Coupled simulation of the influence of austenite deformation on the subsequent isothermal austenite–ferrite transformation

Acta Materialia 54 (2006) 1265–1278 www.actamat-journals.com

Coupled simulation of the influence of austenite deformation on the subsequent isothermal austenite–ferrite transformation Namin Xiao, Mingming Tong, Yongjun Lan, Dianzhong Li *, Yiyi Li Institute of Metal Research, Chinese Academy of Sciences, No. 72 Wenhua Road, Shenyang 110016, PR China Received 11 April 2005; received in revised form 17 October 2005; accepted 25 October 2005 Available online 22 December 2005

Abstract The influence of austenite deformation on the subsequent isothermal austenite–ferrite transformation in binary Fe–C alloys is simulated by coupling a Q-state Potts Monte Carlo (MC) method with a crystal plasticity finite element method (CPFEM). The initial deformed microstructure characteristics induced by the plane strain hot compression are simulated using the CPFEM. Based on a linear interpolation approach, these characteristics, which include the stored energy and the orientation, are mapped onto a regular hexagonal lattice as the initial parameters of the MC simulation. The simulation results reveal that the plastic deformation increases the equilibrium ferrite volume fraction and accelerates the transformation kinetics. The regions with high stored energy at austenite grain interiors and boundaries and the extended austenite grain boundary area induced by plastic deformation can increase the effective ferrite nucleation sites. The effects of plastic deformation on the long-range diffusion of carbon atoms in austenite and short-range diffusion of iron atoms across the austenite/ferrite interface are investigated. This simulation technique provides a novel way for investigating the austenite– ferrite transformation under deformation conditions on a mesoscale.
2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hot deformation; Phase transformation; Monte Carlo; Crystal plasticity finite element; Mesoscale simulation

1. Introduction The mechanical properties of structural materials are largely dependent on their microstructure characteristics, e.g. the grain size, the microstructure morphology and their spatial distribution. In the modern steel industry, thermomechanical processing (i.e. controlled rolling) is an important method used to control the microstructure. Generally speaking, the austenite deformation at non-recrystallization temperatures has a significant influence on the subsequent austenite–ferrite transformation [1]. In the past 30 years, many experiments have been carried out and phenomenological models applied [1–6] in an attempt to understand clearly the essential mechanism of the austenite–ferrite transformation after hot plastic deformation. Most metallographic experiments [1–3] reveal that plastic deformation *

Corresponding author. Tel.: +86 242 397 1973; fax: +86 242 389 1320. E-mail address: [email protected] (D. Li).

can markedly accelerate the austenite–ferrite transformation and refine the ferrite grains. Microstructural defects introduced by plastic deformation such as dislocations, annealing twin boundaries and deformation bands are observed to act as additional nucleation sites during the transformation in deformed specimens [1–6]. Some researchers attempted to develop analytical models to investigate the nature of the ferrite transformation from deformed austenite [5,6]. However, limited by the absence of effective tools for describing the effects of plastic deformation on a mesoscale, the effects of deformed microstructure of prior austenite and the spatial distribution of the stored energy were not considered. Mesoscopic computer simulation provides an alternative approach to experimentation and phenomenological modeling for the study of the local effects of plastic deformation on the microstructure evolution during the transformation. Over the past decade, the crystal plasticity finite element method (CPFEM) has been widely used to simulate the heterogeneous deformation behavior in polycrystalline

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2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2005.10.055

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aggregates [7–9]. According to crystal plasticity theory [10–12], deformation behavior in polycrystalline materials mainly depends on microstructural features such as grain orientations and motion of crystal slip systems. Consequently, the mechanical responses of materials to deformation, such as stress, strain and stored energy, are usually heterogeneous on a mesoscale. As opposed to the classic continuum FEM which is based on the assumption of material homogeneity, the effects of microstructural characteristics including the grain orientations and the motion of crystal slip systems are all taken into account in the CPFEM during deformation. Recently, as an algorithm that describes the discrete spatial and temporal microstructure evolution by applying local or global deterministic or probabilistic transformation rules, the cellular automaton (CA) method has been applied to simulate the austenite–ferrite phase transformation [13–15]. Lan et al. [15] described the deformed austenite–ferrite transformation during continuous cooling by coupling the CA method with the CPFEM. Nevertheless, a phenomenological criterion for ferrite nucleation within austenite grain interiors (i.e. at least 75% of the maximum of the stored energy was deemed to be the additional ferrite nucleation sites) had to be adopted in the model of Lan et al. because of the lack of a nucleation model for defects. Naturally, however, the nucleation activation energy is dependent on the summation of the chemical free energy, stored energy and some other energies rather than on only the maximum of the stored energy. In addition, the effects of plastic deformation on carbon diffusion rate and interface mobility in deformed austenite were neglected. Compared with simulation method discussed above, the Monte Carlo (MC) method provides an alternative approach for modeling the austenite–ferrite transformation. The MC method is a stochastic method widely used to model the microstructure evolution in materials such as grain growth [16,17], recrystallization [18,19] and phase transformation [20,21]. In contrast to the CA method in which the transformation rules rely on the description of phenomenological models for physical phenomena, in MC models the microstructure evolution accompanied by carbon diffusion proceeds in the manner of Metropolis sampling. The characteristics such as temperature, chemical composition, interface energy and stored energy have their effects on the phase transformation via the Hamiltonian of the system rather than the phenomenological transformation rules. Thus the influence of deformation on the transformation, including nucleation, growth and carbon diffusion, can be integrated into the MC model even in the absence of phenomenological models of defects. The principal purpose of this paper is to present a mesoscale simulation approach to simulate the influence of plastic deformation on the subsequent austenite–ferrite phase transformation by coupling a MC method with the CPFEM. The deformed microstructure of prior austenite and the heterogeneous distribution of stored energy are simulated by the CPFEM. This provides initial parameters

for the MC simulation of the subsequent austenite–ferrite transformation. The influence of deformation on the transformation kinetics, the microstructure evolution and the diffusion process is investigated. Similar to other MC microstructure simulations [16–21], the Q-state Potts model is adopted. In the Hamiltonian of the MC system, the effects of the chemical free energy, interface energy and the stored energy are taken into account. 2. CPFEM The details of the CPFEM formulation used for simulating hot deformation have been described in a previous paper [22]. Only the main features of this method are described here. The simulation of hot compression in the present paper is performed using the commercial ABAQUS finite element code with the crystal plasticity constitutive law programmed in the user material subroutine UMAT [23]. The model follows the works of Asaro and Needleman [10] and Peirce et al. [11]. The velocity gradient, L, is decomposed as follows: L ¼ D þ X;

ð1Þ

where D and X are the symmetric rate of the stretching and spin tensors, respectively. D and X may be decomposed into plastic parts (DP, XP) and elastic parts (D*, X*) as follows: D ¼ DP þ D
;

ð2Þ




P

X¼X þX .

ð3Þ

Since plastic deformation is caused by dislocation slip, the following equations can be obtained: N X DP þ XP ¼ ð4Þ c_ ðaÞ m
ðaÞ  n
ðaÞ ; a¼1 N N X 1X DP ¼ ðm
ðaÞ  n
ðaÞ þ n
ðaÞ  m
ðaÞ Þ ¼ P ðaÞ c_ ðaÞ ; 2 a¼1 a¼1

ð5Þ

N N X 1X ðm
ðaÞ  n
ðaÞ  n
ðaÞ  m
ðaÞ Þ ¼ XðaÞ c_ ðaÞ ; 2 a¼1 a¼1

ð6Þ

XP ¼

where c_ ðaÞ is the slipping rate of the slip systems, m*(a) is the vector of the crystallographic slip direction, n*(a) is the vector of the crystallographic slip plane normal and N is the number of the active slip systems. It should be noted that the slip vector evolution (i.e. the lattice rotation) during the deformation is calculated according to m
ðaÞ ¼ F
mðaÞ ;

ð7Þ

n
ðaÞ ¼ nðaÞ F
1 ; (a)

ð8Þ (a)

where m and n are the unit vectors of the crystallographic slip direction and the crystallographic slip plane normal, respectively. F* is the lattice contribution to the total deformation gradient which is associated with stretching and rotation of the lattice. As described by Pierce et al. [11], the Jaumann rate of Kirchhof stress can be expressed as

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^s ¼ C : D 

N X

c_ ðaÞ RðaÞ ;

ð9Þ

a¼1

where C is a fourth-order tensor based on the anisotropic elastic modulus, D the symmetric part of the velocity gradient (i.e. the rate of deformation tensor), and Ra a tensor depending on the current slip plane normal and direction, the Kirchhof stress and the elastic modulus. In the rate-dependent constitutive formulation, the slip rate on a slip system is assumed to be related to the resolved shear stress s(a), which is defined as sðaÞ ¼ P ðaÞ : s; through a rate-dependent power law relation:
ðaÞ 1=m ðaÞ s ðaÞ ; c_ ¼ c_ 0 gðaÞ

ð10Þ

ð11Þ

where g(a) is the slip system strength or resistance to shear, ðaÞ m is the strain sensitivity exponent and c_ 0 is the reference shear rate. During hot compression, part of the deformation energy is stored in the form of dislocations within the material. The average resolved shear stress at the integration point of each element,
s, can be expressed by [24]
s ¼

N 1 X sa . N a¼1

ð12Þ

As proposed by Mecking and Kocks [25], the average shear stress can be expressed as the square root of the dislocation density, q: r ð13Þ s ¼ ¼ albq1=2 ; M where l is the shear modulus of the material, b is the magnitude of the Burgers vector, r is the flow stress, M is the Taylor factor and a is the material constant. For austenite (face-centered cubic crystal lattice), M = 3.11, a = 0.15 [2]. The stored energy per unit volume, Gdef, can be expressed as Gdef ¼ bqlb2 ;

ð14Þ

where b is the constant with a value in the range 0.5 to 1. Combining of Eqs. (13) and (14) leads to the stored energy in terms of the average resolved shear stress and the shear modulus: Gdef ¼

bs2 . a2 l

ð15Þ

The stored energy simulated by the CPFEM is transferred to the MC method as an initial condition for subsequent phase transformation simulation. 3. MC model Tong et al. [20] adopted a MC method to simulate deformation-induced ferrite transformation in a Fe–C binary alloy system. The present MC simulation is based on

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TongÕs model. The microstructure is mapped onto a twodimensional hexagonal lattice. Each hexagonal cell may belong to either the ferrite phase or the austenite phase. The description of the state of a MC cell i requires four parameters: the orientation Si, the carbon concentration Ci, the order parameter gi and the stored energy Gdef i . The orientation is the same as that defined in the traditional MC simulation of grain growth [16]. The order parameter is defined as 0 representing the austenite phase and 1 representing the ferrite phase. The evolutions of these four parameters correspond to the evolution of the microstructure, the stored energy field and the carbon concentration field. This means that there are four procedures coupled in the simulation of the transformation, which are the orientation transition, the carbon diffusion, the transition of the order parameter and the evolution of the stored energy. In this paper, the chemical free energy, interface energy and stored energy terms are introduced into the Hamiltonian of the system in the Q-state Potts model as follows [20]: H¼

n X m n n X X 1X GbS i S j þ Gci þ Gdef i ; 2 i j i i

ð16Þ

where Gci is the chemical free energy of cell i; GbS i S j is the interface energy between MC cell i and cell j is the interface energy between MC cell i and cell j with their corresponding orientations; Gdef is the stored energy of cell i. Clearly, i the chemical free energy and stored energy terms are the driving forces for phase transformation while the interface energy term is the resistance to the transformation. 3.1. The MC transition rules with random jump-based mesoscopic diffusion model In traditional mesoscale simulation methods such as the CA method and phase field method, the analytical solution or the deterministic numerical solution of FickÕs second law (such as finite difference scheme and finite element scheme) are adopted for dealing with the solute diffusion. Different from above, a probabilistic manner of Metropolis sampling is introduced for reproducing the effect of solute diffusion in the present MC model [21]. In each MC step (MCS), cell i has to be selected at random and then one of its nearest neighbors is determined by random selection as cell k. Then, the carbon diffusion attempt between cell i and cell k proceeds as follows: a random number that ranges between zero and the sum of the carbon concentration of cell i and cell k is added to the carbon concentration of cell i. Then, the same value is subtracted from the carbon concentration of cell k. This redistribution of the carbon concentration between cells i and k by means of the random jump must obey the solute conservation rule. The Hamiltonians of the system before and after the above event are stored as H1 and H2, respectively. If H2 6 H1, the redistribution is definitely accepted. Otherwise, the redistribution

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occurs with a probability of W = exp{(H1  H2)/kBT}, where kB is the Boltzmann constant and T is the absolute temperature. A simple verification for the action of the MC diffusion model is carried out as follows. A one-dimensional domain with 1000 MC cells (corresponding to a length of 100 lm) is composed of a single austenite phase domain. At the beginning of the MC simulation, the carbon concentration of the central 16 cells in this one-dimensional austenite domain is set as 0.49 wt.% and that of other cells is set as 0.01 wt.%. Then the system is held isothermally at 1200 K to allow the carbon atoms in the austenite to diffuse along the length direction in this MC simulation. As a comparison, the standard Gaussian function solution of FickÕs second law in this one-dimensional austenite domain can be written as
 S x2 X c ¼ pffiffiffiffiffiffiffiffiffiffi exp  ; ð17Þ 4Dt 4pDt Where Xc is the carbon concentration in austenite, D is the diffusion coefficient of carbon atoms in austenite and S is the total solute atom flux in the domain. Figs. 1(a)–(d) show the comparison between the simulated carbon concentration distribution by the MC model (represented by the discrete points) and the Gaussian function solution of FickÕs second law (represented by the continuous curve) in this one-dimensional austenite domain. It should be noted that the time t in Fig. 1 is the normalized time. The results confirm the reliability and accuracy of the MC simulation results for one-dimensional geometry and lend confidence to the extension of the simulation approach to two and three dimensions. When the carbon diffusion is accompanied by phase transformation, i.e. the diffusion is an inter-phase diffusion, the above MC transition attempt requires a minor change as follows: cell i has to be selected at random and then two cells of its nearest neighbors are determined by random selection as cells j and k. j is allowed to be equal to k. Then, an attempt is made to change the orientation, the order parameter and the stored energy of cell i to the same as

those of cell j. After that, the carbon diffusion attempt between cell i and cell k is performed in the way described above. The Hamiltonians of the system before and after the above transition attempts (order parameter transition, orientation transition, stored energy transition and the carbon concentration redistribution) are stored as H1 and H2, respectively. If H2 6 H1, all the above attempts are definitely accepted. Otherwise, all the attempts occur with the probability of W = exp{(H1  H2)/kBT}. It is clear that this inter-phase MC carbon diffusion model can be applied for modeling the intra-phase carbon diffusion because the intra-phase carbon diffusion is a special inter-phase carbon diffusion with gi = gj. From the above description of the MC rules, it can be seen that the microstructure evolution (corresponding to the order parameter transition attempt and orientation transition attempt), the carbon diffusion (corresponding to the carbon concentration redistribution attempt) and the stored energy evolution (corresponding to stored energy transition attempt) are coupled into the same MC transition step. So the influence of plastic deformation on the transformation characteristics has been involved directly in the present model rather than by use of an additional phenomenological model. 3.2. Chemical free energy model In this paper, the regular solution free energy model is adopted [26]. For the Fe–C binary system, the chemical free energy model of ferrite and austenite can be expressed, respectively, as Gam ðT Þ ¼ xFe ðla0Fe þ RT ln aaFe Þ þ xC ðlgraphite þ RT ln aaC Þ; 0C þ RT ln acC Þ; Gcm ðT Þ ¼ xFe ðlc0Fe þ RT ln acFe Þ þ xC ðlgraphite 0C ð18Þ where xC and xFe are the mole fractions of carbon and iron, respectively; acC and aaC are the activities of carbon element

Fig. 1. (a–d) The simulated carbon concentration distribution by MC diffusion model and a comparison with the Gaussian function solution of FickÕs 2nd Law at 1D scale.

N. Xiao et al. / Acta Materialia 54 (2006) 1265–1278

in austenite and in ferrite, respectively; and acFe and aaFe are the activities of iron element in austenite and in ferrite, respectively. The activities acC , aaC , acFe and aaFe can be determined by the KRC model [27,28]. lgraphite , la0Fe and lc0Fe are 0C the molar Gibbs free energy of carbon element, a-Fe and cFe, respectively, which can be obtained from Ref. [29]. Then the chemical free energy of cell i can be written as Gci ¼ Gm  M i , where Mi is the mole fraction of the substance of cell i. 3.3. Interface energy model If the interface is a grain boundary phases, the Read–Shockley interface adopted [30]:  i ( h h hij ij 1  ln c when
mh h
GbS i S j ðhij Þ ¼ cm when

between the same energy model is ðhij < h
Þ; ðhij P h
Þ;

ð19Þ

where hij is the misorientation between cells i and j, and h* is the misorientation limit for low-angle boundaries which is set as 15. If cells i and j both belong to the ferrite phase, cm = ca = 0.80 J/m2; if cells i and j both belong to the austenite phase, cm = cc = 0.79 J/m2 [31]. If the interface is a grain boundary between different phases, the interface energy is considered as a constant [31]: Gbsi sj ðhÞ ¼ cac ¼ 0:56 J=m2 .

ð20Þ

3.4. Nucleation model When the temperature is lower than the transformation start temperature, nucleation is expected to happen at some sites on austenite grain boundaries and interiors. In this paper, the site saturated nucleation model is adopted. A number of ferrite nuclei N are introduced into the system at the beginning of the simulation. The stored energy of all these nuclei is assumed to be zero and the location of the nuclei is chosen at random. According to the MC rules, those nuclei that are located at a situation of energy advantage (i.e. the addition of the nuclei will decrease the Hamiltonian of the system) will be kept by the system and then start to grow. In contrast, those nuclei that are located at a situation of energy disadvantage (i.e. the addition of the nuclei will increase the Hamiltonian of the system) will disappear automatically at a probability determined by the change of the free energy. This means that the nucleation location is not artificially set but is the result of MC selection. The ferrite nucleation is dominated by the interaction among the chemical free energy, interface energy and stored energy. The deformation takes effect on the ferrite nucleation via the Hamiltonian in Eq. (16) and the MC transition rule rather than by dint of any phenomenological nucleation criterion [15,32]. This nucleation model which is based on thermodynamics principles and the MC rules is proposed as more reasonable than the phenomenological criterion.

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4. Coupling the MC model with the CPFEM The initial parameters required in the MC simulation for deformed austenite decomposition into ferrite include the crystal orientations and the stored energy. The CPFEM provides these parameters at the spatial coordinates of the integration points of the distorted finite element mesh. However, the lattice used in the MC model is regular and equally spaced and thus these two meshes are not matched. Consequently, it is essential to map the crystal orientation and the stored energy data from the irregular points in the CPFEM onto the regular hexagonal cells in the MC lattice. Several mapping approaches have been investigated in the literature [18,32]. In the present paper, the mapping procedure is realized by interpolating linearly the finite element data on the two-dimensional hexagonal MC lattice [15]. The interpolation function is chosen as: f(x, y) = a + bx + cy. This means that the state variables including orientation and stored energy will be expressed as a function of the coordinate values of points. This interpolation procedure can be realized in three steps as follows. First, for every given interpolation point (i.e. the central point of every hexagonal cell) on the regular MC lattice, the corresponding three nearest points in the distorted finite element mesh can be obtained. Second, according to the state variables data at the three nearest finite element points, the coefficients a, b and c in the interpolation function can be obtained. Finally, the orientation and stored energy variables at the interpolation point in the regular MC lattice can be calculated using the coordinates of the interpolation point based on the obtained coefficients, a, b and c. Figs. 2(a) and (b) illustrate the result of the present linear interpolation approach. Fig. 2(a) shows the contour of calculated average shear stress values using the CPFEM by ABAQUS. Fig. 2(b) gives the contour of interpolated average shear stress values which are obtained by the present linear interpolation approach and have been mapped onto the MC lattice. From Fig. 2(a) and (b), it is clear that the interpolated values agree well with the values by ABAQUS. 5. Results of simulation A sample of a Fe–0.2 wt.% C alloy is first austenitized at 1273 K and held for 10 min. Then the sample is cooled to 1153 K followed by isothermal hot deformation in the austenite single-phase region. During the isothermal deformation, the sample is subjected to a simple plane strain hot compression up to a series of logarithmic strains of e = 0.23, 0.5 and 0.9 with a strain rate of 10.0 s1, corresponding to nominal strains of 20%, 40% and 60%, respectively. Once the hot deformation is finished, the sample is cooled quickly to 1070 K and held isothermally at this temperature. This paper focuses on the transformation that occurs during the isothermal holding of the sample at the soaking temperature (1070 K) and the effects of the

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Fig. 2. The calculated average shear stress (MPa) (a) on the finite element meshes by ABAQUS and the linearly interpolated shear stress (b) on the Monte Carlo lattices.

deformation on it. The simulation of deformed austenite decomposition into ferrite is divided into two stages. First, the simulation of hot deformation at different strains is implemented with the CPFEM; second, the subsequent austenite–ferrite transformation is simulated with a Q-state Potts MC method. At the beginning of the MC simulation, the initial crystallographic orientation and the stored energy in each MC cell are extracted from the CPFEM calculation results based on the linear interpolation approach as described in Section 4. 5.1. CPFEM simulation In the CPFEM, the microstructure of the undeformed prior austenite sample which is created using the MC normal grain growth simulation is mapped onto an appropriate finite element mesh. The initial mesh uses a bilinear element with four nodes and four integration points. The finite element sample before hot compression is discretized onto a mesh of 30 · 60 rectangular elements corresponding to 164 · 328 lm2 in the material. Then simple plane strain hot compressions up to logarithmic strains of e = 0.23, 0.5 and 0.9 are applied. Figs. 3(a)–(d) show the distribution of the stored energy (J/mol) at three different logarithmic strains. From Fig. 3(d), it can be seen that the shapes of the statistical distribution curves at different strains are roughly similar but the whole curve moves to the right along with the energy axis as the magnitude of plastic deformation increases. Thus, as expected, the stored energy of the deformed samples increases with the increased strain. Figs. 3(a)–(c) indicate that the distribution of stored energy is highly heterogeneous. Usually the stored energy of grain boundaries is high compared to the grain interiors. This can be easily understood because the grain boundary is often taken as an obstacle to dislocation motion and therefore results in the accumulation of dislocations. However, the stored energy is also concentrated within some grain interiors. This effect is caused by the interaction of active slip systems and the neighboring grains with different misorientation. Another phenomenon is the appearance of shear bands with increased strain (Figs. 3(b) and (c)). It is obvious that the shear bands are also the dislocation

accumulation regions and hence the stored energy is higher there. It should be noted that the austenite grains are compressed in the vertical direction and elongated normal to the compression (i.e. horizontal) direction in Fig. 3. Consequently, the volume of the sample after deformation remains invariable whereas the austenite boundary area is changed by the plastic deformation. The increase of austenite grain boundary area Sd/S0, where S0 is the original austenite grain boundary area before deformation and Sd is the austenite grain boundary area after deformation, as a function of logarithmic strains is shown in Fig. 4. It can be seen that the effect of deformation on the increase of austenite grain boundary area is very marked at large strains. When the strain is 0.23, the grain boundary area increases to only 1.07 times whereas at a strain of 0.9, the grain boundary area increases to almost 1.8 times. Other researchers have developed geometric analytical models for describing the effect of deformation on grain boundary area. In these models, the grain shapes are supposed as regular geometric shapes such as a sphere or tetrakaidecahedron and the deformation of all parts of the grain structure is assumed to be uniform [1,33]. Fig. 4 also shows the comparisons between the result of the CPFEM and that of geometric analytical models of uniform deformation indicating that the plastic deformation in polycrystalline aggregates appears to be systematically higher than that predicted by a uniform deformation model. The CPFEM simulations by Bate et al. also give similar results [34]. In reality, the deformation is expected to be highly heterogeneous due to complex intergranular effects. These will include the different mechanical responses resulting from different grain orientations, substructures and textures. All these influences will result in a higher rate of increase of grain boundary area. 5.2. Influence of deformation on the transformation kinetics Fig. 5 shows the transformation kinetics of systems at different logarithmic strains. The dashed line represents the volume fraction of ferrite predicted by the Fe–C binary

N. Xiao et al. / Acta Materialia 54 (2006) 1265–1278

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Fig. 3. (a–c) Stored energy field (J/mol) simulated by CPFEM at different logarithmic strains. The white lines represent the austenite grain boundaries. (d) The statistical distribution of stored energy at different logarithmic strains.

phase diagram. The calculated equilibrium ferrite volume fraction of the undeformed sample agrees well with the predicted value from the phase diagram. Without deformation, the volume fraction of ferrite reaches its saturation value 45.3% at 6000 MCS. However, when e = 0.9, the equilibrium ferrite volume fraction increases to 56.6% and the time elapsed before the volume fraction of the ferrite saturates decreases to about 2000 MCS. It is clear that the hot plastic deformation accelerates the transformation kinetics and increases the equilibrium volume fraction of the ferrite. These simulation results show the effects of the plastic deformation of the prior austenite on the transformation kinetics and the saturation value of the ferrite volume fraction.

In actual thermomechanical processing, the transformation always lies in the metastable state. The austenite decomposition is always accompanied by the ferrite-to-austenite back transformation. During the transformation, the austenite decomposition increases the ferrite volume fraction whereas the ferrite-to-austenite back transformation decreases the ferrite volume fraction. Therefore the ferrite fraction transformed must adhere to the equilibrium value when the transformation finishes. In the MC model of this paper, the ferrite-to-austenite back transformation has already been taken into account. It is just a transition case of gi = 1 and gj = 0 as mentioned in Section 3.1. However, the recovery and recrystallization of retained austenite after deformation are not considered in the present MC

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Fig. 4. The increase of austenite boundary area Sd/S0 simulated by CPFEM and a comparison with other geometrical model.

Fig. 5. The transformation kinetics of the system at different logarithmic strains. The dash line represents the volume fraction of ferrite predicted by the Fe–C binary phase diagram.

model. The essence of recovery and recrystallization is the exhaustion of dislocation and hence the stored energy of the system. This means that the stored energy of the retained austenite does not vary with time in the present simulation. In other words, the thermodynamic equilibrium state of the present simulation results is not equal to that of realistic systems because of the neglect of recovery and recrystallization in retained austenite. This therefore results in the prediction of an excessive ferrite fraction compared with real systems. The simulation in this paper is only focused on the effects of the plastic deformation of the prior austenite on the isothermal austenite decomposition. The recovery and recrystallization of the retained austenite will be considered in a future model. 5.3. Influence of deformation on the microstructure Fig. 6 shows the microstructure evolution during the undeformed austenite decomposition. It can be seen that

most ferrite grains nucleate at austenite grain boundaries being, as perhaps expected, the preferable nucleation sites because of the effect of interface energy. The evolutions of the microstructure and the stored energy during the deformed austenite decomposition into ferrite at different strains (e = 0.23, 0.5, 0.9) are shown in Figs. 7–9. From Figs. 7–9 some interesting phenomena become apparent. First, the numbers of ferrite nuclei in the simulation domain evidently increase with increasing strain (Figs. 7(a), 8(a) and 9(a)). Because the volume of the simulation domain remains invariable during the deformation, it implies that the ferrite nucleation density increases with increasing strain. According to the locations of ferrite nuclei, the ferrite nuclei can be classified into two types: one is the ferrite nuclei at austenite grain boundaries; the other is the nuclei at the austenite grain interiors. For those nuclei at the austenite grain interiors should be attributed the plastic work stored in the material in the form of dislocations during the hot deformation. The thermodynamic nucleation theory proposed by Christian [35] reveals that those additional energies of atoms at defects will effectively decrease the nucleation activation energy for the nucleation of ferrite. Thus the ferrite nucleation becomes easier in deformed austenite than in the undeformed austenite. Consequently, the area with high stored energy at austenite grain interiors becomes the preferable nucleation site. For those nuclei at austenite grain boundaries, there are two factors enhancing the numbers of ferrite nuclei. (1) The area of austenite grain boundary increases with increasing strain. As discussed above, because of the effect of interface energy, the extended austenite grain boundaries provide additional ferrite nucleation sites. (2) However, it should be noted that the increase of the amount of ferrite nuclei at austenite boundaries is not proportional to the increase of austenite grain boundary area. This difference is shown in Fig. 10. Nd is the amount of ferrite nuclei of the system at the corresponding strain, N0 is the amount of ferrite nuclei of the undeformed system, Sd is the grain boundary area of the system at the corresponding strain and S0 is the grain boundary area of the undeformed system. It can be seen that the increase in the number of ferrite nuclei at austenite grain boundaries is greater than that of the austenite grain boundary area. This implies that the amount of ferrite nuclei per unit area of austenite grain boundary increases with increasing strain. According to the analysis of Umemoto et al. [1] and Khlestov et al. [36], because of the non-uniform slips occurring inside the austenite grains, the smooth grain boundaries become wavy and irregular and some small regions with high stored energy around grain boundaries are produced. The ferrite nuclei formed at austenite grain boundaries decrease not only the interface energy but also the stored energy. As a result, the decreased energy barrier leads to the increase of the amount of ferrite nuclei per unit area at deformed austenite grain boundaries. It should be noted that the spatial distribution and grain-size distribution of the ferrite grains are

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Fig. 6. The simulated microstructure evolution during the undeformed austenite decomposition into ferrite. The black lines represent the austenite grain boundaries. The white regions denote austenite grains and the colored regions denote new formed ferrite grains. Different colors denote different orientations. [For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.]

inhomogeneous (Figs. 8(b) and 9(b)). This inhomogeneous distribution comes from the heterogeneous distribution of the stored energy. The other result of relevance in Figs. 7–9 is that when the ferrite volume fraction saturates, the ferrite grain size DF decreases while the amount of ferrite grains per unit area NF increases with increasing strain. The resulting values for DF and NF are represented as a function of the logarithmic strain in Fig. 11. Because of the increased ferrite nuclei density, the ferrite grains are easier to impinge, resulting in the refinement of the ferrite grains. Clearly plastic deformation is an effective technique for grain refinement. The essence of the grain refinement effect of plastic deformation is the increased ferrite nuclei density arising from the stored energy. However, the feature that should be emphasized is that the nuclei density is not the only factor affecting grain size. Some other factors, e.g. grain coarsening after impingement, also have significant influence on the grain size. Whether the plastic deformation is implemented or not, the ferrite grain coarsening after impingement will result in a larger grain size unless a sufficient fraction of second phase particles is present to limit the rapid migration of ferrite grain boundaries [37]. This paper only focuses on investigating the effects of plastic deformation on the ferrite grain size and other factors are neglected. More comprehensive simulations and analyses of grain refinement will be implemented in future investigations. 5.4. Influence of deformation on the diffusion Figs. 12(a)–(c) indicate the evolution of the carbon concentration field during the deformed austenite decomposition into ferrite in the case of e = 0.23. A significant phenomenon in Fig. 12 is that the carbon concentration distribution in austenite is non-uniform and there exists a

carbon concentration gradient from the interface to the austenite interiors. The carbon concentration in the austenite at the austenite/ferrite interface is not equal to the equilibrium carbon concentration predicted by the phase diagram but varies continuously during the transformation. Generally speaking, the carbon concentration in the austenite at the austenite/ferrite interface is lower than the equilibrium carbon concentration while higher than the initial carbon concentration of 0.2 wt.%. During the austenite–ferrite transformation in the Fe–C alloy, there are two mechanisms dominating the transformation at the same time: the long-range diffusion of carbon atoms in the austenite phase ahead of the moving austenite/ferrite interface and the short-range diffusion of the matrix atoms across the interface which results in the face-centered cubic/body-centered cubic lattice transition (so-called interface migration) [35,38,39]. The plastic hot deformation has a great effect on both the long-range carbon diffusion and the short-range diffusion of matrix atoms. Fig. 13(a) shows the carbon diffusion kinetics in austenite at different logarithmic strains. Clearly, the plastic deformation increases the equilibrium carbon concentration of austenite. This can be understood from the thermodynamic principle shown schematically in Fig. 13(b). The free energy of deformed austenite is higher than that of undeformed austenite because of the effect of stored energy. This then results in the equilibrium carbon concentrations of both austenite and ferrite under deformation conditions being higher than that under non-deformation conditions. Another result from Fig. 13(a) is that the plastic deformation accelerates the carbon diffusion kinetics in austenite. The time elapsed before the carbon concentration in austenite reaches the saturation values decreases with increasing strain. Commonly, the carbon diffusion in austenite is determined by the diffusion activation energy of carbon in austenite. Under deformation conditions, a

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where f * is the characteristic frequency of iron atoms, R the gas constant, Q the activation energy and DGac the Gibbs free energy difference between the austenite and ferrite phases. Then if the distance across the interface is dB, the interface velocity v can be expressed as
 Q B ð22Þ v ¼ j  d  M 0 exp  DGac ; RT using the condition that DGac  RT. M0 = f*dB/RT is the pre-exponential factor. Eqs. (20) and (21) show that the Gibbs free energy difference DGac is the driving force of the short-range diffusion of iron atoms across the austenite/ferrite interface. In the case of deformed austenite decomposition into ferrite, the driving force DGac is expressed as the sum of chemical free energy DGchem and the additional driving force arising from the stored energy DGdef [6]: DGac ¼ DGchem þ DGdef .

ð23Þ

Fig. 14 clearly indicates that the driving force DGac increases with increasing strain. This means that increasing the strain can speed up the short-range diffusion of iron atoms at the interface during the austenite–ferrite transformation under deformation conditions. The austenite–ferrite transformation is more controlled by the slower mechanism between the long-range diffusion of carbon atoms and the short-range diffusion of matrix atoms [40,41]. As described in Ref. [40], the relationship between the carbon diffusion in austenite and the interface migration can be characterized by a mixed-control parameter Y: Y ¼

Fig. 7. The simulated microstructure evolution (left) and the stored energy field (J/mol) evolution (right) during the deformed austenite decomposition into ferrite at e = 0.23.

great number of dislocations and other defects are introduced into the materials. Those crystal lattices with defects possess higher energy than those perfect crystal lattices, resulting in the decreased diffusion activation energy of carbon atoms (i.e. increased carbon diffusion coefficient in austenite). Considering the thermal activation growth theory [35], the short-range diffusion of iron atoms across the austenite/ferrite interface can be considered as the jump of iron atoms from the austenite phase to the ferrite phase. The net jump frequency j can be derived as follows:

 Q DGac
j ¼ f exp  1  exp ; ð21Þ RT RT

xc;ca  xcc c ; c;eq xc  xcc

ð24Þ

is the equilibrium carbon concentration of auswhere xc;eq c tenite predicted by the phase diagram, xcc;ca is the carbon concentration in the austenite at the austenite/ferrite interface and xcc is the average carbon concentration in austenite. Obviously, if xc;ca ¼ xc;eq (Y = 1), the transformation is c c purely diffusion controlled; if xc;ca ¼ xcc (Y = 0), the transc formation is purely interface controlled. Fig. 15 shows the evolution of Y for systems with different strains at T = 1070 K. It can be seen that the isothermally deformed austenite decomposition into ferrite is neither purely diffusion controlled nor purely interface controlled. During the transformation the austenite/ferrite interface is at a nonequilibrium state. Generally speaking, the Y values increase at the early stage of transformation because of the plentiful ferrite nucleation and the finite carbon diffusion rate in austenite. Then because of the balance between the carbon diffusion and the interface migration the Y values reach a plateau value. The dramatic drop of Y values at the last stage of the transformation is perhaps related to the grain ripening mechanism of ferrite at high ferrite fractions transformed [21]. The evolution of Y shows that the isothermally deformed austenite decomposition into ferrite is a diffusion-interface mix-controlled transformation. It

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Fig. 8. The simulated microstructure evolution (left) and the stored energy field (J/mol) evolution (right) during the deformed austenite decomposition into ferrite at e = 0.5.

should be noted that when the magnitude of strains increases from e = 0 to 0.9 the plateau values of Y decrease from 0.41 to 0.19. This implies that with increasing strain the phase transformation inclines to be more interface controlled. In other words, the short-range diffusion of iron atoms across the austenite/ferrite interface becomes the slower mechanism with increasing strain. However, the above analysis has revealed that both the long-range diffusion of carbon atoms in austenite and the short-range diffusion of iron atoms across the interface are accelerated by the plastic deformation. Thus the evolution of Y values implies that the acceleration influence of deformation on the long-range diffusion of carbon atoms is greater than that on the short-range diffusion of iron atoms across the austenite/ferrite interface. 6. Summary and discussion The influence of austenite deformation on the subsequent isothermal austenite–ferrite transformation is investigated by coupling the CPFEM with a Q-state Potts MC

method. The CPFEM provides detailed quantitative information about the austenite hot deformation at a mesoscale. Based on a linear interpolation approach, the crystal orientations and the stored energy obtained at the integration points of the distorted finite element mesh are mapped on the regular hexagonal MC lattice as the initial parameters for the MC simulation of the subsequent austenite–ferrite transformation. (1) The simulation results of the CPFEM show that because of the interaction of active slip systems and neighboring grains with different misorientation, the distribution of stored energy is highly heterogeneous and it is possible for both the grain boundary and the grain interior to become high stored energy regions. Moreover, the plastic deformation can increase the austenite grain boundary area. (2) The plastic deformation accelerates the austenite– ferrite transformation kinetics and increases the equilibrium ferrite volume fraction. Because the recovery and recrystallization in retained austenite during the

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Fig. 9. The simulated microstructure evolution (left) and the stored energy field (J/mol) evolution (right) during the deformed austenite decomposition into ferrite at e = 0.9.

Fig. 10. Comparison between the increase of the amount of ferrite nuclei at austenite grain boundaries (Nd/N0) and the increase of austenite grain boundary area (Sd/S0).

transformation are not considered in this paper, the prediction of an excessive ferrite fraction compared with the real systems is obtained. The effects of them on the transformation kinetics need to be further investigated. (3) The simulated microstructure evolution indicates that the plastic deformation can markedly increase the ferrite nucleation density. This can be attrib-

Fig. 11. The calculated average ferrite grain size DF and the ferrite grain density of the system NF at different logarithmic strains.

uted to three reasons: (i) the increased austenite grain boundary area due to the deformation, (ii) the increased number of ferrite nuclei per unit area at austenite grain boundaries and (iii) the formation of high stored energy regions at austenite grain interiors induced by deformation. The spatial distribution of ferrite grains is inhomogeneous because of the heterogeneous distribution of stored energy.

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Fig. 12. The evolution of simulated carbon concentration field during deformed austenite to ferrite transformation at e = 0.23. The carbon-poor areas where carbon concentration is lower than 0.022 wt.% indicate the newly formed ferrite grains and the carbon-rich regions where carbon concentration is not less than the initial content of 0.2 wt.% represent retained austenite grains.

Fig. 13. Carbon diffusion kinetics in austenite (a) at different logarithmic strains and the free energy vs. carbon concentration schematic curves (b) showing the effect of austenite deformation on the equilibrium carbon concentration in ferrite and austenite.

Fig. 14. The driving force for interface migration at different logarithmic strains.

Fig. 15. The mode parameter Y as a function of ferrite fraction with different logarithmic strains at T = 1070 K and the average bulk carbon concentration C = 0.2 wt.%.

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(4) Plastic deformation can accelerate both the longrange diffusion of carbon atoms in austenite and the short-range diffusion of iron atoms across the austenite/ferrite interface during the transformation. The analysis shows that the acceleration influence of deformation on the long-range diffusion of carbon atoms is greater than that on the short-range diffusion of iron atoms across the interface. Acknowledgements The authors are grateful to Prof. Zhufeng Yue at Northwestern Polytechnical University in PR China for his help concerning the crystal plasticity finite element simulation. The authors also gratefully acknowledge the financial support from the National Science Foundation of China (NSFC) under Grant No. 50471073. References [1] Umemoto M, Ohtsuka H, Tamura I. Trans ISI Japan 1983;23:775. [2] Bengochea R, Lopez B, Gutierrez I. Metall Trans A 1998;29A:417. [3] Inoue T, Torizuka S, Nagai K, Tsuzaki K, Ohashi T. Mater Sci Technol 2001;17:1580. [4] Aaronson HI, Enomoto M, Furuhara T, Reynolds WT. Thermec 88, International conference on physical metallurgy of thermomechanical processing of steels other metals, vol. 1. Tokyo: Iron and Steel Institute of Japan; 1988. p. 80. [5] Umemoto M, Hiramatsu A, Moriya A, Watanabe T, Nanba S, Nakajima N, et al. ISIJ Int 1992;32:306. [6] Hanlon DN, Sietsma J, van der Zwaag S. ISIJ Int 2001;41:1028. [7] Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S. Acta Mater 2001;49:3433. [8] Choi SH. Acta Mater 2003;51:1775. [9] Sarma GB, Dawson PR. Acta Mater 1996;44:1937. [10] Asaro RJ, Needleman A. Acta Metall 1985;33:923. [11] Peirce D, Asaro RJ, Needleman A. Acta Metall 1983;31:1951. [12] Asaro RJ. J Appl Mech 1983;50:921.

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