Localization conditions and diffused necking for damage plastic solids

23 Pages·1.015 MB·English

by  Xue L.

EngineeringFractureMechanics77(2010)1275–1297
ContentslistsavailableatScienceDirect
Engineering Fracture Mechanics
journal homepage: www.elsevier.com/locate/engfracmech
Localization conditions and diffused necking for damage plastic solids
Liang Xue*
DepartmentofMechanicalEngineering,NorthwesternUniversity,2145SheridanRoad,Evanston,IL60208,USA
a r t i c l e i n f o a b s t r a c t
Articlehistory: WestartbygeneralizingtheonedimensionalConsidèreconditiontoamaximumpower
Received15September2009 condition for arbitrary three dimensional loadings. In particular, proportional loadings
Receivedinrevisedform14December2009 with arbitrary confining pressures andLode angles are considered. It is shown that the
Accepted21December2009
maximumpowerlocalizationcriterionforJ2materialdoesnotagreewithexperimental
Availableonline4January2010
observations.Theconstitutiverelationshipofdamageplasticsolidsisthenintroducedto
themaximumpowerconditionforthediffusednecking.Weshowthattheonsetoflocal-
Keywords:
izationisaspontaneousconsequenceoftheevolutionofplasticdamageastheweakening
Damageplasticitytheory
rateincreases.Emphasisisgiventotheextensivelystudiedsheetmetalforming.Weshow
Localization
thatthegoverningfactorofthelocalizationconditionfordamageplasticsolidsisnotthe
Maximumpowercriterion
Diffusednecking damage itself, but the resulting effect of the rate of the weakening from the plasticity
Sheetmetalforming induceddamage,which isafunction ofthe stressstates onthe loading path. Effectsof
Forminglimitcurve thepressuresensitivity,theLodeangledependence,thedamageaccumulationandweak-
ening are explored through parametric studies. Examples are given for several metallic
alloysthatexhibitdifferentshapesoftheforminglimitcurve(FLC).Thepredictedstrain
componentsattheonsetoflocalizationagreewellwiththeexperimentalresults.
(cid:2)2009ElsevierLtd.Allrightsreserved.
1. Introduction
Neckingandfracturearetwoimportantthemesforductilemetalsundersignificantplasticdeformation.Thelocalization
problemisofgreatinteresttoengineersbecauseitisoftenrelatedtopoorproductqualityforplasticformingandisanearly
indicationofultimatestructuralfailures.Forductilematerials,localizationatamaterialleveloftenoccursafterconsiderable
plasticdeformation.Theoreticalworkinthisregardshasbroughtremarkablescientificthoughtsintothephysicalunder-
standingoftheneckingphenomena[1–8].Differentapproacheshavebeenpursuedfromtheirperspectiveviews,e.g.peak
principalforces[3],zeroextensionline[4],bifurcationfrompointvertexontheyieldsurface[6]orpre-existingimperfec-
tionsandanisotropy[8,9].Ontheotherhand,modelingofthefractureofductilesolidshasattractedagreatdealofefforts
frommacrophenomenologicaldescriptions[10–17]tomicrostructuralaspectsinordertopredictthecrackformationand
propagation[18–25].Agoodpredictivemodelforlocalizationreliesonboththeassumptionsmadeforlocalizationandthe
constitutivemodels used for the material. A unified method to predict the localization and the ductile fracture has been
sought from theoretical and numerical approaches. For instance, damage and fracture models have been used to study
theapplicabilityandtoidentifygoverningfactorsoflocalizationinthesheetmetalformingindustry.Forinstance,Ozturk
andLeeshowedthatthepredictionusingexitingductilefracturemodelsisnotsatisfactory[26].Inthepresentstudy,an
attemptismadetolinkadamageplasticitymodel(DPM)tothelocalizationconditioninthesenseofenergydissipation.
Inthepresentstudy,wefocusontheonsetoflocalization,whichisoftenrelatedtotheso-called‘‘diffusednecking”[1,3].
Upon further deformation, the necking region further localizes until an infinitesimal band, which is often related to the
* Tel.:[phone].
E-mailaddress:[email protected]
0013-7944/$-seefrontmatter(cid:2)2009ElsevierLtd.Allrightsreserved.
doi:10.1016/j.engfracmech.2009.12.008
1276 L.Xue/EngineeringFractureMechanics77(2010)1275–1297
materialinstabilityortheso-called‘‘localizednecking”,e.g.[2,5].Thediffusedneckdiffersfromthelocalizedneckinthatits
immediatepost-neckingdeformationmodedoesnotpresumeaninfinitesimalband,ratheritpresumesastrainrateofgreat-
ermagnitudeinthesamedirectioninaportionoftheuniformlydeformedregion,suchasintheConsidère’sconditionwhere
thestrainratebifurcatesatamaterialpointwhenthemaximumforceisreachedinasimpletensionbar.Herein,weadopt
theonsetofneckingasthebifurcationfromauniformdeformationmode,whichisthefirsttraceofalocalizeddeformation
mode.Inaseparatepaper,thematerialinstabilityorthe‘‘localizednecking”intonarrowshearbandsintheHadamardsense
willbediscussed[27].
Inarecentdevelopmentforductilefracturemodeling,athreedimensionaldamageplasticitymodel(DPM)isformulated
usingthe‘‘cylindricaldecomposition”conceptandthecouplingofthedamageinducedweakeningwiththeplasticity[16].
Withthisdevelopment,inthepresentpaper,weproposeathreedimensionallocalizationconditionsimilartothatofthe
Considèrecondition[1]intheonedimensionalcase.
Inthepresentpaper,thebifurcationpointforthelocalizeddeformationisconsideredfromthesenseofenergydissipa-
tion.Itisunderstoodthatthelocalizationmodetakestheleastenergyforthesameprescribedboundarycondition,whichin
thepresentcaseisunderstoodastheonsetofthediffusednecking.Weadoptedthedamageplasticitytheoryandshowthat
thegoverningfactorfortheshapeofforminglimitcurves(FLC)isnotthedamageitself,buttherateofdamageandtherateof
weakening(seeSection4.1).Therefore,theplasticdamageinducedweakeningshouldnotbeoverlooked.
Thispaperisorganizedasfollows.InSection2,ageneralizationoftheonedimensionalConsidèreconditionfortheonset
ofneckingforarbitrarythreedimensionalloadingcasesispresented.InSection3,theplasticdamageisintroducedtothe
constitutiveequations.Asaspontaneousconsequenceoftheplasticdamageevolution,theeffectofdamageinducedweak-
eningingeneralizedlocalizedneckingconditionisdiscussed.InSection4,theinfluencingparametersofdamageandweak-
eningandtheirsignificanceontheneckingconditionsarediscussed.InSection5,wediscussthesignificanceofthematerial
parametersforthepressureandtheLodeangledependenceofthefractureenvelopeontheshapeofFLC.InSection6,we
utilizetheexistingexperimentalresultsforforminglimitcurvesinsheetmetalformingandusethedamageplasticitymodel
to predict the FLC and the fracture curve at the same time. Good agreement on the predicted and experimental FLC’s is
achievedforseveralaluminumalloysandastainlesssteel.
2. Conditionsforlocalization
Weareconcernedwithalocalbifurcationconditionforplasticallydeformedsolidsintheenergysense.Theprincipleof
minimumwork,whichstatesthedeformationshouldtakeplacewheretheresistanceistheminimum.Adirectcorollaryis
thefollowing:forproportionallyloadedmaterials,whentheincrementalenergyrequiredtodeformthematerialinauni-
formmannerisgreaterthantodeformitinalocalizedmatter,localizationoccursregardlessofthenatureofthedeformation
beingelastic,plasticorvisco-plastic,etc.Intheabovestatement,boththeuniformandthenon-uniform(localized)defor-
mationarekinematicallyadmissible.Forfixedloadingrateintime,theincrementalenergycanbeinterpretedasthepower.
Forsimplicity,wewillusepowerinourderivation,butbearinmind,thetimeonlyservesasatime-likevariableinourder-
ivationsincethematerialisassumedtoberateindependent.TheConsidère’sconditionisaspecialembodimentofthiscor-
ollarywhereonlyonedirectionofthematerialissubjectedtotension,orinotherwords,doesthework.
LocalizationphenomenadonotonlyoccurinroundbarsundersimpletensionasConsidèrehasstudied;inmanyother
applicationswheremultiaxialstressesexist,localizationcanalsooccurandcanbesignificantandeventuallyleadtocata-
strophicfracture.Therefore,amoregeneralizedlocalizationconditionisinneed.
Inthepresenttheoreticalanalysis,weconsidertheductilesolidsasrigidplasticsincethetotalelasticdeformationisusu-
allynegligibleforductilemetalscomparedwiththeplasticdeformationwhenlocalizationisthesubjectofstudy.Themate-
rialisassumedtobeisotropic,strainhardenableandthevolumeconservationisenforcedfortheplasticdeformation.Inthis
section,thematerialisassumedtoobeyrateindependentJ2plasticityandassociatedflowrule.Inthenextsection,wewill
extendthisstudytoplasticallydamageablematerials.
Weworkintheco-rotationalframework,wheretherigidbodyrotationisremovedfromthesystemduetoobjectivity.
Proportionalloadingpathsareconsideredandtheprincipaldirectionsdonotrotatewithrespecttothebody.Foranarbitrary
stressstate,theorderedprincipalstressesðr ;r ;r Þcanberepresentedinacylindricalcoordinatesystemðp;r ;h Þ,where
1 2 3 eq L
p¼(cid:2)ðr þr þr Þ=3isthehydrostaticpressure,r isthevonMisesequivalentstressand
1 2 3 eq
!
1 27J
h ¼(cid:2) sin(cid:2)1 3 ð1Þ
L 3 2r3
eq
is the Lode angle, where J ¼s s s is the third stress invariant and s ; s ; ands are the ordered principal deviatoric
3 1 2 3 1 2 3
stresses.
Let us consider a Lagrangian body, which is subjected to multiaxial stretching. The velocity is fixed in the maximum
stretching direction, such that the velocitiesin the other two principal directions can be determined from the deviatoric
stressstate,i.e.theLodeangle.Thesimplestcaseisthesimpletensionofaroundbar.Togainperspective,webeginwith
reviewingthewell-knownConsidède’scriterionforlocalizationinasimpletensionroundbar[1].Inthisonedimensional
case, it is shown that the localization condition is influenced by both the material (the current hardening capacity) and
L.Xue/EngineeringFractureMechanics77(2010)1275–1297 1277
thedeformedgeometry(thecurrentcross-sectionalarea).Wewillfurthergeneralizeittothreedimensionalcasesthattake
accountsboththematerialaspectsandthegeometryaspects.
2.1. Considèreconditionforsimpletension
Considèreisthefirsttoproposedthattheonsetofneckingofatensileroundbartobetheattainmentofthemaximum
pulling force based on experimental observations [1]. The load condition he considered can be formally written as
p¼(cid:2)r =3andh ¼(cid:2)p=6.
eq L
Themagnitudeoftheinstantaneouspullingforceoverthecross-sectionalareaisaffectedbytwofactorssimultaneously:
(1)theworkhardening–thestrainhardeningtendstoincreasetheforceand(2)thegeometricalsoftening–theshrinkageof
thecross-sectionalareatendstodecreasetheforce.Neckingoccurswhenapeakforceisreached,i.e.thesetwoinfluencing
factorsreachabalance.Inrealworld,therealwaysexistcertainmaterialorgeometricalimperfectionsinthetensilebarthat
couldtriggertheneckingattheweakestpointwhenConsidère’sconditionismet.Usingthelogarithmicdefinitionofstrain,
theConsidèreconditionstates
@F
¼0 ð2Þ
@e
eq
attheonsetoflocalization,whereFistheaxialforcefortheuniaxiallystretchedroundbarand,inthiscase,theequivalent
strainisthesameastheplasticstrain,i.e.e ¼e ¼e ande isthemajorstraininthepullingdirection.Beforenecking,the
eq p 1 1
stress distribution is uniform across the cross-sectional area, therefore, the pulling force F¼r A, where A is the current
1
cross-sectionalarea.Bytheproductruleofderivatives,Eq.(2)canbere-castedas
@F @r @A (cid:2)@r (cid:3)
0¼ ¼ 1Aþr ¼ eq(cid:2)r A ð3Þ
@e @e 1@e @e eq
eq eq eq p
wherer istheequivalentstressandr ¼r fortheuniaxialtensioncase.Note,A¼A expð(cid:2)e Þ,whereA istheoriginal
eq eq 1 0 1 0
cross-sectionalareaoftheroundbar.
Forthesimpletensioncase,whenincompressibilityisenforcedandelasticityisignored,theConsidèreconditionisthe
sameas
@r
eq(cid:2)r ¼h(cid:2)r ¼0 ð4Þ
@e eq eq
eq
whereh¼@r =@e isthestrainhardeningmodulusandr istheuniaxialyieldstressthattakesintoaccountforthestrain
eq p eq
hardeningande istheplasticstrain.
p
Itistrivialtoshowthatthedissipationpowerreachesmaximumwhentheaxialforceisthemaximum.Letusconsider
theprincipalstresscomponents.Forthreedimensionalloadingcases,notonlythestresscomponentintheaxialdirection
(i.e.themaximumprincipalstressdirection)doesthework,butalsotheothertwoprincipalcomponentsofstress.Therefore,
weextendthemaximumforceconditionasConsidèretothemaximumpowerconditionfortheonsetoflocalization,i.e.
@P
¼0 ð5Þ
@e
eq
wherethepoweratamaterialpointisdefinedasP¼r:e_ ¼r e_ .
eq eq
Whenthedissipationpowerreachesitsmaxima,theuniformdeformationmodenolongerdissipatestheminimumen-
ergyfortheincrementaldeformationoftheLagrangianbodyand,therefore,localizationtakesplace.
2.2. Generalizationtoarbitraryconfiningpressure
AmoregeneralcasetothesimpletensiondiscussedinSection2.1canbeobtainedbyaddinganarbitraryconfiningpres-
sure, p , in the lateral direction of the one dimensional bar, i.e. by removing the constraint of p¼(cid:2)r =3 and retaining
c eq
h ¼(cid:2)p=6forthe simpletensioncase.Roundbars justafter theonsetofneckingare shownin Fig.1for threecases:(a)
L
nolateralpressure;(b)withconfiningpressureonthelateralsurfacebuttheendsurfacesarenotsubjectedtothesuperim-
posedpressureand(c)theentirespecimenissubmergedinthepressurizedfluid,i.e.thepressureisall-round.Note,ina
laboratorysetupforhighpressuretestsusingahydro-pressurizedchamber,theendsurfacesofthetensilespecimenareusu-
allynotsubjectedtothesuperimposedpressure.
Thesuperimposedpressureeffectondiffusedneckingisaclassicproblem.However,acorrectderivationseemsstilllack-
inginliterature.HillconsideredtheconfiningpressureonatensileroundbarusingthesameapproachasConsidèreinthefirst
editionofhisclassicbookin1950[28,p.12].Heusedthenetaxialforceinthebar.Beforelocalizationoccurs,theaxialforceis
F0¼F ¼r A0¼ðr (cid:2)p ÞA ð6Þ
true 1 eq c
wherer isthetrueaxialstress,p istheconfiningpressure,A0¼Aistheuniformcross-sectionalareaandF0istheaxialforce
1 c
attheendandF istheaxialforceattheminimumcross-section(beforeneckingoccurs,F ¼F0).Byusingthemaximum
true true
forceintheaxialdirection(i.e.case(b)inFig.1),thelocalizationconditionis
1278 L.Xue/EngineeringFractureMechanics77(2010)1275–1297
Fig.1. Aschematicdrawingofneckedroundbarrightaftertheonsetoflocalization:(a)simpletensionofaroundbarunderzeroambientpressure,(b)a
tensilebarunderlateralconfiningpressureand(c)atensilebarsubjectedtoall-roundhydrostaticpressure.Thenecksareexaggeratedinthesefigures.
@r
eq¼r (cid:2)p ð7Þ
@e eq c
eq
wherep istheconfiningpressureandtakenasapositivevalueforcompression.Hillconcludedthattheconfiningpressure
c
canhaveabeneficialeffecttoextendtheuniformdeformation[28].Hillremovedhisderivationinlatereditionsofhisbook
withoutgivinganewinterpretation.Morerecently,Wuetal.[29]showedthesamederivationasHill’s.However,thisder-
ivationisinfacterroneousforhydrostaticpressureexertedbysurroundingfluids,asshownbyXue[30],whichisgivenbe-
low.Inamorerecentpaper[69],Wuetal.includedthisderivationfrom[30]tocorrect[29].
Infact,withthearbitraryconfiningpressureaddedtothesystem,thelocalizationconditiondoesnotchange.Thisiswell-
knowntotheexperimentalists.Withthesuperimposedhydrostaticpressure,thelocalizationconditioncanbederivedunder
Considère’sframeworkbyconsideringthecontributionoftheconfiningpressuretotheaxialforceattheincipientoftheneck
[30].WeadoptthesamehypothesisasConsidèrethattheonsetofneckingcanbedeterminedbytheattainingofthemax-
imumoftheremotelyappliedforce,whichcanbereadfromanexternalaxialloadcell.
FromFig.1b,thetrueforceactingontheneckisdenotedasF .Attheincipientneck,theremotelyappliedforceisthen
true
F0¼F (cid:2)p dA¼ðr (cid:2)p ÞA(cid:2)p dA ð8Þ
true c eq c c
wheredAistheshrinkageofcross-sectionalarea(apositivescalarwhenshrinking,i.e.neckingoccurs)andF ¼r A.Note
true 1
theminimumcross-sectionalareaAis(seeFig.1)
A¼A0(cid:2)dA ð9Þ
andthecross-sectionalareaoutsidetheneckA0remainsconstantafterlocalizationoccurs.ComparingwithEq.(6),apressure
dependentterm((cid:2)p dA)isaddedinEq.(8)totheaxialforceaftertheincipientneckoccurs.Thistermisintroducedbythe
c
pressureontheincipientareareduction,whichisessentialinthefollowingderivationsincethepost-neckloadingpathwill
havetoincludethispart.Ifthistermismissing,afictitiousincreaseinthecriticalstrainatlocalizationisconcluded.This
hydrostaticpressuretermontheincipientneckisvitalinthederivationforthecriticalstrainfordiffusedneck.
InthespiritofConsidère,substitutingEq.(8)intoEq.(2),wehave
@F0 @F @dA (cid:4)(cid:2)@r (cid:3) (cid:5)
0¼ ¼ true(cid:2)p ¼ 1(cid:2)r (cid:2)p A ð10Þ
@e @e c@e @e 1 c
eq eq eq eq
Note,@A=@e ¼(cid:2)@dA=@e isusedinderivingEq.(10).Knowingr þp ¼r ,Eq.(10)canberearrangedas
eq eq 1 c eq
@ðr (cid:2)p Þ @r
0¼ eq c (cid:2)r ¼ eq(cid:2)r ð11Þ
@e eq @e eq
eq eq
ForJ2materials,thelocalizationconditionforaroundbarunderarbitraryconfiningpressureisthen
h
61 ð12Þ
r
eq
L.Xue/EngineeringFractureMechanics77(2010)1275–1297 1279
Theconditionr ¼r ¼(cid:2)p remainsconstantisusedinthederivationofEq.(11),whichisthesameasEq.(4).Eq.(11)
2 3 c
remainstrueforpositivetractiononthelateralsurfaceoftheroundbar(justanegativevalueofp ),providedthatr >0.
c 1
Therefore,Eq. (12) covers allgeneralizedtension scenarios where the Lode angleis (cid:2)p=6. The above derivationassumes
the material is not permeable, such that the superimposed hydrostatic pressure can apply on the outer surface of the
round bar only. For all-round hydrostatic pressures (Fig. 1c), the necking condition is the same. This is because outside
the neck, the post-necking cross-sectional area does not change, such that the difference between F00 and F0 is merely a
constant p A0. Drawing a free-body-diagram of the dash-line boxed region can show these two cases Fig. 1b and c are
c
identical.
Now,letusconsiderthesameproblemintheenergysense.Theworkdonebyr isreducedcomparingwiththesimple
1
tensioncasefortheincrementalelongation,becauser decreaseswithincreasingconfiningpressure.However,theconfin-
1
ingpressureonthelateralsurfacedoesadditionalworktothetensilebar.ThetotalexternalpowerPofaunitvolumeis
(cid:2) 1 1 (cid:3)
P¼r e_ þð(cid:2)p Þðe_ þe_ Þ¼r e_ þð(cid:2)p Þ (cid:2) e_ (cid:2) e_ ¼ðr þp Þe_ ¼r e_ ð13Þ
1 1 c 2 3 1 1 c 2 1 2 1 1 c eq eq eq
TakingderivativeofbothsidesofEq.(13)withrespecttoe ,weobtainthesameconditionasEq.(12).Eq.(12)essen-
eq
tially means the confining pressure will not retard necking in an axially pulled round bar, hence, no benefit in forming
underhighpressureforJ2materials.Intheenergysense,thisisbecausetheexternalworkfromallmediatothesystem
should be considered. In this case, additional work is done by the surrounding fluid. The onset of localization is solely a
function of the stress–strain curve of the material for generalized tension condition, and is independent of the superim-
posed pressure.
This agrees generally with the experimental observations by Bridgman [31], who found the maximum axial load was
independent of the confining pressure, e.g. for armor steels and is confirmed on Brass [32] and on other metals and
alloys [33,34]. A collection of experimental evidences on high pressure experiments are shown in a review paper by
[35].
Ontheotherhand,forsomematerials,thehydrostaticpressuredoeshaveaneffectofincreasinguniformelongationto
someextent.Theincreaseinuniformstrainingismainlyduetotwoeffects:(1)Thehydrostaticpressureoftenincreasethe
workhardening,thoughsmallamountformetals.Thispressureinducedhardeningtendstoincreasetheneckingstrain.Ala-
dagetal.reportedthataslightincreaseintheneckingstrainforBerylliumunderhighpressureanditwasattributedtothe
increaseinthehardeningcapabilityunderhighpressure[36].(2)Thematerialdeteriorationduetothedamageisreducedor
inhibitedbythesuperimposedhydrostaticpressure.Liuetal.showedthatifthelocalizedshearistriggeredbysomedilatant
process,thepressurebecomesmoreimportanttothelocalizationstrainsincethedilatantprocesswillbesuppressedbythe
superpositionofpressure[37].Inreality,thesetwoeffectsbothexistforductilemetals.However,theirinfluencescannotbe
capturedusingpressureindependentplasticitytheorieswithoutdamage.Thedamageeffectwillbediscussedinthenext
section.
Itshouldalsobenoted,theneckingstraincanbeincreasedwhenthehydrostaticpressureappliesonlywhendeformation
takesplaceandisnotpresentattheincipientneck,suchasusingalinearpaddleformingdevicedescribedbyAllwoodand
Shouler[38].Theyfoundtheuniformtensilestraincanbesignificantlyincreasedwhenthesheetislaterallypressedbya
hardtool.SeealsoananalysisgiveninRef.[39].
2.3. GeneralizationtoarbitraryLodeangle
Themaximumpowerconditioncanbereadilytobeappliedtothreedimensionalloadingconditions.Forproportional
loadingpathsandassociatedflowrule,wehavetherelationship
r_ r_ r_ r_ e_ e_ e_ e_
eq¼ 1¼ 2¼ 3 and eq¼ 1¼ 2¼ 3 ð14Þ
r r r r e e e e
eq 1 2 3 eq 1 2 3
Therefore,thepartialdifferentialsofstrainrateare
@e_ @e_ @e_ @e_
eq¼ 1¼ 2¼ 3 ð15Þ
@e @e @e @e
eq 1 2 3
Theminimumcross-sectional areaoccursin theloadingdirectionof e .Wefix thepullingvelocityinthemajorprin-
1
cipaldirectionsuchthatthestrainratesintheothertwoprincipaldirectionscanbedeterminedsolelyfromtherelative
stressratiov,whichisathreedimensionalgeneralizationofthein-planestressratiointheplanestresscaseandisdefined
as
s (cid:2)s
v¼ 2 3 ð16Þ
s (cid:2)s
1 3
Atamaterialpoint,thepowerdensityP is
P¼r:e_ ¼r e_ ð17Þ
eq eq
1280 L.Xue/EngineeringFractureMechanics77(2010)1275–1297
andthepartialdifferentialofPwithrespecttothedeformatione is
eq
@P @r @e_
¼ eqe_ þr eq ð18Þ
@e @e eq eq@e
eq eq eq
@e_
¼he_ þr 1 ðusingproportionalityÞ ð19Þ
eq eq@e
1
¼he_ (cid:2)r e_ ðusinglogarithmicstraindefinitionÞ ð20Þ
eq eq 1
h (cid:6) p(cid:7)i
¼ h(cid:2)r cos h þ e_ ð21Þ
eq L 6 eq
Therefore,themaximumplasticdissipationcriterionforlocalizationcanbewrittenas
(cid:6) p(cid:7)
h(cid:2)r cos h þ 60 ð22Þ
eq L 6
Theleft-hand-sideoftheinequality(22)isamonotonicincreasingfunctionwithrespecttothecosinefunctionoftheLode
angleh ,whereh 2½(cid:2)p=6;p=6(cid:3).Therefore,theLodeanglestabilizesthedeformation(notallowingthedeformationrateto
L L
localizebyapplyingadditionalconstraint).
Forpowerlawstrainhardeningmaterial,r ¼Ken,theplasticstrainattheonsetofnecking,e ,canbederivedfromEq.
y p n
(22)
n
en¼cos(cid:8)h þp(cid:9) ð23Þ
L 6
Note,fromEq.(23),forequi-biaxialtensioncase,theequivalentneckingstraine equalsto2n,whichisthesameasHill’s
n
localized necking solution for simple tension [4]. It is trivial to show that for Swift stress–strain relationship,
r ¼r ð1þe =e Þn,Eq.(22)yields
y y0 p 0
n
en¼cos(cid:8)h þp(cid:9)(cid:2)e0 ð24Þ
L 6
Eqs.(23)and(24)canbeplottedonatriaxialstraincomponentplane,asshowninFig.2formaterialswithhardening
exponentn¼0:2.Itcanbeseenthatthemaximumpowercriterionforpowerlawmaterialisaperfecttriangleforisotropic
materials.Theverticesofthetrianglealignwiththegeneralizedcompressiondirection.ForSwifthardeningmaterials,the
FLC is inside the FLC predicted by the power law material of the same hardening exponent n. The pre-strain e results a
0
uniformdecrease(samemagnitude)fromtheperfecttriangleforpowerlawmaterial.Sincethesamemagnitudeyieldsa
differentpercentageoftheneckingstrainfordifferentLodeangle,thetriangleforpowerlawmaterialsbecomesathree-
pointstarwhenthepre-strainingbecomesmoresignificantcomparedwiththeneckingstrain.ThisisshowninFig.2for
threedifferentpre-strainlevelse ¼0:02,0.05and0.1.InplottingFig.2,thestressmagnitudeKforthepowerlawrelation-
0
shipandtheinitialyieldstressr fortheSwiftrelationshipdonotchangetheFLC’s.
y0
n=0.2 ε (MPa)
2
90
0.4
120 60
0.3
150 0.2 30
0.1
180 0
ε 210 330ε
3 1
power law
240 300 Swift (ε0=0.02)
Swift (ε=0.05)
270 0
Swift (ε=0.1)
0
Fig.2. TheFLCpredictedbythemaximumpowercriterionforpowerlawandSwiftstress–strainmaterials.
L.Xue/EngineeringFractureMechanics77(2010)1275–1297 1281
Example:transverseplanestrainLetusconsideraspecialcaseoftransverseplanestraininagroovedplate,wherer ¼2r
1 2
andr ¼0.Forthetransverseplanestraincondition,thestraincomponentintheintermediatestressdirectionremainszero,
3
whichdoesnotdoanywork.Therefore,thelocalizationconditionEq.(2)remainsthesame.However,
p
ffiffiffi
2 3
eeq¼pffi3ffiffie1 and req¼ 2 r1 ð25Þ
Thepartialdifferentialoftheplasticdissipationpoweris
@P @P @e pffi3ffiffi !
¼ 1 ¼ h(cid:2) r e_ ð26Þ
@e @e @e 2 eq eq
eq 1 eq
Therefore,theonsetoflocalizationconditionis
p
ffiffiffi
3
h(cid:2) r ¼0 ð27Þ
2 eq
Inthetransverseplanestrain,onlythemaximumprincipalstressdoestheworkand,therefore,themaximumpowerdis-
sipationconditionisreachedwhentheforceinthemaximumprincipalstressdirectionreachesthemaxima.Itistrivialto
showthatthemaximumforceisreachedwhenconditionEq.(27)issatisfied.
2.4. Planestress–sheetmetalforming
Themaximumpowercriterionisderivedforarbitrary3Dloadingcases.However,experimentalstudyforthreedimen-
sionallocalizationisscarceanddoesnotallowfurtherdetailedanalysis.
Thelocalizationinsheetmetalforminghasattractedagreatdealoftheoreticalandexperimentalefforttoreducetherisk
offailureinthemanufactureprocess.Duringsheetmetalforming,thethroughthicknessstressisusuallynegligibleandthe
loadingconditionissimplifiedasplanestress.Whenaneckformsinthesheetmetal,thequalitycontroloftheproductislost
andfurtheroverallformabilityofthesheetislimitedbeforeultimatefractureoccurs.Here,thediffusedneckingisconsidered
asthefirsttraceoffailure.
Weconfineourselvestoproportionalloadingpathsanddenotetheratiooftheminortomajorin-planeprincipalstressesto
beaforbiaxialtensionðr Pr P0; andr ¼0Þ,whichisequaltotherelativestressratiovforthethreedimensionalcases
1 2 3
r (cid:2)r r
v¼ 2 3¼ 2¼a ð28Þ
r (cid:2)r r
1 3 1
Withanassociatedflowrule,theevolutionoftheplasticstrainandtheprincipalcomponentsfollow:
e e e e
2pffi1ffiffiffi(cid:2)ffiffiffiffipffiaffiffiffiffiþffiffiffiffiffiaffiffiffi2ffiffi¼2(cid:2)1a¼2a(cid:2)2 1¼(cid:2)1(cid:2)3 a ð29Þ
ApplyingEqs.(29)and(22)canalsobeshownforbiaxialplanestresscasessuchthattheexistingexperimentalresultsin
thesheetmetalformingindustrycanbeutilized.TheFLCforproportional2DstretchingisplottedinFig.3.Forpowerlaw
materials,themajorprincipalstrainremainsconstantformaximumpowercriterion;whileasthepre-straine makesthe
0
0.4
0.35
0.25
0.3
0.2
0.25
ε1 0.15 ε n 0.2
0.15
0.1
power law 0.1 power law
Swift (ε=0.02) Swift (ε=0.02)
0.05 0 0
Swift (ε=0.05) 0.05 Swift (ε=0.05)
0 0
Swift (ε=0.1) Swift (ε=0.1)
0 0
0 0
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0 0.2 0.4 0.6 0.8 1
ε χ
2
Fig.3. TheFLCpredictedbythemaximumpowercriterionfor2DbiaxialstretchingcaseforpowerlawandSwiftstress–strainrelationshipmaterials.
1282 L.Xue/EngineeringFractureMechanics77(2010)1275–1297
majorprincipalstraingreaterfortheequi-biaxialtensioncasethanthesimpletensioncase.Italsopredictedthattheplane
straintensioncasehasagreatermajorstrainthanthesimpletensioncase,whichdoesnotagreewithcommonlyobserved
experimentalresults.ThisnonconformitywillbeaddressedinSection4whenthedamageplasticitymodelisintroduced.
Insummary,forafixedpullingvelocityinthemajorprincipaldirection,theConsidère’sconditionofmaximumforcecan
begeneralizedasthemaximumpowercondition.Neglectingtheelasticity,theindependenceofthemechanicalpowerwith
respecttothesuperimposedhydrostaticpressureseemsobviousbecausetheplasticdeformationisisochoric,i.e.thepres-
sureeffectiscanceledoutbysummationoverallthreeprincipaldirections.
Applyingthemaximumpowerconditionfor3Dlocalization(Eq.(24))tothesheetformingunderbiaxialstretching,the
forminglimitcurveisplottedforaluminumalloy2024-T3.TheexperimentalresultsaregiveninVallellanoetal.[40].The
stress–straincurvecanbefittedusingSwiftrelationshipr ¼330ð1þe =0:025Þ0:245MPa.TheFLCpredictedbyaconstant
M p
hardeningmodulusforsimpletensionisalsoplottedinFig.4(curvedenotedbyh¼r ).Itcanbeseenthattheconstant
M
hardeningmoduluscriterionunder-predictstheFLCintheright-hand-sideoftheFLCwheree >0andover-predictsthe
2
FLCintheleft-hand-sideoftheFLC,wheree <0.
2
TheshapeoftheFLCobtainedbythesheetmetalformingindustryisdifferentfromthatthemonotonicallyincreasing
majorstrainpredictedbythemaximumpowercriterion(e.g.Lankfordetal.[43],Keeler[41],Goodwin[42],Emburyand
Duncan[44]).Thesheetformingmeasurementsshowtheonsetofneckingisastrongfunctionoftheratioofthein-plane
principalstresscomponents–anon-monotonicshapeofthethemajorstrainisobservedformanymetals.Theexperimental
FLC’susuallyformavalleyaroundzerominorstrain,whichrepresentstheplanestraincondition.Ontheleft-handsideof
FLC ðe <0Þ, the major strain decreases and on the right-hand side of the FLC ðe >0Þ, the major strain increases with
2 2
increasinga.ThisshapeofFLCbecomeswell-knowntothecommunitysincethemidoflastcentury.
Theoretical efforts have been made to understand this phenomenon. Swift derived the localization condition for dif-
fused necking as both of the in-plane principal stresses peaked Swift [3]. Hill proposed the localized necking occurs on
zero extension line in the plane of the sheet, which is only valid for the left-hand side of the FLC [4]. On the right-hand
side of FLC, no such zero extension line exists. Stören and Rice used a vertex point yield surface to derive a localization
condition for the full range of FLC [6]. Marciniak and Kuczynski analyzed a through thickness defects and normal aniso-
tropicmaterialandshowedtheright-handsideofFLCissignificantlyinfluencedbytheanisotropy[9].Theseanalysesin-
volve the conventional constitutive laws and impose constraints in the stress and strain rates such that a bifurcation
condition can be derived.
3. Localizationconditionfordamageablesolids
3.1. Constitutivemodelingofdamageplasticity
WeadoptanewdamageplasticitytheoryproposedbyXueforductilematerials[16,17].Itisparticularlysuitableforduc-
tilefractureproblems.ThisnewdamageplasticitymodelisanI1,J2,J3theorythattakesintoaccountallthreestressinvari-
AA 2024−T3 AA 2024−T3
0.4 0.5
h=σM 0.45 h=σM
0.35 max power criterion max power criterion
Experiment, Vallellano et al. 2008 Experiment, Vallellano et al. 2008
0.4
0.3
0.35
0.25 0.3
ε2 0.2 εf 0.25
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0 0
−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0 0.2 0.4 0.6 0.8 1
ε χ
1
Fig.4. Predictionsofforminglimitcurvesbymaximumpowerandhardeningcriteriaforaluminumalloy2024-T3.Plottedalsoaretheexperimentalresults
(after[40]).
L.Xue/EngineeringFractureMechanics77(2010)1275–1297 1283
ants.Specifically,thedamageportionofthemodelincludesathreedimensionalfractureenvelopeandnonlineardamage
evolutionandweakeninglaws.
Ajointprogramofexperimentalinvestigationandnumericalsimulationforaluminumalloy2024-T351hasshownthat
the model predicts crack paths in common loading cases, such as cup-cone fracture of round bar, slant crack in doubly
groovedplatesandshearcrackinupsettingofcylinders,andincaseswherecrackmodesarewell-knowntoexperimentalist
butunabletopredictinsimulationpreviously,e.g.themodetransitioninductileplate,formationofshearlips,shearlocal-
izationinpipes,etc.[45–48].Wesummarizethekeyingredientsofthistheoryhere.
Tobeginwith,weadoptthevonMisesyieldconditionforthematrixmaterialanddefinethemacroscopicequivalent
stressastheproductofthematrixyieldstress(r ,afunctionofstrainhardening)andtheweakeningfactor(w,afunction
M
oftheplasticdamage).Theyieldconditionis
U¼r (cid:2)wðDÞr 60 ð30Þ
eq M
wherer isafunctionoftheplasticstraine ofthematerialandwðDÞisascalarweakeningfunctionoftheplasticdamageD
M p
tocharacterizethematerialdeterioration.ApowerlawformoftheweakeningfunctionwðDÞ¼1(cid:2)Db isassumedinXue
[16],wherebisamaterialconstantandfurtherextendedtoapower-exponentforminXue[49].
The plastic damage is calculated from the so-called ‘‘cylindrical decomposition” where the pressure sensitivity and the
Lode angle dependence of the fracture strain and the nonlinear damage accumulation process are described. The plastic
damageprocessischaracterizedinrateform
(cid:2)e (cid:3)ðm(cid:2)1Þ1
D_ ¼m p e_ ð31Þ
e e p
f f
where
e ¼e l ðpÞl ðhÞ ð32Þ
f f0 p h
wherel isthepressuredependencefunction,l istheLodeangledependencefunctionande isareferencefracturestrain
p h f0
andmisthedamageexponent.Thesetwofunctionsarephenomenologicalinnatureandtherearemanychoicesforl and
p
l .Forinstance,twokindsofLodeangledependencefunctionsareproposed[16].Inparticularwechoosealogarithmicpres-
h
suredependencefunction
(cid:2) p (cid:3)
l ðpÞ¼1(cid:2)qlog 1(cid:2) ð33Þ
p p
lim
todepicttheincreaseinductilitywhenthematerialissubjectedtosuperimposedhydrostaticpressure.Wealsoadoptthe
secondkindofLodeangledependencefunction
(cid:2)6jh j(cid:3)k
l ¼cþð1(cid:2)cÞ L ð34Þ
h p
wherej(cid:4)jdenotestheabsolutevalue,q; p ; candkarematerialconstants.Thematerialconstantcisdefinedastheratioof
lim
thefracturestrainatv¼0:5(generalizedshear)andv¼0(generalizedtension)underthesamepressure.
Comparedwiththewell-knownpressuresensitivityofductilefracture,theLodeangledependenceofductileisarecent
development.Inamacroscopictreatment,alessthanunitycvalueisintroducedbyobservingtheexperimentaldatafrom
Clausing[50],McClintock[51],BaoandWierzbicki[52],BarsoumandFaleskog[53],XueandWierzbicki[48].Microscopi-
cally,thetraditionalmicro-mechanicalmodelsuchastheGurson-typemodelisnotapplicableinthelowstresstriaxiality
regime,wherevoidsdonotgrowduetothelackofmeanstress.Numericalsimulationofunitcellmodelalsoshowedthe
Lode angle of the stress state has significant influence on the void coalescence [54,55]. To fix this drawback in the Gur-
son-typemodel,particularlytomodelthematerialfailuresduetoshearbands,Xue[25,56]modifiedtheGurson-typemodel
byadoptingBerg’sviscoussolution[18]andintroducedanaddictiveJ3dependentvoidshearterminthevoidevolution.The
evolutionofthevoidsheartermdependsonthecurrentvoidvolumefraction,theequivalentplasticstrainandtheLodean-
gleofthestressstate.Inthisway,themodifiedGursonmodelcanbeusedtopredictfailuresinlowstresstriaxialityzone,
suchasasimpleshearfracture[25].NahshonandHutchinsonmodifiedthesheartermtostudylocalizedneckingusingM-K
approachbyintroducingthroughthicknessdefects[57].Here,weapplythemaximumpowercriteriontothemacroscopic
damageplasticmodel.
3.2. Maximumpowercriterionfordamageplasticsolids
ThemaximumpowercriterioncanbederivedinthesamewayasSection2.3.Similarsolutioncanbeobtainedforthe
presentmaterialmodel.Atanarbitrarilydamagedstate,theequivalentstressatyieldingis
r ¼r wðDÞ ð35Þ
eq M
Uponfurtherstretching,theinstantaneouspoweratthedamagedstateremains
1284 L.Xue/EngineeringFractureMechanics77(2010)1275–1297
P¼r:e_ ¼r e_ ð36Þ
eq eq
andthepartialdifferentialofthepowerwithrespecttotheequivalentstrainis
@P @r (cid:2)@w @r (cid:3) @e_
¼ eqe_ þr e_ ¼ r þw M e_ þwr eq ð37Þ
@e @e eq eq eq @e M @e eq M@e
eq eq eq eq eq
@e_
¼ðwDDerMþwhÞe_eqþwrM@e1 ðusingproportionalityÞ ð38Þ
1
¼ðwDDerMþwhÞe_eq(cid:2)wrMe_1 ðusinglogarithmicstrainrelationshipÞ ð39Þ
h (cid:6) p(cid:7)i
¼ ðwDDerMþwhÞ(cid:2)wrMcos hMþ6 e_eq ð40Þ
wherewD¼@w=@DandDe¼@D=@ep.
Tomaintainanincreasingpowerwithrespecttotheequivalentstrain,theterminthebracketofEq.(40)hastobegreater
thanzero,i.e.
(cid:6) p(cid:7)
ðwDDerMþwhÞ(cid:2)wrMcos hLþ6 >0 ð41Þ
Localizationoccurswhenthederivativeofthepowerdropsequaltoorbelowzero.Therefore,wehavethelocalization
conditionofdiffusedneckingfordamageplasticsolidsasthefollowing:
h þwDDe6cos(cid:6)h þp(cid:7) ð42Þ
r w L 6
M
Note,atadamagedstate,wD<0; De>0andw>0.
Fromtheinequality(42),theeffectofdamageistakenintoaccountinthesecondtermontheleft-handside,whichis
wDDe=w.Iftheweakeningeffectisneglectedðw(cid:5)1andwD(cid:5)0ÞorthedamagingeffectisnotconsideredðDe(cid:5)0Þ,thegen-
eralizedmaximumpowerconditionisrecoveredforJ2plasticitymaterial(seeEq.(22)).Forrealmaterials,DeandwDcanbe
negligibleattheonsetofnecking,buttheyarenotzero.Note,De>0andwD<0.Therefore,thesecondtermhasadestabi-
lizingeffectforthesolid.Formaterialswithlowcvalues,thedamagingderivativetermDe ismoresignificantwhenhL is
closetozero.
Experimentsontheneckinghaveshownthatthedamage(e.g.thevoidingorthereductionofYoung’smodulus)areneg-
ligible.Inequality(42)showsthedamageeffectonthelocalizationconditionisthejointeffectoftheweakeningfactorwand
therateofchangeofthedamageaccumulationDeandthatoftheweakeningwD.Evenwhenthedamageinducedweakening
isnegligiblew(cid:6)1,theeffectofdamageonthelocalizationmaynotbenegligiblewhenDeandwDaresignificant.Anexample
isgiveninSection4.1forductilealuminumalloy2024-T351.Anotherexampleisaluminum–coppercrystal,whencoarseslip
bandsappearindicatingachangeinthedamagerate,smallamountofneckingcanoccurwhentheload-elongationcurveis
stillrisingandcausinganabruptfallinload[58].Forincreasingdamagerateintroducedbyincreasinginclusionvolumefrac-
tion,aloweredFLCisfoundexperimentally[59].
3.3. Evolutionofmaterialresistance
TheyieldsurfaceofthematerialisassumedtobevonMisesatadamagedstate,whichmeanstheyieldsurfacedepends
onJ2only.Ontheotherhand,theJ3dependenceonthedamageimpliesthematerialresistancedependsontheJ3.Thiseffect
isimportantinpredictingtheFLC.Barlatshowedthattheratioofthemajorstresscomponentattheplanestrainconditionto
thatatthebiaxialstretchingconditionplaysanimportantroleintheshapeoftheFLC[60].Themacroscopicyieldstresscan
beplottedforaseriesofplasticallydeformedstatetoaconstantplasticstrainlevelatdifferentLodeangles.Inthiscase,the
meanstressisfixedsuchthatitdoesnotchangethematerialresistanceatadamagedstate.
ThesecondkindofLodeangledependencefunctionreducestoaJ3independentmodelwhenthematerialconstantc¼1.
Foraluminumalloy2024-T351,theevolutionoftheyieldstresswithrespecttotheplasticstrainatdifferentLodeanglesare
plottedinFig.5.ItshouldbenotedthatthecurvesplottedinFig.5areforproportionalloadingsatdifferentstrainstates,
ratherthantheyieldconditionformultiaxialloadingsatadamagedstate.Inallcases,thereductionofthematerialresis-
tanceintheplanestrainloadingsissignificantlygreaterthanthoseinthegeneralizedtensionandthegeneralizedcompres-
sionconditions.Thisismoresignificantatalaterstageofdeformation.SeeFig.5b.
4. Maximumpowercriterionwithdamageplasticitymodel
4.1. Aluminumalloy2024
Thematerialparametersforaluminumalloy2024-T351arecalibratedinRef.[48]forthedamageplastictheory.Thematrix
stress–straincurveisfittedusingtheSwiftrelationshipr ¼r ð1þe =e Þn.ThesematerialparametersarelistedinTable1.
M y p 0
Wefirstshowthecorrelationofthelocalizationconditionandthefracturecondition.Usingthedamageplasticitymodel,
theforminglimitcurvefortheonsetoflocalizationandthefracturecurveforaluminumalloy2024-T351underplanestress