A. Yamanaka, T. Takaki and Y. Tomita / Proceedings, AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

ISBN 0-9780479

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

Numerical Modeling of Microstructural Formation
and Deformation Behavior of Fe-C Alloy
A. Yamanaka 1*, T. Takaki 2 and Y. Tomita 3
1

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

Abstract
We develop a numerical model for theprediction of the microstructure formation anddeformation behaviour of an Fe-C alloy bycoupling the phase-field (PF) method with thefinite element method based on thehomogenization method (FEH). The PF
simulation clarifies the growth of the ferrite
phase during the JD transformation in the
Fe-C alloy. The FEH analysis employs themorphology of the D phase and is used toevaluate the mechanical properties of the Fe-C
alloy. Furthermore,  the FEH analysis isperformed to investigate the effects of the© AES-Advanced Engineering Solutions (Ottawa, Canada)
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configuration of microstructures on the
deformation behaviour. The results show thatthe deformation behaviour dependence on themorphology of the D phase is quantitatively inagreement with experimental results. This study
suggests the applicability of the proposedmodel, which enables us to systematicallyevaluate the mechanical properties of the Fe-C
alloy accompanying with the morphologicalchange of the microstructures.

Keywords
Phase-Field Method, Homogenization Method,
Fe-C alloy, Numerical Simulation.

A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

1 Introduction
The mechanical properties of steels, such as
tensile strength, ductility and toughness, arerelated to their inhomogeneous microscopic
deformation behaviour. The deformation
behaviour in the microscopic region is largely
characterized not only by that volume fractionof the constituent phases, but also themorphology of the microstructure that isformed during phase transformations or
precipitation. For simple carbon steel or an
Fe-C alloy, it is well known that the
microstructure can consist of any of theaustenite, ferrite, pearlite, bainite andmartensite. These component microstructures
exhibit various morphologies and affect themechanical behaviour [1]. For example, in theferrite-pearlite steel, the configuration of asofter ferrite (D) phase strongly influences theductility and toughness of the steel. Because of
the above reasons, to efficiently control themechanical properties of steels or alloys, it isessential to develop a numerical model topredict the microstructural formation andsystematically clarify the relationship between
the morphology of the microstructure and themicro- and macroscale deformation behaviour.
Time-dependent Ginzburg-Landau (TDGL)
theory and phase-field (PF) theory have been
proposed as powerful tools for the prediction of
microstructural evolution [2]. The PF approach
has an advantage that it can simulate theevolution of microstructures without explicit
tracking of the position of the interface.
Furthermore, in terms of the evaluation of themechanical properties of materials, thetwo-scale finite element method using
homogenization theory (FEH) has attracted
attention as an excellent numerical techniquefor the investigation of the micro- and214© AES-Advanced Engineering Solutions (Ottawa, Canada)
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macroscopic deformation behaviour ofmaterials [3, 4]. These numerical models havebeen applied to Fe-based alloys. However,according to the author’s knowledge, there has
been no integrated numerical study using boththe PF method and FEH analysis for the
microstructure and mechanical modelling of
Fe-based alloys.
In the present study, the main purpose is todevelop an integrated numerical model for the
microstructure design of the Fe-C alloy bycoupling the PF method with FEH analysis.
Using our proposed simulation model, we
systematically investigate the mechanicalproperties of the Fe-C alloy taking into accountthemorphologicalchangeofthemicrostructure.
We first simulate theevolution of the Dphase due to theaustenite-to-ferrite (JD) transformation below
the Ae3 temperature by the PF method. Then,elastic-plastic FEH analysis is performed using
an unit-cell that represents the simulated
microstructure in the PF simulation. Throughthis procedure, we investigate the distributionof the internal stress-strain field in the
microstructure and the effects of themorphological change of the D phase on themechanical properties of the Fe-C alloy.

2 Phase-Field Method of the Austenite
to Ferrite Transformation in Fe-C Alloy
For the formation of the microstructure in the
Fe-C alloy,the authors proposed the PF modelfor the JD transformation and themorphological change of the ferrite phase [5].
The PF model for the JD transformation issummarized below.
The TDGL equation, which describes theevolution of microstructure during the JD
transformation, is derived by assuming that theA. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

total free energy of the Fe-C alloy decreases
monotonically with time. The total free energyof the Fe-C alloy is defined by the
Ginzburg-Landau-type Gibbs free energyfunctional as follows:G­³ ®¯ g I , u ,T 

H T

C

V

2

2

2½
I ¾dV ,¿(1)

where the first and second terms of Eqn.(1) are
the chemical free energy density and thegradient energy density, respectively. Here, Iisthe nonconserved order parameter, phase field,
defined as I = 1 in the D phase and I = 0 in theaustenite phase. uC is the carbon concentration
related to the normal mole fraction of a carbon
atom. T denotes temperature.
The chemical free energy density of the Fe-C
alloy is postulated to be of the following form.
g I , u C ,TD

p I g u C ,Tᇫᇫᇫᇫᇫ
 1  p I g J u C ,T  Wq I

(2)Here, gD(uC, T) and gJ(uC, T) are the chemical
free energy densities of the pure Dphase andpure austenite phase, respectively. Thesechemical free energy densities are obtained
from the literature [6]. Also, p(I) = I2(10-15I+6I2), q(I) = I2 (1-I)2 and W are the freeenergy density function, the double-wellpotential function and the height of thepotential, respectively. In the gradient term of
the total free energy, H(T) is the gradientenergy coefficient, which describes theinterfacial anisotropy and is related to theinterfacial energy V and the interfacial
thickness Gas follows. 
HT

3VG^1  [ cos k T  T 0 `{ H Tb(3)

Here, [, k, TandT0 are the strength of theanisotropy, the mode number of the anisotropy,
the angle between the interfacial normal andthe x-axis and the preferential growthorientation, respectively. In this study, we
employ the following regularized gradientenergy coefficient in the case of a highinterfacial anisotropy such as [ > 1/(k2-1) [7]:

HT

­
3VG
^1 [ cos k T T 0 `
°
b
°
. (4)
2Si / k T m d T T 0 d 2S i 1 / k T m )°( forᇫᇫ
®
H T m T 0°
cos T T 0
cosT m°°¯ ( forᇫᇫ
2Si / k T m d T T 0 d 2Si / k T m )Here, Tm is defined as the first missingorientation.
The governing equations for the phase field I
and the carbon concentration uC are derived
from the TDGL equations using the total freeenergy of the Fe-C alloy as follows:
wI
wt

wg
ª
2
M I «  H T I 
w
I
¬
w ­
wH wI ½ w ­wH w I ½ º ®H Tᇫᇫᇫᇫᇫᇫ
¾  ®H T¾
wx ¯wx wy ¿ wy ¯
wy wx ¿»¼

^

`

(5)

and
wu C
wt

ª
½º
­ w2g
w2g
  « DC u C ,T ®
u C 
I ¾ » ,2
wu C wI¿¼»¯ wu C¬«(6)

where MI is the kinetic parameter for I whichis related to the mobility of the D/J interface M
DJas [8]:
MI

2W .
6H 0

M D /J(7)

DC(uC, T) is the parameter for the carbon
concentration, which is related to p(I) and theconcentration- and temperature-dependent
mobilities of the carbon atoms in theconstituent phases as follows [9, 10]:
DC uC , T

­
½
§ u ·
vmuc ® p I ¨1 C ¸  1 p I 1 uC ¾ .
3 ¹
©
¯
¿
DpI

uMC
ᇫᇫᇫᇫᇫᇫ

(8)

J 1 p I

MC

3 Finite Element Method Based on the
Two-Scale Homogenization Method
In order to investigate the deformation

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A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

properties of the Fe-C alloy foe different
microstructures,large-deformation
FEH
analysis is performed in two-dimensions. In
this analysis, we consider the Fe-C alloy as aferrite-pearlite two-phase alloy. The microscale
distribution of the constituent phases in thesteel is determined from the PF simulations and
is described using the unit-cell, as mentioned in
the following section.
The Fe-C alloy is considered to be an isotropic
elastic-plastic material and its deformation
behavior is characterized by J2 flow theory.
Therefore, we employ the Plandtl-Reuss
equation as the constitutive equation, that is,
the relationship between the Jaumann rate of

the Kirchhoff stress

S ij and the strain

rate Hij [11]:½­ e
3GV ijc V klc
S ij ® Dijkl
 2
¾H ,
V eq H c 3G 1 ¿ kl¯

(9)where Deijkl, V’ij and Veq are the elastic
coefficient matrix, the deviatric stress tensor
and the equivalent stress, respectively. H’ is theplastic tangent modulus. Using this constitutive
equation, we solve the two-scale elastic-plastic
boundary value problem by the FEH method[4].

4 Computational Model
Figure 1 (a) shows the computational model,
initial condition and boundary condition for the
PF simulation. The size of the computational
domain is 5 × 6 Pm2 and the zero Neumannboundary condition is employed. In the PF
simulation, we simulate the evolution andmorphological change of the preexisting D
phaseat the austenite grain boundary, i.e., thegrain boundary allotriomorph ferrite (DA).

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Therefore, assumingthat the DA is nucleated at
the hexagonal austenite grain boundary, we set
the initial D phase in the computational domain.
The initial concentration is supersaturated in
the austenite matrix andat equilibrium inthe Dphase. Since we simulate the isothermaltransformation, the temperature is set to beconstant at T = 1000 K. Although the growth
direction of the D phase depends on the
Kurdjumov-Sachs orientation relationshipbetween the D phase and the J phase, we define
the growth directions of the D phase with thepreferential  growth orientation T0. Other
interfacial parameters and physical values inthis studyare given as follows [5]: the averageinterfacial energy is V = 1.0 J/m2, the mode
number of interfacial anisotropy is k = 2 andthe interfacial thickness is G = 93.7 nm. Figures
1 (b) and (c) illustrate the computationalmodels for the FEH analysis.xi and yi (i = 1, 2)
indicate the coordinates in the macro- andmicroscopic regions, respectively. As shown in
Fig. 1 (b), the microstructure in the Fe-C alloyis represented by the periodic arrangement ofhexagonal unit-cells and the periodic boundary
condition is employed for the unit-cells [12].
The unit-cell is modeled from the PF
simulations. In order to study the deformation
behavior under tensile deformation, a uniform
tensile deformation is applied to the Fe-C alloy = 10-3 /s.
at a strain rate of (

A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

Figure 1. Computational models and initialconditions for (a) PF simulation and FEH
analysis in (b) micro- and (c) macroscopic
regions.

5 Results and Discussion
5.1 Microstructural evolution duringisothermal J/ D transformation
The evolution of the D phase due to theisothermal J/D transformation is simulated.
Here, we investigate the effects of interfacial
anisotropy on the morphological change of theD phase.
Figure 2 shows the distribution of carbon
concentration during the evolution of the Dphase for [ = 0.2. We can observe that theinitial D/J interface migrates with carbon
diffusion to the austenite matrix. It is also
found that low interfacial anisotropy induces
the growth of the DA. We note that theinterfacial migration behaviour with carbon
diffusion and the build-up of carbon
concentration in front of the moving interface
can be simulated by the PF method.
On the other hand, as shown in Fig.3, many
Widmanstätten ferrite (Dw) plates nucleate andgrow from the initial DA in the case of thehigher strength of anisotropy [ = 0.5. Here, thepreferential growth direction of the D phase,T is the same as the case of Fig.2. From thisresult, the Dw growth mechanism can be
described as follows: (i) the initial Dw tips are
nucleated on the interface of theDA, as shownin Fig.3 (a). In particular, the convex part of theinitial interface is the preferential growth partof Dw. (ii) Some initial Dw tips coalesce intoneighbouring tips and grow competitively(iii) After the
toward the Dw plate.
coalescence, each Dw plate evolves at aconstant rate. Additionally, the rate of DJ
interface migration during the growth of Dw
plates is clearly faster than that of DA formation.
Note that the simulated growth mechanism of
the Dw plates is similar to the diffusional
ledge-wise
growthmechanism[13].
Furthermore, the experimentally observed
morphology of the Dw plate, such as the parallel
broad sides and the fine shape of the tip, can be
simulated by employing the gradient energy
coefficient [14].
Figure 4 shows the evolution of the Dw plate for[ = 0.4. We define Tso as to simulate thegrowth of the Dw plate from one part of theinterface on the DA. It is clearly observed thatthe coarse-grain Dw plate is so large that it
grows across the J grain. The growth rate of Dw
plates decreases with decreasing the strength ofanisotropy.

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A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

Figure 2. Growth of allotriomorph ferrite forstrength of anisotropy [ = 0.2.

Figure 3. Growth of Widmanstätten ferritefor strength of anisotropy [ = 0.5.

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5.2 Unit-cell models for FEH analysis
Figure 5 illustrates the unit-cell models withdifferent morphologies and volume fractions of
the D phase. The total numbers of nodes andcrossed-triangle elements of the unit-cell arealso indicated. The unit-cell model is producedby employing the digital–image basedmodelling method using the image-dataobtained by the PF simulation [15]. Here, (a)PFA-26, (b) PFW-57 and (c) PFL-54
correspond to the microstructures shown in
Fig.2 (d) (DA structure), Fig.3 (d) (fine-grained
Dw structure) and Fig.4 (d) (coarse-grained Dw
structure), respectively. As mentioned above,
we consider the Fe-C alloy as theferrite-pearlite two-phase steel. Therefore, we
assume that the untransformed austenite phasein the PF simulation decomposes into pearlite
during the subsequent cooling process. The
distribution of the constituent phase isdetermined using the profile of the phase fieldI= 0.5, and the physical parameters aredefined for each crossed-triangle finite element.
The white and black areas in the unit-cellcorrespond to the pearlite and the D phases,
respectively. The mechanical property of thepearlite depends on the ferrite-cementite
lamellar spacing. However, in this study, we
assume that the pearlite has an uniform
structure. Accordingly, the initial yield stress ofthe constituent phase is V y = 409 MPa for theD phase and Vy = 490 MPa for the pearlite.
The Young’s modulus of the Fe-C alloy is E =206 GPa and the Poisson’s ratio is Q = 0.3 [16].
5.3 Effects of morphology of ferrite phase on
mechanical properties of Fe-C alloy
We perform FEH analysis using the unit-cellmodels to clarify the effects of themorphological change of the microstructure on

A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

Figure 4. Growth of Widmanstätten ferrite
across the pior-austenite grain for strength ofanisotropy [ = 0.4.

Figure 5. Unit-cell models with different
morphologies and volume fractions of ferrite
phase: (a) allotriomorph ferrite, (b) fine- and(c) coarse-grained Widmanstätten ferrite plates.

the mechanical properties of the Fe-C alloy.
Figure 6 illustrates the macroscopic nominal
stress-nominal strain relationships for  the
Fe-C alloy containing different simulated
microstructures. Although the strain hardening
behaviour of all alloys is similar, we canobserve that the maximum stress decreases
with increasing volume fraction of the softer D
phase.
This
volume-fraction-dependent
behaviour is similar to that found in theexperimental study [16]. However, the increase
the yield stress and the strain hardening
properties due to the formation of afine-grained D structure cannot be simulated,because we do not consider the grain-size
effect in this analysis. In a future study, we will

simulate the size-dependent deformation
behaviour using crystal plasticity theory [17].
Figure 7 shows the evolution of the microscaleequivalent plastic strain in the unit-cell for (a)PFA-26, (b) PFW-57 and (c) PFL-54, for themacroscopic nominal strains (n =0.1, 0.2 and0.3. As shown in the figure, it is clear that theplastic deformation distribution is rather
inhomogeneous and depends on themorphology of the D phase. In the case that theD phase exhibits an allotriomorphic structure,
as for PFA-26, the plastic strain tends to beconcentrated on the DA formed along theprior-austenite grain boundary. Therefore, if theDA phase grows in a coarse-grained
morphology, the DA phase may play a key role219
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A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221

in the fracture behavior of the Fe-C alloy, asreported in the literatures [18, 19].
For PFW-57, on the other hand, we canobserve that the Dw plates in the unit-cell
mainly plastically deform. That is, the plasticstrain is not localized on the grain boundary,but is distributed almost uniformly in theunit-cell along the Dw plates. Furthermore, themagnitude of plastic strain in the unit-cell for
PFW-57 is lower than that for PFA-26.

Figure 7.
Figure 6. Macroscopic nominal stress vsnominal strain relationship.

Distribution of equivalent plasticstrain in the unit-cell.

Acknowledgments
From these results, we conclude that the strain
redistribution due to the formation offine-grained Dw plates causes the increase in
toughness, as shown in the experimental
studies [18]. On the other hand, in PFL-54,
when the coarse-grained Dw structure is present
in parallel formations across the austenite grain
and the spacing between the plates is large, theplastic strain is predominantly distributed on
the plates even though the Dphase exhibits the
Dw structure.

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Financially supported from the Ministry of
Education, Culture, Sports, Science andTechnology of Japan though Grand-in-Aid for
Scientific Research (B), 18360061, 2007 andfor JSPS Research Fellow is acknowledged.

References[1] Krauss, G. and Thompson, S. W., (1995),
Ferritic Microstructures in Continuously
Cooled Low- and Ultralow-carbon steels,
ISIJ Int., 35, 937-945.
[2] Kobayashi, R., (1993), Modeling andNumerical Simulations of Dendritic
Crystal Growth, Phys. D, 63, 410-423.

A. Yamanaka, T. Takakiand Y. Tomita / AES – ATEMA’ 2007 International Conference,Montreal, Canada, August 06 -10, 2007, pp. 213 - 221[3] Grudes, J. M. and Kikuchi, N., (1990),
Preprocessing and Postprocessing for
Materials Based on the Homogenization
Method with Adaptive Finite Element
Method, Comp. Meths. Appl. Mech. Engn.,
83, 143-198.
[4] Higa, Y. and Tomita, Y., (1999),
Computational Prediction of Mechanical
Properties of Nickel-Based Superalloy
with Gamma Prime Phase Precipitates,
Advances Materials and Modeling of
MechanicalBehaviour,
Vol.
III,
1061-1066.
[5] Yamanaka, A. Takaki, T. and Tomita, Y.,(2006),
Phase-field
SimulationofAustenite to Ferrite transformation andWidmanstätten ferrite formation in Fe-C
Alloy, Mater. Trans. JIM, 47, 2725-2731.
[6] Gustafson, P., (1985), Thermodynamic
Evaluation of the Fe-C System, Scand. J.
Metall., 14, 259-267.
[7] Takaki, T. Hasebe, T. and Tomita, Y., (2006),
Two-dimensional Phase-field SimulationofSelf-assembled
Quantum
Dot
Formation, J. Crys. Grow., 287, 295-299.
[8] Hillert, M., (1975), Diffusion and Interface
Control of Reaction in Alloys, Metall.
Trans. A, 6, 1350-1355.
[9] Ågren, J., (1986), A Revised Expression for
the Diffusivity of Carbon in Binary Fe-C
Austenite, Script. Mater., 20, 1507-1510.
[10] Ågren, J., (1982), Computer Simulation oftheAustenite/Ferrite
Diffusional
Transformations in Low Alloyed Steels,
Acta Metall., 30, 841-851.
[11]
Tomita,Y.,(1990),
Numerical
Elasto-Plastic Mechanics, YOKENDO
Ltd.
[12] Ohno, T., Matsuda, T. and Wu, X. (2001),

A
Homogenization
Theoryfor
Elastic–viscoplastic Composites with
Point Symmetry of Internal Distributions,Int. J. Solids. Struct., 38, 2867-2878.
[13] Enomoto, M., (1994), Thermodynamics
and Kinetics of the Formation of
Widmanstätten Ferrite Plates in Ferrous
Alloy, Metall. Mater. Trans. A., 25A,
1947-1955.
[14] Yang, Z-G.. Wang, J-J., Li, C-M., Fang,
H-S. and Zheng, Y-K., (1998), Ledges in
Widmanstätten Ferrite Observed by
Scanning Tunneling Microscopy, J. Mater.
Charac., 17, 331-333.
[15] Terada, K. and Kikuchi, N., (1998),
Microstructural Modeling Technique for
Homogenization Analysis Using Digital
Images, Trans. JSMS, 64, 170-177.
[16] Ishikawa, N. Parks, D. M., Socrate, S. andKurihara, M. (2000), MicromechanicalModeling of Ferrite-Pearlite Steels Using
Finite Element Unit Cell Models, ISIJ Int.,40, 1170-1179.
[17] Higa, Y. and Tomita, Y., (2003),
Computational
Simulationof
Characteristic Length Dependent Behavior
of Polycrystalline Metals, Trans. JSMS, 69,
23-29.
[18] Bodnar, R. L. and Hansen, S. S., (1994),
Effects of Widmanstätten Ferrite on the
Mechanical Properties of a 0.2 Pct C-0.7
Pct Mn Steel, Metall. Mater. Trans. A, 25A,
763-773.
[19] Gulyaev, A. P. and Guzovskaya, M. A.,(1977), Resistance to Fracture of Weldable
Carbon Structural Steel with FerritePearlite and Widmanstätten Structures,Met. Sci. Heat Treat., 19, 1020-1024.

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