figures 2-9 and 2-10 showthethicknessstrainspredictedbythetwosimulationsalongtheOBdiagonaloftheblank,seeFig. 2-5 ,andweconcludethatthePoly4simulationhasbetteragreementwiththeexperimentaldatathantheisotropic(vonMises)simulation.Thisisaneectoftheplasticanisotropyofthesheet. 59

PAGE 60

Yoonetal. ( 2000 ),seeFigure 2-11 .Theotherparametersoftheformingprocessweregivenasfollows: Thesmallfrictioncoecientsimulatesawelllubricatedcontactbetweenblankandtools.Also,theforceontheholderissmallenoughsothatitdoesnotaddadditionalyieldinginthematerial,butbigenoughtopreventthewrinklingoftheblank. Duetotheorthotropicsymmetryofthematerial,onlyaquarteroftheblankwassimulated.ThetypicalmeshusedontheblankisshowninFig. 2-12 .Forplanestressanalysisitconsistedofonelayerofcontinuumshellelements,SC8R,with21integrationpointsacrossthickness(1806nodes,854elements),whereasfor3Danalysisitconsistedofvelayersoflinearcontinuum(brick)elements,C3D8R(5418nodes,4270elements).For3Danalyzes,thetwocoecientsk13andk23in 2{73 werecomputedusingdataforyieldinginsimpleshear: Sincethisdataisdiculttoobtainforthinsheetsbyusingmechanicaltests,onecanestimateitusingpolycrystalsimulationsbasedontexturedata.Thisapproachhasbeenadoptedin Barlatetal. ( 2005 ),whereitisfoundthat Figure 2-13 showsthenalcongurationofatypicalfulldrawncup.Figure 2-14 presentstheprolesofthenalcupforplanestressand3DPoly4simulationsandcomparisonwithYld96prole.TheinplanematerialcoecientsweusedwerethoselistedinTable 2-2 forAA2090-T3.Threeobservationsareimportanttobemadehere. 60

PAGE 61

InFigure 2-15 theprolesforthePoly6andPoly83D-simulationsareshown,togetherwiththeYld2004-proleobtainedfrom Yoonetal. ( 2006 ).TheAA2090-T3in-planecoecientsusedforPoly6arethoseinTable 2-3 andforPoly8thoseinTable 2-4 .TheoverallPoly6andPoly8prolepredictionsareverygood(correctheightandcorrectpositioningandnumberofears,six)anddierverylittlefromtheYld2004prole.Thisdierencemightcomefromthedierentniteelementusedinthecitedpaper.ThedierencebetweenthePoly6andPoly8prolesisalmostnegligible.Thisshowsthatacorrectdescriptionoftheanisotropicplasticpropertiesissucientinordertoobtainthecorrectprole,andthusthepredictedproleisyield-criterion-independent. 61

PAGE 62

=d=0isequivalentwith @xdx @ydy @xydxy whichcanbefurthertransformedintotheequality cos2sin2 ifoneneglectstheelasticpartfromthestrainincrement.Thelastequalityexpressestheconditionforthestressandstrainincrementtensorstobecollinear.Next,aninnitesimalelementattherimatananglefromtherollingdirectionisinastateofuniaxialcompressionalongthehoopdirection.Sincetheyieldfunctionisassumedorthotropicandsymmetricwithrespecttotensioncompressionyieldingproperties,itfollowsthatthelocationswhereearsandhollowsdevelop(stressandstrainincrementarecollinear)areat90oi,whereiarethestationarydirectionsofthedirectionalyieldstrength.Inparticular,duetotheorthotropicsymmetry,the=0oand=90odirectionsarealwaysstationarypointsforthedirectionalyieldstrengthandthereforetheabovetheorypredictsthatcupsdrawnfromorthotropicblanksalwayshaveearsorhollowsat0oand90o.Thisisconrmedbyvirtuallyallexperimentalprolesofdrawncylindricalcups,anditalsoexplainsthehollowsalongtherollingandtransversedirectionsforthesquarecupproblem. However,theabovetheorydoesnotpreciselypredictwhetheraearorahollowwilldevelopatthecorrespondinglocation.InFig. 2-16 theevolutionofthetopologyoftherimduringcupdrawingsimulationisshownforAA2090-T3sheet,wherethedistancebetweenthepointsontherimandthecenterofthebottomofthecupisplottedversus 62

PAGE 63

2-17 and 2-18 ,wherethedatatforthedirectionalyieldstrengthandr-values,anditscorrespondingyieldsurfacearedrawn.Table 2-3 liststhecorrespondingcoecientsofPoly6forMat1.ThepredicteddirectionalyieldstrengthforMat1isvirtuallyconstant(withavariationlessthan0.0005),whereasthebiaxialpredictionsforMat1are 2-19 .ThepredictedproleofMat1haseightearslocatedat0o,45o,90o,135o,etc,withrespecttotherollingdirection.Thisresultagainunderlinestheinuenceofthedirectionalvariationofther-valueonthelocationoftheearsandhollows. Aftercarefulexaminationsofmanyexperimentaldataforyieldstrengthandr-valueandtheircorrespondingproles, Yoonetal. ( 2006 )reachtheconclusionthatther-valuedataisingeneralresponsiblefortheshapeoftheprole,whilethevariationoftheyieldstrengthdictatesthemagnitudeoftheears.AcloserlookatthepredictedproleforMat1ontheinterval[0o;90o]revealsthat,withtheexceptionofthesmallearat90o,theshape 63

PAGE 64

Chungetal. ( 1996 ) wherer,andtarethestrainintheradial,circumferentialandthicknessdirections,andintherightmemberoftheaboveequationuseoftheincompressibilityofthestrainincrementhasbeenmade(theelasticpartofstrainisneglected).LetRdandRbdenotetheradiiofthedieopeningandoftheblank.ThenaringofradiusR,withRdRRb,iscompressedtotheradiusRdinthenaldrawncup,andthereforeonecanapproximatethecircumferentialstrainincrementforthisringwith=ln(Rd=R).Solvingforrineq( 2{77 ),theheightofthecupisthenobtainedintheform(Yoonetal.,2006): (2{78) whereRpdenotestheradiusofthepunchshoulder.ThetermRp+(RbRd)correspondstothe"initial"heightofthecup,theonethatwouldbeobtainedbypurerigidmotion(anidealbendingoftheblankintoacup). Formula( 2{78 )isentirelybasedonr-valuedataandthereforeonecannotexpectittooutputaccuratequantitativeproleprediction.However,itisproposedasaquickand 64

PAGE 65

2{78 ),weobtaintheprolemarkedasYMAT1inFig. 2-19 .YMAT1hassixearsanditisapredictionforbothAA2090-T3andMAT1proles,sincetheyhavethesamer-valuedescriptionbyPoly6.Atthesametime,thedierencebetweentheshapeoftheAA2090-T3sheetFEprole,Poly6curveinFig. 2-15 ,andthatofMAT1Poly6FEproleinFig. 2-19 isclear:onehassixears,theotherhaseight.Thisdierencecanbeagainpushedatextremebyconsideringanimaginarymaterialthathasuniformdirectionalr-value(data:rb=rj=1:000,j=1;:::;7),andyieldstrengthvariationidenticalwiththeoneofAA2090-T3,includingthebiaxialvalue.Poly6descriptionofthisctitiousmaterialcalledMAT2islistedinTable 2-3 andplottedinFigs. 2-17 and 2-18 .Poly6biaxialpredictionsforMat2are 2-20 .InthesamegureisplottedtheYMAT2prolepredictedbyformula( 2{78 )ifthePoly6descriptionforMAT2r-valueisusedasinput.Thetopologicaldierencebetweenthetwoprolesisself-evident:theYMAT2proleisuniform,whiletheFEPoly6MAT2prolehasfourbigears. Itfollowsthenfromtheabovediscussionthatabasictheoryofearingmusttakeintoaccountboththedirectionalvariationoftheyieldstrengthandr-value,andalsothehardeninglawofthematerial.Furtherimprovementinearingprolepredictionisnotpossiblewithouttakingintoconsiderationeitherananisotropichardeninglaw,orthepossibilitythattheblankmightinitiallypossesonlymonoclinicsymmetry,orboth.Indeed,itcanbenotedthattheexperimentalproleforAA2090-T3issymmetriconlywithrespecttotherollingdirection.Thiscanbeexplainedeitherbythereorientation 65

PAGE 66

Dafalias ( 2000 ).Or,byassumingfromthebeginningthattheinitialblankhasonlyoneplaneofsymmetry,theonenormaltotheblankandparallelwiththerollingdirection.Theseaspects,however,arebeyondthescopeofthistextandconstitutetopicsoffurtherresearch. 66

PAGE 67

ExperimentaldataforAA2090-T3,AA2008-T4andNUM'93mildsteel. 1.00000.96050.91020.81140.80960.88150.91021.03502008-T4 1.00000.99630.98350.94590.93030.91710.90440.9010NUM'93 1.0000--1.0569--1.03290.21150.32690.69231.57691.03850.53840.69230.67002008-T4 0.86740.80770.61880.49150.49550.51140.5313-NUM'93 1.79--1.51--2.27Table2-2. Poly4coecientsforAA2090-T3,AA2008-T4andNUM'93mildsteel. 1.0000-0.69841.4969-2.38381.45684.8808-1.01502008-T4 1.0000-1.85792.9549-2.07421.49466.5600-4.1447NUM'93 1.0000-2.56633.6988-2.43920.87845.7851-7.6630 8.709523.44982008-T4 7.94908.1031NUM'93 5.84358.2863 Table2-3. Poly6coecientsforisotropicMises,AA2090-T3,Mat1,andMat2. 1.0-3.06.0-7.06.0-3.01.09.02090T3 1.0000-1.10592.5255-5.19146.1458-4.32541.775314.190Mat1 -18.027.0-18.09.027.027.027.027.02090T3 -4.9759-4.39263.465215.8060.0000-9.491686.661116.42Mat1 67

PAGE 68

Poly8coecientsforisotropicMisesandAA2090-T3.Forbothwehavea1=1:0000. -410-1619-1610-412090T3 -1.33762.1967-5.786712.312-16.00013.260-7.04152.1508 12-3672-8472-3612542090T3 12.697-4.271974.294-31.487-18.387-11.09628.553116.65 -108162-10854108-108108812090T3 -238.83158.71-57.545172.28-0.2559-8.2036558.15543.50 Table2-5. Draw-inpredictionsofPoly4FEsimulationandexperimentaldata. Punchtravel(mm) DX(mm) DY(mm) DD(mm) FEExp. FEExp. FEExp. 15 5.945.64 6.066.63 3.283.48 40 28.2427.89 28.6829.24 15.1015.95 68

PAGE 69

ProjectionsonthebiaxialplaneofthePoly4yieldsurfaceforAA2090-T3andAA2008-T4 Figure2-2. Poly4materialcharacterizationofAA2090-T3.Directionalyieldstrengthandr-value;comparisonwithYld96isalsoshown. 69

PAGE 70

Poly4materialcharacterizationofAA2008-T4.Directionalyieldstrengthandr-value;comparisonwithYld96isalsoshown. Figure2-4. Poly4materialcharacterizationoftheNUM'93mildsteel.Directionalyieldstrengthandr-value. 70

PAGE 71

ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA2090-T3. Figure2-6. Poly6andPoly8materialcharacterizationofAA2090-T3.Directionalyieldstrengthandr-value;comparisonwithYld2004isalsoshown. 71

PAGE 72

Tooldimensionsforthesquarecupproblem. Figure2-8. Proleofthedrawnsquarecup,at40mmpunchstroke,anddenitionsofdraw-in(dimensionsinmm). 72

PAGE 73

ThicknessstrainvariationalongtheOBdiagonalaspredictedbyFEsimulationswithPoly4andVonMises,andasmeasuredat15mmpunchstroke. Figure2-10. ThicknessstrainvariationalongtheOBdiagonalaspredictedbyFEsimulationswithPoly4andVonMises,andasmeasuredat40mmpunchstroke. 73

PAGE 74

Geometricalsettingandtooldimensionsforthedeepdrawingwithcylindricalpunchsimulation. Figure2-12. Typicalmeshusedontheblank. 74

PAGE 75

Fullydrawncup(nalconguration).Onlyaquarteroftheblankwassimulated,thefullcupbeingconstructedbysymmetryconsiderations. Figure2-14. ProlesofAA2090-T3drawncupssimulatedwithPoly4,2Dand3D.Yld96simulationisalsoshown,dataafter Yoonetal. ( 2006 ). 75

PAGE 76

ProlesofAA2090-T3drawncupssimulatedwithPoly6-3DandPoly8-3D.TheYld2004simulationisalsoshown,withdataafter Yoonetal. ( 2006 ). Figure2-16. EvolutionoftheproleofthecupduringthedrawingoftheAA2090-T3blank,simulatedwithPoly6. 76

PAGE 77

ProjectionsonthebiaxialplaneofthePoly6yieldsurfacesforMat1andMat2. Figure2-18. Poly6materialcharacterizationofMat1andMat2.Directionalyieldstrengthandr-value. 77

PAGE 78

ProlesofctitiousmaterialMAT1cupaspredictedbyPoly6(FEM),andbyanalyticalformula( 2{78 )proposedby Yoonetal. ( 2006 ).Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. Figure2-20. ProlesofctitiousmaterialMAT2cupaspredictedbyPoly6(FEM),andbyanalyticalformula( 2{78 )proposedby Yoonetal. ( 2006 ).Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. 78

PAGE 79

Theidenticationofthecoecientsofthepolynomialcriteriastudiedinthepreviouschapterwasdonebyminimizingaquadraticfunctionofthesecoecients,theminimizationprocessbeingsubjectedtoquadraticconstraints.Itishoweverpossibletoformulatetheyieldfunctioninsuchawaythattheidenticationprocessreducestotheminimizationofaquadraticwithlinearconstraints.Thenewformulationwillalsoallowfortheintroductionofthestrengthdierentialeectintotheyieldsurface,thatis,theyieldsurfacenolongerhasacenterofsymmetry. theconvexityconditionwascastintheform foranyplanecurveontheunitspherejjxjj=1,seethediscussiononconvexityinthepreviouschapter.Forthepolynomialcriteriaconsideredthere,theyieldfunctionwasintheformf=P1=nnandthisleadtotheconvexityconditions 2{4 ,whicharequadraticinequalitieswithrespecttothecoecientsofthepolynomialPn.Linearinequalitieswouldthenbeobtainedifthesupportfunctionhdependedlinearlyonthematerialparameters.Inparticular,itwouldalsobeimportanttoretaintheconvenienceofthepolynomialformulationoftheyieldfunction.Thesetwoconsiderationsthenleadtothefollowingformula 79

PAGE 80

Moregenerally, Thisformula,obviously,cangeneratelevelsurfacesthatnolongerpossesacenterofsymmetry. Inthecontextofsheetforming,weintendx1=x,x2=y,andx3=xy.However,inthisform,theisotropicrestrictionofformula 3{3 is(wemusthaveh=constant) ThisrestrictiondoesnotcoincidewiththeplanestressrestrictionofvonMises'scriterion: Tosolvethisissueweconsiderthefollowinglineartransformation Imposingthecondition wearriveatasimplequadraticsystemofequationsforthecoecientsdiwiththegeneralsolution (3{12) 80

PAGE 81

Theproposedyieldfunctionsarethen withnaneveninteger,and inwhichu=(u1;u2;u3)isdenedby withconstantsdigivenby 3{13 .Since 3{14 ,or 3{15 ,isobtainedfrom 3{3 ,or 3{5 ,byalineartransformation,itfollowsthat 3{14 ,or 3{15 ,ispositiveandconvexifandonlyif 3{3 ,or 3{5 ,ispositiveandconvex. Theconstantcischosensuchthatboth 3{14 anditsgeneralization 3{15 willreducetovonMises'scriterionwhenallthepolynomialcoecientsin 3{4 aresettozero.Thus 3{4 .ItcanbeeasilyshownthatforagivenpositiveintegerNwehave 3{14 or 3{15 arepositive 81

PAGE 82

3{14 weobtainanewfourthorderpolynomialyieldfunction,withsimplerconvexityrestrictionsuponitscoecients.Thiscaseisfurtherpursuedin SoareandYoon ( 2007 ). Anothermetalwithstrengthsimilartothatofsteelistitanium.Yet,ithashalfthespecicweightofsteel.Italsohashighresistancetocorrosion,andhighermeltingpoint(1725oC)thanbothsteel(about220oCmore)andaluminum(about1000oCmore).Thesepropertiesmaketitaniumidealforuseintoughanddemandingenvironments(in1970,theGrummanCorporationmadethewing-boxofitsF-14ghterjetentirelyof 82

PAGE 83

HullandBacon ( 2001 ),itsucestosaythatbesidesdislocationmodeitalsofavorsplasticdeformationbytwinning,adeformationmodesensitivetotheorientationoftheappliedstress.Magnesium,zinc,zirconiumarealsometalswithHCPlattice. Phenomenologicalmodelsthatincludethisasymmetricbehaviorinplasticdeformationarescarceintheliterature.Therstattemptinthisdirectionseemstobethatof Hosford ( 1966 )whoconsideredthefollowingadaptationofHill'squadraticcriterion However,thisapproachisequivalentwithtranslatingthecenteroftheyieldsurface,leavingthesurfaceitselfunchanged.Asimilarattemptisdescribedin Yoonetal. ( 2000 )whereYld96yieldsurfaceisused,instead,withcentertranslation.Thisapproachhaslimitedcapabilitiesleavingthetopologyofthedirectionalpropertiesintensionandcompressionvirtuallythesame. Usingthemethodofgeneralizedinvariants, CazacuandBarlat ( 2004 )proposedthefollowingcriterion whereJo2andJo3aregivenby 1{36 and 1{37 ,respectively.Inbroadlines,thecritiqueofthemethodbasedongeneralizedinvariantssketchedintherstchapterappliestothisyieldfunctionaswell.Duetothesmallnumberofparameters,10forplanestress,the 83

PAGE 84

Plunkett ( 2005 )proposedthefollowingisotropicyieldfunctionwithstrengthdierential whereiaretheprincipalvaluesofthestresstensor,andkandaarematerialparameters.TheabovefunctionisageneralizationofKaralisandBoyceisotropicfunctionin 1{50 anditisconvexforanyk2[0;1]andanya1.Togeneralizehiscriteriontoorthotropicsymmetrytheauthorthenfollowsthelineartransformationapproachandsubstitutesintheaboveformulatheprincipalstressesofthetransformedstresstensor 1{46 (withanonsymmetrictransformationtensor).Thenumberofmaterialparametersisthus9,for3Dstressstates,and7forplanestressstates,limitingthecriterion'sabilitiestobiaxialyieldcurvepredictionsonly.Toextendthemodelingcapabilitiesofhiscriterion, Plunkett ( 2005 )proposestheuseoftwolineartransformations,similarto Barlatetal. ( 2005 )approachforYld2004: Theanisotropiccriterionnowhas18materialparametersfor3Dstresses,and14forplanestress.Itisappliedin Plunkett ( 2005 )tothemodelingoftheAA2090-T3describedin Yoonetal. ( 2000 ),wheredatafortheyieldstressincompressionisalsoprovided(nodata,however,forr-valueincompression).Thisistherstattemptto 84

PAGE 85

AmoreaccuratecharacterizationoftheAA2090-T3describedin Yoonetal. ( 2000 )isreportedin Soareetal. ( 2007b ),wheretheauthorsgeneralizedtheheuristicapproachof CazacuandBarlat ( 2004 )tothefollowingpolynomialcombinations,forplanestress: withPngivenby 3{4 .ThesucientnumberofmaterialparametersforthePoly65combination,28,allowedtheauthorstoincludeintheirmodelingthecompressionr-valuesalso.Asshownin Soare ( 2006 ),thepredictedproleforthecorrespondingcup-drawingproblemisalmostidenticalwiththeoneobtainedusingasymmetriccriterion.Inparticular,ithassixears. Inthisworkweprefertopursuethemoregeneralapproachproposedintheprevioussection.Overformulas 3{23 and 3{24 ishastheadvantageofamuchsimplerwaytoachievethedesiredconvexity,anditalsoallowsforasimplerimplementationofanisotropichardening. TotestthegeneralapproachproposedintheprevioussectionweconsiderthemodelingofamaterialthathasisotropicdirectionalpropertiesincompressionandthesamebehaviorasAA2090-T3intension.Thisisnotarealmaterial.However,itsmodelingisproposedherefortworeasons.First,asabenchmarkmaterialfortestingthecapabilitiesofthemethod.Second,asseeninthepreviouschapterduringthediscussiononearing,thefewtheoreticalattemptstoexplaintheapparitionandlocationoftheearsstartwiththeassumptionthatthenonuniformproleismainlygeneratedbytheuniformcompressionoftheanisotropicrim,orrings,oftheangearea.Accordingtothesetheories,ifthesheethasisotropicpropertiesincompressiontheproleofthe 85

PAGE 86

Theisotropicdirectionalr-value,incompression,canhavebutonevalue: Thesameistrueforthebiaxialvalue.Thusrcb=1:0.Weconsidertwomaterials,Mat1andMat2,withdierentisotropicdirectionalyieldstrength.IncompressionMat1willhave and and 2-1 Tomodelthesetwomaterialsweusedthefollowingyieldfunction whichwillbereferredtoasPoly87.Theidenticationprocedureanditsimplementationdetailsarepresentedinthenextsection.TheresultsofPoly87modelingofMat1andMat2areshowninFigures 3-1 3-2 ,andFigures 3-3 3-4 ,respectively.ThecorrespondingcoecientsarelistedinTables 3-1 and 3-2 .Poly87has45materialscoecients.Thebasicinputdataconsistedof32datapoints:16yieldstresses( 86

PAGE 87

ThedeepdrawingofMat1andMat2blanksheetswasthensimulatedusingAbaqus/Standard.TheparametersofthesimulationswereidenticalwiththeonesfortheAA2090-T3simulationdescribedinthepreviouschapter,theonlydierence,ofcourse,beingthematerialofthesheet.TheresultingprolesarepresentedinFigure 3-5 .Contrarytowhatwouldnormallybeexpected,theprolesarefarfromuniform.ThecorrespondingevolutionoftheMat2proleisalsoshowninFigure 3-6 .Fourearslocatedat45o,135o,etc,fromRD,startdevelopingrightfromthebeginningofthedrawingprocess.ThenalearingproleofMat2hassixears,whereasthatofMat1hasfour.ThetwoextraearsofMat2proleareverysmall,arelocatedat0oand180ofromRD,andstartdevelopingquitelateintheprocess,after50%punchtravel. Sincetheearingprolesarenotuniform,itfollowsthatthecompressivehoopstresswithintheangeareadoesnotentirelyexplaintheearingprole,andthereforethetensileradialcomponent,duetothepullingofthepunch,mustalsobeconsidered.Thepurelykinematicapproachof Yoonetal. ( 2004 )canbeastartingpointforamorecompletemodelingoftheproblem.Consideringtheradialstresswillallowforincludingtheeectsoftheblank-holdingforcetogetherwiththefrictioncoecientbetweenblankandtools.Thisisatopicoffutureresearch. 87

PAGE 88

LetTdenotetheyieldstressforauniaxialtensiontestatananglefromtherollingdirection.Thestresseldwithinthesamplehaswithrespecttothesymmetryaxesthecomponents []=T264cos2cossincossinsin2375(3{30) andthepredicteddirectionalyieldstressis where,accordingto 3{16 DenotingwithCtheyieldstressfortheuniaxialcompressiontest,itspredictedvaluewillbe withubeingthesameasin 3{32 .Similarly,forbiaxialtension/compressiontests,withTBandCBtheyieldstresses,respectively,thecorrespondingstresseldsare(withrespecttothesymmetryaxes) []=TB2641001375;[]=CB2641001375(3{34) andthepredictedvaluesare 1 88

PAGE 89

Thepartofthemeritfunctionregardingtheyieldstressesisthen 1 2wTB[ 2XwT[ wherethew'sareweights,andtheexpexponentindicatesexperimentaldata. Asimilarreasoningisrepeatedforthepredictionofthedirectionalandbiaxialr-valuesintensionandcompression,wherethecomputationswilltakeintoaccountthefollowingrelations ThegradientofMises'snormis Toshortenthenotation,inwhatfollowswewillalsousetheindexing andthenthegradientoftheyieldfunctionwillbe @i=gih+jjjjM@h @i(3{43) with,summationimplied, @i=@P1 89

PAGE 90

TheHessianoftheyieldfunctionisneededinboththematerialidenticationandusermaterial(UMATinAbaqus)subroutines(theHessianofhisactuallyneededfortheconvexityconstraints).Inourcaseittakestheform @i@j=@gi @j+gj@h @i+jjjjM@2h @i@j(3{46) Wehave @jgigj(3{47) andthus Finally,forthehessianofhwehave(summationimplied) @i@j=@2P1 with (3{50) 1

PAGE 91

(3{51) 1 "@2u3 (3{52) 1 Forconvenience,thepolynomialexpressionsarealsoprovidedhere(withadded"optimized"variantsfornumericalimplementation). {z }C0+ {z }C1u23 {z }C2u43+(a19u1+a20u2)| {z }C3u63

PAGE 92

SimilarlyforthegradientandhessianofP7wehave:

PAGE 99

Poly8coecientsforthetwomaterialsMat1andMat2. Table3-2. Poly7coecientsforthetwomaterialsMat1andMat2. 99

PAGE 100

Poly87materialcharacterizationofMat1.Directionalyieldstrengthintensionandcompression,andr-valueintensionandcompression. Figure3-2. ProjectiononthebiaxialplaneofMat1yieldsurface. 100

PAGE 101

Poly87materialcharacterizationofMat2.Directionalyieldstrengthintensionandcompressionandr-valueintensionandcompression. Figure3-4. ProjectiononthebiaxialplaneofMat2yieldsurface. 101

PAGE 102

ProlesofthetwoctitiousmaterialsMAT1andMat2cupsaspredictedbythePoly87simulation.Forcomparison,theexperimentalproleofAA2090-T3isalsoshown. Figure3-6. EvolutionoftheproleoftheMat2cupduringdrawingprocess(simulatedwithPoly87). 102

PAGE 103

Avastproportionofthescrapsandrebutsinthemetalformingindustryisgeneratedbyfailureofthemetalsheetduringformingprocessesthatinvolvebiaxialstretchingofthesheet.Thisfailureusuallyappearsintheformoflocalizedfracture,splitting,tearingandsoon.Allthesetypesoffailureareprecededbylocalizednecking:somesmallareaofthesheetmakesarapidtransitionfromtheintendeddeformationmodetoamostlythinningdeformationmode(planestrain).Alocalizedbandofrapidlydecreasingthicknessthenstartspropagatinginthesheet,reducingitsloadbearingcapacityandultimatelyleadingtofracture,splitting,etc.Thematerialpointinthesheetattheoriginofthisinstabilitymaybeanythingfromasmallzoneinthesheetwithresidualstresses(leftbyincompleteannealing,forexample),oramaterialimperfection(abiggerthanusualconcentrationofmicro-voids),orevenaverysmallvariationinthicknessduetothefabricationprocess. Forthedesigningofaspecicformingprocessitisthenessentialtoknowhowmuchstrainthemetalsheetcanendureuntilthelocalization(necking)setsin.Thislead KeelerandBackofen ( 1963 ),andlater Goodwin ( 1968 )toproposetheconceptofaforminglimitdiagram,orinshort,FLD.Onseveralblanksheetsgridsofgaugecirclesaremarkedandtheneachblanksheetissubjectedtoaspecicin-planebiaxialloading,withaconstantstrainratio,say=d2=d1duringtheentireloadinghistory.Thenforeachstrainratio,thelimitingstrainsaremeasured(bymonitoringthegaugecirclesattheinstantofneckinitiation)andthenplottedona(2;1)diagram.Theresultingcurveinthe(1;2)strainspaceistheforminglimitcurve(FLC)beyondwhichthesheetshouldnotbeloaded.ThistypeofFLDiscurrentlyknownasthelinearFLD,duetotheproportionalstrainpathsthatareusedtogenerateit.However,inpracticemostoftheformingoperationsareperformedinmultiplestepswithdierentloadingconditions,withtheconsequencethatabruptchangesinthestrainpathatamaterialpointusuallytakeplace.Inthiscasetheforminglimitdiagramistermednonlinearanditusuallydiersconsiderablyinshapeandmagnitudefromthelinearone,seeforexample BaratadaRochaetel. ( 1984 ).Thus,the 103

PAGE 104

Regardingthetheoreticalanalysisofnecking, Swift ( 1952 )wasthersttoproposeafailurecriterion.However,sincehiscriterionisconcernedonlywithdiusenecking(aprecursorstatetothelocalizednecking,ofverymildinstability)weshallnotpursueitsdetailshere.Thersttheoreticalanalysisoflocalizedneckingisproposedin Hill ( 1952 ).Althoughthistheoryislaterrenedin Hill ( 2001 ),Hill'smethodofanalysiscannotproducepredictionsfortheentirestrainratiorangeoftheFLC,andthereforeitwillnotbepursuedhereeither.Itistheworkof MarciniakandKuczynski ( 1967 )thatrstproposedageneraltheoreticalmethodforpredictingtheentireFLC,andithasbecomethestandardtoolformostofthepresenttheoreticalworkonFLD's. TheFLC'saspredictedusingtheMarciniakandKuczinskitheorydependstronglyontheshapeoftheyieldsurface,seeforexample Barlat ( 1987 ),inthesensethattwoyieldsurfacesthatdierverylittleinshapeandmagnitude,canpredictFLC'sthatdierremarkablyinmagnitude.However,thisfacthasoftenbeen(mis)interpretedasdependenceontheyieldcriterion,e.g., KurodaandTvergaard ( 2000 ), FriedmanandPan ( 2000 ),theissueactuallybeinganimproperorincompletecharacterizationoftheexperimentaldataregardingtheyieldsurface.ItisalsoworthmentioningthatmostoftheexperimentallyproducedFLC'sexhibitawidescatteringamongtheirdatapoints,andsomescatteringisalsopresentamongthemeasureddatapointsontheyieldsurfaceitself.Thispromptssomeresearcherstoconsidertheconceptofforminglimitbandswithappropriatebandpenetrationfactors. ThischapterstartswiththedescriptionoftheMarciniakandKuczynskimodelastheanalysistoolforFLCprediction.Itthencontinueswithapplicationsofthepolynomial 104

PAGE 105

4-1 .TheinitialanalysisofMarciniakandKuczynskiassumedthegrooveparallelwithoneofthesymmetryaxes.However,toaccountformoregeneralitywewillfollowherethesuggestionof HutchinsonandNeale ( 1978 )andassumetheinitialgrooveorientedatananglewithrespecttotherollingdirection.WeshallrefertothezoneoutsidethegrooveaszoneA,whereasthegrooveitselfwillbereferredtoaszoneB.ForanyfunctiondenedonthedomainofthesheetwewillthendistinguishitsrestrictiontozoneA,oraparticularvalueatalocationinzoneA,byattachingtoitasuperscriptA,andsimilarlyitsvaluestakeninzoneBwillbedistinguishedbythesuperscriptB.AglobalCartesiancoordinatesystemisalignedwiththesymmetryaxes:thex-axisisalongtherollingdirection,andsoon.Alocalorthonormalframe(n;t)formedbytheexteriornormalandthetangentialdirectiontothegroovewillalsobeconsidered.Theboundaryofthesheet,assumedatafardistancefromthegroove,isthensubjectedtomonotonicproportionalstrainingparallelwiththesymmetryaxes 105

PAGE 106

Theinitialgeometricalimperfection,thegroove,ischaracterizedbytheratio wherehA0andhB0aretheinitialthicknessesofzoneAandzoneB,respectively.Howtochoosethisinitialvalueorexplainitsphysicalrelevanceseemstobe,still,asubjectofdebate.In Barlat ( 1987 ),forexample,itissuggestedthattheinitialimperfectionratiobeconnectedwiththeinternaldamageduetosomerandomlocalaccumulationofmicro-cavities.Inthecitedworkaxedvalueof0:996isthenrecommendedforaluminumsheets.Theopiniontrendtoday,however,seemstobethattheimperfectionratioshouldbeviewedasaninternalparameterofthemodel, NarasimhanandWagoner ( 1991 ),andthereforeitcanbeadjustedtobetterttheexperimentaldata.Thenf0canalsotakevaluesaround0:98forsomealuminumsheets, Banabicet.al ( 2005b ),orevenaround0:97forsomesteels, BaratadaRochaetel. ( 1984 ). Asthestrainingattheboundaryincreases,thecontinuouslyreducingthicknessofzoneBhastobearincreasinglyhigherstressesthanthoseinzoneA.ThenduringthedeformationhistoryapointisreachedwhenthematerialinsidethegroovestartsdeformingatmuchfasterratesthanthebulkmaterialinzoneA,signalingtheinitiationofunstabledeformation,ornecking,seeFigure 4-2 .Thefailurecriterionisthen B ANf(4{3) thelimitingstrainsinthesheetforthecorrespondingstrainratiobeingthestrainsinzoneAatthemomentoffailure. 106

PAGE 107

( 2005b ),itcouldbe10, Caoetal. ( 2000 ),oreven300, Wuetal. ( 2003 ).ThepointisthatitshouldanumberbigenoughsothatzoneBapproachesaplanestrainstateofloading(seenextsection).Oncetheinstabilityistriggered,theaboveratioincreasesextremelyfastwithverysmallstrainincrementsinzoneA(actually,afterlocalizedneckinghasdeveloped,zoneAhasanalmostrigidbodymotion, Hill ( 2001 )). Tocompletethemodel,equilibriumandcompatibilityconditionsattheinterfacebetweenthetwozonesaretobeimposed.EquilibriumattheinterfacebetweenzoneAandzoneBrequires (B:n)hB=(A:n)hA(4{4) whereistheCauchystresstensor,andhisthecurrentthicknessofthesheet.CompatibilityofdeformationrequiresthatattheinterfacebetweenzoneAandzoneBthestrainincrementalongtheinterfacebecontinuous HutchinsonandNeale ( 1978 ), Hill ( 2001 ), Wuetal. ( 2003 ),amoregeneralformofthecompatibilityconditionispreferred whereListhevelocitygradient,andvanundetermined(velocity)vector,anddenotesthetensorialproduct,thus(vn)ij=vinj.Itiseasytoseethatequation 4{5 iscontainedintheaboverelation,forwehave 2(vn+nv)(4{7) withDbeingtherateofdeformationtensor(thesymmetricpartofL),andthen 2n(vt)=)DB:tt=DA:tt(4{8) 107

PAGE 108

4{6 togetherwiththerateformoftheequilibriumequation 4{4 andtheobliqueorientationofthegroovearethebasicingredientsforthemoreformalapproachof HutchinsonandNeale ( 1978 )totheM-Kmodel.Thisapproachhasitsconvenienceswhenmoresophisticatedconstitutivemodelsareusedtosimulatetheplasticityofthesheet,e.g., Dafalias ( 2000 ), Leeetal. ( 1995 ), Wuetal. ( 1997 ).However,fortheclassicalconstitutivemodelusedhere,thereisnogainintheaccuracyofthepredictedFLD'soverthesimpler,initialM-Kformulationadoptedhere,andalsoin,e.g., Butucetal. ( 2003 ), Banabicet.al ( 2005b ). Equations 4{4 and 4{5 ,togetherwiththeconstitutiverelationsforthemetalsheet,allowforthecompletedeterminationofthestrainsinthesheetbyusinganincrementalapproachasfollows.DuetothelargeamountofplasticdeformationinvolvedinanFLDpredictionproblem(usuallybetween20%-50%),weneglecttheelasticstrainsandthusassumethematerialofthesheetasbeingrigid-plastic.WealsoassumethatthedeformationinzoneAishomogeneous.Thisiscertainlynottrueintheimmediatevicinityofthegroove,when<90o.However,thewidthsofthenecksobservedinpracticeareverysmall,ofthesameorderofmagnitudeasthethicknessofthesheet,andthereforethewidthofthegrooveintheM-Kmodelcanbeconsideredtobeverysmall,ergothedeformationinzoneAcanbeconsideredhomogeneous.Then,takingasinputequation 4{1 andusingtheconstitutiverelations,thestressesinzoneAarecomputed.ThenassuminghomogeneousdeformationinzoneB(thistimeduetoitsverysmallwidth),usingtheequilibriumequations 4{4 ,thecompatibilityequation 4{5 andtheconstitutiverelations,thestrainsinzoneBarecomputed.Incrementsintheboundarystrainsareaddeduntilinequality 4{3 isveried,whichsigniesfailure,thenalstrainsinzoneAbeingthelimitingstrainsforagiveninitialangle0.Foragivenstrainratio,theprocessisrepeatedforalltheinitialangles02[0o;90o](ofcourse,inpracticeonlyadiscretegridofvaluesisused),thusobtainingafunctionAx(0).ThelimitingstrainAxis 108

PAGE 109

4{1 .Theprocesscontinuesuntilfailureaccordingtoformula 4{3 isreachedinsidezoneB. Next,sincethematerialisorthotropic,andsincethestrainincrementsareparallelwiththesymmetryaxes,itfollowsthatthestresstensorisparallelwithwiththesymmetryaxes,andthustheshearcomponentiszero.Indeed,fromthenormalityrulewehave @(4{11) andsincedAxy=0,itfollowsthat@f=@xy=0.Assumingtheyieldsurfaceisnotacylinderparallelwiththexy-axis(thatis,theyieldfunctiondependsexplicitlyonthesheartermxy),theorthotropicsymmetryoftheyieldsurfaceisequivalentwiththesymmetryofthesurfacewithrespecttothexy=0plane,andthenthelocusoftheyieldstresseswith@f=@xy=0istheplanexy=0.Thus 109

PAGE 110

A)(4{13) withd Adenotingtheincrementinequivalentplasticstrainduetothecurrentstrainincrement,andHdenotingthehardeningcurveofthematerial.Also,atthestresspointontheyieldcurve(inthebiaxialplane)theexteriornormaltothetheyieldcurvemustbecollinearwiththestrainincrement: @()dAx=d@f @x;dAy=d@f @y=)dAy Thus @y=@f @x(4{15) Finally,theincrementinequivalentplasticstrainiscomputedfromtheworkequivalenceprinciple, 1{6 : Af(A)dA:A=0(4{16) Equations 4{13 4{15 and 4{16 forma33nonlinearsystemtobesolvedforthethreeunknownsAx,Ay,andd A.Itcanbereducedtoonlyonenonlinearequationasfollows. Forsomet0wecanwriteAy=tAx,andequation 4{15 canbewrittenas @y(Ax;tAx)=@f @x(Ax;tAx)(4{17) Sincetheyieldfunctionfisrstorderhomogeneous,itsrstorderpartialderivativesarezerothorderhomogeneous: @()=@f @();(8)0(4{18) andthenfactoringouttheAxcomponentinequation 4{17 ,weobtainanonlinearequationfortheunknownproportionalityfactort @f @y(1;t)=@f @x(1;t)(4{19) 110

PAGE 111

2{5 ,equation 4{19 takestheform andsimilarly,forasixthorderpolynomialyieldsurfacethelefthandsideoftheequationissixthorderpolynomialint,etc. Next,usingtherstorderhomogeneityoffweobtainfrom 4{13 A)=)Ax=H( A) andsubstitutingintoequation 4{16 weobtain A=1+t f(1;t)dAx(4{22) Foreachstrainratioequation 4{19 needstobesolvedonlyonce.Itcanbeeasilysolvedwithafew(alwaysconvergent)Newtoniterations.For>0agoodinitialguessist0=1:0,whereasfor0theinitialguessshouldbetakent0=0:0.Thentheincrementoftheequivalentplasticstrainiscomputedusingformula 4{22 ,andnallythestresses A)=f(1;t) (4{23) B@f @B(4{25) 111

PAGE 112

4-3 .Then,if(x0;y0)denotestheinitialpositionofanarbitrarypointontheboundarybetweenzoneAandzoneBintheinitialconguration(theinitialun-stretchedsheet),and(x;y)denotesitscurrentpositionduringthedeformationprocess,wehave x0;Ay=lny y0(4{26) fromwherewegetAxAy=lnx yy0 y=exp(AxAy)x0 Wuetal. ( 2003 ), tan=exp[(1)Ax]tan(0)(4{27) Thisspeciestheorientationofthelocalframe(n;t)ofthegrooveatanymomentduringthedeformationprocess: Fromnowonthecurrentmomentwillbedenotedtn+1,thestrainincrementdAxtakingplaceinthetimeinterval[tn;tn+1].Theevolutionoftheimperfectionfactorf=hB=hAisdescribedasfollows.Forthethickness(logarithmic)straininzonesAandBwehave 112

PAGE 113

Theaboveformulais"implicit":itdependsontheyetunknowncurrentvalueofBz(n+1).However,usingasmallstrainincrementdAxallowsustouseinsteadthefollowingexplicitformulatoupdatef: Att1,f1=f0.Thethicknessstrainsarecomputedusingtheincompressibilitycondition: ThusinzoneA whereasinzoneB Withrespecttothelocalframe(n;t)ofzoneB,theequilibriumequation 4{4 atthecurrentmomenttn+1takestheform fromwhereweget withfgivenby 4{32 .Thecomponentsonthe(x;y)axesofthestresstensorinzoneAatthemomenttn+1areknown,andthenits(n;t)componentsare []A(nt)=[Q]T[]A(xy)[Q];with[Q]=264n1n2n2n1375(4{38) 113

PAGE 114

B.Atwo-by-twononlinearsystemofequationsforBtandd Bisformedbyusingtheyieldconditionandthecompatibilitycondition 4{5 Althoughfromaglobalperspectivethegrooverotateswithrespecttotheglobal(x;y)frame,anyinnitesimalelementinzoneBisactuallysubjectedtoinplanestretchingandtranslation.Thus,atanylocationwithinzoneBthesymmetryaxesremainxedwithrespecttotheirinitialposition,thexandy-axes,duringtheentiredeformationhistory.ThentoexpressthestateofyieldinginzoneBweneedthecomponentsofthestressBwithrespecttothesymmetryaxes.Similarlywith 4{38 Theyieldconditionprovidestherstequation B):=f(Bx;By;Bxy)H( B)=0(4{40) with [dB](xy)=d B@f @B(xy)(4{41) Thelefthandsideoftheaboveequationcontainsthecomponentsofatensorialquantity,whereaswehavetorecallthatthelistofargumentsoftheyieldfunctiondoesnotcoincidewiththeentirelistofcomponentsofthestresstensor,thatis,insteadoftakingboththeshearcomponentsxyandyxasarguments,forbrevityweonlytakexy.Thereforeinthe 114

PAGE 115

2@f @Bxy(4{42) Takingthisintoaccountandusingaformulaoftype 4{38 weobtain B@f @Bxn22+@f @Byn21@f @Bxyn1n2(4{43) SinceinzoneAwehaveAxy=0andsincetheyieldfunctionisorthotropic,wemusthave@f=@Axy=0(forpolynomialtypecriteriathisisevident,sincexyappearsonlyatevenpowers).Then A@f @Axn22+@f @Ayn21(4{44) Finally,thesecondequationisobtainedfrom 4{5 B):=dBtdAt=0(4{45) withtherighthandsidecomputedaccordingto 4{43 and 4{44 B@f @By=d A@f @Ay=)@f @By=d A B@f @Ay(4{46) Takingintoaccountthat@f=@Ayisalwaysanitenumber,theaboveformulaandthefailurecriterion 4{3 implythatzoneBreachestheunstableregimeofdeformationwhen@f=@By!0,thatiswhendBy!0.Thusatfailure(necking)zoneBstopsdeformingalongitslength:planestrain.Thesameistrueforthegeneralcasewhenthegrooveisoblique,thedeformationinthelongitudinaldirectionapproachingzeroneartheinstabilitymoment. Thesystemofequations 4{40 and 4{45 isnonlinearandthereforeitissolvednumericallyusingtheNewton-Raphsoniterativemethod.Thus,withthenotationsF=(F1;F2),andv:=(Bt;d B),thesequenceofiterationsisdenedbyvk+1=vk+v, 115

PAGE 116

TheJacobianinthelefthandsideoftheaboveequationiscomputedinAppendix D .Asinitialguessforthesequenceofiterationsweusedthevaluesfromthepreviousincrement B)n(4{48) andfortherstincrement(whennostressorstrainisyetavailableinzoneB)weusedthecorrespondingvaluesinzoneA A)1=(Axn22+Ayn21;d A)1(4{49) Besidesthematerialparametersoftheconstitutivemodel,itremainstospecifythesizeoftheinitialimperfectionf0.Thisparameterwillbechosentobestttheplanestrain(=0)datapointontheFLC.Thus,theFLDproblemcanberestatednowas:topredicttheentireFLCknowingtheconstitutiveparametersofthematerialandtheplanestraindatapointontheFLC. 4-2 ,thehardeninglawofthematerialplaysanessentialroleinthedeterminationoftheFLC.Anotherequallyimportantparameteristheyieldsurface.InthissectionweapplythepolynomialcriteriadescribedinChapter2tothepredictionofFLD'sfortwoaluminumalloys. Banabicet.al ( 2005b ).Thedirectionalyieldandr-valuedataarelistedinTable 4-1 .Theisotropichardeninghardeninglawisgivenintheform 116

PAGE 117

Poly4andPoly6characterizationsofthisalloysareobtainedfollowingtherecipeinChapter2.ThecoecientsarelistedinTables 4-2 and 4-3 .ForPoly4theweightsusedwerew11=1:00;w12=0:50;;w21=1:00;;w22=0:50 Forthe"in-between"dataweused1=15o,2=75o,andin 2{45 and 2{46 weused1=0:7;2=0:3;1=0:8;2=0:3 ForPoly6thein-betweendatawasgeneratedbyPoly4yieldstrengthandr-valuevaluesatthecorrespondinglocations(=15o;30o;60o;75o).ResultsareshowninFigures 4-4 and 4-5 .ItisremarkablethatbothPoly4andPoly6yieldsurfacestexactlytheyieldstrengthalongRDandTDdirectionsandalsotheequi-biaxialyieldstrength 4-6 Figure 4-7 showstheevolutionofthefailurecriterion 4{3 withrespecttotheequivalentstraininzoneAforplanestrainloading(=0).Thisevolutionseemstobeindependentoftheyieldfunctionused.Indeed,itisclearrstthatthelimitstrainisreachedfortheorientationofthegroove=0o,sinceforthisorientationbothzoneAandzoneBarefromthebeginninginplanestrainstate,whereasalltheotherorientationsofthegrooverequireadditionalstraininginzoneA.Thenthefailurecriterion 4{3 issatisedonlywhenzoneBispulledbyzoneAsucientlyhighonthehardeningcurve.ThustheplanestrainstrainpointontheFLCismostlydeterminedbythehardeninglaw.Several 117

PAGE 118

4-8 ,thelimitstrainAxcanbepositionedinthiscasewithintheinterval[0:15;0:2).Thiswouldrequireaninitialimperfectionf00:96.However,thisvalueprovestobetoosmall,foritgenerateslimitstrainsthataretooconservative.TooconservativeestimationsoftheFLCconstraintdrasticallythedesigningprocessbylimitingtoomuchtheamountofstrainthatcanbecountedon,whereastoo"liberal"(over)estimationsoftheFLCcanleadtodesignsthatinpracticebreaktheblanksheet.Weadopttheinitialvaluef0=0:98,asrecommendedin Banabicet.al ( 2005b ).But,asnotedabove,thehardeninglawforthismaterialmightcontainsomeerrorandthereforefurtherdiscussionwouldneedtotakeintoaccountthiscorrection. Figure 4-8 showsthepredictedFLC'susingPoly4andPoly6yieldfunctionsinconjunctionwiththeM-Kmodel.Forvericationpurposes,planestressniteelementcalculationswiththegrooveorientedalongtheTDdirection(=0o)havealsobeenconducted(usingABAQUS/Standard).Duetosymmetryonlyaquarterofthesheetwassimulated.Besidesthesymmetryconditions,theconstantstrainratio,forthecasewhen0,wassimulatedbyimposingthedisplacementboundaryconditions whereLwasthelengthofthesquareblanksheetusedinthesimulation(u1onlywasimposedononeside,andu2onlyontheother).Onereferencepoint,A,farfromthegroovewaschoseninzoneAtorecordthecurrentstrain,andanother,B,waschoseninthecenterofthegroove.WhenthefailurecriterionwasmetatpointB,theUMATsubroutinereportedthecurrentstrainatpointAandthencausedaforcedendingofthesimulation.ThelefthandsideoftheFEpredictedFLCwasdenedasthelinearsegmentconnectingtheplanestrainandsimpleuniaxialtensionlimitpoints.ThusonlyoneFEsimulationwasperformedfornegativeminorstrains,withtheboundarycondition 118

PAGE 119

@y(x;0)@f @x(x;0)=r0 Inparticular,foranisotropicmaterialr0=1andtheuniaxialtensiontestcorrespondsto=0:5.However,imposingtheaboveratioattheboundaryisnotequivalentwiththeuniaxialtensiontestwhereaboundaryisfreeofconstraints.Nevertheless,ifthisratioisusedintheM-Kmodelwith=0oandwithoutthecompatibilitycondition,thesamelimitingstrainvalueasintheFEsimulationisobtained.ThisthencanbethebasisforasimpliedM-Kmodelinwhichonlythebiaxialyieldcurveisused. TherstthingtobenoticedinFigure 4-8 ,however,isthescatteringoftheexperimentaldatapoints.Especiallythewiderangespannedbythenearplanestraindatapoints,(0:15;0:2),suggeststhedicultiesinvolvedintheprocessoftesting.ThenextnoticeablefeatureisthedierenceinthetwoFLC'saspredictedbyPoly4andPoly6.Forthebiaxialregime(2(0;1]),theminimumin 4{9 wasalwaysattainedfor=0o.Sincefor=0onosheartermsareinvolvedineitherstressorstrain,itfollowsthatthebiaxialyieldcurveonlywasenoughtodeterminetheentirerighthandsideoftheFLC.Atarstglance,thetwobiaxialyieldcurvesinFigure 4-6 areveryclosetoeachotherparticularlyinthestressrangesinvolvedinbiaxialstretching.AnexplanationofthedierenceinthetwoFLC'spredictionscouldbesuggestedbynotingthepositionsofthetwostressstatescorrespondingto=0and=1oneachcurve, SowerbyandDuncan ( 1971 ).Duringequi-biaxialstretching,=1,thestressstateinzoneBstartsfromtheupperpointonitsyieldcurveand,hardeningadded,itthentravelsdownontheyieldcurvetowardstheplanestrainpoint,=0.Ifthen,forthemoment,weimaginetwoyieldcurves(1)and 119

PAGE 120

y)and( y),with (2)x,thecurve(1)willpredictalowerlimitstrainthantheonepredictedby(2).ThisisbecauseittakeslowerstresslevelsinzoneA,forcurve(1),toreachthestress( y)inzoneB.LowerstresslevelsinzoneAmeanlesshardeningandthuslessstraininzoneA.However,thisargumentneedsfurtherrenementsinceinourcasePoly4couldplaytheroleof(1),seeFigure 4-6 ,whereasPoly6couldtaketheroleof(2),andasweseeinFigure 4-8 themonotoniccharacterofthecorrespondencebetweenthebiaxialcurveanditspredictedFLCisbrokeninthiscase.Nevertheless,onecangeneralizetheaboveargumentas:theloweristhepositionofequivalentstressoftheplanestrainloadingonthehardeningcurve,thelowerthelimitstrainpredicted. Theleft-handsideoftheFLCisalmostastraightline.Thisisageneralcharacteristicnomatterwhattypeofmodelisusedforitsprediction,M-KorHill.However,fortheM-Kmodeltheorientationofthegrooveisessential.Forexample,withPoly4asyieldsurface,forthesimpletensiontest,inwhichcase=0:391,theangleforwhichtheminimuminequation 4{9 isattainedis=18o,forwhichthelimitstrainisAx=0:3332.Thesameistrueforanynegativestrainratio,thatis,theminimumstrainisachievedforgrooveorientationsotherthan=0o. Wuetal. ( 2003 )fromwheredatainTable 4-1 istaken(theextension(W)isusedtodistinguishthisdatasetfromtheoneusedintheprevioussubsection).Inthecitedworkthehardeninglawisprovidedinthedierential(incremental)form =K21 K1Nd (4{54) 120

PAGE 121

wheretheadditionalconstantistheinitialyieldstressalongtheRDdirection,0=120:0MPa. TheexampleintheprevioussubsectionconrmsthatthepredictedFLCisverysensitivetotheshapeoftheyieldfunction,andinparticulartotheshapeofthebiaxialyieldcurve,e.g., Barlat ( 1987 ), Butucetal. ( 2003 ), Wuetal. ( 2003 ).Therefore,inordertoobtainasaccurateaspossibledescriptionsoftheyieldsurfaceweshallusefromnowonPoly6andPoly8asyieldfunctions.Thiswillallowustofurtherrestricttheshapeofthebiaxialyieldcurvebytakingintoaccountadditionaldatapointsinthebiaxialplane,besidesthethreepoints( b),and(0; 90).Thus,letusassumegiventhedatapoint( y):=(x=0;y=0)inthebiaxialplane.Denotingwithgtherestrictiontothebiaxialplaneoftheyieldfunction,intheidealcasethebiaxialyieldcurvepassesthroughthisdatapointif0=g(x;y)()1=gx y) Inpracticewetrytominimizethedistancebetweentheyieldcurveandapossiblesetofbiaxialdatapoints MinNXk=1wkg( (k)y)12(4{56) wheretheoptimizationtakesplaceinthespaceofthematerialparameters,andwkweighstheclosenesstothedatapoint( (k)y).TheabovedistancefunctionisaddedtothegeneralobjectivefunctioninChapter2,equation 2{53 YieldfunctionslikeYld96areprimarilydesignedforthemodelingofaluminumanditsalloys, Barlatetal. ( 1997 ).AsdiscussedinChapter1,FCCmetalswithrandom 121

PAGE 122

Wuetal. ( 2003 ).Seegure 4-14 .TogetherwiththedirectionaldatathesedatapointswereincludedinourPoly6andPoly8modelingofthisalloy.TheidentiedcoecientsarelistedinTables 4-3 and 4-4 ,respectively,onthe5182-O(W1)lines.InFigure 4-9 bothPoly6andPoly8predictionsforthedirectionalpropertiesareshowntoagreeverywellwiththeexperimentaldata.Figure 4-10 showsthecorrespondingyieldsurfacesandFigure 4-14 showstherstquadrantofthecorrespondingbiaxialyieldcurves.TheYld96biaxialpointsarewellcapturedbybothPoly6andPoly8.Actually,thetwobiaxialyieldcurvesalmostoverlapeachother.Still,therearedierencesinthevariationofthelocalframe(tangentandnormal)ofthetwocurves:asshowninFigure 4-14 ,whilethetwo=1pointsalmostcoincide(withtheequi-biaxialyieldstress),the=0pointsareclearlydistinct. Aftertestingseveralvaluesforthesizeoftheinitialimperfection,wefoundthatf0=0:9995(alsoadoptedin Wuetal. ( 2003 )),orf0=0:9996werethebestchoices,seeFigure 4-11 .Thesevaluespredictapproximately0:192forthelimitplanestrain.Theexperimentalvalue,however,isaround0:215.Thisvalueclearlycannotbematchedforanyf0.Likeinthecaseof Banabicet.al ( 2005b )data,thiscanbeexplainedeitherbythefactthattheanalyticalexpressionofthehardeninglawdoesnotoerasatisfactorydescriptionoftheexperimentalhardeningcurve,orbythefactthatthehardeninginthisalloyisanisotropic,causednotbytexturereorientationbutbyanisotropicductiledamageinthematerial.AnisotropicductiledamageseemsmoreplausibleheresincetheexperimentalFLCinFigure 4-16 hasthebiaxiallimitstrainashorizontalasymptote,whichwouldimplyacceleratedhardening(bynucleationandcoalescenceofmicro-voids). 122

PAGE 123

4-16 .Theirrighthandsidewasagaincompletelydeterminedbythebiaxialyieldcurve,thatisforeachstrainratiotheminimumstrainwasattainedfor=0o.Poly8FLCisidenticalwithYld96FLC.ThedistancebetweenthePoly6andPoly8FLC'sislessthan0:025.Thisisabout6%oftheequi-biaxiallimitstrainofPoly8,closetoanacceptableerrorfromthepracticalpointofview.Nevertheless,takingintoaccountthatthebiaxialPoly6andPoly8yieldcurvesalmostoverlapeachotherinFigure 4-14 ,thedierencebetweenthetwoFLC'sisstillremarkable. Returningtotheidenticationofthebiaxialyieldcurve,wesuggestnowthattheadditionaldatapointsinthebiaxialplanebetakenas (k)y=(1k)( b);0
PAGE 124

4-17 theFLC'spredictionsareshown.Theinitialinhomogeneitywastakenf0=0:9996.TheresultsaresimilarwiththeonesinFigure 4-16 ,whereYld96biaxialpointswereusedinstead.ThePoly6FLCnowcoincideswiththeYld96FLC,whereasthePoly8FLChasthesatisfactoryconservativeaspect:nolimitpointisoverestimatedbutinthesametimetheunderestimationismild. Wuetal. ( 2003 ).TheexperimentaldataislistedinTable 4-1 .Thehardeninglawwasprovidedintheform Thedirectionaldataat=15o;30o;60o;75owasgeneratedusingPoly4.Withthesamew-weightsasforAA5182-O,butwith1=1:0;2=0:5;1=0:8;2=0:3 weobtainthePoly4coecientslistedinTable 4-2 Forthismaterialthecitedpaperalsoprovidesasetofexperimentaldatapointsinthebiaxialplane.Therefore,asforAA5182-O(W),usingPoly6andPoly8weattempttwodescriptionsofthismaterial.Onebasedontheexperimentaldirectionalandbiaxialdata,theotherbasedonexperimentaldirectionaldataandoptimizationofthebiaxialcurvetowardtheanisotropicinscribedhexagon.ThematerialcoecientsforbothdescriptionsarelistedinTables 4-3 and 4-4 onthelinesidentiedas3104-H19(1)and3104-H19(2),respectively.Figures 4-18 4-19 and 4-20 presenttheresultsoftheoptimizationusingexperimentalbiaxialdata,andFigures 4-24 4-25 and 4-21 presenttheresultsoftheoptimizationusingtheinscribedanisotropichexagonasinput. Theinitialsizeoftheimperfectionhastottheplanestrainstraindatapoint.Foralimitof0:04thebestchoiceisf0=0:992,seeFigure 4-26 .Figures 4-22 and 4-23 presentthecorrespondingFLC'spredictionsforeachidenticationprocedure.Assumingthe 124

PAGE 125

ThesharposcillationofthePoly8FLCneartheplanestrainpoint,Figure 4-22 ,mightbeduetosomenumericalerror,althoughthiskindofdiscontinuitydidnotappearonanyoftheotherPoly8orPoly6FLC'spredictions(ofcourse,acoarsergridonthestrainratiointervalwouldremoveit;thegridweusedon[0:5;1]hada0:05step).ItshouldberemarkedthattheYld96FLCalsohasaverystirighthandsideslopeattheplanestrainpoint,2=0,suggestingthatthePoly8behaviorinthiscasemightalsobeaconsequenceoftheM-Kmodelitself.Indeed,whileforPoly6thelimitstrainwasreachedinbothcasesforthegrooveorientationwith=0o,forthePoly8FLCcurveinFigure 4-22 evenfor=0:05thelimitstrainisreachedforanangleotherthanzero(4o).ItisprobablethatinthecitedworkforYld96toothelimitingstrainnear2=0wasreachedwithanangleotherthanzero,probablyhigherthan4owhichledtofasterdecreaseratethanPoly8.However,everylimitpointonthePoly8FLCinFigure 4-23 wasreachedforthe=0ogrooveorientation.SincethedirectionalpropertiesweresimilarlyttedbyPoly8inbothcases,itfollowsthattheuseoftheM-KmodelwithvariablegrooveorientationforthepredictionoftherighthandsideoftheFLCisofquestionablerelevance. Besidesadeeperanalysisofthevariablegrooveorientationassumption,furtherrenementoftheM-Kmodelmustbeconsideredatconstitutivelevelstartingwiththehardeninglaw,which,consideringthehighlevelofstraininginvolvedinFLCprediction,mustbeanisotropic.Evenso,wehaveprovedthatpolynomialtypecriteriacanbe 125

PAGE 126

126

PAGE 127

ExperimentaldataforAA5182-O,(B), Banabicet.al ( 2005b ),AA5182-O,(W), Wuetal. ( 2003 ),andAA3104-H19.Notethatforallthreecasesonlydataat0o,45oand90oisexperimental.ThedatainbetweenwasgeneratedwithPoly4.Thebiaxialyieldstrengthandr-valuedata(whereavailable)isexperimental. 1.0000.99110.97550.96860.97400.98620.99301.0005182-O(W) 1.0000.99230.97760.97010.97410.98200.98550.9859AA3104 1.0001.00051.00121.00461.01501.02801.03380.9609 0.6420.72810.91111.0391.01740.89680.8291.1265182-O(W) 0.72000.75710.83930.90000.89690.85920.8400-AA3104 0.43000.47910.62680.84001.04641.19891.2600Table4-2. Poly4coecientsforAA5182-O(B),AA5182-O(W),andAA3104-H19 1.0000-1.56392.4000-1.86441.02846.0894-5.57615182-O(W) 1.0000-1.67442.6087-1.93601.06026.1742-5.18573104-H19 1.0000-1.20282.4527-1.95240.87555.1585-4.7251 6.400310.26125182-O(W) 6.40299.61593104-H19 5.75818.3445 Table4-3. Poly6coecientsforAA5182-O(B),AA5182-O(W1),AA5182-O(W2),AA3104-H19(1)andAA3104-H19(2) 1.0000-2.34594.9308-6.75215.9608-2.83661.04309.23605182-O(W1) 1.0000-2.51147.6942-11.4508.2627-2.99021.09159.26405182-O(W2) 1.0000-2.51098.0490-12.3608.8226-2.99961.09609.25183104-H19(1) 1.0000-1.80176.4068-10.2767.7932-2.73920.82037.74463104-H19(2) 1.0000-1.79866.2062-10.6858.4902-2.76150.82837.7391 -16.58825.892-19.5989.807330.085-25.54131.97231.2365182-O(W1) -21.00033.345-22.7749.891131.439-24.20832.65027.0835182-O(W2) -21.72634.567-23.9239.876231.809-23.71433.16926.4113104-H19(1) -16.17732.035-21.3948.090124.661-25.90827.80424.2103104-H19(2) -15.86132.732-22.7378.031624.447-25.71928.47123.986 127

PAGE 128

Poly8coecientsforAA5182-O(W1),AA5182-O(W2),AA3104-H19(1)andAA3104-H19(2).Forallwehavea1=1:0000. -3.348911.759-23.01828.280-23.33412.759-4.10481.12405182-O(W2) -3.346913.494-29.62039.035-31.95315.519-4.13061.13153104-H19(1) -2.399510.217-22.06229.961-25.90813.100-3.42020.76743104-H19(2) -2.402410.095-22.40330.133-26.00613.653-3.45640.7760 12.211-37.75876.414-88.63176.930-42.28913.51962.9445182-O(W2) 12.112-40.66182.777-91.97785.040-46.81413.27366.0143104-H19(1) 10.207-29.05382.450-88.25683.696-40.1659.973948.7903104-H19(2) 10.201-28.72782.777-88.33283.682-40.9159.860248.739 -103.11166.04-102.6765.763111.44-95.900114.4195.9345182-O(W2) -104.60163.84-103.4270.571104.35-95.953104.59106.103104-H19(1) -109.18160.83-108.4954.1295.204-106.0795.33986.1003104-H19(2) -109.22160.75-108.5154.73794.833-106.1295.33086.300 Figure4-1. GeometricalsettingforM-Kanalysis. 128

PAGE 129

Positioningonthehardeningcurveofthetwozonesofthesheet.Deformationinstability(necking)istriggeredwhenzoneBapproachestheatportionofthehardeningcurve. Figure4-3. Rotationofthegrooveduringthestretchingofthesheet. 129

PAGE 130

Poly4andPoly6materialcharacterizationofAA5182-O,withdatafrom Banabicet.al ( 2005b ).Directionalyieldstrengthandr-value. Figure4-5. ProjectionsonthebiaxialplaneofthePoly4andPoly6yieldsurfacesforAA5182-O(B).. 130

PAGE 131

ThePoly4andPoly6biaxialyieldcurves(xy=0)forAA5182-O(B) Figure4-7. Theevolutionoftheratioin 4{3 duringplanestrainloading. 131

PAGE 132

ForminglimitdiagramfortheAA5182-O(B)alloy. Figure4-9. Poly6andPoly8materialcharacterizationofAA5182-O,withdatafrom Wuetal. ( 2003 ).Directionalyieldstrengthandr-value. 132

PAGE 133

ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA5182-O(W). Figure4-11. Theevolutionoftheratioin 4{3 duringplanestrainloadingforAA5182-O(W). 133

PAGE 134

Poly6andPoly8materialcharacterizationofAA5182-O,withdatafrom Wuetal. ( 2003 )andoptimizationtowardtheinscribedanisotropichexagon.Directionalyieldstrengthandr-value. Figure4-13. ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA5182-O(W):optimizationtowardtheinscribedanisotropichexagon. 134

PAGE 135

ThePoly6andPoly8biaxialyieldcurves(xy=0)forAA5182-O(W);withYld96biaxialpointsasinput Figure4-15. ThePoly6andPoly8biaxialyieldcurvesforAA5182-O(W);thePoly6andPoly8biaxialyieldcurvesareoptimizedtowardtheinscribedanisotropichexagon. 135

PAGE 136

ForminglimitdiagramfortheAA5182-Oalloydescribedin Wuetal. ( 2003 );Yld96pointsinthebiaxialplaneusedasinput. Figure4-17. ForminglimitdiagramfortheAA5182-O(W).ThePoly6andPoly8biaxialyieldcurvesareoptimizedtowardtheinscribedanisotropichexagon. 136

PAGE 137

Poly6andPoly8materialcharacterizationofAA3104-H19.Directionalyieldstrengthandr-value.Biaxialdataincludedinoptimization Figure4-19. ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA3104-H19. 137

PAGE 138

ThePoly6andPoly8biaxialyieldcurves(xy=0)forAA3104-H19;biaxialpointsfrom Wuetal. ( 2003 )wereusedasinputforoptimization. Figure4-21. ThePoly6andPoly8biaxialyieldcurvesforAA3104-H19;optimizedtowardstheanisotropicinscribedhexagon. 138

PAGE 139

ForminglimitdiagramforAA3104-H19.ComparisonwithYld96alsoshown. Figure4-23. ForminglimitdiagramforAA3104-H19.Biaxialyieldcurvesoptimizedtowardtheanisotropicinscribedhexagon. 139

PAGE 140

Poly6andPoly8materialcharacterizationofAA3104-H19.Directionalyieldstrengthandr-value.Optimizedtowardtheinscribedhexagon. Figure4-25. ProjectionsonthebiaxialplaneofthePoly6andPoly8yieldsurfacesforAA3104-H19.Optimizedtowardtheinscribedhexagon 140

PAGE 141

Theevolutionoftheratioin 4{3 duringplanestrainloadingforAA3104-H19. 141

PAGE 142

Theunitcircleintheplaney=0isparameterizedintheformx=cos!andxy=sin!.TherestrictionofP4tothiscircleisthen andeq.( 2{4 )becomes where Wenoterstthatfor!=0;=2;;or3=2wehaveA=C=0,andthereforewemusthavea60.Excludingthesecases,eq.( A{2 )isaquadraticinequalityfora6.TakingintoaccountthesignsofA,BandCweseethat,foranyxed!theequationg(!;a6)=0alwayshastworoots where:=(B=2)2AC>0.Thus,a6mustsatisfy 0a6min!2[0;]x2(!)(A{5) Solvingforx2weobtain wherek6:=12a1a9,and 142

PAGE 143

andnallyweobtainfora6: 0a6f(!6)+[f2(!6)+k6]1=2=6p Anentirelysimilarreasoningleadstothesecondinequalityineq.( 2{19 ).ForbothrestrictionsofP4totheplanesx=0andy=0thepositivityconditioniseasilyshowntoreducetoweakerinequalitiesthana80anda60,andthereforeinequalities( 2{19 )arealsosucientforthesetworestrictionstobereal-valuedfunctions.Finally,afterparameterizingtheunitcircleintheplanex=yintheformx=y=(cos!)=p Thisexpressionissimilarwiththeoneinequation( A{1 ),andthus: 0B56p Recallingfromequation( 2{17 )thedenitionofB5,wethenobtainthefollowingtwoinequalities 2B1(2= 45)4 Theleft-handsideinequalityin( A{12 )givesthefollowingboundonr45 143

PAGE 144

45)4=B1=(2 45)4.Usingtheformulafora9,equation( 2{16 ),andaftersomealgebra,theright-handsideinequalityof( A{12 )isequivalentwith (9t+8)r245+2(5t+8)r45+9(t=21)20(A{14) Regardingr45asvariable,theaboveinequalityhasthesolutionintheformr452(;x1][[x2;+1),wherexiaretherootsoftheassociatedequation.Sincex1<0andr45>0,onlytheright-handintervalisacceptedassolution.Thisleadstotheboundsineq.( 2{20 ). 144

PAGE 145

Letusconsidertheorthotropicfourthorderpolynomial 2{48 ),weobtain 145

PAGE 147

ThisAppendixpresentssomedetailsofthereturnmappingalgorithmusedintheUMAT(ABAQUS)subroutinesforthenumericalintegrationoftheconstitutiveequations. Anyanisotropicyieldfunctiontakesintoaccount,inamoreorlessevidentway,theorientationofthestresstensorwithrespecttothesymmetryaxes.Forpolynomialtypecriteriathisisalmostatautologysincetheexpressionoftheyieldfunctiontakesasinputdirectlythecomponentsoftheyieldstresswithrespecttothesymmetryaxes.Fortheprincipalstresses-basedcriteria,likeYld2004,thisorientationistakenintoaccountbymeansofthelineartransformations.Thereforeateachmomentofthedeformationprocessknowledgeoftheorientationoftheaxesofsymmetryisnecessaryinordertocharacterizethestateofyielding.Forprocessesinvolvingsmalldeformationsonly,thesymmetryaxesatamaterialparticlearealwaysassumedxed.Thesameisingeneralacceptedformoderateplasticdeformationinvolving,possibly,largerotations(thesocalledco-rotationalformulation).Inthiscasetheonlydierenceovertheinnitesimalcasebeingthatthespatialpositionofthesymmetryaxesisupdatedwiththeamountofrigid-bodyrotationinvolvedinthemotion(inotherwords,thesymmetryaxesrotatewiththematerial,theirpositionwithinthematerialremainingxed).However,whenlargeplasticdeformationsareinvolvedthe(orthotropic)symmetryaxesofthematerialmayrotatewithinthematerialor,evenmore,theoriginalorthotropicmaterialsymmetrymaybedestroyed.Inwhatfollowsweassumethatthesheetmetalretainsitsorthotropicsymmetryduringtheentiredeformationhistory,andalsothatthesymmetryaxesofthesheetremainxedwithinthematerial.Thisisaconstitutiveassumption. InwhatfollowsweadoptanupdatedLagrangiandescription,whichisadequateforsimulatinglargeplasticdeformationbehavior.First,theinitialcongurationistakenasreferenceconguration.Then,thereferencecongurationisupdatedtothelastconvergedcongurationinastep-by-stepprocedure. 147

PAGE 148

Inmetalformingapplications,theelasticstrainsareusuallymuchsmallerthantheplasticstrains(twotothreeordersofmagnitude)andthismotivatesanadditivedecompositionoftherateofdeformationintoelasticandplasticparts.Thus,weshallwrite:d=de+dp _=RT_R;(C{1) whichshowsthat_isalsotherotatedJaumannrateoftheCauchystresstensor,andsoitisobjective. Inthespaceofstresstensors,ayieldfunctionintheformf=f(;)= K()isintroduced,satisfyingtherequirementsthatitbepressureinsensitive,convex,andcontaintheorigin(zerostress)insideitsinitiallevelset.Theyieldfunctiondenestheboundaryoftheelasticdomain:fj

PAGE 149

= ()willdenetheshapeoftheelasticdomaininthestressspace(seenextsection),whereasthehardeningfunctionKwilldeneitssize. Wefurtherassumeanassociateowruleandisotropichardening.Thehardeningparameterisidentiedbytheconditionthattheplasticworkrateassociatedwithathree-dimensionalstress-strainstatebeequivalentwiththeplasticworkrateassociatedwiththeone-dimensionalexperimentalhardeningcurve(usuallyauniaxialstress-straincurve): _Wp=Dp=_@f @= _)_=_(C{2) Thentheconstitutiveequationstaketheform: K()0(C{3) _=C[De]=C[DDp](C{4) @(C{5) where,above,Cisthefourthordertensorofelasticity. Themultiplier_issubjecttotherestrictions: _8><>:=0;iff<0;oriff=0and@f=@0>0;iff=0and@f=@>09>=>;(C{6) andisdetermined,asusual,fromtheconsistencycondition_f=0. Finally,wenoticethattheentiresetofevolutionequationscanberecastintotheform:_=Cep[D] 149

PAGE 150

@C@f @ @@f @+K0;if_>0:(C{7) whererepresentsthetensorialproduct,andthescalarproduct. Theabovesystemofevolutionequations(eqs. C{3 C{4 C{5 ),hastobeintegratedinordertoprovidestressupdatesatmaterial(orintegration)pointsassociatedwithameshinaniteelementsolution.ThisisdonenumericallyusingabackwardEulerintegrationschemeoveragenerictimeinterval[tn;tn+1].Ift=tn+1tndenotesthetimeincrementinaniteelementstep,thenthestrainincrementoverthestepiscomputedas=Dt.Then,aftereliminatingDp,theintegratedformweshalluseforstressupdateis: n+1K(n+)0(C{8) @n+1=0(C{9) whereandaredenedbyn+1=n+,n+1=n+,andSisthecompliancetensor.Inaddition,itcanbeproved, SimoandHughes ( 1999 ),thatthenumericalcounterpartoftheloading-unloadingconditions,equation( C{6 ),takesthesimpleform wheretrial:=n+C:.Itfollowsthatforanelasticstepthesolutionofsystem( C{8 ),( C{9 ),issimply=C[],=0,whichfurtherimpliesn+1=trial,andn+1=n. 150

PAGE 151

C{8 ),( C{9 )becomes @n+1 andhastobesolvedforstressandhardeningparameterincrements,and,respectively.Itisa77nonlinearsystem.Forquadraticyieldfunctions(likeMisesorHill48),thissystemcanbereducedtoasinglenonlinearequation(withimportantconsequencesintermsofaccuracyandeciency).Forthenon-quadraticcasethisisnot,ingeneral,possibleandgeneralapproachesforsolvingnonlinearsystemshavetobeemployed.Inourcase(staticanalysis)thealgorithmathandisNewton-Raphson.If,say,x:=[;]T,andx:=[2;2]T,thenthesequenceofiterations(xi)iapproachingthesolutioniscomputedasxi+1=xi+x;where@F WithFdenedinequation( C{11 ),wehave(thesubscriptn+1willbedroppedfromnowonsinceitisclearthatallthecomputationsareperformedwithinthetimestep[tn;tn+1]):@F @@@f @@f @K03775=2641@f @@f @K0375 @i=Ri@f @i2(K0)i2=ri

PAGE 152

@ii[Ri] @ii"@f @i#+(K0)i @iri= iK(n+i) Thestressandhardeningparametercannowbeupdatedi+1=i+2;i+1=n+i+1i+1=i+2;i+1=n+i Forstabilityandaccuracy,thealgorithmcanbeenhancedwithabacktrackingstrategysuchasline-searches.Thatis,insteadofacceptingtheentireincrementxcomputedbyaNewton-Raphsoniteration,onlyafractionofit,x:=x,with2(0;1]istakenasthenewincrement.TheparameteriscomputedfromtheconditionthatthemagnitudeofFreachitsminimumalongthesegmentx(thedirectionxascomputedbyNewton-Raphson,isindeedaminimizingdirectionforthefunctionFF=2).Forasimpleandecientstrategyofcomputingwereferthereaderto Pressetal. ( 1996 ).Itmustbepointedout,however,thatiftherstincrementdidnotdecreasejjFjj,then0:5. 152

PAGE 153

andtakingintoaccountthatdp=d(@f @)=d()@f @+@2f @@d @](C{12) Next,fromthealgorithmicconsistencyconditiondf=0()@f @dK0d()=0 Fromequation( C{12 )wealsohave@f @d=@f @[dd()@f @] Combiningthelasttwoequationsonecansolveford():d()=@f @[d] @[@f @]+K0 C{12 )nallygetsthedesiredformulad=8>><>>:Cd;ifelasticstep([@f=@])([@f=@]) 153

PAGE 154

Thenf=P1=n=)@f @i=1 @i=1 @i @i=1 P@P @i(C{14) @i@j=1 P@2P @i@j11 @i@P @j(C{15) where Then whereGandHarethegradientandhessianoftheyieldfunctionf. Leftforinputarethentheexpressionsofthepolynomial,itsgradientanditshessian.ForPoly6theywillbelistednexttogetherwiththeiroptimized(toreducethenumberofmultiplications)formulas.ForPoly8theyarelistedinthenextchapter. 154

PAGE 157

2{73 ,weextendrsttheindexingofthestresstensoras Thenwehavehave 157

PAGE 158

@i=1 @4=@f @1+@f @2(C{22) @5=2k135 @6=2k136 @i@j=P2=n1n P2=nn@f @i@f @j;i;j=1;2;3(C{24) @i4=@2f @i@1+@2f @i@2;i=1;::;6(C{25) @i@5=1 @i@f @5;i=1;2;3;@2f @5@5=1 @52#(C{26) @i@6=1 @i@f @6;i=1;2;3;@2f @6@6=1 @62#(C{27) 158

PAGE 159

ThisappendixprovidestheJacobianofthefunctionF=(F1;F2),withF1andF2denedby 4{40 and 4{45 ,respectively. @Bxn22+@f @Byn21@f @Bxyn1n2(D{1) B=H0( @Bx@Bxn22+@2f @Bx@Byn21@2f @Bx@Bxyn1n2n21+@2f @By@Bxn22+@2f @By@Byn21@2f @By@Bxyn1n2n21 @Bxy@Bxn22+@2f @Bxy@Byn21@2f @Bxy@Bxyn1n2n1n2 B=@f @Bxn22+@f @Byn21@f @Bxyn1n2(D{4) 159

PAGE 160

Arminjon,M.,Bacroix,B.,Imbault,D.,Raphanel,J.L,1994.Afourth-orderplasticpotentialforanisotropicmetalsanditsanalyticalcalculationfromtexturefunction.ActaMechanica,107,33. Banabic,D.,Balan,T.,Comsa,D.S.,2000.Anewyieldcriterionfororthotropicsheetmetalunderplanestressconditions.In7thColdMetalFormingConferenceProceedings,ClujNapoca,Romania,217. Banabic,D.,Aretz,H.,Comsa,D.S.,Paraianu,L.,2005.Animprovedanalyticaldescriptionoforthotropyinmetallicsheets.InternationalJournalofPlasticity,21,493. Banabic,D.,Aretz,H.,Paraianu,L.,Jurco,P.,2005.ApplicationofvariousFLDmodellingapproaches.ModellingandSimulationinMaterialsScienceandEngineering,13,759. BaratadaRocha,A.,Barlat,F.,Jalinier,J.M.,1984.Predictionoftheforminglimitdiagramsofanisotropicsheetsinlinearandnon-linearloading.MaterialsScienceandEngineering,68,151. Barlat,F.,1987.Crystallographictexture,anisotropicyieldsurfacesandforminglimitdiagrams.MaterialsScienceandEngineering,91,55. Barlat,F.,Lege,D.J.,Brem,J.C.,1991.Asix-componentyieldfunctionforanisotropicmaterials.InternationalJournalofPlasticity,7,693. Barlat,F.,Maeda,Y.,Chung,K.,Yanagawa,M.,Brem,J.C.,Hayashida,Y.,Lege,D.J,Matsui,K.,Murtha,S.J.,Hattori,S.,Becker,R.C.,Makosey,S.,1997.Yieldfunctiondevelopmentforaluminumalloysheets.JournaloftheMechanicsandPhysicsofSolids,45,1727. Barlat,F.,Brem,J.C.,Yoon,J.W.,Chung,K.,Dick,R.E.,Lege,D.J.,Pourboghrat,F.,Choi,S.H.,Chu,E.,2003.PlaneStressYieldFunctionForAluminumalloySheet-PartI:Theory,InternationalJournalofPlasticity,19,1297. Barlat,F.,Aretz,H.,Yoon,J.W.,Karabin,M.E.,Brem,J.C.,Dick,R.E.,2005.Lineartransformationbasedanisotropicyieldfunction.InternationalJournalofPlasticity,21,1009. Barlat,F.,Yoon,J.W.,Cazacu,O.,2007.Onlineartransformationsofstresstensorsforthedescriptionofplasticanisotropy.InternationalJournalofPlasticity,23,876. Bishop,J.F.W.,HillR,1951.Atheoryoftheplasticdistorsionofapolycrystallineagregateundercombinedstresses.PhilosophicalMagazine,Vol42,pp414. BoehlerJ.P.,1987ApplicationofTensorsFunctionsinSolidsMechanics,CISMCoursesandLectures,Vol.292,Springer,Berlin. 160

PAGE 161

Bron,F.,Besson,J.,2004.Ayieldfunctionforanisotropicmaterials-Applicationtoaluminumalloys.InternationalJournalofPlasticity,20,937. Butuc,M.C.,Gracio,J.J.,BaratadaRocha,A.,2003.Atheoreticalstudyonforminglimitdiagrampredictions.JournalofMaterialsProccessingTechnology,142,714. Cao,J.,Yao,H.,Karalis,A.,Boyce,M.C.,2000.Predictionoflocalizedthinninginsheetmetalusingageneralanisotropicyieldcriterion.InternationalJournalofPlasticity,16,1105. Cazacu,O.,Barlat,F.,2001.GeneralizationofDrucker'syieldcriteriontoorthotropy.MathematicsandMechanicsofSolids,6,613. Cazacu,O.,Barlat,F.,2004.Acriterionfordescriptionofanisotropyandyielddierentialeectsinpressure-insensitivemetals.InternationalJournalofPlasticity,20,2027. Chung,K.,Lee,S.Y.,Barlat,F.,Keum,Y.T.,Park,J.M.,1996.Finiteelementsimulationofsheetformingbasedonaplanaranisotropicstrain-ratepotential.InternationalJournalofPlasticity,12,93. Dafalias,Y.F.,2000.Orientationalevolutionofplasticorthotropyinsheetmetals.JournaloftheMechanicsandPhysicsofSolids,48,2231. Danckert,J.,1995.Experimentalinvestigationofasquare-cupdeepdrawingprocess.JournalofMaterialsProcessingTechnology,50,375. Davis,C.,1957.Allconvexinvariantfunctionsofhermitianmatrices.ArchivderMathematik,8,276. Drucker,D.C.,1949.Relationofexperimentstomathematicaltheoriesofplasticity.JournalofAppliedMechanics,16,349. Friedman,P.A.,Pan,J.,2000.Eectsofplasticanisotropyandyieldcriteriaonpredictionofforminglimitcurves.InternationalJournalofMechanicalSciences,42,29. Gambin,W.,2000.Plasticityandtextures.KluwerAcademicPublishers. Goodwin,G.M.,1968.Applicationofstrainanalysistosheetmetalformingproblemsinthepressshop.LaMetallurgiaItaliana,8,767. Gotoh,M.,1977.Atheoryofplasticanisotropybasedonayieldfunctionoffourthorder(planestress)-partIandII.InternationalJournalofMechanicalSciences,19,505. Groemer,H.,1996.Geometricapplicationsoffourierseriesandsphericalharmonics.CambridgeUniversityPress. 161

PAGE 162

Hill,R.,1948.Atheoryofyieldingandplasticowofanisotropicmetals.ProceedingsoftheRoyalSocietyofLondon,193A,281. Hill,R.,1950.Themathematicaltheoryofplasticity.ClarendonPress,Oxford. Hill,R.,1952.Ondiscontinuousplasticstates,withspecialreferencetolocalizedneckinginthinsheets.JournaloftheMechanicsandPhysicsofSolids,1,19. Hill,R.,1979.Theoreticalplasticityoftexturedaggregates.MathematicalProceedingsoftheCambridgePhilosophicalSociety,75,179. Hill,R.,1990.Constitutivemodellingoforthotropicplasticityinsheetmetals.JournaloftheMechanicsandPhysicsofSolids,38,405. Hill,R.,2001.Onthemechanicsoflocalizednecking.JournaloftheMechanicsandPhysicsofSolids,49,2055. Hershey,A.V.,1954.Theplasticityofanisotropicaggregateofanisotropicfacecenteredcubiccrystals.JournalofAppliedMechanics,ASME21,241. Hosford,W.F.,1966.Texturestrengthening.MetalsEngineeringQuarterly,6,13. Hosford,W.F.,1972.Ageneralizedisotropicyieldcriterion.JournalofAppliedMechanics,Trans.ASME39,607. Hosford,W.F.,1979.Onyieldlociofanisotropiccubicmetals.7thNorthAmericanMetalworkingConference,Dearborn,MI,191. Hosford,W.F.,Caddell,R.M.,1983.Metalforming:mechanicsandmetallurgy.Prentice-Hall,Inc.,EnglewoodClis,N.J. Hosford,W.F.,1993.Themechanicsofcrystalsandtexturedpolycrystals.OxfordUniversityPress,NewYorkOxford. HullD.,Bacon,D.J.,2001.Introductiontodislocations,4thedition.Butterworth-Heinemann. Hutchinson,J.W.,Neale,K.W.,1978.Sheetnecking-II.Timeindependentbehavior.InKoistinen,D.P,Wang,N.M.,eds,Mechanicsofsheetmetalforming.PlenumPress,NewYork. Karallis,A.P.,Boyce,M.C.,1993.Ageneralanisotropicyieldcriterionusingboundsandatransformationweightingtensor.JournaloftheMechanicsandPhysicsofSolids,41,1859. Keeler,S.P,Backofen,W.A.,1963.Plasticinstabilityandfractureinsheetsstretchedoverrigidpunches.ASMTransactions,56,25. 162

PAGE 163

Kuroda,M.,Tvergaard,V.,(2000).Forminglimitdiagramsforanisotropicmetalsheetswithdierentyieldcriteria.InternationalJournalofSolidsandStructures,37,5037. Lee,H.,Im,S.,Atluri,S.N.,1995.Strainlocalizationinanorthotropicmaterialwithplasticspin.InternationalJournalofPlasticity,11,423. Liu,I-S,1982.Onrepresentationsofanisotropicinvariants.InternationalJournalofEngineeringSciences,20,1099. Logan,R.,Hosford,W.F.,1980.Upper-boundanisotropicyieldlocuscalculationsassuming<111>-pencilglide.InternationalJournalofMechanicalSciences,22,419. Malvern,L.E.,1969.Introductiontothemechanicsofacontinuummedium.Prentice-Hall,EnglewoodClis,NJ. Marciniak,Z.,Kuczynski,K.,1967.Limitstrainsintheprocessofstretch-formingsheetmetal.InternationalJournalofMechanicalSciences,9,609. Mellor,P.B.,Parmar,A.,1978.Plasticityanalysisofsheetmetalforming.InMechanicsofsheetmetalforming,Kostinen,D.andWang,N-M.eds,PlenumPress. Narasimhan,K.,Wagoner,R.H.,1991.Finiteelementsimulationofin-planeforminglimitdiagramsofsheetscontainingnitedefects.MetallurgicalTransactionsA,22,2655. Pearce,R.,1968.Someaspectsofanisotropicplasticityinsheetmetals.InternationalJournalofMechanicalSciences,10,995. Plunkett,B.W.,2005.Plasticanisotropyofhexagonalclosedpackedmetals.Doctoraldissertation,UniversityofFlorida. Press,W.H.,Teukolsky,S.A.,Vetterling,W.T.,FlanneryB.P.,1996.NumericalRecipesinC.CambridgeUniversityPress. Schitkowski,K.,1986.NLPQL:AFortransubroutinesolvingconstrainednonlinearprogrammingproblems,AnnalsofOperationalResearch,5,485. Simo,J.C.,Hughes,T.J.R.,1999.ComputationalInelasticity,SpringerVerlag,Berlin. Soare,S.,2006.UnpublishedworkregardingtheapplicationofPoly65tothesimulationofearinginAA2090-T3sheets. Soare,S.,Yoon,J.W.,Cazacu,O.,Barlat,F.,2007.Applicationsofarecentlyproposedyieldfunctiontoshhetforming.InD.Banabic(ed.):Advancedmethodsinmaterialforming.Springer. Soare,S.,Yoon,J.W.,2007.Afourthorderpolynomialyieldfunction.Workinprogress. 163

PAGE 164

Sowerby,R.,Duncan,J.L.,1971.Failureinsheetmetalinbiaxialtension.InternationalJournalofMechanicalSciences,13,217. Swift,H.W.,1952.Plasticinstabilityunderplanestress.JournaloftheMechanicsandPhysicsofSolids,1,1. Taylor,G.I.,1938.Plasticstrainsinmetals.JournaloftheInstituteofMetals,62,307. Tugcu,P.,Neale,K.W.,1999.Ontheimplementationofanisotropicyieldfunctionsintonitestrainproblemsofsheetmetalforming.InternationalJournalofPlasticity,15,1021. Wang,C.C.,1970.Anewrepresentationtheoremforisotropicfunctions,partsIandII.ArchiveofRationalMechanicsandAnalysis,36,166. Woodthorpe,J.,Pearce,R.,1970.Theanomalousbehaviorofaluminumsheetunderbalancedbiaxialtension.InternationalJournalofMechanicalSciences,12,341. Wu,P.D.,Neale,K.W.,vanderGiessen,E.,1997.OncrystalplasticityFLDanalysis.ProceedingsoftheRoyalSocietyofLondon,A,453,1831. Wu,P.D.,Jain,M.,Savoie,J.,MacEwen,S.R.,Tugcu,P.,Neale,K.W.,2003.Evaluationofanisotropicyieldfunctionsforaluminumsheets.InternationalJournalofPlasticity,19,121. Yoon,J.W.,Barlat,F.,Chung,K.,Pourboghrat,F.,Yang,D.Y.,2000.Earingpredictionbasedonasymmetricnonquadraticyieldfunction.InternationalJournalofPlasticity,16,1105. Yoon,J.W.,Barlat,F.,Dick,R.E.,Chung,K.,Kang,T.J.,2004.Planestressyieldfunctionforaluminumalloysheets-partII:FEformulationanditsimplementation.InternationalJournalofPlasticity,20,495. Yoon,J.W.,Barlat,F.,Dick,R.E.,Karabin,M.E.,2006.Predictionofsixoreightearsinadrawncupbasedonanewanisotropicyieldfunction.InternationalJournalofPlasticity,22,174. Zhou,Y.,Jonas,J.J,Savoie,J.,Makinde,A.,MacEwen,S.R.,1998.Eectoftextureonearinginfccmetals:niteelementsimulations.InternationalJournalofPlasticity,14,117. 164

PAGE 165

StefanC.SoarewasborninBucharest,Romania,onFebruary2nd,1974.HegraduatedwithaBachelorofScienceinAppliedMathematicsfromtheUniversityofBucharest,FacultyofMathematics,inJune2001.AftergraduationheworkedfortwoyearsasteachingassistantwithintheDepartmentofMechanicsandEquationsoftheFacultyofMathematics,UniversityofBucharest.In2003hestartedhisPhDstudiesatUniversityofFloridawhereheoccupiedaresearchassistantpositionwithintheDepartmentofMechanicalandAerospaceEngineering. 165