XXII ICTAM, 25–29 August 2008, Adelaide, Australia

ELASTOPLASTIC PHASE-FIELD SIMULATION OF CUBIC-TETRAGONAL MARTENSITIC
TRANSFORMATION IN POLYCRYSTALS
Akinori Yamanaka∗ , Tomohiro Takaki∗∗ & Yoshihiro Tomita∗∗∗∗ Graduate School of Science and Technology, Kobe University, Kobe, 657-8501, Japan∗∗ 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 perform an elastoplastic phase-field simulation of the cubic → tetragonal martensitic transformation with plastic deformation in a two-dimensional polycrystal. The results shows that the proposed phase-field model can reproduce the cubic → tetragonalmartensitic transformation in the polycrystalline material including not only evolution of stress-accommodating multi-variant martensitic structure from grain boundary of parent phase but also plastic deformation reducing the transformation-induced stress.

INTRODUCTION
Recently, a phase-field (PF) method has been extensively studied as a powerful tool for predicting microstructural evolution [1] and has been applied to martensitic transformation. Khachaturyan and other researchers have proposed the phasefield microelasticity (PFM) models of the martensitic transformation [2, 3]. In the previous study [4], we have proposed
the elastoplastic phase-field model (EPFM) of cubic → tetragonal (CT) martensitic transformation based on the PFM theory and simulated self- and plastic accommodation behaviors in a single crystal. The results show that plastic deformation
during the CT transformation largely accommodates the transformatio-induced stress and arrests growth of the martensite
phase. In this study, the CT martensitic transformation in a two-dimensional polycrystal is simulated using the EPFM andstudy the effects of plastic deformation on the evolution of the martensite phase in the polycrystal.
PHASE-FIELD MODEL
The EPFM of the CT martensitic transformation is constructed by combining the PFM model of the martensitic transformation [3] with the TDGL equation of the plastic strain proposed by Guo et al [5]. To simulate the CT martensitic
transformation using the PF theory, the total free energy of the system G is described by the Ginzburg-Landau-type Gibbs
free energy functional, which is defined as the sum of chemical free energy, gradient energy and elastic strain energy. The
chemical free energy is defined as the Landau polynomial expansion with respect to the order parameters. In this model,
the order parameter φi describes the continuous distribution of the i-th orientation variant of the martensite phase whose
tetragonality axes are along the three <100> directions in the cubic phase. φi gradually changes from φi = 1 in the i-th
variant to φi = 0 in the other variants of tetragonal and parent phases. The gradient energy is defined as the sum of gradientenergies due to the inhomogeneity of order parameters.
The elastic strain energy density of a system with an arbitrary mixture of the cubic and the tetragonal phases can beevaluated using the micromechanical approach proposed by Khachaturyan, i.e., the PFM theory [2] as,1
gel = Cijkl (²cij − ²0ij )(²ckl − ²0kl ),(1)2
where Cijkl , ²cij and ²0ij are the elastic coefficient matrix, the total strain tensor and the total eigen strain tensor, respectively. In this study, to simulate the CT martensitic transformation with elastoplastic deformation in two-dimensions, thetotal eigen strain is defined as [5],2∑p²0ij =Rik (θ)²00(2)kj (m) + ²ij ,m=1

where the first and second terms are the transformation-induced eigen strain ²tij and the plastic strain ²pij , respectively.
Here, to describe the CT martensitic transformation in a polycrystalline material, a rotation matrix Rij (θ) is introduced[3]. ²00ij (m) is the transformation-induced eigen strains of the m-th (m = 1 and 2) tetragonal variants in two-dimensions.
To calculate the distribution and magnitude of plastic strain during the CT martensitic transformation, the evolutionof plastic strain is evaluated by the PF methodology. According to the Von Mises yield criterion, because the plastic
deformation takes place when the shear strain energy reaches a certain value related to the yield stress σy , the yield
criterion can be described in energetic form as, Fy = Gshear− σy2 /6µ = 0, where the shear strain energy, Gshear, is alsoelelevaluated by the PFM theory. We assume that the evolution of plastic strain is governed by minimization process of theshear strain energy and can be described with the following TDGL equation [5]:∂²pijδGshear= −Kijkl elp ,∂t
δ²kl

(3)

where Kijkl is the fourth-order kinetic coefficient tensor for plastic strain and is assumed to be Kijkl = (KCijkl )−1 withconstant K for simplicity [5]. Note that since this model does not describe deformation hardening, Eqn. (3) is valid in thecase of an elastic-perfectly plastic material. And, the Eqn. (3) is solved only in the region satisfyng the yield criterion.

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450(e)(f)(g)(h)Figure 1. Evolution of tetragonal phase in elastic material((a)-(d)) and elastoplastic material ((e)-(h)) at different dimensionless times t = 30, 40, 48 and 56.

Figure 2. Evolution of equivalent plastic strain in elastoplasticmaterial. Figures (a)-(h) correspond to different dimensionlesstimes t = 24, 30, 34, 40, 44, 48, 56 and 60.

With the above equations, the time evolution of the tetragonal phase, i.e., the order parameters, is described by thefollowing TDGL equation:)(∂φi∂gel∂gch020 δG= −L+− κ∇ φi ,(4)= −L∂tδφi∂φi∂φiwhere L0 and κ are the kinetic parameter for the phase field and the gradient energy coefficient, respectively.
RESULTS AND CONCLUSIONSThe CT martensitic transformation in elastically isotropic polycrystal is simulated in two dimensions. The polycrystalline
structure with seven parent phase grains is produced by an ordinary multi-phase-field simulation of grain growth. We
use a square 256 × 256 nm2 computational domain and apply periodic boundary conditions. Equations (3) and (4) aresolved by the finite difference method and the fast fourier transformation with the plane strain condition. The magnitudeof chemical driving force is set to be constant at ∆f = 321 J/mol. The transformation-induced eigen strains are given
000000by ²0022 (1) = ²11 (2) = 0.1 and ²11 (1) = ²22 (2) = −0.1. The i-th variant of tetragonal phase is defined as the domain ofφi ≥ 0.6, and the yield stress is chosen as σy = 250 MPa for the cubic matrix and σy = 800 MPa for the tetragonal phase,respectively. It is well known that the martensite phase often nucleates on the grain boundary of the parent phase by local
elastic stress due to crystal defect and dislocation. Therefore, to simulate the evolution of the tetragonal phase from thegrain boundary, we distribute random shear plastic strain around the grain boundary as a potential nucleation site.
Figure 1 shows the evolution of the tetragonal phase in the elastic and elastoplastic materials. In Fig. 1(e), the number inthe grain indicates the crystal orientation of parent phase grain, which is defined as a rotation degree between the [100]axis of each parent phase grain and the horizontal direction. Blue, red and yellow domains correspond to the parent phase,the first and variant second variant of the tetragonal phases, respectively. The randomly distributed plastic strain generates
local stress-field in the polycrystal. The distribution of the stress is inhomogeneous and concentrated around the grainboundary of the parent phase by the elastic coupling between different parent grains [6]. Thus, as shown in Fig. 1 (a) and(e), it can be observed that the tetragonal phase preferentially starts to develop around the grain boundary of the parentphase so as to reduce the transformation-induced stress. Furthermore, as shown in Figs. 1 and 2, we can find that in
the elastoplastic material the growth of the tetragonal phase is suppressed by the plastic deformation accompanying themartensitic transformation and thus larger fractions of the parent phase is retained.
From above results, we conclude that the present EPFM can describe the CT martensitic transformation in the polycrystalline material including the evolution of multi-variant martensitic structure from the grain boundary of the parent phaseand the plastic deformation to reduce the transformation-induced stress. In future work, the extention of this EPFM will bea useful tool predinting for the formation of the martensitic microstructure affected by self- and plastic accommodations,such as lath martensite.
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