A constitutive model accounting for strain ageing effects on work-hardening. Application to a C–Mn steel

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A constitutive model accounting for strain ageing effects on work-hardening. Application to a C–Mn steel

Sicong Ren, Matthieu Mazière^∗,Samuel Forest,Thilo F. Morgeneyer, Gilles Rousselier

MINESParisTech,PSLResearchUniversity,MAT–Centredesmatériaux,CNRSUMR7633,BP87,91003Évry,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received19April2017 Accepted18September2017 Availableonline6October2017

Keywords:

Portevin–LeChateliereffect Dislocationdensity Identiﬁcation FEMsimulation Plasticity C–Mnsteel

One of the most successful models for describing the Portevin–Le Chatelier effect in engineering applications is the Kubin–Estrin–McCormick model (KEMC). In the present work, the influence of dynamic strain ageing on dynamic recovery due to dislocation annihilationisintroducedinordertoimprovetheKEMCmodel.Thismodificationaccounts foradditionalstrainhardeningrateduetolimiteddislocationannihilationbythediffusion ofsolute atomsand dislocationpinning atlow strainrate and/orhightemperature. The parameters associated with thisnovel formulation are identified basedon tensile tests foraC–Mnsteelatseventemperaturesrangingfrom20^◦Cto350^◦C.Thevalidityofthe modelandtheimprovementcomparedtoexistingmodelsaretestedusing2Dand3Dfinite elementsimulationsofthePortevin–LeChateliereffectintension.

©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Serrated flow andpropagating plastic strain localisation bands are salient features of the Portevin–Le Chatelier (PLC) effect, which isobserved during tensiletestsin many industrialmetallic alloyswithin a given rangeof strain rates and temperatures.Ithasbeenreportedinalloyscontainingsubstitutionalatoms,suchasAl–Cualloys[1–4],Al–Mgalloys[5–8], orinvolvinginterstitialelementslikecarboninsteels[9,10].Ithasalsobeenobservedinnickel-[11,12]andtitanium-based [13]alloys.ThePLCeffectisusuallyassociatedwithanegativestrainratesensitivity(nSRS)oftheflowstress,whichmeans thatthelatterdecreaseswhentheprescribedstrainrateincreases.Besidestheabnormalflowstress,thelocalisationofstrain intoLüdersorPLCbandsandthenegative strainratesensitivity,some otherseverechangesinmechanicalpropertieshave beenreportedwithin thePLCactivedomain,suchaslossoftoughnessandductility[9,14–19].Someauthorshaveshown that the strain localisationassociated withthePLC effectpromotes theinitiation ofcracks andconsequently resultsina reducedmaterial toughnessinfracture mechanics CTspecimens[10].Incontrast,theauthorsin [20] noticeda reduction of thestrain for initiationofnecking insmooth tensilespecimens, butthey didnot report anydecrease ofthe strain to failure. Themainparametercontrollingtheoccurrenceofneckingandthesubsequentfinalfailureisthehardeningrate.A properdescriptionofthecouplingbetweenstrain ageingandhardeningisthereforenecessarytoexplainthedeterioration inmechanicalpropertieswithinthePLCdomain.

*Correspondingauthor.

E-mailaddress:[email protected](M. Mazière).

https://doi.org/10.1016/j.crme.2017.09.005

1631-0721/©²⁰¹⁷Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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At themicroscopic scale,it iswidely acceptedthat thePLC effectinalloysis causedby dynamic strain ageing(DSA), whichreferstotheinteractionbetweenmobiledislocationsandthediffusionofsolute atoms.CottrellandBilby[21] were the firstto interpretPLCeffects basedon theDSAmechanism. The interactionwas assumedto take placebetweencon- tinuouslymoving dislocations anda solute atmosphere. Later,Penning [22] investigatedseveral importantaspects ofthe PLCeffectfroma mathematicalpointof view,includingthepropagativelocalised plasticdeformationband, theinfluence of machine stiffness and, mostimportantly, the necessityof negative strain rate sensitivity.These two pioneering works setthebasisforthedevelopmentofDSAmodelsaccountingforthePLCeffect.VandenBeukel[23]took intoaccountthe discontinuous motionofdislocationsinthe DSAmodelby introducingthewaiting timetw ofthedislocationsarrested at obstacles.Duringthewaitingtime,soluteatomsdiffusetowardsthearresteddislocations,andanover-stressisrequiredto triggerfurtherdislocation motion.Themovementofan unpinneddislocationtothenextobstacleisassumedtobealmost instantaneous:thisflyingtime isnegligible comparedwiththewaiting time.KubinandEstrin[24,25]proposeda relation betweenthewaitingtimeandthestrainrate,whichhasbeenwidelyusedinworks[26,27].Thecriticalconditionsforthe onsetofthePLCeffectwerediscussedindetailbytheseauthors.McCormick[28]furtherdevelopedacompletemechanical modelaccountingforthetransient behaviourobservedduringstrainratejumptestsbyunderlyingthedifference between theeffectiveageingtimetaandwaitingtimetw.

The KEMCmodelasproposed by McCormick[28] andKubinandEstrin [24] isbasedon theintroduction inthetotal flowstressofanover-stressduetodynamicstrainageinganddependingontheageingtimeta.Thisover-stressismaximal forlowstrainrates,hightemperatures,orlargevaluesoftheinitialageingtime.Sincetheageingtime evolutiondepends onthewaiting timet_w,whichdependsitselfontheplasticstrain rate,theageingover-stressisanimplicitfunctionofthe latter,allowingthemodellingofthenegativestrainratesensitivity.TheKEMCmodelpredictsbothLüdersandPortevin–Le Chatelierinstabilities,asit canbe noticed innumerousworks usingthismodelforfinite elementsimulations [3,9,26,27, 29–31]. Inthe original version of the KEMCmodel [28] and in manyof the following studies [26,29], thewaiting time tw isalsoassumedto be afunction oftheplastic strain.Since thenumber ofobstacles increaseswithplasticstrain, the waiting timeshould growaswell. Onthecontrary,themaximumover-stressleveldueto strainageing isassumedtobe independent of the plasticdeformation. Finally, in the KEMC model,dynamic strain ageing only has a limitedinfluence onthehardeningratewithevolving prescribedstrain rate.Insome alloys,however,suchasC–Mnsteels,theinfluenceof DSAon thestrain hardening rateis ratherstrong [9]. Thisinteraction hasalso beenobserved inmagnesium alloys [32], cobalt-basedsuperalloys[33],orinnickel-basedsuperalloys[34].

Inaddition tothe original KEMCmodel,dislocation dynamics-basedmodelsforDSAandPLC havebeenrecently proposed[35–37].Strain hardeningis thencontrolled bythe dislocationdensityevolution,divided intoa productionandan annihilation term. In [36], the modelaccounts fora single population ofdislocations. It israther similar to the original KEMC model,since the over-stress is addedto the total flow stress andincreases consequently, but DSAhas almost no influenceon thehardening rate. Onthecontrary, in[35] and[37],an indirectcoupling betweenstrain ageing andstrain hardeningisproposedinamodelbasedontwodifferentdislocationpopulations(mobileandforestdislocations).Thecou- plingbetweenstrain ageingandstrain hardeningisintroducedbyanincrease oftheforest dislocationdensityinorderto representthe“probabilityofdislocationpinningbysoluteatomsatsomeobstaclessuchasforestdislocations”.Themodel proposed in[35] and[37] is,however,rathercomplex,withalargenumberofparameterstoidentify.Ontheother hand, theoriginalKEMCmodelwasmodifiedbyBöhlkeetal.in[3]toaccountfortheinfluenceofDSAonthehardeningrateby introducingastrain-dependentmaximal valueoftheover-stressduetostrain ageing(σB0 in[3]).Thismodificationofthe originalKEMCmodelispurelyphenomenologicalandmainlymotivatedbyabetterdescriptionofthecriticalplasticstrain forPLC inaluminiumalloys. The same approachhasbeen usedin some followingarticles to accountfor theinteraction betweenstrainageingandstrainhardening[31,33].

To achieve the goalof a better description of the experimental hardening ratewithin a simpleand physicallysound model,theideaofthepresentworkistoproposeamodiﬁcationofthehardeningratebytheimpedimentoftheannihilation ofdislocation segments by solute atoms. This dislocation mechanismhasactually been observedin TEMobservations as shownin[38,39].TheobjectiveofthepresentworkisthereforetoformulateamodiﬁedKocks–Meckingmodelintowhich DSAeffectsareintroducedintheyieldfunctionlikein[36],butalsointhedynamicrecoverytermofthedislocationdensity evolution.Theadvantagesofthisconstitutivemodelwillbehighlighted.

The workisorganisedasfollows.The theoreticalconsiderations aboutthenewmodelare presentedinsection 2.The dislocationevolutionlawisseparatedintothemultiplicationandannihilationparts.Theinfluenceofstrainageingisintro- ducedinthe dislocationannihilation term. Asystematicidentificationprocedureis providedin section 3tocalibrate the material parameters fora C–Mnsteel testedat severalstrain rates fromroom temperatureto 350^◦C. However, a model accountingforDSAisvalidonlyifserrationsarepredictedwithintheaccuratestrain ratedomain.Forthatpurpose,finite elementsimulationsmustbeperformedforthewholespecimensandnotonlyatthematerialvolumeelementlevel.Insec- tion4,validationtestsarethencarriedoutandpresentedusingtheidentifiedmaterialmodelfor2Dand3Dfiniteelement meshes.

2. Formulationoftheconstitutivemodel

The proposed constitutive model is presentedwithin thefinite strain framework usingthe concept of localobjective framesfollowing[40].Thesecondandfourthordertensorsaredefinedbyasingletilde_∼ ândâ^double^tilde_≈^,respectively.

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Observerinvariant stressandstrainratemeasures σ_∼ andε_∼˙ aredeﬁnedbythetransformationoftheCauchystress tensor T_∼ andtheEulerianstrainratetensor D_∼ intothecorotationalframecharacterizedbytherotation Q

∼ (x,t)ateachmaterial point:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ σ_∼=Q

∼.T_∼.Q

∼ T

˙ ε_∼₌Q

∼.D_∼.Q

∼ T

Q∼ such as Q˙

∼ T.Q

∼ =_∼ (corotational)

where D_∼ and_∼ respectivelyarethesymmetricandskew-symmetricpartsofthevelocitygradient.

2.1. Constitutiveequations

Thestrainrateisthendecomposedintoelasticandplasticparts:

˙

ε_∼₌ε_∼˙^e₊ε_∼˙^p, σ_∼₌C

≈:ε_∼^e (1)

whereC

≈ istheHooketensorofelasticity.Theplasticﬂowisdescribedbythenormalityrule

˙

ε_∼^p_{= ˙}pn_∼, n_∼=∂f

∂σ_∼ ⁽²⁾

where pisthecumulativeplasticstrainandtheyieldfunction, f,istakenintheform

f(σ_∼,ρ,ta)=σeq(σ_∼)−^R(ρ)−^Ra(ta) (3)

In thepreviousequation, σeq istheequivalent stressmeasure.Thethresholdforyieldingisthesum R(ρ)+^Ra(ta)where R(ρ) is the usual yield stress depending on the total dislocation density ρ and Ra(ta) is the yield stress enhancement (over-stress)duetoageing.

In addition, an hyperbolic ﬂowrule is introduced inthe currentmodel that accountsforthe thermalactivation phe- nomenon:

˙

p= ˙ε0exp

− ^E^a kBT

sinh

Va^f kBT

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where^f=^max(f,0).Thethresholdstrainrateatwhichthermalactivationtakesplaceisε˙0 andkB andT arerespectively theBoltzmannconstantandthetemperatureinKelvin.V_aandE_aarerespectivelytheactivationvolumeandtheactivation energy.

ThelasttermRa(ta)ontherightsideofEq.(3)istheyieldover-stressduetostrainageing:

Ra=^P1φ (ta) (5)

with

φ (t_a)=¹−^exp(−(t_a/t₀)ⁿ) (6)

whereφ (t_a)istherelativeconcentration ofsolute atomspinningthedislocations. Itvariesbetween0and1.φ=^{1 means} thatdislocationsaretotallypinned,whileφ=0 correspondstotheunpinnedcase. P1 isthemaximalstressdropmagnitude fromthepinnedstateφ=^{1 to}^the^unpinned^state φ=^0.^The^parameter^t0 isrelatedtothediffusivityofsoluteatoms,see [36].

Theageingtimetaisdeﬁnedbythefollowingevolutionlaw:

˙

t_a=¹− ^t^a

t_w, t_w=^wp˙, t_a(t=⁰)=^ta0 (7)

where t_w denotes theaverage waitingtime ofdislocationsatobstacles. Itis worth notingthat t_a isequal tothewaiting time tw onlyforthesteadystate [28].Theinitialtime ta0 isrelatedtothepresenceofstaticageing.Aproperinitialtime value allows a gooddescription of Lüderspeak andplateau. The parameter ω representsthe strain increment produced whenallarresteddislocationsovercomelocalobstaclesandmoveforwardtothenextpinnedconﬁguration.

The secondterm R(ρ) ontheright-handsideofEq.(3)representsisotropichardening,whichisexpressedintermsof thetotaldislocationdensity ρ:

R(ρ)=σ0+γ μb√

ρ, R₀=σ0+γ μb√

ρ0 (8)

where μistheshearmodulus; γ isamaterial-scaleparameter[41];bisthemagnitudeoftheBurgersvector.Forthesake ofsimplicity,wedonotdistinguishbetweenforestandmobiledislocationdensities.

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TheevolutionofdislocationdensityisbasedonamodiﬁedKocks–Meckingequation[41]:

˙ ρ=

a₀√

ρ−^b0(1−ζ φ)ρ ^p˙ ⁽⁹⁾

wherethe ﬁrstterma0√

ρ onthe right-hand sideof Eq.(9)describesthe multiplicationof thedislocations. The second termb0(1−ζ φ)ρ^p˙ ^is^related^to ^the annihilation(dynamicrecovery) process fordislocation dipoles,which isconsidered to be affected by strain ageing. This coupling with strain ageing is absent in earlier formulations of the KEMC model.

Physically,theshort-rangemotionofdislocationsrequiredforannihilationordynamicrecoveryisimpededbythetrapping of solute atomsat dislocations(especially at low strain rates orhightemperatures). Consequently, strain ageing reduces theabilityofthematerialtoexhibitdynamicrecovery.InEq.(9),thestrainageinginfluenceondynamicrecoverydepends on the current value of the function φ (ta). When φ (ta) is maximal, i.e.for low strain rates or high temperatures, the dynamicrecoveryprocessissignificantly reducedbystrainageing,whichresultsinalargerdislocationdensityproduction rateρ˙, and, accordingly, ina larger hardening rate. Onthe contrary, when φ (ta)is minimal, i.e.for highstrain rates or lowtemperatures,strainageinghasnoinfluenceondislocationannihilationallowing forstandarddynamicrecoveryanda reduceddislocationdensitygrowthrateρ˙ andassociatedhardeningrate.Finally,thehardeningratewillthenbelargerfor lowstrainratesthanforlargestrainrates,asexperimentallyobserved[9].Theparameterζ,0≤ζ≤^1,^has^to^beîdentified inordertocontrol thelevelofinfluenceofstrainageing onstrainhardening (ζ=^{0 means} ^noînfluence,^whileζ=^{1 the} maximalinfluence).

2.2. Newfeaturesinthemacroscopicresponseofthenewmodel

Afterintroducingthe ageingeffectondislocation annihilationandconsequently intheisotropichardening term R(ρ), Raisnolongertheonlyterminthemodelresponsiblefordynamicstrainageingeffects.Thisnewfeatureofthemodelis nowillustratedinthecaseoftheC–Mnsteel,forwhichtheKEMCmaterialparameterswerepreviouslyidentiﬁed,see[42].

The modified modelhas been identified anew based on the same experimental curves asin [42].This identification procedureispresentedinsection3.Inthepresentsubsection,themainfeaturesofthemodelareillustratedbysettingζ to ζ=^{1 to}ênhance^theêffect.Âctually,^theôptimal^valueζ=⁰.2 wasfoundforthecurrentidentificationasshowninTable 2.

Thedetailsabouttheidentiﬁcationwillbepresentedinthenextsection.

The strain rate sensitivity of the models is investigated in Fig. 1, which shows the evolution of the 1D steady-state stress asa function of the plastic strain rate atthree different plastic strain levels (0.05, 0.10, and 0.15) for the C–Mn steelusingthe original KEMCmodelandthe novelformulation. The originalversion of themodelis retrievedforζ=^0.

The curvesofFig. 1(a)display theusual S-shape forstrain-ageingmaterials. The roleofthe parameter P1,see Eq.(5),is investigated.Foursituations areconsideredinFig. 1: P1=⁰,ζ=0 (originalKEMCmodel), P1=⁰,ζ=0 (truncatedKEMC model), P1=⁰,ζ=^{1 (new}^model), ^P1=⁰,ζ=1 (truncatednewmodel).SettingP1=^{0 amounts}^to^killing^theover-stress Ra(ta).

Anegative strainratesensitivity(nSRS)domaincan beobservedfortheoriginal aswell asforthenewmodelforthe cases P1=^0,^as^shownⁱⁿ^{Fig. 1(a)}^and^(b).

For P₁=^0, ^see^{Fig. 1(c),} â ^slight^positive sensitivity canbe observed, whichindicates that the dynamic strain-ageing effectiscompletelyeliminated for P₁=^{0 in} ^theôriginal ^KEMC^model.Ôn^the^contrary,^the^new^model^can ^still^produce a negative strain-rate sensitivity, even for P₁=^0, âs ^shown ⁱⁿ ^{Fig. 1(d).} ^Such â ^feature ôf ^the ^modified ^model ^can ^be particularlyusefulsince,intheoriginal KEMCmodel,theparameter P1=0 controlssimultaneouslythe amplitudeofthe serrationsandtheamplitudeofthenegativestrain-ratesensitivitydomain.Thesetwoamplitudesbeingforsomematerials (especiallysteels)differentbyoneorderofmagnitude,acompromisehastobemadeintheparameteridentificationofthe KEMCmodel.Inthenewmodel,thenegativestrainratesensitivityisnotonlycontrolledbytheRaterm,butisalsoinduced bythecouplingbetweenstrainageingandstrainhardeninginEq.(9).Thematerialparameter P1 canthenbedevotedto anaccuratedescriptionofserrationsonly.

InFig. 2,tensilestress–straincurvesundervariousglobalprescribedstrainratespredictedbythetwomodelsarecom- pared. Thered arrowindicates thedirectionofincreasingapplied strain rates.Bothfigures show thestrain ratesensitive behaviour inthenSRSzone (i.e.between10⁻⁵ and10⁻¹ s⁻¹).It canbe noticedthat thecurvesobtainedby theoriginal modelatdifferentstrainratesare almostparallel toeach other.Thisindicates that theevolutionofthehardening rateis the sameateach strain rateduring straining(see Fig. 2(a)). Incontrast,some experimental observations onC–Mnsteels show that the distance betweenthe curves increaseswith strain [9].A better modellingof this effectwas proposed by Böhlkeetal.[3],whoprescribedalinearincreasingevolutionfor P1 withrespecttothecumulativeplasticstrain p.From a physicalpoint of view,it isexpected that theapparent hardening rateshould also be influenced by strain ageingdue totheinteraction betweendislocationsperturbedby thepinningofsegments. Theproposed newmodelincludessuch an influenceofstrainageingonthestrainhardeningrate,viathefunctionζ φ inEq.(9).Thestress–straincurvesproducedby themodifiedmodelareclosetoeachotheratlowstrain,andspreadathigherstrainlevels,whichshowsastrongsimilarity totheexperimentalresults(seeFig. 4(a))thatwillbepresentedinthenextsection.

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Fig. 1.StrainratesensitivityaccordingtotheoriginalKEMCandnewmodelsforuniaxialtensionunderprescribedplasticstrainrate.Thecorresponding stressisgivenatthreedistinctplastic-strainlevels.

Fig. 2.Comparison of the stress–strain curves for different applied strain ratesE.˙