X International Conference on Computational Plasticity
COMPLAS X
E. Oñate and D.R.J. Owen (Eds)c°CIMNE,Barcelona, 2009

CRYSTAL PLASTICITY PHASE-FIELD SIMULATION OF
DEFORMATION BEHAVIOR AND MICROSTRUCTURE
EVOLUTION IN POLYCRYSTALLINE MATERIAL
Akinori Yamanaka∗† , Tomohiro Takaki†† ,Yoshihiro Tomita††† and Masahiko Yoshino†∗,† Graduate School of Science and Engineering, Tokyo Institute of Technology

Ookayama, Meguro-ku, Tokyo, 152-8552, Japane-mail: ayamanaka@mes.titech.ac.jp, myoshino@mes.titech.ac.jp
Web page: http://www.3mech.titech.ac.jp/†† Graduate School of Science and Technology, Kyoto Institute of Technology

Matsugasaki, Sakyo-ku, Kyoto, 606-8585, Japane-mail: takaki@kit.ac.jp, Web page: http://www.cis.kit.ac.jp/ takaki†† Faculty of Engineering, Fukui University of Technology

3-6-1, Gakuen, Fukui-shi, Fukui 910-8505, Japane-mail: tomita@fukui-ut.ac.jp, Web page: http://www.fukui-ut.ac.jp/ut/english/
Key words: Phase-Field Method, Crystal Plasticity Theory, Microstructure Evolution.
Summary. A new crystal plasticity phase-ﬁeld model is developed to simulate microstructure
evolution with elastoplastic deformation. In this paper, three-dimensional simulation of thegrowth of precipitates during solid phase transformation is performed. And, it is demonstrated
that the distribution of stress and dislocation density around the growing precipitates can bepredicted by the developed phase-ﬁeld model.

1INTRODUCTION

Recently, the phase-ﬁeld (PF) method has been actively studied as a powerful tool to simulate
microstructure evolutions in various materials. In particular, Guo et. al. [1] proposed theelastoplastic PF models to simulate the microstructure evolution with both elastic and plastic
deformations. However, since their elastoplastic PF model is based on the J2 ﬂow theory and,therefore, can not describe changes of crystal orientation and dislocation density during themicrostructure evolution. In order to conduct precise numerical simulation of the microstructure
evolution with the elastoplastic deformation, such as formation of lath martensite structure,using the PF method, it is essential to enable us to evaluate the changes of crystal orientationand dislocation density due to the microstructure evolution in frame of the PF theory.
In this study, a new crystal plasticity phase-ﬁeld model (CP-PFM) is developed by coupling
the phase-ﬁeld microelasticity (PFM) theory [2] with the crystal plasticity theory. Furthermore,using the developed CP-PFM, the three-dimensional simulation of the growth of precipitates
1Akinori Yamanaka, Tomohiro Takaki, Yoshihiro Tomita and Masahiko Yoshino

during solid phase transformation is performed. And, it is demonstrated that the CP-PFM canpredict the evolution of the precipitates and investigate the plastic deformation behavior aroundthe precipitates including the change of dislocation density.
2CRYSTAL PLASTICITY PHASE-FIELD MODEL

The total free energy of the system is deﬁned by the Ginzburg-Landau free energy functional,
which is the sum of chemical free energy Gchem , elastic strain energy Gelastic and gradient energy
Ggrad as, Gtotal = Gchem + Gelast + Ggrad . Here, the chemical free energy Gchem is represented
by the Landau polynominal expansion of the phase-ﬁeld variable φi . φi is deﬁned as φi = 1
in the i th precipitate and φi = 0 in the parent phase and other precipitates. In the interface
region, φi gradually changes from 1 to 0.
In this CP-PFM, the elastic strain energy is evaluated by the PFM theory proposed by
Khachaturyan [2]. According to the PFM theory, the elastic strain energy can be described by
the following equation.
Z1elCijkl ²elkl ²ij dV
2 VZ³
´³
´
1
Cijkl ²ckl − ²0kl ²cij − ²0ij dV
2 V

Gelast ==(1)c0Here, ²elij , ²ij , ²ij are the elastic strain, the total strain and the eigen strain, respectively. Thetotal strain is deﬁned as the sum of the homogeneous strain ²̄cij and heterogeneous strain δ²cij as,²cij = ²̄cij +δ²cij . The homogeneous strain is determined from the macroscopic boundary condition.
On the other hand, the heterogeneous strain is calculated by solving the mechanical equilibrium
equation using the Fourier transformation. And, the heterogeneous strain can be obtained by
the inverse Fourier transformation of the following heterogeneous strain in the Fourier space.

δ²̂cij (~k) =10(nj Ωmi (~n) + ni Ωmj (~n)) nn σ̂mn(~k)
2(2)

¯ ¯

¯ ¯Here, Ω−1n) = Cijmn nn nj and ~n = ~k/ ¯~k ¯ is the unit vector in the Fourier space.
mi (~The eigen strain in Eqn. (1) is assumed to be the sum of the transformation strain ²transandijpp0
transthe plastic strain ²ij evaluated by the crystal plasticity theory as, ²ij = ²ij+ ²ij . The transformation strain is deﬁned as a linear function of the phase ﬁeld variable and the transformation
P00
strain due to formation of m th variant of the precipitates as, ²trans= N
m=1 ²ij (m)φm . The plasij

tic strain is calculated by time integration of the following plastic strain rate; ²̇pij =Pn(a) (a).
a=1 Pij γ̇(a)Here, γ̇ (α) and Pij are shear strain rate and the Schmid tensor on the a th slip plane. In thisstudy, we employ the following equation [3].
¯γ̇(a)¯1(a) ¯ τ (a) ¯ m −1(a) τ¯¯= γ̇0 (a) ¯ (a) ¯¯¯gg(3)Here, the evolution of the critical resolved shear stress g (a) is assumed to be the modiﬁed BaileyHarsh equation based on the strain-gradient crystal plasticity theory [4].
2Akinori Yamanaka, Tomohiro Takaki, Yoshihiro Tomita and Masahiko Yoshino

Precipitate 3(φ3 > 0.6)

[001]

50s

72s

[100]

[010]

36s

Precipitate 1 Precipitate 2(φ1 > 0.6)

(φ2 > 0.6)

Figure 1: Evolution of the precipitates during phase transformation.

The evolution of the precipitates is described by solving the following time-dependent GinzburgLandau (TDGL) equation for the phase ﬁeld variables.
µ∂φiδGtotalδGchem δGelast δGgrad= −Mφ= −Mφ++∂t
δφi
δφi
δφiδφi3¶(4)

SIMULATION RESULTSIn this study, the three-dimensional simulation of the growth of precipitates with the elastoplastic deformation is performed. The cubic computational domain of 100 × 100 × 100 µm3
is meshed with 643 regular computational grids. Thus, the mesh size is ∆x = 1.56 µm. Three
initial nuclei of the precipitate are set to be formed in the computational domain randomly. It
is assumed that the phase transformation induces isotropic dilatational transformation strain.
Furthermore, we assume that the precipitate and parent phase have FCC crystal structure and,thus, the twelve slip systems are considered. The TDGL equation is solved with the ﬁnite difference method and the fast Fourier transformation under the periodic boundary condition. For
the numerical simulation, the following physical values and parameters are used: the mobility ofphase ﬁeld Mφ = 1.0, Young’s modulus E = 77.0 GPa, Poasson’s ratio ν = 0.375, the referencial(a)shear strain rate γ̇0 = 0.1 s−1 , the strain rate sensitivity factor m = 0.05, the initial dislocation
density ρ0 = 1010 1/m−3 and time increment ∆t = 0.002 s.
Figure 1 shows the growth of the precipitates during the phase transformation. The precipitate i is visualized by the proﬁle of φi ≥ 0.6 (i = 1, 2, 3). All precipitates grow with time andthe grain boundary is formed between diﬀerent precipitates. Figure 2 indicates the evolution ofequivalent stress with the growth of the precipitates on the {100} cross-section for x = 40∆x.
It is shown that the stress concentrates in the parent phase near the interface and the highstress region migrates with the growth of the precipitates. Furthermore, since we employ thestrain-gradient crystal plasticity theory, the changes of SS and GN dislocation densities in theP(n) aroundmicrostructure can be evaluated. The evolution of the dislocation density ρa = 12
n=1 ρ(n)the precipitates is shown in Fig. 3. Here, ρis calculated by the sum of the SS dislocation
(n)
(n)
(n)
density and the GN dislocation density as, ρ = ρSS + kρGN k. From the Fig. 3, the plastic
deformation occurs in the high stress region between the precipitates. Thus, it can be observedthat the high dislocation density is exhibited in the vicinity of the grain boundary.

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Akinori Yamanaka, Tomohiro Takaki, Yoshihiro Tomita and Masahiko Yoshino

[001]

50s

72s

[100]

[010]

36s

Precipitate 3
Precipitate 2

3103
204
Equivalent stress σ [MPa]Figure 2: Evolution of equivalent stress on the {100} cross-section for x = 40∆x.
[001]

50s

72s

[100]

[010]

36s

Precipitate 3
Precipitate 2

0.01
0.54
1.1
Dislocation density ρa [ 1013 m-3 ]Figure 3: Evolution of dislocation density on the {100} cross-sections for x = 40∆x.

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CONCLUSIONSIn this study, a new CP-PFM was developed by coupling the PFM theory with the crystal
plasticity theory to simulate the microstructure evolution with the elastoplastic deformations.
And, the three-dimensional simulation of the growth of the precipitates was performed by using
the developed CP-PFM. The simulation results demonstrated that the high stress region isformed around the precipitates due to the dilatational transformation strain. And, the highdislocatin density was exhibited near the grain boundary. From these results, it can be concluded
that the developed phase-ﬁeld model can simulate not only the microstructure evolution during
the phase transformation, but also the plastic deformation behavior during the microstructure
evolution including the changes of the crystal orientation and the dislocation density.
REFERENCES[1] X. H. Guo, S-Q. Shi and X. Q. Ma, Appl. Phys. Lett., 87, 221910, (2005).
[2] A. G. Khachaturyan, Theory of Structural Transformations in Solids, Willey and Sons,NY., (1984).
[3] J. Pan and J. R. Rice, Int. J. Solid. Struct., 19, 973-987, (1983).
[4] T. Ohashi, Philos. Mag., 70, 793-803, (1994).

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