Numerical simulation of dynamic strain-induced austenite–ferrite transformation in a low carbon steel

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Acta Materialia 57 (2009) 2956–2968 www.elsevier.com/locate/actamat

Numerical simulation of dynamic strain-induced austenite–ferrite transformation in a low carbon steel Chengwu Zheng, Namin Xiao, Luhan Hao, Dianzhong Li *, Yiyi Li Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China Received 4 November 2008; received in revised form 28 February 2009; accepted 1 March 2009 Available online 30 March 2009

Abstract A modified cellular automaton modeling has been performed to investigate the dynamic strain-induced transformation (DSIT) from austenite (c) to ferrite (a) in a low carbon steel. In this modeling, the c–a transformation, ferrite dynamic recrystallization and the hot deformation were simulated simultaneously. The simulation provides an insight into the mechanism of the ferrite refinement during the DSIT. It is found that the refinement of ferrite grains derived from DSIT was the result of the increasing ferrite nuclei density by the ‘‘unsaturated” nucleation, the limited ferrite growth and the ferrite dynamic recrystallization. The effects of prior austenite grain size and strain rate on the microstructural evolution of the DSIT ferrite and the characteristics of the resultant microstructure are also discussed. Crown Copyright Ó 2009 Published by Elsevier Ltd. on behalf of Acta Materialia. All rights reserved. Keywords: Dynamic strain-induced transformation; Ferrite refinement; Cellular automaton; Mesoscopic modeling; Low carbon steel

1. Introduction An ongoing drive within the steel industry is to produce high strength steels with lower cost and improved properties. Among the various applied techniques, refinement of the ferrite grains provides a promising approach to improving the strength without sacrificing toughness. Both theory and experiment [1–3] have proved that the strength of some low carbon steels would be doubled when the ferrite grains were refined to 3–5 lm. Therefore, much effort has been put into refining the ferrite grains in carbon steels. Recently, the dynamic strain-induced transformation (DSIT) [4] from austenite (c) to ferrite (a) in low carbon steels (i.e. transformation occurs during deformation applied within an appropriate temperature range) has received much wider attention since it has been proved to be a most effective *

Corresponding author. Tel.: +86 24 23971281; fax: +86 24 23891320. E-mail addresses: [email protected] (C. Zheng), [email protected] (D. Li).

way to produce ultra-fine-grained ferrite in steels. Most experimental investigations have reported that a fine microstructure of equiaxed ferrite grains with an average grain size of 1–3 lm can be produced by DSIT [5–12]. Furthermore, this approach is a simple and potentially commercially exploitable route. Therefore, an understanding of the mechanisms for the refinement of ferrite grain is desirable for making further improvements. However, in contrast to the conventional strain-assisted transformation (e.g. the controlled rolling process), DSIT occurs during rather than after the deformation. It is a complex mechanism involving a number of physical processes including diffusion of solute atoms, evolution of dislocation, propagation of grain boundaries, phase transition, dynamic recrystallization and their interactions. Naturally, this makes it difficult to carry out any direct experimental observations to study the DSIT mechanisms. At present, the most popular method to study DSIT is the characterization of microstructure on a sample waterquenched immediately after the finish of hot deformation. However, the microstructure formed during the dynamic

1359-6454/$36.00 Crown Copyright Ó 2009 Published by Elsevier Ltd. on behalf of Acta Materialia. All rights reserved. doi:10.1016/j.actamat.2009.03.005

C. Zheng et al. / Acta Materialia 57 (2009) 2956–2968

deformation is not easily preserved because the unstable deformed austenite could continue to transform to ferrite during quenching. Recently, a novel quantitative dilatometric analysis [13] was developed to measure the ferrite fraction formed during DSIT with better reliability than the quenching technique. Nevertheless, even this method was unable to study the microstructure development during DSIT. Yada et al. [14] presented direct evidence of DSIT by in situ X-ray diffraction, which clearly showed that austenite–ferrite transformation occurred during deformation. However, quantitative study is presently almost impossible on account of the low density of available X-ray beams. As a result, the DSIT mechanism and its kinetics still remain largely ambiguous due to the lack of appropriate and effective experimental methods. In the last decade, with the development of numerical algorithms and computer power, it is now possible to model the ferrite transformation on a mesoscale [15]. Numerical modeling has thus emerged as an alternative and powerful tool to investigate the mechanism of DSIT in carbon steels. In the previous work, a Q-state Pottsbased Monte Carlo (MC) model has been developed to simulate DSIT above the Ae3 temperature [16]. Although some interesting results have been obtained, MC technique has the drawback of being unable to describe the temporal evolution of DSIT microstructure due to the fact that the MC time step used in the calculation is not correlated with real time. Furthermore, it is not possible to predict the transformation kinetics quantitatively because of the limitation of the probabilistic transition rules. In contrast to the MC technique, cellular automaton (CA) can provide an alternative approach, describing the spatial and temporal microstructure evolution at a mesoscale. It has been applied successfully to simulate the c–a transformation [17–20] in steels in real time and space. However, for DSIT, the microstructure evolution and the hot deformation happen simultaneously and affect each other, involving significantly more complex physical metallurgy processes than the individual c–a transformation modeling. Thus those conventional CA models with a single transition rule and incomplete descriptions of microstructure characteristics cannot deal with this complex physical procedure. In this paper, we use a modified CA model to study the integrated microstructural evolution and the transformation kinetics for DSIT in a low carbon steel. By coupling the fundamental physical metallurgical principles, concurrent modeling of the metallurgical phenomena during the dynamic deformation has been achieved. These features include the evolution of dislocation density, topology deformation, austenite–ferrite transformation and ferrite dynamic recrystallization (DRX). The model permits the mechanisms for the refinement of the DSIT ferrite to be studied, together with the effects of the prior austenite grain size (PAGS) and the deformation strain rate.

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2. Model description and numerical methods 2.1. Ferrite nucleation and growth during DSIT During DSIT, the transformation kinetics is mainly dominated by nucleation and growth of the newly formed ferrite grains. Classical nucleation theory indicates the nucleation rate for DSIT ferrite formation Is can be described by [21]: ! K2 12 ð1Þ I s ¼ K 1 Dc ðkT Þ exp  2 kT ðDGÞ where K1 is a constant related to the nucleation site density (2.07  1011 J1/2 m4) [21], K2 is a constant related to all the interfaces involved in nucleation, i.e. the c/c interface and the c/a interface and the shape of the critical nucleus (2.5  1018 J3 m2) [22], Dc is the carbon diffusion coefficient in austenite, k is the Boltzmann constant, T is the temperature in Kelvin and DG is the driving force for ferrite nucleation. During DSIT, continuous deformation introduces deformation defects with high stored energy within the deformed austenite phases. These regions then become preferred sites for the generation of DSIT ferrite nuclei and provide additional driving forces for ferrite nucleation. Hence the driving force for DSIT ferrite nucleation, DG, is reduced to the contribution of both the chemical driving force, DGchem, and the stored deformation energy, Edef: DG ¼ DGchem þ Edef

ð2Þ

where DGchem is provided by the difference of Gibbs chemical-free energy between the ferrite and austenite phases. Its estimation is described in detail in Ref. [23]. The kinetics of the ferrite growth during DSIT is assumed to be determined both by the carbon diffusion in austenite and by the c/a interface mobility. As the c/a interface moves during the transformation, carbon atoms transfer into the austenite at the c/a interface due to the lower carbon solubility in ferrite and then diffuse into the austenite grain interior. Consequently, the c/a transformation on the moving c/a interface is described as a free boundary problem for carbon diffusion in austenite and the dynamics of the interface. The kinetics of the moving c/a interface is determined by V ac ¼ M ac F

ð3Þ

where Vac is the velocity of the c/a interface, Mac is the effective mobility of the moving c/a interface which can be described as   Q ð4Þ M ac ¼ M 0 exp  RT where M0 is the pre-exponential factor, Q is the activation energy for atom motion at the interface, R is the universal gas constant, T is the absolute temperature, and F is the driving pressure for interface motion. During DSIT, the

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driving pressure is provided both by the chemical driving force, Fchem and by the stored deformation energy, Edef: F ¼ F chem þ Edef

ð5Þ

Fchem can be derived from Svoboda et al. [24]: F chem ¼ lcFe  laFe

ð6Þ

where lcFe , laFe are the chemical potentials of iron in austenite and ferrite, respectively, determined by the local carbon concentration on either side of the c/a interface. Here we calculate the chemical potentials of iron in austenite and ferrite phases using the regular solute sub-lattice model [25]. The carbon diffusion in austenite phase is governed as follows: @c ¼ r  ðDc rcÞ @t

ð7Þ

  Q _ e; T Þ ¼ C_eg exp  N nð_ RT

ð9Þ

where e_ is the strain rate, C is a constant, QN is the activation energy for nucleation and the exponent g is set to 1 in the present simulation. R and T have their usual meanings. The critical dislocation density for nucleation on grain boundaries is calculated by considering the energy change, as proposed by Roberts and Ahlblom [28]:  1=3 20cm e_ ð10Þ qc ¼ 3blM aa s2 where s = lb2/2 is the dislocation line energy, l is the shear modulus of the material, cm is the grain boundary energy and Maa is the grain boundary mobility. The dislocation mean free path l is determined by the flow stress, r [29]: lb r

where c is the carbon concentration (wt.%). Dc can be calculated using the following relation [26]:

l ¼ 10:5 

Dc ¼ 4:7  105 exp½1:6c  ð37000  6000cÞ=RT 

The velocity of the recrystallization front, Vaa, moving into the deformed ferrite can be expressed as

ð8Þ

Fig. 1 shows the influence of the stored deformation energy on the driving force for the c–a transformation at 780 °C. It can be seen that the driving force is markedly increased by the stored deformation energy for a given carbon concentration. Additionally deformation increases the equilibrium carbon concentration. Of course, Edef is not only influenced by the deformation conditions (i.e. deformation temperature and strain rate), but also dependent on the spatial sites within the specimen because of the heterogeneous nature of the deformation. 2.2. Ferrite dynamic recrystallization In the present model, the ferrite DRX nuclei are assumed to be generated on the ferrite grain boundaries once the dislocation density increases to the critical value, qc. The nucleation rate for ferrite DRX as a function of temperature and strain rate becomes [27]:

V aa ¼ M aa P

ð11Þ

ð12Þ

where P is the driving pressure for the recrystallization front movement derived from the difference of stored energy between the recrystallized ferrite grains and the deformed ferrite matrix, Maa is the boundary mobility of high angle grain boundary, which can be estimated by [30]   D 0 b2 Q exp  b ð13Þ M aa ¼ kT RT where Qb is the activation energy for grain-boundary motion and D0 is the boundary self-diffusion coefficient. 2.3. Hot deformation The current model simulates the hot deformation in three parts: (1) modeling the change of dislocation density with the continuous deformation to determine the stored deformation energy; (2) describing the discrete distribution of the stored deformation energy within deformed grains; (3) the simulation of the plastically deformed grain structure. 2.3.1. Distribution of stored deformation energy and its calculation During hot deformation, part of the deformation energy is stored in the form of dislocations. In this case the stored deformation energy is related to the dislocation density, q: Edef ¼ bqlb2

Fig. 1. Driving force for transformation as a function of the carbon concentration in the low carbon steel at 780 °C.

ð14Þ

where b is a constant with a value in the range of 0.5 to 1, b is the magnitude of the Burgers vector. According to Kocks and Mecking [31], the variation of the dislocation density with respect to the deformation strain is given by the K– M model,

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dq pffiffiffi ¼ k1 q  k2q de

ð15Þ

where k1 = 2h0/(alb) is the constant representing work hardening and k2 = 2h0/rs is the softening parameter that represents recovery of dislocations. The hardening rate h0 is independent of the strain rate, and only dependent on temperature through the influence of temperature on shear modulus. rs is the steady state stress. Conventionally, h0 and rs can be determined from the experimental flow stress–strain curves. However, the experimental curves for either pure austenite or pure ferrite are not easily obtained. In the present model, the values of the parameters h0 and rs for both pure austenite and pure ferrite phases are estimated from a stress–strain model developed by Hatta et al. [32]:   Q n0 ð16Þ e_ ¼ A0 ½sinhðars Þ exp  A RT where A0, n0 , a and QA are constants available from the original authors [32]. In general, the distribution of dislocations is heterogeneous within grains. In fact, higher levels of stored deformation energy in the vicinity of the grain boundaries and triple junctions have been reported both from the measurement based on the image quality of the electron back-scatted diffraction (EBSD) patterns and in the modeling using the crystal plasticity finite element method (CPFEM). [33] It can thus be reasonably assumed that local stored deformation energy associated with the spatial sites has its maximum near the grain boundary and minimum at the grain center. In the present paper, a simplified analytical approach was used to describe the discrete distribution of stored deformation energy within the deformed grains. Details of the model can be found in Ref. [34]. According to the model, the stored deformation energy in the cell Si is given by the following relation: ES i ¼ ðf ðLÞ þ P ðS i ÞÞH MAX

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2.3.2. A uniform topology deformation model During the plastic deformation, the grain topology changes with increasing deformation strain. In this model, a uniform topology deformation based on a vector operation is assumed. Each point in two-dimensional space can be considered as a vector based on a certain origin. Thus each vector can be operated in turn to generate a set of new vectors defining the new shape [37]. The deformation can be handled mathematically as a vector operation. For the 2D model, the deformation itself can be described by a 2  2 deformation matrix. Thus an original vector u becomes a new vector v as a result of a uniform deformation S: v ¼ Su The form of the matrix can be written:      lx 0 ux mx ¼ 0 ly u y my

ð18Þ

ð19Þ

where ui (i = x, y) are the components of the original vector u and vi (i = x, y) are the components of the new vector v after deformation. li (i = x, y) are the principal nominal deformation at two directions (the ratios of the final to initial lengths of vectors along two principal axes). Therefore ei = lnli (i = x, y) are the true strains along the two principal axes of deformation. It is, of course, assumed that there is no change in volume during deformation so that the determinant of S must be unity. It means that lxly = 1. 2.4. Cellular automaton modeling The cellular automaton model for the present DSIT study is a modification of that developed for a previous study [38]. In this CA model, hexagonal cells and Von Neumann’s neighbor rule, which considers the nearest six neighbors, are applied. To simulate the microstructural

ð17Þ

where HMAX is the maximum possible value of stored deformation energy in the whole lattice at a given deformation strain, and the quantity f is a factor depending on the distance from the grain boundary L which decreases from 1.0 to 0.2 within a distance range of 1 lm to quantify the stored energy gradient from the grain boundary to the grain interior. P(Si) is assigned by a random number between 0 and 1 which is independent of L with the purpose of quantifying the high energy defects in the intragranular regions. Clearly, the quantity of the intragranular high energy defects described in this model is affected by the prior grain size. The larger the prior grain size, the higher the volume fraction of the intragranular deformation defects for a given set of deformation conditions, which has been validated experimentally [35,36]. The schematic illustration of the distribution of the stored deformation energy across grain in a deformed microstructure is presented in Fig. 2.

Fig. 2. Schematic illustration of the inhomogeneous stored deformation energy distribution across grain in a deformed microstructure.

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evolution during the whole DSIT transformation, each cell in the lattice is assigned seven state variables: (i) the phase state that denotes whether the cell belongs to the ferrite, austenite or c/a interface; (ii) the ferrite DRX status that indicates whether it belongs to deformed ferrite, DRX ferrite or interface; (iii) the orientation variable which is assigned an integer representing the crystallographic orientation of the ferrite nucleus; (iv) the carbon concentration variable; (v) the dislocation density variable; (vi) the ferrite transformation fraction variable that indicates the ferrite fraction transformed; and (vii) the ferrite DRX fraction variable. The kinetics of the automaton is realized by updating the state variables according to a deterministic transformation rule. For the cell (h, k) in which the status variable belongs to the moving c/a interface or the interface between the DRX ferrite and the deformed ferrite, the moving distance of the interface at time, dt can be described as: Z t vdt ð20Þ lth;k ¼ 0

where v is the velocity of the boundary movement. Then t the transformation fraction in (h, k), fh;k can be calculated by: t fh;k ¼ lth;k =LCA

ð21Þ

where LCA is the distance between two neighboring cells. If the value of the accumulated transformation fraction variable is greater than 1.0, the interface cell should switch into ferrite state or recrystallized state. The procedure to simulate the microstructure evolution during DSIT in one time step is as follows: (1) Calculate the evolution of the dislocation density from the deformation strain according to the K–M model, and then determine the distribution of the stored deformation energy. The deformed microstructure is obtained by implementation of the topology deformation model. The simulated results of the hot deformation, including the grain topology structure, the distribution of the stored deformation energy and the stored energy level, become the initial state for the subsequent microstructure simulation during the c–a transformation or ferrite DRX. (2) DSIT ferrite nucleation occurs at preferred nucleation sites based on the distribution of the stored deformation energy as described in Section 2.1. Once a nucleation take place in a cell, its state changes immediately into the ferrite state, and all of its neighboring austenite cells change their states from austenite to c/a interface. The rejected carbon atoms from the ferrite phase are transferred onto the interface and the stored deformation energy of the cell drops to zero. The carbon atoms diffuse into the surrounding austenite regions at a rate given by Fick’s second law. This is numerically solved using a finite volume method.

(3) Calculate the interface velocity at the c/a interface cells according to Eq. (3) and then determine the ferrite transformation fraction. If the transformation fraction is not less than 1, the phase state of the cell switches to ferrite and all of its neighboring austenite cells change to a c/a interface. The excess carbon atoms are rejected in the surrounding interface cells and the stored energy drops to zero. (4) If the dislocation density within the cells of the ferrite state located on the ferrite grain boundaries is greater than the critical DRX threshold, ferrite DRX nuclei are generated in these cells according to Eq. (9). The growth of the DRX ferrite nuclei is then calculated. The numerical implementation of the ferrite DRX nucleation and growth are similar to that described in Step (2) and Step (3). (5) The simulation continues until the pre-strain is reached. 3. Details of simulation The investigated steel in this study is a plain low carbon steel with a chemical composition (wt.%) of 0.13C–0.19Si– 0.49Mn. Its transformation temperature Ae3 is calculated as 848 °C by Thermo-Calc. For the investigation of DSIT, the sample is subjected to a uniaxial compression to a reduction of 70% at an isothermal temperature of 780 °C. The initial simulated domain is discretized by a CA lattice of 600  1200 which corresponds to a real domain of 120 lm  240 lm in the sample. The initial microstructure is created by a simulation of the normal grain growth process. In order to investigate the effect of the PAGS on DSIT, simulations on three different PAGS levels of r0 = 10 lm, r0 = 20 lm and r0 = 30 lm in radius have been performed. To investigate the influence of strain rate three values e_ ¼ 5 s1 , e_ ¼ 10 s1 and e_ ¼ 20 s1 have also been implemented. Furthermore, to clarify the mechanism of the ferrite refinement during DSIT, another transformation process of isothermal decomposition of the undeformed austenite, during which the sample is isothermally held for a long time at the same temperature of 780 °C till the ferrite transformation is completed, was also considered. Details about the isothermal c–a transformation are given in Ref. [39]. The key parameters used in the present modeling are listed in Table 1. An equivalent grain radius Ri is used to define the grain size of each ferrite grain, and is defined as follows: rffiffiffiffiffi Ai ð22Þ Ri ¼ p where Ai is the area of ferrite grain i, which can be calculated by totalizing the included cells of each ferrite grain.  is calculated Then the average diameter of ferrite grains hRi as PN a  ¼ ð i¼1 Ri Þ ð23Þ hRi Na where Na is the total number of ferrite grains in the system.

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Table 1 Key parameters used in simulations [32,40,41]. l (GPa)

b (m)

A0 (s1)

n0

a (MPa1)

QA (kJ mol1)

50 32

2.48  1010 2.58  1010

1.2  1012 8.2  1016

5.5 4.9

7.1  103 6.3  103

289 325

M0 (mmol J1 s1)

Q (kJ mol1)

Qb (kJ mol1)

QN (kJ mol1)

D0 (m2 s1)

cm (J m2)

0.5

147

120

170

8.9  105

0.56

c a

4. Results 4.1. Formation of ultra-fined ferrite during DSIT With a purpose of apperceiving the formation of the ultra-fined ferrite during DSIT, microstructural evolution of the DSIT ferrite at a strain rate of 10 s1 is also investigated experimentally by hot compression test on a Gleeble 3500 machine. The cylindrical specimens used were uniax-

ially compressed to serial true strains of 0.3, 0.5 and 0.7 at an isothermal temperature of 780 °C and then subjected to immediate quenching by 20% ice-brine to preserve the microstructure formed during hot deformation. The microstructure of a section in the middle of the specimen was observed under an optical microscope. Fig. 3 shows the microstructure derived both from the simulation (Fig. 3a–c) and from the optical microscopic observation (Fig. 3d–f). In the simulated microstructure,

Fig. 3. Microstructures derived from the simulation at different strains ((a) e = 0.3, (b) e = 0.5, (c) e = 0.7) and from the optical micrographs ((d) e = 0.3, (e) e = 0.5, (f) e = 0.7) for the deformation of e_ ¼ 10 s1 , T = 780 °C.

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the gray regions are retained austenite phases and the blue areas are DRX ferrite. The white areas are the newly formed DSIT ferrite without DRX happening. Different grains are separated by the black lines. Under the microscope, the quenched microstructure is seen to be composed of DSIT ferrite (white areas) and lath martensite or pearlite (black areas). From Fig. 3, it can be seen that DSIT ferrite grains nucleate preferably at the prior austenite grain boundaries at the early stage of transformation and then originate close to the c/a interface with the increased deformation either in the simulation or in the OM observations, as indicated by the arrows in the figures. A larger deformation strain leads to more ferrite transformed. The evolution of the simulated microstructure agrees well with that observed from the experiment in quenched samples. It is known that, for the decomposition of undeformed austenite, ferrite nuclei prefer to originate on the prior austenite grain boundaries because of the effect of grain boundary energy. The nucleation sites become ‘‘saturated” once the c/c boundaries are entirely occupied. While for DSIT, the austenite areas adjacent to the newly formed c/a interface are produced with high stored energy due to the subsequent heterogeneous deformation and thus become preferable for ferrite nucleation. Therefore, DSIT ferrite nuclei can originate not only at the prior austenite grain boundaries but also in front of the c/a interface. New ferrite layers are seen to be formed close to the c/a interface and spread away into the austenite interior, as indicated in Fig. 3. On the other hand, the subsequent deformation introduces intragranular defects such as deformation bands on which ferrite grains can also nucleate within the prior austenite grains. Because these highly

deformed regions are produced continuously during the deformation, ferrite grains nucleate repeatedly. The nucleation sites remain ‘‘unsaturated” till the phase transformation is completed during the deformation process. This kind of ‘‘unsaturated” nucleation leads to a higher nucleation density compared to isothermal transformation. This seems to be an important aspect of ferrite refinement by DSIT. Fig. 4a and b depicts the comparison of the simulated carbon concentration field between DSIT and the isothermal c–a transformation in an undeformed sample. Fig. 4c shows the corresponding distributions of carbon concentration along the black straight lines in Fig. 4a and b. Because of the lower solubility of carbon in ferrite compared to austenite, carbon atoms are rejected continuously from the ferrite into the austenite trough the c/a interface, which increase the carbon concentration at the c/a interface. Meanwhile, the carbon atoms at the interface diffuse away into the austenite grain interior. As a result the ferrite grain grows steadily till the carbon concentration in the overall retained austenite increase to the equilibrium value, as shown by Fig. 4a. For the DSIT, a faster kinetics is obtained on the moving c/a interface due to the role of deformation and thus more carbon atoms are rejected to the c/a interface. However, the time for DSIT is so limited that the carbon atoms are unable to diffuse efficiently into the grain interior. As a result, they remain concentrated near the c/a interface within a very thin carbon-rich layer. The high carbon concentration on the c/a interface will in turn decrease the driving force for the interface migration and then suppress the growth of ferrite grains. It indicates that the growth of

Fig. 4. Simulated carbon concentration fields (wt.%) during (a) the isothermal decomposition of the undeformed austenite at the time of 10 s under the isothermal temperature T = 780 °C, (b) the DSIT when e = 0.4 under a deformation of e_ ¼ 10 s1 and T = 780 °C and (c) the carbon concentration distributions of DSIT (upper) and isothermal c–a transformation (lower) along the black straight lines shown in (a) and (b).

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ferrite has been limited during DSIT. This kind of limited growth is considered to be another factor for the ferrite refinement during DSIT. The ‘‘unsaturated” ferrite nucleation at the c/a interface can not only increase the ferrite nucleation density, but also facilitate the soft or hard impingement among the neighboring ferrite grains as shown in Fig. 4b. It is seen that some small-sized islands/flakes of the retained austenite are isolated with high carbon concentration. Within these austenite regions, the driving force for ferrite growth is decreased to zero when the carbon concentration reaches the equilibrium value. Then the growth of ferrite grains gets fully retarded and thus they are retained during the subsequent transformation. The small-sized islands/flakes of retained austenite can also be found in the optical micrograph observations as shown in Fig. 3e and f. These fine islands/flakes are expected to retard the coarsening of the DSIT ferrite grains through pinning, which plays an important role in preserving the fine-grained DSIT ferrite during the transformation. From Fig. 3, we can see that with increasing deformation strain the whole simulated domain is compressed and elongated normal to the compression direction. Thus the DSIT ferrite generated at an early stage should be also noticeably elongated due to the continuous deformation, whereas, in fact, the ferrite maintains its equiaxed morphology even at large deformation strains observed in most DSIT experiments [5–14]. This behavior definitely suggests that another mechanism of ferrite DRX should be involved other than just c–a transformation during DSIT. From Fig. 3a–c, it can be seen that ferrite DRX has introduced more ferrite nuclei in the sample. These newly formed DRX ferrite nuclei not only raise the density of ferrite grains, but also subdivide the elongated prior-originated ferrite grains and so maintain the equiaxed morphology. Therefore, ferrite DRX is an additional contributor to the refinement of the DSIT ferrite. From this analysis we can reasonably conclude that the ferrite refinement by DSIT is the result of the increasing ferrite nuclei density by ‘‘unsaturated nucleation”, limited ferrite growth as well as ferrite DRX.

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defects can be induced within the larger prior austenite grains. Fig. 6 depicts the evolution of the volume fraction of the DSIT ferrite with the deformation strains and its variations with PAGS. It can be seen that the fraction of DSIT ferrite is larger for fine PAGS at the early stage of DSIT. However, the difference between the fine and coarse PAGS becomes less pronounced at a later stage. According to Section 4.1, the DSIT ferrites are formed preferentially along the prior austenite grain boundaries and advance in the form of rafts in the area ahead of the c/a interface. Accordingly, the distribution and amount of the prior austenite grain boundaries determine the spatial distribution and transformation fraction of the DSIT ferrite at an early stage of the transformation. In comparison with the larger prior austenite grains, the finer grains provide more boundaries to act as preferred nucleation sites for the DSIT ferrite and hence lead to a larger volume fraction at the first stage of the deformation. This conclusion has been verified by numerous experimental observations [10,42,43]. However, with the development of the high energy intragranular defects at increasing strain, large quantities of DSIT ferrite nucleate at the deformation defects (e.g. deformation bands) in the grain interiors of coarse austenite grains. Therefore, the effect of PAGS on the transformation kinetics becomes less pronounced at a late stage. Fig. 7 describes the size distribution of the resultant DSIT ferrite at a volume fraction of 90%. It is seen that PAGS has little effect on the frequency distribution of the DSIT grain size, whereas the PAGS does affect the spatial distribution of the DSIT ferrite nucleation sites, this becoming more homogeneous from the fine austenite compared with the coarse variety as indicated in Fig. 5. As the strain is increased, the effect of PAGS on the spatial distribution is reduced due to the development of intragranular nucleation. In summary it is reasonably concluded that if the prior austenite grains are refined, e.g. by recrystallization in the individual austenite region, a larger volume fraction of fine-grained and homogeneously distributed ferrite is expected to be produced, even at low deformation strains.

4.2. Influence of PAGS on DSIT 4.3. Influence of the strain rate on the DSIT Fig. 5 shows the influences of PAGS on the simulated microstructure (left) and the distribution of the stored energy (right) when e = 0.5 and e_ ¼ 10:0 s1 . It is shown that the DSIT ferrite initially nucleated at the prior austenite grain boundaries at an early stage of transformation followed by nucleation in the grain interiors for both coarse and fine prior austenite grains. However, the ferrite layers cannot travel far into the interior of the coarse austenite grains within the limited period of transformation, therefore large areas of the retained austenite remain untransformed. Another result of relevance in Fig. 5 is that, for the coarse austenite, the intragranular ferrite nucleation is more developed than that for the finer prior austenite. That is because larger amounts of intragranular deformation

Fig. 8 shows the simulated microstructure and the stored energy field of the DSIT at the deformation e = 0.5 for three different strain rates (_e ¼ 5 s1 , 10 and 20 s1). Distinct differences can be observed in the morphology of the DSIT ferrite at the different rates. It can be seen clearly that the DSIT ferrite is much finer at the higher strain rate with minor growth and small transformation fraction. During the DSIT, increasing the strain rate will reduce the time for transformation to take place. Similarly, carbon atoms rejected from the DSIT ferrites are more suppressed to diffuse away from the moving c/a interface as mentioned in Section 4.1. As a result, the growth of the DSIT ferrites

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Fig. 5. Simulated microstructure and stored energy field (J mol1) of the DSIT at a deformation of e = 0.5, e_ ¼ 10 s1 for different PAGS of (a) r0 = 30 lm, (b) r0 = 20 lm, (c) r0 = 10 lm. The black lines in the figures indicate the grain boundaries. The white areas in the left images are the newly formed DSIT ferrite grains and the gray regions are retained austenite phase. The areas in blue in the figures indicate the DRX ferrite grains. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

will be more retarded. Another factor, as described in the stored energy field in Fig. 8, is that deformation at a high strain rate inputs an increased stored energy, raising the DSIT nucleation level and introducing more nucleation sites both in front of the c/a interface and in the intragranular areas. As a result of all these factors, the mean grain size of the DSIT ferrite is reduced as the strain rate is increased, as shown in Fig. 8. A similar result can be obtained in Fig. 9, which describes the influence of the strain rate on the size distribution of the resultant DSIT ferrite. It shows clearly that the ferrite grains after higher strain-rate deformation are mainly distributed at the side of small size magnitudes. The results also imply that the distribution peak for higher strain rate is narrower than that for lower strain rate, which means that the ferrite size distribution at large strain

rate is more homogeneous. Decrease in the transformation time due to the increasing strain rate allows less growth for the DSIT ferrite grains. It reduces the tendency for grain coarsening and thus leads to a more homogeneous size distribution of the final DSIT ferrite. Fig. 10 shows the changes of the average grain size of the DSIT ferrite with the deformation strain. It is worth noting that the mean ferrite grain size is nearly constant during transformation. This indicates that the ferrite coarsening is nearly exactly counteracted by the continuous introduction of the newly formed ferrite grains owing to unsaturated nucleation. Meanwhile, the elongated and coarsened ferrite grains are subdivided by the formation of the ferrite DRX. As a result, the average ferrite grain size remains unchanged at a late stage of the DSIT. This fixed mean grain size of the DSIT

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ferrite fraction. Conversely, if the strain rate is low, the DSIT ferrite will have time to grow. Meanwhile, some deformation defects for potential ferrite nuclei are expected to be consumed by the growth of the DSIT ferrite, leading to a coarse grain size. Therefore, it seems reasonable to conclude that there is an optimum strain rate range for the formation of ultra-fine ferrite with a large volume fraction. 5. Summary and discussion

Fig. 6. Volume fraction of the DSIT ferrite as a function of deformation strain for different PAGS.

Fig. 7. Frequency distribution of the DSIT ferrite grain size at the volume fraction of F = 0.9 for various PAGS.

has been reported by Beladi et al. [10] in their hot torsion experiments. Fig. 11 illustrates the transformation kinetics of the DSIT at different strain rates. It can be seen that increasing strain rate leads to faster transformation at the same deformation temperature due to the larger stored deformation energy. However, at a given deformation strain, the volume fraction of the DSIT ferrite is reduced owing to the reducing transformation time, as seen in the inset of Fig. 11. It suggests that the influence of the increasing strain rate causing the decrease of the transformation time might be countered by increasing the deformation strain to obtain similar volume fraction of the DSIT ferrite. Experimental results in agreement with this conclusion have been reported by Mintz [44] and Yang et al. [45]. Actually, changing the strain rate has a complex effect on the DSIT. Increasing the strain rate can induce more DSIT ferrite nuclei, which favors the formation of ultra-fine ferrite, but the decreasing deformation time leads to a small DSIT

In the present study, the dynamic strain-induced austenite-to-ferrite transformation in a low carbon steel has been investigated using a modified CA model in which the interactions of the concurrent ferrite transformation and deformation during the DSIT are addressed. This modeling provides an alternative methodology of computational simulation to an insight of the ferrite refinement during DSIT on a mesoscale. The simulated results indicate that the refined ferrite grains derived from DSIT are achieved by ‘‘unsaturated” ferrite nucleation and ‘‘limited” growth. Furthermore, ferrite DRX is occurring to subdivide the DSIT ferrite grains formed at the early stage of transformation and maintain their equiaxed morphology. The formation of the fine-sized islands/flakes of the retained austenite also contributes to retarding the subsequent DSIT ferrite coarsening. The results also indicate that the processing variables (e.g. deformation strain, strain rate and PAGS in the present modeling) influence the evolution of the DSIT ferrite and the characteristics of the resultant microstructure markedly: (1) A larger deformation strain leads to more ferrite formation, whereas the mean ferrite grain size is nearly constant during straining. (2) The PAGS does affect the spatial distribution of the DSIT ferrite nucleation sites at an early stage of the transformation. For fine prior austenite, the DSIT ferrite grains distribution is more homogeneous compared with the coarse austenite. And the reduction of PAGS intends to induce larger amount of the DSIT ferrite. It is indicated that refinement of the prior austenite grains is a possible way to achieve refined and homogeneously distributed ferrite grains with a larger volume fraction. (3) Finer ferrite grains can be achieved by increasing the strain rate, while the transformation fraction will decrease at the same strain level. Thus, a larger deformation strain is needed at higher strain rate in order to produce the same fraction of the DSIT ferrite. In addition, increasing the strain rate will decrease the transformation time, allowing less growth for the DSIT ferrite grains, which is expected to reduce the tendency of grain coarsening and result in a more homogeneous size distribution of the final DSIT ferrite.

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Fig. 8. Simulated microstructure and stored energy field (J mol1) of the DSIT for a deformation of e = 0.5 at different strain rates of (a) e_ ¼ 5 s1 , (b) e_ ¼ 10 s1 , (c) e_ ¼ 20 s1 . The black lines in the figures indicate the grain boundaries. The white areas in the left images are the newly formed DSIT ferrite grains and the gray regions are retained austenite phase. The areas in blue in the figures indicate the DRX ferrite grains. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

In the current modeling, the ‘‘dynamic” transformation is the issue to be addressed, during which the geometrical interaction of grains and the concurrent deformation have both been considered. During DSIT, the intragranular ferrite nucleation occurred rapidly and extensively on the dislocation substructures. This seems fundamental for the ferrite refinement [46,47]. In the present work, a phenomenological deformation model was applied to describe qualitatively the heterogeneous distribution of the stored deformation. However, for a more precise understanding of the mechanism of DSIT, a quantitative description of the sites and amounts of the intragranular deformation features such as the deformation bands as well as the nature and evolution of these as a function of the deformation condition within the austenite grains would be necessary. Fortunately, many researchers have

paid attention to the heterogeneity of deformation in polycrystalline materials and proposed some meaningful deformation models based on crystal plasticity theory (e.g. CPFEM) [48–50]. Local distribution of the mechanical responses of materials to deformation, such as stress, strain and stored deformation energy, can now be described quantitatively on a mesoscale. Coupling the modeling incorporated CPFEM with some mesoscopic simulation techniques such as Monte Carlo [51] or cellular automaton [52] has resulted in successful sequentially integrated models investigating the effect of austenite deformation on the subsequent c–a transformation in steels. If incorporated into this present synchronous deformation/transformation model, these improvements would provide a complex but welcome development for future DSIT simulation.

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Acknowledgements The authors are grateful to Prof. John Campbell at the University of Birmingham for language polish and valuable discussion. The authors also gratefully acknowledge the financial support from National Natural Science Foundation of China (NSFC) under Grant No. 50871109. References

Fig. 9. Frequency distribution of the DSIT ferrite grain size at the volume fraction of F = 0.85 at different strain rates.

Fig. 10. Changes of the simulated average grain size of the DSIT ferrite with deformation strain at different strain rates.

Fig. 11. Simulated transformation kinetics at different strain rates.

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