K. Tanaka, T. Takaki and Y. Tomita / Proceedings, AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 205 - 212

ISBN 0-9780479

International Conference on Advances and Trends in Engineering Materials and their Applications(AES – ATEMA’ 2007)

2D Phase-Field Simulation of Static Recrystallization
K. Tanaka1*, T. Takaki2, Y. Tomita3
1*Graduate School of Science and Technology, Kobe University, 1-1, Rokkodai, Nada, Kobe
657-8501, Japan(Email:tanaka@solid.mech.kobe-u.ac.jp)
2
Faculty of Maritime Sciences, Kobe University, 5-1-1, Fukaeminami, Higashinada, Kobe, 658-0022,
Japan
3(Email: takaki@maritime.kobe-u.ac.jp)
Faculty of Engineering, Kobe University, 1-1, Rokkodai, Nada, Kobe 657-8501, Japan(Email: tomita@mech.kobe-u.ac.jp)*Corresponding Author

Abstract
In this study, we have performed a series ofnumerical
simulationsofthestaticrecrystallization process using the numericalmodel developed in our previous study. The
numerical model enables us to continuously
simulate the process from the deformation of apolycrystal metal to the annealing by coupling
the crystal plasticity finite element method andthe phase-field method. The simulations werecarried out under the conditions of two different
grain sizes and two different nucleation models,
i.e.,
site
saturatednucleation
andcontinuous-nucleation rate models. As a result,the numerical results for different grain sizes
indicate that the size fluctuation ofrecrystallized grains is larger and the© AES-Advanced Engineering Solutions (Ottawa, Canada)
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recrystallization kinetics is slower for thepolycrystal with large grains than that withsmall grains. These phenomena werereproduced by introducing strain-gradienttheory into the crystal plasticity finite element
method to simulate the size-dependent work
hardening. Furthermore, we point out theimportance of the nucleation model and thepossibility of the development of a novel
numerical model taking the spontaneous
nucleation formed during recovery into
account.

Keywords
Static Recrystallization, Phase-Field Method,
Crystal Plasticity, Nucleation, Recovery.

K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

1Introduction
The establishment of a numerical model andsimulation technique will enable us to obtain
the optimum design and perform precise
predictions of material microstructures formed
during annealing [1]. Therefore, much effort
has been devoted to developing numericalmodels of recrystallization [2], and various
phenomena have been investigated from theviewpoints of numerical simulations [3, 4].
It is widely recognized that the microstructures
formed during annealing are significantly
affectedbythepre-deformation
microstructures [5], since the recrystallization
originates from dislocation cells or subgrainsthat appear after deformation, and subsequently,
the recrystallized grain growth is driven by thestored energy resulting from the dislocation
accumulated during deformation. Recently,
numerical studies using the Monte Carlo (MC)Potts model [6] and the cellular automata (CA)model [7] based on the data measured byelectron back-scatter diffraction (EBSD)
analyses have been performed for static
recrystallization. However, to enable more
systematic investigations of recrystallization
textures, it is essential to develop a
computational procedure that does not use
experimental data.
In this study, we perform a series of numericalsimulations of the static recrystallization
process. The effects of the grain size and thenucleation model on the recrystallization
microstructure and kinetics are investigated.
Here, we employ the coupled numerical model
developed in Ref. [8], in which the deformationof polycrystalline metal is simulated using thefinite element (FE) method based on
strain-gradient crystal plasticity theory, and thenucleation and growth of the recrystallized

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grains are simulated by the phase-field method.

2 Numerical Procedure
In this study, we employ the numerical
procedure for static recrystallization proposedby Takaki et al. [8]. Figure 1 shows thecomputational sequence composed of thefollowing three steps: the deformation
simulation, the data mapping and therecrystallization simulation. Next, we briefly
explain each step.

Figure 1.

Numerical procedure.

2.1 Crystal plasticity FE simulation
The crystal plasticity FE method based on thestrain-gradient theory of rate-dependent
plasticity [9] is used to examine themicrostructure and dislocation distribution
during the deformation of polycrystalline
metals. Here, the critical resolved shear stressg(a) on slip system (a) is assumed to be
expressed by a Bailey-Hirsch function:g (a)g 0( a )  aPb ¦ Zab U ( b ) ,

(1)

(b)

where g0(a) is the initial value of g(a), a is a

K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

constant, P is the elastic shear modulus, b is themagnitude of the Burgers vector, Zab is theinteraction matrix and U b) is the accumulated
dislocation density, which is the sum of thedensities of the statistically stored dislocations(SSDs) and the geometrically necessity
dislocations (GNDs). The evolution of theSSDs is expressed by the balance between the
production rate of dislocations and theannihilation by dynamic recovery. The densityof the GNDs is calculated from the gradient of
the shear strain on slip system (a). Since thehardening equation, Eq. 1, includes thestrain-gradient term through the dislocation
density, it is possible to express the effects ofthe grain size.
2.2 Data mapping
The data calculated by the crystal plasticity FE
simulation, such as the dislocation density andcrystal orientation, are mapped onto a regularlattice used in the phase-field simulation. The
stored energies calculated from the dislocation
density are smoothed on the lattice, because thephase-field
methodrequiresspatially
continuous driving forces.
2.3 Phase-field simulation
We generalize the phase-field model proposedby Kobayashi, Warren and Carter, i.e., KWC
model [9,10,11] to simulate the migration of
the recrystallized grain boundary driven by thestored energy and the impingement between
the recrystallized grains. In this model, twoorder parameters, phase field I andcrystallographic orientation T, are used. The
time evolution equations for I and T are as
follows:

wI
wt

ª
º
wf I
M I «D 2  2I 
 2Is T »
wI
¬
¼

wT
wt

MT

§ 2 T ·
¨I s
¸,


T ¸¹
I 2 ¨©
1(2)(3)

where D is the gradient coefficient, s is aconstant, MI is the mobility of I and MT is themobility of T. The parameters are selected as D
= (3GV)0.5, s = D(2W) 0.5/S, MI = m(2W) 0.5/6Dand MT = (1-p(I))MI, where V is the interface
energy, G is the interface thickness and m is themobility of the grain boundary migration. The
free energy density f(I) is expressed byfollowing double-well function:f (I )(1  p (I )) Estore  Wq(I ) .

(4)

Here, p(I) = I3(10-15I+6I2) and q(I)=I2(1-I)2.
Estore is the stored energy calculated by Estore =0.5UPb2, where U is the dislocation density. W is
the barrier height expressed by W = 6V/G.

3 Simulation Results
3.1 Deformation simulation
Figure 2 shows the polycrystal model
consisting of 23 regular grains. This model isdivided into 64x64 regular crossed-triangle
elements and is compressed at a strain rate 10-3
[1/s] as a 2-slip plane-strain problem. Figures 3(a) and (b) illustrate the numerical results ofthe crystallographic orientation at acompression of 50% thickness for 23-and
77-grain models, respectively. From Fig. 3, we
can observe the misorientation inside some ofthe original grains.
The dislocation densities and crystal
orientations are utilized as basic data in thestatic recrystallization simulation using thephase-field method.

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K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

conditions that must be considered from acomputational viewpoint, the following criteria
are defined: (1) a high-angle grain boundary of
more than 15 degrees, (2) a stored energy ofmore than 0.6 MPa, (3) the orientation of thenucleus is present in the deformed structureand (4) the minimum distance between two
neighboring nuclei is 10 'x, where 'x is thegrid size. The initial radius of the nucleus is set
to 3 'x. Here, we choose the following
material parameters: 'x = 0.4 [Pm], G= 0.8
Figure 2.

Polycrystal computational modelwith 23 regular grains.

[Pm], dt = 0.1 [s], V= 0.6 [J/m2], D= 1.14
10-3 [(J/m)0.5], m = 4.1110-14 [m3/Ns], W =4.94106 [J/m3] and MI= 1.810-8 [m3/Js].
Next, we consider two types of nucleation
models, one is the site saturated nucleation(SSN) model and the other is theconstant-nucleation-rate (CNR) model. In the
CNR model, we consider the nucleation rate to

Figure 3. Crystallographic orientation at 50%
compression for (a) 23-grain model and (b)77-grain model.
3.2 Static recrystallization simulation
The nucleation and growth of recrystallized
grains during annealing are simulated by thephase-field method based on the data
calculated by the crystal plasticity FE
simulation. Although the grain boundary
migration of the recrystallized grains follows
Eqs. 2 and 3, the nucleation of therecrystallization is not defined yet. It is themost difficult process to simulate because of
the difficulty of its experimental observation.
Although there are still uncertainties about
nucleation mechanisms, some points have beenclarified [5]. On the basis of the points and the208© AES-Advanced Engineering Solutions (Ottawa, Canada)
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be N (1  X ) , where N

is the constantnucleation rate and X is the recrystallized
volume fraction. This means that the nucleation
rate gradually decreases with recrystallization[5, 12].
Figure 4 shows the initial nucleation sites of (a)23-grain and (b) 77-grain models using SSN. It
is observed that there are many nuclei aroundthe original and newly developed grainboundaries; thus, is confirmed that nucleation
can occur on the basis of the defined nucleation
criteria. It is widely recognized fromexperimental observations that the grainboundaries and the misoriented transition
bands inside the grains are active nucleationsites [5, 12].

K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

Figure 4. Initial nucleation sites of (a)23-grain model and (b) 77-grain model.
Figures 5 and 6 illustrate the recrystallized
grain growth process of the 23-grain modelusing SSN and CNR, respectively. The white
region indicates the deformed material and thegrain boundary between recrystallized grains,
and the black region shows the recrystallized
grains. From Fig. 6, we can observe continuous
nucleation during the recrystallization.
Figure 7 shows the final recrystallized
microstructures. Comparing the 23-and
77-grain models, it is observed that the sizefluctuation of the recrystallized grains is larger
for the 23-grain model than the 77-grain one.

Figure 6. Recrystallized microstructure
evolution for 23-grain model using CNR. t =(a) 100, (b) 500, (c) 1000 and (d) 3000 s.
This fluctuation depends on the distribution of
the nucleation sites, as shown in Fig. 4. Since
the nucleation sites are determined from thedeformed microstructures, it is confirmed thatthe recrystallized microstructures are largely
dependent on their location. Furthermore, from
Fig. 7, we can see the recrystallized grains withthe characteristic shape elongated in thevertical direction.
Figure 8 illustrates the variation of the

Figure 5. Recrystallized microstructure
evolution for 23-grain model using SNR. t =(a) 100, (b) 1000 and (c) 3000 s.

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K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

recrystallization kinetics, and its gradient is
known as the Avrami exponent. The theoreticalvalues of the Avrami exponent in 2D are 2 andFigure 8. Recrystallization volume fraction.
Figure 7. Final recrystallized microstructuresof (a) 23-grain model using SSN, (b) 77-grainmodel using SSN, (c) 23-grain model using
CNR and (d) 77-grain model using CNR.
recrystallization volume fraction. From Fig. 8,
it can be seen that the recrystallization rate of
the 77-grain model is faster than that of the
23-grain model for both nucleation models.
This tendency is identical with the reported
experimental result that the recrystallization
rate of large pregrains is slower than that ofsmall pregrains [5]. This is because that thestored energy inside grains is higher for small
pregrains than large ones, and by introducing
strain-gradient theory to crystal plasticity, thisphenomena can be reproduced. In the CNRmodel, the typical S-shaped curves can beobserved for both 23-and 77-grain models.
Figure 9 shows the Jhonson, Mehl, Avrami andKolmogorov (JMAK) plots corresponding to
Fig. 8 [5]. The JMAK plot characterizes the210© AES-Advanced Engineering Solutions (Ottawa, Canada)
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Figure 9.

JMAK plots.

3 for SSN and CNR, respectively. However, as
shown in Fig. 9, our calculations indicate
markedly lower values than those of the theory(the Avrami exponents for the 23 grain (SSN),
77 grain (SSR), 23 grain (CNR) and 77 grain(CNR) models are 0.8, 1.0, 2.0 and 2.3,
respectively). This difference is due to theK. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

random-nucleation
assumption
inthetheoretical calculation. In fact, it has beenreported that the values of the Avrami exponent
obtained from experiments are lower thanthose based on theory [5].

4 DiscussionAsmentioned
above,
thesimulatedrecrystallized microstructures are largely
affected by the definition of the nucleation
model. Therefore, the development of a precise
nucleation model based on the actual
nucleation mechanism will be indispensable.
Although abnormal grain growth is recognized
as one of the possible nucleation mechanisms
and is reproduced by MC and CA simulations,the effects of the rotation of grains or theircoalescenceduringrecoveryon therecrystallization nucleation is still hardly
understood. The KWC phase-field model
employed in this study can simulate the grainboundary migration and the grain rotation
simultaneously. Therefore, we believe that, bygeneralizing the numerical procedure used inthis study, we will be able to construct a static
recrystallization model of spontaneous
nucleation including both abnormal grain
growth and coalescence and taking thedeformation microstructure into account.

5 Conclusions
In this study, we have performed a series ofnumerical simulations of static recrystallization,in which the numerical model coupled with the
crystal plasticity finite element method and thephase-field method was employed. The
numerical results for different grain sizes
indicate that the size fluctuation ofrecrystallized grains is larger and therecrystallization kinetics is slower for thepolycrystal with large grains than that withsmall grains. These tendencies are similar to
the reported experiment results. Our results
were obtained by introducing strain-gradienttheory into the crystal plasticity finite element
method to simulate size-dependent work
hardening. Therefore, we anticipate thepractical application of the proposed numerical
procedure to the optimum design of material
microstructures by the integrated evaluationfrom plastic deformation to annealing. In orderto improve the simulation accuracy, it will be
indispensable to establish a numerical model
including the spontaneous nucleation. We leavethis as a future work.

Acknowledgments
This research was partially supported by the
Ministry of Education, Culture, Sports, Scienceand Technology Grant-in-Aid for Scientific
Research (B), 18360061, 2007.

References[1] Raabe, D., Hntcherli, L. (2002) Cellular
automata in materials science withparticular reference to recrystallization
simulation,
Computational
Materials
Science, 32, 53-76.
[2] Rios, P.R., Siciliano, Jr.F., Sandim, H.R.Z.,
Plaut, R.L., Padilha, A.F., (2005)
Nucleation
andgrowthduringrecrystallization, Materials Research., 8,225-238.
[3] Simmons, J.P., Wen, Y., Shen, C., Wang,
Y.Z. (2004) Microstructural development
involving
nucleation
andgrowthphenomena simulated with Phase Field
method,Materials
ScienceandEngineering, A365, 136-143.
[4] Humphreys, F.J. (1992)
Modelling211© AES-Advanced Engineering Solutions (Ottawa, Canada)
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K. Tanaka, T. Takaki and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,2August 06 -10, 2007, pp. 205 - 212

mechanisms and microstructures ofrecrystallization, Materials Science andTechnology, 8, 135.
[5] Humphreys, F.J., Heatherly, M. (2004),
Recrystallization and related annealing
phenomena, Elsevier (Springer).
[6] Choi, S., Cho, J.H. (2005)
Primary
recrystallization modeling for interstitial
free steels, Materials Science andEngineering., A405, 86.
[7] Hesselbarth, H.W., Gobel, I.R. (1991)
Simulation of recrystallization by cellular
automata, Acta Metallurgica et Materialia.,
39, 2135.
[8] Takaki, T., Yamanaka, A., Higa, Y., Tomita,
Y. (2007) Phase-field simulation during
static recrystallization based on crystal
plasticity theory, J. Computer-Aided
Material Design, (in press).
[9] Higa, Y., Tomita, Y., Sawada, Y. (2003)
Computational simulation of characteristic
length dependent deformation behavior,Trans.JSME, A69, 523-529.
[10] Warren, J.A., Kobayashi, R., Lobkovsky,
A.E., Carter, W.C. (2003)
Extendingphase field models of solidification topolycrystalline materials, Acta materialia.,
51, 6035-6058.
[11] Lobkovsky, A.E., Warren, J.A. (2001)
Sharp interface limit of phase-field modelof crystal grains, Physycal Review E, 63,051605.
[12]Doherty, R. D., Hughes, D.A., Humphreys,
F.J., Jonas, J.J., Jensen, D.J., Kassner,
M.E., King, W.E., McNelley, T.R.,
McQueen, H.J., Rollett, A.D. (1997)
Current issues in recrystallization: areview,
Materials
ScienceandEngineering., A238, 219-274.

212© AES-Advanced Engineering Solutions (Ottawa, Canada)
All rights are reserved[13] Upmanyu, M., Srolovitz, D.J., Lobkovsky,
A.E., Warren, J.A., Carter, W.C. (2006)
Simultaneous grain boundary migrationand grain rotation, Acta Materialia, 54,
1707-1719.
[14] Kobayashi, R., Warren, J.A., Carter, W.C.
(2000) A continuum model of grainboundaries, Physica D, 140, 141-150.
[15] Kobayashi, R., Warren, J.A., Carter, W.C.
(2000)
Grain boundary model andsingular diffusivity, Mathematical Sciences
and Applications, 14, 283-294.
[16] Read, W.T., Shockley, W. (1950)
Dislocation Models of Crystal Grain
Boundaries, Physics Review, 78, 275.