Modeling Dynamic Recrystallization Using Cellular Automata

Scripta Materialia, Vol. 38, No. 3, pp. 405– 413, 1998 Published by Elsevier Science Ltd Printed in the USA. All rights reserved. 1359-6462/98 $0.00 1 .00

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MODELING DYNAMIC RECRYSTALLIZATION USING CELLULAR AUTOMATA R.L. Goetz1 and V. Seetharaman1 Materials Directorate, Wright Laboratory, WL/MLLM, Wright-Patterson AFB, OH 45433-7817, U.S.A. 1 UES, Inc., 4401 Dayton-Xenia Road, Dayton, Ohio 45432-1894, USA (Received June 5, 1997) (Accepted October 28, 1997) Introduction Dynamic recrystallization (DRX) and dynamic recovery (DR) are two competing processes by which metals and alloys undergo a reduction in the density of accumulated dislocations during plastic deformation at elevated temperatures. Certain high stacking-fault energy materials, such as Al, a-iron, and their single phase alloys, do not readily undergo DRX because DR by cross-slip and climb processes is sufficiently rapid to prevent dislocation densities from reaching levels critical for the formation of DRX nuclei. With low stacking fault energy materials, such as copper, nickel, g-iron, and their single phase alloys, dissociation of dislocations into widely separated partials renders cross-slip and climb processes very difficult. This effectively reduces the rate of DR, thereby allowing dislocation densities to attain high values required for the nucleation of dynamically recrystallized, strain free crystals. It is important to note that DR occurs either separately or in conjunction with DRX. The study of dynamic recrystallization at high temperatures and strain rates higher than 1023 s21 began with the work of Rossard and Blain [1], and Luton and Sellars [2]. It was found that the flow curves for metals undergoing DRX at high strain rates, or low temperatures, exhibited a single peak before dropping to a lower steady state value. At low strain rates, or high temperatures, the curves exhibited multiple peaks before damping out to a steady state value. This was attributed to deformation controlled growth at high strain rates resulting in fine recrystallized grains, while growth at lower strain rates is impingement controlled with larger grains. Concurrent deformation at high strain rates reduces the driving force for growth by creating new dislocations in the recrystallized material at a rate faster than the rate of grain boundary migration. At lower rates, growth is faster than the reduction in driving force and the material fully recrystallizes, which then causes a drop in the flow curve. When recrystallization is completed, the flow curve rises again until dislocation densities reach levels for a second nucleation wave, followed by complete recrystallization and another drop in flow stress. Hence, the oscillations in the flow curves. At high rates, single peak curves result because growth is limited, and several waves of nucleation and growth take place simultaneously. Sakai and coworkers [3,4] showed that single peak flow curves for materials that recrystallize at the grain boundaries occur when DO/DSS . 2 and multiple peaks occur when DO/DSS # 2, where DO is the initial grain size and DSS is the recrystallized grain size in the steady state microstructure. The modeling of DRX began with Luton and Sellars [2]. Their model correctly predicted a transition from single to multi-peak flow curves with a reduction in the strain rate. However, the model produced 405

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multi-peak curves with continuous oscillations of equal amplitude, whereas experimentally, multi-peak curves rapidly dampen out to reach a steady state value of the flow stress. The model developed by Sandstro¨m and Lagneborg [5] used subgrain dislocation density to determine flow stress and the dislocation density in the subgrain boundaries as the driving force for grain growth. The model correctly predicted the transition from single to multi-peak flow curves for nickel, although the strain at the peak stress for the highest strain rate was much larger than that found experimentally. This was caused by the model neglecting the grain boundary nucleation behavior characteristic of DRX. The model put forth by Roberts et al. [6] dealt with deformation limited growth in the high strain rate/low temperature regime with single peak flow curves. This work examined the kinetics of DRX in terms of a modified Johnson-Mehl-Avrami- Kolmogorov (JMAK) model [7,8,9], with Avrami time exponents of approximately 1.3. Rollet et al. [10] and Peczak and Luton [11] used the Monte Carlo (MC) method to study DRX in detail and predict the transition from single to multi-peak flow curves, the effect of initial grain size, the peak strain dependence on peak stress, and the steady state grain size dependence on steady state stress. It is important to point out that the MC method uses the DR model of Mecking and Kocks [12], which has been shown by Roberts [13] to underestimate the peak stress due to a linear strain hardening rate (u 5 ds/de). Also, strain rate sensitivity was assumed and predefined (m5 0.2) for DR. The objective of the present work was to model DRX under isothermal and constant strain rate conditions using the cellular automata (CA) technique [14]. CA is similar to the MC method. However, fundamental differences exist due to the use of different phenomenological relationships, MC’s probabilistic method of grain boundary motion, and its use of the Potts model. This study is a sequel to the authors work on static recrystallization at grain boundaries using CA [15]. The application of CA to DRX was practicable because DRX also occurs through grain boundary nucleation. The model will be used to examine the effect of initial grain size on the transition from single to multi-peak flow curves, nucleation density, recrystallization kinetics, and the strain hardening rate.

Cellular Automata The CA technique for recrystallization developed by Hesselbarth and Go¨bel [14] was used by Goetz and Seetharaman [15] to model static, grain boundary initiated recrystallization of single phase materials. CA consists of a grid of uniform cells, which are updated every time step with a finite value or state using deterministic rules. Every cell is connected to a neighborhood of surrounding cells which control its evolution in time. These attributes are completely deterministic; random updating of cells renders the CA approach probabilistic. The alternating 7-cell neighborhood was used, which produces an octagon-shaped grain before its impingement with neighboring grains. An unrecrystallized cell becomes recrystallized if it is randomly chosen as a nucleus, or if a member of its neighborhood is recrystallized by a growing grain during the preceding time step. Impingement occurs when the neighbors of a recrystallized cell are already recrystallized by another advancing grain, thereby halting boundary migration at this cell for both grains. A 4903490 grid of cells was used in this work. Modeling of DRX by CA is a two-step process in which an initial microstructure is first created. For DRX, the dislocation density of a cell, rC, is tracked and determines nucleation and growth. If a critical dislocation density is reached in any cell located on a grain boundary, DRX begins. The following four processes take place sequentially: 1) cell dislocation input, 2) DR, 3) nucleation of recrystallized grains, and 4) growth of recrystallized grains. Cell dislocation density is incremented uniformly at a fixed rate, dr/dt, for each time step. Strain, e, and strain rate, e5de/dt, are assumed to be directly proportional to r and dr/dt, respectively. Also, the flow stress at each time step, s, is taken to be proportional to the square root of the average dislocation density (rA)1/2, with rA 5 ((rC)/(4902).

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The DR model of Mecking and Kocks [12] used by Peczak and Luton [11] was not used in this work because it was shown by Roberts [13] that the Mecking-Kocks model does not reproduce the experimental characteristic of an inflection point in the u vs. s curve. Dynamic recovery is modeled here by randomly choosing a certain number, N, of cells at each step and reducing the dislocation density in these cells by a certain fraction, typically one-half, i.e., riC5(ri21C)/2. This results in a nonuniform dislocation distribution. N varies as a function of dr/dt as well as m, i.e., N53170(dr/ dt)(122m). The derivation of N is based on

sDR,SS 5 (rT)1/2

rT 5 (490)2)rDR

(1)

where sDR,SS is the flow stress at steady state with DR only, rT is the total dislocation density, and 4902 is the total number of CA cells. At steady state (with DR only) there is no net accumulation of dislocations. Therefore, (4902)(dr/dt) 5 (N)(rDR)/2

(2)

In eqn. (2), the left-hand side represents the number of dislocations added by deformation during dt, while the right-hand side refers to the number lost due to recovery during this period. Solving for rDR, equation (2) becomes

r DR 5 (2)(4902)(dr/dt)/N

(3)

where rDR is the average steady state dislocation density attainable if DR were to operate alone, without DRX. With rDR, equation (1) becomes

sDR,SS 5 (4902)(2)1/2(dr/dt)1/2/N1/2

(4)

Combining the power law equation for sDR,SS (eqn. 5)

sDR,SS 5 K(dr/dt)m

(5)

with equation (4) and solving for N yields N 5 [(4902)(2)1/2/K]2(dr/dt)(122m)

(6)

where K56030. Thus N 5 3170(dr/dt) . The value of the strain rate sensitivity of the system is kept constant (m50.2) making DR strain rate sensitive with the same m assumed by Peczak and Luton. Nucleation of DRX is handled by randomly choosing a certain fraction of the total grain boundary cells present at each step. The fraction of grain boundary cells randomly checked for nucleation is a function of the dislocation input rate; (dr/dt)/1000. If the dislocation density of a chosen cell exceeds rDR, the cell becomes a nucleus for DRX and rC is reset to zero. Whenever a cell recrystallizes, rC becomes zero. Thus, from equation (3), rDR 5 (2)(4902)(dr/dt)/N. Nucleation of DRX may occur either at the parent grain boundary or at the boundary of recrystallized grains, if rC $ rDR. Recrystallized grains grow until they impinge with another recrystallized grain, or the rC of the nucleus becomes $ rCR. This approach is similar to the concept used by Sandstro¨m and Lagneborg [16]. The value of rCR controls the steady state grain size and is a function of dr/dt; rCR 5 15.88(dr/dt)0.7. In this manner, growth is either impingement controlled or deformation controlled, depending on the magnitude of dr/dt. The derivation of rCR was done using (122m)

DSS 5 2(G)(tG)

(7)

tG 5 rCR/(dr/dt)

(8)

DSS 5 2(rCR)/(dr/dt)

(9)

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Figure 1. Flow stress curves for dr/dt 5 10, 100, and 1000; a) s vs. Time, b) s vs. e.

where DSS is the steady state grain size, G is the radial grain boundary velocity (G is arbitrarily set to equal 1), and tG is the total time a grain is allowed to grow. Derby [17] has shown that DSS is also related to the steady state flow stress, sSS 5 (DSS)22/3. Using Derby’s equation and the power law relation for steady state flow stress, sSS 5 (dr/dt)0.2, DSS is a function of dr/dt, DSS 5 (dr/dt)20.3

(10)

By combining equations (9) and (10), rCR 5 15.88(dr/dt)0.7, where tG 5 2 with dr/dt51000. Results and Discussion Simulations were run for three dislocation input rates, dr/dt 5 10, 100, 1000. The stress vs. time and stress vs. strain curves obtained from these simulations are shown in Fig. 1. The strain rate sensitivity, m 5 dlog(s)/dlog(e), for the peak stresses is ' 0.2, which is to be expected due to the rate sensitivity of DR being set at 0.2. The rate sensitivity for steady state stresses increases from 0.185 to 0.20 as the strain rate decreases from 1000 to 10. Moreover, the ratio of the steady state to the peak stresses, sSS/sP, is equal to 0.76, 0.76, and 0.72 for dr/dt 5 10, 100, and 1000, respectively. It was determined that the nucleation rate for dr/dt 5 1000 was the cause for the lower ratio of sSS/sP, with the nucleation rate being too high. However, every boundary cell is checked for dr/dt 5 1000. Changing the DR parameters was found to lower the number of boundary cells meeting the critical value for nucleation, rDR. The number of cells randomly chosen for recovery was doubled and the drop in rC was changed from 1/2 to 1/4, i.e., N 5 6340(dr/dt)0.6 and riC 5 (3/4)ri21C. This is analogous to retarding DR, which is a characteristic of dynamically recrystallizing metals. Since N was doubled, the factor ‘29 in the equation (3) was also doubled (rDR 5 (4)(4902)(dr/dt)/N) to keep rDR constant. With these changes, the ratio of sSS/sP becomes 0.76 for dr/dt 5 1000. The ratio of the critical strain, eCR, for the onset of recrystallization to the strain, eP, associated with the peak stress, eCR/eP, decreased with increasing rate. Values of eCR/eP equal to 0.88, 0.78, and 0.57 were obtained for dr/dt 5 10, 100, and 1000, respectively. The preceding values of eCR/eP are in good agreement with those found experimentally [18].

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Figure 2. Avrami plot showing Log(-Ln(1-F)) vs. Log(t 2 tRX) for dr/dt 5 10, 100, and 1000.

The recrystallization kinetics are presented in Fig. 2 in the form of Avrami plots. The abscissa for this plot is the effective time, t-tRX, where tRX corresponds to the onset of DRX. Best fit straight lines result in slopes of k ranging from 1.4 to 1.6, where F 5 1 2 exp[2Bztk]. These are typical values of k for metals undergoing DRX [6]. The volume fraction recrystallized, F, at the peak stress increased from 0.02 to 0.12 with an increase in dr/dt from 10 to 1000. Typical microstructures obtained using the CA model of DRX are shown in Fig. 3 for strain rates of 10 and 1000. These structures correspond to fixed volume fractions (F50.1 & 0.5) of the recrystallized material. With an increase in dr/dt from 10 to 1000, the nucleation rate increases rapidly and grain growth becomes increasingly deformation limited resulting in a fine grained, necklace type microstructure. Strain Hardening Rate The DRX behavior of stainless steels has been extensively studied by McQueen et al. [18,19]. They have shown that the initiation of DRX distinctly affects the strain hardening rate (u 5 ds/de) vs. stress plots. Figure 4a shows a plot of ds/dt vs. s for DR alone as well as DR 1 DRX. The initiation of DRX is clearly marked by the departure of the ds/dt curve from that of the DR curve. The DR curve was generated by using the DR model discussed above (N 5 3170(dr/dt)0.6, riC 5 (ri21C)/2). The peak stress (sP) associated with DRX was 7700, while the steady state flow stress, sDR,SS, attributed to DR alone was 9500. The work of Poliak and Jonas [20] applied the principles of irreversible thermodynamics to relate the initiation of DRX to the minimum observed in the plot of -du/ds vs. s. Figure 4b shows such a plot for dr/dt 5 10. This plot reveals a minimum at s 5 7500, which corresponds to the deviation point for the DRX1DR curve from the DR curve in Fig. 4a. Poliak and Jonas show that the critical state for the initiation of DRX must occur before the peak stress, sP, which is also a characteristic of the present work and the work of Peczak and Luton [11]. Poliak and Jonas state that early models for DRX and DR [5,10,12] initiate recrystallization at the peak stress and deviations from linearity, such as the inflection

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Figure 3. CA microstructures; a) dr/dt 5 10 (F 5 0.10), b) dr/dt 5 1000 (F 5 0.10), c) dr/dt 5 10 (F 5 0.50), d) dr/dt 5 1000 (F 5 0.50).

point in Fig. 4a, are not inherent. These models use the upper limit of the average dislocation density that survives after DR for the initiation of recrystallization. This occurs only at the peak stress where du/ds is zero. The present work also uses the upper limit of the average steady state dislocation density during DR (rDR) as the critical density for nucleation. However, nucleation is only considered at the grain boundary cells. Also, DR is random and creates a nonuniform dislocation density throughout the cell grid. Thus, some grain boundary cells reach or exceed rDR before the peak stress. The fraction of grain boundary cells reaching rDR is very small as shown in Fig. 5 for dr/dt 5 1000, where the ratio of nuclei to grain boundary cells (P) is plotted vs. time. The fraction is 0.055 with the initial structure and 0.025 at steady state, even though every grain boundary cell is checked for dr/dt 5 1000. The steady state and initial fractions differ by 2.2, which correlates with the work of Sakai et al. [3,4]. They derived a relationship which showed that for annealed materials the probability of a site becoming a nucleus per unit grain boundary surface area, PO, was twice the nucleation probability during steady state, PS; PO 5 2PS.

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Figure 4. Strain hardening rate (u) curves for dr/dt 5 10; a) u vs. s, b) du/ds vs. s.

Single To Multi-Peak Transition One of the characteristics of DRX mentioned earlier is the transition from single to multiple peak flow stress curves when the strain rate is lowered or when the temperature is raised. Numerous researchers have also shown that the initial grain size also effects the transition to multiple peak behavior. Sakai et al. [3,4] have shown that the transition occurs at DO/DSS 5 2; single peak curves occur when DO/DSS . 2 and multi-peak curves when DO/DSS # 2, where DO is the initial grain size and DSS is the recrystallized grain size at steady state. A series of four CA simulations were conducted at a fixed rate, dr/dt 5 10, but with different values of the initial grain size, DO. It is well established that the steady state grain size is only a function of

Figure 5. Probability for nucleation (P) vs. time for dr/dt 5 1000. Arrow denotes initiation of DRX.

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Figure 6. Flow stress curves for dr/dt 5 10 with four different initial grain sizes (DO) and fixed steady state grain size (DSS).

the strain rate and temperature, and is independent of the initial grain size, DO [3,4]. Therefore, these simulations can be interpreted in terms of the variation in DO/DSS from 0.6 to 4.7. Figure 6 illustrates the transition from single to multi-peak flow curves as a function of DO/DSS. The observed transition at DO/DSS 5 2 correlates with Sakai et al. [3,4]. Conclusions The cellular automata method was successfully used to model DRX in single phase alloys. This model has comprehensively predicted the following experimentally observed features of DRX: (1) necklace type microstructures at high strain rates, (2) kinetics of DRX obeying the JMAK expression with an Avrami exponent of k51.4 –1.6, (3) an increase in the volume fraction recrystallized at the peak stress with increasing strain rates, (4) the ratio of the strain at the onset of recrystallization to the strain at the peak stress, eCR/eP, varied from 0.57 to 0.88, (5) a deviation in the strain hardening rate vs. stress curve at the onset of DRX, and (6) a transition from single to multi-peak flow curves at DO/DSS 5 2. Also, the probability of nucleation at the onset of DRX is twice the probability for nucleation at steady state. Acknowledgments This work was performed as part of the in-house research activities of the Processing Science Group, Materials Directorate, Wright Laboratory, Wright-Patterson AFB, OH. The authors were supported by Air Force Contracts No. F33615–92-C-5900 and F33615–96-C-5251. The authors wish to thank Dr. S.L. Semiatin for his enthusiastic support of this work and for valuable comments on the manuscript. References 1. 2.

C. Rossard and P. Blain, Me´m. Scient. Revue Me´tall. 56, 285 (1959). M. J. Luton and C. M. Sellars, Acta Metall. 17, 1033 (1969).

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