Polycrystal Plasticity - Multiple...
Transcript of Polycrystal Plasticity - Multiple...
Polycrystal Plasticity -Multiple Slip"
27-750Texture,Microstructure&Anisotropy
A.D.Rolle<,Lecturenotesoriginallyby:H.Garmestani(GaTech),G.Branco(FAMU/FSU)
Lastrevised:23rdFeb.2016
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Objective"■ The objecKve of this lecture is to show how plasKc
deformaKon in polycrystals requiresmul<ple slip in eachgrain. This iscommonlyreferredtoasthe“Taylormodel”intheliterature.
■ Further,toshowhowtocalculatethedistribuKonofslipsineach grain of a polycrystal (principles of operaKon of LosAlamos polycrystal plasKcity, LApp; also the ViscoplasKcSelfconsistent code, VPSC; also “crystal plasKcity”simulaKonsingeneral).
■ DislocaKoncontrolledplasKcstrain■ MechanicsofMaterials,or,micro-mechanics■ ConKnuumMechanics
Requirements:
Questions"■ WhatisthekeyaspectsoftheTaylormodel?■ WhatisthedifferencebetweensingleslipandmulKpleslipintermsofboundarycondiKons?■ Whatis“deviatoricstress”andwhydoesithave5components?■ HowdoesthevonMisescriterionforducKlityrelatetothe5componentsofdeviatoricstress
andstrain?■ HowdoestheBishop-Hilltheorywork?Whatistheinputandoutputtothealgorithm?Whatis
meantbythe“maximumwork”principle?■ WhatistheTaylorfactor(bothdefiniKonandphysicalmeaning)?■ Whyistherate-sensiKveformulaKonformulKpleslipusefulaboveandbeyondwhattheBishop-
Hillapproachgives?■ Whatisitthatcauses/controlstexturedevelopment?■ OnwhatquanKKesisla\cereorientaKonbased(duringmulKpleslip)?■ Howcanwecomputethemacroscopicstrainduetoanygivenslipsystem?■ Howcanwecomputetheresolvedshearstressonagivenslipsystem,starKngwiththe
macroscopicstress(tensor)?■ WhatdoesBishop&Hillstateofstressmean(whatisthephysicalmeaning)?[EachB&Hstress
state(oneofthe28)correspondstoacornerofthesinglextalyieldsurfacethatacKvateseither6or8slipsystemssimultaneously]
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References"■ KeyPapers:
1.TaylorG(1938)"PlasKcstraininmetals",J.Inst.Metals(U.K.)62307;2.BishopJandHillR(1951),Phil.Mag.421298;3.VanHou<eP(1988),TexturesandMicrostructures8&9313-350;4.LebensohnRAandTomeCN(1993)"ASelf-ConsistentAnisotropicApproachfortheSimulaKonofPlasKc-DeformaKonandTextureDevelopmentofPolycrystals-ApplicaKontoZirconiumAlloys"ActaMetall.Mater.412611-2624.
■ Kocks,Tomé&Wenk:Texture&Anisotropy(Cambridge);chapter8,1996.DetailedanalysisofplasKcdeformaKonandtexturedevelopment.
■ Reid:Deforma<onGeometryforMaterialsScien<sts,1973.Oldertextwithmanyniceworkedexamples.BecarefulofhisexamplesofcalculaKonofTaylorfactorbecause,likeBunge&others,hedoesnotusevonMisesequivalentstress/straintoobtainascalarvaluefromamulKaxialstress/strainstate.
■ Hosford:TheMechanicsofCrystalsandTexturedPolycrystals,Oxford,1993.Wri<enfromtheperspecKveofamechanicalmetallurgistwithdecadesofexperimentalandanalyKcalexperienceinthearea.
■ Khan&Huang:Con<nuumTheoryofPlas<city,Wiley-Interscience,1995.Wri<enfromtheperspecKveofconKnuummechanics.
■ DeSouzaNeto,Peric&Owen:Computa<onalMethodsforPlas<city,2008(Wiley).Wri<enfromtheperspecKveofconKnuummechanics.
■ GurKn:AnIntroduc<ontoCon<nuumMechanics,ISBN0123097509,AcademicPress,1981.
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Output of Lapp*"
■ FigureshowspolefiguresforasimulaKonofthedevelopmentofrollingtextureinanfccmetal.
■ Top=0.25vonMisesequivalentstrain;0.50,0.75,1.50(bo<om).
■ Notetheincreasingtexturestrengthasthestrainlevelincreases.
*LApp=LosAlamospolycrystalplasKcity(code)
Increasingstrain
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Development"
ThemathemaKcalrepresentaKonandmodelsq IniKallyproposedbySachs(1928),CoxandSopwith(1937),andTaylorin1938.ElaboratedbyBishopandHill(1951),Kocks(1970),Asaro&Needleman(1985),Canova(1984).q Self-ConsistentmodelbyKröner(1958,1961),extendedbyBudianskyandWu(1962).q FurtherdevelopmentsbyHill(1965a,b)andLin(1966,1974,1984)andothers.
TheTheorydependsupon:Ø ThephysicsofsinglecrystalplasKcdeformaKon;Ø relaKonsbetweenmacroscopicandmicroscopicquanKKes(strain,stress...);
• Read Taylor (1938) “Plastic strain in metals.” J. Inst. Metals (U.K.) 62, 307; � available as: Taylor_1938.pdf
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Sachs versus Taylor!SachsModel(previouslectureonsinglecrystal):- Allsingle-crystalgrainswithaggregateorpolycrystalexperiencethesamestateofstress;- EquilibriumcondiKonacrossthegrainboundariessaKsfied;- CompaKbilitycondiKonsbetweenthegrainsviolated,thus,finitestrainswillleadtogapsandoverlapsbetweengrains;- GenerallymostsuccessfulforsinglecrystaldeformaKonwithstressboundarycondi<onsoneachgrain.
TaylorModel(thislecture):- Allsingle-crystalgrainswithintheaggregateexperiencethesamestateofdeformaKon(strain);- EquilibriumcondiKonacrossthegrainboundariesviolated,becausethevertexstressstatesrequiredtoacKvatemulKpleslipineachgrainvaryfromgraintograin;- CompaKbilitycondiKonsbetweenthegrainssaKsfied;- Generallymostsuccessfulforpolycrystalswithstrainboundarycondi<onsoneachgrain.
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Sachs versus Taylor: 2"
■ DiagramsillustratethedifferencebetweentheSachsiso-stressassumpKonofsingleslipineachgrain(a,cande)versustheTaylorassumpKonofiso-strainwithmulKpleslipineachgrain(b,d).
iso-stress iso-strain
10 Sachs versus Taylor: 3 Single versus Multiple Slip"
ExternalStressorExternalStrain
Smallarrowsindicatevariablestressstateineachgrain
SmallarrowsindicateidenKcalstressstateineachgrain
MulKpleslip(with5ormoresystems)ineachgrainsaKsfiestheexternallyimposedstrain,D
EachgraindeformsaccordingtowhichsingleslipsystemisacKve(basedonSchmidfactor)
Increasingstrain
D=ETdγ
€
στ
=˙ γ ˙ ε
=1
cosλcosφ
€
DC = ˙ ε 0m(s) :σ c
τ (s)
n ( s )
m(s) sgn m(s) :σ c( )s∑
Taylor model: uniform strain"11
AnessenKalassumpKonoftheTaylormodelisthateachgrainconformstothemacroscopicstrainimposedonthepolycrystal
12 Example of Slip Lines at Surface (plane strain stretched Al 6022)"T-Sampleat15%strain
PD//TDPSD//RD
■ Notehoweachgrainexhibitsvaryingdegreesofsliplinemarkings.
■ Althoughanygivengrainhasonedominantslipline(traceofaslipplane),morethanoneisgenerallypresent.
■ TakenfromCMUPhDresearchofYoon-SukChoi(PusanU)onsurfaceroughnessdevelopmentinAl6022
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Notation: 1"
■ Strain,local:Elocal;global:Eglobal■ SlipdirecKon(unitvector):bors■ Slipplane(unit)normal:n■ Slip,orSchmidtensor,mij=binj=Pij■ Stress(tensororvector):σ■ Shearstress(usuallyonaslipsystem):τ■ Shearstrain(usuallyonaslipsystem):γ■ Stressdeviator(tensor):S■ RatesensiKvityexponent:n■ Slipsystemindex:sorα■ Notethatwhenanindex(e.g.ofaSlipsystem,b(s)n(s))isenclosedin
parentheses,itmeansthatthesummaKonconvenKondoesnotapplyeveniftheindexisrepeatedintheequaKon.
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Notation: 2"■ Coordinates:current:x;referenceX■ Velocityofapoint:v.■ Displacement:u■ Hardeningcoefficient:h (dσ = h dγ )■ Strain,ε
◆ measuresthechangeinshape■ Workincrement:dW
◆ donotconfusewithlaTcespin!■ InfinitesimalrotaKontensor:Ω■ ElasKcSKffnessTensor(4thrank):C■ Load,e.g.onatensilesample:P
■ donotconfusewithsliptensor!
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Notation: 3"
■ PlasKcspin:W◆ measurestherotaKonrate;morethanonekindofspinisused:
◆ “Rigidbody”spinofthewholepolycrystal:W◆ “grainspin”ofthegrainaxes(e.g.intorsion):
Wg◆ “la\cespin”fromslip/twinning(skewsymmetricpartofthestrain):Wc.
■ RotaKon(small):ω
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Notation: 4"
■ DeformaKongradient:F ◆ Measuresthetotalchangeinshape(rotaKonsincluded).
■ Velocitygradient:L ◆ Tensor,measurestherateofchangeofthedeformaKongradient
■ Time:t ■ Slipgeometrymatrix:E(donotconfusewithstrain)■ Strainrate:D
◆ symmetrictensor;D = symm(L)
€
≡ ˙ ε ≡ dεdt
€
Fij =∂xi∂X j
Basic Equations"Sachsmodel:iso-stress:IdenKfytheindex,s,oftheacKvesystem(s)fromkavailablesystemsfromthemaximumSchmidfactor:maxs(b(s) σn(s) ).Ifstrainisaccumulatedcomputetheslip(shearstrain)fromthemacroscopicappliedstrain.IfmorethanonesystemisacKve(e.g.primary+conjugate)dividetheshearstrainsequally.Taylormodel:iso-strain(Bishop&Hillvariantforfcc/bcconly):IdenKfytheindex,r,acKve(mulK-slip)stressstate(fromlistof28)fromthemaximuminnerproductbetweenthevertexstressstateandtheappliedstrain:maxr(σ(r)dε).EachpossiblevertexstressstateacKvates6or8slipsystems;eithermakeanarbitrarychoiceof5tosaKsfytheexternalslipor,moretypicallycomputethesoluKontothe"rate-sensiKveslipequaKon",below,i.e.thestressthatsaKsfiestheimposedstrainrate.ThesliprateonthesthsystemisgivenbytheexponenKatedexpression.La\cespiniscomputedfromtheskew-symmetricversionofthesameexpression.
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€
DC = ˙ ε 0m(s) :σ c
τ (s)
n ( s )
m(s) sgn m(s) :σ c( )s∑
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Dislocations, Slip Systems, Crystallography"
q ThissecKonisprovidedtoremindstudentsaboutthebasicgeometryofslipviadislocaKonglide.Fulldetailscan,ofcourse,befoundinstandardtextbooks.
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Dislocations and Plastic Flow"q AtroomtemperaturethedominantmechanismofplasKcdeformaKonisdislocaKonmoKonthroughthecrystalla\ce.
q DislocaKonglideoccursoncertaincrystalplanes(slipplanes)incertaincrystallographicdirec<ons(//Burgersvector).
q AslipsystemisacombinaKonofaslipdirecKonandslipplanenormal.
q Asecond-ranktensor(mij = binj )canassociatedwitheachslipsystem,formedfromtheouterproductofslipdirecKonandnormal.TheresolvedshearstressonaslipsystemisthengivenbytheinnerproductoftheSchmidandthestresstensors:τ = mij σij.
q ThecrystalstructureofmetalsisnotalteredbytheplasKcflowbecauseslipisasimpleshearmodeofdeformaKon.Moreovernovolumechangeisassociatedwithslip,thereforethehydrostaKcstresshasnoeffectonplasKcity(intheabsenceofvoidsand/ordilataKonalstrain).ThisexplainstheuseofdeviatoricstressincalculaKons.
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Crystallography of Slip"
Slipdirection–istheclose-packeddirectionwithintheslipplane.
Slipplane–istheplaneofgreatestatomicdensity.
Slipoccursmostreadilyinspeci:icdirectionsoncertaincrystallographicplanes.
Slipsystem–isthecombinationofpreferredslipplanesandslipdirections(onthosespeci:icplanes)alongwhichdislocationmotionoccurs.Slipsystemsaredependentonthecrystalstructure.
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Example:Determinetheslipsystemforthe(111)planeinafcccrystalandsketchtheresult.
Theslipdirectioninfccis<110>Theproofthataslipdirection[uvw]liesintheslipplane(hkl)isgivenbycalculatingthescalarproduct:hu+kv+lw=0
Crystallography of Slip in fcc"
Slip Systems in Hexagonal Metals"
Basal (0002) <2 -1 -1 0>
Pyramidal (c+a) (1 0 -1 1) <1 -2 1 3> Pyramidal (a) (1 0 -1 1) <1 -2 1 0>
Prism {0 -1 1 0}<2 -1 -1 0> Also: (2 -1 -1 0)
Pyramidal (1 0 -1 2)
22Berquist&Burke:Zralloys
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Slip Systems in fcc, bcc, hexagonal"TheslipsystemsforFCC,BCCandhexagonalcrystalsare:
ForthislecturewewillfocusonFCCcrystalsonly
InthecaseofFCCcrystalswecanseeinthetablethatthereare12slipsystems.Howeverifforwardandreversesystemsaretreatedasindependent,therearethen24slipsystems.
Note:
Also: Pyramidal (c+a) (1 0 -1 1) <1 -2 1 3>
Schmid / Sachs / Single Slip"
■ ThissecKonisincludedasareminderofhowtoanalyzesingleslip.SinceitassumesstressboundarycondiKonstheanalysisisstraighzorward.Moredetailisprovidedinthelecturethatexplicitlyaddressesthistopic.
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Schmid Law"q Initialyieldstressvariesfromsampletosampledependingon,amongseveralfactors,therelationbetweenthecrystallatticetotheloadingaxis(i.e.orientation,writtenasg).q Theappliedstressresolvedalongtheslipdirectionontheslipplane(togiveashearstress)initiatesandcontrolstheextentofplasticdeformation.q Yieldbeginsonagivenslipsystemwhentheshearstressonthissystemreachesacriticalvalue,calledthecriticalresolvedshearstress(crss),independentofthetensilestressoranyothernormalstressonthelatticeplane(inlesssymmetriclattices,however,theremaybesomedependenceonthehydrostaticstress).q Themagnitudeoftheyieldstressdependsonthedensityandarrangementofobstaclestodislocation:low,suchasprecipitates(notdiscussedhere).
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Understressboundaryconditions,singleslipoccursUniaxialTensionorCompression(where“m”isthesliptensor):
PisaunitvectorintheloadingdirecNon
The(dislocation)slipisgivenby(where“m”istheSchmidfactor):
γ=ε
cosλ cosφ=εm
Minimum Work, Single Slip (Sachs)"
Thisslide,andthenextone,areare-capofthelectureonsingleslip
=b,or,s
=n
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ApplyingtheMinimumWorkPrinciple,itfollowsthat
€
στ
=˙ γ ˙ ε
=1
cosλcosφ= 1m
σ = τ(γ)m
= τ (ε m)m
Note:τ(γ) describesthedependenceofthecriKcalresolvedshearstress(crss)onstrain(orslipcurve),basedontheideathatthecrssincreaseswithincreasingstrain.TheSchmidfactor,m,hasamaximumvalueof0.5(bothangles=45°).Iffinitestrainisimposed,theshearstrain(slip)incrementisgivenbythemacroscopicstraindividedbytheSchmidfactor,dγ=dε÷m.A}ereachincrement,theSchmidfactormustberecalculatedbecausethela\ceorientaKonhaschanged(inrelaKontothetensilestressaxis)
Minimum Work, Single Slip"
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Elastic vs. Plastic Deformation"SelectionofSlipSystemsforRigid-PlasticModels
Assumption–Forfullyplasticdeformation,theelasticdeformationrateisusuallysmallwhencomparedtotheplasticdeformationrateandthusitcanbeneglected.
Reasons:
Theelasticstrainislimitedtotheratioofstresstoelasticmodulus
Perfectplasticmaterials-equivalentstress=initialyieldstress
Formostmetals,theinitialyieldstressis2to4ordersofmagnitudelessthantheelasticmodulus–ratiois<<1
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Macro Strain – Micro Slip"SelectionofSlipSystemsforRigid-PlasticModels
Oncetheelasticdeformationrateisconsidered,itisreasonabletomodelthematerialbehaviorusingtherigid-plasticmodel.TheplasticstrainrateisgivenbythesumoftheslippingratesmultipliedbytheirSchmidtensors:
€
D = Dp = mαα=1
n
∑ ˙ γ αwhere
nis≤to12systems(or24systems–)forwardandreverseconsideredindependent
Note:Dcanexpressedbysixcomponents(SymmetricTensor)Becauseoftheincompressibilitycondition–tr(D) = Dii = 0,only:iveoutofthesixcomponentsareindependent.
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Von Mises criterion"SelectionofSlipSystemsforRigidPlasticityModels
Asaconsequenceoftheconditionthenumberofpossibleactiveslipsystems(incubicmetals)isgreaterthanthenumberofindependentcomponentsofthetensorstrainrateDp,fromthemathematicalpointofview(under-determinedsystem),soanycombinationof:iveslipsystemsthatsatisfytheincompressibilityconditioncanallowtheprescribeddeformationtotakeplace.Therequirementthatatleast<iveindependentsystemsarerequiredforplasticdeformationisknownasthevonMisesCriterion.Iflessthan5independentslipsystemsareavailable,theductilityispredictedtobelowinthematerial.Thereasonisthateachgrainwillnotbeabletodeformwiththebodyandgapswillopenup,i.e.itwillcrack.Caution:evenifamaterialhas5ormoreindependentsystems,itmaystillbebrittle(e.g.Iridium).
0D=tr(D) ii =
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Minimum Work Principle"
v ProposedbyTaylorin(1938).v TheobjecKveistodeterminethecombinaKonofshearsorslipsthatwilloccurwhenaprescribedstrainisproduced.v Statesthat,ofallpossiblecombinaKonsofthe12shearsthatcanproducetheassignedstrain,onlythatcombinaKonforwhichtheenergydissipaKonistheleastisoperaKve.v ThedefectintheapproachisthatitsaysnothingabouttheacKvityorresolvedstressonother,non-acKvesystems(ThislastpointwasaddressedbyBishopandHillin1951).
Mathematicalstatement:
€
τ cα=1
n
∑ ˙ γ α ≤ τα*
α=1
n
∑ ˙ γ α*
BishopJandHillR(1951)Phil.Mag.42414;ibid.1298
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Here,
-aretheactuallyacKvatedslipsthatproduceD.-isanysetofslipsthatsaKsfytr(D)=Dii = 0,butareoperatedbythecorrespondingstresssaKsfyingtheloading/unloadingcriteria.-isthe(current)cri<calresolvedshearstress(crss)forthematerial(appliesonanyoftheαthacKvatedslipsystems).-isthecurrentshearstrengthof(=resolvedshearstresson)theαthgeometricallypossibleslipsystemthatmaynotbecompaKblewiththeexternallyappliedstress.
€
˙ γ α
€
˙ γ α*
MinimumWorkPrinciple
€
τ c
€
τα*€
τ cα=1
n
∑ ˙ γ α ≤ τα*
α=1
n
∑ ˙ γ α*
Minimum Work Principle"
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RecallthatintheTaylormodelalltheslipsystemsareassumedtohardenatthesamerate,whichmeansthat
€
τ c = τα*
andthen,
Notethatwenowhaveonly12operaKveslipsystemsoncetheforwardandreverseshearstrengths(crss)areconsideredto
bethesameinabsolutevalue.
Minimum Work Principle"
€
˙ γ αα=1
n
∑ ≤ ˙ γ α*
α=1
n
∑
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ThusTaylor’sminimumworkcriterioncanbesummarizedasinthefollowing:Ofthepossible12slipsystems,onlythatcombinaNonforwhichthesumoftheabsolutevaluesofshearsistheleastisthecombinaNonthatisactuallyoperaNve.Theuniformityofthecrss(sameonallsystems)meansthattheminimumworkprincipleisequivalenttoaminimummicroscopicshearprinciple.
€
˙ γ αα=1
n
∑ ≤ ˙ γ α*
α=1
n
∑
Minimum Work Principle"
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Stress > CRSS?"
■ TheobviousquesKonis,ifwecanfindasetofmicroscopicshearratesthatsaKsfytheimposedstrain,howcanwebesurethattheshearstressontheother,inacKvesystemsisnotgreaterthanthecriKcalresolvedshearstress?
■ ThisisnotthesamequesKonasthatofequivalencebetweentheminimum(microscopic)workprincipleandthemaximum(macroscopic)workapproachdescribedlaterinthislecture.
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Stress > CRSS?"■ Theworkincrementisthe(inner)productofthestress
andstraintensors,andmustbethesame,regardlessofwhetheritiscalculatedfromthemacroscopicquanKKesorthemicroscopicquanKKes:
��Fortheactualsetofshearsinthematerial,wecanwrite(omi\ngthe“*”),�
��wherethecrssisoutsidethesumbecauseitisconstant.
[Reid:pp154-156;alsoBishop&Hill1951]
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Stress > CRSS?"■ NowweknowthattheshearstressesonthehypotheKcal(denotedby“*”)setofsystemsmustbelessthanorequaltothecrss,τc,forallsystems,so:Thismeansthatwecanwrite:
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Stress > CRSS?"
■ HowevertheLHSofthisequaKonisequaltotheworkincrementforanypossiblecombinaKonofslips, δw=σijδεij whichisequalto τc Σαδγα, leavinguswith:
Sodividingbothsidesbyτc allowsustowrite:
€
δγα
∑ ≤ δγ*
α
∑ Q.E.D.
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Multiple Slip"
■ ThissecKonanalyzesthegeometryofmulKpleslip,allinthecrystalframe.ThissetsthesceneforthetreatmentoftheproblemintermsofsimultaneousequaKons.
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Generalcase–DØ Only:iveindependent(deviatoric)components
Ø Deformationrateismulti-axial
Crystal-FCC
Sliprates-,,....,ontheslipsystemsa1,a2,a3...,respectively.
γa1 γa2 γa3
Multiple Slip"
€
101[ ]
NotecorrecKontosystemb2 Khan&Huang
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Usingthefollowingsetofrelationscanbeobtained
2 6Dxy = 2 6ex ⋅D ⋅ey=- γa1 + γa2 − γb1 + γb2 + γc1 − γc2 + γd1 − γd2
2 6Dyz = 2 6ey ⋅D ⋅ez =- γa2 + γa3 + γb2 − γb3 − γc2 + γc3 + γd2 − γd3
2 6Dzx = 2 6ez ⋅D ⋅ey=- γa3 + γa1 + γb3 − γb1 + γc3 − γc1 − γd3 + γd1
Note:ex,ey,ezareunitvectorsparalleltotheaxes
Multiple Slip"D = Dp = mα
α=1
n
∑ γα
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= 66
=
66
= 66
d2d1c2c1b2b1a2a1
d1d3c1c3b1b3a1a3
d3d2c3c2b3b2a3a2
γγγγγγγγ
γγγγγγγγ
γγγγγγγγ
−+−+−+−
⋅⋅=
−+−+−+−
⋅⋅=
−+−+−+−
⋅⋅=
zzzz
yyyy
xxxx
eDeD
eDeD
eDeD
Multiple Slip"
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Toverifytheserelations,considerthecontributionofshearonsystemc3asanexample:
Given:
Slipsystem-c3; c3γ
Unitvectorintheslipdirection–
€
n =13
(-1,1,1)
Unitnormalvectortotheslipplane– (1,1,0)21
=b
Thecontributionofthec3systemisgivenby:
011120102
62)(
21 c3
c3
!!!
"
#
$$$
%
&−
=+γ
γ
nbbn
Multiple Slip"
Khan&Huang
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FromthesetofequaKons,onecanobtain6relaKonsbetweenthecomponentsofDandthe12shearratesonthe12slipsystems.BytakingaccountoftheincompressibilitycondiKon,thisreducestoonly5independentrelaKonsthatcanbeobtainedfromtheequaKons.So,themaintaskistodeterminewhichcombinaKonof5independentshearrates,outof12possiblerates,shouldbechosenasthesoluKonofaprescribeddeformaKonrateD.ThissetofshearratesmustsaKsfyTaylor’sminimumshearprinciple.Note:Thereare792setsor12C5combinaKons,of5shears,butonly384areindependent.Taylor’sminimumshearprincipledoesnotensurethatthereisauniquesoluKon(auniquesetof5shears).
Multiple Slip"
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Multiple Slip: Strain"
■ Supposethatwehave5slipsystemsthatareprovidingtheexternalslip,D.
■ Let’smakeavector,Di,ofthe(external)straintensorcomponentsandwritedownasetofequaKonsforthecomponentsintermsofthemicroscopicshearrates,dγα.
■ SetD2 = dε22,D3 = dε33,D6 = dε12,D5 = dε13,andD4 = dε23.
D_2&=[m_{22}^{(1)}&m_{22}^{(2)}&m_{22}^{(3)}&m_{22}^{(4)}&m_{22}^{(5)}]\cdot[d\gamma_1&\\d\gamma_2&\\d\gamma_3&\\d\gamma_4&\\d\gamma_5&]
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Multiple Slip: Strain"■ ThisnotaKoncanobviouslybesimplifiedandallfive
componentsincludedbywriKngitintabularormatrixform(wheretheslipsystemindicesarepreservedassuperscriptsinthe5x5matrix).Thisissimilartothe"basis",bp,describedbyVanHou<e(1988).
\begin{bmatrix}D_2&\\D_3&\\D_4&\\D_5&\\D_6&\end{bmatrix}=\begin{bmatrix}m_{22}^{(1)}&m_{22}^{(2)}&m_{22}^{(3)}&m_{22}^{(4)}&m_{22}^{(5)}\\m_{33}^{(1)}&m_{33}^{(2)}&m_{33}^{(3)}&m_{33}^{(4)}&m_{33}^{(5)}\\(m_{23}^{(1)}+m_{32}^{(1)})&(m_{23}^{(2)}+m_{32}^{(2)})&(m_{23}^{(3)}+m_{32}^{(3)})&(m_{23}^{(4)}+m_{32}^{(4)})&(m_{23}^{(5)}+m_{32}^{(5)})\\(m_{13}^{(1)}+m_{31}^{(1)})&(m_{13}^{(2)}+m_{31}^{(2)})&(m_{13}^{(3)}+m_{31}^{(3)})&(m_{13}^{(4)}+m_{31}^{(4)})&(m_{13}^{(5)}+m_{31}^{(5)})\\(m_{12}^{(1)}+m_{21}^{(1)})&(m_{12}^{(2)}+m_{21}^{(2)})&(m_{12}^{(3)}+m_{21}^{(3)})&(m_{12}^{(4)}+m_{21}^{(4)})&(m_{12}^{(5)}+m_{21}^{(5)})\end{bmatrix}\begin{bmatrix}d\gamma_1&\\d\gamma_2&\\d\gamma_3&\\d\gamma_4&\\d\gamma_5&\end{bmatrix}
or, D=ETdγ
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Multiple Slip: Stress"■ Wecanperformtheequivalentanalysisforstress:justaswecanformanexternalstraincomponentasthesumoverthecontribuKonsfromtheindividualsliprates,sotoowecanformtheresolvedshearstressasthesumoverallthecontribuKonsfromtheexternalstresscomponents(notetheinversionoftherelaKonship):
Or,
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Definitions of Stress states, slip systems"
Kocks: UQ -UK UP -PK -PQ PU -QU -QP -QK -KP -KU KQ
■ Nowdefineasetofsixdeviatoricstressterms,sinceweknowthatthehydrostaKccomponentisirrelevant,ofwhichwewillactuallyuseonly5:A:=(σ22-σ33) F:=σ23B:=(σ33-σ11) G:=σ13C:=(σ11-σ22) H:=σ12
■ Slipsystems(asbefore):
Note:thesesystemshavethenegaKvesoftheslipdirecKonscomparedtothoseshowninthelectureonSingleSlip(takenfromKhan’sbook),exceptforb2.
[Reid]
51
Multiple Slip: Stress"
■ Notethatitisfeasibletoinvertthematrix,providedthatitsdeterminantisnon-zero,whichitwillonlybetrueifthe5slipsystemschosenarelinearlyindependent.
■ Equivalent5x5matrixformforthestresses:
σ =E-1τ
52
Multiple Slip: Stress/Strain Comparison"■ ThelastmatrixequaKonisinthesameformasforthestraincomponents.■ WecantestfortheavailabilityofasoluKonbycalculaKngthedeterminantofthe“E”
matrix,asin:τ=Eσ or,D=ETdγ
■ Anon-zerodeterminantofEmeansthatasoluKonisavailable.■ Evenmoreimportant,thedirectformofthestressequaKonmeansthat,ifweassume
afixedcriKcalresolvedshearstress,thenwecancomputeallthepossiblemulKslipstressstates,basedonthesetoflinearlyindependentcombinaKonsofslip:σ=E-1τ
■ Itmustbethecasethat,ofthe96setsof5independentslipsystems,thestressstatescomputedfromthemcollapsedowntoonlythe28(+and-)foundbyBishop&Hill.
■ TheTaylorapproachcanbeusedtofindasoluKonforasetofacKveslipsystemsthatsaKsfiestheminimum(microscopic)workcriterion.ThemosteffecKveapproachistousethesimplexmethodbecausethemulKplepossiblesoluKonsmeanthattheproblemismathemaKcallyunderdetermined.AcompletedescripKonisfoundinthe1988reviewpaperbyVanHou<e[TexturesandMicrostructures8&9313-350].
■ ThesimplexmethodisalsousefulforanalyzinggeometricallynecessarydislocaKon(GND)content,seeEl-Dasheretal.[Scriptamater.48141(2003)].
Bishop and Hill model"
■ ThissecKondescribesthealternateapproachofBishopandHill.Thisenumeratesthecorners(verKces)ofthesinglecrystalyieldsurfacethatpermitmulKpleslipwith6or8systems.
53
54
Maximum Work Principle"■ BishopandHillintroducedamaximumworkprinciple,whichinturnwasbased
onHill'sworkonplasKcity*.Thepapersareavailableas1951-PhilMag-Bishop_Hill-paper1.pdfand1951-PhilMag-Bishop_Hill-paper2.pdf.
■ Thisstatesthat,amongtheavailable(mulKaxial)stressstatesthatacKvateaminimumof5slipsystems,theoperaKvestressstateisthatwhichmaximizestheworkdone.
■ InequaKonform,δw=σijdεij ≥σ*ijdεij ,wheretheoperaKvestressstateis
unprimed.■ Forcubicmaterials,itturnsoutthatthelistofdiscretemulKaxialstressstatesis
quiteshort(28entries).ThereforetheBishop-HillapproachismuchmoreconvenientfromanumericalperspecKve.
■ Thealgebraisnon-trivial,butthemaximumworkprincipleisequivalenttoTaylor’sminimumshear(microscopicwork)principle.
■ Ingeometricalterms,themaximumworkprincipleisequivalenttoseekingthestressstatethatismostnearlyparallel(indirecKon)tothestrainratedirecKon.
*Hill, R. (1950). The Mathematical Theory of Plasticity, Clarendon Press, Oxford.
55
Yield surfaces: introduction"■ BeforediscussingtheB-Happroach,itishelpfultounderstandtheconceptofayieldsurface.
■ ThebestwaytolearnaboutyieldsurfacesisthinkofthemasagraphicalconstrucKon.
■ AyieldsurfaceistheboundarybetweenelasKcandplasKcflow.
Example: tensile stress σ=0 σelastic plastic
σ= σyield
56
2D yield surfaces"■ Yieldsurfacescanbedefinedintwodimensions.■ ConsideracombinaKonof(independent)yieldontwodifferentaxes.
The material is elastic if σ1 < σ1y and σ2 < σ2y 0 σ1
σ2
elastic
plastic
plastic
σ= σ1y
σ= σ2y
57
Crystallographic slip: a single system"
■ Nowthatweunderstandtheconceptofayieldsurfacewecanapplyittocrystallographicslip.
■ TheresultofsliponasinglesystemisstraininasingledirecKon,whichappearsasastraightlineontheY.S.
■ ThestraindirecKonthatresultsfromthissystemisnecessarilyperpendiculartotheyieldsurface
[Kocks]
58
A single slip system"■ Yieldcriterionforsingleslip:
biσijnj ≥τcrss■ In2Dthisbecomes(σ1≡σ11:
b1σ1n1+ b2σ2n2≥τcrssThe second equation defines a straight line connecting the intercepts
0 σ1
σ2
τcrss/b1n1
τcrss/b2n2
elastic
plastic
59
Single crystal Y.S."
■ WhenweexamineyieldsurfacesforspecificorientaKons,wefindthatmulKpleslipsystemsmeetatver<ces.
■ Cubecomponent:(001)[100]
8-fold vertex
The 8-fold vertex identified is one of the 28 Bishop & Hill stress states (next slides)
Backofen Deformation Processing
60
Definitions of Stress states, slip systems (repeat)"
Kocks: UQ -UK UP -PK -PQ PU -QU -QP -QK -KP -KU KQ
■ Asetofsixdeviatoricstresstermscanbedefined.AspreviouslyremarkedweknowthatthehydrostaKccomponentisirrelevantbecausedislocaKonglidedoesnotresultinanyvolumechange.Thereforewewilluseonly5outofthe6:A:=(σ22-σ33) F:=σ23B:=(σ33-σ11) G:=σ13C:=(σ11-σ22) H:=σ12
■ Slipsystems(asbefore):
Note:thesesystemshavethenegaKvesoftheslipdirecKonscomparedtothoseshowninthelectureonSingleSlip(takenfromKhan’sbook),exceptforb2.
[Reid]
61
Multi-slip stress states"
Example:�the 18th multi-slip stress state:�A=F= 0�B=G= -0.5�C=H= 0.5
EachentryisinmulKplesof√6mulKpliedbythecriKcalresolvedshearstress,√6τcrss
[Reid]
62
Work Increment"■ Theworkincrementiseasilyexpandedas:
SimplifyingbynoKngthesymmetricpropertyofstressandstrain:
ThenweapplythefactthatthehydrostaKccomponentofthestrainiszero(incompressibility),andapplyournotaKonforthedeviatoriccomponentsofthestresstensor(nextslide).
63
Applying Maximum Work"
■ Foreachof56(withposiKveandnegaKvecopiesofeachstressstate),findtheonethatmaximizesdW:
€
dW = −Bdε11 + Adε22 +
2Fdε23 + 2Gdε13 + 2Hdε12Reminder:thestrain(increment)tensormustbeingrain(crystallographic)coordinates(seenextpage);alsomakesurethatitsvonMisesequivalentstrain=1.
64
Sample vs. Crystal Axes"■ ForageneralorientaKon,onemustpaya<enKontotheproductofthe
axistransformaKonthatputsthestrainincrementincrystalcoordinates.Althoughoneshould,ingeneral,symmetrizethenewstraintensorexpressedincrystalaxes,itissensibletoleavethenewcomponentsasisandformtheworkincrementasfollows(usingthetensortransforma<onrule):
Notethattheshearterms(withF,G&H)donothavethefactoroftwo.ManyworkedexampleschoosesymmetricorientaKonsinordertoavoidthisissue!
€
deijcrystal = gikg jldεkl
sample
BecarefulwiththeindicesandthefactthattheaboveformuladoesnotcorrespondtomatrixmulKplicaKon(butonecanusetheparKcularformulafor2ndranktensors,i.e.T’=gTgT
66
Taylor factor"■ FromthisanalysisemergesthefactthatthesameraKocouplesthemagnitudes
ofthe(sumofthe)microscopicshearratesandthemacroscopicstrain,andthemacroscopicstressandthecriKcalresolvedshearstress.ThisraKoisknownastheTaylorfactor,inhonorofthediscoverer.Forsimpleuniaxialtestswithonlyonenon-zerocomponentoftheexternalstress/strain,wecanwritetheTaylorfactorasaraKoofstressesofofstrains.IfthestrainstateismulKaxial,however,adecisionmustbemadeabouthowtomeasurethemagnitudeofthestrain,andwefollowthepracKceofCanova,Kocksetal.bychoosingthevonMisesequivalentstrain(definedinthenexttwoslides).
■ Inthegeneralcase,thecrssvaluescanvaryfromonesystemtoanother.ThereforeitiseasiertousethestrainincrementbaseddefiniKon.
€
M =στ crss
=
dγ (α )α
∑dε
=dW
τ crssdεvM
67
Taylor factor, multiaxial stress"■ FormulKaxialstressstates,onemayusetheeffecKve
stress,e.g.thevonMisesstress(definedintermsofthestressdeviatortensor,S=σ -(σii/3),andalsoknownaseffec<vestress).NotethattheequaKonbelowprovidesthemostself-consistentapproachforcalculaKngtheTaylorfactorformulK-axialdeformaKon.
σvonMises ≡ σvM =32S : S
€
M =σ vM
τ=
Δγ (s)
s∑dεvM
=dW
τ c dεvM=σ : dετ c dεvM
68
Taylor factor, multiaxial strain"■ Similarlyforthestrainincrement(wheredεpisthe
plasKcstrainincrementwhichhaszerotrace,i.e.dεii=0).
€
dεvonMises ≡ dεvM =23dεp : dεp =
23
12 dεij : dεij =
29$
% & '
( ) dε11 − dε22( )2 + dε22 − dε33( )2 + dε33 − dε11( )2{ } +
13dε23
2 + dε312 + dε12
2{ }
Compare with single slip: Schmid factor = cosφcosλ = τ/σ
€
M =σ vM
τ=
Δγ (s)
s∑dεvM
=dW
τ c dεvM=σ : dετ c dεvM
***
***Thisversionoftheformulaappliesonlytothesymmetricformofdε
69
Polycrystals"
■ Givenasetofgrains(orientaKons)comprisingapolycrystal,onecancalculatetheTaylorfactor,M,foreachoneasafuncKonofitsorientaKon,g,weightedbyitsvolumefracKon,v,andmakeavolume-weightedaverage,<M>.
■ Notethatexactlythesameaveragecanbemadeforthelower-boundorSachsmodelbyaveragingtheinverseSchmidfactors(1/m).
70
Multi-slip: Worked Example"
[Reid]
ObjecKveistofindthemul<slipstressstateandslipdistribu<onforacrystalundergoingplanestraincompression.QuanKKesinthesampleframehaveprimes(‘)whereasquanKKesinthecrystalframeareunprimed;the“a”coefficientsformanorienta<onmatrix(“g”).
71
■ ThisworkedexampleforabccmulKslipcaseshowsyouhowtoapplythemaximumworkprincipletoapracKcalproblem.
■ Importantnote:Reidchoosestodividetheworkincrementbythevalueofδε11.ThisgivesadifferentanswerthanthatobtainedwiththevonMisesequivalentstrain(e.g.inLApp).Insteadof2√6 asgivenhere,theansweris√3√6 = √18.
Multi-slip: Worked Example"
InthisexamplefromReid,“orientaKonfactor”=Taylorfactor=M
72
Bishop-Hill Method: pseudo-code"■ HowtocalculatetheTaylorfactorusingtheBishop-Hill
model?1. IdenKfytheorientaKonofthecrystal,g;2. Transformthestrainintocrystalcoordinates;3. Calculatetheworkincrement(productofoneofthe
discretemulKslipstressstateswiththetransformedstraintensor)foreachoneofthe28discretestressstatesthatallowmulKpleslip;
4. TheoperaKvestressstateistheonethatisassociatedwiththelargestmagnitude(absolutevalue)ofworkincrement,dW;
5. TheTaylorfactoristhenequaltothemaximumworkincrementdividedbythevonMisesequivalentstrain.
€
M =σ : dετ c dεvM
Note:giventhatthemagnitude(inthesenseofthevonMisesequivalent)isconstantforboththestrainincrementandeachofthemulK-axialstressstates,whydoestheTaylorfactorvarywithorientaKon?!Theansweristhatitisthedotproductofthestressandstrainthatma<ers,andthat,asyouvarytheorientaKon,sothegeometricrelaKonshipbetweenthestraindirecKonandthesetofmulKslipstressstatesvaries.
73
Multiple Slip - Slip System Selection"■ So,nowyouhavefiguredoutwhatthestressstateisinagrainthatwillallow
ittodeform.Whatabouttheslipratesoneachslipsystem?!■ TheproblemisthatneitherTaylornorBishop&Hillsayanythingabout
whichofthemanypossiblesoluKonsisthecorrectone!■ ForanygivenorientaKonandrequiredstrain,thereisarangeofpossible
soluKons:ineffect,differentcombinaKonsof5outof6or8slipsystemsthatareloadedtothecriKcalresolvedshearstresscanbeacKveandusedtosolvetheequaKonsthatrelatemicroscopicsliptomacroscopicstrain.
■ ModernapproachesusethephysicallyrealisKcstrainratesensiKvityoneachsystemto“roundthecorners”ofthesinglecrystalyieldsurface.ThiswillbediscussedinlaterslidesinthesecKononGrainReorientaKon.
■ Evenintherate-insensiKvelimitdiscussedhere,itispossibletomakearandomchoiceoutoftheavailablesoluKons.
■ ThereviewofTaylor’sworkthatfollowsshowsthe“ambiguityproblem”asthisisknown,throughthevariaKoninpossiblere-orientaKonofanfcccrystalundergoingtensiledeformaKon(shownonalaterslide).
BishopJandHillR(1951)Phil.Mag.42414;ibid.1298
74
• Thiswasthefirstmodeltodescribe,successfully,thestress-strainrelaKonaswellasthetexturedevelopmentofpolycrystallinemetalsintermsofthesinglecrystalconsKtuKvebehavior,forthecaseofuniaxialtension.
• TaylorusedthismodeltosolvetheproblemofapolycrystallineFCCmaterial,underuniaxial,axisymmetrictensionandshowthatthepolycrystalhardeningbehaviorcouldbeunderstoodintermsofthehardeningofasingletypeofslipsystem.Inotherwords,thehardeningrule(a.k.a.consKtuKvedescripKon)appliesattheleveloftheindividualslipsystem.
Taylor’sRigidPlas<cModelforPolycrystals:HardeningandReorienta<onoftheLaTce
75
Taylor model basis"■ IflargeplasKcstrainsareaccumulatedinabodythenitis
unlikelythatanysinglegrain(volumeelement)willhavedeformedmuchdifferentlyfromtheaverage(aspreviouslydiscussed).Thereasonforthisisthatanyaccumulateddifferencesleadtoeitheragaporanoverlapbetweenadjacentgrains.OverlapsareexceedinglyunlikelybecausemostplasKcsolidsareessenKallyincompressible.GapsaresimplynotobservedinducKlematerials,thoughtheyareadmi<edlycommoninmarginallyducKlematerials.Thisthenisthe"compaKbility-first"jusKficaKon,i.e.thattheelasKcenergycostforlargedeviaKonsinstrainbetweenagivengrainanditsmatrixareverylarge.
76
Uniform strain assumption"
dElocal = dEglobal,�wheretheglobalstrainissimplytheaveragestrainandthelocalstrainissimplythatofthegrainorothersubvolumeunderconsideraKon.ThismodelmeansthatstressequilibriumcannotbesaKsfiedatgrainboundariesbecausethestressstateineachgrainisgenerallynotthesameasinitsneighbors.ItisassumedthatreacKonstressesaresetupneartheboundariesofeachgraintoaccountforthevariaKoninstressstatefromgraintograin.
77
Inthismodel,itisassumedthat:
v TheelasKcdeformaKonissmallwhencomparedtotheplasKcstrain.
v EachgrainofthesinglecrystalissubjectedtothesamehomogeneousdeformaKonimposedontheaggregate,
deformaNon
Infinitesimal-
Large-
€
εgrain = ε , ˙ ε grain = ˙ ε
€
Lgrain = L , Dgrain = D
TaylorModelforPolycrystals
Taylor Model: Hardening Alternatives"
■ ThesimplestassumpKonofall(rarelyusedinpolycrystalplasKcity)isthatallslipsystemsinallgrainshardenatthesamerate,h.
78
€
dτ = h dγ polyxtal
■ ThemostcommonassumpKon(o}enusedinpolycrystalplasKcity)isthatallslipsystemsineachgrainhardenatthesamerate.Inthiscase,eachgrainhardensatadifferentrate:thehighertheTaylorfactor,thehigherthehardeningrate(becausethelargeramountofmicroscopicslip).ThesumiisoveralltheacKveslipsystems.
d⌧ = hX
i
d�(i)
Taylor Model: Hardening Alternatives, contd."
■ ThenextlevelofcomplexityistoalloweachslipsystemtohardenasafuncKonofthesliponalltheslipsystems,wherethehardeningcoefficientmaybedifferentforeachsystem.ThisallowsfordifferenthardeningratesasafuncKonofhoweachslipsysteminteractswitheachothersystem(e.g.co-planar,non-co-planaretc.).Notethat,toobtainthecrssforthejthsystem(intheithgrain)onemustsumupoveralltheslipsystemacKviKes.
79
€
dτ j(i) = h jkdγ k
(i)
k∑
Taylor Model: Work Increment"
■ Regardlessofthehardeningmodel,theworkdoneineachstrainincrementisthesame,whetherevaluatedexternally,orfromtheshearstrains.TheaverageoverthestressesineachgrainisequivalenttomakinganaverageoftheTaylorfactors(andmulKplyingbytheCRSS.
80
dW = σpolycrystal
dε = τ k (γ )k∑ dγ k
polycrystal
81
Note:Circles-computeddataCrosses–experimentaldata
TaylorModel:ComparisontoPolycrystal
Thestress-straincurveobtainedfortheaggregatebyTaylorinhisworkisshowninthefigure.Althoughacomparisonofsinglecrystal(undermulKslipcondiKons)andapolycrystalisshown,itisgenerallyconsideredthatthegoodagreementindicatedbythelineswassomewhatfortuitous!
TheraKobetweenthetwocurvesistheaverageTaylorfactor,whichinthiscaseis~3.1
82
Taylor’sRigidPlas<cModelforPolycrystals
AnotherimportantconclusionbasedonthiscalculaKon,isthattheoverallstress-straincurveofthepolycrystalisgivenbytheexpression
€
σ = M τ(γ)
ByTaylor’scalculaKon,forFCCpolycrystalmetals,
Where,τ(γ) isthecriKcalresolvedshearstress(CRSSasafuncKonoftheshearstrain)forasinglecrystal,assumedtohaveasinglevalue;<M>isanaveragevalueoftheTaylorfactorofallthegrains(whichchangeswithstrain).
€
M = 3.1
Updating the Lattice Orientation"
■ ThissecKonanalyzesoneapproachtocompuKngthechangeinla\ceorientaKonthatresultsfromslip.ThenumberofslipsystemsisnotrestrictedtoanyparKcularvalue.
83
84
Fortexturedevelopmentitisnecessarytoobtainthetotalspinfortheaggregate.Notethatthesinceallthegrainsareassumedtobesubjectedtothesamedisplacement(orvelocityfield)astheaggregate,thetotalrotaKonexperiencedbyeachgrainwillbethesameasthatoftheaggregate.Theqintroducedherecanbethoughtofastheskew-symmetriccounterparttotheSchmidtensor.
Foruniaxialtension 0=W*=WThen,
€
dWe = −dWC = q(α )dγ (α ) α=1∑
Note:
€
W e =W −W C
TaylorModel:GrainReorienta<on
€
qij(α ) =
12
ˆ b i(α ) ˆ n j
(α ) − ˆ b j(α ) ˆ n i
(α )( )
Note:“W”denotesspinhere,notworkdone
85
Taylor model: Reorientation: 1"
■ ReviewofeffectofslipsystemacKvity:■ SymmetricpartofthedistorKontensorresulKngfromslip:
■ AnK-symmetricpartofDeformaKonStrainRateTensor(usedforcalculaKngla\cerotaKons,sumoveracKveslipsystems):
mij(s) =
12
ˆ b i(s) ˆ n j
(s) + ˆ b j(s) ˆ n i
(s)( )
qij(s) =
12
ˆ b i(s) ˆ n j
(s) − ˆ b j(s ) ˆ n i
(s)( )
86
Taylor model: Reorientation: 2"
■ Strainratefromslip(addupcontribuKonsfromallacKveslipsystems):
■ RotaKonratefromslip,WC,(addupcontribuKonsfromallacKveslipsystems):
DC = ˙ γ (s)m(s)
s∑
€
W C = ˙ γ (s)q(s)
s∑
87
Taylor model: Reorientation: 3"
■ RotaKonrateofcrystalaxes(W*),whereweaccountfortherotaKonrateofthegrainitself,Wg:
■ RatesensiKveformulaKonforsliprateineachcrystal(solveasimplicitequaKonforstress):
W* =Wg −WC
€
DC = ˙ ε 0m(s) :σ c
τ (s)
n ( s )
m(s) sgn m(s) :σ c( )s∑
=τ(s)
Crystalaxesgrainslip
88
Taylor model: Reorientation: 4"
€
˙ γ (s) = ˙ ε 0m(s) :σ c
τ (s)
n ( s )
sgn m(s) :σ c( )
=τ(s)=τ(s)
■ Theshearstrainrateoneachsystemisalsogivenbythepower-lawrelaKon(oncethestressisdetermined):
89
Iteration to determine stress state in each grain"
■ AniteraKveprocedureisrequiredtofindthesoluKonforthestressstate,σc,ineachgrain(ateachstep).Notethatthestrainrate(asatensor)isimposedoneachgrain,i.e.boundarycondi<onsbasedonstrain.OnceasoluKonisfound,thenindividualslippingrates(shearrates)canbecalculatedforeachofthesslipsystems.TheuseofaratesensiKveformulaKonforyieldavoidsthenecessityofadhocassumpKonstoresolvetheambiguityofslipsystemselecKon.
■ WithintheLAppcode,therelevantsubrouKnesareSSSandNEWTON
90
Update orientation: 1"
■ GeneralformulaforrotaKonmatrix:
■ Inthesmallanglelimit(cosθ~1,sinθ~θ):
€
aij = δij cosθ + eijk nk sinθ+ (1− cosθ)nin j
€
aij = δij + eijk nk θ
91
Update orientation: 2"
■ Intensorform(smallrotaKonapprox.): R = I + W*
■ GeneralrelaKons:ω= 1/2 curlu = 1/2 curl{x-X} �- u := displacement
€
ω i =12eijk∂uk /∂X j
Ω jk = −eijkω i
ω i = −eijkΩ jk
Ω:= infinitesimal�rotation tensor
92
Update orientation: 3"
■ TorotateanorientaKon: gnew = R·gold � = (I + W*)·gold,�
�or, ifno“rigidbody”spin(Wg = 0), Note:morecomplexalgorithmrequiredforrelaxedconstraints.
gnew = I + ˙ γ sqss∑
#
$ %
&
' ( ⋅gold
93
Combining small rotations"
■ ItisusefultodemonstratethatasetofsmallrotaKonscanbecombinedthroughaddiKonoftheskew-symmetricparts,giventhatrotaKonscombineby(e.g.)matrixmulKplicaKon.
■ ThisconsideraKonreinforcestheimportanceofusingsmallstrainincrementsinsimulaKonoftexturedevelopment.
94
Small Rotation Approximation"R3 = R2R1
⇔ R3 = I + ˙ γ 2q2( ) I + ˙ γ 1q1( )⇔ Rik
(3) = δij + ˙ γ (2)qij(2)( ) δ jk + ˙ γ (1)qjk
(1)( )⇔ Rik
(3) = δijδ jk + δij ˙ γ (1)qjk(1) + δ jk ˙ γ (2)qij
(2) + ˙ γ (2 )qij(2) ˙ γ (1)qjk
(1)
≈ Rik(3) = δ ik + ˙ γ (1)qik(1) + ˙ γ (2)qik(2)
⇔ R3 = I + ˙ γ ( i )q ( i)
i∑
Q.E.D.
Neglect this second�order term for�small rotations
95
TaylorModel:Reorienta<oninTension
IniNalconfiguraNon
FinalconfiguraNon,acer2.37%ofextension
Texturedevelopment=mixof<111>and<100>fibers
NotethattheseresultshavebeentestedinconsiderableexperimentaldetailbyWintheretal.atRisø;althoughTaylor’sresultsarecorrectingeneralterms,significantdeviaKonsarealsoobserved*.
*WintherG.,2008,Slipsystems,la\cerotaKonsanddislocaKonboundaries,MaterialsSciEng.A483,40-6
EachareawithinthetrianglerepresentsadifferentoperaKvevertexonthesinglecrystalyieldsurface
Final Texture"■ ItisnotparKcularlyclearfromthepreviousfigurebutthe
Taylortheory(iso-strain)foruniaxialtensioninfccmaterialspredictsthatthetensileaxiswillmovetowardseitherthe111or100corner.Thismeansthatthefinaltextureispredictedtobeamixof<111>and<100>fibers.Thisis,infact,whatisobservedexperimentally.
■ ContrastthisresultfortheTaylortheory(iso-strain)withthatofthesingleslipsituaKon(previouslecture,iso-stress)inwhichthetensileaxisendsupparallelto112.NotethatthisrequirestwoslipsystemstobeacKve,theprimaryandtheconjugate.Thusthepredictediso-stresstextureisa<112>fiber,whichisnotwhatisobserved.
96
97
Taylor factor: multi-axial stress and strain states"
■ Thedevelopmentgivensofarneedstobegeneralizedforarbitrarystressandstrainstates.
■ Writethedeviatoricstressastheproductofatensorwithunitmagnitude(intermsofvonMisesequivalentstress)andthe(scalar)criKcalresolvedshearstress,τcrss,wherethetensordefinesthemulKaxialstressstateassociatedwithaparKcularstraindirecKon,D.S = M(D) τcrss.
■ Thenwecanfindthe(scalar)Taylorfactor,M,bytakingtheinnerproductofthestressdeviatorandthestrainratetensor:S:D = M(D):D τcrss = M τcrss.
■ Seep336of[Kocks]andthelectureontheRelaxedConstraintsModel.
98
Summary"■ MulKpleslipisverydifferentfromsingleslip.■ MulKaxialstressstatesarerequiredtoacKvatemulKpleslip.
■ Forcubicmetals,thereisafinitelistofsuchmulKaxialstressstates(56).
■ Minimum(microscopic)slip(Taylor)isequivalenttomaximumwork(Bishop-Hill).
■ SoluKonofstressstatesKllleavesthe“ambiguityproblem”associatedwiththedistribuKonof(microscopic)slips;thisisgenerallysolvedbyusingarate-sensiKvesoluKon.
Self-Consistent Model"■ FollowingslidescontaininformaKonaboutamore
sophisKcatedmodelforcrystalplasKcity,calledtheself-consistentmodel.
■ Itisbasedonafindingamean-fieldapproximaKontotheenvironmentofeachindividualgrain.
■ ThisprovidesthebasisforthepopularcodeVPSCmadeavailablebyToméandLebensohn(Lebensohn,R.A.andC.N.Tome(1993)."ASelf-ConsistentAnisotropicApproachfortheSimulaKonofPlasKc-DeformaKonandTextureDevelopmentofPolycrystals-ApplicaKontoZirconiumAlloys."ActaMetallurgicaetMaterialia412611-2624).
100
101
Kröner,BudianskyandWu’sModel
Taylor’sModel-compaKbilityacrossgrainboundary-violaKonoftheequilibriumbetweenthegrains
BudianskyandWu’sModel
-Self-consistentmodel-EnsuresbothcompaKbilityandequilibrium
condiKonsongrainboundaries-BasedontheEshelbyinclusionmodel
102
Kröner,BudianskyandWu’sModel
Themodel: v Sphere(singlecrystalgrain)embeddedinahomogeneouspolycrystalmatrix.
v CanbedescribedbyanelasKcsKffnesstensorC,whichhasaninverseC-1.
v Thematrixisconsideredtobeinfinitelyextended.
v TheoverallquanKKesandareconsideredtobetheaveragevaluesofthelocalquanKKesandoverallrandomlydistributedsinglecrystalgrains.
v ThegrainandthematrixareelasKcallyisotropic.
**,εσ p*εεσ , pε
Khan&Huang
103
Kröner,BudianskyandWu’sModel
TheiniKalproblemcanbesolvedbythefollowingapproach
1–splittheproposedschemeintotwootherasfollows
Khan&Huang
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Kröner,BudianskyandWu’sModel
1.a–TheaggregateandgrainaresubjecttotheoverallquanKKesand.InthiscasethetotalstrainisgivenbythesumoftheelasKcandplasKcstrains:
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σ ,ε
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ε p
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ε = C-1 :σ +ε p
Khan&Huang
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Kröner,BudianskyandWu’sModel
1.b–Thesphere
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" ε = ε p -ε pcompositev hasastress-freetransformaKonstrain,ε’,whichoriginatesinthedifferenceinplasKcresponseoftheindividualgrainfromthematrixasawhole.
v hasthesameelasKcpropertyastheaggregate
v isverysmallwhencomparedwiththeaggregate(theaggregateisconsideredtoextendtoinfinity)
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ε =S : # ε =S : (ε p -ε p)
Kröner,BudianskyandWu’sModel
ThestraininsidethesphereduetotheelasKcinteracKonbetweenthegrainandtheaggregatecausedbyisgivenby
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" ε
Where,SistheEshelbytensor(notacompliancetensor)forasphereinclusioninanisotropicelasKcmatrix
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Kröner,BudianskyandWu’sModel
ThentheactualstraininsidethesphereisgivenbythesumofthetworepresentaKons(1aand1b)asfollows
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ε = C-1 :σ +ε p + S : (ε p -ε p) Giventhat,
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S : (ε p -ε p) = β(ε p -ε p) where
( ) )1(15
542=νν
β−
−
Itleadsto
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ε = C :σ +ε p + β(ε p -ε p)
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Kröner,BudianskyandWu’sModel
FromthepreviousequaKon,itfollowsthatthestressinsidethesphereisgivenby
=−= )(::C= pεεεσ Ce
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=σ − 2G(1− β)(ε p -ε p)
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Kröner,BudianskyandWu’sModel
Inincrementalform
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˙ σ = ˙ σ − 2G(1− β)( ˙ ε p - ˙ ε p)where
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σ = (σ)ave, ˙ σ = ( ˙ σ )ave
ε p = (ε p)ave, ˙ ε p = ( ˙ ε p)ave
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Equations"Slide31:\tau=m_{11}\sigma_{11}+m_{22}\sigma_{22}+m_{33}\sigma_{33}+(m_{12}+m_{21})\sigma_{12}\\+(m_{13}+m_{31})\sigma_{13}+(m_{23}+m_{32})\sigma_{23}
\begin{bmatrix}-C\\B\\F\\G\\H\end{bmatrix}=\begin{bmatrix}m_{22}^{(1)}&m_{22}^{(2)}&m_{22}^{(3)}&m_{22}^{(4)}&m_{22}^{(5)}\\m_{33}^{(1)}&m_{33}^{(2)}&m_{33}^{(3)}&m_{33}^{(4)}&m_{33}^{(5)}\\(m_{23}^{(1)}+m_{32}^{(1)})&(m_{23}^{(2)}+m_{32}^{(2)})&(m_{23}^{(3)}+m_{32}^{(3)})&(m_{23}^{(4)}+m_{32}^{(4)})&(m_{23}^{(5)}+m_{32}^{(5)})\\(m_{13}^{(1)}+m_{31}^{(1)})&(m_{13}^{(2)}+m_{31}^{(2)})&(m_{13}^{(3)}+m_{31}^{(3)})&(m_{13}^{(4)}+m_{31}^{(4)})&(m_{13}^{(5)}+m_{31}^{(5)})\\(m_{12}^{(1)}+m_{21}^{(1)})&(m_{12}^{(2)}+m_{21}^{(2)})&(m_{12}^{(3)}+m_{21}^{(3)})&(m_{12}^{(4)}+m_{21}^{(4)})&(m_{12}^{(5)}+m_{21}^{(5)})\end{bmatrix}\begin{bmatrix}\tau_1&\\\tau_2&\\\tau_3&\\\tau_4&\\\tau_5&\end{bmatrix}
SLIDE34:\begin{bmatrix}\tau_1&\\\tau_2&\\\tau_3&\\\tau_4&\\\tau_5&\end{bmatrix}=\begin{bmatrix}m_{11}^{(1)}&m_{22}^{(1)}&m_{33}^{(1)}&(m_{23}^{(1)}+m_{32}^{(1)})&(m_{13}^{(1)}+m_{31}^{(1)})&(m_{12}^{(1)}+m_{21}^{(1)})\\m_{11}^{(2)}&m_{22}^{(2)}&m_{33}^{(2)}&(m_{23}^{(2)}+m_{32}^{(2)})&(m_{13}^{(2)}+m_{31}^{(2)})&(m_{12}^{(2)}+m_{21}^{(2)})\\m_{11}^{(3)}&m_{22}^{(3)}&m_{33}^{(3)}&(m_{23}^{(3)}+m_{32}^{(3)})&(m_{13}^{(3)}+m_{31}^{(3)})&(m_{12}^{(3)}+m_{21}^{(3)})\\m_{11}^{(4)}&m_{22}^{(4)}&m_{33}^{(4)}&(m_{23}^{(4)}+m_{32}^{(4)})&(m_{13}^{(4)}+m_{31}^{(4)})&(m_{12}^{(4)}+m_{21}^{(4)})\\m_{11}^{(5)}&m_{22}^{(5)}&m_{33}^{(5)}&(m_{23}^{(5)}+m_{32}^{(5)})&(m_{13}^{(5)}+m_{31}^{(5)})&(m_{12}^{(5)}+m_{21}^{(5)})\end{bmatrix}\begin{bmatrix}\sigma_{11}\\\sigma_{22}\\\sigma_{33}\\\sigma_{23}\\\sigma_{13}\\\sigma_{12}\end{bmatrix}\begin{bmatrix}\tau_1&\\\tau_2&\\\tau_3&\\\tau_4&\\\tau_5&\end{bmatrix}=\begin{bmatrix}m_{22}^{(1)}&m_{33}^{(1)}&(m_{23}^{(1)}+m_{32}^{(1)})&(m_{13}^{(1)}+m_{31}^{(1)})&(m_{12}^{(1)}+m_{21}^{(1)})\\m_{22}^{(2)}&m_{33}^{(2)}&(m_{23}^{(2)}+m_{32}^{(2)})&(m_{13}^{(2)}+m_{31}^{(2)})&(m_{12}^{(2)}+m_{21}^{(2)})\\m_{22}^{(3)}&m_{33}^{(3)}&(m_{23}^{(1)}+m_{32}^{(3)})&(m_{13}^{(3)}+m_{31}^{(3)})&(m_{12}^{(3)}+m_{21}^{(3)})\\m_{22}^{(5)}&m_{33}^{(4)}&(m_{23}^{(1)}+m_{32}^{(4)})&(m_{13}^{(4)}+m_{31}^{(4)})&(m_{12}^{(4)}+m_{21}^{(4)})\\m_{22}^{(5)}&m_{33}^{(5)}&(m_{23}^{(5)}+m_{32}^{(5)})&(m_{13}^{(5)}+m_{31}^{(5)})&(m_{12}^{(5)}+m_{21}^{(5)})\end{bmatrix}\begin{bmatrix}-C\\B\\F\\G\\H\end{bmatrix}
SLIDE37\deltaw=\sigma_{11}d\epsilon_{11}+\sigma_{22}d\epsilon_{22}+\sigma_{33}d\epsilon_{33}+2\sigma_{12}d\epsilon_{12}+2\sigma_{13}d\epsilon_{13}+2\sigma_{23}d\epsilon_{23}
SLIDE53:\Omega_{ij}^{(\alpha)}=\frac{1}{2}(b_i&^{(\alpha)}n_j&^{(\alpha)}-b_j&^{(\alpha)}n_i&^{(\alpha)})