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

MULTI-PHASE-FIELD MODELING AND SIMULATIONFOR STATIC RECRYSTALLIZATIONT. Takaki∗ and Y. Tomita†∗ Graduate School of Science and Technology

Kyoto Institute of Technology
Matsugasaki, Sakyo, Kyoto, Japane-mail: takaki@kit.ac.jp, web page: http://www.cis.kit.ac.jp/˜takaki/† Department of Mechanical Engineering

Fukui University of Technology
3-6-1, Gakuen, Fukui, Japane-mail: tomita@fukui-ut.ac.jp
Key words: Static Recrysatallization, Phase-Field Method, Crysatl Plasticity, Microstructure
Evolution.
Summary. The static recrystallization simulations by driving force model and subgrain growthmodel are performed and both results are compared in detail. As a result, the similar recrystallized microstructures are observed for both models.

1INTRODUCTION

It is well known that the static recrystallization (SRX) which occurs during post deformation annealing is largely aﬀected by deformation microstructure formed during cold working.
Therefore, to investigate and predict SRX microstructure and texture numerically with highaccuracy, it is necessary to simulate SRX process by taking the deformation microstructure intoconsideration. A coupling of deformation simulation by crystal-plasticity ﬁnite element (CP-FE)
method and microstructure evolution simulation by phase-ﬁeld model, Monte Carlo model orcellular automata model is believed to be the most promising approach for SRX microstructure
design. In this coupling model, there are two kind of approach in the evolution simulation ofrecrystallized grain. One is a grain growth model driven by stored energy or elastic energy ofdislocation [1, 2]. The other model is subgrain growth model where the microstructure evolves
so as to keep the balance of grain boundary energies and the SRX simulation starts from thedeformation microstructure predicted from the results of CP-FE simulation [3, 4]. Here, we callthe ﬁrst model driving force model and second model subgrain growth model . Figure 1
schematically shows the growth of recrystallized grain (RG) in deformed grain (DG).
In this study, we compare the recrystallization microstructures calculated by driving force
model and subgrain growth model. Although the subgrain growth model enables a spontaneous
nucleation of recrystallized grain through abnormal grain growth, the driving force model needs

1

T. Takaki and Y. Tomita(b)(a)

RG

RG

DG

DG

Figure 1: Two recrystallization models: (a) driving force model and (b) subgrain growth modela nucleation criterion. Therefore, the recrystallization simulations from a shear band are performed, where the shear band is assumed as a region formed by many subgrains with arbitrary
crystal orientation.
2

PHASE-FIELD MODEL

A multi-phase-ﬁeld (MPF) model proposed by Steinbach and Pezzolla is employed here [5].
Let us consider a polycrystalline system including N grains. The ith grain is indicated by thephase ﬁeld φi , where φi takes a value of 1 inside the ith grain, 0 inside the other grains and0 < φi < 1 at the grain boundary. The evolution equations of phase ﬁeld φi are expressed byφ̇i = −n 2M φ
ij
j=1n

 n 



(Wik − Wjk ) φk +k=1

1 28
aik − a2jk ∇2 φk −φi φj ΔEij ,2π(1)where, n is the number of phase ﬁeld with a value of 0 < φi ≤ 1 on a grid point and aij , Wijand Mijφ , are respectively related to the grain boundary thickness δij , grain boundary energy γij
and grain boundary mobility Mij byaij =22δγij ,πWij =4γij,
δMijφ =

π2Mij .
8δ(2)

ΔE√ij is the diﬀerence in stored energy between grains i and j and 8/π is obtained from1
φ1 φ2 dφ = 01 φ (1 − φ)dφ = π8 .
0From a numerical point of view, the MPF model has a high computational cost, because
usually N phase ﬁelds must be saved and N evolution equations must be solved at all gridpoints. However, Steinbach and Pezzolla’s MPF model has some computational merits that wedo not have to solve Eq. (1) at the grid points with n = 1 and it is suﬃcient to save not N but
n phase ﬁelds at a grid point. To further improve the computational eﬃciency, we introduce thealgorithm proposed by Kim et al. [6] and modify it to achieve more accurate computation.
3NUMERICAL CONDITIONS

Figure 2 shows the initial structures for (a) driving force model and (b) subgrain growthmodel. The computational domain with 150 μm × 150 μm is divided into 1500 × 1500 ﬁnite2

constant stored energy

constant stored energy

T. Takaki and Y. Tomita(b) subgrain growth model(a) driving force model

Figure 2: Initial structures

diﬀerence grids. The central region consisting of small subgrain with crystal orientation of0∼90◦ is shear band and the both sides of shear band are deformed grains. The deformedgrain is modeled by subgrains with misorientation of 50±10◦ for subgrain growth model anda grain with crystal orientation of 50◦ and stored energy of 0.5 MPa for driving force model.
For high angle grain boundary of more than 15◦ , grain boundary energy γ and grain boundary
mobility M are set to γ = γ0 = 0.6 J/m2 and M = M0 = 4.0741×10−14 m4 /Js, respectively.
For low angle grain boundary,those are set to γ = γ0 (Δθ/15) {1 − ln (Δθ/15)} and M =
4M0 1 − exp −5 (Δθ/15)
where Δθ is misorientation of 0∼45◦ . By setting above condition,
spontaneous nucleation from shear band occurs through abnormal grain growth.
4NUMERICAL RESULTSFigures 3 and 4 show the microstructure evolutions during static recrystallization simulation
for driving force model and subgrain growth model, respectively. Here, the low angle and highangle grain boundaries are indicated by white and black lines, respectively. At the beginningof simulation, the grain growth inside shear band is observed and the size of subgrains in shear
band increases. After a period of time, some grains surrounded by high angle grain boundarywith high mobility grow abnormally into the deformed grain. Comparing driving force modeland subgrain growth model, the size of recrystallized grain of driving force model is relatively
smaller than that of subgrain growth model. This is thought to be caused by the diﬀerence of theamount of energy stored in the deformed grain. In the driving force model, because the stored
energy inside deformed grain is constant, almost ﬂat grain boundaries in front of recrystallized
grain are observed. However, because we can see similar nucleation and growth of recrystallized
grain for both models, it is concluded that the both model can be used in the coupling SRX
model with CP-FE method and phase-ﬁeld method.
Although we need much more numerical experiments and discussions, it may be said that thedriving force model and subgrain growth model can be replaced each other when the nucleationsite is clear as a shear band. This brings us computational beneﬁts, because the computational
cost needed in driving force model is lower than that of subgrain growth model. These areachieved by employing Steinbach’s MPF model, because this MPF model enables the simulations
by driving force model and subgrain growth model.

3T. Takaki and Y. Tomita(a)

(b)

(c)

(d)

Figure 3: Microstructure evolutions during SRX for driving force model: (a) 10000, (b) 20000, (c) 50000
and (d) 100000 steps

(a)

(b)

(c)

(d)

Figure 4: Microstructure evolutions during SRX for subgrain growth model: (a) 100000, (b) 200000, (c)300000 and (d) 400000 steps

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[3] B. Radhakrishnan, G. Sarma, H. Weiland, P. Baggethun, Phase-Field Model during StaticRecrystallization based on Crystal-Plasticity Theory Modelling Simul. Mater. Sci. Eng.,8, 737–750, (2000).
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