Static Recrystallization Simulation Using Phase-Field Model
Based on Crystal Plasticity Theory
Tomohiro Takaki1, Akinori Yamanaka2, Yoshikazu Higa3, Yoshihiro Tomita2
1

Faculty of Maritime Sciences, Kobe University, 5-1-1, Fukaeminami, Higashinada,
Kobe 658-0022, Japan, takaki@maritime.kobe-u.ac.jp;
2
Graduate School of Science and Technology, Kobe University,1-1, Rokkodai, Nada, Kobe 657-8501, Japan;
3
Department of Mechanical System Engineering, Okinawa National College ofTechnology, 905, Henoko, Nago 905-2192, Japan.

ABSTRACTA numerical model and computational procedure for static recrystallization are developedusing a phase-field model coupled with crystal plasticity theory. The microstructure andaccumulated dislocation density during deformation of a polycrystalline metal are simulatedusing finite element method based on the strain gradient crystal plasticity theory. Phase-field
simulation of the nucleation and growth of recrystallized grain is performed using thecrystallographic orientation and stored energy calculated by crystal plasticity finite element
simulation. Through this computational procedure, we can get the final recrystallization
microstructure taking the deformation microstructure into consideration.

1. IntroductionThe microstructures formed during annealing are significantly affected by the predeformation microstructures, since the recrystallization originates from dislocation cells orsubgrains which appear after deformation and subsequently the recrystallized grain growthoccurs driven by the stored energy resulted from the dislocation accumulated during
deformation. Recently, the numerical studies using Monte Carlo Potts model [1] and cellular
automata model [2] based on the data measured by EBSD analysis are made on static
recrystallization. However, to enable more systematic investigations for recrystallization
texture, it is key to develop the computational procedure without using experimental data.
In this study, we develop a phase-field model which can simulate the nucleation andgrowth of recrystallized grain. Here, the crystallographic orientation and dislocation density at
the deformation of polycrystalline metal are simulated using finite element method based on
the strain gradient crystal plasticity theory [3].

2. Numerical Procedure
The numerical procedure developed here consists of following three steps:< Step. 1> The crystallographic orientation and dislocation density after deformation ofpolycrystalline metals are calculated by crystal plasticity finite element simulation.
< Step. 2 > The calculated data are mapped onto a regular lattice used in phase-fieldsimulation. The stored energies calculated from dislocation density are smoothedon the lattice, because phase-field method requires the continuous driving forces.

< Step. 3 > Phase-field simulation during recrystallization is performed in which thenucleation and growth of recrystallized grain are reproduced.

3. Models and Results
By following the procedure shown in previous chapter, crystal plasticity theory, data mapping
method and phase-field method employed in this study are briefly explained together withnumerical results.

3.1 Strain gradient crystal plasticity
The crystal plasticity finite element method based on a strain gradient theory of ratedependent plasticity [3] is used to examine the microstructure and dislocation distribution
during deformation of polycrystalline metals. Here, the critical resolved shear stress on slip
system (a) is assumed to be a Bailey-Hirsch type function:~g (a ) = g 0(a ) + aµb ∑ϖ ab ρ (b )
(b )

(1)

~
where, g 0(a ) initial value of g (a ) , a constant, m elastic shear modulus, b the magnitude ofBurgers vector, ϖ ab interaction matrix, and ρ (b ) accumulated dislocation density. Theaccumulated dislocation density is the sum of the densities of SSD and GND. The evolutionof SSD is expressed by the balance between the production rate of dislocations and theannihilation by dynamic recovery. The density of GND is calculated from the gradient ofshear strain on slip system (a). Since the hardening equation, Eqn. 1, includes the strain
gradient term through the dislocation density, it is possible to express the grain size effects.
Figure 1(a) shows the polycrystal model with 23 grains where the average grain diameter is115.5 µm. This model is divided into 64x64 regular crossed-triangle elements and iscompressed at a strain rate 10-3 s-1 as a 2-slip plane strain problem. Figure 1(b) and (c)illustrate the crystallographic orientation and stored energy after 50% compression,~respectively. The stored energy Estore is calculated by E store = 0.5 ρµb 2 , where ρ is the total
dislocation density of all slip systems.

Figure 1. Crystallographic orientation (a) before deformation and (b) at 50%
compression, and (c) stored energy. Solid lines indicate initial grain boundaries.
3.2 Data mapping

The data computed by crystal plasticity finite element simulation, i.e. crystallographic
orientation and stored energy, are mapped onto the regular lattice of phase-field simulation.
The relationship between triangle elements used in finite element simulation and lattice usedin phase-field simulation is shown in Fig. 2. The regular lattice size is 0.4 µm and triangle
elements are a part of Fig. 1 (b). First, the crystallographic orientation and stored energy on
lattice are determined as a value inside an element. In other words, the values of lattice
located inside a triangle element are all identical. Next, the misorientation on lattice isdetermined as a maximum value among the four misorientations between the two neighboring
lattices, i.e., ∆θ1, ∆θ2, ∆θ3, and ∆θ4, represented in Fig.2. Then, the stored energies mapped
onto the regular lattice are smoothed using Winslow’s smoothing method:mEstore = ∑ E j w jj =1m∑wj =1(2)jwhere, Estore smoothed stored energy on lattice i, Ej stored energy before smoothing on latticej, wj weight represented as an inverse of distance between lattice i and lattice j, and m thenumber of lattice inside a considering circle with radius r. In this study, r = 3∆x is selected
where ∆x is lattice size. Finally, the crystallographic orientation, misorientation and smoothed
stored energies on all regular lattices are determined. Figures 3 (a) and (b) show thedistributions of smoothed stored energy and misorientation, respectively, for the regions 1(332.8 x 64 µm2) and 2 (268.8 x 76.8 µm2) illustrated in Fig.1.
(i,j+1)ri∆θ1(i+1,j) (i,j)

∆θ4
∆θ3
∆θ2

(i-1,j)(i,j-1)Figure 2. Triangle elements for crystal plasticity finite element simulation and regularlattice for phase-field simulation.

Figure 3. (a) smoothed stored energy and (b) misorientation. 1 and 2 indicates region 1and 2 of Fig.1, respectively.
3.3 Phase-field simulation
By using the data mapped onto regular lattice, the nucleation and growth of recrystallized
grain are simulated using phase-field method. In the present phase-field model, two orderparameters, i.e., phase field φ which equals zero in the deformed matrix and unity inrecrystallized grain and crystallographic orientation θ, are employed. The time evolved
equations for the order parameters are as following:∂φ∂f (φ )
= M φ α 2∇ 2φ −− 2φs ∇θ ∂t∂φ

where,∂θ∇θ 
1= M θ 2 ∇ ⋅  φ 2 s∂t∇θ φ(3)(4)

f (φ ) a free energy density expressed by double well type function:()f (φ ) = (1 − p(φ ))Estore + Wq(φ ) with p(φ ) = φ 3 10 − 15φ + 6φ 2and q(φ ) = φ 2 (1 − φ ) ,W = 6σ δ energy wall height related to interface energy σ and interface thickness δ,
2α = 3δσ gradient coefficient, s = α 2W π , M φ = m 2W 6α mobility for φ where m is a
mobility of grain boundary migration, and M θ = (1 − p(φ ))M φ mobility for θ. Equations (3)and (4) are solved by an adaptive finite element method [4].
Figure 4 shows the initial nucleus sits, the growth process, and final recrystallization
microstructure. The site saturated nucleation model is used, in which the following nucleation
criteria are assumed: (1) the high angle grain boundary of more than 15 degree, (2) the stored
energy of more than 0.6 MPa, (3) the orientation of the nucleus is presented in the deformed
structure, and (4) the minimum distance between the two neighboring nuclei is 10∆x. The
initial radius of nucleus is set to 3∆x. From Fig.4, it is observed that, in region 1, thenucleation and growth of recrystallized grains occur inside the grain before deformation,
while, in region 2, the recrystallized grains originate only from grain boundary.

Figure 4. Time evolutions of recrystallized grain growth and final recrystallization
microstructures. Adaptive meshes are illustrated only for results at 500s.
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Mater. Sci. Eng. A405, 86, (2005).
[2] D. Raabe and L. Hntcherli, “2D cellular automaton simulation of the recrystallization
texture of an IF sheet steel”, Compu. Mater. Sci. 34, 299, (2005).
[3] Y. Higa, Y. Sawada and Y. Tomita, “Computational simulation of characteristic length
dependent deformation behavior of polycrystalline metals”, Tran. JSME 69, 523, (2003).
[4] T. Takaki, T. Fukuoka and Y. Tomita, “Phase-field simulation during directional
solidification of a binary alloy using adaptive finite element method”, J. Crystal Growth 283,263, (2005).