T. Takaki, A. Yamanaka and Y. Tomita / Proceedings, AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 223 - 231

ISBN 0-9780479

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

Phase-Field Modeling for Dynamic Recrystallization
T. Takaki 1*, A. Yamanaka 2, Y. Tomita 3
1

Faculty of Maritime Sciences, Kobe University, 5-1-1, Fukaeminami, Higashinada, Kobe, 658-0022, Japan(Email: takaki@maritime.kobe-u.ac.jp)
2
Graduate School of Science and Technology, Kobe University, 1-1, Rokkodai, Nada, Kobe,
657-8501, Japan(Email: yamanaka@solid.mech.kobe-u.ac.jp)
3Faculty of Engineering, Kobe University, 1-1, Rokkodai, Nada, Kobe, 657-8501, Japan(Email: tomita@mech.kobe-u.ac.jp)*Corresponding Author

Abstract
Dynamic recrystallization (DRX) exhibits very
complicated phenomena including both
hardening due to the accumulation of
dislocations and softening due to the
nucleation and growth of recrystallized grains.
In other words, the mechanical behaviour
during the DRX process is closely related to theevolution of microstructures and dislocations.
In this study, the phase-field model, whichenables the simulation of microstructuralvariation during the DRX process, is developedby generalizing the multi-phase-field method
proposed by Steinbach et al. The hardening due
to the accumulation of dislocations and thenucleation criteria of recrystallization are
expressed theoretically by employing themethod proposed by Guo et al. In order toconfirm the basic performance of the© AES-Advanced Engineering Solutions (Ottawa, Canada)
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developed model, a simulation of single-grain
growth is performed. As a result, thecharacteristic variation of the grain boundary
migration rate due to the dislocation-densitychange and a good agreement with thetheoretical result were observed. Furthermore,
the DRX process including nucleation issimulated for a regular-hexagonal grain
structure. It is confirmed that a typical stress –strain curve with multiple peaks can be
reproduced by the prediction of microstructural
evolution.

Keywords
Phase-Field Method, Modeling, Simulation,
Dynamic Recrystallization, Microstructure
Evolution, Dislocation.

T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231

1 Introduction
It is well known that when a low-to-medium
stacking fault energy (SFE) metal is deformed
under a high-temperature environment,
dynamic recrystallization (DRX) occurs [1, 2].
DRX
isdistinguishedfromstaticrecrystallization (SRX), which occurs inpost-deformation annealing. Although bothrecrystallization processes have many common
characteristics, the most different point is thatfor DRX, the dislocation density in therecrystallized grains increases with continuous
deformation, while for SRX, it is constant.
Since DRX exhibits complicated phenomena
such as hardening due to deformation andsoftening due to recrystallized grain growth, orthe simultaneous evolution of dislocations andmicrostructures over time, the construction of anumerical model that enables the evaluation of
microstructural evolution is essential, in
addition to experimental observation.
As a numerical model for DRX, the cellular
automaton (CA) method [3-9] has beensuccessfully applied to the microstructuralinvestigation of the DRX process. Guo et al.
[7-9] have proposed a simulation method that
couples the CA method to a theoretical model,
in which the hardening by the accumulation of
dislocations is expressed by equations and thesoftening by grain boundary migration due torecrystallization is simulated by the CA method.
However, since the CA method describes thestates in terms of discrete variables, it isproblematic when applied to modelingcurvature-driven growth [10].
The phase-field method was first used to model
dendrite formation [11, 12], and over the past
decade, it has been applied to various problems
of material science. The phase-field method
can easily reproduce complicated shapes and224© AES-Advanced Engineering Solutions (Ottawa, Canada)
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morphologies without tracking the location of
the interface, since the interface migration is
described by the time evolution of anadditional order parameter, or the phase field.
In this model, the time and scale can be treatedas the real values, and the effect of curvature is
explicitly included.
In this study, we establish a phase-field modelfor the DRX process, in which themulti-phase-field
methodproposedby
Steinbach et al. [14] is generalized to simulate
the grain boundary migration driven by stored
energy, or the softening process. The hardening
process is modeled by the equations employed
by Guo et al. [6, 7].

Figure 1.

Schematic stress-strain curves
under hot working.

2 Dynamic Recrystallization
Figure 1 schematically illustrates typicalstress-strain curves under hot working. In
metals with a high-SFE, such as Al and D-Fe,
the stress increases monotonically withincreasing strain, because the dynamic
recovery is rapid and the dislocation density
does not achieve a critical value that originates
the nucleation of recrystallization. On the other
hand, the low- or medium-SFE metals, such as
Cu, Ni and J-Fe, exhibit DRX. The stress-strain
curve for the low- or medium-SFE metals ischaracterizedbytheZener-Holloman
parameter Z, defined by the following
equation:

T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231(1)
Z H exp Q RT ,where H is the strain rate, T is thedeformation temperature, Q is the activationenergy and R is the gas constant. As shown in
Fig. 1 (b), under the conditions of low Z, i.e.,low H or high T, the stress-strain curve has
multiple peaks at low strain, and under high Z,
i.e., high H or low T, it exhibits a single peak.
For both cases, when the stress attains a criticalvalue Vc which is somewhat less than the
maximum stress Vmax, the nucleation ofrecrystallization is generated. Since thedislocation density of the recrystallized grain isconsiderably smaller than that of the deformedmaterial, softening occurs with the growth of
the new grain. Finally, the steady-state stress Vs
is achieved.

3 Theoretical ModelWe employ the model of dislocation evolution
and nucleation used in Refs. [6] and [7].
3.1 Dislocation evolution model
The variation of the dislocation density U withrespect to the strain H is given bydU(2)
k1 U  k2 U .
dHHere, the first term of the right hand side
expresses the work hardening, and k1 is aconstant that represents hardening. The second
term is the dynamic recovery term, and k2 is afunction of temperature T and strain rate H .
The flow stress V is related to the dislocation
density as follows:~(3)
V DPb U ,where D is a dislocation interaction coefficient~of around 0.5, P is the shear modulus, and b
is the magnitude of the Burgers vector.
In our numerical model, the local dislocation
evolution is expressed by Eq. 2, and themacroscopic true stress is calculated from Eq.
3 by changing U to an average dislocation

density Uave over a numerical region.
The following formulations are performed inorder to identify k1 and k2 in Eq. 2. From Eqs. 2
and 3, we can obtain the following equation,

dV
dH

~ §
1
V ·
¸
DP b k1 ¨¨ 1 
2V s ¸¹©

(4)

where Vs is the steady-state stress and isexpressed as~(5)
V s DP b k 1 k 2
From the gradient of the stress-strain curve at V
= 0, k1 can be determined. The steady-state
stress Vs is related to the strain rate H andtemperature T as
§ Q ·
(6)
H A V sm exp ¨  act ¸© RT ¹
where A and m are constants and Qact is theactivation energy. When Vs is determined from
Eq. 6, k2 can be obtained from Eq. 5.
~(7)
k 2 DP b k1 V s
3.2 Nucleation model
Since the nucleation originates at high-angle
boundaries, it is assumed that the nucleation of
DRX only occurs at the original andrecrystallized grain boundaries. In this case, thenucleation mechanism reduces to bulging
nucleation. Considering that the nucleation
originates when the dislocation density reaches
a critical value Uc, Uc can be calculated on thebasis of the bulging mechanism as follows[13]:§ 20JH · ,
(8)
¨ ~
2 ¸
© 3b LMW ¹
where J is the grain boundary energy, L is the13

Ucmean free path of the dislocation, M is the grainboundary mobility and W is the dislocation line
energy calculated by W cPb~ 2 , in which c is aconstant of the order of 0.5. The mean free path L
is calculated from L K D U , where K is aconstant of about 10.

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T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231

4 Phase-Field Modelvariables.

4.1 Phase-field equations for DRX
We generalize the multi-phase-field model
proposed by Steinbach et al. [14] to simulate
the DRX phenomenon, in which therecrystallized grain boundary migration isdriven by the stored energy.
We consider a system containing N different
grains I1,I2,....IN. Phase field ID takes a valueof 1 inside the Dth grain, and is 0 inside othergrains, and 0 <ID < 1 at the grain boundary. ID
is not an independent variable and must satisfy
following condition:

­ § n
·½
(14)
F  ³ ®O ¨ ¦ Ii  1¸¾dV
V¹¿¯ ©i 1Since Ii is a non-conserved order parameter,
the time evolution equation of Ii can be
expressed as follows:
$
GF
(15)
Ii 
O,

N

ID
¦
D

1.

(9)

1

Here, we use the free-energy functional³ f dVFV

2ª N N § aDEº·¨ID  IE  WDE ID IE ¸  f e »dV¦³V ««¦¨¸¹¬ D 1 E D 1 © 2
¼»

,

(10)
where aDE and WDE are the gradient coefficientand the barrier height, respectively, which are
related to the interface energy Jij and theinterface thickness Gij. fe is the bulk free-energy
density and a function of I1,I2,....IN, i.e.,f e I1 , I2 , I N .

feNow, we define a step function Va as­1 0 I D  1
.
(11)
VD ®
¯0 elsewhere
Using this step function, the number of locallypresent phases is expressed asNn

V D x, t .
¦
D

(12)

1

Using Eq. 12, Eq. 9 reduces to
n

¦I

i

1.

(13)

i 1

By introducing the Lagrange multiplier O,
I1,I2,....IN can be treated as independent

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*

GIi
$

where I i denotes the time evolution without
considering a specific time scale.
Furthermore, an interface field \ij is newly
defined as follows:(16)\ ij Ii  I j (i < j),where \ij = –\ij. Substituting Eq. 16 into Eq.
13, we obtain
·
1§ n
(17)
¨ ¦\ ij  1¸ .
Ii¸¨
n© j 1¹
By the definition of the interface field \ij, orusing Eq. 16, the time evolution of \ij becomes$$$GF GF .
(18)
\ ij I i  I j 


GIi

GI j

We can see from Eq. 18 that the time evolutionof \ij is independent of the Lagrange multiplierO. From Eq. 17,
$
1 n $
(19)
Ii
¦\ ij .
nj1

Substituting Eq. 18 into Eq. 19, we have$
1 n § GF GF ·¸.
(20)I i  ¦ ¨¨
n j 1 © GIi GI j ¸¹
The functional derivative GF/GIi is calculated
as
GF wf
wf

GIi wIi
wIi
.
(21)
n
§
aik2 2 · wf e¸¨¦ ¨WikIk  2  Ik ¸  wI
k 1 ©
i
¹
k zi

Taking into account Eq. 21 and the mobility
MIij, the time evolution equation of Ii isobtained as

T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231
Ii

M ijI § GF GF ·¸¨¨GI j ¸¹
j 1 n © GI iI

M ijI ª n ­
wf ·º
1 2
½ § wf2
2«¦ ® Wik  W jk I k  a ik  a jk  I k ¾  ¨¨ e  e ¸¸»2II j ¹¼»w
w¯¿
1
j 1 n ¬k«© iwhere I = 1/2 at x = 0. Solving Eq. 29 for x,a(30)
sin 1  2I  1 .
x
2W
Since the interface region is 0 < I< 1, theinterface thickness G is calculated asaS.                         (31)G2W
The grain boundary energy J is obtained from
Eq. 28 by setting fe = 0 as

n

¦

(22)
Here, we selectwf e wf e
8
I I 'E ,wIi wI j
S i j ij

(23)

where 'Eij is the difference in bulk free-energy
density between grains i and j. The factor 8/S is1needed to satisfy 8 S ³ I 1  I dI
0

1 . Finally,

ªn ­
1 2
½º
2
2
® Wik  W jk Ik  aik  a jk  Ik ¾»
M ij «¦
2
¿» .
k 1¯
«
¦
n
«
»
8
j 1
«¬ S IiI j 'Eij
»¼
n

I

(24)
4.2 Phase-field parameters
The parameters aij, Wij and MIij in Eq. 24 are
related to the material constants. Let us
consider the case of N = 2 and n = 2. For I1 = I,
I2 = 1–I, a = a12 = a21, W = W12 = W21, MI =MI12 and 'E = 'E12, the time evolution
equation of I1 = I is written as
MI ª 2 28ºa  I  W 1  2I I
I 1  I 'E » .
«
2 ¬
S
¼
(25)
In same way, from Eq. 10, we obtain

F

­ a2

(29)

a W
(32)
S.
4 2
Considering the condition that a grainboundary with the equilibrium phase-field
profile migrates with a constant velocity V, weobtain following equation from Eq. 27:
4a
(33)
V MI
'E .
S 2W
Comparing Eq. 33 and the grain boundary
mobility M,
4a
.
(34)
M MI
S 2W
From Eqs. 31, 32 and 34, the phase-fieldparameters are related to the material
parameters as follows:
2
4J
S2
(35)
, a
W
2GJ , M I
M.
S
G
4G
Generalizing Eq. 35 to the multi grain problem,
4J ij
2
S2
, aij
2GJ ij , M ijIWij
M ij .
S
G
4G(36)
where aij = aji, Wij = Wji, MIij = MIji, Jij = Jji and
Mij = Mji, and we set G ij = G.
The driving force of the migration of therecrystallized grain boundary is expressed by
the dislocation density difference between
grains i and j, or(37)'Eij W U j  Ui ,J

we obtainIi§ 2W ·º1ªx ¸¸» ,
«1  sin ¨¨
2 ¬«
¹¼»
© a

n

¦

½

³ ®¯ 2 I  WI 1  I  f ¾¿dV .
2

e

(26)

In a one-dimensional problem, Eqs. 25 and 26
respectively reduce toº8
M I ª 2 w 2II
I 1  I 'E »
 W 1  2I a«2
2 ¬ wxS¼(27)­° a 2 § wI · 2½°(28)
F ³ ® ¨ ¸  WI 1  I  f e ¾dV .
°̄ 2 © wx ¹°¿
From Eq. 27, the phase-field profile under theequilibrium condition can be obtained by
setting I  = 0 and 'E = 0 as follows:

where 'Eij = –'Eji. The dislocation density
increases with continuous deformation by Eq. 2.

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T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231

When nucleation occurs, the dislocation
density in the new grain is set to an initialvalue U0. We assume that the dislocation
density in a grain is uniform, i.e., Ui is
uniformly distributed in the region of Ii > 0
and Ui = U0 when I = 0.

5 Numerical Results
On the basis of Refs. [15] and [7], we employ
the following material parameters for OFHC~
copper: D = 0.5, P = 42.1 GPa, b = 0.256 nm,
A = 2.78×10-45, m = 7.58, Qact = 261 KJ/mol, R= 8.314 J/mol·K, k1 = 3.71×108 1/m, J = 0.625
J/m2, c = 0.5 and K = 10. The grain boundary
mobility M is expressed by~b Gb D§ Q ·exp¨  b ¸ ,             (38)M
kT
© RT ¹
where Gb is the characteristic grain boundary
thickness, D is the boundary self-diffusion
coefficient, k is the Boltzmann constant and Qb
is the boundary diffusion activation energy. We
use GbD = 5.0×10-15 m3/s, k = 1.381×10-23 J/K
and Qb = 104 KJ/mol [7].
Equation 24 is solved by the finite difference
method, and the grid size 'x = 5 Pm and theinterface thickness G = 7'x are employed. The
simulations are carried out under T = 775 K,
H = 2×10-3 1/s, and U0 = 109 1/m2.
5.1 Single-grain growth
During the DRX process, the driving force ofgrain boundary migration changes with
continuous deformation. Here, we performfundamental simulations in which one
recrystallized grain grows in a deformedmaterial with a crystal orientation, and wecompare numerical and theoretical results toclarify the validity of the proposed phase-fieldmodel for DRX.
A nucleus is placed at the origin of the square

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numerical model. The radius of the nucleus is
set to R0 = 5.5 Pm, because it must satisfy R0 >J/Ec to avoid the shrinking of the grain due tothe curvature effect. The critical dislocation
density Uc calculated by Eq. 8 is 8.82×1014
1/m2, then, the stored energy Ec = 0.12 MPa.
Figure 2 shows the variation of grain boundary
migration velocity with time. The result
denoted by a broken line is for a constant
stored energy Ec. This corresponds to the SRX
process. In this case, the velocity is slow at the
beginning of the growth due to the curvatureeffect then gradually become faster. On theother hand, the result for DRX indicates a peakvelocity at approximately 18 s. Figure 3 shows
the variation of dislocation densities in the
deformed material and recrystallized grain, andthe driving pressure with time. It can beobserved that the dislocation density in the
deformed material increases slightly, whereasthat in the recrystallized grain increases rapidly
from U0. Consequently, the driving pressure
calculated by the difference in the dislocation
densities also reaches a peak. At the balance
between the driving pressure and the curvatureeffect, the peak value of the migration velocity
is determined. The solid circles in Fig. 2
indicate the theoretical values calculated fromV = M('Estore – J/R). It is, therefore, confirmedthat the numerical result perfectly agrees withthe theoretical value.
5.1 Multi grain nucleation and growth
The DRX simulation for a polycrystal metal
with regular hexagonal grains and a mean grain
size is 78 Pm is performed here. The size of thecomputational domain is 223.5×129.0 Pm(447×258 grid) and the number of original
grains is six. Periodic boundary conditions are
employed at all boundaries.

T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231

Figure .

Variations of grain boundary

migration velocity with time.

Figure .

Variations of dislocation densities

and driving pressure with time.
First, a preliminary calculation is carried out tocreate the initial polycrystal structure. Figure 4
shows time slices during the preliminary
calculation. The six nuclei are placed in thecalculated positions and grow inside a basematerial driven by a uniform driving force.
Since the grain boundary energies are allidentical, six regular-hexagonal grains can beobtained.
The DRX simulation considering nucleation is
performed using the created regular-hexagonal
grain structure. The nuclei are generated on thegrid satisfying the conditions U > Uc andbe 22.2 /s in the computational area. Here, as
mentioned in the previous section, the required
size of the nucleus is much larger than that ofan actual nucleus. Therefore, we use MI = MI/4
and Estore = 4 Estore at all points and set theradius of the nuclei to R0 = 5'x. Figure 5
demonstrates the time evolution of the
microstructure. When the dislocation density
reaches the critical value Uc, the recrystallized
grains are nucleated at the grain boundaries andgrow toward the center of the original grains.
Figure 6 shows the variation of dislocation
densities with the progress of deformation. It
can be seen that the dislocation density inside
the recrystallized grains is lower than that in
the original grains and increases with thecontinuous deformation. The stress – strain
curves are illustrated in Fig. 7. The broken line
shows the result obtained using Eqs. 2 and 3
and the solid line shows the present result. The
open circles in Fig. 7 correspond to those in
Figs. 5 and 6. We can observe typical multiple
peaks caused by the softening due to thegrowth of recrystallized grains and thehardening due to the accumulation of
dislocations. As a result, it is concluded that thedeveloped phase-field model can simulate the
DRX process including the softening andhardening.

n¦ I  0.6 . The nucleation rate is assumed to
2
i

i 1

229
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T. Takaki, A. Yamanaka and Y. Tomita / AES – ATEMA’ 2007 International Conference, Montreal, Canada,

August 06 -10, 2007, pp. 223 - 231

Figure .

Time slices during computation to

Figure .

Stress-strain curves.

create initial polycrystal structure.

6 Conclusions

Figure . Microstructure evolutions H = (a)0.102, (b) 0.114, (c) 0.126 and (d) 0.138.

We have established a phase-field model forthe
DRX
process,
in
which
the
multi-phase-field
methodproposedby
Steinbach et al. was generalized to simulate thegrain boundary migration driven by stored
energy. The hardening was modeled using thetheoretical equations employed by Guo et al.The basic single-grain growth and the DRX
process resulting from the regular hexagonal
grain structure were simulated. It was
confirmed that the developed phase-fieldmodel can simulate the DRX process.

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

References
Figure .

Variations of dislocation density.

230© AES-Advanced Engineering Solutions (Ottawa, Canada)
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