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

2

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]

3

4

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.

Background, Concepts"

5

6

Output of Lapp*"

■  FigureshowspolefiguresforasimulaKonofthedevelopmentofrollingtextureinanfccmetal.

■  Top=0.25vonMisesequivalentstrain;0.50,0.75,1.50(bo<om).

■  Notetheincreasingtexturestrengthasthestrainlevelincreases.

*LApp=LosAlamospolycrystalplasKcity(code)

Increasingstrain

7

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

8

Sachs versus Taylor!SachsModel(previouslectureonsinglecrystal):- Allsingle-crystalgrainswithaggregateorpolycrystalexperiencethesamestateofstress;- EquilibriumcondiKonacrossthegrainboundariessaKsfied;- CompaKbilitycondiKonsbetweenthegrainsviolated,thus,finitestrainswillleadtogapsandoverlapsbetweengrains;- GenerallymostsuccessfulforsinglecrystaldeformaKonwithstressboundarycondi<onsoneachgrain.

TaylorModel(thislecture):- Allsingle-crystalgrainswithintheaggregateexperiencethesamestateofdeformaKon(strain);- EquilibriumcondiKonacrossthegrainboundariesviolated,becausethevertexstressstatesrequiredtoacKvatemulKpleslipineachgrainvaryfromgraintograin;- CompaKbilitycondiKonsbetweenthegrainssaKsfied;- Generallymostsuccessfulforpolycrystalswithstrainboundarycondi<onsoneachgrain.

9

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

13

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.

14

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!

15

Notation: 3"

■  PlasKcspin:W◆ measurestherotaKonrate;morethanonekindofspinisused:

◆  “Rigidbody”spinofthewholepolycrystal:W◆  “grainspin”ofthegrainaxes(e.g.intorsion):

Wg◆  “la\cespin”fromslip/twinning(skewsymmetricpartofthestrain):Wc.

■  RotaKon(small):ω

16

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.

17

€

DC = ˙ ε 0m(s) :σ c

τ (s)

n ( s )


18

Dislocations, Slip Systems, Crystallography"

q  ThissecKonisprovidedtoremindstudentsaboutthebasicgeometryofslipviadislocaKonglide.Fulldetailscan,ofcourse,befoundinstandardtextbooks.

19

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.

20

Crystallography of Slip"

Slipdirection–istheclose-packeddirectionwithintheslipplane.

Slipplane–istheplaneofgreatestatomicdensity.

Slipoccursmostreadilyinspeci:icdirectionsoncertaincrystallographicplanes.

Slipsystem–isthecombinationofpreferredslipplanesandslipdirections(onthosespeci:icplanes)alongwhichdislocationmotionoccurs.Slipsystemsaredependentonthecrystalstructure.

21

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

23

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.

24

25

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).

26

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

27

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"

28

Elastic vs. Plastic Deformation"SelectionofSlipSystemsforRigid-PlasticModels

Assumption–Forfullyplasticdeformation,theelasticdeformationrateisusuallysmallwhencomparedtotheplasticdeformationrateandthusitcanbeneglected.

Reasons:

Theelasticstrainislimitedtotheratioofstresstoelasticmodulus

Perfectplasticmaterials-equivalentstress=initialyieldstress

Formostmetals,theinitialyieldstressis2to4ordersofmagnitudelessthantheelasticmodulus–ratiois<<1

29

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.

30

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 =

31

Selection of Active Slip Systems:Taylor’s Minimum Work Principle"

32

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

33

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*


34

RecallthatintheTaylormodelalltheslipsystemsareassumedtohardenatthesamerate,whichmeansthat

€

τ c = τα*

andthen,

Notethatwenowhaveonly12operaKveslipsystemsoncetheforwardandreverseshearstrengths(crss)areconsideredto

bethesameinabsolutevalue.


€

˙ γ αα=1

n

∑ ≤ ˙ γ α*

α=1

n

∑

35

ThusTaylor’sminimumworkcriterioncanbesummarizedasinthefollowing:Ofthepossible12slipsystems,onlythatcombinaNonforwhichthesumoftheabsolutevaluesofshearsistheleastisthecombinaNonthatisactuallyoperaNve.Theuniformityofthecrss(sameonallsystems)meansthattheminimumworkprincipleisequivalenttoaminimummicroscopicshearprinciple.

€

˙ γ αα=1

n

∑ ≤ ˙ γ α*

α=1

n

∑


36

Stress > CRSS?"

■  TheobviousquesKonis,ifwecanfindasetofmicroscopicshearratesthatsaKsfytheimposedstrain,howcanwebesurethattheshearstressontheother,inacKvesystemsisnotgreaterthanthecriKcalresolvedshearstress?

■  ThisisnotthesamequesKonasthatofequivalencebetweentheminimum(microscopic)workprincipleandthemaximum(macroscopic)workapproachdescribedlaterinthislecture.

37

Stress > CRSS?"■  Theworkincrementisthe(inner)productofthestress

andstraintensors,andmustbethesame,regardlessofwhetheritiscalculatedfromthemacroscopicquanKKesorthemicroscopicquanKKes:

��Fortheactualsetofshearsinthematerial,wecanwrite(omi\ngthe“*”),�

��wherethecrssisoutsidethesumbecauseitisconstant.

[Reid:pp154-156;alsoBishop&Hill1951]

38

Stress > CRSS?"■  NowweknowthattheshearstressesonthehypotheKcal(denotedby“*”)setofsystemsmustbelessthanorequaltothecrss,τc,forallsystems,so:Thismeansthatwecanwrite:

39

Stress > CRSS?"

■  HowevertheLHSofthisequaKonisequaltotheworkincrementforanypossiblecombinaKonofslips, δw=σijδεij whichisequalto τc Σαδγα, leavinguswith:

Sodividingbothsidesbyτc allowsustowrite:

€

δγα

∑ ≤ δγ*

α

∑ Q.E.D.

40

Multiple Slip"

■  ThissecKonanalyzesthegeometryofmulKpleslip,allinthecrystalframe.ThissetsthesceneforthetreatmentoftheproblemintermsofsimultaneousequaKons.

41

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

42

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

43

= 66

=

66

= 66

d2d1c2c1b2b1a2a1

d1d3c1c3b1b3a1a3

d3d2c3c2b3b2a3a2

γγγγγγγγ

γγγγγγγγ

γγγγγγγγ

−+−+−+−

⋅⋅=

−+−+−+−

⋅⋅=

−+−+−+−

⋅⋅=

zzzz

yyyy

xxxx

eDeD

eDeD

eDeD

Multiple Slip"

44

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

45

FromthesetofequaKons,onecanobtain6relaKonsbetweenthecomponentsofDandthe12shearratesonthe12slipsystems.BytakingaccountoftheincompressibilitycondiKon,thisreducestoonly5independentrelaKonsthatcanbeobtainedfromtheequaKons.So,themaintaskistodeterminewhichcombinaKonof5independentshearrates,outof12possiblerates,shouldbechosenasthesoluKonofaprescribeddeformaKonrateD.ThissetofshearratesmustsaKsfyTaylor’sminimumshearprinciple.Note:Thereare792setsor12C5combinaKons,of5shears,butonly384areindependent.Taylor’sminimumshearprincipledoesnotensurethatthereisauniquesoluKon(auniquesetof5shears).

Multiple Slip"

46

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&]

47

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

48

Multiple Slip: Stress"■  Wecanperformtheequivalentanalysisforstress:justaswecanformanexternalstraincomponentasthesumoverthecontribuKonsfromtheindividualsliprates,sotoowecanformtheresolvedshearstressasthesumoverallthecontribuKonsfromtheexternalstresscomponents(notetheinversionoftherelaKonship):

Or,

49

Multiple Slip: Stress"■  Pu\nginto5x6matrixform,asforthestraincomponents,yields:

or, τ=Eσ

50

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

Taylor Factor"

■  ThissecKonexplainswhattheTaylorisandhowtoobtainit,withaworkedexample.

65

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


■  Strainratefromslip(addupcontribuKonsfromallacKveslipsystems):

■  RotaKonratefromslip,WC,(addupcontribuKonsfromallacKveslipsystems):

DC = ˙ γ (s)m(s)

s∑

€

W C = ˙ γ (s)q(s)

s∑

87


■  RotaKonrateofcrystalaxes(W*),whereweaccountfortherotaKonrateofthegrainitself,Wg:

■  RatesensiKveformulaKonforsliprateineachcrystal(solveasimplicitequaKonforstress):

W* =Wg −WC

€

DC = ˙ ε 0m(s) :σ c

τ (s)

n ( s )


=τ(s)

Crystalaxesgrainslip

88


€

˙ 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


■  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


■  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.

99

Supplemental Slides"

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

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Themodel: v Sphere(singlecrystalgrain)embeddedinahomogeneouspolycrystalmatrix.

v CanbedescribedbyanelasKcsKffnesstensorC,whichhasaninverseC-1.

v Thematrixisconsideredtobeinfinitelyextended.

v TheoverallquanKKesandareconsideredtobetheaveragevaluesofthelocalquanKKesandoverallrandomlydistributedsinglecrystalgrains.

v ThegrainandthematrixareelasKcallyisotropic.

**,εσ p*εεσ , pε

Khan&Huang

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TheiniKalproblemcanbesolvedbythefollowingapproach

1–splittheproposedschemeintotwootherasfollows

Khan&Huang

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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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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)


ThestraininsidethesphereduetotheelasKcinteracKonbetweenthegrainandtheaggregatecausedbyisgivenby

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" ε

Where,SistheEshelbytensor(notacompliancetensor)forasphereinclusioninanisotropicelasKcmatrix

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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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FromthepreviousequaKon,itfollowsthatthestressinsidethesphereisgivenby

=−= )(::C= pεεεσ Ce

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=σ − 2G(1− β)(ε p -ε p)

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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)})

polycrystal plasticity - multiple...

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