Influence of M23C6 carbides on the heterogeneous strain development in annealed 420 stainless steel

Delft University of Technology

Influence of M23C6 carbides on the heterogeneous strain development in annealed 420

stainless steel

Hidalgo , J.; Vittorietti, M.; Farahani, H.; Vercruysse, F.; Petrov, R.; Sietsma, J.

DOI

10.1016/j.actamat.2020.08.072

Publication date

2020

Document Version

Final published version

Published in

Acta Materialia

Citation (APA)

Hidalgo , J., Vittorietti, M., Farahani, H., Vercruysse, F., Petrov, R., & Sietsma, J. (2020). Influence of

M23C6 carbides on the heterogeneous strain development in annealed 420 stainless steel. Acta Materialia,

200, 74-90. https://doi.org/10.1016/j.actamat.2020.08.072

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

J.

Hidalgo

a,∗

_,

_M.

_Vittorietti

b,c

_,

_H.

_Farahani

a,d

_,

_F.

_Vercruysse

e

_,

_R.

_Petrov

a,e

_,

_J.

_Sietsma

a

a Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, Delft 2628 CD, the Netherlands b Department of Applied Mathematics, Delft University of Technology, Van Mourik Broekmanweg 6, Delft 2628 XE, the Netherlands c Materials Innovation Institute (M2i), Van Mourik Broekmanweg 6, Delft 2628 XE, the Netherlands

d Now at Tata Steel Research Development and Technology, IJmuiden, the Netherlands

e Department of Electromechanical, Systems & Metal Engineering, Ghent University, Technologiepark, 46, 9052 Ghent, Belgium

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 6 March 2020 Revised 7 July 2020 Accepted 27 August 2020 Available online 2 September 2020

Keywords:

Stainless Steel M 23 C 6 carbides plastic strain gradient representative volume element finite element crystal plasticity

a

b

s

t

r

a

c

t

Understanding thelocal strainenhancementand latticedistortionresulting fromdifferent

microstruc-turefeaturesinmetalalloysiscrucialinmanyengineeringprocesses.Thedevelopmentofheterogeneous

strainnotonlyplaysanimportantroleintheworkhardeningofthematerialbutalsoinotherprocesses

suchasrecrystallizationanddamageinheritanceandfracture.Isolatingthecontributionofprecipitatesto

thedevelopmentofheterogeneousstraincanbechallengingduetothepresenceofgrainboundariesor

othermicrostructurefeaturesthatmightcauseambiguousinterpretation.Inthisworkastatistical analy-sisoflocalstrainsmeasuredbyelectronbackscatterdiffractionandcrystalplasticitybasedsimulations arecombinedtodeterminetheeffectofM23C6carbidesonthedeformationofanannealedAISI420steel.

Resultssuggestthatcarbidesprovideamoreeffectivehardeningatlowplasticstrainbyapredominant

long-rangeinteractionmechanismthanthatofapureferriticmicrostructure.Carbidesnotonlyinfluence

localstraindirectlybyelasticincompatibilitieswiththeferriticmatrix,butalsothespatialinteractions

betweenferritegrains.Carbidesplacedatthegrainboundariesenhancedthedevelopmentofstrainnear

ferritegrain boundaries.Howeverthepositiveeffectofcarbidesand grainboundaries todevelophigh

localstrainsismitigatedatregionswithhighdensityofcarbidesandferritegrainboundaries.

© 2020ActaMaterialiaInc.PublishedbyElsevierLtd. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

It is well knownthat strain developmentin metallic alloys is critically affected by the microstructural characteristics such as grainsize ofmatrixphasesaswellassize, densityandnature of existingprecipitates.Thesecharacteristicsinfluencethedislocation motionin the structure andplay a fundamentalrole inthe me-chanicalbehaviourofthemetallicalloys.

Understanding the evolution of complicateddislocation struc-turesinmetalsandtheireffectonthehardeningbehaviourofthe materialsduringdeformationisamajorissueinmaterialsscience. Dislocations are commonly categorized into redundant and non-redundantdislocations,respectivelycalledStatisticallyStored Dis-locations(SSDs)andGeometricallyNecessary Dislocations(GNDs). GNDsshare similarBurgers vector andthey allow the accommo-dationoflattice curvatureduetonon-homogeneousdeformation.

∗ _{Corresponding author.}

E-mail address: J.HidalgoGarcia@tudelft.nl (J. Hidalgo).

TheexactmannerinwhichGNDscontribute tothestrengthening of materials is not completely understood. Existing GNDs locally interact with moving dislocations by forming jogs that provide macroscopic isotropic hardening under strain development [1,2]. Pile-upsofGNDsalsoleadtothedevelopmentoflong-rangeback stresses,whichresultinkinematichardening[3].Therelative sig-nificanceofeachmechanismvarieswiththeoverallimposedstrain andsizeofmicrostructureelementsrelatedtomaterial strengthen-ing[2].

Theliterature relatingthemicrostructure tothe propertiesvia thedevelopmentofGNDsinmetalalloysisabundant[4–9].GNDs typicallyaccumulateatthegrainboundariesduestrain incompat-ibilities of grains with different orientation or constituents with dissimilarpropertiessuchashardprecipitatesinasoftmetal ma-trix.The relationbetweenGNDsandgrainsizehasboth theoreti-calandexperimentalvalidationsanditcanbelinked tothe well-knownHall-Petcheffect[9,10],consideringthatatsmallgrainsize, thegrainboundarylayer inwhichGNDs typicallyaccumulate en-compasses a relatively large volume fraction of the material [6]. TherelationbetweenhardprecipitatesandGNDsinmetalalloysis

https://doi.org/10.1016/j.actamat.2020.08.072

still unclear.Karamchedetal.[5]showedforsinglecrystal super-alloythat GNDdensity increaseswiththemagnitude ofimposed strain.IntheirstudytheyobservedthattheGNDdensitywas sig-nificantly highernearthe elongated carbideparticles than inthe matrix.However,inmostcommonpolycrystallinemetalswith dis-persecarbides,theevaluationoftherelationbetweencarbidesand GNDsisnottrivialbecausethepresenceofgrainboundariesmight causeambiguousinterpretation.

It was extensively reportedthat large-size carbides and other brittle particles have a notably adverse effect on the low-temperature toughness because they improve the formation and propagation ofmicro-cracks [11,12]. DeCock etal. [13]proposed that coarse cementite carbides develop a deformation zone with a highdislocation density,whichpromotestheformationofa re-covered andequiaxedferritic matrix.Vivasetal.[14]suggesteda similar mechanisminduced by coarse M23C6 carbides to explain

theformationoffineequiaxedferrite grainsinferritic/martensitic chromium steelduring creeptests. These regions withhigh pop-ulation oflargeM23C6 carbides andsmall sizeferrite grainswith

lowangleboundariestendtodevelopmicrocavities,worseningthe material performanceto creep.Itis,therefore,necessaryto study andunderstandthelocalstrainenhancementandlatticedistortion resulting fromdifferent microstructure features, i.e.grain bound-aries,precipitatesoracombinationofboth,becausetheynotonly play animportantrole intheworkhardeningofthematerial but alsoinotherprocessessuchasrecrystallizationanddamage inher-itanceandfracture.

In this work, we study the effect of large M23C6 carbides in

thelocalstrain developmentandstrain hardeningofan annealed AISI420 stainless steel. Toisolate the effect of carbidesfrom fer-rite grainboundaries, two approachesarefollowed.As a first ap-proach, real material is subjected to interrupted tensile tests at different strain levelsandthe developmentoflocal strain arising fromdifferentmicrostructurefeatures ischaracterizedbyelectron backscatterdiffraction(EBSD).The secondapproach usesadigital recreation of the microstructure with different 3D representative volume elements(RVE) andcrystal-plasticity based simulationof the strain developmentinthe microstructureunder uniaxial ten-siledeformationusingDAMASKsoftware[15].

2. Microscopicmodellingbasedoncrystalplasticity

The strain and stress development in different digital recre-ations of the microstructure was simulated combining a crystal plasticitymodelandthespectralsolverbasedonFFT(FastFourier Transform) provided byDAMASK software.Here, only the consti-tutiveequationsfortheelasticandplasticdeformationarebroadly presented.Foracompletedescriptionofsimulationprocedure,the readerisreferred toRefs. [15,16].The intricatestress interactions betweenthegrainsofa polycrystallinematerial aremodelled nu-merically using the spectral element (SE) method. Each grain is represented by one or more finite elements, and the polycrystal is subjected to boundary conditions that simulate the deforma-tion underspecificconstraints.The singlecrystalplasticitymodel iscombinedintotheSEframeworktodefinetheconstitutive rela-tion ateach integrationpoint oftheelement.The deformationin thecontinuum theoryofcrystalplasticityisdescribedasa multi-plicativedecompositionintoelastic, F e,andplastic, F p,partsofthe

deformationgradient F ,wheretheelasticpartaccountsforlattice distortionandrotation,andplasticdistortionarisesduetoslip:

F=FeFp (1)

The stress atthe elasticstrain regime isexpressed informof the 2nd _{Piola-Kirchhoff stress S ,and dependsonly on the elastic}

strain expressed as the Green–Lagrange strain tensor E and the materialspecificstiffnessC,accordingto

S=C :E (2)

E=1/ 2



FeFeT− 1



(3) Forcubic crystals in this study, the elastic stiffness matrix is composedofthreeindependentterms,C11,C12andC44.Itisworth tonotethatreversibledislocationglide,i.e.dislocationanelasticity, isnot considered inthe model.The evolution of plastic strain is givenby:

˙

Fp=LpFp (4)

where L p isthe plastic velocity gradient. A widely adopted phe-nomenologicaldescriptionforthehardeningisusedinthepresent work,whichisbasedonlyonslipofmultipleslipsystems

β

i.The evolutionofcriticalshearstress,

τ

˙_Cβ,i.e.thehardening,of individ-ualslipsystemsinasinglecrystalisgivenby:

˙

τ

Cβ= 

η

h_βη

γ

˙η (5)

The instantaneous slip-system hardening moduli h_βη, in gen-eral,dependonthehistoryofslipandprovidesinformationabout additionalhardeningcausedbyinteractionsoffixedslipsystems

β

andactiveslipsystems

η

.h_βη isdeterminedby

h_βη=qβη



h0



1−

τ

η C/

τ

sat



a



(6) The parameters h0,

τ

Cη and

τ

sat are respectively the reference hardening, the critical slip resistance and the saturation shear stress, and depend on the crystal structure and the slip system. Theparameter a,typically a≥ 1, hasnota directphysical mean-ing,but hasa directinfluence on thedevelopmentof hardening. The latent hardening parameter, qβη, defines the interaction

be-tween system

β

and

η

andis set to 1, if

β

and

η

are coplanar, otherwiseqβη =1.4. The shearstrain rate

γ

˙η ofthe system

η

is restrictedbyitresolvedshearstress,

τ

η,and

τ

_Cη:

˙

γ

η₌

_γ

_˙

0

τ

η/

τ

Cη



1/n

sign

(

τ

η

)

(7)

wheren is relatedto the strain rate sensitivityof slip and

γ

˙0 is

thereferenceshearrate,beingbothmaterial-specificvariables.The exponentnisusuallyusedasanumericalvariabletoapproximate theSchmidtlawandhasnophysicalmeaning.The shearratesof allslipsystemscanbethenusedtodeterminetheplasticvelocity gradient: L p= N  β=1 ˙

γ

η_m βnβ (8)

where N denotes the number of slip systems (N = 12 for iron {110}_bccbasedon[17,18]and12forM23C6carbide{111}fcc), m the

normalizedslipdirectionand n theunitnormaloftheslipplane.

3. Experimentalprocedure

The AISI 420 steel used in this study contains 0.32 wt.% C, 0.2wt.%Si,0.3wt.%Mn and13.7wt.% Crandit wasreceived in theformoffully annealedsheets of0.45mmthickness.Sub-size tensiletest specimens following the ASTME8/E8M−13astandard

[19]and miniature tensile test specimens,with dimensions shown in Fig.1, were machined withthe long axis (gauge section) ori-ented along the sheet rolling direction. Sub-size specimens were testedinanInstron5500Relectromechanicaltensiletestmachine, withloadcellof50kN,atroomtemperatureandinextension con-trolmode.Aclip-onextensometerwithknife-edges,agaugelength of 7.8mm anda maximum extension of ±2.5 mm wasused to

Fig. 1. Drawings of the different tensile specimens dimensions, (a) sub size specimen according to the ASTM E8/E8M −13a [19] , (b) miniature specimen in wich indentations to delimit inspection area and determine plastic strains are indicated.

record the elongation during the tensile test. Three strain rates, 0.01s−1, 0.001 s−1 and0.0001 s−1, were appliedto evaluate the strainratesensitivity.

Miniaturespecimensweresubjectedtointerruptedtensiletests toapproximately 0.5mm, 10mm and15 mm ofcross-head dis-placement in a Deben micromechanical tester. The plastic strain achievedineachdeformationstephasbeenderivedbythechange in length between the centre of two indents placed along the gaugeandseparatedby1mmdistance,asshowninFig.1b.These indentsweremadebyaposition-controlledVickershardnesstester applying a load of 1 N. Extra indents were made to delimit a squareareaofapproximately60× 50

μ

m2_{uponwhichEBSD}

anal-ysiswasconductedintheunstrainedconditionandaftereach de-formationstep.Previous tothe interruptedtest andaftermaking theindents,thetop surfaceofthetensilespecimenwasprepared followingthe same procedure asfor microstructure characteriza-tionbyEBSD.Thisprocedureconsistsofgrindingandpolishingthe specimenswith a final polishing step to 0.03

μ

m colloidalsilica solutionduring60 min to mitigatethe plastic strainsintroduced atthesurfaceduringtheprocess.

AFEIQuanta450scanningelectronmicroscopeequippedwith aFieldEmissionGun (FEG-SEM)andEDAX-TSL, OIM Data Collec-tionsoftwarewere usedtoobtainEBSDpatterns.The set-up con-ditionsaredetailedas:accelerationvoltageof20kV,spotsize#5 correspondingto beamcurrentof2.4nA, workingdistanceof16 mm,tiltangleof70°,andstepsizeof50nminahexagonalscan grid.Plasma cleaningstep wascarriedout before each EBSDtest tomakesurethatnochangesinindexationwascausedby pollu-tion/damageofthesampleduetoanearlierscan.Post-processing andanalysisoftheorientationdatawasperformedwithTSLOIM® Analyses6.0 software.A grain confidence index(CI) standardiza-tionwasapplied totheraw data,withaminimum tolerance an-gle andgrain size of 5° and 6pixels respectively. It was consid-ered that grains are formed by multiple pixel rows. Thereafter, neighbour-orientationcorrelationwithatoleranceangleof5° and aminimumconfidenceindexof0.1wasimplemented.

Microstructureofthesampleswasadditionallycharacterizedby aFieldEmissionGun(FEG-SEM)JEOLJSM-6500FScanningElectron Microscope(SEM)usingtheSecondary ElectronImaging(SEI) de-tection mode. Specimens were polished to 1

μ

m diamond paste and etched by waterless Kalling’s reagent [20] for SEM charac-terization.ABruker D8 Advancediffractometer inBragg-Brentano geometryandwithgraphite monochromatorandVantecposition sensitivedetectorwasusedforidentifying andcharacterizingthe precipitatesobservedinthemicrostructure.CoK_α radiation,45kV and35mAwereusedinthe2

θ

scanfrom30° to130° withastep sizeof0.021° andcountingtimeperstepof3s.Thespecimenwas

rotatedat 30rpm duringthe measurement to minimisepossible textureeffects.

4. Results

4.1. CharacterizationoftheinitialmicrostructureandRVEGeneration

InordertorecreatesimulationRVEsascloselyaspossible rep-resentingthe realmaterial, the initialmicrostructure of thesteel wasextensivelystudied andquantified usingEBSDandXRD. The microstructure of annealed AISI420 consists of ferrite with var-ious precipitate particles, predominantly M23C6 carbides and MX

carbonitrides[21].Fig.2ashowsasecondaryelectronimageofthe microstructurebefore deformation inwhichferrite grainssmaller than 10

μ

m are highlypopulated with precipitates.Large round precipitatesare identifiedasM23C6 Fe-Crcarbidesby energy

dis-persive X-ray spectroscopy (EDS). This was confirmed by X-ray diffraction(XRD) analysis.M23C6 diffractionpeaks are clearly

re-solvedinthediffractogramofFig.2b.M23C6 carbidesmainly

pre-cipitate along prior austenite grain boundaries and boundaries withalargemisorientation angle[22–24].M23C6carbidestendto

coarsen easily because the solubility of iron and chromium, the major constituents in this carbide, is high. Homogeneously dis-tributed nanometre size precipitates, likely MX nitrides, can be alsodistinguishedwithinferritegrains.TheMXcarbonitrides typi-callyprecipitatefinelyanddenselyinthematrixbccphase,mainly alongdislocations,anddonotgrowsignificantlyathigh tempera-tures [25].Although not confirmedby EDS, their presence inthe microstructureisdetectedbyXRD(Fig.2b).

TheoverlappedphaseandimagequalityEBSDmapsareshown in Fig. 3. White lines delimit ferrite grain boundaries with mis-orientation angles larger than 10°. Image quality map reveals some ferrite grains sharing boundaries with angles smaller than 10°. Ferrite exhibits a broad grain size distribution, with clus-ters of small equiaxed grains surrounding regions of elongated larger grains which might be a reminiscence of a recrystallized rolled microstructure.M23C6 carbides, identified asan FCC phase

in EBSD, are stochastically distributed along the ferrite matrix. Dark spots, with low image quality, can be appreciated in the image quality map indexed as BCCphase. These spots might be emerging carbides situated in the range of the depth penetra-tion of the electron beam or carbides that were wipedout dur-ing the sample preparation process. To account for these car-bidesinthequantificationprocess,pointswithimagequalitylower thanathresholdbaseduponbimodaldistributionare filteredand added to the phase quantification. A carbide fraction of 0.032 is measured by EBSD, which is a small fraction compared to the

Fig. 2. (a) Secondary electron SEM image of AISI 420 in which rolling direction (RD) and normal direction (ND) is indicated. (b) XRD diffractogram in which characteristic peaks of each constituent are marked.

Fig. 3. (a) EBSD phase map overlapin image quality map in which white lines delimits ferrite grains with boundary angles higher than 10 °. (b) ND inverse pole figure map fot the ferrite phase; black regions represent M 23 C 6 carbides.

Table 1

Average grain size and shape parameters of the ferrite and carbide phases. with the corresponding standard error of the mean.

Fraction Grain size μ σ2 _{Grain size (Area) D}

b / D a Dc / D a μm μm Ferrite 0.968 2.58 ±0.05 0.93 ±0.04 0.55 ±0.02 4.43 ±0.07 0.58 ±0.01 0.51 ±0.01 Carbide 0.032 0.45 ±0.03 0.56 ±0.03 0.067 ±0.003 0.70 ±0.03 0.60 ±0.01 0.53 ±0.01 0.11 ∗ IG 0.47 0.26 ±0.01 0.38 ±0.02 0.028 ±0.001 GB 0.53 0.54 ±0.03 0.48 ±0.06 0.039 ±0.003 ∗ _{Measured by XRD.}

0.11±0.01estimatedby XRD.MXprecipitateshaveFCC latticeand might be confounded with M23C6 carbides. However, the

esti-matedsizebySEManalysisfallsbelowthe50nmEBSDstepsize, and presumably cannot be resolved by this technique. A thresh-old of 6 kernels (for a size larger than 100 nm) is adopted for grainidentificationtoavoidaccountingtheseprecipitatesasM23C6

carbides.

The size distributionsof ferritegrainsandM23C6 carbides are

showninFig.4andtherelevantstatisticsarecollectedinTable1. ThesevalueswereobtainedfromseveralEBSDmapsfromsurfaces

perpendicular to the normaland transversedirection. Morethan 1000 grains were included in the analysis. The results were fit-tedto a lognormaldistribution withcharacteristic

μ

and

σ

2

pa-rameters,whichrepresent,respectively, themeanandvariance of thenaturallogarithmofthegrainsize.TheaspectratioDb/Daand

Dc/Da oftheparticles weremeasuredfromnormalandtransverse images,whereDa,Db,Dc representthedimensionsintherolling, normalandtransversedirections,respectively. Carbidesizeswere discretized accordingto carbidesin the grain boundaries(GB) or inside (IG) ferrite grains as shown in Fig. 4b. There is a similar

Fig. 4. (a) Histogram showing ferrite grain size distribution (b) Histogram showing carbide size distribution in which every bin is split according to the carbide is inside the ferrite grain or at the ferrite grain boundary.

Fig. 5. Inverse pole figure map of (a) RVE CARB . and (b) RVE ∗_{CARB . in which grain limits are lined in black and M 23 C 6 carbides are filled in black colour.}

fractionof carbides placed at GBand IG, butcarbides at GBare significantlylarger.

TheferriteinversepolefiguremapinFig.3bindicatesa prefer-ential orientation of {111}bcc parallel to the normal direction, i.e.

the

γ

fibre, which is typical for rolled steels [26]. Texture was translatedin termsof an orientation distributionfunction (ODF), whichalong withthemicrostructurestatistics wasusedto gener-atea representativevolumeelementofAISI420steel microstruc-turebyStatsGeneratorfilterofDREAM.3Dsoftware[27,28].The al-gorithmfirstcreatesacollectionofidealizedellipsoidalgrains hav-ingadistributionofsize,shapeandshapeorientationequivalentto thoseobserved in theexperimental microstructure. Secondly,the generatedgrainsareplacedinsidearepresentativevolume.A num-berofconstraintsareusedtodeterminethearrangementand spa-tiallocationofthe grainsinside thisvolume.Thereafter, the con-strainedVoronoitessellationmethodisemployed forthecreation ofa Voronoi tessellation that represents the grain structure pro-ducedinpreviousstepsinordertoreducethenumberofelements and possibly improve the accuracy of the boundary representa-tion.Finally,crystallographyorientationsareassignedtothegrains, suchthattheorientationandmisorientationfunctionsare statisti-cally equivalentto the experimental dataset. TwoRVEs with dif-ferentresolutionandnumberofgrainsweregeneratedforspecific purposes.RVECARB., corresponding to Fig.5a, with50 × 50 × 50

voxels,502 ferrite grainsand 174carbides, wasused for calibra-tionofmaterialproperties.RVE∗CARB.,representedinFig.5a,with

100× 100× 100voxels, 268ferritegrainsand109carbides,was

usedfortheassessmentoflocalstraindevelopmentduring uniax-ialdeformation.

4.2. Calibrationofmaterialsparameters

Thecalibration ofthecrystalplasticityconstitutiveparameters forferrite,

τ

C,0,

τ

sat,h0,aandn, wasperformedbasedon strain-stressdatafromtensiletest(seeFig.6)andtheimplementationof amodifiedNedler-Mead(NM)simplex algorithmfollowinga sim-ilar procedure as described in [29]. NM simplex algorithm [30], outstandsforitssimplicityandeasyimplementation.Its determin-istic character and independence on gradient information makes itsuitable fortherelativelylow dimensionaloptimizationinverse problem. Thealgorithm iterativelyadjusts theparameters by per-formingcrystalplasticitysimulationsoftheuniaxialtensile defor-mation of RVEand comparing the resulting strain-stress data to theexperimental reference. When thedeviation, evaluated inthe presentworkbytheerrorsumofsquares,meetsagiventolerance, thealgorithmfinalises.Thefittingprocesswasperformedforthree strain ratestobestoptimisethen parameter.Theboundsof cali-bratedparametersweredefinedbasedontypicalvaluesforferrite presentedinTable2.Thereisalackofdedicateddataofferrite pa-rametersinstainlesssteelandingeneralforsteelsystems. There-fore,theboundswere largelyexpanded,maintainingthelimitsof typicalvaluesinmetals.Inthecrystalplasticitymodel,plastic de-formationinitiatesonlybyslipandisregulatedforasinglecrystal by

τ

C,0.It canbeconsidered thatthe calibratedvalueof

τ

C,0 will

Table 2

Material parameters of BCC iron and M 23 C 6 carbides in single crystals.

Ferrite M23 C 6

Parameter Unit Ref. [ 7] Ref. [ 37] Ref. [ 38] Adopted Adopted

C11 , C 12 , C 44 GPa 233, 135, 128 - 233, 135, 128 472 a, 216 a, 135 a ˙ γ0 s −1 1 × 10 −6 1 × 10 −6 1.07 × 10 −6 1 × 10 −6 1 × 10 −6 nslip - 20 10 – 65∗ 200 τC,0 MPa 95 200 14 77∗ 1200 τsat MPa 222 400 132 226∗ 2000 h0 GPa 1 0.8 0.7 2.53∗ 20 a – 2.25 1.5 1.5 1.75∗ _1.1 a Ref. _{[31] ,}

∗ _{Calibrated in for AISI 420 of the present study.}

Fig. 6. Experimental (open symbols) and simulated (lines) true strain stress curves obtained at different strain rates.

effectivelyaccount forother strengtheningmechanisms,e.g. pres-enceofgrainboundaries,dislocations,precipitates,etc.,whichalso affect the material yielding. Therefore, in the polycrystal and for uniaxial tension, estimationof

τ

C,0 was madeby connecting this parameter withthe yield stress (

σ

y) andthe arithmetic meanof theTaylorfactor,M,by:

σ

y=M

τ

C,0 (9)

Theexperimental

σ

y for0.0001s−1 strainrate, determinedby the0.2% offsetmethod,resultedto be222MPa.Themeasured M

for the ferrite in present steel is 2.9, which leads to a value of 76.5MPafor

τ

C,0,applyingEq.(9).Thenparametercanbeworked outdirectlyfromrelativecomparisonofthetensilecurvesat differ-entstrains[31].Consideringthevirtual-workprinciple

τ

d

γ

₌

σ

d

ε

applies[32],thenfromEq.(7): ˙

ε

=

ε

˙0



σ

(

ε

˙

)

σ

(

ε

˙0

)

n ⇒

σ

(

ε

˙

)

σ

(

ε

˙0

)

=



˙

ε

˙

ε

0

1/n (10) Applying Eq.(10)toexperimentaltensilecurves,n=65is ob-tained.

Finding experimental measurements of the mechanical prop-erties ofsingle crystalofM23C6 carbideis morechallenging and

thus, a ratherqualitative choice of material parameters hasbeen made.Ab-initiocalculationofelasticpropertiesofdifferentM23C6

carbides canbe found inLiuetal.[33].M23C6 carbidescould be

consideredasrigidparticlescomparedtotheferritematrix.There is evidencethat M7C3 can yieldto plasticdeformation at

moder-atetemperatures,wheredislocationglidinganddeformationtwins mechanisms operate[34],however noinformation wasfound on its yield stress. Inoueet al.[35] determined thehardnessof (Fe-Cr)23C6dependingonseveralalloyingelementsandconcludedthat

Mn,presentinAISI420composition,hasaminoreffect.Valuesof

1100 HV0.3 were reported, which can be translated to 10.8 GPa

[19].Assumingthattheyieldstrengthisonethirdofthehardness Vickersvalue[36]andconsideringM=3,

τ

C,0resultsin1200MPa. Therestoftheparameterswereselectedinordertoemulatea par-ticleexhibitinghigh hardening.Simulations revealed thatdespite thestressbeingpartitionedtocarbides,itdidnotdevelop to val-ueshighenough toinitiate plastic deformation.Hence, the hard-eningrelatedparameters,except

τ

C,0,canbeconsideredanecdotic forM23C6 intheframeofthepresentstudy.

Fig. 6 shows the experimental (open symbols) and simulated (lines)truestrain-stresscurvesobtainedatdifferentstrainratesfor optimised simulation parameters. The experimental strain-stress curves at different strain rates are well predicted by the crys-tal plasticity model using the calibrated ferrite parameters in

Table2andRVECARB.ComparingtheferriteparametersforAIS420

steelwith the values in literature, similarities are found for

τ

C,0 and

τ

satwithRef. [7], whichwere optimisedforferrite ina dual phasesteel.

τ

C,0 and

τ

sat,alsoadoptedforferriteinadual phase steel,arehigherinRef.[37],whichindicatesthatanycomparison shouldbedonewithcare.Anyhow,thelow

τ

C,0 and

τ

satvaluesin Ref. [38] areconsistent with thefact that they are forplainiron ferrite.Itis worthto notethat thecalibrated

τ

C,0 forthepresent steelisclosetothevalueestimatedbyEq.(9).Thereference hard-ening,h0,issignificantlyhigherinthepresentworkcompared to all consulted references, whereas a takes an analogous value to mostofthem.The strainratesensitivityobtainedforthe studied rangeofstrainrateisconsiderablylower thanforferriteinother studiesasitisdeducedfromthehighnvalue.

4.3.LocalstraindevelopmentbyEBSDanalysis

The local evolution of strain during tensile deformation of AISI420 microstructurewas evaluated by theanalysis ofthe Ker-nelAverage Misorientation(KAM)parameter fromEBSD scansas creditedinanumberofstudies[39,40].KAMaccountsforthe lo-calaveragecrystalmisorientation

<θ>aroundthedistance

x

froma measurement point [41], which can be connected to the lattice curvaturetensor (

κ

) by the Nye tensorassociated to GND density(

α

).Inasimplifiedone-dimensional(scalar)representation andassumingonlyparalleledgedislocation ofthesamesign,this tensorrelationisexpressedas[42]:

κ

=d

θ

/ dx=b

α

/ c (11)

where d

θ

/dx can be approximated as

<θ> /

x andb is the modulus of the Burgers vector, which is (a/2)<111> for bcc lat-tice.Thelatticeparameter,a,oftheferritephaseinAISI20steelis 2.8723_±0.0001 ˚A, whichwascalculated fromXRDpeaks andthe Nelson-Ridleymethod[43].cisaconstantthatdependsonthe ge-ometryoftheboundaries,havingvaluesof2and4forpuretiltand puretwistboundaries,respectively.Inreference[40],itis demon-strated thatusing Nye’s tensor,

α

= 3,which representa mixed-type boundaries and is selected forthe presentstudy. More

so-Fig. 7. (a) KAM map of ferrite phase for the 0.066 plastic strain condition. White lines delimits a threshold grain boundary misorientation of 10 ° and black ellipsoids represent M 23 C 6 carbides. The map is parcelled in 25 × 25 (b) and 50 × 50 (c) grid for statistical analysis.

Fig. 8. (a) Average <θ( x ) > with 5 ° upper threshold as a function of the kernel radius (1 st , 3 rd , 5 th and 7 th neighbours) calculated on an EBSD map of AISI 420 deformed to different strains. (b) Variation of average KAM with strain. Original and corrected KAM are presented and fitted to a linear equation (dashed lines).

phisticatedmethodsconsidering extendedcomponentsoftheNye tensor[44,45]andcross-correlation EBSDtechniquesforthe esti-mationofGNDdensitycanbefoundinliteraturee.g.[5,9,46]. Al-thoughpresentresultscouldpotentially differfromthesestudies, itislikelythattheywouldfollowthesametrends.

StrainheterogeneitieswererevealedbyKAMmapsatall defor-mation levels, asrepresentatively shown inFig. 7a for the 0.066 plasticstraincondition. HighKAMvaluesareobservedneargrain boundaries(delimitedbywhitelines)andparticularlycloseto car-bides (black ellipsoids), while low KAM values are rather more spreadthroughoutthematerial.KAMisverysensitivetotheEBSD stepsize[44,46],andtheoverestimationduetothemeasurement noise is drasticallyincreased when decreasing the step size. The methodproposedby Kamaya[47],illustratedinFig.8a,isusedto estimatethe measurementerror. <θ(x)>values havebeen calcu-latedforeachpixelofthemap usingtheith _{neighbours(i =1,3,} 5,7).Theindividualvaluesof<θ(x)>arethen averagedtoobtain arepresentativevalueofthewholescan.<θ(x)>valuesbelow5° areconsideredinthecalculationtodiscountgrainboundariesfrom theanalysis.The average value of<θ(x)> increases linearlywith thekernelradius,asshowninFig.8a.Thisisan indicationofthe

misorientationgradientbeingconstantintheexplored neighbour-hoodaroundeach pixel.Theincrease ofthe slopewiththe strain isexplainedbylargermisorientationgradientsasthesteelismore plasticallydeformed.Inabsenceofmeasurementnoise,the extrap-olated <θ(x)> values tox = 0 should tendto zero.In Fig. 8a, it can be observed that <θ(x = 0)> ranges between0.4 ° and 0.6 ° at different strains,which can be considered as an estimate of the measurement noise. Assuming that under very small (shear) strainsd

γ

= tg(d

θ

) ≈ d

θ

and theestimatedmeasurement noise, the shear plastic strain detection limit is in the range of 0.007 - 0.01, which falls below the chosen macroscopic plastic strains. Hence,itisconsideredthatmeasurementnoiseisnotsignificantly affectingevaluationoflocalstrainsbyanalysingKAMmapsexcept fortheunstrainedcondition.

Fig. 8b shows an ascending linear relation between the plas-tic strain and the average KAM values (or equivalent GND den-sity by applying Eq. (10)), which up to the strain level analysed here (0.139) well reproduces the Ashby model [42] and is con-sistent withtheobservationsreportedby otherstudieson ferritic steels[44,45].StandarddeviationinthedistributionsofKAMwas used to generate the error bars. The increase of standard

devia-Fig. 9. KAM distribution at different macroscopic plastic strain levels for AISI 420 steel.

tion withincreasing plastic strain is explained by thestrain het-erogeneities present in the specimens. This is evidenced by the evolutionofKAMdistributionwithplasticstrain inFig.9a,which appears highlyskewed with a pronounced tail to the highKAM sideofthepeak.ItisworthnotingthattheKAMdistribution (in-cluding measurement noise) in the unstrained conditionalready exhibits significantly high values compared to the strained spec-imens. In the unstrained specimen, high KAM values are mainly concentrated at M23C6/ferrite interfaces(see Fig. 10), which may

be explained by thermal strains arising from differences in the thermal expansion coefficientsduring cooling [5] or from lattice mismatch [48]. The average KAM value for the carbide phase is higherthanforferriteintheunstrainedcondition.Thisfactmight beexplainedbyapoorimagequalityofcarbides.Nevertheless, av-erage KAM of carbides remained nearly unaltered regardless the appliedmacroscopicstrain,whichsuggestsanegligibleplastic de-formation.

The locationswhere strain preferentially developsare qualita-tivelyanalysedbycomparingKAMandimagequality(IQ)mapsin

Fig. 10atfour differentlevels ofdeformation insameEBSD scan region. High KAM values locallydevelop primarily at carbides at lowmacroscopicstrains.Athigherstrainlevels,localizedstrain at ferrite grain boundaries and triple junctions becomes more evi-dent,butregions withhighKAMfurtherextendto graininterior, connectingothercarbidesandformingacellstructure.However,it isdifficulttojudgetheindividualeffectofdifferentmicrostructure featuresduetomutualinteractionandthefactthatinformationis missingfrommicrostructurebelowthespecimensurface.

TheformationofshearbandsisdetectedinIQmapsfrom0.066 plastic strain. The developmentofshear bandsis lessevident by analysing KAM maps. Only fora few cases, the observationof a shearbandinIQcorrespondstoanobservationofhighKAMatthe surroundings.ThecorrelationbetweenlowIQandhighKAMinthe surrounding ofshear bandsbecomes moreevident as theplastic strain isincreased. Shear bandsare oftencharacterizedby strong latticecurvatureandveryhighdensitiesofgeometricallynecessary dislocations at the interior ofthe shear band and the surround-ing matrix.However, it should be considered that a free surface canrelaxhighlocalshearsbyproducingasurfacerelief,excluding theneedto accommodateheterogeneousstrain bythegeneration ofgeometricallynecessarydislocations.Shearbandsstillconsistof verysmalldislocationcellsandcontainveryhighdislocation den-sities.TheIQwillbemoreaffectedbysurfacereliefandhigh den-sityofSSDsthantheKAMaslongasthepatternisstillsufficiently strongtoresultintrustworthyorientationdata.

Astatisticalanalysiswasperformedforan in-depthanalysisof the microstructure features that mostinfluence the development oflocalstrain withdeformation intheAISI420steel microstruc-ture.EBSDmapsareparcelledindifferentareascontaining differ-entmicrostructure features asshown inFig. 7band7c. 25× 25 and50× 50gridsarecomparedtoassesstheinfluenceofthe par-celarea on thestatistical analysis. The parcelarea ofunstrained conditionis4

μ

m2_and1

_μ

_m2 _{forthe25× 25and50× 50grids,}

respectively,whichareontheorderoftheaverageferritegrainand carbidesizes. KAMisusedasaresponsevariablefortheanalysis, whichisaproxyofthelocalstrainsanddensityofGND,andis re-latedtothepresenceofgrainboundaries,carbidesoracombined effect.Twopossibleexplanatoryvariablesare initiallyconsidered: numberofgrains (NG) andnumberof carbides(NC). The average KAMofeach parceliscalculatedalong withNGandNC.A consis-tent population of points, in particular in measurements at high plastic strains,exhibit KAMvalues near zero(see Fig. 9a). These pointsarerelatedtopoorindexingandarefilteredoutbycreating areduced image withasignificant andrepresentative numberof data.

TheKAMboxandwhiskerplotsofNC andNGexplanatory vari-ablesatdifferentplasticstrainsareshowninFig.11.Thebox rep-resentsthequartiles25–75%.Thewhiskersextendupto1.5times theinterquartilerange.Theoutliersarerepresentedbyhollow di-amonds,withtheminimumandthemaximumofthedistribution indicated as solid diamonds. Solid squares represent the average KAMofthepopulationanddottedhorizontallinesdenotethe av-erageKAMofthewholemap.

Focusingontheunstrainedconditionandcomparingtheboxes ofthe explanatoryvariables, it can be clearlyseen that KAM in-creases with the number of grains and the number of carbides. Inabsenceofcarbideorwhenthereisnograin boundary,i.e.NG equalsone,theaverageKAMofthepopulationisbelowthe aver-ageKAM value ofthe scan. Theseeffects are more prominentin thecaseof50× 50gridandindicatethelocalizationofhighKAM valuesaround ferritegrainsandcarbides.Aspreviously discussed, theseobservationsareexplainedbystrainsinducedduringcooling duetodifferencesinthecoefficientofthermalexpansionofferrite andM23C6 carbides anddue to a mismatch betweenthe grains’

crystallography[5,48].

KAMincreaseswithplasticstrain. Thisstatementholdsforall populationsashighlightedinFig.11.Thepresenceofcarbide typ-ically results on higherKAM compared to the condition without carbides,whichisequivalenttoorexceedstheaveragevalueofthe scan.KAMincreaseswiththenumberofcarbides.Thesame state-ment applies for parcels withhigh population of grains, but the effectislessnoticeable.Thedifference betweenthe averageKAM ofdifferentpopulationsofNGandNC decreaseswithplasticstrain. This fact is more obvious in the results of 50 _{× 50} grid. More-over, the increase ofKAM with plastic strain is lesspronounced for highnumbers of NG and NC. The influence of individual ex-planatory variables on the evolution of KAM with plastic strain compared to others is difficult to assess by only looking at the plotsinFig.11becauseofthecombinedeffectsofNG andNC.The followingmultivariate linearregressionmodel,whichincludesthe macroscopicplasticstrain

ε

P,isfitted:

KAM = K 0+K 1

ε

P+K 2

_N G A



+K 3

_N C A



+K 4

(

N GN C/ A 2

)

(12) withK_i/i ₌0,1,2,3,4arethe modelcoefficientsandAthe par-cel area. K0 is the intercept and represents the average KAM in

thecaseofunstrainedconditionfortheferritephasewithout car-bides or grain boundary contribution.The interaction of NC and NG is introduced by the last term. Table 3 collects the resulting coefficientsafterapplyingmultilinearregressionanalysisbasedon

Fig. 10. KAM and IQ at different macroscopic plastic strains for the same region of AIS 420 steel.

Fig. 11. Box and whisker plots of different explanatory variables, (a) number of carbides and (b) number of grains corresponding to 50 × 50 grid.

Table 3

Coefficients after applying multilinear regression analysis of Eq. (11) .

K0 (Intercept) K1 ~εP K2 ~N G /A K3 ~N C / A K4 ~N G N C / A 2 ( °) ( °) ( μm 2 ) _{( μm} 2 ) _{( μm} 4 ) 50 × 50 Estimate 0.320 2.59 0.041 0.072 −0.010 Std. Error 0.004 0.03 0.001 0.004 0.001 t value 78.6 85.9 34.2 15.9 −8.1 Pr ( > |t|) ∗ _{< 2 •10} −16 _{< 2 •10} −16 _{< 2 •10} −16 _{< 2 •10} −16 _{< 2 •10} −16 25 × 25 Estimate 0.360 2.60 0.078 0.09 −0.04 Std. Error 0.007 0.05 0.006 0.02 0.01 t value 54.8 55.6 13.9 5.05 −3.1 Pr ( > |t|) ∗ _{< 2 •10} −16 _{< 2 •10} −16 _{< 2 •10} −16 _{5 •10} −7 _0.002 ∗ _{Pr ( > |t|) evaluates the significance of the explanatory variable on the model. Values} approaching zero indicates high significance. R 2

Table 4

Codes for the variable CP which classify EBSD grid parcels according to the observed microstructure features.

0 No carbides

1 Inner carbide

2 Carbide in the boundary between two grains

3 Carbide in the boundary between three grains, i.e. a triple point

12 Two carbides: one inner carbide and one in the boundary between two grains

13 Two carbides: one inner carbide and one in the boundary between three grains

23 Two carbides: one carbide in the boundary between two grains and one in the boundary between three grains

123 Three carbides: one inner carbide, one in the boundary between two grains and one in the boundary between three grains

Fig. 12. Box and whisker plots showing the evolution of the mean kernel average misorientation (KAM) with plastic strain an as a function carbide position in the microstructure as defined in Table 4 .

model ofEq.(11), along withthestandard errorandsignificance parameters.

From thelinearregressionmodelitcanbeobservedthatK0is on theorder ofthemeasurement noise,which means thatmean KAMshouldtendto0inabsenceofgrainboundariesandcarbides, whichiswhatisphysicallyexpected.ThehighvaluesofK1indicate that KAM isprimarily influenced by strain, andthevalue of 2.6° matcheswellwiththeslopeoflinearrelationofKAMwithplastic strain presentedinFig.11.TheincreaseofKAMwiththenumber of carbides (i.e. carbide density) is more effective than withthe number of grains(i.e. grain boundary density),as resulting from comparingK2 andK3,butthisisonlyevidentinthe50× 50grid. Thepositive effectofthesevariables wasexpectedfromthe anal-ysis of box and whisker plots. However, the negative sign asso-ciated with the interaction parameter K4 means that the regions withhighdensityofgrainboundariesandcarbideswillcounteract thedevelopmentofKAMbythepositiveeffectofK2 andK3.

Formeranalyses disregardtheinfluenceofthepositionof car-bides inthemicrostructureon theintensityofKAM.Gridparcels inthe EBSDmapwere classifiedaccordingtothepositionof car-bidesby acategoricalvariable,CP,whichiscodedasdescribedin

Table4.Fig.12showstheKAMboxandwhiskerplotsofCP.There is a cleardependenceon carbide positionin thedevelopmentof KAM.AverageKAMisenhancedwhenthecarbidesarepositioned atferrite grain boundariesandspecially attriple points(CP ₌ 2, 3) compared to carbides inside ferrite grains (CP = 1). This fact points to a magnified effect on the development oflocal strains duetoacombinedactionofcarbidesandgrainboundaries.Taupin etal.[49] discussedthe thicknessofa layerwithincreasedGND density in a ferritic steel with spheroidized cementite. Their re-sultsshowedthatwhenthecementitediameterisatmicronscale, thethicknessofGNDlayerisindependentofcarbidesize.Maetal.

[50] argued,however,that whencementitediameterisat submi-cron size,asitisthe caseofthepresentstudy,a ratioshouldbe maintainedbetweenthethicknessofGNDlayerandthediameter oftheparticle.AveragesizeofintragrainM23C6carbidesissmaller

Fig. 13. True strain true stress curves obtained from experimental (Exp.) and mod- elled (RVE CARB. and RVE FERR. ) tensile tests.

thanthatofcarbidesatgrainboundaries,whichmightexplainthe lowervaluesofaverageKAM.Nevertheless,averageKAMinparcels containing carbides at grain boundaries is typically higher than thatinparcelsinwhichonlygrainboundarieswererecorded.From

Fig.12,itisalsoobservedthattheaverageKAMofdifferentCPat highplasticstrainstendstoequalwithNGandNC.

4.4.Isolationofcarbideeffectbymodelmicrostructures

Theeffectofcarbidesonthemechanicalresponseisevaluated byanewrepresentativevolumeelement,RVEFERR.,sharingthe

mi-crostructuretopologyandtexturewithRVECARB.,butinwhich

ma-terialpropertiesfortheM23C6carbidearesubstitutedforthoseof

ferritephase.Anydifferenceinthecrystalplasticitymodelling be-tween RVEFERR and RVECARB. will consequently beattributed only

toanincompatibility arisingfromamismatchinthestrength be-tweentwophases.

Fig.13 showsthe strain and stress curves after tensile defor-mation ofRVECARB. andRVEFERR. withastrain rateof0.0001 s−1.

Differences in theyielding of RVECARB., RVEFERR. are insignificant.

However, the initial hardening rateof RVECARB. is slightly higher

than that of RVEFERR., which results in a lower ultimate tensile

strengthin RVEFERR. Fig.14a showsthe distribution ofequivalent

Von Mises strain for the ferrite phase in RVECARB. and RVEFERR.

atmacroscopic plastic strainssimilar to those of the experimen-taltests.Fortheinitialunstrainedcondition(notshowninFig.14) thestrain distribution isrepresentedby a Diracdeltafunction. A broadening of strain distribution is observed in both RVEs with increasing strain. Broadening is more pronounced in RVECARB. at

all strain levels, indicating a more heterogeneous strain, by de-velopmentofregions withhighstrain levelsandothers inwhich strain development is hindered compared with fully ferritic mi-crostructure.Itisnotpossibletoestablishadirectcorrespondence betweenexperimental andsimulatedresults,consideringthat the formeronlyaccountsforplasticdeformationleadingtoGNDs(but

Fig. 14. (a) Histograms of the local strain distribution obtained from RVE CARB. and RVE FERR. at differen levels of plastic deformation. The number fraction corresponds to 50 bins. For shake of a better comparison, a continuous representation by connected lines is shown. (b)–(d) show the fitting results to asymmetric double sigmoidal funtion, AsyS, described in Eq. (12) , of diferent contributions to local strain g CARB. and f FERR. . They also shows the cumulative of both fitting curves, h CARB. ∗_{FERR . Diamonds represent the}

bins of histogram distribution of local strain in RVE CARB. .

notduetothemotionandgeneration ofSSDs),andthelatter in-cludesinthestatistics thedeformation atevery pointinthe RVE regardlessthe operatingmechanism. Nevertheless,it can be con-cludedthat the simulations areconsistent withthe experimental measurements.

The ferritestraindistributionfunction resultingfrom deforma-tionofRVECARB.,h.,canbe decomposedintwoparts: (1)f,which accounts for texture and strain incompatibilities between ferrite grainsaffecting thedeformation offerrite,i.e.the strain distribu-tion function resulting from RVEFERR. and (2) g, which considers

theeffectsofcarbides, includingtheinfluence ofcarbidesonthe ferrite-ferritegrain interactions. Thefollowing procedurewas fol-lowedinordertoobtain g..First,since h=f∗ g,where∗ stands forconvolution, gwasobtainedfromhofRVECARB. by

deconvolu-tionoff.Theshapesoffandgcanbeconsideredrepresentativeof eachindividualeffecttotheglobalstrainheterogeneitiescaptured bythe shape ofh. However, thesummation ofthe fandg areas doesnot correspondto thearea ofh.Inorderto circumventthis issue,fandgwerefittoanasymmetricdoublesigmoidalfunction thatwasfoundtobestfitbothcurves,definedas

y =





1+e −

(

x−xc+w21

)

/w2



−1



1−

1+e −

(

x−xc+w21

)

/w3



−1

(13)

where



represents the amplitudeof the curve, xc the strain at curve maximum and w1, w2, w3 are parameters controlling the shapeofthecurve.hwasthenfitconsideringthatthiscurveisthe convolutionoftwoasymmetricsigmoidalfunctionshaving respec-tively the xc, w1,w2,w3 valuesobtained fromprevious fittingto

fandg.TheAparameters ofbothcurveswereestimatedtomake the cumulative curve fit best to h. The Levenberg–Marquardt al-gorithmbasedonthe

χ

-squareminimisationmethodwasusedin theNLfittoolinOriginPro9® software.Theresultingfittingofthe differentcurvesat differentstrain ratesisshownin Fig.14b,14c and14d. andtheir integrated area andfull widthhalf maximum (FWHM)inFig.15.TheFWHMofallcurvesincreaseswith increas-ing strain, but significantly stronger for thecurve accountingfor the solecontribution ofcarbides.This indicates a main contribu-tionofcarbidesinthedevelopmentofheterogeneousstrainsinthe ferritephase.Thecontributionofcarbidesalonetoglobalstrainis similartothatoftheothereffectsinviewofthesimilarintegrated areasoftheircorrespondingcurves.

Fig. 16 shows equivalent Von Mises strain maps of the same sectionofRVE∗CARB.andRVE∗FERR.atdifferentstrainlevels.Adirect

comparisonbetweenexperimentalKAMandIQmapsandthe sim-ulatedstrain map should be done withcare.For example, trans-latingKAMmapsintostrainmapswillrequiretochooseastarting pointandintegrateoverthetotaldomaintogenerateastrainmap. However,KAMandIQmapsgiveareasonableindicationofregions whereheterogeneousstrain stronglydevelops.Uponthese

consid-Fig. 15. Full width half maximum (FWHM) and integrated area of g CARB. , f FERR. and

hCARB. ∗_{FERR . variation with macroscopic strain.}

erations,experimentalobservationsaretosomeextentreproduced

inRVE∗CARB..Carbidesbarelydeformcomparedtoferriteand

con-centrate highstrain levels intheir surroundings. High strain also develops at ferrite boundaries and triple junctions compared to grain interior. The effect of carbides is clearly evidenced when comparing RVE∗CARB. to RVE∗FERR..Strain developsmore

heteroge-neously in RVE∗CARB., with regions exhibiting higher strains than

theirRVE∗FERR.counterparts,butalsoregionsthatdeformless.This

fact was alreadypredicted upon analysing the strain distribution curvesinFig.14a.Features resemblingslipbandsarerecognizable from0.0384plasticstraininRVE∗CARB.andacellstructureformed

by band intersectionsdevelopswithincreasing strainina similar fashionasexperimentallyobserved.Itshouldberemarked,though, thatcellstructureinexperimentalspecimenscanalsobea conse-quence ofsmallprecipitates,which areabsent inthe microstruc-tureoftheRVEs.StraingradientsarelesspronouncedinRVE∗FERR.

andfeaturesresemblingdeformationbandsarenotclearlyformed, although highstrain values concentrateat ferriteboundaries and triple junctions. Equivalent regions in RVE∗CARB. develop higher

strainlevelsregardlessthepresenceorabsenceofcarbides,which indicates that carbides not only influence local strain directly by

elasticincompatibilitieswiththeferriticmatrix,butalsothe spa-tialinteractionsbetweenferritegrains.

5. Discussion

5.1. Assessmentofmicromechanicalmodel

Theconclusionsextractedfromacomparativeanalysisof mod-elleddeformedmicrostructures aresubjectedto theintrinsic lim-itationsof therepresentative volumesand theconstitutive equa-tionsadopted.Firstremarkableissueisthat thelinearelasticpart in the simulated curves in Fig. 13 exhibits an abnormally high Young’s modulus of 275 GPa compared to the general accepted Young’smodulusofsteel.ThedirectiondependenceoftheYoung’s modulusofcubiccrystalsystemcanbeexpressedas

A hkl= h 2_k 2_+l 2_k 2_+h 2_l 2



h 2_+k 2_+l 2



2 (14) 1/ E hkl=S 11− 2



S 11− S12− 1 2S 44



A hkl (15)

whereAhkl istheelasticanisotropyfactor,Ehklistheelastic modu-lusofthecrystallographicplane_<hkl_>andS_ijaretheelastic com-pliances. The equation relating the elastic compliances with the elasticstiffnessmatrixcomponentscanbe foundin,e.g.,Knowles etal.[51].Ahighpopulation ofgrainsorientedwith<111> nor-mal to the load directionin the RVE, with E111 = 306 GPa,

ex-plainsthehighYoung’smodulusobtainedinthesimulatedtensile curves.Fig.5showsthatasthenumberofferritegrainsdecreases

inRVE∗CARB., thegrain orientation along the{111} becomesmore

dominant, an unintended result of the algorithm used to assign texturetograinsbasedonODF.InthecaseofRVECARB.,thenumber

offerritegrainsmightbe notenoughto capturelessintense tex-turecomponents,andtexturealong{111}manifests moreintense thanthat oftherealmaterial.Thisfactrevealstheimportance of theselectionofan adequatenumberofgrainswitha representa-tive texture, whichwas confronted inthisstudy withlimitations incomputational power.SimulationswithRVEswithahigh num-berofcarbidesandgrain boundariesarerecommendedforfuture studies.Nevertheless,simulationswitharandomtextureRVEleads

Fig. 17. Instantaneos work hardening rate, , vs. stress curves obtained from ex- perimental (Exp.) and modelled (RVE CARB. and RVE FERR. ) tensile tests.

toE ₌ 216 MPa,whichis still higherthanthat forthe alloy,but creditsthechoiceofCij.

Anotherissuearisesduringthetransitionfromelasticto plas-ticdeformation.Theinstantaneousworkhardening,

= d

σ

/d

ε

,is plottedasa function ofstress in Fig.17 fora better assessment ofelastictoplastictransitionaccordingtoArechabaletaetal.[52]. Intheexperimental plot,an abruptmonotonic decreaseof

be-low the material yield stress indicates an anelastic regime, con-sequenceofdislocationbow-outbyOrowan mechanism,butwith limited dislocation glide. This trend is more progressive in AISI 420steelthaninpureironandlowalloyferriticsteel[52],which pointstoaneffectofthecarbidesorthereducedferritegrainsize. Initial instantaneous modulus ofaround 275 GPa is attributedto inaccuraciesin the strain measurement at low deformation. This valuerapidlyevolvestoamorereasonablevalueof200GPa.Above 220MPastress, achange intrendis observedand

slightly in-creases to subsequently smoothly decrease to

₌ 0 GPa when maximum uniform elongation is achieved. Cheng et al. [53] ob-servedthisbehaviourforoveragedaluminiumalloysasthe precip-itatesreachedanaveragesizeatwhichdislocationscannotlonger shearit,whichitisassumedthecaseforlargeM23C6 carbidesin

AISI420.

The modelled curvesdonot accurately capturethereal mate-rial hardening behaviour until approximately 275 MPa, well into theplastic regime.Fig. 17showsa hardening rateinthe simula-tionthatsignificantlydeviatesfromtheexperimentalresultinthe strainregionbelow2%.Inthesimulation,theinitialstatedoesnot includeany internal stress. However, as shown inthe KAM map ofFig.10,theinitialstate, withoutappliedload,doesinclude lat-ticedistortionaround theboundaries, whichcan beattributedto amisfitstrainduetothermalstressduringcoolingintheprocess. Theinitialresidualstressmayassistinaplasticityinitiationby dis-locationnucleation/multiplicationasindicatedbyShimokawaetal.

[54].Thiscan beareasonforthelowerflowstressleadingtothe lowerhardeningrateasshowninFigs.13and17.Sincethe simu-lationisoperatedindisplacement-controlledmode,agivenstrain is compensated by plastic and elastic strains. If the microstruc-turedoes not allow the developmentof plastic strain, the strain isaccommodatedelastically, leadingto ahigherapplied stressby Hooke’s law. In the model, the early hardening behaviour is ad-dressedbytheh0 parameterorinitialhardening,whichis consid-eredconstantalongtheindividualgrains.Infuturework,itwould be interesting to incorporate a third phase with different h0 in theRVE along thegrain boundariesto consider theeffect of ini-tialstrainattheselocations.

However,theydorecreateanabruptdecreasein

aftera pro-longedplateaumaintainingtheinitialinstantaneousmodulus.This decreasecannot be attributedto a reversible anelasticbehaviour,

σ

y

higher in RVECARB. (208 MPa) compared to RVEFERR. (203 MPa).

However,itcannotbeexplaineddirectlybyclassicalstrengthening mechanismsinthesyntheticmicrostructures.

Phenomenological constitutive formulation suffers from the drawbackthatthematerialstateisonlydescribedintermsofthe critical resolved shear stress,

τ

_Cη, andnot in terms oflattice de-fects population. The presence, motions, and interactions of dis-locations are not explicitly identified by the model. Any specific physicalcontribution tohardening ofGNDsis disregardedby the local crystalplasticity model.Nevertheless, ifa carful interpreta-tionismade, usefulinformationcanbe extractedfromthe appli-cationofthismodel.Basedon thehardeninglawadoptedinthis work,AcharyaandBassani[57]formulated thath_βη dependsboth ontheslipsandtheir gradientsviatheincompatiblelattice defor-mations,i.e.,Nye’sGNDdensityevolutionwithstrain,

α

˙i j:

˙

α

i j= e i jkl N  β=1 ˙

γ

η_m βnβ (16)

where e_ijkl denotes the alternating vector. It should be stressed herethat Eq.(16) only providesa relationofthe strain gradients arising bythe localplasticitytheorywithGNDs.Undermulti-slip deformation offerriteinthisstudy,a distinctionshould bemade for short-and long-rangecharacter of the interaction between a mobile dislocation andtheGNDcontent. Short-rangeinteractions maybefullyaccountedforinthehardeningratematrix,h_βη.The contribution tostress of long-rangeinteractions dueto the pres-enceofGNDscanbeconsidered toarisesolelyfromthepresence ofan incompatibility.Incompatibilitywilldevelopinferrite-ferrite andferrite-M23C6 boundariesasadirectconsequenceofthe

indi-vidualorientationsofgrains,combinedwiththeanisotropyofthe elastic stiffnesstensor.Under theseassumptions,and considering thattheferriteparameters arethesame,thedifferencesobserved betweenRVECARB. andRVEFERR. wouldbe dueto thecontribution

ofM23C6 carbidestohardeningduetobothlong-rangeand

short-rangeinteractions.

The results should be also interpreted considering the limita-tionsinthemodelwhentacklingspatialgraininteractions,which affect the manner in which grains deform and rotate influenc-ing the development of texture. An advantage of crystal plastic-ity finite element predictions is that they can reproduce experi-mental findings well. However, they are restrained to the use of a proper RVE and constitutive model as inputs. For the valida-tion ofthesimulation, texturesofexperimental andartificial mi-crostructuresforthe unstrained and0.14plasticstrain conditions are illustrated by

ϕ

2 = 45° sections of the orientation

distribu-tionfunction(ODF)inFig.18andFig.19,respectively. Unstrained AISI420steel(Fig.18a)exhibitsatypicaltextureofrolledand an-nealed iron bcc, with most intense components located approx-imately at {111}<112> and {554}<225> [58–60]. These orienta-tions ofthe recrystallizedgrains arepreferredfor nucleationand

Fig. 18. ϕ2 = 45 ° sections ODF of AISI 420 obtained from EBSD data and TSL OIM® Analyses 6.0 software, (a) εP = 0, (b) εP = 0.139.

Fig. 19. ϕ2 = 45 ° sections ODF of obtained after DAMASK simulation and MTEX Toolbox, (a) εP = 0 (RVE CARB , RVE FERR ), (b) εP = 0.143, RVE CARB. , (c) εP = 0.143, RVE FERR. growth from deformed grains with orientation {112}<110> and

{111}<110>.Furtheruniaxialdeformation rotates{111}<112> and {554}<225> components preferentially to {111}<110>, but maxi-mum intensity decreases compared to the unstrained state. Arti-ficial microstructuretexture (equal for RVECARB. and RVEFERR.),

il-lustrated in Fig. 19a, reproduces AISI 420 steel texture reason-ably well, but exhibits a more intense texture at {111}<112>. The {554}_<225_> component is less intense and texture is more pronounced towards {111}<110>. The deformed RVECARB. and

RVEFERR. (Fig. 19b and 9C, respectively) exhibit very similar

tex-ture, which indicates a minor effect of the relatively small frac-tionofM23C6carbides.Texturesevolvesimilarlytothatof

experi-mentalspecimenat0.14plasticstrain,inwhichintensitydecreases compared to the unstrained condition. Assuming the abovemen-tioned limitations, and in view of good agreement between ex-perimental and predicted curves and textures at high plastic strains, it can be stated that observations made in RVECARB. and

RVEFERR. wouldbe reasonablywell reproduced in equivalentreal

microstructures.

5.2. Effectofcarbidesonplasticflow

TheworkhardeningofAISI420annealedmicrostructure, mag-nitude 330 MPa, is higher than that for other annealed ferritic low carbonand stainless steels,which are in the range between 100 MPa to200 MPa [61].This fact isexplained by highdensity

oflargeM23C6 andsmallMXprecipitates.Theeffectofsmall

fer-ritegrainsisalsosignificant,followingfromsimulationswith mod-elledfullferriticmicrostructureresultinginonlylessthan20MPa ofworkhardeningdifferencewithrespecttomodelled microstruc-turecontaininglarge M23C6 carbides. However, thecalibration of

parameters for ferrite also accounted forthe effect of small MX precipitates,whichwerenotincludedinthesimulated microstruc-tures.Therefore,theeffectofsmallferritegrainsizeshouldbe cor-roboratedinothercarbidefreeferriticmicrostructures.

LargeM23C6carbideswouldalsoresultinhighinitialhardening

rates,whicharetypicallyobservedformetalalloyswithalow vol-umefractionofnon-shearableprecipitates[53].Thishasbeen at-tributedtotwomechanisms:(1)thestorageofgeometrically nec-essarydislocations,(2)thestorageofelasticenergyofthesecond phase.Thecalibratedh0 hardeningparameteris highercompared toother ferritic steels[7,37,38]. High h0 is typicalformartensitic steels with high dislocation density and small grain size [7,62]. Alow initial dislocationdensityis assumedinannealed AISI420 steel.Hence,highh0mightbe relatedtopresenceofcarbide pre-cipitates in present study. Further analysis was made recalibrat-ingtheferritepropertiestodescribethebehaviourofrealmaterial containingcarbides,usingthistimeRVEFERR..Theonlyremarkable

differenceintherecalibratedhardeningmodelparametersfor fer-ritewasobtainedforh0 withavalue of2660MPa.Thisvariation of130MPainthehardeningparameterisinterpretedasanexcess ofinitialhardening,e.g.duetoahigherinitialdensityofslips

sys-α

willberelaxedbycrossslipandtheprincipalsourceofhardening becomesisotropichardening associated withshort-range disloca-tioninteraction.In thehardening model,h_βη and

τ

_Cη interdepen-dentlyevolvewithmacroscopicplasticstrainaccordingtoEqs.(5)–

(7)which complicates discerningthe character of carbides interac-tion. However, the saturation stress is the same in RVECARB. and

RVEFERR sincethis representsthe physical limit to thedensity of

dislocationsthat mightbe storedinthe ferrite crystal. Therefore, thedifference of20MPaintheultimatetensilestrengthbetween

RVECARB. andRVEFERR arises onlydue todifferences inhardening

belowastrainof0.05.Consideringthath0forferriteisthesamein bothRVEs,itcanbedeductedthatobserveddifferencesin harden-ingarisesolelyfromstrainincompatibilities.Thelocalcrystal plas-ticitymodeldoesnotconsiderextrahardeningduetoalong-range interactionmechanism,howeveritcanbeassumedtobebasedon existingtheory that long-range interactions willdominate atlow strainsandwillbe morepronounced dueto thestrain incompat-ibilitiesintroduced by carbides compared to ferrite grain bound-aries.The0.05straincoincideswiththeobservationofshearbands indeformedtheAISI420microstructureasshowninFig.10.Wang etal.[63] demonstratedby 3D dislocationdynamicsthat the un-derlyingmechanismforslipbandformation isthroughcross-slip. Hence,theobservationofshearbandscanbeconnectedtoashift inthegoverninghardeningmechanism.Thedevelopmentof cross-slipmightbeevidencedinRVECARBbytheformationofbandswith

highlocalstrain(Fig.16), whichwouldbeinagreementto obser-vationofshearbandsintherealmicrostructure.Byrneetal.[64], observedthatsinglecrystalscontainingnon-shearableprecipitates of the Al-Cu system immediately began deformation by polyslip ratherandsingleslip. Thiscan be extrapolatedfromRVECARB

re-sultsin which the activation ofmultiple slip systems is demon-stratedbytheformationofintersectingdeformation bandsin fer-ritegrains.ThiscontrastswithdeformedRVEFERRgrains, inwhich

activationofsingleslipseemstobepredominantandferritegrains deformedmorehomogeneously.

KAM continuously increases with macroscopic strain follow-inga linear trend(see Fig.8b), consistent withtheAshby model

[42] and the observations reported by other researchers in fer-ritic steels[44,45]. Acompetition between dislocation accumula-tion and dynamicrecovery at high plastic strains would explain thehighinitial instantaneous work hardeningdespite the rateof increase of GND density with strain remaining constant. Results withmodelledmicrostructuresinFig.15,showabroadeningofthe straindistribution alsofollowing alineartrend withamain con-tributionofcarbidestothedevelopmentofstrain heterogeneities. Itisalsoobservedthatcarbidesenhancethecontributionofferrite grainboundariestotheincreaseoflocalstrain.Thestatistical anal-ysisofevolutionoflocal strain indeformedAISI420 microstruc-tureevidencesapositiveeffectofcarbidesandgrainboundariesin KAM(Fig.11),inparticularatlowmacroscopicplasticstrain,which canberelatedtodevelopmentofGNDs.Athighplasticstrain,

aver-whichevolveinlow-anglesubgrainboundaries,e.g.during anneal-ingtreatmentsorcreep[13].Thisstudyalsosuggestsaninfluence of carbide position in the development of heterogeneous strains inthemicrostructure.Generationofartificialmicrostructureswith controlleddistributionofcarbidesandcarbide/ferritegrainsize ra-tiowillbeapowerfultoolinfuturestudiesofAISI420steelsand othermaterialswithlargeprecipitates.

6. Conclusions

The influence of large M23C6 carbides on the heterogeneous

strain development in annealed 420 stainless steel was studied combiningexperimentalresults andcrystal-plasticity based simu-lationsonanartificialrecreationofthemicrostructure.Simulations of two different 3D representative volume elements (RVE), one mimickingrealferrite-carbidesmicrostructureandanother consist-ing only offerrite phase, were compared to experimental results andthefollowingmainconclusionsareextracted:

(1) Modelledtensilecurves,obtained afterthecalibration of con-stitutivemodel parameters,accurately reproduceexperimental curves,althoughtheyshowlimitationsonaccuratelycapturing therealhardeningbehaviouratstrainsbelow0.02,likelydueto residualstressesobservedinthematerialarenotconsideredin theRVEs.Theevolutionoftextureandlocalstainwasalsowell predictedindeformedRVEs,whichemphasisestheimportance ofhaving agood RVErepresentationand confirmstherole of differentmechanismsofstraindevelopmentinpresenceor ab-senceoflargeM23C6carbides.

(2) Results suggest that carbides provide a more effective hard-ening at low plastic strain as they introduce stronger strain heterogeneities compared to a purely ferritic microstructure, likelyby a predominantlong-rangeinteraction mechanism. At strains around 0.05, cross-slip is evidenced in real and arti-ficial microstructures containing carbides indicating a change in the hardening mechanism dominated by short range dis-location interactions. The hardening behaviour of ferritic and carbides-containing microstructures become similar at high plasticstrain.

(3) Straindevelopedmore heterogeneouslyinRVEcontaining car-bidescomparedtoferriticRVE.UnlikeferriticRVE,acell struc-turedelimitedbyhighlocalstrainsisformedintheRVEwith carbidessimilartoexperimentalobservations.Carbidesnotonly influencelocal strain directlyby elastic incompatibilities with theferriticmatrix,butalsothespatialinteractionsbetween fer-ritegrains.

(4) Carbidesplacedatgrain boundariesenhance thedevelopment of strain near ferrite grain boundaries. A statistical linear re-gressionmodelshowsthepositiveeffectofferritegrain bound-aries and carbides in the development of high local strains. However, the model also reflects that development of local