MULTI-PHASE-FIELD RECRYSTALLIZATION SIMULATION BASED ON DEFORMATIONMICROSTRUCTURE
Tomohiro Takaki∗ & Yoshihiro Tomita∗∗∗ Graduate School of Science and Technology, Kyoto Institute of Technology, Kyoto, 606-8585, Japan∗∗ Graduate School of Engineering, Kobe University, Kobe, 657-8501, Japan

Summary We developed a static recrystallization model which enables natural nucleation and grain growth from deformation microstructure. In this model, the subgrain microstructure at large deformation is predicted from crystal orientation and stored energy
calculated by crystal plasticity finite element simulation. The nucleation and growth of recrystallized grains from the predicted microstructure is simulated by multi-phase-field method with computationally efficient algorism. From the results of simulation employing the developed model, we could confirm that the proposed model can reproduce the plausible recrystallization process.

INTRODUCTION
The development of recrystallization model is essential to predict the recrystallization texture, microstructure and kinetics
during annealing. There are two types of recrystallization model: One is the grain growth model driven by the stored
energy [1], in which the deformation microstructure is characterized by the crystal orientation and stored energy. Since thismodel needs nucleation criteria, the calculated recrytallization kinetics and structure change depending on the predefinednucleation conditions. The other model describes grain growth by the balance of grain boundary energies of the subgrain
structure [2]. This model enables the natural nucleation of recrystallized grains through the abnormal grain growth byintroducing the misorientation dependence of boundary energy and mobility [3], although, in order to express subgrain
microstructures, it requires smaller computational grids and much computational cost comparing to the first model.
In this study, we develop a coupled model for static recrystallization of a phase-field (PF) method that describes second
model mentioned above and crystal plasticity finite element (CP-FE) method. Since the normal CP-FE simulation nevergive subgrain microstructures, the subgrain microstructures are predicted from the results calculated by the CP-FE deformation simulation. The multi-phase-field model (MPF) of Steinbach et al. [4] is employed as a grain growth model andthe misorientation dependency of grain boundary energies and mobility are introduced into the PF model.
RECRYSTALLIZATION MODEL
The procedure of recrystallization model proposed in this study is as follows:(1) Crystal orientation and stored energy of deformed polycrystalline metal are calculated through CP-FE simulation.
(2) The crystal orientation and stored energy are mapped onto the regular grids for PF simulation and smoothed, because
finite elements used in CP-FE simulation and finite difference
grids used in PF simulation are different in shape and size.
22(∂θ/∂x) + (∂θ/∂y) is calculated on all grid points.
γm
Kusing the smoothed stored(4) The misorientation ∆θ is calculated by ∆θ = ∆θm exp 1 − 1 |∇θ| Estoer∆θmenergy Estoer and local orientation gradient |∇θ|. This equation is derived by assuming ∆θ = |∇θ| Ds .
Kγs , where γs is misorientation dependent grain(5) Subgrain size Ds at all grid points is determined using Ds = Estore


∆θ
∆θ
boundary energy expressed by γs = γm ∆θm 1 − ln ∆θm .

(3) The amount of local orientation gradient |∇θ| =(6) Nuclei to generate the subgrain structure are seeded on the regular grids for PF simulation taking into account thecalculated subgrain size.
(7) Subgrain structures are prepared by performing a normal grain growth MPF simulation from the nuclei.
(8) Recrystallization PF simulation is carried out from the initial subgrain structure.
In grain growth simulation during recrystallization, the MPF model of Steinbach et al. [4] is employed. The time evolution
equation of phase field φi which indicates i-th grain is obtained asφφnnn
 2MijMijδFδF1 22a − ajk ∇ φk ,−φ̇i = −=−(Wik − Wjk ) φk +(1)
n
δφi
δφjn
2 ik
j=1
j=1
k=1

φwhere i, j and k are the local grain number from 1 to n at a numerical grid point. aij , Wij and Mij
are the gradientcoefficient, the height of double well potential and the phase field mobility between i-th and j-th grains and are related
24γ
to the material parameters as follows : Wij = δij , aij = π2 2δγij and Mijφ = π4δ Mij , where δ is the grain boundary
thickness and γ ij and Mij are the grain boundary energy and mobility. To introduce the misorientation dependency ofgrain boundary energy and mobility, the following equations are used [3].
4∆θij∆θij∆θij.
(2)γij (∆θij ) = γm
1 − ln, Mij (∆θij ) = Mm 1 − exp −5∆θm∆θm∆θm

Figure 1. Resutls of CP-FE simulation: (a) crystal orientation θ, (b) stored energy Estore and (c) misorientation ∆θ. (d) Nuclei sowed
randomly to generate deformation subgrain microstructure and (e) predicted subgrain microstructuresFigure 2. Time evolution during static recrystallization simulation at (a) 300, (b) 1500, (c) 3000, (d) 4500 and (e) 6000 sHere, ∆θij is the misorientation between i-th and j-th grains, γ m and Mm are respectively the boundary energy and mobility at ∆θij = ∆θm which is the misorientation when the boundary becomes a high angle boundary. The computationaldifficulties caused by using many phase field variables are overcome by employing an efficient algorism of Kim et al. [5].
NUMERICAL RESULTS
Figures 1 (a-c) show a part of results calculated by CP-FE simulation, in which the polycrystal metal is compressed until
50%. The created 2229 nuclei determined from the results is shown in Fig. 1 (d). From these nuclei, the normal grain
growth is simulated to obtain the subgrain structures. Figures 1 (e) is the predicted subgrain structures. From Fig.1 (e),the crystal orientations are almost same with Fig. 1 (a). The small subgrains are observed at the area where the stored
energy is high. Therefore, it is confirmed that the plausible subgrain structures can be predicted by the proposed model.
Figure 2 show the time evolution of subgrain structures and the growth of recrystallized grains during annealing. At
the beginning of simulation, the high-angle grain boundaries become smooth owing to their curvature, as can be seenin Fig. 2 (a) with comparing to Fig. 1 (e). After then, some grains with high-angle boundaries grow abnormally andbecome recrystallized grains. In particular, large recrystallized grains grow into upper and lower grains, because their
two grains are deformed intensely and have small subgrain structures. The large deformation area on the diagonal line ofcomputational domain is also preferred nucleation site. However, since the deformation is localized around the diagonalline, the growth rate of recrystallized grains is slower than those of upper and lower regions. In the large initial subgrain
regions in Fig. 2 (d), no recrystallization occurs in this simulation period. As a result, the spontaneous nucleation byabnormal grain growth and the recrystallized grain growth depending on the deformation structures are observed in themodel developed here.
CONCLUSIONS
We developed a coupled model for static recrystallization with the MPF and CP-FE methods. In this model, the subgrain
structures are predicted from the results calculated by CP-FE simulation, and the MPF model is employed as a grain
growth model. By performing a series of procedure, we could confirm the plausible recrystallization process.
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