A Monte Carlo study of the influence of dynamic recovery on dynamic recrystallization

Acta metall, mater. Vol. 41, No. I, pp. 59-71, 1993

0956-7151/93 $5.00 + 0.00 Copyright © 1992 Pergamon Press Ltd

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A MONTE CARLO STUDY OF THE INFLUENCE OF DYNAMIC RECOVERY ON DYNAMIC RECRYSTALLIZATION P. P E C Z A K and M. J. L U T O N Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale, NJ 08801, U.S.A. (Received 2 December 1991; in revised form I July 1992)

Abstract--A Monte Carlo simulation technique has been used to study dynamic recrystallization in a polycrystalline matrix. The model employed, maps the microstructure onto a discrete, two dimensional lattice with use of a magnetic analog, the Q-state Potts model. The response of the material to hot deformation is simulated by adding recrystallization nuclei and stored energy continuously with time. The simulations presented in this work use an energy storage rate procedure that models stage III hardening in metals and thereby incorporates both work hardening and dynamic recovery. This model reproduces many features found recently by Rollett et aL in a more simple model that did not include an explicit dynamic softening mechanism. It is found, that the type of work hardening law assumed results in relationships between transient and steady state recrystallization parameters which more closely approach those observed in physical experiments. In this view, the roles of the controlling mechanisms of dynamic recrystallization in real polycrystalline materials are discussed. R~um~--On utilise une technique de simulation de Monte-Carlo pour &udier la recristallisation dynamique dans une matrice polycristalline. Le module utilis~ dresse des cartes de la microstructure sur un rtseau discret ~ deux dimensions en utilisant un analogue magn&ique, le module de Potts de l'ttat-Q. La r~ponse du mat~riau ~i la dfformation ~i chaud est simul~e en ajoutant continuement des germes de recristallisation et de l'tnergie stock~e au cours du temps. Les simulations pr~sent~es dans ce travail utilisent un mode de stockage d'tnergie qui mod~lise le durcissement de stade III des mttaux et donc incorpore ~i la fois l'~crouissage et la restauration dynamique. Ce module reproduit plusieurs aspects trouvts r~cemment par Rollett et al. avec un module plus simple qui ne tenait pas compte d'un m~canisme explicite d'adoucissement dynamique. On trouve que le type de loi d'~crouissage suppos~ conduit ~i des relations entre les param~tres de recristallisation des 6tats transitoire et permanent, qui s'approchent de plus pros des relations observres exprrimentalement. Dans cette hypoth~se, les rrles des mr:anismes contr61ant la recristallisation dynamique darts des matrriau polycristallins r661s sont discutrs.

Zusmmenfmung--Die dynamische Rekristallisation in einer polykristallinen Matrix wird mittels eines Monte-Carlo-Verfahrens simuliert. Das benutzte Modell bildet die Mikrostruktur auf ein diskretes zweidimensionales Gitter mittels eines magnetischen Analogons, dem Modell des Q-Zustandes yon Potts, ab. Das Verhalten des Materials bei Hochtemperatur-Verformung wird simuliert, indem Rekristallisationskeime und gespeicherte Energie in Abh/ingigkeit vonder Zeit zugefiihrt werden. Die in dieser Arbeit vorgelegten Simulationen nutzen eine Energiespeicherrate, welche die Verfestigung in Bereich III von Metallen nachbildet und somit sowohl Verfestigung wie auch dynamische Erholung beriicksichtigt. Das Modell stellt viele der kiirzlich von Rollett et al. mit einem einfacheren Modell gefundenen Eigenschaften dar; dieses Modell enth/ilt keinen expliziten dynamischen Entfestigungsmechanismus. Es ergibt sich, dab die Art des angenommenen Verfestigungsgesetzes zu Zusammenh/ingen zwischen transienten und station/iren Rekristallisationsparametern fiihrt, die nahe bei den in physikalischen Experimenten beobachteten Parametern liegen. In dieser Hinsicht werden die Rollen der die dynamische Rekristallisation steuernden Mechanismen in realen polykristallenen Materialien diskutiert.

mechanical energy is stored in the material in the form of dislocations. The rising dislocation density is gradually compensated by the processes of dynamic recovery and dynamic recrystallization. The first of these two softening processes occurs by the annihilation of pairs of dislocations and the redistribution of dislocations into more stable structures (sub-boundaries) within existing grains. By contrast, recrystallization eliminates the dislocation content by the

1. INTRODUCTION It is now well established that certain polycrystalline materials like copper and nickel can recrystallize dynamically, during deformation to large strains at elevated temperatures [1-9]. The changes that occur in the microstructure of a deforming sample result from a balance between the work hardening and dynamic softening. During straining, part of the 59

60

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION

formation of new strain-free nuclei which grow and consume the surrounding deformed material. In high stacking fault energy materials, the dynamic recovery process can completely balance the effect of work hardening. On the other hand, in materials of moderate to low stacking fault energy, dynamic recovery is less effective and the dislocation density increases to appreciably higher levels. Eventually the local differences in dislocation density are high enough to allow the nucleation and growth of dislocation-free regions into the surrounding matrix. Dynamic recrystallization and dynamic recovery are the subject of many studies conducted over the last two decades [1-30] and also have been the focus of many recent reviews [12, 24, 29, 31-33]. Despite this large body of literature, our understanding of the mechanisms whereby both phenomena occur is still incomplete. Particularly, little is known of the correlation between the dynamic recovery and dynamic recrystallization. It is commonly accepted that both these processes are competitive in the removal of stored dislocations [31]. However, recent investigations demonstrate that dynamic recrystallization is aided by a high, although limited, recovery rate, which arranges the dislocations in a subgrain structure with locally high stored energy [34, 35]. These local fluctuations of stored energy provide nuclei and driving force for recrystallization. The major aim of this study is to examine the influence of the dynamic recovery on the dynamic recrystallization. The approach is to employ an extension of the Monte Carlo computer simulation techniques used to account for the microstructural evolution during grain growth [36-40], static recrystallization [41-43] and dynamic recrystallization [44]. In an earlier Monte Carlo study of dynamic recrystallization, the work hardening was assumed to follow a simple athermal parabolic law so that the rate dependence of dynamic recovery was not expliticly included. In the model the stored energy within the grains was increased at a fixed rate 0 and new nuclei were continuously added at a constant rate, ti. Within the framework of this simple model the researchers were able to demonstrate the characteristic temporal oscillations in the stored energy or flow stress and the mean grain size. The observed oscillations damp out over time periods that decreased with increasing storage rate and increasing nucleation rate. Examination of the flow curves and microstructures that were formed under the same values of t] and h but with different initial grain sizes showed the existence of three distinct stages in the evolution of the dynamically recrystallizing system. These stages were characterized as an initialmicrostructure dependent transient region, an initialmicrostructure independent transient regime and a steady state stage. The simulations also indicated that while necklacing occurred under some circumstances it was not a necessary condition for a steady state grain refinement. This model [44] represents a

significant advance in the description of the phenomenon of the dynamic recrystallization. One particular virtue of the model is that it does not require the assumption of any empirical parameters, such as a critical strain for the onset and end of the recrystallization, which are integral parts of recent theoretical approaches. The present paper reports an investigation designed to evaluate the effect of the dynamic recovery on overall process of the dynamic softening. The nucleation aspect of the dynamic recrystallization behavior will be presented in the forthcoming work on this topic [45]. The layout of the rest of the paper is as follows. In Section 2 the method and the model are described, while Section 3 presents the results of the simulations along with detailed comparison with experimental and another studies. Section 4 summarizes the main results of the study.

2. S I M U L A T I O N

PROCEDURE

2.1. Nucleation and growth The computer model employed in this study is very similar to that used by Rollett et al. in a previous study of dynamic recrystallization [44]. The complexity of the grain structure is approximated by mapping a continuum microstructure onto an L x L two dimensional triangular lattice. A triangular lattice was selected since an earlier study had shown that zero temperature simulations with such systems exhibit the same kinetics as other lattice geometries at high temperatures [36]. Each lattice site is assigned a number S, where 1 ~< S ~< Q, that represents the orientation of the grain in which it is imbedded. In the present simulations the total number of grain orientations Q was taken to be 48 which is sufficiently large to ensure the grains of like orientation impinge infrequently. A grain boundary segment is defined as the line between two sites of unlike orientation. The Hamiltonian describing this interaction is the Q-state Potts model [46] which is defined for the case of isotropic grain boundary energy as

H = Jgb ~ (1-6s, sj) + ~ Us~ (U>

(1)

i

where S~ is one of the Q orientations on site i, the first sum is taken over all nearest neighbor site pairs ( / j ) and 6s, sj is the Kronecker delta function. The positive constant, Jsb, is proportional to the grain boundary energy per unit length. The second sum is taken over all sites of the lattice associated with orientation S~ and stored energy Us~. This term accounts for work hardening within the grains by assigning each site the stored energy per unit area, U, which is proportional to the local dislocation density p, that is U = kup [47]. The kinetics of the boundary motion is simulated by a zero temperature Monte Carlo method which

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION corresponds to selecting a site at random, reorientating it to a randomly chosen, different orientation and evaluating the change in energy associated with this reorientation. If the change in energy is less than or equal to 0, the reorientation is accepted; otherwise the old orientation is retained. A generalization of the "n-fold" or continuous time method of Bortz et al. [48] was employed to increase the efficiency of the Monte Carlo algorithm. It is well established [49] that this technique is valuable particularly when most of the sites are not on grain boundaries, which is generally the case in the present work. In the following, N reorientation attempts per site are referred to as one Monte Carlo Step (MCS), where N = L × L is the number of the lattice sites. To reduce finite size effects, the simulations were carried out on a triangular lattice with L = 200, since this lattice size is always much larger than the size of the domains of interest. Periodic boundary conditions were employed in all of the simulations. The data presented in this paper represent averages over at least five simulation runs. The starting microstructures used in the present study were obtained from normal grain growth simulations. In most of the simulations an initial mean grain size R0 = ~ = 6.5 was used, where A is the number of lattice sites associated with a particular grain. Also, a limited number of simulations were performed with R0 = 12.0 and 4.7 to investigate the influence of starting microstructures on the transient and steady state properties of the model. The nucleation of recrystallized grains is represented by placing, on the lattice, a fixed number of new embryonic grains after a suitable number of Monte Carlo Steps. Each embryo consists of a group of three sites arranged in a triangle. The embryo is given a new orientation S > Q, such that no two nuclei have the same value of S. When such a nucleus is placed in the system it is initially assigned a stored energy of 0, that is Us = O. Most of the simulations were performed at nucleation rate of h = 0.2/MCS, which corresponds to adding one nucleus every five MCS. Eight strain increment rates, g - d E / d t , were used; namely 0.001, 0.002, 0.004, 0.007, 0.01, 0.02, 0.04 and 0.2/MCS, to calculate rate of increase in the stored energy of grains 0 = dU/dt. (The relation between strain increment dE and stored energy increment dU will be discussed in the next section.) Additional simulations were performed at ~ = 0.004/MCS and for nucleation rates of ri = 0.05, 0.1, 0.5, 1, 4 and 10/MCS. It should be noted that the nucleation rates given tit is worth noticing, that given few experimental data (grain boundary energy density, energy of dislocation of a unit length, critical dislocation density p) and assuming that recrystallization occurs when energy stored within a grain reaches a value comparable with grain's boundary energy, one can deduce dimension of the smallest recrystallized grain: l =O.023/(pb) which is about 9 pm.

61

represent only the rates at which potential nuclei were randomly introduced to the lattice. For a three-side embryo to grow inside a grain, the stored energy of the grain must be greater than 2 Jsb, otherwise the "tension" of the boundary between the embryo and surrounding matrix will cause it to shrink away [41-43]. By contrast, when a nucleus is introduced at a grain boundary, a minimum stored energy U = 1 Jgb is required for growth. In this case, however, it is found that growth proceeds along the grain boundary. At this level of stored energy, only embryos introduced at grain vertices grow out into the grain interior. It follows that recrystallization will occur generally only when the stored energy exceeds 1 Jgb' The consistency of this estimate with experimental results can be checked by first noting that the smallest size of a grain on the lattice is just one site. A typical fine grain size,/, in copper is about 10/~m or 4 x 104b where b is the Burgers vector [25]. Since copper has a grain boundary energy density Ygb, of 0.63 J/m 2, i.e. 0.27eV/b 2 (Ref. [50]), the total grain boundary energy for the minimal grain, Egb = 7gb12, is 4.3 × 108 eV. This is equivalent to the unit of boundary energy, Jgb, in the lattice model. It has been reported [18] that the dislocation density present at the initiation of the dynamic recrystallization in copper, Per, is typically 1013m 2. Also it is known that [51] the average energy of a dislocation, ed, is of order of 12 eV/b, so the total energy of the dislocations stored in a grain of size of 1 is edp, l 3, which is about 5.0 × 108 eV or 1.2 Jgb. The 20% difference is small considering the approximations implicit in the estimate.t It should be noted that the simulation scheme does not provide an explicit criterion for nucleation based on a critical parameter value. In this way it contrasts with many earlier studies on the subject where a critical value of strain is assumed for the initiation of recrystallization [3, 8, 11, 24]. Nevertheless, the model dictates that successful nucleation of a recrystallized grain is only possible if the stored energy of the matrix is above a certain minimum ( ~ 1 J,b). Such a stress based criterion is more consistent with the assumption that the driving force for recrystallization is the stored energy difference across grain boundaries [18, 30]. 2.2. Work hardening and recovery In order to model the effect of recrystallization during hot deformation successfully it is necessary to account for the effect on the stored energy of the interplay between work hardening and dynamic recovery. This can be accomplished by adopting the phenomenological approach to stage III hardening of metals presented by Kocks and Mecking [52-57]. Their model is based on the assumption that the kinetics of flow is determined by the zero temperature flow stress 6 which represents the strength of the obstacles to dislocation glide. The structure

62

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION

parameter d is related to the dislocation density through the relation d = a/tb x/P

(2)

where b is the magnitude of the Burgers vector, # is the shear modulus and the term a is a constant of order of 0.5 that averages all the possible interactions between the dislocations. At a finite temperature, T, and strain rate, ~, the strain dependent dislocation-component of the flow stress, a, is given by

cr = ~t(£, T)kib v/~

(3)

where ~t(£, T) is a function that goes to ~ as T - , 0. It follows that the evolution equation for d can be derived by specifying the dislocation density p. This, in turn, can be obtained by assuming that the process of dislocation storage (hardening) is independent of the processes of dislocation rearrangement or removal (softening) so that they superimpose in an additive manner, that is

dp

(dp~

d--~= \ d---~/h.,d

+(dp)

~ soft"

(4)

If the length of dislocations stored per unit strain is proportional to the inverse mean free path between existing dislocations 2 ~ 1/x/~, then

where k~ is the athermal storage constant. Equations (3) and (5) can be combined to show that equation (5) is descriptive of linear hardening and is thus associated with stage II hardening in single crystals [57]. The recovery part of equation (4) can be modeled by assuming that mobile dislocations, sweeping through the dislocation network, aid other nearlyfree dislocation segments in breaking past the obstacles. Then the contribution due to recovery is given as

Equations (4)-(6) imply that the evolution of the dislocation content of the material is given by the relation dp

d--~= ki w/p - kEp.

(7)

In turn, combination of equations (3) and (7) leads to the evolution equation for or, viz.

where Oo=(~#b/2)kl and a,=(~gb/ks)kl. Here as is the steady state value of the hardness parameter. Equation (8) is descriptive of stage III hardening in metals [52-57]. The experimental evidence [54-57] suggest that the temperature and strain rate dependence of the saturation stress ~s can be represented by o --

=

.

(9)

O'so At low temperatures, the exponent m is temperature dependent, that is m = k a T / W , where W is the activation energy. At high temperatures, however, m is a constant and typically takes a value of 0.2. This suggests that within the framework of the model for hot deformation, ks should vary as (~/d0)-°s. Equation (7) implies that, under dynamic loading at a constant strain rate, ~, an incremental change of the stored energy dU is related to an incremental change of strain dE = ~ dt, through the relation

dU =kvdp =(Kl~"-O-k2U)de

(10)

where K ~ - k l x / ~v and k v ~ 12 eV/b is the stored energy per unit length of dislocation. In the simulations, equation (10) is used to govern the rate of increase in the local value of the stored energy of a grain associated with each orientation number, S. Since the stored energy increment depends on an instantaneous value of energy stored in a given grain, once the first nucleus is created the stored energy of the lattice becomes inhomogeneous. To further simplify the calculations, the strain units were selected where KI = 1, and U had units of Jgb' Furthermore, to allow for the experimentally observed variation in the steady state stress as with the strain rate ~, the softening constant ks was chosen to have the following dependence on strain rate

{~-o.s k2 = k20 t~0)

•

(I1)

Clearly a suitable reference level must be chosen so that work hardening and recovery are properly balanced. With the reference level for the softening coefficient set to 0.71, a change in strain rate from 0.001 and 0.2/MCS leads to a decrease in the softening coefficient k s from 0.71 to 0.25. The appropriateness of this choice of the range of k 2 can be checked by considering experimental data obtained on copper tested at 0.5 Tm [57, 58]. At a strain rate of 4 x 10-4s -1 the athermal hardening rate, 00/ix, is of the order of 8.0 x l0 -3 and the steady state stress as/# is 1.4 x 10 -3. Accordingly ks = 200/~ - 11

(12)

and K, = - 20° - ~ x/~u-- 1.7 x 1 0 4 ~ .

(13)

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION Thus equation (10) can be written as dU = (1.7 x 104v/-U - ] l U) de

(14)

where the stored energy density has units of J/m 3. Since a two dimensional lattice model corresponds to a thin layer of material of thickness l (the diameter of a smallest grain) the above equation can be rewritten in terms of d U t - ! dU, the stored energy increment per unit area dUt = (54x/~t - l l Ut) dE

(15)

63

transient regime where oscillations in microstructure were most pronounced, samples of these properties were taken more often than at later times. Typically, five times as many measurements were made during the early stages of evolution than in the asymptotic regime while keeping the total number of measurements to about 1000. In order to present the data in a manner that is directly comparable with experiment, flow stress is used rather than the stored energy. It is assumed that the flow stress is related to the total stored energy of the lattice U through the relation, tr = ( l / N ) x / ~ , where O U/Jgbt. Since the measured level of the flow stress, tr, is related to parameters 00, ct, p, b, p and kv which are not specified in the model, it is only possible to estimate the value of the unit of stress calculated by this relationship. This estimate can be obtained by reasoning similar to that carried out at the end of the previous section. From equations (3) and (13) it can be shown that =

where UI has units of J/m 2. Alternatively, the stored energy can be expressed in units of 0.63 J/m 2, that is, the grain boundary energy per the diameter l of a smallest grain, as discussed above. Then dU~ = ( 6 8 x / ~J - 11Uj) de

(16)

where Us-= U/(0.63 j/m2). With a suitable change of the time (or strain) units d ~ - 68 de, the most convenient form of equation (10) is obtained dUj = (x//~j - 0.17Uj) de.

~ (20ox/~'sb///Ki)x/~

(17)

It should be observed that this equation is true only for a material of thickness l. Since the diameter of smallest recrystallized grain I is inversely proportional to the critical dislocation density Per, any variation in Per can change value of parameter in equation (17). For example, assuming a reasonable half order of magnitude variation in the critical dislocation density, P'~r=PPcr, P ~ 5, equation (17) takes the form

2 2 . 4 x 1 0 - 4 p x / ~ -~- l l M P a x / - ~

where the value of 00 is obtained from the data for reported in Refs [58, 59]. The stress-strain curves simulated, using the stage III hardening law, for a microstructure with the initial grain size R o = 6.5 are displayed in Fig. l(a). (a)

STAGE

dU = (x/~J - 0.17p-,/2 Us) d(p

NUCLEATION

RATE

O.2/MCB

Initial Grain

4

0.38Uj)d~,

IN

CONBTANT it .

(~j-

81za

-

6.5

0.2

(18)

where d~p -= 68.v/p de ~ 48 de, and from which it is concluded that it is reasonable to choose the value of the parameter k2 to be in a range 0.25 ~
(19)

o04 OO2 o.9, aoo7 oct, ooo2

..

0 0

i

i

10

20

|

$0

h

40

STRAIN

(b)

PARABOLIC CONSTANT ~

NUCLEATION

RATE

0.21MCS

Initial Grain

Size

-

6.8

4

oo,

002

3. RESULTS AND DISCUSSION

2

The measurements made during the course of the simulation were the stored energy U~ and the average grain area at constant Monte Carlo time intervals. In order to assure the accuracy of measurements, in the

0

tAn alternative way of defining the flow stress is to consider that a = (l/N) Ei or, where a i - ~ is the stress contribution due to the ith grain and the summation is carried out over all grains of the matrix.

o.o 1 ooe~ o c,e,, 0o02 ooo,

i 10

i 20

i 30

J 40

STRAIN

Fig. 1. Stress-strain curves for a microstructure, with initial grain size R0 ~ 6.5, obtained for a constant nucleation rate of ti =0.2/MCS and strain rates in the range 0.001 ~
64

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION STAGE Ill CONSTANT STRAIN RATE ¢ - O.OO41MCR Initial Grain Size

0

10

20

30

-

6.8

40

STRAIN

Fig. 2. Stress-strain curves for a microstructure, with initial grain size R0 -~ 6.5, obtained for a constant strain rate of ~ = 0.2/MCS and nucleation rates in the range 0.05 ~
A fixed nucleation rate ti = 0.2/MCS was used for all the curves shown which are parameterized by strain rate ~ in the range 0.001 ~<~ ~< 0.2/MCS. In order to allow a ready comparison with the data obtained in the earlier study [44], Fig. l(b) presents a similar plot obtained using a linear ramp in stored energy which corresponds to a parabolic hardening lawt. It should be noted that the range of strain rates (0.001 ~< ~ ~< 0.04/MCS) covered in Fig. l(b) is slightly different from that in Fig. l(a). The flow stress curves generated using the two hardening laws show a n u m b e r of similarities, but they also exhibit qualitative differences in the way work hardening and dynamic recrystallization are correlated. In both cases the flow stress initially rises with increased strain. After reaching a maximum, the flow stress begins to saturate and display distinct oscillations, the amplitude of which decreases with increasing strain rate. The level of damping increases with the strain rate, so that for a high enough value of ~ the oscillations become overdamped and disappear. Thus the stress converges quickly toward its steady state level. Although small oscillations in flow stress could be seen at higher strain rates, they appear to be aperiodic and of decreasing amplitude when more Monte Carlo runs were used in the data averaging. This strongly suggests that the aperiodic oscillations are merely fluctuations in the local level of the stored energy and that they would disappear if more runs and/or larger lattices were used in the simulation. The simulations using stage III hardening not only reproduce flow curves with the features described above but also capture many of the properties observed in the numerous experimental studies of

dynamic recrystallization, see for example Refs [7, 11]. First, there is a clear "fanning-out" of the bundle of flow curves in the low strain region of Fig. l(a). This result should not be surprising, since the dynamic evolution of dislocation density in the model includes an explicit strain rate dependence of the recovery process, see equation (11). Accordingly, the self-similarity of the initial portion of the flow curves, found in the previous study and shown in Fig. l(b), is absent. Secondly, the flow curve oscillations appear to be less regular; having larger periods which are generally more damped. It will be demonstrated in the following, that this effect is due to the contraction of the range of stored energy present in the system due to the explitic inclusion of dynamic recovery. It is well known from experimental observations that the level of nucleation in a deformed matrix is strongly correlated with the a m o u n t of energy stored in the dislocation network [31]. Nevertheless, although a considerable number of studies have been devoted to the problem of the interrelation between these two processes, a complete understanding of underlying physical mechanisms is still lacking. Therefore it was decided that these two degrees of freedom should be purposely decoupled, so that their individual influences could be studied. To this end, flow curves for a constant strain rate were simulated for a range of nucleation rates; namely, 0.02 < ri < 10/MCS. In each case, an initial grain size

(a) STAGE

Initial Grain

Size

-

6.8

lS

E

0C02

r

OO04

_

0.007 O.C1 0.02

8

10

20

30

40

STRAIN

Co)

r

PARABOLIC CONETANT NUCLEATION RATE n - 0.21MCS Initial Grain S i z e - 6.8

16

e

o~T

tl-

E

o~r o.o~

~

0

10

tThere is slight inconsistency in the computer code that Rollett et al. used to calculate the time elapsed after each successful Monte Carlo flip which causes their Monte Carlo time unit to be smaller than it should be. In turn, this makes their nucleation rates higher than the corresponding values used in the present study so that the two sets of flow curves are not numerically compatible.

nl

CONSTANT NUCLEATION RATE It - O .2 1 M CS

20

30

40

STRAIN

Fig. 3. Plot of grain size vs strain for a constant nucleation rate of ti =0.2/MCS and strain rates in the range 0.001 ~<~ ~<0.2/MCS. The curves were obtained for microstructure with initial grain size R0 ~ 6.5. (a) Curves for the stage III hardening law. (b) Curves for the parabolic hardening law.

PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

R 0 of 6.5 was used and the Monte Carlo runs were continued to between 3 x 103 and 1.2 x 105MCS. The results of these experiments are presented in Fig. 2. These curves can be compared with an analogous set of flow curves obtained by Rollett et al. [44]. Once again, after reaching a peak, the amplitude of which increases as the nucleation rate increases, the flow stress exhibits oscillations. As the nucleation rate decreases, the degree of damping initially declines then rises once more at small values of nucleation rate. Since this type of damping occurs regardless of the type of hardening law used it is reasonable to assign its origin to the underlying process of the normal growth [36-40]. A visible sign of the effect of recovery is the stronger attenuation of the oscillations in flow stress which causes all the curves to lie in a region below the stored energy ramp defined by equation (18) which is, effectively, a description of the limit of the stored energy as set by dynamic recovery. It is interesting to compare the behavior of the mean grain size Ravg as a function of strain corresponding to the two sets of the flow stress data described above. The evolution of grain size developed during the simulations are shown in Figs 3 and 4. Figure 3(a) shows the data obtained for the stage III hardening law for the strain rate range (0.001 ~<~ ~<0.2/MCS) and a constant nucleation rate of 0.2/MCS. These data can be compared with those obtained when the parabolic hardening law is assumed, which are presented in Fig. 3(b). In addition, the data obtained for nucleation rates in the range 0.05 ~< ri ~< 10/MCS, at a constant strain rate of 0.004/MCS are shown in Fig. 4. It is evident that the grain size evolution process is both strain rate and nucleation rate sensitive. This point was discussed at some length by Rollett et al. [44]. Indeed, there clearly exists a transient strain (time) region followed by a regime of oscillating grain size which gradually changes to a steady state regime. These curves show much more scatter than corresponding flow stress plots but the periodicity of both types of curves appear to be closely related. 8TAGE III CONSTANT STRAIN RATE I~ - O.O041MC8 I l d t l a l G r a i n 81ze ~ 6 . 5

18

4

S ~

0

o~

10

20

30

40

STRAIN

Fig. 4. Plot of grain size vs strain for a constant strain rate of ~ = 0.004/MCS and nucleation rates in the range 0.05 ~
65

(a)

d

S T A G E III i - O.O04/MCS .....

IOE(IOE(II(1-F))) .

STRESS

i

. . . . . . . 2

I

I

10

20

STRAIN

(b) :c ~

PARABOLIC c - O.OOtlMCS - 0.21M0$ Inltl|l Grain Size -

log(log( 1/( 1 -F)))

6.8

o 1

4

8

STRAIN

Fig. 5. A combined semi-logarithmic plot of stress, grain size and log{log[I/(1 -F)]} as a function of strain for a nucleation rate of ti = 0.2/MCS and strain rate ~ = 0.004/MCS. The curves were obtained for microstructure with initial grain size R0 -~ 6.5. (a) Curves for the stage III hardening law. (b) Curves for the parabolic hardening law. Broken and solid vertical lines denote the onset of, correspondingly heterogeneous and homogeneous growth of nuclei. In order to better understand the meaning of the dependencies of Ravg and a on the applied strain E, combined plots were prepared of the flow stress tr, the mean grain size Ravg and the function l o g { l o g [ 1 / ( l - F)]}, where F = N x / N and N~ is the number of sites that have recrystallized at least once. Such plots are presented in Figs 5(a) and (b) for the stage III and parabolic hardening laws, respectively. Examination of Fig. 5 reveals significant differences between the evolution of these parameters under the two work hardening laws. In both cases the three parameters all initially increase with strain. For tr this is simply due to the imposed work hardening and for Ravg the increase is due to normal grain growth. The slight increase in l o g { l o g [ 1 / ( l - F ) ] } is due to the periodic generation of shrinking nuclei. The increase in these parameters continues up to the point at which a weak peak in mean grain size occurs, indicated by the first broken vertical line in Fig. 5(a). Here, there is a corresponding inflection in the tr curve and the value of log{log[I/(1 - F)]} begins to increase rapidly. This transition takes place at the point at which Uj reaches a value of unity, that is the condition for the onset of heterogeneous growth of new embryos. Clearly, this point corresponds to the critical strain q for the initiation of dynamic recrystallization in the original microstructure. This signature of the beginning of the first recrystallization cycle

66

PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

is experimentally found to occur at approximately constant ratio Ec/% the value of which is observed to vary between 0.65 and 0.85 [20, 31]. The effect of the two work hardening laws becomes apparent once this critical condition has been reached. In the case of stage III hardening, the stored energy in the unrecrystallized regions of the lattice increases less rapidly than in the freshly recrystallized regions due to the effect of dynamic recovery. This effect combined with the continuous removal of stored energy by the continuous creation of new nuclei causes a stagnation of the total stored energy with a consequent flattening of the stress-strain curve. During this intermediate stage, recrystallization occurs exclusively by the heterogeneous formation of nuclei since the stored energy nowhere exceeds 2 Jgb, the condition for homogeneous recrystallization. The competition between grain refinement due to the formation of nuclei and growth produces little change in the average grain size. Finally, the stored energy in the unrecrystallized regions of the lattice is sufficient to cause rapid, homogeneous growth of nuclei. This leads, initially, to a sharp drop in the mean grain size and flow stress. After a short interval, the mean grain size increases rapidly due to the growth of these homogeneously nucleated grains. The contrasting behavior under the influence of the parabolic hardening law is shown in Fig. 5(b). Here, the inflection in the stress-strain curve occurs near the initial maximum in the mean grain size curve and corresponds to the beginning of homogeneous nucleus formation. As the strain is increased, the stored energy in both the unrecrystallized and freshly recrystallized region increases at the same rate. Accordingly, the condition for homogeneous nucleation is reached quickly enough to mask out the effects of the initial period of heterogeneous nucleus formation. This causes more regular, shorter oscillations in the stress-strain curve, the inflection points of which correspond to the onset of homogeneous growth of nuclei. It is readily seen that, for both hardening laws, during the first and subsequent recrystallization cycles each inflection point on the tr vs e curve corresponds to a peak on the Ravs vs E curve. This observation can be also made when studying the parabolic hardening flow diagrams, Fig. 4, and as such constitutes a more precise and general description of the observation made by Rollett et al. [44]; namely, that there is a one quarter period shift between stored energy and average grain size oscillations. Hence it is concluded that each flattened, double-peak region of the flow stress curve corresponds to exactly one period of recrystallization which, in the case of stage III hardening, is made up of a slow, heterogeneous sub-sweep followed by a rapid, homogeneous growth of new grains. It is well known from experimental Observations that the dynamic recovery leads to creation of subgrains whose walls are made up of discrete

dislocations or networks. At high temperatures, the recovery processes not only remove dislocations from cell interiors but increasingly cause the cell walls to condense which eventually transforms them into mobile boundaries. It is generally thought that this process sets up the conditions necessary for nucleus formation, the subsequent rapid growth of which leads to the recrystallization of the lattice. In this sense recovery enhances the propensity of a microstructure to undergo recrystallization [31, 34, 35]. In the present model the intra-domain distribution of dislocations is uniform so that this constructive effect of recovery is absent and is merely reflected by the constant nucleation rate imposed on the system. It is realized that a constant nucleation rate mechanism is a great simplification but it overcomes the constraint imposed by the presence of a uniform dislocation distribution which is known to prevent recrystallization [63] because of the lack of heterogeneities required for nucleation. The determination of the strain at the onset of heterogeneous recrystallization, where the Avrami (i.e. JMAK) plots of the recrystallized volume fraction ratio against strain exhibit an abrupt change in slope gives values of F which are typically of the order of 0.1-1.0%, see Fig. 6. The corresponding values of Ec were slightly strain rate dependent. For example, the results of simulations with constant nucleation rate of 0.2/MCS yielded values of the strain at the beginning of recrystallization which vary

l O ~ o.¢¢4

ool

STAGE nl NUCLEATIORATE CONSTANT N

~

o2 o.ool

-

0.1

-

0.001 1

Initial Grain .....

;O

.......

;+00

0.2/Me8

Size- 6.5 ....

10OO

STRAIN

STAGEIII IO CONSTANT STRAINRATE - 0.0041MCS ~A~ ~ 0Initial. OGrainOSize1 - 06.5. 1

0.1

1 I~© STRAIN

o~

10

Fig. 6. Avrami plots for microstructure with an initial grain size R0 ~ 6.5 (a) for a nucleation rate of ti = 0.4/MCS and strain rates in the range 0.001 ~<~ ~<0.2/MCS, (b) for a strain rate of ~ = 0.004/MCS and nucleation rates in the range 0.05 ~
PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

CONSTANT NUCLEATION RATE 100

o

"

0.2/MCS

10

o

!

o

o

o

•

•

o o

1 S.5

1

3

STRESS TO THE PEAK

Fig. 7. Logarithmic plot of strain to the peak, %, against stress to the peak, ap, obtained for initial microstructure with R0 ~6.5, a nucleation rate ti =0.2/MCS, and strain rates in the range 0.001 ~<~ ~<0.2/MCS. Stage III hardening---open symbols, parabolic hardening--solid symbols.

between 0.3(1) % for ~ = 0.02/MCS and 0.6(1) Ep for = 0.002/MCSt. It is evident from Fig. 6(b) that the quoted values are nucleation rate independent, although volume fraction of new grains at the beginning of recrystallization does obviously depend on the nucleation rate. A more accurate determination of Ecis difficult, especially for large strain rates so that the present results can be only qualitatively compared with results of the careful metallographic studies of hot worked steels by Rossard [8], who concluded that Ec~- 0.8 %. Experimental studies of dynamic recrystallization [24, 29] have shown that the strain to the initial peak in stress is dependent on the value of the peak stress. This observation is typically represented by a relation of the form % ~ try, where fl takes values between 0.8 and 1.2 [24]. The data obtained from the simulations, using the two hardening laws are plotted logarithmically in Fig. 7. It is evident that, unlike the data obtained within the framework of parabolic hardening, the data corresponding to the stage III hardening model cannot be represented by a straight line over the entire range investigated. The characteristic flattening of the low stress portion of the plot is absent in the data set corresponding to the parabolic hardening. The upward curvature of the data is, however, similar to the experimental data presented in Fig. 13(b) of Ref. [44]. If a linear fit is forced through the low stress portion of the data, the slope is reasonably consistent with experimentally observed values. Since a change of nucleation mechanism did not significantly alter the shape of % vs ap curve [45], it is concluded that this is a direct consequence of softening mechanism that is built into the simulation scheme. The behavior of the flow stress a and the mean average grain size Ravg as a function of strain, discussed above, clearly indicates the existence of a strong correlation between these two variables. The tThe figures in parenthesis indicate the level of error associated with the quoted values.

67

question arises, however, to what extent any set of values a and Ravgis meaningfully determined, that is, what is the influence of the initial microstructure on or, Raysand other quantities describing the flow curves and particularly their steady state values. In order to investigate this correlation, two additional sets of simulations were performed with identical dynamic recrystallization parameters, ti = 0.5/MCS and ~ = 0.004/MCS, but with two different initial grain sizes R0= 4.6 and 12.0. Figure 8(a), (b) and (c) show, respectively, the strain dependence of flow stress, grain size and log[l/(1 - F)] for the first recrystallization cycle. While the parameters describing transient behavior of the microstructure: strain to the onset of recrystallization E~, strain to the peak %, stress to the peak ¢rp, and period of oscillations depend on the initial grain size R0, the steady state properties

(a)

STAGE III CON8TANT RECRYSTALLIZATION PARAMETER8 - O.O04/Idce. n ~ 0.6/UC$ 1.s

~: t . o

initial Grain Size

o

4,7

0.5 10

20

30

40

STRAIN

(b)

STAGE

+s

III

CONSTANT RECRYeTALLIZATION PARAMETER8 ¢ - 0+0041MC$,

n - O.5111CS

lo

~, I n i t l l ol

$

Grain Size 1 42 .. 70

65

o

10

2S

30

40

STRAIN

(c) 1o

STAGE III CONSTANT RECRYSTALLIZATION PARAIIETER8 - O.O04/MCS, n - 0.S/MCS

= ~" ~ 0.+

0.001 0.2

1

10 STRAIN

Fig. 8. Dependence of the strain evolution of the measured parameters on the initial grain size R0. The simulations were performed with the same recrystallization parameters, ri =0.5/MCS and ~ =0.004/MCS but with three different initial grain sizes of 4.7, 6.5 and 12.0. (a) Stress vs strain. (b) Average grain size vs strain. (c) log(l/(1- F)) vs strain.

68

PECZAK and LUTON: MC STUDY OF DYNAMIC RECRYSTALLIZATION

are clearly initial configuration independent. Indeed, a series of careful measurements of asymptotic stresses and grain sizes produced, for a lattice with initial median grain sizes of 4.6, 6.4 and 12.0, very similar results: ao~ = 1.213(7), 1.215(6), 1.214(5) and Ro~ = 9.038(47), 9.050(52), 9.042(64), respectively. In this respect, the results of the present study confirm the earlier simulation results [44] which indicated that the choice of initial grain size distribution only influences the early history of deformed structure. As mentioned earlier, Rollett et al. [44] described the microstructural evolution occurring during dynamic recrystallization with use of three distinct stages: a first stage that was initial-microstructure dependent, an initial-microstructure independent stage wherein the grain size oscillates and a steady state stage. The results of the present study suggest that the distinction between the first and second stage is somewhat artificial. Although the evolution of the grain size indicates that during the first recrystallization cycle there is, indeed, a rapid and irregular change in grain size, the flow stress remains smooth and insensitive to initial grain size distribution. Furthermore, analysis of Fig. 6(a,b) provides convincing evidence that subsequent developments over a time period, corresponding to one oscillation period, are self similar and consequently lead the system towards a steady state regime. In physical experiments on dynamic recrystallization, the strain rate sensitivity of the flow stress is typically similar to that observed in materials that

(a)

CONSTANT 7

=

5

o

3

NUCLEATION

It "

RATE

0.21MC8

•

o

o •

o

o o

o •

o

O,O1

O.OO1

0.1

STRAIN

(b)

RATE

CONSTANT 7

NUCLEATION -

RATE

O.2/MCS

•

o

o

•

o o

•

o

o

o o i i:l 0.001

. . . . . . . .

I 0.O1 STRAIN

. . . . . . . .

I 0.1

. . . .

RATE

Fig. 9. Comparison of the relationship between stress and strain rate for the two work hardening laws in a microstructure with an initial grain size R0 ~ 6.3 and with the nucleation rate, ri, set to 0.2/MCS. (a) Asymptotic stress. (b) Stress to the peak.

CONSTANT

NUCLEATION

20

"

RATE

O.21MCS

o o o o

10

•

o o •

o o

n"

41

0.001

0.01 STRAIN

0.1 RATE

Fig. 10. Log plot of average grain size Ravgagainst strain rate obtained for an initial microstructure with R0 ~ 6.5 and with a nucleation rate, h, set to 0.2/MCS. The data for the stage III law are represented with open symbols while closed symbols are used for those of parabolic law.

exhibit only dynamic recovery during deformation, that is in the range of 0.15 to 0.25 [20]. The asymptotic stress and the stress to the peak are displayed as a function of the strain rate in Fig. 9. The measured strain rate sensitivity of the peak and the steady state flow stress, for simulations using the stage III hardening law, were found to be about 0.3. This, value is slightly smaller than the value found for the parabolic work hardening law. Since a strain rate sensitivity of 0.2 was assumed for the dynamic recovery part of the stage III hardening law, the new result suggests that proposed dynamic recrystallization mechanism is largely consistent with the nature of this hot working phenomenon. Moreover, if a suitable coupling between nucleation rate and stored energy is allowed, the rate sensitivity of the flow stress can more closely approach that observed in physical experiments [43]. Figure 10 shows the dependence of the asymptotic grain size Ro~ on the strain rate. The behavior of Roo as a function of ~ is opposite to the behavior of the asymptotic stress as a function of strain rate, as shown in Fig. 9. Over the range of ~ of interest log R~ can be related to log g by a linear function, i.e. R~o ~ ~ ~. The present data yields a value of the exponent of 0.2, which is very close to that obtained from simulations using the parabolic hardening law. Accordingly, the most appropriate set of parameters which uniquely determines the effect of deformation are go~ and R~o. The asymptotic grain size R~ is plotted as a function of the asymptotic stress ~rooin Fig. 11 for the entire range of ~i and studied. This figure shows very different behavior when the nucleation rate ri is varied at fixed strain rate than when the strain rate is varied at fixed nucleation rate. Specifically, for ti = (const) the grain size Ro~ is a decreasing function of stress try, while the converse applies if ~ = (const). In addition, the variation in grain size with the strain rate for the case of constant nucleation rate has almost the same form irrespective of the work hardening law that is assumed. Discrepancies between these two data sets

PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

69

nucleation mechanisms in dynamic recrystallization [451. ", pAradox.it. ~ = o 2 m l c s The final point of discussion regards the observations of Sakai and co-workers [21-23] who associated oscillations in the flow curves with the __0 coarsening of the grain size. Their detailed metallo~10 -.Ao ~ . • o graphic studies have shown that the transition from single to multipeak behavior depends on the relationship between the initial grain size R0 and that estab& 3 lished under steady state flow conditions, R~. They 0.5 1 3 5 showed that flow stress oscillations occur when ASYMPTOTIC 8TRESS Fig. l 1. Plot of the asymptotic grain size vs the asymptotic Ro~ > vRo, where v = 0.5. These observations have stress for a microstructure with an initial grain size R0 =~6.5. been interpreted in terms of a nucleus selection model The entire range of nucleation rates and strain rates used in by Sakai and Jonas [23, 24]. Although the above the study are represented. The data for the parabolic law are criterion appears to work well for carbon steels represented by the full symbols while the data for stage III [21, 22], it does not provide a good description of hardening are indicated by the open symbols. some observations in copper and copper alloys [25, 26], and microalloyed steels [27, 28]. The present arise in the region of (R~, a~) phase space which simulations show a weak consistency with the model correspond to large strain rates, of order of 0.2/MCS. of Sakai and Jonas [23,24]. Close examination of In this regime, however, the stored energy flows into Fig. l(a) shows that, for the case of constant nuclethe system at such high rates that, in just a few Monte ation rate, oscillations in the flow stress occur at Carlo steps, new nuclei acquire almost the same level the smallest strain rates (i <0.01/MCS). At these of stored energy as the surrounding matrix. This rates, the asymptotic grain size Ro~ is larger than effectively removes the driving force for recrystalliza- 1.53 R0. Furthermore, the curves generated using tion. As a result the microstructure evolves by quasi- the parabolic hardening law, Fig. l(b) exhibit oscilnormal grain growth, so the model does not provide lations at strain rates below 0.02/MCS. In this case a proper simulation of microstructural evolution R~ > 1.69 R0. This latter observation differs from during hot working. that of Rollett et al. [44] because here the aperiodic A numerical fit to the data in Fig. 11 shows that oscillations in flow stress are discounted for the dependence between the asymptotic grain size and reasons discussed above. It is notable that the stage asymptotic stress can be represented as R~ ~ tr~ d~ III hardening law yields a value of the parameter v with dg -- 1.0(1). The data was measured with a high which is closer to v = 0.5 than the result from simuprecision so that any deviation from linearity in the lations using the parabolic law. The discrepancy of log-log plot would be clearly seen. It is evident that order 3 which still exists may, in part, be related to all the constant nucleation rate data appear to have the decoupling of the nucleation rate and level of the same or very similar form irrespective of the stored energy in the lattice. Indeed, the results of hardening law that is used. Accordingly, the value of recent studies [45], using more realistic nucleation the slope quoted is in no way unique. Instead, the mechanisms, show that v ~ 1.0 and are, therefore, data supports the contention that for a constant more consistent with experiment. nucleation rate, the mean asymptotic grain size is inversely proportional to the mean stored energy in the lattice, independent of the path taken to achieve 4. CONCLUSIONS this level. This results in a weaker dependence than it is observed in experiments. For example, in a recent A Monte Carlo model of dynamic recrystallization review of available experimental data Derby [64] has been investigated in which the stored energy quoted the value d~ _-__1.5. Accordingly, the simu- within individual grains is increased at a continuously lation results suggest that a very good fit to the changing rate due to presence of an explicit recovery experimental relationship between stress and grain mechanism. Recrystallization is stimulated by the size could be obtained for a particular choice of the addition of new nuclei at a constant rate. Rate functional dependence of nucleation rate on the dependent dynamic recovery is explicitly included in storage rate, namely, a weak positive dependence of the present model. In this way it departs from an earlier study [44] where dynamic recovery was only nucleation rate on storage rate. This means that dynamic recovery, which acts to inhibit recrystalliza- implied by a negative departure from linear work tion, does not significantly influence the steady state hardening behavior. This model provides a comprebehavior of the microstructure. A strain or stored hensive description of the dynamic recrystallization energy dependent nucleation mechanism is necessary process and requires one strain rate dependent recovto explain the experimentally observed relation ery parameter, k2, the value of which is set to the between R~ and tr®. This observation was recently experimentally observed range. The results obtained confirmed in a Monte Carlo study of the role of with this study lead to the following conclusions: o

STAGE

~Jl, ~ = 0.2a¢<:S

STAC~ al. ~ = OX~O4a~CS

70

PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

1. The general behavior of the stress-strain curves is insensitive to the type of hardening law that is applied. The stress-strain curves are characterized by an oscillatory behavior at low strain rates and a single peak in stress at higher rates. Three distinct stages of microstructural evolution can be identified, an initial microstructure dependent stage, an initial microstructure independent transient stage and a steady stage. In this regard the results are similar to those obtained by Rollett et al. [44]. It it notable that the intrinsic mechanism that leads to the occurrence of each of these stages is the same. 2. The explicit inclusion of rate dependent dynamic recovery in the model gives rise to a marked flattening of the oscillatory behavior displayed by the flow curves. This latter, commonly reported experimental observation was absent in the data generated by a more simple M o n t e Carlo model used in an earlier study [44]. Typical plots of transient and steady state values of the recrystallization parameters show that the stage III hardening law is necessary for the proper description of microstructural evolution under conditions of dynamic recrystallization. 3. It is demonstrated that dynamic recovery represented by a strain dependent one variable (i.e. average dislocation density) function cannot significantly influence the relationships between steady state and transient regime parameters. In order to remove this limitation of the model the nucleation rate has to be sensitive to the degree of deformation imposed onto the system. This is clearly demonstrated in another recent study [45]. Acknowledgements--The authors would like to thank A. D. Rollett, G. Grest and M. P. Anderson for stimulating discussions and useful comments.

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PECZAK and LUTON:

MC STUDY OF DYNAMIC RECRYSTALLIZATION

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AM 41/I--F

71

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