Analysis of surface roughness evolution of ferritic stainless steel using crystal plasticity finite element method

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University of Wollongong

Research Online

Faculty of Engineering and Information

Sciences - Papers: Part B

Faculty of Engineering and Information

Sciences

2019

Analysis of surface roughness evolution of ferritic stainless steel using

crystal plasticity finite element method

Xiaoguang Ma

University of Wollongong, [email protected]

Jingwei Zhao

University of Wollongong, [email protected]

Wei Du

Baoshan Iron and Steel

Xin Zhang

Baoshan Iron and Steel, [email protected]

Zhengyi Jiang

University of Wollongong, [email protected]

Follow this and additional works at: https://ro.uow.edu.au/eispapers1

Part of the Engineering Commons, and the Science and Technology Studies Commons

Recommended Citation

Ma, Xiaoguang; Zhao, Jingwei; Du, Wei; Zhang, Xin; and Jiang, Zhengyi, "Analysis of surface roughness

evolution of ferritic stainless steel using crystal plasticity finite element method" (2019). Faculty of

Engineering and Information Sciences - Papers: Part B. 2944.

https://ro.uow.edu.au/eispapers1/2944

Research Online is the open access institutional repository for the University of Wollongong. For further information

contact the UOW Library: [email protected]

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plasticity finite element method

Abstract

In order to evaluate the surface quality of ferritic stainless steel (FSS) sheets tensile deformation, a

crystal plasticity (CP) model, in which the constitutive laws were incorporated with the consideration of

the heterogeneous distribution of the properties of grains, was established to analyse the effect of

texture, grain sizes and initial surface roughness on the surface roughness evolution of FSS sheets. The

electron backscatter diffraction (EBSD) tests were performed to characterise the texture and the grains. A

tensile test of the represent volume was simulated and further verified by experimental results. The

numerical simulation results indicate that the surface roughness is dependent almost linearly on the

average grain size. The {001}(110) and the {112}(110) components induce remarkable undulation on the

surface of FSS sheets during uniaxial tension. The surface topology of FSS sheets after tensile

deformation are obtained using 3D laser scanning microscope, which shows an agreement with the

simulated results.

Disciplines

Engineering | Science and Technology Studies

Publication Details

Ma, X., Zhao, J., Du, W., Zhang, X. & Jiang, Z. (2019). Analysis of surface roughness evolution of ferritic

stainless steel using crystal plasticity finite element method. Journal of Materials Research and

Technology, 8 (3), 3175-3187.

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j materres technol.2019;8(3):3175–3187

w w w . j m r t . c o m . b r

Availableonlineatwww.sciencedirect.com

Original

Article

Analysis

of

surface

roughness

evolution

of

ferritic

stainless

steel

using

crystal

plasticity

ﬁnite

element

method

Xiaoguang

Ma

a,∗

_,

_Jingwei

_Zhao

a,∗

_,

_Wei

_Du

b

_,

_Xin

_Zhang

b

_,

_Zhengyi

_Jiang

a,∗

a_School_of_Mechanical,_Materials,_Mechatronic_and_Biomedical_Engineering,_University_of_Wollongong,_Wollongong,_NSW_2522,_Australia b_Stainless_Steel_Technical_Centre,_Baosteel_Research_Institute_(R&D_Centre),_Baoshan_Iron_&_Steel_Co.,_Ltd.,_Shanghai_200431,_China

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received21September2018 Accepted27March2019 Availableonline21May2019

Keywords:

Ferriticstainlesssteel Surfaceroughnessevolution Surfacetopography Crystalplasticitymodel.

a

b

s

t

r

a

c

t

Inordertoevaluatethesurfacequalityofferriticstainlesssteel(FSS)sheetstensile defor-mation,acrystalplasticity(CP)model,inwhichtheconstitutivelawswereincorporated withtheconsiderationoftheheterogeneousdistributionofthepropertiesofgrains,was establishedtoanalysetheeffectoftexture,grainsizesandinitialsurfaceroughnessonthe surfaceroughnessevolutionofFSSsheets.Theelectronbackscatterdiffraction(EBSD)tests wereperformedtocharacterisethetextureandthegrains.Atensiletestoftherepresent volumewassimulatedandfurtherveriﬁedbyexperimentalresults.Thenumerical simula-tionresultsindicatethatthesurfaceroughnessisdependentalmostlinearlyontheaverage grainsize.The{001}110andthe{112}110componentsinduceremarkableundulation onthesurfaceofFSSsheetsduringuniaxialtension.ThesurfacetopologyofFSSsheets aftertensiledeformationareobtainedusing3Dlaserscanningmicroscope,whichshowsan agreementwiththesimulatedresults.

1. Introduction

For ferriticstainlesssteels(FSSs),surface rougheningis an undesirablephenomenoninmanufacturingengineering. Dur-ing the forming process, the FSS sheets show profound undulationsatdifferentpositionswithoutﬂuctuationsinthe thicknessdirection. Theincreasedsurfaceroughness dete-riorates the appearance of the formed product, and even affectsthesurfacepropertiessuchasthereﬂectivity,lubricant transport, weldability, adhesionand mechanicalproperties due to the strain localisation [1]. In order to reduce the

∗ _{Corresponding}_authors.

E-mails:[email protected](X.Ma),[email protected](J.Zhao),[email protected](Z.Jiang).

surface roughness and improve the surface quality of the formedproducts, thesurfaceroughening phenomenonhas beenextensivelystudiedbyresearchersusingbothnumerical simulationandexperimentalmethods.Takechietal.[2] pro-posedanumericalmodelwithcrystalplasticity(CP)theories, andthe effectsofdifferentshearstrainsbetweenRD//110 ﬁbreswereinvestigatedbasedonbothnumericaland exper-imental results.Wright [3] studiedthe plastic–strain ratios between the {111}112matrix and the {001}110 band.It wasfoundthatthedifferentplasticstrainratiosbetweenthe

{111}112andthe{001}110componentscontributedtothe

https://doi.org/10.1016/j.jmrt.2019.03.017

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surfacerougheningduringtheformingprocess.Shinetal.[4]

investigatedthesurfaceroughening evolution ofFSSwhen subjected totensile deformation and foundthat the lower plasticstrainratioofthe{001}110componentsanddifferent sheardeformationsofthe {111}110or {112}110 compo-nentscontributedtothedevelopmentofsurfaceroughening ofFSSsheets.Anewfiniteelementmethod(FEM)was pro-posedbyKnutsen andWittridge [5]inordertoaccountfor theoccurrenceofparallelcorrugationsonthesurfaceofFSS duringuniaxialtensilestraining.Theresultsindicatethat par-allelsurfacecorrugationprofilewasmainlydependentonthe local anisotropy in plastic behaviour. Jung et al. [6] evalu-atedtheeffectofcoarseprecipitatesonsurfaceroughening of aluminiumalloys using crystal plasticity finite element method(CPFEM).Thenumericalsimulationresultsshowthat theprecipitatesactdifferentlydependingonthespatial dis-tributionofkinematicallyweakandstrongorientations.Shi et al.[7] quantitativelyassociatedsurface roughening with singleCPtheoriesandspatialgrainorientationdistribution, andfoundthatthebandingoftheCubeandtheGoss compo-nentsareresponsiblefortheprofoundsurfacerougheningof aluminiumalloyAA6xxx.Anewsurfacerougheningmodel wasdevelopedbyEngler[8]tostudytheeffectof recrystallisa-tiontextureorientationsonsurfacerougheningevolution.The band-likeclustersofgrainswithsimilarcrystalorientations wasfoundtopromoteundulationson thesurface.In addi-tion,Huhetal.[9]proposedaCPFEMinordertoquantitatively describethe surface roughingofFSS430 aftertensile test. Bothsimulatedand experimentalresultsindicatedthatthe outofplaneshearandspatialvariationsinthrough-thickness strainswerethemostcommoncausesoftheobserved ridg-ingbehaviour.Withdifferentplastic–strainratiosandTaylor factors,theneighbouringgrainsinpolycrystallinematerials tendtoshowdifferentplasticdeformationsaftertensionor deep-drawn,leadingtoremarkablesurfacecorrugationsand surfaceroughening.

Exceptfortexture,itiscommonlyacknowledgedthatthe grain size also affects the surface roughness after plastic deformation[10–13].StoudtandRicker[10]investigatedthe relationshipbetweenthegrainsizeandthesurface roughen-ingbehaviourofAl–Mgalloys,andstatedthattheroughening ratewasdependentonthegrainsize.Inaddition,Stoudtetal.

[11]studiedthecorrelationbetweendeformation-induced sur-faceroughnessandplasticstraininAA5754.Itwasfoundthat theﬁnegrainsizeproducedalinearrelationshipbetweenthe surfaceroughnessandplasticstrain.Theevolutionofsurface roughnessduringdeformationofpolycrystallinealuminium alloyswassystematicallystudiedbyWoutersetal.[12]using aheight–heightcorrelationtechnique.Theresultsindicated thattheobservedroughnesswasinducedbyacombination ofself-afﬁnerougheningonasubgrainscaleandagrainscale rougheningcausedbyorientationdifferencesbetween neigh-bouringgrains.Alinearrelationshipbetweentheroot-mean squaresurfaceroughnessandgrain sizewasalsofoundin experiments.Theeffectofaveragegrainsizeonsurface rough-nessevolutionduringtensiontestwasanalysedbyMaetal.

[13]usingbothnumericalsimulationandexperimental meth-ods.Thecoarsergrainsizewasfoundtoinduceremarkable undulationsonthesurfaceofFSSaftertensionperformance. Asthegrainsizeincreased,theirregularitygeneratedfrom

rotationandslipofgrainscanbedominant,inducingthe sur-facedefectionsontheformedproducts.

Inordertoimprovethesurfacequalityoftheformed prod-ucts, understanding of the surface roughness evolution is consideredtobecriticalindevelopingstrategiestoenhance theresistanceagainstthesurfaceroughening.Lefebvreetal.

[14]evaluatedtheeffectofthespatialdistributionof crystallo-graphicorientationsonropingamplitudeandwavelengthin FSSusingtheviscoplasticfastFouriertransformmodel.Saleh etal.[15]analysedthemicrostructureandtextureevolution inatwinning-induced-plasticitysteelduringuniaxialtension withproposedvisco-plasticself-consistentmodel,andfound that the effectofthe perfect slipwas dominantcompared withtwinningonthe textureformation.Amovingwindow approachwasproposedbyQinetal.[16]tounderstandthe macroscopicsurfacerougheningbasedonthespatial distri-butionofgrainsbelongingtospeciﬁcindividualorientations. It was stated that the difference in the r-value between neighbouring grainscontributed tothesurfaceroughening. Astep-by-stepmethodwasproposedbyRomanovaetal.[17]

toevaluatethesurfacehardeningeffectonthe deformation-inducedrougheninginpolycrystals.Theauthorsstatedthat thepresenceofthesurface-hardenedlayerinpolyscrystalline structurecausedadelayinthedevelopmentofsurface rough-nessandreduceditsgrowthrate.

Although a number ofmodels were built to investigate thesurfacerougheningevolutionofpolycrystallinematerials aftertensiledeformation,theeffectofinitialsurface rough-nessandorientationdistributionarenotyetcomprehensively investigated.Inthisstudy,boththemeasuredinitialsurface profileandthekeytexturecomponentsareincorporatedinto theFEmodellingtoenhancetheaccuracyofsimulatedresults. Inordertofurtherinvestigatetherelationshipbetweenthe surfacerougheningandgrainpropertiesofFSSsheets,a crys-talplasticityfiniteelementmodelisestablishedandadopted tosimulatetheevolutionofthesurfaceroughnessofFSS430 duringuniaxialtensiletests.Allthesimulatedresultsare ver-ifiedbytheuniaxialtensiletests.

2. FE

simulation

models

2.1. Crystalplasticitytheory

Thelocaliseddeformationinductilesinglecrystalswas anal-ysedbyPierceetal.[18].Itiscommonlyacknowledgedthatthe plasticdeformationofpolycrystallinematerialsisaffectedby acombinationofcrystallographicdislocationslip,twinning, diffusionandgrainboundarysliding.Incomparisontoslip, however,theeffectoftwinning,diffusionandgrainboundary slidingonplasticstraincanbenegligible.Forthisreason,the totaldeformationgradientcanbedescribedbythe combina-tionofelasticandplasticcomponents:

F=Fe_·_Fp ₍₁₎

whereFe_represents_the_plastic_shear_of_the_material,_and_Fp

meansthestretchingandrotationofthelattice.Itisassumed thattheelasticpropertiescannotbeaffectedbyslip,thestress,

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j mater res technol.2019;8(3):3175–3187

3177

therefore,isdeterminedbyFp_._The_velocity_gradient_in_this

stateis:

Lp=FpFp−1=

˛S˛₀ (2)

S˛₀=m˛⊗s˛ (3)

where˛ _is_the_slipping_rate_of_the_˛_slip_system,_and_S˛ 0 is

theSchmidtensor.Inallactivatedslipsystems,m˛_and_s˛_are

theslipdirectionandthenormaltoslipplaneonthe˛thslip system,respectively.Theslipdirectionandthenormaltothe slipplaneoftheslipsystem˛aredeﬁnedas:

m∗˛=Fem˛ (4)

s∗˛=((Fe)−1)Ts˛ (5)

For rate-sensitive slip systems, the slip rates can be describedasfunctionsofplasticshearingrateandresolved

shearstressesbasedonSchmidlaws[19].Theshearingrateis displayedas: ˛₌˛ 0 ˛ 0 g˛

˛ 0 g˛

(1/m)−1 (6)

where₀˛,g˛and₀˛refertothereferenceshearrate,slip resis-tanceandresolvedshearstresson˛slipsystem,respectively. Itisassumedthatthecurrentstrainhardenedstateofcrystal isonlydependentonsipstrain.Theselfandlatenthardening functiong˛_is_{characterised}_by_the_strengths_as_follows:

g˛₌

ˇh

˛ˇ

ˇ

₍₇₎

h˛ˇ₌_q˛ˇ_hˇ ₍₈₎

whereh˛ˇ_are_the_slip_hardening_moduli_of_active_slip_systems.

h˛ˇ=

q+ (1−q) ı˛ˇ

hˇ (9)

Fig.1–Voronoistructuresandgrainmodels:(a)VoronoimodelsgeneratedwithMATLAB,(b)meshofgrainmodelwith

idealﬂatsurface,(c)theinitialsurfacetopographyobservedusingmicroscope,and(d)themodelwiththeconsiderationof

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where q denotes the latent hardening parameter, and hˇ

denotesthehardeningrate.Thehˇ_can_be_calculated_with_the

followingequation: hˇ=h0

1−g ˇ s

˛ (10)

whereh0,sand˛aretheslipsystemhardeningparameters

usedinthisstudy.h0denotestheinitialhardeningmodulus,s

denotesthesaturationstresswherelargeplasticﬂowinitiates and˛representsthereferencestrainrate.Alltheequations havebeen implementedinto a user subroutineUMAT,and theprogrammeisimportedintotheFEsoftwareABAQUSfor furthercalculation.

2.2. Voronoimodels

Inordertoestablishgrainmodelswithdesiredshapesand sizes,voronoitessellationisutilisedtodescribethe proper-tiesofpolycrystallineaggregates.Beinggivenasetofpoints

E={Gi(xi)}intoaregionofa3DspaceD∈R3,anarbitrarypoint

PwithintheareaR(i)isclosertopointithananyotherpoint, andeachpointGi formsaVoronoipolyhedronbasedonthe

abovementionedlaws[13].ThepointGi isassociatedwith

polyhedronCias: Ci=

P (X) ∈D|d (P,Gi) <d (P, ) Gj∀j /=i

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whereCi istheVoronoipolyhedron,the normcorresponds

with Euclidean distance.In grainmodelling, the generated polyhedronscorrespondtotherecrystallised,equixedgrains. EachVoronoicellwasassignedwithaspeciﬁccrystal orienta-tionanduniqueresponsetodeformation.Consequently,the surfacerougheningofestablishedmodelcanbeaffectedbyall grains.

Four surfaceroughness parameters,including the arith-meticaverageroughness(Ra),therootmeansquareroughness

(Rq),theskewness(Rsk)andthekurtosisroughness(Rku),are

utilised inthe present studyto characterisethe 3D rough-nesstopographyonthesurfaceofFSS.Theaboveroughness parameterscanbeexpressedby:

Ra=

Si

Zi−

s_iZ_i

s_i

Si (12) Rq=

Si

Zi−

s_iZ_i

s_i

2

Si (13) Rsk= 1 Rq3

⎡

⎢

⎣

Si

Zi−

siZi

s_i

3

Si

⎤

⎥

⎦

(14) Fig.2–ODFsofFSS430.

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3179

Fig.3–Thestrain–stresscurvesoftensiletestsand

numericalsimulation. Rku= 1 R4 q

⎡

⎢

⎣

Si

Zi−

siZi

s_i

4

Si

⎤

⎥

⎦

(15)

whereZi istheaverageasperityofheightofaVoronoicell,

Si isthe sample area, and Si is individual Voronoi cell.

Inthisstudy,the surfacetopologywasmeasuredusingthe KeyenceVK-X100-3DLaserScanningMicroscope.Threetests wererepeatedforeachgroupofexperimentsandthemean value was calculated and utilised to evaluate the surface roughness.

TheestablishedVoronoistructuresandgrainmodelsare showninFig.1.Bothaveragegrainsizeandcrystalorientation distributionfunction (ODF) were considered whenbuilding theFEmesh.Arepresentativevolumeofasolidrectangular

Table1–Keyparametersusedfornumericalsimulation.

Materialproperties Valuesusedinsimulation Young’smodulus E=1.94×105_MPa

Poissonratio =0.29 Strainratesensitivity m=0.02 Initialvalueofslipresistance

parameter

g0=110MPa Referencestrainrate ˛=0.0001 Initialhardeningmodulus h0=1700MPa StageIstress g=252MPa Latenthardeningparameter q=1.4

Table2–Roughnessparametersindifferentcases.

Case Ra(␮m) Rq(␮m) Rsk(␮m) Rku(␮m)

1 0 0 0 0

2 0.51 0.70 1.53 10.57

3 0.82 1.07 −0.38 3.86

4 0.97 1.25 −0.04 3.70

wasformedtoevaluatethesurfacetopographyofFSS.The modelcontained100–1000Voronoipolyhedronswithan arbi-traryshapegeneratedbycommercialsoftwareMATLAB.The grainsweremeshedwithfour-nodetetrahedralelementswith linearinterpolationandfullintegration(C3D4elements).

Inordertoenhancetheaccuracyofthenumerical simu-lation,crystallographicorientationdistributionisconsidered in this model. It is commonlyacknowledged that the cor-rugation on the surfaceis induced bythe different plastic deformationsofneighbouringgrainswithdifferentcrystal ori-entations[20–23].Themajortexturecomponents, including the{001}110,Cube,Gossand{112}110components,have beenassignedtothegrainsinestablishedmodel.InABAQUS usersubroutine,thecrystalorientationiscalculatedbasedon thematrixbelow:

g=

⎛

⎝

coscos−sinsincos sincos+cossincos sinsin −cos sin−sincoscos −sinsin+coscoscos cossin

sin sin −cos sin cos

⎞

⎠

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Fig.5–Surfacetopographiesoftheestablishedmodelwithdifferentinitialsurfaceroughness:(a,c,e,g)theinitialsurface

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Fig.6–TherelationshipbetweenR2

qfand(R 2 q0+R2qd).

whereϕ, and are the Eulerangles ineach grain. To describethemicrostructureinproposedmodel,thefractions ofmajortexturecomponentsofFSS430isobtainedfromEBSD tests.Fig. 2shows theODFs ofFSS430. Inthe established model,thecrystalorientationsareassignedtoallgrainsbased onthefractionsofmajortexturecomponentsobtainedfrom EBSDtests.

2.3. Boundaryconditions

Inordertoassessthesurfaceroughnessevolutionduring plas-ticdeformation,aseriesoftensiletestsweresimulatedalong RD.AsshowninFig.1(b),axesX,YandZrepresentRD,TD andNDoftheFSSspecimen,respectively.ThesurfacesX1,Y0 andZ1aredisplayedinFig.1(b)andtheoppositesurfacesare deﬁnedtobeX0,Y1andZ0.ThemodelisstretchedalongRD andthetopsurfaceZ1isusedforsurfacetopography observa-tion.Theperiodicboundaryconditionsareappliedonsurfaces X0,X1,Y0andY1.Fivedifferentaveragegrainsizesareutilised inthismodelsothattheeffectoftheaveragegrainsizecan beanalysed.Finally, twostepsare appliedinthis modelto ensurethe accuracyof results.In step1, anominal strain ε=0.003isassigned.Instep2,aspeciﬁcdisplacementisset alongtheRD.50incrementspersimulationstepwereusedin thismodeltoachievetheconvergenceandprovidenumerical stability.

FSS430isutilisedinthepresentstudy.24slipsystemsare assumedtobeactivatedduringtheformationprocess.Each grainwasassignedwithaspeciﬁcorientationandthewhole modelwasmeshedwithC3D4(3-D,four-node,Tet)elements. Theprogramrun oncomputer withthree microprocessors inparallel,over10000stepswererequiredtoreachthe con-vergence.Fig.3showsthestress-straincurvesobtainedfrom numerical simulation and experiments. Theresults match wellwitheachother,indicatingthattheestablishedmodelcan beutilisedtopredictthesurfaceroughnessevolutionofFSS 430.Thekeyparametersinproposedmodelaresummarised inTable1.

3. Results

and

discussion

3.1. Effectsofinitialsurfaceroughness

During theplasticdeformationofpolycrystalline materials, the initialsurfaceroughness isconsidered tobecritical in determiningthesurfacetopography.Toevaluatetheeffectof theinitialsurfaceroughnessonsurfaceroughnessevolution duringdeformation,aseriesofmodelswithdifferentinitial surfaceroughnessaresimulated.Theroughnessparameters indifferentcasesarelistedinTable2.Theotherkey parame-tersremainthesameinthisgroup:(i)theaveragegrainsizeof grainsis10␮m,and(ii)themodelisstretchedalongRDwith 20%extension.

Fig.4showstherelationshipbetweenthesurface rough-nessandstrainalongRD.Thevariationofsurfaceroughness Raand Rqaredeﬁnedasthesurfaceroughnessvalueafter

plasticdeformationminustheinitialsurfaceroughness.As canbeseeninFig.4(a)and(b),aquadraticcorrelationbetween Ra( Rq)andtensionstrainisfoundtodescribethesurface

roughnessevolutionofFSS430aftertensionperformance.For specimenswithsignificantinitialsurfaceroughness,the vari-ationsofsurfaceroughnessaresmaller.Itisknownthatthe surfaceroughnessevolutioncanbeattributedtothe flatten-ingandthemovementofgrainsduringuniaxialtension.For specimenswithlargeinitialsurfaceroughness,theeffectof flatteningonsurfaceroughnessevolutioncanbedominant, causinglimitedincreaseofsurfaceroughnessduringtension performance[24].Ontheotherhand,thesurfaceroughness evolutionofspecimenswithflatinitialsurfaceissignificantly affected bythe deformation ofindividual grains, therefore leadstotheremarkableincreaseofthesurfaceroughnessafter extension[13].

Fig.5showsthe surfacetopographiesoftheestablished model with different initialsurface roughness (cases 1–4). It can be seen from Fig. 5(a) and (b) that even for spec-imens with ideal flat plane, the final surface topography can be inhomogeneous after tension performance due to the movement oflattice. Withdifferent plasticstrain ratio and plastic deformation of neighbouring grains, the mis-orientation between neighbouring grains is considered to be responsible for the surface roughness development of FSS during uniaxial tension [25–27]. Therefore, the evolu-tionofsurfaceroughnessduringtensiledeformationcanbe representedbythehomogeneousflatteningandthe inhomo-geneouslatticemovement.

To further evaluate the surface roughness evolution of FSS430duringtensiledeformation,amathematiccalculation wasmadetoquantitativelydescribethesurfaceroughness. ForhomogeneousspecimensstretchedalongRD,thesurface roughnesscanbeexpressedas:

Raf= c1f+c2f+···+cnf n = (c01+cd1)+ (c02+cd2)+···+ (c0n+cdn) n =Ra0+Rad (17)

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Fig.7–Thesurfacetopographiesofestablishedmodelswithdifferentaveragegrainsizes:(a)10␮m,(b)12␮m,(c)14␮m,(d)

16␮mand(e)18␮m,and(f)thefractionoftexturecomponentsindifferentmodels.

Rqf =

c2 1f+c22f+···+c2nf n =

(c01+cd1)2+ (c02+cd2)2+···+ (c0n+cdn)2 n

R2 q0+R2qd+2 c01∗cd1+c02∗cd2+···+c0n∗cdn n ≈ki∗

R2_q0+R2_qd (18)

whereki isa coefﬁcient.Therelationship betweenR2_qf and

(R2

q0+R2qd)isshowninFig.6.Ascanbeseenfromthecurve,

kiequalsto1.0176inthepresentwork.BasedonEqs.(17)and

(18),theﬁnalsurfaceroughness(RafandRqfcanbecalculated

bytheinitialsurfaceroughness(Ra0andRq0)andthedifference

ofsurfaceroughness(RadandRqd).Forhomogeneousmaterial

sheetstretchedalongRD,thedatumplanecoefﬁcientsofthe homogeneousdeformationsurfacecanbeexpressedas[27]:

Rad=

cd1+cd2+···+cdn

n =e

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3183

Fig.8–Therelationbetweentheaveragegrainsizeandthesurfaceroughnesswiththeextensionof20%:(a)Raand(b)Rq.

Rqd=

c2

1d+c22d+···+c2nd

n =e−εxRq0 (20)

whereisthePoisson’sratio.CombiningEqs.(18)and(20),the finalsurfaceroughnesscanbecalculatedbytheinitialsurface roughness.Itcanthereforebeconcludedthattheinitial sur-faceroughnessplaysasignificantroleindeterminingthefinal surfaceroughnessofFSSafterformingprocess.

3.2. Effectsofgrainsize

Thesurfacedeformationofpolycrystallinematerialscanbe described by the geometric non-uniform surface deforma-tion and the anisotropic mechanical deformation. For FSS, crystalsundergorotationandmovementduringloading, caus-ingthe irregularityonthe surface[18].During theforming process, signiﬁcant variations in the amount of localised deformationcan beproduced withinthe individual grains. Theoverlapandgapregionsinducedbythevaryingamounts oflocalised slip are accounted for through rearrangement ofstored geometrically necessary dislocations[10]. Conse-quently,theneighbouringgrainswithdifferentorientations exhibit different deformation during tension performance, causingcorrugationsonthesurfaceofFSSsheets.The defor-mationofanindividualgrainissigniﬁcantlydependenton grain sizeand orientationduring loading. Thegrain sizes, therefore,playanimportantroleinsurfaceroughness evo-lutionunderthesubsequentformingprocess.

Agroupofmodelswithdifferentaveragegrainsizesare simulatedtoexploretheeffectoftheaveragegrainsizeonthe surfaceroughnessdevelopmentofFSS.TheresultsofEBSD testsindicatethat theaveragegrainsizeofcold-rolled FSS 430rangesfrom10to18␮m,theaveragegrainsizesin mod-els,therefore,areconsideredtoberangingfrom10to18␮m. Theinitialsurfaceroughnessisproposedtobezeroinallthe modelsinthisgroup.Inaddition,allthemodelsareassigned thesamefractionsoftexturecomponentsandﬁvemodelsare simulatedinthepresentstudy(asshowninFig.7).

Table3–Valuesofsurfaceroughnessparameters

obtainedafterregressionanalysis.

Parameters k1s C1 k2s C2

Values 0.106 0.086 0.34 0.12

Therelationshipbetweenthesurfaceroughnessandthe averagegrainsizewiththeextensionof20%isshowninFig.8. Forthecold-rolledandannealedFSS430specimens,equixed grainsareformedafterrecystallisation.Theaveragegrainsize, therefore,canbeutilisedtodescribethesizeofgrainsinFSS. Ascanbeseenfromthecurve,alinearrelationshipbetween thesurfaceroughnessandtheaveragegrainsizeisobtained, whichcanbeexpressedas:

Raf0=(k1s·D+C1)∗ε2x (21)

Rqf0=(k2s·D+C2)∗ε2x (22)

whereDistheaveragegrainsizeofFSS430,andCisa con-stant.Theslopeofthecurveisdeﬁnedasks.Thevaluesofks

andCareshowninTable3.BasedonEqs.(21)and(22),the sur-faceroughnessaftertensiledeformationcanbeevaluatedin combinationwiththeaveragegrainsizeofcold-rolledFSS430. Thegrainreﬁnement,therefore,isconsideredtobeeffective toreducethesurfaceroughnessofFSS430duringtheforming process.

3.3. Effectsoftextures

Duringtheformingprocess,thedeformationofgrainsofFSS canbesigniﬁcantlyaffectedbythecrystallographic orienta-tions[1–3].Toanalysetheeffectsoftexturesonthesurface roughnessevolution, grainswithsixdifferenttexture com-ponents aresimulatedafter tensiledeformation (asshown in Fig.9). Theresultsindicate that the strain distributions andthedeformationofgrainsare affectedbytextures.The strainsarehomogeneouslydistributedinmodels,thismeans thatthesurfaceroughnessafterplasticdeformationiscaused

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Fig.9–Thedeformedandundeformedshapesandstraindistributionsofmodelswithdifferenttextures:(a)brass,(b)cube,

(c)copper,(d)Goss,(e){112}110,and(f){001}110.

bythedifferentialofcrystalorientationsbetween neighbour-inggrains.Forgrainswiththesameorientations,thestrain localisation is eliminated on the free surface after tensile deformation.Thus,thehomogeneityofcrystallographic ori-entationisconsideredtohaveasigniﬁcantinﬂuenceonthe surfaceroughnessevolutionofFSSaftertensiledeformation. Aftertensile deformation, differenttexture components caninducevariousyieldingbasedonHall–Petcheffect[13].

Thus, afurtherdiscussionismadetoidentifytheeffectof majortexturesonthesurfaceroughnessevolution.Asshown inFig.9(a),signiﬁcantshrinkcanbeobservedonthesurfaceof grainswiththe{001}110component.Comparedwithother typicalBCCtexturecomponents,{001}110orientationshows theminimumstoredenergyandlowerplasticresistance[28]. Asaconsequence,thedeformationofthe{001}110 compo-nentcanbecritical,causingirregularityonthesurfaceofFSS.

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Table4–CalculatedR-valueoftexturecomponents.

Texturecomponent R-value(RD) r-value(45◦) r-value(TD)

{001}110 0.00 1.00 0.00

{011}100 1.00 0.40 Inﬁnite

{001}100 1.00 0.00 1.00

{112}110 0.11 6.00 1.00

Copper 1.00 5.99 0.11

Giventhatthedeformationofgrainswiththe{001}110 com-ponentismuchlargerthanthatofadjacentgrains,remarkable corrugationisgeneratedonthesurfaceofFSSduringthe form-ingprocess.Unlikethe{001}110components,theeffectsof theGossandtheCubecomponentsonsurfaceroughness evo-lutionarequitesmall(asshowninFig.9(e)and(f)).TheGoss grainsaremainlygeneratedcombiningwiththeshearbands thatdevelopedinthelatestagesoftherollingprocesses[29], andtheformationofshearbandsmayleadtoshear deforma-tion,preventingtheformationofcorrugationonthesurfaceof FSS.Inordertoevaluatetheplasticresistanceoftexture com-ponents,the planestrainratio (R-value) ofcrystallographic orientations was calculated. Table 4 shows the R-value of majortexturecomponents. Itcanbeseen thattheR-value

ofthe{001}110and{112}110componentsstretchedalong RDislower than thatofother majorcomponents, causing differentplasticdeformationsofgrainswithdifferentcrystal orientations.Itcanthereforebeconcludedthatthe{001}110

and{112}110componentscontributetothesurface rough-eningofFSS.

For evaluating the plastic resistance of texture compo-nents,the planestrainratioofcrystalline orientationswas calculated.Theresultsindicatethatthestrainratioofmajor

Table5–Surfaceroughnessandaveragegrainsizeof

thespecimensfortensiletests.

Sample Cold-rolledFSS430 Initialsurfaceroughness(Ra/␮m) 0.82

Initialsurfaceroughness(Rq/␮m) 1.07

Initialsurfaceroughness(Rsk/␮m) −0.38

Initialsurfaceroughness(Rku/␮m) 3.86

Averagegrainsize(␮m) 14.2

texturecomponentsstretchedalongRDislowerthanthatof othermajorcomponents,inducingdifferentplastic deforma-tionsofgrainswithdifferentcrystalorientations.Thetexture components,therefore,contributetothesurfaceroughening ofFSS.

4. Surface

roughness

of

FSS

after

tension

performance

TheuniaxialtensiletestswereperformedusingINSTRON ten-siletestingmachineandthesampleswerestretchedalongRD withthestrainrateof6.6×10−4s−1.Thetensiletestsfollow ASTM-E8MStandard[30],andthedimensionsofthetensile specimensareshowninFig.10.Theinitialsurfaceroughness andtheaveragegrainsizeofthespecimensweremeasured beforetensiletests,andtheresultsarelistedinTable5.

Acomparisonwasmadebetweentheresultsfrom numer-icalsimulationand practicaltensiletestsinorder toverify the conclusions made inprevious section. Combining Eqs.

Fig.10–Dimensionsofthespecimenusedfortensiletests.

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Fig.12–Evolutionofsurfacetopographyofspecimen:(a)withouttensiledeformation,and(b)after20%extension.

(17)–(22),aquantitativedescriptionofthesurfaceroughness aftertensiletestinRDisobtained,whichcanbeexpressedas:

Raf = (k1s∗D+C) εx2+e−εx∗Rq0 (23) Rqf =

(k2s∗D+C) ε2x

2 +[ki

e−εx∗_R_q0

_]2 ₍₂₄₎

BasedonEqs.(23)and(24),thesurfaceroughnessofFSS430 aftertensiletestscanbeevaluated.Fig.11comparesthevalues ofsurfaceroughnessthatare obtainedbynumerical simu-lationand experimentaltests.Inaddition, the evolutionof surfacetopographyisshowninFig.12.Itcanbeseenfromthe curvesthatthesimulatedvaluesofsurfaceroughnessmatch wellwiththeexperimentalresults.Theproposedmodeland equations,therefore,canbeutilisedtoevaluatethesurface roughnessofFSS430afteruniaxialtensiletests.

5. Conclusions

Inthiswork,CPFEMhasbeenutilisedtoevaluatethesurface roughnessofFSS430aftertensile deformationalongRD.A quantitativeequationwasproposedtoevaluatethesurface roughnessofFSS430, andthe effectsofthe initialsurface roughness,averagegrainsizeandtexturecomponentsonthe

surfacerougheningarenumericallyandexperimentally inves-tigated.Severalconclusionscanbedrawnasfollows: (1) Theﬁnalsurfacetopographyaftertensiletestsismainly

dependent on the homogeneous deformation and the inhomogeneous strain localisation. Thecorrugation on the surface of polycrystalline materials is signiﬁcantly affected by the misorientations between neighbouring grains.

(2) A quantitative description of surface roughness evolu-tionduringtensionperformanceisestablished.Basedon theinitialsurfaceroughness,extensionandtheaverage grainsize,theﬁnalsurfaceroughnessofFSS430after ten-siledeformationcanbeevaluatedbasedontheproposed model.Theexperimentalresultmatcheswellwiththe pre-dictedvalues,indicatingthatthemodelcanbeutilisedto evaluatethesurfaceroughnessafteruniaxialtension. (3) Therelationshipbetweenthe grainsizes,initialsurface

roughness,texturecomponentsandﬁnalsurface rough-nessareanalysedbasedonproposedmodel.Aquadratic correlationbetween Ra( Rq)andtensionstrainisfound

to describethe surface roughness evolutionof FSS430 withidealflatsurface.Inaddition,theaveragegrainsize is found to be in the order of the surface roughness. Theresultsalsoindicatedthattherefinementoftexture benefitsthefinalsurfacetopography.Therefinementof microstructure,therefore, isconsidered tobecriticalin

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3187

improvingthesurfacequalityofFSS430duringthe form-ingprocess.

Conﬂicts

of

interest

Theauthorsdeclarenoconﬂictsofinterest.

Acknowledgements

Theﬁrstauthorwouldliketothanktheﬁnancialsupportby IPTAscholarshipfromUniversityofWollongongforhisPhD study.ThestudyisalsosupportedbyBaosteel-AustraliaJoint ResearchandDevelopmentCentre(BAJC).

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