Microstructure and Property Modifications of Cold Rolled IF Steel by Local Laser Annealing

17 Pages·2017·2.28 MB·English

by  Håkan Hallberg

Microstructure and Property Modifications of Cold
Rolled IF Steel by Local Laser Annealing
HÅKAN HALLBERG, FRÉDÉRIC ADAMSKI, SARAH BAÏZ,
and OLIVIER CASTELNAU
Laser annealing experiments are performed on cold rolled IF steel whereby highly localized
microstructureandpropertymodificationareachieved.Themicrostructureisseentodevelopby
strongly heterogeneous recrystallization to provide steep gradients, across the submillimeter
scale,ofgrainsizeandcrystallographictexture.Hardnessmappingbymicroindentationisused
to reveal the corresponding gradients in macroscopic properties. A 2D level set model of the
microstructure development is established as a tool to further optimize the method and to
investigate,forexample,thedevelopmentofgrainsizevariationsduetothestrongandtransient
thermalgradient.Particularfocusisgiventotheevolutionofthebeneficialγ-fibertextureduring
laser annealing. The simulations indicate that the influence of selective growth based on ani-
sotropic grain boundary properties only has a minor effect on texture evolution compared to
heterogeneous stored energy, temperature variations, and nucleation conditions. It is also
shown that although theα-fiber hasan initial frequency advantage,thehigher probability ofγ-
nucleation, in combination with a higher stored energy driving force in this fiber, promotes a
stronger presence of the γ-fiber as also observed in experiments.
DOI: 10.1007/s11661-017-4210-y
© The Author(s) 2017. This article is an open access publication
I. INTRODUCTION Themethodwillbepossibletoapplyinallareasofsheet
metal modifications, such as deep drawing and folding
W
ITH few exceptions, the present use of laser
as well as stamping, blanking, and punching. It is
annealing is related to modification of the structure
envisaged that the method will permit reduced tool
and properties of thin films. Examples include perfor-
forces and tool wear, and less risk of fracture and that
mance optimization of thin film cathodes in lithium-ion
finer geometrical tolerances can be met.
batteries,[1] improvement of the properties of thin films
By local laser annealing of the material, a grain size
for photovoltaic applications,[2] fabrication of thin film
gradient will develop in the microstructure and alter
diodes,[3] and control of grain growth in thin films.[4]
macroscopic material properties accordingly. As de-
Studies on laser annealing of larger specimens are
scribed by the classical Hall–Petch relation, the macro-
scarcer. One example related to local modification of a
scopic yield stress of the material is related to the
thin Ni-Ti shape memory alloy strip with 0.5 mm
averagegrainsize,andsincethegrainsizecanbealtered
thickness is given in Reference 5 and surface laser
by annealing—in the irradiated zone—laser annealing
annealingofstainlesssteelisstudiedinReferences6and
can be used to modify the yield stress of the material in
7. Even less attention is given to the possibility of using
selectedregions.Thelaserannealingprocessalsocauses
laserannealingonlargersamplesofrolledsheetmetalin
a reduction of the stored energy in the material by
order to manufacture functionally graded materials
growth of new grains from (nearly) dislocation-free
throughlocalmodificationsofthemicrostructure,which
recrystallization nuclei. In addition, annealing also
is not achievable by standard annealing methods. This
provides a characteristic recrystallization texture. In
kind of applications are in focus of the present study.
thecaseoftheinterstitialfree(IF)steel,presentlyunder
consideration,coldrollingyieldsatexturedominatedby
two components comprising the α-fiber, having the
HA˚KAN HALLBERG is with the Division of Solid Mechanics,
h110iorientationsparalleltotherolling direction(RD),
LundUniversity,P.O.Box118,22100Lund,Sweden,andalsowiththe
LabProcessandEngineeringinMechanicsandMaterials(PIMM-UMR and the γ-fiber having the h111i orientations parallel to
8006), ENSAM, CNRS, CNAM, Hesam, 151 Bd de l’Hoˆpital, 75013 the normal direction (ND). In Reference 8, the initial
Paris,France.Contacte-mail:[email protected]´DE´RIC fractions of the α- and γ-fibers after 75 pct cold rolling
ADAMSKI,SARAHBAI¨Z,andOLIVIERCASTELNAUarewiththe
are found to be 41 and 27 pct, respectively, clearly
LabProcessandEngineeringinMechanicsandMaterials(PIMM-UMR
showing the texture to be dominated by these compo-
8006),ENSAM,CNRS,CNAM,Hesam.
ManuscriptsubmittedJanuary11,2017. nents. Similar observations are made in Reference 9
ArticlepublishedonlineJuly17,2017
4786—VOLUME48A,OCTOBER2017 METALLURGICALANDMATERIALSTRANSACTIONSA
where,directlyfollowing90pctcoldrolling,thevolume tallization,[33,34]Areviewofdifferentmethodsformesoscale
fractions of the α- and γ-fibers are found to be 50 and modeling of recrystallization is provided in Reference 35.
40 pct, respectively. The ratio between the fractions is, Regardless of which modeling approach is adopted, it is
however,observedtochangeduringsubsequentanneal- emphasizedin,forexample,Reference36thatinordertobe
ing as the γ-fiber grows at the expense of the α-fiber abletopredictrecrystallizationtexturesinsimulations,both
duringrecrystallization.Theincreasedpresenceoftheγ- a correct description of grain growth as well as a non-
fiber is also observed in References 10 and 11 and is of randomnucleustexturearerequired.Bothoftheseaspects
primary concern in practical applications as it signifi- aretakenintoaccounthere.
cantly enhances the ductility and hence the formability Grain growth in the presence of a thermal gradient
of the material.[12–15] has been studied in numerical models previously, for
The development of the characteristic texture compo- example by Monte Carlo Potts and phase field models
nents has been explained in terms of oriented nucleation relatedtoweldinginReferences37and38,respectively,
and/orselectivegrowthduringrecrystallization,[9,11,13,16–19] andbyMonteCarloPottsmodelingofUO nuclearfuel
2
or as size or frequency advantage for grains with γ- pellets in Reference 39. Similar to welding, the present
orientations.[20] Preferential nucleation of grains with local laser annealing is characterized by a rapidly
orientations belonging to the γ-fiber during annealing of varying and steep transient thermal gradient which is
IF steel is noted in a number of studies, for example in accounted for in the numerical simulations. To provide
References 8, 9, 17, 19 through 22. The conditions under the spatial and temporal temperature variation during
which the recrystallization texture develops are addressed laserannealing,astandardthermalfiniteelementmodel
in the numerical simulations in the present work along of the macroscopic annealing process is considered
with the evolution of grain structure gradients. initially. This permits the temperature gradients in the
In the present work, laser annealing experiments are material to be identified for subsequent employment in
performed on cold rolled IF steel. It is shown that local themicrostructureRVEs,whicharesmallersubdomains
microstructural modifications through recrystallization of the macroscopic specimen. Use of level set-based
andgraingrowthcanbeachievedandrelatedmacroscopic RVEs in such modeling framework has not been
propertiessuchasductilityandhardnesscanbemodified considered previously in the literature.
in confined regions of the material. This makes laser This paper is structured into sections, beginning with
annealing a promising method for manufacturing of Section II, discussing the laser annealing experiments.
functionally graded materials. To further optimize the Toprovidethespatialandtemporaltemperaturehistory
method, additional studies on the correlation between in the workpiece during laser irradiation, the thermal
processing conditions and microstructure evolution are finite element model is introduced in Section III. The
required. This is addressed in the present work by numerical simulations of microstructure evolution are
establishingalevelset-basednumericalmodelinorderto performed using a level set formulation which is
study the key mechanisms behind microstructure evolu- established in Section IV, with subsections providing
tionduringlaserannealing.Themodelcapturesthecentral details of recrystallization and grain growth modeling.
aspects of microstructure development at the mesoscale, The details of the adopted simulation model are
while also considering the transient thermal gradient presented in Section V, followed by a presentation of
conditionswhicharepresentduetothelaserprocessing. the results and a discussion in Section VI. Some
Modeling of rolling and recrystallization texturesin IF concluding remarks close the paper in Section VII.
steel—considering oriented growth relative to oriented
nucleation—is performed using a statistical model in
References11and23.Butsuchanapproachprovideslittle
II. LASER ANNEALING EXPERIMENTS
information on the microstructure morphology or on the
influence of any thermal gradient conditions. As an TheexperimentswereperformedonaTi-stabilizedIF
alternative, a number of different numerical techniques steel sheet which had been cold rolled from an initial
areavailableforperformingmesoscalesimulationsofgrain thickness of 4.5 mm down to a final thickness of
structure evolution in polycrystals. These include, for 1.263mm,resultingina72pctthicknessreduction.The
example,cellularautomataalgorithms,MonteCarloPotts average grain size in the cold rolled material was 59 μm
models, phase field models, and level set formulations. along RD, 8.7 μm along ND, and 31.7 μm along the
Considering applications involving IF steel, Monte Carlo transverse direction (TD), similar to what was found in
PottsmodelsareusedinReferences24and25andacellular Reference 40 for ultralow carbon steel at a correspond-
automataformulationinReference14tostudymicrostruc- ing thickness reduction.
tureevolutioninthismaterial,althoughnotrelatedtolaser Specimens measuring approximately 100 9 100 mm2
annealingandunderisothermalconditions.Inthepresent were taken from the rolled sheet for the experiments. A
work, a mesoscale model of microstructure evolution is Nd-YAGlasersourcewith1.06μmwavelengthwasusedto
formulated in terms of a representative volume element provide a square 12 9 12 mm2 beam cross-section with
(RVE),basedonalevelsetformulationwhichissolvedina uniform “top-hat” distribution of the laser intensity.
finiteelementsetting.Thelevelsetmethodwasintroduced Differentlaserpowersandirradiationtimeswereinvesti-
inReference26andprovidesaframeworkfortracingthe gated in order to find the processing conditions which
evolutionofinterfacesindifferentphysicalsettingsandthe provided appropriate annealing temperatures without
methodhasbeenemployedinstudiesongraingrowth,[27–31] melting the material. For the present study, the results
primarystaticrecrystallization,[32]andondynamicrecrys- obtained with a laser power of P = 320 W and an
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME48A,OCTOBER2017—4787
irradiation time of 9 s are considered. To measure the correspondinggradientsinmacroscopicmaterialprop-
evolution of the thermal field, thermocouples were spot erties were also measured by mapping of the Vickers
weldedtothebottomsurfaceofthespecimen,facingaway hardnessinaregionaroundthefocalpointofthelaser
from the laser beam. The temperaturemeasurements are beambymeansofmicroindentation,cf.Figure4.Itcan
used to calibrate the thermal finite element model in be seen that the laser annealing has resulted in a
SectionIII.Furtherdetailsontheexperimentalprocedure significantdecreaseofhardnessatthecenterofthelaser
and the influence of laser process parameters on the irradiatedzone.LookingattheEBSDmapsinFigure2,
obtained microstructure will be published elsewhere.The obtainedintheRD–TDplaneonthetopsurfaceofthe
microstructureobtainedherewasinvestigatedbeforeand specimen,theareafractionsofthetwodominanttexture
after laser annealing using a combination of optical fibers after annealing can be distinguished. An angular
microscopy,cf.Figure1,andEBSDmapping,cf.Figure2. toleranceof±13.5degaroundtheidealorientationswas
Apronouncedgradientinthegrainsizewasobtained,as used to identify the fiber components. Red and blue
showninFigure1.Figure1alsoshowshowthecoldrolled colorsindicateα-andγ-grains,respectively.Figure2(a)
laminargrainstructureisgraduallyreplacedby arecrys- showsanEBSDmapatpointA,closetothelasercenter,
tallized grain structure of nearly equiaxed grains in the where the α- and γ-fractions are 17.7 and 31.1 pct,
annealedregion.Thegrainsizegradientcanalsobeseenin respectively.Figure2(b)showsanEBSDmapatpointB,
Figure 3, showing the variation of the average grain size locatedfurtherawayfromthelasercenter,wheretheα-
across the laser irradiated zone. It can be noted from and γ-fractions are found to be 20.4 and 29.4 pct,
Figure3thatthegrainsizeisslightlylargeratthecenterof respectively.Itisevidentthattheγ-fiberdominatesover
the annealed zone compared to at the edges of the zone. the α-fiber after recrystallization, as also reported in
This is most likely due to that the temperature has been theliterature,forexampleinReferences9and11.This
higherclosertothelaserbeamcenterwhichhaspermitted trend becomes increasingly pronounced closer to the
more growth of the recrystallized grains than in the less laser beam centerline, i.e., deeper into the region
heatedregionsattheedgesoftheannealingzone.Follow- which has been exposed to the highest annealing tem-
ing the variations in grain size in the workpiece, the peratures.
Laser beam
Workpiece
ND
Point of observation RD 100 μm
Fig. 1—Microstructure in the RD–ND plane at the point indicated. A clear gradient in grain size and structure has developed due to the laser
annealing.
(a) Point A (b) Point B Laser center
Point A
Point B
12 mm
RD RD
TD TD
Fig. 2—Presence of the α-fiber (red) and γ-fiber (blue) at the two locations indicated in the recrystallization zone, after laser annealing. The
EBSD maps are made at the top surface of the specimen, looking down along the ND direction. (a) Point A, located close to the laser center.
(b)PointB,locatedfurtherawayfromthelasercenter(Colorfigureonline).
4788—VOLUME48A,OCTOBER2017 METALLURGICALANDMATERIALSTRANSACTIONSA
III. THERMAL MODEL ergy input from the laser irradiation is modeled as a
surface heat flux q, normal to the surface. The square
l
The temporal and spatial evolution of the thermal
laser beam profile with a uniform intensity distribution
fieldintheworkpieceduetolaserirradiationismodeled
which was used in the experiments, can be modeled by
using the heat equation:
the heat flux
qcT_ þdivðqÞ(cid:2)Q¼0; ½1(cid:3) 8< P
ð1(cid:2)RÞ if r(cid:4)r
where T is the absolute temperature. Further, qlðrÞ¼: r2l l ; ½2(cid:3)
q¼7870 kg/m3 and c¼432 J/kg/K are the mass den- 0 if r>rl
sity and specific heat, respectively. A superposed dot
whereRisthesurfacereflectivity,estimatedtobe0.5inthe
indicates derivation with respect to time. Also found
model,Pthelaserbeampower,andr ¼12mmisthelaser
in Eq. [2], q¼(cid:2)krT denotes the heat flux vector as l
given by Fourier’s law, with k¼88 Wm(cid:2)1K(cid:2)1 being beamwidth.TheformatofthelaserheatfluxinEq.[2]is
also considered in, e.g. References 41 through 43. A
the isotropic thermal conductivity. The quantity Q rep-
schematic illustration of the thermal model is shown in
resents any internal heat source, such as the latent heat
associated with phase transformations.[41] In the pre- Figure 5, also defining the coordinate r. Convective
boundaryconditionsareprescribed alongthesectionof
sent case, however, the material remains in the solid
thetopsurfacewhichisnotexposedtolaserirradiationas
state at all times and it is assumed that Q¼0. The en-
well as along the right and bottom boundaries. The
convectivefluxboundaryconditionsapplynormaltothe
surfaces and have the form q ¼aðT(cid:2)T Þ, where
c 1
45 a¼350 W/m2/K is the convection coefficient and
T ¼298K(25°C)istheambient(room)temperature,
1
m] 40 alsosetastheinitialtemperatureinthesample.Advan-
μ tageistakenoftheverticalsymmetryplane,indicatedin
[ 35
e Figure5.Thedimensionsofthemodelarew¼50mmand
z
si 30 h¼1:263mm,asfortherealspecimen.
n
ai The thermal model is solved in a 2D finite element
gr 25 settingandthetemperatureevolutionatthepointsA–C
ge in Figure 5 is traced throughout the solution. These
a 20
r pointscorrespondtothelocationsofthethermocouples
e
v
A 15 which were used during the experiments. Figure 6(a)
shows the temperatures at the three points as obtained
10 in the simulation (solid lines) compared to the temper-
6 4 2 0 2 4 6
aturesregisteredduringtheexperiment(symbols).Itcan
Distance from laser center [mm] be noted that the temperature rises rapidly during the
initial laser irradiation, and a sharp peak occurs at the
Fig.3—Variationoftheaveragegrainsizeacrossthelaserirradiated point in time when the laser is switched off (after
zone.
9 seconds) and the specimen is allowed to cool.
Laser beam width = 12 mm
HV
220
m 220
m
2 200 200
1
=
180 V] 180
h
H
t
d [
i 160 s 160
w s
e
n
m d
a 140 ar 140
e H
b
120
er 120
s
a
L 100
100 6 3 0 3 6
Distance from center [mm]
Laser beam width = 12 mm
(b)
(a)
Fig.4—(a)MapoftheVickershardness,measuredbymicroindentation,overthespecimensurfaceinaregionsurroundingthelaserbeamcen-
ter.(b)Hardnessvariationalongalinethroughthelasercenter.
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME48A,OCTOBER2017—4789
Considering Figures 3 and 4, it is seen that the most of precision. Hence, it is sufficient to record the
significant influence of the laser annealing is found temperature history at the left and right edges of the
within a region of 4 mm from the laser center, in the individual microstructure RVEs during the thermal
present case. Three locations in the macroscopic spec- simulation, and subsequently use each of these pairs of
imen are considered for the microstructure simulations, time-varyingdatatoobtainthetemperatureatanypoint
indicated by the microstructure representative volume ineachoftheRVEsthroughlinearinterpolationatany
elements (RVEs) shown in Figure 5. The RVEs are time during the annealing process. The temperatures
located along the mid-section of the workpiece at the fromthethermalmodelattheleftandrightedgesofthe
distancesr¼0,r¼2mm,andr¼4mm fromthelaser individual RVEs are shown in Figure 6(b). It can be
beam center, respectively (measured to the left edges of noted that the maximum temperature variation across
the RVEs). Each RVE has the dimensions 800 9 200 any of the RVEs occurs at the end of the irradiation
μm2.Usingdifferentlocationsallowsinvestigationofthe phase, just as the laser is switched off and the cooling
influence of the severity of the thermal gradient and to phase begins.
obtaintheannealedmicrostructureatdifferentdistances
away from the laser center.
Duetothelimitedthicknessofthespecimen,thereare
IV. LEVEL SET MODELING OF
only very small variations in the temperature distribu-
RECRYSTALLIZATION AND GRAIN GROWTH
tion along the normal direction (ND), cf. Figure 5.
Neglecting these small variations, the microstructure A level set approach is taken to modeling of the
RVEs are assumed to be subject to temperature gradi- microstructure evolution and advantage is taken of the
ents alongthe RD direction only. Further,astheRVEs framework previously established in References 31 and
areofverysmalldimensions,thistemperaturevariation 33. The main features of the level set formulation are
canbeassumedtobelinearwithoutanyappreciableloss summarized here for completeness and in order to
Laser ND
beam
Microstructure RVEs
Max Min
RD
A B C
Thickness=h
Thermocouples
r
w
Symmetry plane
Fig.5—Schematicillustrationofthe2Dthermalmodelofthelaserannealingprocess(notdrawntoscale).PointsA–Ccorrespondtotheplace-
mentofthermocouplesduringtheexperiments.Notethatadvantageistakenofsymmetryalongtheleftboundary,wherer¼0.
1200 1200
Point A
RVE 1
Point B
1000 Point C 1000 RVE 2
K] K] RVE 3
[mperature 680000 mperature[ 680000
Te Te
400 400
200 200
0 10 20 30 40 0 10 20 30 40
Time[s] Time[s]
(a) (b)
Fig. 6—(a) Temperaturehistoryatthe pointsA–C,showninFig. 5. Thesolidlinesshowthesimulatedtemperatureevolutionandthe symbols
are values measured during the experiments. (b) Temperature variation at the left and right edges, indicated by solid and dashed lines, respec-
tively,ofthemicrostructureRVEs,cf.Fig.5.
4790—VOLUME48A,OCTOBER2017 METALLURGICALANDMATERIALSTRANSACTIONSA
identify quantities that are central to the present study. where v¼v n was introduced to denote the local
n
Foradditionaldetailsonthelevelsetformulationandthe interfacevelocityvectorwithmagnitudev .Inaddition,
n
numericalimplementation,pleaserefertoReference33. /0ðxÞ represents the initial positions of the individual
i
Centraltothelevelsetmethodistheintroductionofa interfaces at time t¼0.
scalar field /ðx;tÞ, i.e.. the level set function, defined
throughout the computational domain X. Here, x and t
A. Level Sets Applied to Recrystallization Modeling
were introduced to denote the spatial coordinates and
the time, respectively. The zero contour, where /¼0, Cold working of metallic materials increases the
represents the spatial discontinuity Γ which in the stored energy, most significantly through an increased
present physical settings traces the location of a grain dislocation density in the crystal structures. The raised
boundaryinthemicrostructure.Thefunction/istaken internal energy leaves the material in a thermodynamic
tobeadistancefunction,meaningthatthevalueof/at non-equilibrium state and recrystallization is one of the
a certain point x corresponds to the closest distance main mechanisms to reduce the stored energy. Recrys-
dðx;t;CÞ between the point and the interface at the tallized grains may grow by migration of high-angle
current time. Inaddition, a sign convention isrespected grain boundaries to consume the high-energy cold
whereby / is defined to be positive inside the domain worked matrix material and the local grain boundary
delimited by the interface, zero at the interface, and velocity v can be written as
negative elsewhere. These preliminaries can be stated as
(cid:2) v¼mpn; ½8(cid:3)
/ðx;tÞ¼dðx;t;CÞ;x2X
½3(cid:3) where m is the boundary mobility and p is the driving
C¼fx2X;/ðxÞ¼0g: pressure acting on the grain boundary.[44] This driving
Since /ðx;tÞ is defined as a signed distance function, it pressure may be due to a number of different factors
holds that but the two main components are
kr/ðx;tÞk¼1; x2X: ½4(cid:3) pq ¼s½½q(cid:3)(cid:3) ; ½9(cid:3)
pc ¼(cid:2)jc
The local interface normal n and curvature j can be
evaluated directly from the level set function by con- where pq is due to a dislocation density jump ½½q(cid:3)(cid:3)
sidering across the interface, with the dislocation line tension
8 given by s¼lb2=2. The magnitude of the Burgers vec-
>><n¼ r/ (cid:5)r/; tor and the shear modulus are denoted by b and μ,
kr/k (cid:3) (cid:4) ½5(cid:3) respectively. In addition, γ is due to the curvature of
>>:j¼rn(cid:5)r krr//k (cid:5)r2/ tthoetalbodurinvdinagrypraensdsurtheeacgtrianignobnouangdraariyn ebnoeurngdyaγr.yTcahne
be stated as
where the last equalities hold as long as the property
kr/k¼1 is valid. During the numerical solution p¼pqþpc: ½10(cid:3)
procedure, the level sets may occasionally overlap at
Working in the level set framework outlined in Sec-
interfaces or separate to create artificial voids. This
tion IV, the local grain boundary velocity in Eq. [8]
requires a level set interaction correction to be per-
can be written as
formed. In addition, there is sometimes a tendency for
the level sets to drift from being proper distance v¼vqþvc; ½11(cid:3)
functions. Adopting the standard approach, the latter
issue is in the present work remedied by performing a where the following quantities were introduced:
reinitialization of the level set functions in each time
step,wherebykr/k¼1isensured.Levelsetinteraction vq ¼vqn; vq ¼ms½½q(cid:3)(cid:3) ½12(cid:3)
correction and reinitialization are in the present imple- vc ¼vcjn; vc ¼(cid:2)mc:
mentation performed as described in Reference 33.
Noting that the interface normal n and the interface
Next, differentiation of Eq. [3] with respect to time
curvature j can be obtained directly from the level set
provides the evolution law for the interface as
formulation according to Eq. [5], the velocity compo-
@/ @x nents in Eq. [12] can be employed together with
þðr/Þ ¼0: ½6(cid:3)
@t @t Eq. [7] to obtain
8
aTrobigtreanreyranliuzemtbheerroefsuilntteinrfaEcqe.s,[6a] ttootsailtunautimonbserwoitfhNan/ <:@@/tiþvTqr/iþvcr2/i ¼0; 8i2f1:::N/g: ½13(cid:3)
level set functions are employed, allowing Eq. [6] to be /ðx;t¼0Þ¼/0ðxÞ;
stated as i i
8 The transient advection/diffusion equation in Eq. [13]
<@/
iþvTr/ ¼0; constitutesthelevel setformulationwhichissolved ina
: @t i 8i2f1:::N/g; ½7(cid:3) finite element setting in the present study. The convec-
/iðt¼0;xÞ¼/0iðxÞ; tive velocity field vq is evaluated according to the
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME48A,OCTOBER2017—4791
methodintroducedinReference32inordertoensurea is allowed to take place. This means, for example, that
continuousvelocityfield.Theadoptedtreatmentofthe nucleation can be controlled to take place in either or
convective velocity is also further discussed in Refer- both of the populations of α- and γ-fiber grains.
ence 33. Different combinations of these nucleation conditions
are investigated in the numerical simulations in
Section V.
B. Nucleation
Thetwofundamentalaspectsofmicrostructuretrans-
formation through recrystallization are nucleation and C. Variation of Properties with Misorientation and
subsequent growth of new grains. As primary static Temperature
recrystallization is under consideration here, nucleation
The orientation of each grain is defined by the three
can be expected to be a process of discontinuous
Euler angles u , U; and u (following the Bunge
subgrain growth.[45–47] 1 2
convention),wherebytherotationmatrixforeachgrain
A successful nucleation event requires the local
can be stated as gðu ;U;u Þ. The misorientation
microstructure conditions to satisfy both a kinetic 1 2
between two crystals with orientations g and g is
criterion, whereby sufficiently mobile grain boundaries i j
found as Dg ¼ggT, providing the misorientation as
should be formed by the nucleation, and a thermody- ij j i
the rotation which rotates one crystal reference frame
namic criterion, according to which enough stored
into that of another crystal. Considering crystal sym-
energy (dislocation density) difference must be present
metries, the minimum misorientation is evaluated from
across the interface to exert a positive drag force. (cid:7) (cid:5) (cid:6)(cid:7)
Mplersiismcerraorspitlrryuocbfotauubrnielditsyiat,leosinnwgghgriarcaihninsianbttoiesufryinodrtahsreiaseets clbaourngtdeairtlisopona,rswtiaictrlhee h¼Oms2iGnc(cid:7)(cid:7)(cid:7)acos 12ftrðOsDgÞ(cid:2)1g (cid:7)(cid:7)(cid:7) ½14(cid:3)
inclusions and along shear bands.[44] Since subgrain where the indices i and j were dropped for convenience
structuresarenotspatiallyresolvedinthepresentmodel and where O is one of the 24 operators in the cubic
s
formulation,itisassumedthatthenecessarydislocation symmetry group Gc. The absolute value j(cid:6)j is taken in
substructures are present after the initial cold working, Eq. [14] to reflect that a negative angle simply indicates
as indeed indicated in Reference 48, and that grain thattherotationaxispointsintheoppositedirection.In
boundariesarethemostlikelynucleationsites,asnoted Eq. [14], trð(cid:6)Þ denotes the trace of a tensor.
for IF steel in, e.g., References 10 and 18. The grain boundary energy is in the present model
In the present model, nucleation is modeled as site allowedtovarywiththelocalgrainboundarymisorien-
saturated. By this approach, a number of nuclei are tationaccordingtotheclassicalRead–Shockleyrelation:
distributed in the microstructure at the onset of the 8 (cid:5) (cid:3) (cid:4)(cid:6)
simulation.Thenumberofnucleiischosentomimicthe <c h 1(cid:2)ln h ifh<h
recrystallization behavior observed in the experimental cðhÞ¼ mh h m ; ½15(cid:3)
: m m
results. Nucleation is in the present model assumed to c ifh(cid:7)h
m m
take place at sites along the grain boundaries and the
nuclei are randomly placed at such locations. In where h ¼15deg is the angle that differentiates
m
addition, different assumptions on nucleation in one or between low- and high-angle boundaries and c is the
m
both of the α- and γ-fibers will be considered. This is energy of general high-angle grain boundaries. The
further discussed in Section V. misorientationdependenceofthegrainboundaryenergy
Similar approaches to modeling of the nucleation isshowninFigure7(a)(blueline).Parametersrelatedto
phase as adopted here are also used in the cellular the grain boundary properties are provided in Table I.
automata model of static recrystallization in IF steel in Migration of misoriented grain boundaries can be
Reference 14 and in the Monte Carlo Potts models in viewed as a thermally activated process and the tem-
References 24 and 25. Site-saturated nucleation of perature dependence of the grain boundary mobility is
primary static recrystallization is also adopted in the given by
MonteCarloPottsmodelsinReferences49and50.The (cid:3) (cid:4)
b2D Q
details on modeling the nucleation event in a finite m ðTÞ¼ gb where D ¼D exp (cid:2) m ;
element-based level set formulation are provided in T kBT gb 0 kBT
Reference 33. ½16(cid:3)
New recrystallization nuclei are added in the present
model by identifying a nodal point in the finite element wherek istheBoltzmannconstantandD isthegrain
B gb
mesh where the nucleation criteria are met and subse- boundarydiffusivitywithD andQ beingthediffusion
0 m
quently adding a circular contour, i.e., a new level set coefficientandtheactivationenergy forgrainboundary
function, centered around that node. The radius of the migration, respectively. The resulting temperature
new zero contour corresponds to the radial size of the dependence of the grain boundary mobility is shown
nucleus. Being strain-free, new recrystallization nuclei in Figure 7(b) (red line). The mobility can be seen to be
will be added with zero initial dislocation density. significantly increased for temperatures above approx-
The present model is formulated in such a way that imately 850 K (577 °C), i.e., at temperatures which are
the nature of the nucleation event can be controlled in reached locally in the material during the present laser
termsoforientationandregardingwherethenucleation annealing, cf. Figure 6.
4792—VOLUME48A,OCTOBER2017 METALLURGICALANDMATERIALSTRANSACTIONSA
Themobilityoflow-angleboundaries,havingh(cid:4)h , step with a one-time computational cost. Should subse-
m
is generally much lower than that of high-angle bound- quent nucleation or grain removal be required, as
aries. Following Reference 46, a misorientation depen- encountered in dynamic recrystallization, the only
dence mh of the mobility can be added to Eq. [16] to added expense is addition and/or removal of singular
provide rows and columns in the matrix. To further optimize
memory usage, the diagonal terms of the matrix in
mðT;hÞ¼mTð"TÞmhðhÞ where !# Figure8canbeusedtostoresomeotherscalarproperty
(cid:2) (cid:8)
h 4 ½17(cid:3) of the individual grains, such as dislocation density.
hðhÞ¼ 1(cid:2)exp (cid:2)5 h :
m
The misorientation dependence of the grain boundary V. NUMERICAL SIMULATIONS
mobility is illustrated in Figure 7(a) (red line).
The model used in the numerical simulations mea-
Also the shear modulus, appearing in the dislocation sures 800 9 200 μm2 and is shown in Figure 9. The
line tension in Eq. [9], has a temperature dependence
initial model grain structure—comprising 130 grains—
whichneedstobetakenintoaccountasitinfluencesthe
was generated by Voronoi tessellation on a square
stored energy driving force for grain growth. The
domain which was then compressed along the vertical
temperature dependence of the shear modulus for α-Fe
direction. Different Voronoi tessellations were tested in
is stated in Reference 52 as
order to achieve a grain structure which after compres-
(cid:3) (cid:4)
T(cid:2)300 sion was representative for the actual cold rolled
lðTÞ¼69:2(cid:8)109 1(cid:2)1:31 Pa; ½18(cid:3) material as observed in the experiments, cf. Section II.
T
m The microstructure is discretized using triangular ele-
where T ¼1810 K (1537 °C) is the melting tempera- mentswithlinearinterpolationandthemeshisupdated
m
ture. The format in Eq. [18] is adopted here and the in each solution step through Delaunay triangulariza-
variation of the shear modulus with temperature is tion in order to maintain a high resolution of the grain
shown in Figure 7(b) (blue line). boundarynetwork.Theelementsizealonggrainbound-
ariesiscontrolledbyaprescribedmaximumelementsize
and a typical mesh is shown in Figure 9. The material
parameters used in the level set model are collected in
D. Efficient Handling of Local Grain Boundary Properties
Table I. Following the observations made during the
in a Level Set Framework
experiments,asdiscussedinSectionsIandII,theinitial
As grain boundary energy and mobility are local rollingtextureisdominatedbytheα-andγ-fibers,witha
quantities, they have to be evaluated along each higherpresenceoftheformer.Thisisinagreementwith
interface in the polycrystal. In addition, during grain other experimental studies on rolled IF steel, such as
growth,anindividualgrainmaybecomeadjacenttoany Reference 9. As the α- and γ-fibers are clearly statisti-
other grain. As each of the grain boundary interfaces cally dominant, the grains in the model microstructure
may comprise hundreds of grain boundary segments, willbelongtoeitherofthesetwopopulations.Tomimic
depending on the degree of mesh refinement, and if the the initially stronger presence of the α-fiber, the texture
evaluation is done in every time step of the simulation, priortolaserannealingisassumedtoconsistof60pctα-
the computational effort in considering Eqs. [14], [15], fiber and 40 pct γ-fiber. The texture is defined by
and mh in [17] may be significant. It can also be noted assigning Euler angles from the appropriate texture
that evaluation of Eq. [14] involves a minimization distributions with a 5 deg random Gaussian spread
wherein several symmetry rotations have to be com- around the ideal orientation to the individual grains,
pared, and the effort depends on the order of crystal providing the initial texture shown in the orientation
symmetry.However,astheanisotropyofthelocalgrain distribution map in Figure 10(a). The initial texture is
boundary properties is presently described by misorien- also indicated in Figure 9,where α-grains are white and
tation only, without any influence of the local orienta- γ-grains are gray. In addition, high-angle grain bound-
tion of the grain boundary plane, computational aries—being boundaries with a misorientation greater
efficiency can be greatly improved by precomputing than 15 deg—are indicated by thicker lines in Figure 9.
the possible values of γ and mh which can occur in the The stored energy in 80 pct cold rolled IF steel is
polycrystal. Such a precomputed look-up table is estab- measured by neutron diffraction in Reference 53. It is
lishedasaN/-by-N/interfacepropertymatrixAhwhere found that, apart from at singular orientations, the γ-
the elements above the diagonal contain the interface fiber generally has a higher stored energy content
energy c and the subdiagonal elements contain the compared to the α-fiber. Attempts to predict the
ij
misorientationdependenceoftheinterfacemobilitymh;ij orientation dependence of the stored energy by means
for the grain boundary separating grains i¼1;...;N/ of micromechanical models in Reference 54 show that
and j¼1;...;N/. This storage format is illustrated in different results may be found depending on how the
Figure 8. It can be noted that the index ij may be mechanical interaction between grains is treated, but a
switched arbitrarily, i.e., ð(cid:6)Þ ¼ð(cid:6)Þ . As site-saturated limited influence on the hardening law is obtained.
ij ji
nucleation is considered, all crystal orientations are Orientation dependence of the stored energy in low-
present from the startandthe interface property matrix carbonsteel,fordifferentrollingdeformationlevelsand
Ah can be conveniently established as a preprocessing measured by X-ray diffraction, has been shown in
METALLURGICALANDMATERIALSTRANSACTIONSA VOLUME48A,OCTOBER2017—4793
Table I. Material Parameters Used in theNumerical Simulations, Representative forIF Steel
Parameter Value Description Source
b 2.480 Å magnitude of the Burgersvector 51
μ 69.2GPa shear modulus 14
μ 2:66(cid:8)10(cid:2)19 J activation energyforgrain boundarymigration 51
D 0:89(cid:8)10(cid:2)4 m2/s grain boundary mobility coefficient 51
0
c 0.79J/m2 energyof high-angle grain boundaries 14
m
T 1810K(1537°C) melting temperature 51
m
1 10 70
0.8 s] 8 60
/
N
T /
m 3
/ 0.6 m 6 50 a]
m P
2
1 G
γ,m 0.4 −10 4 40μ[
/ ×
γ [
T
0.2 m 2 30
m/m
T
γ/γ
m
0 0 20
0 5 10 15 20 300 440000 500 660000 700 880000 900 11000000 1100
θ [◦] T [K]
(a) (b)
Fig. 7—(a) Misorientation dependence of the grain boundary mobility and energy, where small-angle boundaries are defined as those having a
misorientation h(cid:4)hm¼15deg. (b) Temperature dependence of the mobility mT for high-angle grain boundaries and for the shear modulus,
respectively.
Reference 55. The stored energy levels in IF steel the γ-fiber after subgrain coarsening.[15] The nuclei are
following 75 pct cold rolling are also estimated in also found to most frequently have their orientations in
Reference 14 and, again, the γ-fiber is found to have a the same fiber as the parent grain. Similar observations
higher stored energy than the α-fiber. The same obser- arealsomadeinReference59,studyingthedevelopment
vation is made in References 56 and 57. Based on these of recrystallization textures in pure iron, where recrys-
studies, grains belonging to the α-fiber are given a tallization is found to occur by subgrain growth and
dislocationdensityof191015m-2,whileavalueof29 grain boundary nucleation with nuclei having approx-
1015m-2isassumedfortheγ-grainsinthepresentwork. imately30degmisorientationwithrespecttotheparent
The stored energy is assumed to be homogeneous in crystals. Regarding IF steel, the study in Reference 17
each grain of the α- and γ-fibers and in the nuclei, indicates preferential nucleation of γ-fiber grains with a
respectively. A higher stored energy in the γ-fiber and a 30 deg misorientation around the h111i//ND axis. In
homogeneous stored energy distribution in the grains Reference 20, it is noted that nucleation in rolled IF
are also considered in the Monte Carlo Potts model of steel, being of BCC structure, is most frequent in the γ-
IF steel in Reference 24 and in the cellular automata fiber which has a higher stored energy than the α-fiber.
modelinReference14.InReference33,itisshownhow This is in contrast to most FCC-structured crystals
stored energy gradients can be considered in a level set where nucleation rather occurs in texture components
framework, but this is beyond the scope of the present with a low stored energy content. Grain boundary
study. Further, recovery is not considered here as the nucleation and a dominating nucleation of the γ-fiber
processing time is very short, only a few seconds. are also observed in Reference 60.
Considerably longer time spans are required for recov- Adopting an approach from studies on FCC materi-
ery to be of any significance during annealing of IF als, the concept of “nucleation factors” is employed in
steel.[56] In passing, it can also be noted that crystal studies on IF steel and low-carbon steel in References 9
plasticity finite element modeling can be employed to and 40. A nucleation factor N is defined as the number
i
simulate the rolling process and provide an estimate of ofrecrystallizedgrainsoffiberi(αorγ)perdeformation
the stored energy distributions, as discussed in Refer- band in the rolled microstructure, observed along ND.
ence 58 related to IF steel. Thisispresentedasameasureoftheabilityofbandsofi
Turningto the nucleation,it iswidelyacceptedin the type to nucleate. Regarding IF steel, the study in
literature that recrystallization during annealing of IF Reference 9 presents nucleation factors in the order of
steel primarily nucleates along boundaries of grains in 0.1 and 0.8 for α- and γ-nucleation, respectively, for
4794—VOLUME48A,OCTOBER2017 METALLURGICALANDMATERIALSTRANSACTIONSA
annealing at temperatures of relevance here. These of the individual nucleation site candidates is identified,
numbers are in the present study adopted as nucleation after which a random number in the interval [0, 1] is
probabilities for each fiber. generated and compared to the nucleation probability
AsdiscussedinSectionIV–B,differentnucleationand for that fiber (0.1 for α and 0.8 for γ). If the random
growthconditionswillbeconsideredinthesimulations, numberislessthanthenucleationprobability,anucleus
guided by the aforementioned observations. The differ- is placed at the site, otherwise the site is discarded.
ent simulation scenarios are summarized in Table II. In Thenumberofnucleitoplaceisdeterminedbasedon
case 3, the temperature dependence of the grain bound- estimates of the number of grains and the grain sizes
arymobilityiskept,whilethemisorientationdependence observed in the experiments, for example in Figure 2.
of both grain boundary mobility and energy is switched The number of nuclei is in this manner controlled to
off.Inallcases,theinitialsizeofthenucleiissetto1μm, provide annealing microstructure morphologies which
in agreement with the subgrain sizes observed in correspond to those obtained in the experiments. In the
deformed IF steel in References 8,18 and 61. present simulations, 670 new nuclei are seeded in each
Considering site-saturated nucleation, a prescribed RVE.Obviously,onlythosenucleiwhichhaveadequate
number of nuclei are distributed in the microstructure localdrivingforces willcontinue togrow asnew grains.
when initializing the simulations as discussed in Sec- Startingwith130coldrolledgrains,eachRVEwillthus
tion IV–B. Beginning with nucleation according to comprise 800 grains/nuclei. Some different RVE sizes
case 1, cf. Table II, all possible nucleation sites—i.e., were investigated but it was found that the relative
allnodesoftheelementscontainingagrainboundaryof presence of different texture fibers was quite indifferent
a γ-grain—are identified. Next a number of these to the choice of RVE size. As the relative stability,
possible nucleation nodes are picked randomly, corre- rather than the absolute presence, of each of the two α-
spondingtothenumberofnucleitobeplaced.Inathird and γ-fibers is under investigation, the adopted grain
step, the nuclei are distributed among these randomly count and RVE size are assumed sufficient.
picked sites. When both α- and γ-nucleation are consid-
ered (cases 2 and 3 in Table II), an additional selection
step is needed. The nodes of all elements containing the
VI. RESULTS AND DISCUSSION
boundaries of both α- and γ-grains are first identified.
But when subsequently distributing the nuclei, the fiber Beginning with the microstructure morphology,
Figure 11 shows the appearance of the microstructure
j = 1...N in each of the RVEs at the end of the annealing period
φ
(35 s), after which the specimen temperature is almost
γ γ ...γ
12 13 1Nφ down to its initial value, as shown in Figure 6, and the
microstructure has reached a quasi-static state. The
m γ ...γ i resultsinFigure11arefoundwhencase2inTableIIis
21 23 2Nφ =
consideredandtheresultsaresimilarfortheothercases.
A =
θ m m ...γ 1 By comparing the RVE microstructures in Figure 11
31 32 3Nφ ..
... .N withtheopticalmicroscopeimageinFigure1,thesame
type of gradient in grain size and grain structure can be
φ
m m ... seentohavedeveloped,withatransitionfromequiaxed
N 1 N 2
φ φ grains in the annealed region towards a maintained
laminar rolling structure further out.
Fig. 8—Layout of the proposedN/-by-N/ interface property matrix In RVE 1 (Figure 11(a)), the recrystallized grain
for precomputed anisotropic grain boundary energy c and mobility
ij structure is relatively constant throughout the RVE.
Manhd;ij,j¼rel1a;t.e.d.;tNo/t.hTehgerationtablonuunmdabreyr obfetwgreaeinnscpryrsetsaelnstii¼n t1h;e..p.;oNly/- This is an indication of the small spatial variation in
crystalisdenotedbyN/. temperature in this particular RVE, located directly
α-fiber γ-fiber
h
ND
l
RD
Fig. 9—Initial grain structure of the microstructure RVEs, with a magnification of the typical mesh resolution of the grain boundary network.
Theinitialcoldworkedmicrostructureconsistsof130grains.Thecolorsshowwhichtexturefiberthegrainsbelongtoandthickergrainbound-
arylinesindicatehigh-anglegrainboundaries.Thedimensionsofthe RVEarel¼800 μmandh¼200 μm.Thedifferentlocationsofthe RVE
intheworkpieceareshowninFig.5.
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