Modelling martensitic transformation in titanium alloys: The influence of temperature and deformation

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Modelling martensitic transformation in titanium alloys:

The influence of temperature and deformation

Pedro E.J. Rivera-Díaz-Del-Castillo, Madeleine Bignon, Emmanuel Bertrand,

Franck Tancret, Pedro Rivera-Díaz-Del-Castillo

To cite this version:

Pedro E.J. Rivera-Díaz-Del-Castillo, Madeleine Bignon, Emmanuel Bertrand, Franck Tancret, Pedro

Rivera-Díaz-Del-Castillo. Modelling martensitic transformation in titanium alloys: The influence of

temperature and deformation. Materialia, Elsevier, 2019, 7, pp.100382. �10.1016/j.mtla.2019.100382�.

�hal-02875249�

ContentslistsavailableatScienceDirect

Materialia

journalhomepage:www.elsevier.com/locate/mtla

Full

Length

Article

Modelling

martensitic

transformation

in

titanium

alloys:

The

influence

of

temperature

and

deformation

Madeleine

Bignon

_,

_Emmanuel

_Bertrand

_,

_Franck

_Tancret

_,

Pedro

E.J.

Rivera-Díaz-del-Castillo

a Institut des Matériaux Jean Rouxel (IMN), Université de Nantes, CNRS, Polytech Nantes, Rue Christian Pauc, Cedex 3, Nantes 44306, France b Department of Engineering, Lancaster University, Lancaster LA1 4YK, UK

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Newtheoryispresentedtodescribetheoccurrenceofplasticity-inducedtransitionsintitaniumalloys.The ap-proachisabletopredictthecompositiondependenceoftransformationinducedplasticity(TRIP),superelasticity, aswellasmartensiteformationuponquenching.Martensiteformationintheabsenceofstressisconsideredas theresultofacompetitionbetweenelasticstrainenergyandchemicaldrivingforce.Assumingthattheformation ofmartensiteistheresultofathermallyactivatednucleationprocessfollowedbyathermalgrowth,a nucle-ationparameterispostulatedtodescribetheconditionsunderwhichmartensiteisformeduponquenching;the parameteraccountsfortheratiobetweentheavailablethermalenergyandanenergybarrierfornucleation, sug-gestingthat𝜔phaseisnotthemainfactorcontrollingmartensiteinhibition.Thisnucleationparameterisable todescribe,forthefirsttime,martensiteoccurrencein130alloysfromtheliterature,quantifyingthemartensite starttemperature(Ms)reportedfor49alloyswithgreatprecision.Anempiricalparameter([Fe]eq)isproposed

and,whencombinedwiththeMsprediction,itallowstodefineregionswithinwhichTRIPandsuperelasticity

occur.BydefiningthresholdvaluesfortheMs,the[Fe]eqandthenucleationparameter,candidatealloyslikely

todisplayTRIP,superelasticityormartensitictransformationuponquenchingcanbeidentified.Asaresult,this methodcanbeadoptedtodesignalloyswithtailoredplasticitybehaviour.

1. Introduction

Titanium alloys are used for various industrial applications in-cluding aircraft, medical devices andmarine structures [1]. Among them,metastable𝛽 alloyshavereceivedspecialattentionduetotheir deformationbehaviour,sometimesdisplayedastransformationinduced plasticity(TRIP),twinninginducedplasticity(TWIP),shapememoryor superelasticityeffects.Superelasticity,observedinavarietyofsystems includingnickelalloys,ferrousalloysandtitaniumalloys[2,3],isthe capacityofanalloytoreturntoitsoriginalshapeuponunloadingeven afterhavingbeendeformedtostrainsofupto7%or8%.Thisproperty isused,forexample,inorthodonticwires[4].Alloyspresentingshape memorycan recover their original shape uponheatingafter having been deformed. Shape memory alloys have been used for various applicationssuch ascivilstructures[5], actuatorsin robotics[6] or biomedical implants [7]. TRIP alloys exhibit excellent ductility and goodworkhardeningrate.Benefitcanbe takenfromtheirrelatively lowinitialyieldstresscombinedwithahighworkhardeningrateto manufacturepartsofintricateshapes;theyareeasilyformableandthe

∗_{Correspondingauthor.}

E-mailaddress:p.rivera1@lancaster.ac.uk(P.E.J.Rivera-Díaz-del-Castillo).

finishedpartscanreachahighyieldstress,whilekeepingsignificant ductility.Automotive industryhasadoptedTRIPsteelsforstructural partssuchasbumpers,andalthoughseveralstudiesonTRIPtitanium alloys insist on their suitability for biomedical implants [8], their propertiesmaketheminterestingcandidatesforstructuralapplications aswell.

Thedeformationbehaviourof𝛽 titaniumalloysisstrongly composi-tiondependentandisgenerallyassumedtobecontrolledbythestability ofthehightemperaturebody-centredcubic(BCC)𝛽 phase[9].This sta-bilitycanbeincreasedby𝛽 stabiliserssuchasFe,Mo,VorCrandit canbedecreasedbyelementsstabilisingthelowtemperature hexago-nalclose-packed(HCP)phasesuchasAlorO.Aminimumquantityof𝛽

stabilisersisnecessarytoretaintheBCCphaseatroomtemperatureand itisreportedthattheleaststable𝛽 alloysexhibitstress-induced marten-sitetransformation,leadingtoTRIPeffectwhenthemartensiteis me-chanicallystableorsuperelasticitywhenitisunstable.Uponincreasing theconcentrationin𝛽 stabilisers,thedeformationmodeisbelievedto switchtotwinning,andwhentheirconcentrationisfurtherincreased, thedeformationonlyoccursbydislocationglide[9]inasinglestable𝛽

https://doi.org/10.1016/j.mtla.2019.100382

Received12June2019;Accepted18June2019 Availableonline6July2019

2589-1529/© 2019ActaMaterialiaInc.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense. (http://creativecommons.org/licenses/by/4.0/)

Fig.1. (a)ReproductionoftheBo–Md map from[10].Thefourcolouredzonescorrespond tothefourexpectedtypesofbehaviour accord-ingtoMorinaga’smethod.(b)Alloysfromthe literaturedisposedonthemapfromMorinaga. Theexperimentallyobservedbehaviourofthe alloysisindicatedbythecolour ofthe bul-lets(orbytheirnumber,withnumbercodeas showninAppendixA.)(Forinterpretationof thereferencestocolourinthisfigurelegend, thereaderisreferredtothewebversionofthis article.).

phase.However,themechanismsinvolvedinthestress-induced trans-formationsarenotfullyunderstoodyet.

AnumberofcriteriahavebeenadoptedtodesignTRIPandTWIP titaniumalloys.Ausefultooltopredict𝛽 phaseretentionisthe em-piricalMoequivalent([Mo]eq)parameter[9].Thecriterionbasedon

thisparameterisreliablebutdiscardsacertainnumberof𝛽 alloysthat havebeenshowntodisplayTRIP.ThemostfamousdesigntoolforTRIP andTWIPalloysisthemethodinitially proposedbyMorinagaetal. [10],sometimescalled“BondOrdermethod”.Itconsistsinmapping al-loysasafunctionoftwocomposition-dependentelectronicparameters: thebondorderbetweenatoms(Bo)andthemetald-orbitallevel(Md). ThepositionoftheTi-alloysonthemapallowstoidentifytheir defor-mationbehaviour.Theoriginalmapin[10]isreproducedinFig.1a. Althoughthis approachhasprovenits effectivenessin designing nu-merousnewTRIP/TWIPcompositions[11–13], itdisplayssome lim-itations.Theso-called phasestabilitymaphasbeendetermined from existingsystems,butsomeTRIPorTWIPalloyslieoutsidethedrawn borders(seeFig.1b),sothatitislikelythatmanyothersexistin un-exploredcompositionranges.Oneaimofthisworkistoproposeanew approachtodesignalloysundergoingmartensitictransformationupon deformation.

Amodel combiningthermodynamics andmicromechanicsis pro-posedhere to describethe martensiteformation. It is assumed that

thenucleationofmartensiteduringquenchingisathermallyactivated process requiring an energy barrier to be overcome. An expression forthis barrierissuggested,leadingtothedefinitionofanucleation parameter that allowstopredict whethermartensitic transformation occursuponquenching.Consideringthatmartensitegrowthistheresult of a competitionbetween elasticstrain energyandchemicaldriving force,amartensitestabilisationtemperature(𝑀∗

𝑠),whichcorresponds

to the experimental Ms when the energy barrier can be overcome,

is calculated. Then,a criterion combining 𝑀∗

𝑠 and a new empirical

parameter([Fe]_eq)isproposedtoidentifythecompositiondependence of TRIP, superelasticity or slip/twinning. This is the first approach capabletopredicttheoccurrenceandcompositiondependenceofsuch widerangeoftransformationanddeformationbehaviours.

2. Microstructureatroomtemperature

Inordertodesignalloysdisplayingtailoreddeformationbehaviour suchasTRIP,TWIP,superelasticityorshapememoryeffect,itisfirst re-quiredtopredicttheirmicrostructureatroomtemperature.Quenched titaniumalloyscanberoughlydividedintotwocategories.Firstly, al-loysthatformmartensiteuponquenching,andwhichmicrostructureat roomtemperature(priortodeformation)iseitherfullymartensiticor amixtureof martensiteand𝛽 phase.Themartensitecaneitherhave

anHCP structure,designatedas𝛼′,oranorthorhombiccrystal struc-ture,designatedas𝛼″[14].Thealloysofthiscategorywillhereonbe referredtoasmartensitic.Thesecondcategoryconsistsofalloysthat exhibitonly𝛽 phaseatroomtemperature.Theywillhereonbereferred

toas𝛽 alloys.ItisgenerallythoughtthatthelatterhaveanM_sbelow

roomtemperature,butnoexperimentalevidencesupportsthis hypoth-esis:somemetastable𝛽 alloyshavebeenquencheddowntocryogenic temperaturesandstilldisplaynomartensite[15,16].Thepurposeofthis sectionistoproposeamethodtopredictwhetheranalloyis marten-siticafterquenching,andtocalculateMsifitismartensitic.Therangeof

factorsinfluencingthetransformationincludetheGibbsfreeenergy,the elasticenergy,thepresenceofmartensitenucleationsites,thegrainsize andthequenchrate.Theseareaccountedforinthetheorypresentedin thisandtheforthcomingsection.

2.1. Martensitestarttemperaturecalculation

2.1.1. Model

InordertoestimatetheMstemperature,thechangeinenergyΔG

associatedwiththemartensiteproduct,treatedhereasanellipsoidal inclusionin theBCC𝛽 phase, is computedasdescribed byWollants etal.[17].

Δ𝐺=2𝜋𝑟2_{𝛾 −}4

3𝜋𝑟

2_𝑐(Δ𝑔

𝑐ℎ−Δ𝑔𝑒𝑙) (1)

wherecisthesemithicknessofthemartensiteellipsoidalinclusion,ris itsradius,Δgch1isthevolumetricchemicaldrivingforce.Δ𝑔𝑐ℎ=𝑔𝛽−𝑔𝛼′

whereg_𝛽 and𝑔𝛼′ arerespectivelytheGibbsfreeenergiesoftheBCC

andtheHCPphases.Thisquantityiscomputedemploying thermody-namicsoftware(Thermo-Calc®_{)and’TCTI1’thermodynamic titanium}

database2_{,andbyassumingthatthefreeenergyoftheorthorhombic}

phase𝛼″canbeapproximatedbythefreeenergyoftheHCPphase𝛼′

(discussedinSection5.3).Δgel isthevolumetricelasticstrainenergy

inducedbythepresenceoftheplate;𝛾 istheinterfacialenergyperunit areaofproductphase.Theinterfacialandelasticenergiesopposethe transformation.

Computationoftheelasticstrainenergy

Theeigenvaluesofthestraintensordescribingthetransformation fromtheBCCstructuretotheorthorhombicorHCPstructureare[14]:

𝜆1= 𝑎𝛼𝑖−𝑎_𝛽 𝑎𝛽 (2) 𝜆2= 𝑏𝛼𝑖−√2𝑎_𝛽 √ 2𝑎_𝛽 (3)

1 _{Theconventionadoptedhereissuchthatthedrivingforceispositivewhen}

thetransformationispromoted

2 _{Calculationswereperformedaswellwiththeconcurrentdatabase’TTTI3’}

whichproducedsimilarresults.Anotherdatabasespecificto𝛽 titaniumalloys labelled’Ti-Gen’hasbeenproposedbyYanandOlson[18].Tovalidatethis database,theseauthorsshowedthattheycouldobtaingoodagreementbetween experimentalvaluesofMs,andvaluesofMsextractedfromTi-Gendatabase.

However,themodeltheyusedtocomputeM_sincludedbothvaluesfromthe Ti-Gendatabaseandfittingparameters.Therefore,potentialinaccuracyinthe databasewouldbecorrectedbythefittingparameters,sothattheadequacyof Ti-GencanonlybeprovenintheframeofthisMsmodel.Assoonasother

equa-tionsareusedtocomputeMs,wethereforehavenoguaranteeoftheaccuracy

ofthedatabase.SincetheapproachusedhereisdifferentthanYan’sapproach, usingTi-Genhasnotbeenattempted.Inaddition,elementspresentineither TTTI3orTi-Genwouldallowtheirapplicationtoamuchreducednumberof alloysthanTCTI1.

𝜆3=

𝑐𝛼𝑖−√2𝑎_𝛽 √

2𝑎_𝛽

(4) wherea_𝛽 isthelatticeparameteroftheBCCstructure,𝑎_𝛼𝑖,𝑏_𝛼𝑖 and𝑐_𝛼𝑖 arethelatticeparametersofthemartensiticphase,where𝛼i_designates 𝛼′ifthemartensiteisHCPand𝛼″ifitisorthorhombic.Theexpression forthelatticeparametersasafunctionof compositioncanbe found intheliteratureforbinaryalloys,andtheelementalcontributionsare assumedtobeadditive.Theexpressionsemployedforlatticeparameters calculationareprovidedinAppendixB.

Theelasticstrainenergyiscomputedbyconsideringanellipsoidal martensiteplateinaninfinitehomogeneousisotropicelasticmedium. Themartensiteisconsideredtobeelasticallyisotropic.Thestraintensor isdividedintoitsdeviatoricandhydrostaticcomponents,assuggested byOlsonandCohen[19],sothattheelasticstrainenergyΔgelcanbe

dividedintoatransformationvolume-changeterm(Δ𝑔_𝑒𝑙𝑑𝑖𝑙)anda shape-changeterm(Δ𝑔𝑠ℎ𝑎𝑝𝑒_𝑒𝑙 ),withΔ𝑔_𝑒𝑙=Δ𝑔_𝑒𝑙𝑑𝑖𝑙+Δ𝑔𝑠ℎ𝑎𝑝𝑒_𝑒𝑙 .

Thevolumechange isapproximated byapuredilatationandthe associatedelasticenergyiscomputedasdescribedbyEshelby[20]: Δ𝑔𝑑𝑖𝑙_𝑒𝑙(𝑇)=2 9 (1+𝜈) (1−𝜈)⋅ 𝜇(𝑇)⋅ (_Δ𝑉 𝑉 )2 (5) where𝜈 isthe Poisson’sratio,taken hereas equalto 1₃,𝜇(T) is the shear modulusattemperatureT, computedasdescribed by Galindo-Nava[21]:

𝜇(𝑇)=𝜇_𝑅𝑇(𝑘1+𝑘2𝑇) (6)

where𝜇RTisthecomposition-dependentshearmodulusatroom

tem-perature(RT),obtained byamixture rule,Tis theabsolute temper-ature,k1 andk2 arethefittedconstants describedin [21]andyield

𝑘1=1+19₂₍₁₊.63689_𝜈) and𝑘2=−0₂₍₁₊.07193_𝜈). Δ𝑉

𝑉 istherelativechangeinvolumeaccompanyingthe

transforma-tion: Δ𝑉 𝑉 = (₁ 3(𝜆1+𝜆2+𝜆3)+1 )3 −1 (7)

Theshapechangeisapproximatedbyapureshearandiscomputed asdescribedbyEshelby[20]: Δ𝑔𝑠ℎ𝑎𝑝𝑒_𝑒𝑙 (𝑇)=𝜇(𝑇)⋅ 𝜋 2 (2−𝜈) (1−𝜈)⋅ 𝜖 2_{⋅ 𝑐} 𝑟 =𝐾(𝑇)𝑐𝑟 (8)

K(T)isintroducedinEq.(8)toshortenthenotation,𝜖 istheshear com-ponentofthestrainassociatedwiththephasetransformation[19]and iscompositiondependentviathelatticeparameters:

𝜖 = √ 1 6 ( (𝜖1−𝜖2)2+(𝜖2−𝜖3)2+(𝜖3−𝜖1)2 ) (9) where𝜖1,𝜖2and𝜖3aretheeigenvaluesofthedeviatoriccomponentof

thestraintensor(𝜖𝑖=𝜆𝑖−31(𝜆1+𝜆2+𝜆3))

ComputationofMs Bycombining(1),(5)and(8): Δ𝐺(𝑇)=2𝜋𝑟2_{𝛾 −}4 3𝜋𝑐𝑟 2(_Δ𝑔 𝑐ℎ(𝑇)−Δ𝑔𝑑𝑖𝑙𝑒𝑙(𝑇) ) +4 3𝜋𝑐 2_{𝑟𝐾(𝑇) (10)}

Above T0, nucleationcannotoccurandthe alloyonly displays𝛽

phase(Fig.2a).Thenucleationofaproductembryoofsemi-thickness

c0andradiusr0occursbelowthetemperatureatwhichtheHCPand

BCCGibbsfreeenergiesareequal(T0).Then,asexplainedbyWollants

et al.[17], whenthe temperatureis suchthat 𝜕Δ𝐺_𝜕𝑟 <0, theembryo growsradially(increaseofr,Fig.2b), untilitisblocked byastrong obstacle (e.g. a grain boundary) andreaches aradius a/2, wherea

isthesizeoftheparentgrain(Fig.2b).Fromthatpoint,theparticle continuestothickeninthedirectionnormaltotheplateplane(increase ofthesemi-thicknessc),untilitreachesastatesuchthatthegrowth does not allow thesystem to further reduce its energyand a local equilibriumisreachedattheinterfacebetweentheparentandproduct

Fig.2. Martensitenucleationandgrowth.(a) BCC𝛽 phaseatatemperatureaboveT0(b)A

martensiteembryonucleatesandstarts grow-ingradially(c)Radialgrowthisstoppedbya grainboundaryandthethickeningstarts.(d)A localthermoelasticequilibriumisreachedand thethickeningstops.Furthercoolingis neces-saryforthethickeningtocontinue.

phases (Fig. 2d).The condition for such anequilibrium is: 𝜕Δ𝐺

𝜕𝑐 =0

[17]. When a product plate is formed from an existing defect, this conditionisobtainedbyderivatingEq.(10):

Δ𝑔_𝑐ℎ−Δ𝑔𝑑𝑖𝑙_𝑒𝑙 −2𝐾𝑐

𝑟=0 (11)

Letuslabelc_dtheminimaldetectablesemi-thicknessfora marten-siteplate.Thisdetectablesemi-thicknessissetto50nm;thisroughly matchestheusualminimalthicknessthatisexperimentallyreported3_.

Then,ifaisthegrainsize,wedefinethemartensitestabilisation tem-perature𝑀∗

𝑠 asthetemperatureatwhichaplateofmartensitereaches

adetectablesize:

Δ𝑔_𝑐ℎ(𝑀_𝑠∗)−Δ𝑔_𝑒𝑙𝑑𝑖𝑙(𝑀_𝑠∗)−2𝐾(𝑀_𝑠∗) 𝑐𝑑

𝑎∕2 =0 (12)

Anyfurthercoolingleadstofurtherthickening,untilthemartensite finishtemperatureisreached(Mf)andthetransformationiscomplete.

Themartensitestabilisationtemperature𝑀∗

𝑠 isdefinedasthehighest

temperatureatwhichmartensitecanreasonablyexperimentallybe ob-servedifnucleationhashappened,andshouldthereforebeveryclose fromtheexperimentalmartensitestarttemperature.

3 _{For example,Gao etal. investigatedthe deformed microstructure of a}

TWIP/TRIPtitaniumalloyatdifferentstrainlevels[12];fortheloweststrain atwhichmartensiteplateswereobserved,theywereapproximately100nm thick(50nmsemi-thickness).Verydetailedmicrostructuralinvestigationwas alsoperformedbyZhangetal.inaTRIP/TWIPalloy;theyreportedthepresence of30–150nmthickmartensiteplates[22].Itthereforeseemsreasonableto as-sumethatinthegeneralcase,thepresenceofafewplatesofmartensiteless than100nmthickmightgounnoticed,whichjustifiesthechosendetectable limit.

𝑀∗

𝑠isobtainedbyiterativelycomputingthequantityonthelefthand

sideofEq.(12)fordifferenttemperatures.Sincetheelasticstrainenergy dependsonthestructureofthemartensite(𝛼″or𝛼′),𝑀∗

𝑠 iscalculated

twiceforeachcomposition,byassumingbothanHCPmartensiteand anorthorhombicmartensite(seeAppendixB).Theretained𝑀∗

𝑠 isthe

highestofboth.AflowdiagramforthiscalculationisshowninFig.3. Themartensitetemperaturecalculatedhereistakentobethatat which a plate of martensite reaches a detectable size. The compu-tation of 𝑀∗

𝑠 is thus based on a criterion regarding the growth of

themartensiteandnotitsnucleation.Indeed,theequationsdescribed above arevalidforanexistingmartensiteplate.𝑀∗

𝑠 iscomputedby

assuming that the nucleationhasalready taken place ata tempera-turesomewhereinbetween𝑀∗

𝑠 andT0.Thislastpointisdevelopedin

Section2.2.

Themethoddescribedabovehasbeenappliedtocalculatethe𝑀∗

𝑠

temperatureofalloysfromliterature.Sincethecalculationdependson the grain sizea, which is not alwaysreported, anaverage valueof 100𝜇mhasbeenassumedforallalloys(thegrainsizedependencywill bediscussedinSection5.3).Fig.4showsthecalculated𝑀∗

𝑠 for

differ-entalloystogetherwiththeirtypeofbehaviourduringplastic deforma-tion.ReferringtoFig.4,whenthe𝑀∗

𝑠 isapproximatelybelow−100°C,

martensiteisnotreported,neitheruponquenchingnortheapplication ofanexternalstress.Thealloysforwhichthe𝑀∗

𝑠 isaroundorjust

be-lowroomtemperaturearemostlysuperelastic.Forthesealloys,whenan externalstressisapplied,thedrivingforcenecessaryforthemartensitic transformationissuppliedbytheexternalwork,butwhenthestress isremoved, themartensiteisnolongerstableandreversesbackto𝛽

phase.TRIPalloysentirelyretainthe𝛽 phasefollowingquenching,but the𝑀∗

𝑠 formostofthemliesintherange100–300°C.Nevertheless,the

calculationof𝑀∗

Fig.3. Flowdiagramfor𝑀∗

𝑠 calculation.

Fig.4. Calculated𝑀∗

𝑠 foralloysgroupedasmartensiticatroomtemperature,TRIP,superelasticorslip/twinning(fromlefttoright,thedottedlinesseparatethe

differenttypesofbehaviour).Thehorizontalaxisisonlytospreadthealloysinthepanel.ThealloyswhichnumberiswritteninagreybackgrounddisplayTRIP, buthavealsobeenseenmartensiticundercertainexperimentalconditions.Whenthecalculationsindicatedthattherewasnotemperatureforwhichmartensiteis thermomecanicallymorestablethan𝛽,the𝑀∗

𝑠 hasbeensetto−200°.ThenumbercodeforeachcompositionisshowninAppendixA. TRIPalloys,whenanexternalstressisapplied,themartensitic

trans-formationisinducedanddoesnotreversewhentheloadisreleased, whichmeansthatthemartensiteis probablystableat room temper-atureandthatitcannotbeassumedthatMsliesbelowroom

tempera-ture.Themodelrightlypredictsthatmartensiteshouldbestableatroom temperature.

In the case of TRIP alloys, it may seem paradoxical that the calculated 𝑀∗

𝑠 liesabove room temperaturewhen the experimental

evidenceshowsthatnomartensiteformsuponquenching.Thereason forthisisthatthe𝑀∗

𝑠 computationimplicitlyconsidersthatmartensite

nucleationhasalready taken placewhenthe𝑀∗

𝑠 is reached.It may

Fig.5. Calculated𝑀∗

𝑠 andexperimentalvaluesforMsobtaineduponquenching.ThenumbercodeforeachcompositionisshowninAppendixA.

quenching, andthatat roomtemperaturetheexternalwork triggers nucleation.Oncethemartensitehasnucleatedduetoexternalwork,the growthcan athermallystart.Asaconsequence,itseemsnecessaryto haveatoolthatallowstopredictifthenucleationofmartensiteislikely tooccuruponquenching,orifthenucleationrequiresexternalwork.

Fig.5showsacomparisonbetweenthecalculated𝑀∗

𝑠 andthe

ex-perimentalMs,whenthisexperimentalMs ismeasuredduringcooling

andthetransformationoccursintheabsenceofexternalstress.Some martensiticalloysappearinFig.4butnotinFig.5,thesecompositions areknowntobemartensiticafterquenchingbutnoinformationwas foundregardingtheirMs.

2.2. Martensitenucleationuponquenching

TRIPalloysdisplay𝛽 microstructureafterquenching,eventhough

𝑀∗

𝑠computationindicatesthatmartensiteshouldbestableatroom

tem-perature(seeFig.4).Thecompetitionbetweenmartensiteand𝜔phase isaplausiblehypothesistoexplaintheabsenceofmartensite[18]. Al-though𝜔formationduringcoolingmayplayaroleinmartensite in-hibition,thermodynamiccalculationssuggestthatitisnotnecessaryto invoke𝜔phaseformationtoexplainthestabilisationof𝛽 phaseatroom temperature,aswillbediscussedinSection5.2.Thepurposeofthis sec-tionistoproposeanexplanationforthestabilisationofthe𝛽 phaseat roomtemperature,andamethodtoquantifyit.

2.2.1. Model

OlsonandCohen[23]haveshownthatthefirststepofthetransition fromtheBCCtoHCPphasecanberationalisedasthedissociationofa

1

2[111]screwdislocationona(1̄10)planeintothreepartialsaccordingto

thereaction1₂[111]=1

8[110]+ 1 4[112]+

1

8[110][24].Assumingthatthe

latticeopposestheglideofthedissociatingpartials,anenergybarrier mustbeovercomeforthedislocationtosplitintopartials(corresponding toahighenergytransitionstatebetweenBCCandHCP,ortotheenergy necessaryforthefirststepsofthedissociationofaperfectdislocation intopartials).Thisenergybarriercould,forexample,besimilartoa Peierlsbarrier(asitrepresentsanoppositiontotheglideofapartial dislocation),andthus,iftheenergybarrierperdislocationis labeled

E∗_{,itcouldhaveaformsuchas[25]:}

𝐸∗_{(𝑇)=𝜒𝑏}3_{𝜇(𝑇) (13)}

where𝜒 isaconstantindependentfromcomposition,bisthemagnitude oftheBurgersvectorofthescrewdislocation,calculatedas𝑏=1

2

√ 3𝑎_𝛽. Thisenergybarrierisassumedtobeovercomebythermalfluctuations. Foracharacteristicdislocationvibrationoffrequencyf₀,theenergy bar-rierissuccessfullyovercomeataratef[25]:

𝑓(𝑇)=𝑓₀⋅ exp(−𝐸∗_{(𝑇)∕(𝑘𝑇)) (14)}

wherekistheBoltzmannconstant.Thenucleationrateofmartensiteis expressedas:

𝑑𝑁

𝑑𝑡 =𝑁0⋅ 𝑓(𝑇)=𝑁0⋅ 𝑓0⋅ exp(−𝐸∗(𝑇)∕(𝑘𝑇)) (15)

whereN₀isthenumberdensityofpotentialnucleationsites.Ifthe cool-ingoccursatacoolingrate𝑄=−𝑑𝑇_𝑑𝑡,fromEq.(16)comes:

𝑑𝑁 𝑑𝑇 =− 𝑁0 𝑄 ⋅ 𝑓(𝑇)=− 𝑁0 𝑄 ⋅ 𝑓0⋅ exp(−𝐸∗(𝑇)∕(𝑘𝑇)) (16)

BydividingthecoolingprocessinelementaryΔTtemperaturesteps, thenumberofembryosthatnucleateateachstepyields:

Δ𝑁(𝑇)=||_||𝑑𝑁

𝑑𝑇Δ𝑇||||=

𝑁0⋅ |Δ𝑇|

𝑄 ⋅ 𝑓(𝑇) (17)

ThenucleationofmartensiteisassumedtostartaroundT₀asthe maximumtemperatureatwhichamartensiticembryocanpotentially form,anditisthereforethetemperatureatwhichthethermalenergy fornucleationismaximumandtheenergybarrierisminimum(the bar-rierisproportionaltotheshearmoduluswhichlinearlydecreaseswith temperature).SincethereisnomartensitenucleusaboveT0,thenumber

densityN(T0)ofmartensiteembryosnucleatingduringthefirststepof

thetransformationcanbeapproximatedby(𝑑𝑁

𝑑𝑇

)

𝑇0

⋅ Δ𝑇.Thus, accord-ingtoEqs.(14)and(17):

Fig.6. Nucleationparameter,calculatedforalloysfromtheliterature.Thehorizontallineisthethresholdthatisfoundtoseparatethemartensiticalloysfromthe

𝛽 alloys.Thehorizontalaxisisonlytospreadthealloysinthechart.Thealloysreferredtoas“martensiticor𝛽” aretheonesthathavebeenreportedmartensiticor

𝛽 dependingontheexperimentalconditions.ThenumbercodeforeachcompositionisshowninAppendixA.

𝑁(𝑇0)=

𝑄0

𝑄 ⋅ 𝑁0⋅ exp(−𝐸∗(𝑇0)∕(𝑘𝑇0)) (18)

where𝑄0=|Δ𝑇| ⋅ 𝑓0

Thenumberofmartensitenucleiappearingatanylowertemperature intervalisassumedtoscalewiththenumberdensityofnucleiatT0,so

thatthetotalnumberofnucleishouldberoughlyproportionaltoN(T0).

Therefore,thereshould exista criticalnumberdensityof martensite nucleiNm,suchthatnucleationcanbeconsideredasoccurringonlyif N(T0)>Nm.FromEqs.(13)and(18),theconditionforthemartensite

nucleationuponquenchingis:

𝑘𝑇0 𝜇(𝑇0)𝑏3 > 𝜒 ln(𝑁0 𝑁𝑚 𝑄0 𝑄) (19)

OnthelefthandsideofInequation(19),thetermkT0istheavailable

thermalenergyfornucleationatthefirststagesoftransformationand theterm𝜇(T0)b3isproportionaltotheenergybarrierfornucleation.The

term 𝑘𝑇0

𝜇(𝑇0)𝑏3thereforescaleswiththeratiobetweenanavailablethermal

energyandarequiredenergyfornucleation;itwillhereonbereferredto as“nucleationparameter”.Thenucleationparameterstronglydepends oncomposition,throughthepartitionlessequilibriumtemperatureT0

between𝛽 and𝛼′,comingfromThermo-Calc®_{,andthroughtheshear}

modulus.Inequation(19)indicatesthatmartensitecannucleateduring coolingonlyifthenucleationparameterexceedsacertainvalue,which willbereferredtohereonas“criticalnucleationparameter”.Thiscritical nucleationparameterdepends onthecoolingratebutiscomposition independent.Byseparatingthealloysfromtheliteratureintotwogroups (martensiticand𝛽),andcomputingforeachofthemthevalueofthe nucleationparameter,thevalueofthecriticalnucleationparametercan thuseasilybefitted.

2.2.2. Applicationtoalloysfromliterature

Thenucleationparameteriscomputedfordifferentalloysfromthe literature asshown in Fig.6, whereitis clearthat acritical nucle-ationparametersetas[ln(𝑁0

𝑁𝑚

𝑄0

𝑄)]−1⋅ 𝜒 =1.24⋅ 10−2allowstoseparate

martensiticfrom𝛽 alloys.Thisvalueisassumedtobevalidfora con-ventionalcoolingrate.Thenucleationparameterisalsocalculatedfor

Fig.7.Calculationofthenucleationparameterforbinarysystems.Thedotted lineshowsthethresholdexpectedtodelimitatethemartensiticalloysfromthe

𝛽 alloysafterquenching.

binarysystems,andshowninFig.7.Theblackdottedlineshowsthe limittoretain𝛽 atroomtemperature.Inordertocomparethiswith theavailableexperimentaldata,thecriticalcompositionsnecessaryto retain𝛽 atroomtemperatureareextractedfromtheliteratureand gath-eredinTable1.Thethresholdof1.24⋅ 10−2_{isalsousedtocalculate,}

foreachbinarysystem,thecriticalcompositionnecessarytoretain𝛽 at roomtemperature.Thesecalculatedvaluesareplottedasafunctionof experimentaldatafromTable1inFig.8.

3. IdentifyingTRIP,superelasticandslip/TWIPalloys

Thissectionproposes parameterstoidentifyTRIP,superelasticity, andalloysdeformingbydislocationsliportwinning.Inthiswork,TRIP effectisdefinedastheformationofmartensiteupondeformation

with-Fig.8. Calculationofthecriticalconcentrationtoretain𝛽 uponquenchingfor differentbinarysystemsasafunctionoftheexperimentaldata.Thehorizontal linesrepresenttheexperimentalrangesofcriticalcompositionsfoundinthe literature,whichreferencesareshowninTable1.Thedottedlineisshownfor reference.

Table1

Criticalcompositionsnecessarytoretain𝛽 phaseuponquenchinginbinary al-loys,extractedfromtheliterature.

Element Minimum concentration necessary to retain 𝛽 upon quenching in wt % Fe 3.5 [26] , 5.8 [27] , between 3.08 and 4.07 [28] Cr 6.5 [26] , 7.6 [27] , between 5.4 and 6.5 [28] Mo 10 [26] , 9.5 [27] V 15 [26] , 15.8 [27] , about 15 [28] W 22.5 [26] , 30 [27] , between 20 and 25 [28]

Nb 29 [29] (high quenching rate) 36 [27]

Ta 45 [26] , 68 [27]

outreversionto𝛽 uponunloading,whereas thealloysreferredtoas superelasticaretheonesinwhichthemartensitethatformsupon de-formationreversesuponunloading.Sofar,no solutionwas foundto predicttwinningoccurrence,so thealloysreferredtoas“TRIP” may alsodisplaytwinning,asoftenobservedexperimentally.Alloysreferred toas“sliportwinning” mayormaynotdisplayTWIPbutdonotdisplay martensitictransformation.

IdentifyingslipandTWIPalloys

Inthepresentsection,wefocusonalloysthatarefully𝛽 atroom tem-perature.Astheconcentrationin𝛽 stabilisersincreases,thedeformation behaviourgenerallyswitchesfromstress-inducedmartensite(TRIPor superelasticity),toTWIPordislocationslip.Thistransitioncannotbe explainedsolelybythevalueof𝑀∗

𝑠(Fig.4).Itisassumedthat,forsuch

alloysthecriticalstressfortriggeringthenucleationof martensiteis toogreatandthattwinningordislocationglideisenergeticallymore favourable.Thecriticalstresstotriggermartensite,dislocationglideor twinningcouldnotbeassessedinthepresentwork.Anempirical cri-terionisinsteadproposed;inspiredbutdifferingfromthe[Mo]eq.The

[Mo]eqhadbeenbuiltthankstoexperimentaldataonbinaryalloysin

order topredict thetendencyofanalloytoretainthe𝛽 phaseupon

quenching.TheproposedFeequivalent([Fe]eq)criterionistailoredto

experimentaldatainorder todescribethetendencytoretain𝛽 upon

deformation.

ThesuggestedcriterionusesFeasareferenceelementbecauseithas thestrongeststabilizingeffectonthe𝛽 phase.Theproposed “[Fe]eq”

criterionisobtainedinthefollowingway:thecoefficienttoeachterm inEq.(20)isobtainedbydividingthecriticalconcentrationofFe

nec-Table2

Criticalcompositionsnecessarytoretainthe𝛽 phaseupondeformationinbinary alloys,calculatedandextrapolatedfromdataavailableintheliterature.

Element Adopted limit wt.% composition for SIM

Alloys used as references

Fe 3.5 Ti–10Mo, Ti–10Mo–1Fe [30]

Cr 9 Ti–10Cr [31]

Mo 14 Ti–13Mo [32] , Ti–14Mo [33]

V 20 Ti–16V [32] , Ti–20V [34]

W 25

Sn 27 Ti–3Mo–6Sn–8Zr at%, Ti–3Mo–6.5Sn–8Zr at%

[35]

Nb 43 Ti–40Nb, Ti–43Nb [36]

Ta 75 Ti–72Ta [37]

Zr 90 Ti–3Mo–7Sn–4Zr at%, Ti–3Mo–7Sn–5Zr at% [35]

Al − 18 Ti–15–Mo–5Zr, Ti–15Mo–5Zr–3Al [34]

essarytoretain𝛽 underdeformationintheTi–Fesystembythecritical concentrationnecessarytoretain𝛽 underdeformationinthegiven bi-narysystem.Whenthiscriticalcompositionwasnotfoundinthe litera-ture,ithasbeenextrapolatedfrommulticomponentalloys.Inthecaseof CrandFebinarysystems,nocompositionhasbeenreportedtodisplay TRIP;therefore,itisassumedthat,forthesestrong𝛽 stabilisers,the crit-icalconcentrationtoretain𝛽 upondeformationisverycloseto,orlower thanthecriticalcompositiontoretain𝛽 atRT.Thecriticalcompositions consideredforthe[Fe]eqexpressionaregatheredinTable2,together

withthecompositionsthatwereusedtodeterminetheirvalues.Sincein thecaseoftheTi–Fesystem,thecriticalconcentrationnecessaryto

re-tain𝛽 upondeformationhasbeenevaluatedto[Fe]=3.5wt%(Table2),

thecompositionssuchthatthe[Fe]eqexceeds3.5%areexpectedto

in-hibitthemartensitictransformationunderdeformation.Theassessment ofthiscriteriononmulticomponentalloyswillbepresentedlater.

[ Fe ] _{𝑒𝑞 = 3 . 5 ⋅} ( [ Fe ] 3 . 5 + [ Cr ] 9 + [ Mo] 14 + [ V ] 20+ [ W ] 25 + [ Sn ] 27 + [ Nb ] 43 + [ Ta ] 75 + [ Zr ] 90 − [ Al ] 18 ) ( wt% ) (20)

IdentifyingTRIPandsuperelasticalloys

Whenmartensitetransformationisinducedbyanexternalwork,the productphasecaneitherremainorreverseto𝛽 afterunloading.The firstcasecorrespondstoTRIPalloys,thesecondtosuperelasticalloys. Oncethealloysthatcannotformmartensiteunderstressareidentified viaan[Fe]eqbelow3.5wt%,the𝑀_𝑠∗computationcanseparate

supere-lasticfromTRIPalloys(Fig.9).Ifthe𝑀∗

𝑠 exceeds90°C,itislikelythat

themartensitereversetemperatureis aboveroomtemperature; once thenucleationistriggeredbyanexternalwork,martensiteisexpected toformandtoremainafterunloading.Thus,thesealloysareexpected todisplayTRIP.If−90°C < 𝑀∗

𝑠 < 90°C,theenergyprovidedbythe

external stressmayenablethegrowthof themartensite,butits sta-bilityatroomtemperatureisnotsufficienttoremainoncetheloadis removed.Thus,thealloyswith−90°C<𝑀∗

𝑠 <90°Careexpectedto

dis-playsuperelasticity.If𝑀𝑠<−90°C,itisunlikelythattheexternalstress

providesenoughworkforthemartensitetogrow.Thus,thealloyswith

𝑀∗

𝑠 <−90°Careexpectedtodisplaysliportwinningonly.Therefore,

combinedwiththe[Fe]_eqcriterion,the𝑀∗

𝑠calculationallowstopredict

thetypeofdeformationbehaviourof𝛽 alloys.

Regionsin𝑀∗

𝑠 -[Fe]eqspaceidentifyingsystemsundergoingslipor

twinning,TRIPorsuperelasticityeffectsareshowninFig.9

4. Criteriafortailoreddeformationbehaviour

Thecriteriadescribedabovecanbecombinedtopredictifa com-positionislikelytodisplaymartensiteuponquenching,superelasticity, TRIPorSlip/TWIPeffect.Thepredictionprocedureissummed upin Fig.10.

Fig.9. Mapofthealloysfromtheliterature.Thethreeregionscorrespondingtotheindicateddeformationbehaviourareoutlinedbytheblackdottedlines.The coloursofthebulletscorrespondtotheexperimentallyobservedbehaviour.Whenthecalculationsindicatedthattherewasnotemperatureforwhichmartensiteis thermomecanicallymorestablethan𝛽,the𝑀∗

𝑠 hasbeensetto−200°.ThenumbercodeforeachcompositionisshowninAppendixA.(Forinterpretationofthe

referencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

Fig.10. Suggestedproceduretopredictthebehaviouroftitaniumalloysas afunctionoftheircomposition;thevalueof𝑀∗

𝑠 istheonethatiscalculated

accordingtotheproceduredescribedinthisarticle

Theperformanceofthemodelindiscriminatingthemicrostructure atroomtemperatureisdisplayedinFig.6,andtheperformancein iden-tifyingthedeformationbehaviourof𝛽 alloysisshowninFig.9.

5. Discussion

5.1. Thermallyactivatedmartensitenucleation

Toexplainwhysomecompositionsretain𝛽 atroomtemperature, theassumptionofathermallyactivatednucleationprocessseemstobe supportedbyexperimentaldata.Jepsonetal.havedemonstratedthat martensitictransformationcanbeinhibitedinTi–Nballoysby quench-ingatextremelyhighcoolingrates[29].Theirobservationsare consis-tentwithathermallyactivatednucleationmechanism;thehigherthe coolingrate,thelesstimeisspentatT0andtheloweristhedensity

numberofmartensitenuclei(Eq.(18)).Thisdependencyonthecooling rateisexpressedinEq.(19).Foragivencomposition,thenucleation parameter isconstantbutthecriticalvaluefornucleationparameter increaseswiththequenchingrate,whichsuggeststhatthemartensitic transformationcanbeinhibiteduponquenchingifthecoolingrateis highenough;thisisconsistentwithJepson’sexperimentalwork[29]. Ontheotherhand,iftheenergybarriertomartensitenucleationistoo high,itmaybethatthemaximalquenchingratethatallowsmartensite nucleationisverylow.Ifitistoolow,diffusioncanoccurduring cool-ing,insuchawaythatnoquenchingrateallowsmartensiteformation uponquenching.

Moreover,thecommonstatementaccordingtowhich𝛽 alloyshave anMsbelowroomtemperatureisnotinagreementwithexperimental

observations.Forexample,theformationofmartensiteinTi–10V–2Fe– 3AlwasinvestigatedbyDuerigetal.[16]onsixspecimensofthesame composition.Afterquenching,onesampleshowedevidenceof marten-sitictransformationandthefiveothersexhibiteda𝛽 microstructureat roomtemperature.These𝛽 alloyswerecooleddownto−170∘_Candstill

ofaround200MPainducedmartensitetransformationthatdidnot re-verseto𝛽 afterunloading.Thedeformedspecimenswerethenheated andthemartensitewasfoundtoreverseto𝛽 around200∘_C.The

vari-abilityofthemicrostructureatroomtemperaturebetweenthe differ-entspecimens(𝛽 ormartensite)forthiscompositionisconsistentwith theproposednucleationparametercriterion;indeed,forthis composi-tion,𝑘𝑇0∕[𝜇(𝑇0)𝑏3]=1.31⋅ 10−2liesveryclosetothecriticalthresholdof

1.24⋅ 10−2_.Then,iftheM

stemperaturewasactuallybelowroom

tem-perature,itwouldhavetoliebelow−170∘_{C.Themartensitereverse}

temperaturehasbeenshownbyexperimenttobearound200∘_C,and

anMsbelow−170∘Cwouldimplyahysteresisofmorethan300∘C

be-tweenthedirectandreversemartensitetransformation.Thisisunlikely, sincethehysteresisformartensitetransformationintitaniumalloys typ-icallyliesaround10∘_{Cwhenitcanbe measured.Otherexperiments}

performedonTRIPalloysshowthatthemartensiteformedduring de-formationreversesto𝛽 attemperaturesexceedingroomtemperatureby morethan100∘_{C[38,39].Moreover,isothermalformationof}

marten-sitethatdoesnotinvolveanydiffusionhasbeenobservedinatleast threedifferenttitaniumalloys[40,41].Forthesethreecases,the tem-peratureatwhichmartensiteisothermallyformedisbeloworaround thecalculatedmartensitestabilizationtemperature𝑀∗

𝑠.Thissupports

thestatementthatmartensiteformationin titaniumalloysisa diffu-sionlessthermallyactivatedprocess.Therefore,thehypothesisthatthe martensitestabilisationtemperature𝑀∗

𝑠 liesaboveroomtemperature

forTRIPalloys,andthatmartensitenucleationrequiresexternalstress, whilegrowthoccursmainlyathermally,seemstobeconsistentwith ex-perimentalevidence.Theexistenceofsuchatransformation,wherethe nucleationisthermallyactivatedbutthegrowthisathermal,hasbeen reportedbyLaughlinetal.[42],whereitisdesignatedasanisothermal. Cottrellalsosupportstheideathatmartensitenucleationisthermally activated[43].

Moreover, this statement allows to explainthe difference of be-haviourbetweensuperelasticandTRIPalloys.Itissometimesbelieved thatthedifferencebetweenTRIPalloysandsuperelasticalloysliesin thefactthatmartensiteissomehowstabilizedbyplasticdeformation (or“stressinduced”)inTRIPalloyswhereassuperelasticityisobserved whenmartensiteformationoccursbeforeplasticdeformation(“strain induced”).However,inTRIPalloys,martensitesometimesformsunder stressbeforeplasticdeformationoccurs,asshownbythe characteris-ticdouble yieldingonthestress–strain curve[44],andstillremains uponunloading.Thismeansthatplasticdeformationisnotnecessary tostabilisethedeformationinducedmartensite.Conversely,a signifi-cantpartofthemartensitethatformsupondeformationin superelas-ticalloysreverseseveninheavilyplasticallydeformedsamples[45], whichmeansthatplasticdeformationaloneisnotsufficienttoexplain thestabilisationofdeformationinducedmartensiteinthecaseofTRIP alloys.Therefore,theconcomitancebetweenplasticdeformationand martensiteformationisnotnecessarytoexplaintheoccurrenceofTRIP effect.Ontheotherhand,thermodynamiccriteria(via𝑀∗

𝑠 calculation)

allowtoseparatesuperelasticfromTRIPalloys.Inbothcases,external worktriggersmartensitenucleationatroomtemperature.For supere-lasticalloys,𝑀∗

𝑠liesaroundroomtemperatureandMfliesbelowroom

temperature;themartensiteisunstableatroomtemperature,which ex-plainsthereversionto𝛽 phaseuponunloading.ForTRIPalloys,𝑀∗

𝑠 is

faraboveroomtemperature,martensiteismorestablethan𝛽 phaseat roomtemperature,andoncethetransformationistriggered,mostofit remainsandisunabletoreverseto𝛽 uponunloading.

5.2. Roleofthe𝜔phase

Thestabilisationof𝛽 phaseisoftenattributedtoathermal𝜔phase [18,46],whichformationduringquenchingwouldoccuraboveMsand

inhibit the martensitic transformation [18]. Assuming that 𝜔 phase formsattemperaturesaboveoraroundMsduringcooling,itwould

in-deedbelikelythatmartensiteformationisinhibited,duetoa competi-tionbetweenbothphases[46].However,theideathat𝜔phase

system-aticallyformsaboveMsinalloysthatretain𝛽 phaseuponquenching

isnotsupportedbythermodynamiccalculations.Thethermodynamic database‘TCTI1’ofThermo-Calc®_{hasbeenusedtocompute,foralloys}

fromliterature,thetemperature𝑇𝜔

0 atwhich𝜔phaseand𝛽 phase

ex-hibitthesamefreeenergy.𝑇𝜔

0 isexpectedtobethehighesttemperature

atwhich𝜔transformationcanpossiblyoccuruponquenching.Formost alloysexperimentallyknowntoretain𝛽 phaseatroomtemperature,the computed𝑇₀𝜔waslowerthanthecalculated𝑀∗

𝑠.Thissuggeststhatthe 𝜔phasenucleatesbelowthetemperatureatwhichmartensiteshould haveformed,andthattheabsenceofmartensitecannotbeattributed tothepriorpresenceof𝜔.Moreover,ithasbeenshownthatmartensite and𝜔phasecancoexistatroomtemperatureafterquenching,andthat the𝜔particlescouldtransformto𝛽 upondeformation,beforefurther transformingintomartensite[47];implyingthatthepresenceof𝜔isnot necessarilyanobstacletomartensiteformation.Therefore,althoughit certainlyplaysaroleinthedeformationbehaviouroftitaniumalloys,it seemsthatthecompetitionbetween𝜔andmartensiteisnotnecessaryto explaintheretentionof𝛽 atroomtemperature.Thenucleation parame-terproposedinthiswork,ontheotherhand,suggestsanexplanationfor

𝛽 retention,quantitativelysupportedbycalculationsandqualitatively

consistentwithexperimentalevidence.

5.3. M_sprediction

The martensite stabilisation temperature 𝑀∗

𝑠, which is the

high-esttemperatureatwhichmartensiteis observedifnucleationis pos-sible,showsgoodagreementwithexperimentalmeasurementsof Ms,

as can be seen in Fig. 5. However, it should be noted that among the54compositionsthataredisplayedinFig.5,43correspondto bi-naryalloysand11tomulticomponentalloys,becauselimited informa-tionisavailableontheMs ofquenchedmulticomponentalloys4. The

modelforthe𝑀∗

𝑠 temperaturedoesnotworkwellforalloys

contain-ingasignificantconcentrationoftin(Fig.5).Thereasonforthisisnot clear.OtherMsmodelsthatincludetinalsohaddifficultieswithsuch

alloys([18]).

Inbinarysystems,themartensitegenerallyhasanHCPstructure un-tilacriticalcompositionatwhichthetransitiontoorthorhombic struc-tureoccurs.The𝑀∗

𝑠 calculationisperformedbyassimilatingthefree

energyoftheorthorhombicphasetotheoneoftheHCPphase.Yanand Olsonmadethisassumptionbefore,andjustifieditbythecontinuity oftheMstemperatureattheHCP/orthorhombictransitioncomposition

[18].Inthepresent𝑀∗

𝑠 calculation,althoughthedifferencebetween

orthorhombicandHCPphasecannotbeconsideredwhenperforming thermodynamiccomputation,itistakenintoaccountinthecalculation oftheelasticstrainenergy.Someauthorshaveempiricallycalibrated thevalueofthefreeenergyinordertoaccountforthedifference be-tweenfreeenergiesoftheHCPphaseandtheorthorhombicphase[18]. Sincegoodagreementbetween experimentalandcalculatedvaluesis obtainedwiththemodelproposedhere(Fig.5),whichisfreeoffitting parameters,wesuggestthatitisreasonabletoapproximatethefree en-ergyof𝛼′′bytheoneof𝛼′.

SeveralattemptshavebeenmadetopredicttheM_stemperaturein Titaniumalloys.Galindo-Navasuggestedamodelfreeoffitting parame-terstocalculatetheMstemperatureinbinaryalloys,butitdoesnot

con-sidermulticomponentalloys[21],andmainlyfocusesonhigh tempera-tures.Neelakantanetal.[48]aswellasYan[18]proposedanapproach basedonGhoshandOlson’stheory[49].These twomodelsprovided goodresults.However,theydidnotcalculatetheelasticstrainenergy, butusedinsteadalinearfunctionoftheatomicfractionsofthealloying elements,supposedtoincludebothelasticstrainenergyandfrictional work,andfittedfromexperimentaldata.Inthepresentwork,the calcu-lated𝑀∗

𝑠 isthetemperatureatwhichtheenergyofamartensiteplate 4_M

sissometimesestimatedwhileapplyingalowstressduringcooling(for

example50MPain[37]).TheMsmeasurementsobtainedwithindirectmethods

ofdetectablesizereachesalocalminimum(Eq.(12)).Thisminimum ofenergy,bydefinition,doesnotdependontheworkdissipatedduring thetransitionfromonestatetotheother,andthereisthusnoneedto computeanyworkrelatedtointernalfrictiontodetermine𝑀∗

𝑠 asitis

definedhere.Iffrictionalworkthatopposesthetransformationistobe consideredinthemodel,itshouldbeincludedbothduringgrowthand nucleationundertheformofanenergybarrier.Inthecaseofmartensite induceduponquenching,themainsourceofenergytoovercomethe nu-cleationbarrieristhermal.Asmartensitegrowthisveryfast,itcannot beassumedthatgrowthrequiressignificantthermalactivation.Thus,if somefrictionalworkopposesmartensitegrowth,itsmagnitudehasto besmallerthanthenucleationbarrier,insuchawaythatovercoming thenucleationenergybarrierbythermalenergyallowstoovercomethe energybarrierformartensitegrowthaswell.Sincetheenergybarrier fornucleationisnotexplicitlycalculatedinthiswork,theproposed nu-cleationparameterobtainedbycollective fittingalreadyincorporates theeffectoffrictionalwork.

The computation of 𝑀∗

𝑠 takes into account the grain size effect

Eq.(12).Notethatthecalculationswereperformedbyassumingagrain sizeof100μm.Aslongasthegrainsizeisintherange20–100μm,the

𝑀∗

𝑠calculationwasfoundnottobeverysensitivetograinsize;the

max-imaldifferencebetweenthe𝑀∗

𝑠 calculatedfor20μmandtheonefor

calculatedfor100μmwas30∘_C.The𝑀∗

𝑠 stronglyvarieswithgrainsize

onlyforsmallergrains.𝑀∗

𝑠 decreaseswhenthegrainsizeisreduced,

becausetheelasticstrainenergyincreases.Thedecreasingof theM_s

withdecreasinggrain sizehasbeenconfirmedinsteels[50,51].This grainsizedependencysuggeststhatmartensitegrowthbecomesmore andmoredifficultifthemicrostructureisrefined.Inparticular,if twin-ning occursinconjunction withmartensitenucleation, thefreepath formartensiteradialgrowth(seeFig.2b)canbeconsiderably dimin-ished.Inthe𝛽 grains containingtwins,the𝑀∗

𝑠 can thereforebe

dra-maticallydecreased.Forexample,inthecaseofTi–12Mo,the𝑀∗

𝑠fora

standardgrainsizeof100𝜇mis437∘_{Candstableplatesofmartensite}

areexpectedtoformupondeformation(whichis consistentwith ex-periments;TRIPeffectisdisplayedinthisalloy[22]).Ifmultipletwins nucleateinagrain,reducingtheapparentgrainsizeto,say,600nm,the

𝑀∗

𝑠 fallsto1∘C.Withsuchavalueof𝑀𝑠∗,anyplateofmartensitethat

wouldformbetweenorinsidethetwinswouldbeexpectedtoreverse uponunloading.ThismighthelpexplainingwhyinTRIP/TWIPalloys, themicrostructureafterdeformationoftenconsistsofamixture of𝛽 andmartensite,andnotof100%martensite,despiteasometimesvery high𝑀∗

𝑠.InalloysdisplayingbothTRIPandTWIP,alimitedamount

ofmartensitehasalreadybeenreportedtoreverseto𝛽 uponunloading, eventhoughmostofthemartensiteremainedafterdeformation[22,52]. Thisseemstobeconsistentwiththegrainsizedependencyadvanced here.

Intheend,theproposedmodelfor𝑀∗

𝑠 providesaratheraccurate

predictionofMs overawiderangeofcompositionsandalloys,while

beingphysically-basedandusingminimal fittingasopposedtomost existingmodels.Fig.5isobtainedwithoutanyadjustableparameter, andtheonlydata-derivedparameteristhethresholdvalueofthe nucle-ationparameter 𝑘𝑇0

𝜇(𝑇0)𝑏3 fromFig.6toaccountforthermallyactivated

nucleation.

5.4.Comparisonwithusualdesigntools

Microstructureprediction

Thetypeofmicrostructureformedatroomtemperaturecanbe pre-dictedwithgoodaccuracy:martensiticmulticomponentalloyscanbe identified,asshowninFig.6.Moreover,theproposednucleation param-etercriterion(kT0/[𝜇(T0)b3])seemstobeaphysicallyrelevantindicator

sinceitallowstopredictforeachbinarysystemthecritical composi-tionnecessarytoretain𝛽 atroomtemperature,asshowninFig.8and Table1.Thecriterionthatisusedintheliteraturetopredictthe possi-bilitytoretain𝛽 atroomtemperatureisthe“molybdenumequivalent”

([Mo]eq)[26]calculatedasfollows:

[Mo]eq =[Mo]+0.67[V]+0.44[W]+0.28[Nb]+0.22[Ta]

+1.6[Cr]+1.25[Ni]+1.7[Co]+2.9[Fe]−1.0[Al](wt.%) (21) Thecoefficientstotheconcentrationsinthe[Mo]eqcalculationhave

beenestablishedinthefollowingway:foreachalloyingelement, the criticalconcentrationtoretain𝛽 atroomtemperatureintheTi–Mo sys-tem(10wt%)isdividedbythecriticalconcentration(experimentally observed)necessarytoretain𝛽 atroomtemperatureforthegiven ele-ment.Itisgenerallyacceptedthatanalloycanretain𝛽 atroom temper-atureif[Mo]eqisgreaterthan10%.

Forbinaryalloys,bothcriteria([Mo]_eq andthenucleation param-eter)-Inequation(19)givesimilar(andcorrect)results.However,the equation ofthe[Mo]eq hasbeendefinedsoastoprovidecorrect

re-sultsforbinaryalloys,sothechallengeremainsonthepredictionfor multicomponentalloys.Forthose,thenucleationparametercriterion maypresentanadvantage.Themicrostructurepredictionissuccessful for92%ofthealloyswiththenucleationparametermethod,andfor 88%ofthealloyswiththe[Mo]_eq method.Thenucleationparameter criterionalsointegratestheeffectofZrandSn,absentinthe[Mo]eq.

Consequently,althoughthe[Mo]eqisareliabletooltopredictthe

mi-crostructure, thenucleationparameterpresentstheadvantagesof of-feringaphysicalexplanationforthe𝛽 retention,ofallowingmore ac-curatepredictionsandofbeingapplicabletoamorediversepaletteof elements.

Deformationbehaviourprediction

Thedeformationbehaviourof𝛽 alloyscanbefairlywellpredicted aswell.Themethod presentedinFig.9tocategorizethedifferent 𝛽

alloysgivesreasonabletrends.Twocompositionshave𝑀∗

𝑠 >90°Cand

stilldisplaysuperelasticity(Ti–31Nb–9Zrat%andTi–3Mo–10Zr-–5.5Sn at%).BothofthemhaveahighconcentrationinZr,andthe𝑀∗

𝑠 may

bemispredictedbecauseofawrongestimationoftheinfluenceofZr on the lattice parameters. Formost of the other wrongpredictions, thecompositionsinvolvedareclose fromthebordersdefinedby the threshold.

Regardingtheretentionof𝛽 upondeformation,theproposed[Fe]eq

assumesthattheeffectofeachalloyingelementinmulticomponent sys-temscanbeextrapolatedfromitsindividualeffectinbinarysystems.It isbuiltinasimilarwaytothe[Mo]eq,butwhilethe[Mo]eqequationis

establishedwiththecriticalconcentrationsnecessarytoretain𝛽 upon

quenchinginbinaryalloys,theproposed[Fe]eqisbasedonthecritical

concentrationsnecessarytoretain𝛽 upondeformation.Twoelements thatarenotincludedinthe[Mo]eqequation,ZrandSn,aretakeninto

accountinthe[Fe]eqcriterion.Thesetwoelementsareoftenconsidered

tobeneutralwithrespecttothe𝛽 stability,becausetheyhavelittle in-fluenceonthe𝛽 transustemperature.However,ithasbeen experimen-tallyshownin [35]thattheconcentrationinZrandSnsignificantly influencesthedeformationbehaviourof𝛽 titaniumalloys.

The method presented here offersa new approach for designing martensitic,TRIP,orsuperelasticalloys,differingfromthetraditional Bo–Mdmethod[10].Todiscussitsaccuracy,literaturedatacollected forthisworkisgatheredonaBo–Mdmap(Fig.1b).The correspond-ingzones aredisplayedin Fig.1a. Itcan beseen thattheboundary betweenmartensiticandnonmartensiticalloysisrelativelywell pre-dictedbyMorinaga’smap,andthatmostofthealloysdisplayingTRIP areindeedinthe“TRIP–TWIP” zone.However,anappreciablequantity ofmartensiticalloysareinthepredictedTRIP–TWIPzone,someTRIP alloysareoutsidetheTRIP–TWIPzone,andsotheboundariesdonot seemtodescribeveryaccuratelythedeformationbehaviour.Moreover, thesignificantnumberofTWIPalloysintheSlipzonedoesnot sup-porttheideathataclearzoneseparatesTWIPfromslip(65%ofthe alloysfromAppendixAknowntodisplayTWIParenotintheTRIP– TWIPzoneoftheBo–Mdmap).Additionally,asdiscussedabove,the

Fig.11. Expectedbehaviourasafunctionof theBo andMd parameters,accordingtothe methoddescribedinthiswork.Morinaga’smap has been superimposed to the chart (plain lines)forcomparison purpose;thebehaviour expectedinthezonesdelimitedbyblacklines, accordingtoMorinaga’s map,isdisplayed in

Fig.1a.

statementthattheboundarybetweenTRIPandmartensiticalloysis de-finedbytheequationMs=RoomTemperature(RT)isquestionable,so

itshouldnotbeassumedthattheM_sisconstantonthelinedisplayed as“Ms=RT” inFig.1a.Thelinesseparatingthetypesofbehaviourhave

beendrawnwithexistingalloys,andnophysicalbackgroundjustifies thepeculiarshapeobtained.

The method proposed in this work on the other hand, is semi-empirical,meaningthatitusesempiricalthresholdswithmodelsthat relyonphysicalbases(for𝑀∗

𝑠andthenucleationparameter).The

quan-titiestobecalculatedinordertopredictthebehaviourcanbecomputed withasimplecodethatrunsefficiently.Althoughthemethoddoesnot relyentirelyonphysics,itgivessome leadstobetterunderstandthe compositiondependencyofmartensiteformation,andallowstopredict withgoodconfidence thebehaviourof thealloys, ascanbe seen in Fig.9andFig.6.Itoffersadvantagescomparedtothecurrent meth-ods.ThecompositionspacetobeexploredisnotlimitedbytheBo–Md combinationsdefinedbytheframeofthemap(Fig.1a)andacriterion isproposed totargetsuperelasticity.Inordertocomparethe predic-tionsobtainedwiththestabilitymapwiththeonesobtainedwiththe presentwork,themethoddescribedherehasbeenappliedtoasetof ap-proximately6000compositionschoseninsuchawaythattheyfillthe spaceoftheBo–Mdmap.Thecorrespondingpredictionsaredisplayed inFig.11.Theoverallshapeofthemartensiticzoneisroughlysimilar totheoneofthephasestabilitymapof[10](Fig.1a),butaccording tothepresentcalculationmethod,thereshouldbemanyTRIPregions inthemiddleofthemartensitearea.Theregionscorrespondingtothe differentdeformationmechanismsassessedwiththemethodpresented hereareveryscatteredintheBo–Mdspace;thissuggeststhatthe val-uesofBoandMdmaynotbethemostphysicallyrelevantparameters todescribethedeformationinducedtransformationsintitaniumalloys. ThemethodproposedinthisworkcanbeusedtodesignTRIP/TWIP alloys;itiscapabletodeterminethecompositionaldependenceofTRIP andsuperelasticityeffects.Itmayconstituteausefultooltodesign𝛽

titaniumalloysexploringnovelrangesofcomposition.

6. Conclusion

Anewmodel isproposed todescribethemartensitic transforma-tionintitaniumalloys-eitheruponquenchingordeformation-asa

functionoftheircomposition.Althoughitusesadhocthresholdsand oneempiricalcriterion([Fe]eq),theproposedmodelreliesonphysical

approachestodeterminemartensitenucleationandgrowth.The nucle-ationparameter (kT0/[𝜇(T0)b3])allowstodeterminethetypeof

mi-crostructurepresentatroomtemperature,the[Fe]eq and𝑀_𝑠∗criteria

thenallowtoidentifythedeformationbehaviourofthe𝛽 alloys.When

𝑘𝑇0∕[𝜇(𝑇0)𝑏3]>1.24⋅ 10−2,martensiteisexpectedtoformupon

quench-ing. Otherwise,thealloyretains𝛽 at roomtemperature.Ifthealloy displaysonly𝛽 phaseatroomtemperature,itdeformsbyslipor twin-ningif[Fe]eq > 3.5orif𝑀_𝑠∗<−90∘C,whereasitcanformmartensite

upondeformationotherwise.Amongthealloysthatcanformmartensite upondeformation,superelasticityisexpectedif−90∘_C<𝑀∗

𝑠 <90∘C,

whereasTRIPisexpectedif𝑀∗

𝑠 >90∘C.

Thisapproachcanprovideexplanationsforsomecharacteristic fea-turesoftitaniumalloysthathadnotquantitativelybeenmodeledsofar. Theassumptionofathermallyactivatedmartensitenucleationmay ex-plainwhysomealloyingelementsallowtoretainthe𝛽 phaseatroom temperature.Thedeterminationofamorepreciseexpressionforthe en-ergybarriertomartensitenucleationcouldconstitutethefocusofsome futurework.The𝑀∗

𝑠 modelrelieson thermodynamicsand

microme-chanics andoffersanexplanationforthecompositiondependenceof TRIPandsuperelasticeffects.Thecombinationofthesemodelsallows todiscriminatemartensitic,TRIP,superelasticandTWIP/Slipalloysand maybeusedasadesigningtoolfortailoreddeformationbehaviour.

Declarationofinterests

Theauthorsdeclarethattheyhavenoknowncompetingfinancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinfluence theworkreportedinthispaper.

Acknowledgment

TheworkofMBhasbeensupportedbyDGA(FrenchMinistryof De-fense).PEJRDDCisgratefultotheEngineeringandPhysicalSciences Re-searchCouncil EPSRChttps://doi.org/10.13039/501100000266.K.for theprovisionoffundingviagrantEP/L025213/1(DesigningAlloysfor ResourceEfficiency),andtotheRoyalAcademyofEngineeringfor fund-inghischair.

AppendixA.Alloysfromtheliterature

n Alloy composition (w %) a _{Microstructure Ms Deformation mechanism Ref}

1 Ti–10Mo–1Fe 𝛽 TWIP {332} [30] 2 Ti–10Mo–3Fe 𝛽 Slip [53] 3 Ti–10Mo–5Fe 𝛽 Slip [53] 4 Ti–15Mo–1Fe 𝛽 Slip [54] 5 Ti–10V–3Fe–3Al–0.27O 𝛽 TRIP [55] TWIP {332} stress-induced 𝜔 6 Ti–8Mo–4Nb–2V–3Al 𝛽 TRIP [56] [57]

7 Ti–9.6Mo–4Nb–2V–3Al–0.25Fe 𝛽 Apparently no TRIP [56]

8 Ti–11Mo–4Nb–2V–3Al 𝛽 No information [56]

9 Ti–29Nb–13Ta–4.6Zr 𝛽 ? [58]

10 Ti–10V–2Cr–3Al 𝛽 and a bit of 𝛼″ ? TRIP [59]

11 Ti–10V–1Fe–3Al 𝛽 and a bit of 𝛼″ TRIP [59]

12 Ti–3Al–5Mo–7V–3Cr 𝛽 TRIP [60]

TWIP {332}

13 Ti–31Nb–9Zr 𝛽 Superelastic (grain size 1 𝜇m ) [61]

14 Ti–38Nb–1.4Al 𝛽 Superelastic [62] 15 Ti–9Mo–6W 𝛽 TRIP [11] TWIP {332} stress-induced 𝜔 16 Ti–32Nb 𝛽[27] or 𝛼″ [36] TRIP [36] [63] [27] 17 Ti–40Nb 𝛽 Superelastic [36] 18 Ti–43Nb 𝛽 Slip [36] 19 Ti–2Mo–37Nb Superelastic [36] 20 Ti–3.3Mo–33Nb Superelastic [36] 21 Ti–5Mo–29Nb Superelastic [36] 22 Ti–30Nb–19Ta Superelastic [36] 23 Ti-34Nb–9Zr Superelastic [36] 24 Ti–15Nb–12Zr at % 𝛽 [64] 25 Ti–16Nb–12Zr at % 𝛽 [64] 26 Ti–17Nb–12Zr at % 𝛽 Superelastic [64] TRIP? 27 Ti–18Nb–12Zr at % 𝛽 Superelastic [64] 28 Ti–19Nb–10Ta at % 𝛽 Superelastic [64]

29 Ti–12Nb–20Ta at % 𝛽 Partial superelasticity [64]

30 Ti–13Nb–20Ta at % 𝛽 Partial superelasticity [64]

31 Ti–62Ta–2Nb 𝛽 TRIP [64] 32 Ti–61Ta–4Nb 𝛽 Superelastic [64] 33 Ti–2Mo 𝛼′ 780 [65] 34 Ti–3Mo 𝛼′ 740 [65] 35 Ti–4Mo 𝛼′ 700 [65] 36 Ti–6Mo 𝛼″ 610 [65] 37 Ti–8Mo 𝛼″ 515 [65] 38 Ti–12Mo 𝛽 TRIP [33] TWIP ({332} and {112})

39 Ti–14Mo 𝛽 pure TWIP [33]

40 Ti–13V–11Cr 𝛽 Slip [66]

41 Ti–5Al–5Mo–5V–3Cr 𝛽 TRIP [67]

42 Ti–5Mo–6V–3Cr–3Al 𝛽 TWIP{112} and {332} [68]

No information regarding TRIP

43 Ti–11.5Mo–6Zr–4.5Sn 𝛽 (quenched -195°C)) TWIP {112} and{332} [34] [15] 44 Ti–13Mo 𝛽 TRIP [32] 45 Ti–16V 𝛽 TRIP [32] [69] TWIP stress-induced 𝜔 46 Ti–13V 𝛽[38] or 𝛼″ [28] TRIP [38] [28,38] 47 Ti–4Al–9V–1.2Fe 𝛼″ 483 [18] 48 Ti–3.2Al–14.6V 𝛼″ 493 [18] 49 Ti–15.15V–3.17Al 𝛼″ 463 [18] 50 Ti–15.6V–3.1Al 𝛼″ 423 [18] 51 Ti–13V-3A–0.6Fe 𝛼″ 453 [18]

52 Ti–13V–3Al–1Fe at 𝛽 + a bit of 𝛼″ TRIP [39]

53 Ti–13V–1Al at 𝛽 + 𝛼″ [38]

( continued on next page )

(continued)

n Alloy composition (w %) a _{Microstructure Ms Deformation mechanism Ref}

54 Ti–13V–3Al at 𝛽 + 𝛼″ over 350 [18,39] 55 Ti–16V–2Fe–2Al 𝛽 Slip [70] 56 Ti–16V–2Fe 𝛽 Slip [70] 57 Ti–16V–2Fe–Al 𝛽 Slip [70] 58 Ti–16V–Fe 𝛽 TWIP {332} [70] stress-induced 𝜔

59 Ti–14V–2Fe–Al 𝛽 TWIP {332} mainly [70]

detection of stress-induced 𝛼″

60 Ti-12V-2Fe-Al 𝛽 TWIP {332} mainly [70]

detection of stress-induced 𝛼″ 61 Ti-16.1V-4Al 𝛽 TRIP [39] [71] 62 Ti-15.4V-4Al 𝛽 + ordered 𝛼″ [71] 63 Ti-13V-3Al-1.5Fe at 𝛽 ? [39] 64 Ti-13V-3Al-0.5Fe at mainly 𝛼″ 𝐴 𝑠 = 240 [39] 𝐴 _{𝑓 = 281} Ms ≈ 250? 65 Ti-25Ta-25Nb 𝛽 Superelastic [72] 66 Ti-25Ta-24Nb 𝛽 Superelastic [3] [73] TWIP ({332} and {112}) 67 Ti-25Ta-30Nb 𝛽 TWIP {332} [73] 68 Ti-15Mo-5Zr 𝛽 TWIP {332} [34] 69 Ti-20V-6Al 𝛽 Slip [34] 70 Ti-20V 𝛽 TWIP {332} [34] 71 Ti-20V-0.15O 𝛽 TWIP {332} [66] 72 Ti-15Mo-5Zr-3Al 𝛽 Slip [34] 73 Ti-3Al-8V-6Cr-4Mo-4Zr 𝛽 Slip [34] 74 Ti-15V-3Cr-3Al-3Sn 𝛽 Slip [34] 75 Ti-8Mo-8V-2Fe-3Al 𝛽 Slip [34] 76 Ti-13V-11Cr-3Al 𝛽 Slip [34] [10] 77 Ti-10V-2Fe-3Al 𝛽 or 𝛼″ + 𝛽 TRIP [16] 78 Ti-12Mo-6Zr-2Fe 𝛽 TWIP [10] 79 Ti-20V-3Sn 𝛽 TWIP [34] 80 Ti-4Al-7Mo-3V-3Cr 𝛽 TRIP [74] TWIP 81 Ti-10Mo 𝛽 + 𝛼″ TWIP {332} [53] (segregation) ( + TRIP?) 82 Ti-7Mo-3Cr 𝛽 TRIP [12] TWIP{332} and {112} 83 Ti-1Cr 𝛼′ 800 [28] 84 Ti-2Cr 𝛼′ 750 [28] 85 Ti-3Cr 𝛼′ 700 [28] 86 Ti-3.5Cr 𝛼′ 650 [28] 87 Ti-4.2Cr 𝛼′ 620 [28] 88 Ti-1.91V 𝛼′ 807 [28] 89 Ti-4.24V 𝛼 727 [28] 90 Ti-6.9V 𝛼 627 [28] 91 Ti-8.5V 𝛼 527 [28] 92 Ti-12.7V 𝛼 327 [28] 93 Ti-22V 𝛽 TWIP {112} [75] 94 Ti-28V 𝛽 TWIP {112} [75] 95 Ti-2.9Al 𝛼 918 [29] 96 Ti-5.9Al 𝛼 960 [29] 97 Ti-9Al 𝛼 1015 [29] 98 Ti-12.3Al 𝛼 1060 [29] 99 Ti-1Fe 𝛼 760 [28] 100 Ti-2Fe 𝛼 675 [28] 101 Ti-2.5Fe 𝛼 630 [28] 102 Ti-3Fe 𝛼 575 [28] 103 Ti-4Fe 𝛼 475 [28] 104 Ti-12W 660 [28] 105 Ti-18W 480 [28] 106 Ti-21.9W 350 [28] 107 Ti-9.1Nb 759 [29] [48] 108 Ti-10Nb 720 [29,48] 109 Ti-13Nb 680 [29] [48] 110 Ti-14Nb 650 [29,48] 111 Ti-17Nb 578 [29,48] 112 Ti-22Nb 485 [29] [48] 113 Ti-25Nb 387 [29,48] 114 Ti-29Nb 𝛽 or 𝛼″ 307 [29,48] 115 Ti 855 [29] [28] 116 Ti-12Mo-5Zr 𝛽 TRIP [76] TWIP {332} 117 Ti-11Mo 𝛼″ [77] or 𝛽 [29]

(continued)

n Alloy composition (w %) a _{Microstructure Ms Deformation mechanism Ref}

118 Ti-15Mo 𝛽 TWIP {332} [54] 119 Ti-10Zr 777 [48] 120 Ti-17.5Zr 727 [48] 121 Ti-24.42Zr 652 [48] 122 Ti-32.52Zr 627 [48] 123 Ti-25Nb-2Ta-3Zr 𝛽 TRIP [78] 124 Ti-25Nb-10Ta-1Zr-0.2Fe 𝛽 TRIP [78] 125 Ti-38Nb-3Zr-2.1Ta 𝛽 Superelastic b _[79] TWIP 332 126 Ti-50Nb 𝛽 TWIP {112} [80] stress-induced 𝜔 127 Ti-15Ta 𝛼 760 [29]

128 Ti-62Ta 𝛼″ or 𝛽 stress-induced Martensite c _{[37] [81]}

129 Ti-72Ta 𝛽 stress-induced Martensite d _[37]

130 Ti-8.5Cr-4.5Sn 𝛽 TRIP [82]

TWIP {332}

131 Ti-10Cr 𝛽 TWIP {332} [31]

132 Ti-25Nb-3Zr-3Mo-2Sn 𝛽 TWIP {332} mainly [83]

TWIP {112} TRIP stress-induced 𝜔 133 Ti-6Cr-4Mo-2Al-2Sn-1Zr 𝛽 TWIP {332} [84] TWIP {112} stress-induced 𝜔 134 Ti-10V-4Cr-1Al 𝛽 TWIP {332} [52] TWIP {112} TRIP (unstable) 135 Ti-5Al-2Sn-4Zr-4Mo-2Cr-1Fe 𝛽 TRIP [85]

136 Ti-3Mo-8Zr-5Sn at% 𝛽 TRIP [35]

137 Ti-3Mo-9Zr-5.5Sn at% 𝛽 Superelastic [35]

138 Ti-3Mo-5Zr-4Sn at% 𝛼″ [35]

139 Ti-3Mo-6Zr-7Sn at% 𝛽 Slip [35]

140 Ti-3Mo-10Zr-6Sn at% 𝛽 Slip [35]

141 Ti-10Ta-4Fe 𝛽 Slip [86] 142 Ti-7Ta-5Fe 𝛽 Slip [86] 143 Ti-12Nb-5Fe 𝛽 Slip [86] 144 Ti-13V at 𝛽 TRIP [38] 145 Ti-16Nb-4Sn at 𝛽 + 𝛼″ 87 [87] 146 Ti-16Nb-5Sn at 𝛽 + 𝛼″ − 53 [87]

147 Ti-24Nb-3Al at mainly 𝛽 − 13 superelastic [62]

148 Ti-35Nb 𝛽 or 𝛼″ 175 TRIP [88] [89]

149 Ti-35Nb-3Sn 𝛼″ 77 [88]

AppendixB. Compositiondependenceofthelatticeparameters

Whenthecompositiondependencywasnotfoundintheliterature, theeffectof thealloyingelementwaseitherignored orobtained by extrapolation(liketheeffectofZrandSnintheorthorhombicphase).

x_iistheatomicfractionofelementi.

BCC(V,Mo,Fe,Crfrom[27],Nbfrom[14],ZrandSnfrom[35]):

𝑎𝛽 =3.274+0.043𝑥Nb−0.1259𝑥Mo−0.2𝑥V−0.55𝑥Fe−0.15𝑥𝑊

+0.044𝑥Ta+0.11𝑥Sn−0.4𝑥Cr+0.31𝑥ZrÅ

HCP(Zrfrom[90],Mo,Nb,Tafrom[91],V,W,Fefrom[92],Cr from[93]): 𝑎𝛼′=2.959−0.083𝑥_Fe−1.3𝑥_𝑊 +0.22𝑥_Zr−0.26𝑥_Cr−𝑥_VÅ 𝑏𝛼′= √ 3𝑎_𝛼′Å 𝑐𝛼′=4.68+0.2381𝑥_Mo+0.5𝑥_Nb+0.5𝑥_Ta−0.4𝑥_𝑊−0.4𝑥_V +0.1𝑥Fe+0.5𝑥Zr−1.6𝑥CrÅ

b _{Thealloyisnotreferredtoassuperelasticin[79]butafterunloading,many}

martensiteplatesreverseto𝛽.

c _{noinformationonwhetheritisTRIPorsuperelastic(mostlikelyTRIP).} d _{noinformationonwhetheritisTRIPorsuperelastic.}

Orthorhombic(ZrandSnfrom[35],TaandNbfrom[73],Mofrom [91],V,Wfrom[92],Crextrapolatedfrom[67]),FeandAlextrapolated from[18]: 𝑎𝛼′′ =2.89+1.37𝑥Nb+3𝑥Mo+0.6𝑥Ta+2.43𝑥𝑊+2.05𝑥Sn+𝑥V +3𝑥Cr+0.9𝑥Zr+0.083𝑥Al+0.0223𝑥FeÅ 𝑏𝛼′′ =3.02 √ 3−1.68√3𝑥Nb−3 √ 3𝑥Mo−1.21𝑥Ta− 1.42 √ 3𝑥_𝑊 −2.64𝑥Sn −0.28𝑥_Zr−√3𝑥V−5.67𝑥Cr−0.018𝑥Al+0.0504𝑥FeÅ 𝑐𝛼′′ =4.734−0.184𝑥Nb−1.5𝑥Mo−0.4𝑥Ta−0.57𝑥𝑊+0.23𝑥Sn +0.4𝑥_V−2.67𝑥_Cr+0.3𝑥_Zr−0.0535𝑥_Al−0.0326𝑥_FeÅ

Asexplainedinthetext,inordertocomputeMs,theelasticenergy

wascalculatedbothfortheHCPandfortheorthorhombicstructure,by assumingthatthecompositiondependenceofthelatticeparametersfor eachstructurecanbeextrapolatedindomainswheretheydonot ex-ist.Foreachalloy,twoMswereobtained(oneforeachstructure)and

theretainedonewasthegreatestofboth.Forallbinarysystemsexcept forTi–V,thisextrapolationofthelatticeparametersallowedtofindthe criticalcompositionatwhichthetransitionfromHCPtoorthorhombic happens.ForalloyscontainingVanadiumhowever,theextrapolationof thecompositiondependencyofthelatticeparametersoftheHCP struc-tureindomainswherethisstructuredoesnotexist(V>6%at[92])did notallowtofindthecriticalcompositionwhereorthorhombicstructure