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Study of the strain reversal effect on the recrystallization and strain-induced precipitation in a Nb-microalloyed steel
Acta Materialia 52 (2004) 333–341 www.actamat-journals.com
Study of the strain reversal effect on the recrystallization and strain-induced precipitation in a Nb-microalloyed steel D. Jorge-Badiola, I. Gutierrez
*
Centro de Estudios e Investigaciones T ecnicas de Gipuzkoa and University of Navarra, TECNUN, P Manuel de Lardizabal, 15, 20018 Donostia-San Sebasti an, Spain Received 17 July 2003; received in revised form 17 September 2003; accepted 18 September 2003
Abstract Efforts to produce more accurate through-process models for hot working need to take into account variations in the deformation conditions. In the present work, the effect of the reversal of the strain for a Nb-microalloyed steel is discussed. It is shown that, depending on the magnitude of the reversal, the static softening kinetics can be accelerated or delayed. The same is true for the case of the strain-induced Nb-carbonitride precipitation, leading to an interaction between both processes and the deformation path. Current models of recrystallization and precipitation have been used to analyze the obtained results. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hot working; Strain path; Recrystallization; Strain-induced precipitation
1. Introduction In industrial hot working local strain reversal, together with strain rate and temperature changes, characteristic of each particular forming process, take place during deformation passes. Efforts have been carried out to investigate the material behavior during transients following instantaneous changes of conditions [1–7]. For the case of hot working, it has been demonstrated that when combining the outputs obtained from a finite element model with empirical equations of static recrystallization, important discrepancies can be found between the predictions of the model and the true behavior of the material [8]. This has been attributed to the way in which the final equivalent strains are computed from the incremental strains. These differences can be considered negligible for most of the cases in materials undergoing fast static recrystallization after deformation. However, when the recrystallization kinetics becomes slower, due to the material and/or to a decrease
*
Corresponding author. Tel.: +34-943-21-28-00; fax: +34-943-21-30-
76. E-mail address: [email protected] (I. Gutierrez).
on the deformation temperature, the effect can be non-negligible. When reversing the strain during deformation, a decrease in stress is generally observed to take place which is very often accompanied by a transient decrease of the work hardening rate [9], leading to the formation of a ‘‘yield stress plateau’’ extending over a relatively large strain interval [4,10–15]. This transient is generally attributed to the partial dissolution, during the first stages after the reversal [10], of the substructure present before changing the direction of deformation. This has been reported to affect the geometrically necessary dislocations [6,16] but to activate recovery processes affecting all the dislocations [17]. Both the static and the dynamic recrystallization depend on the amount and distribution of the stored energy, which is directly related to the local dislocation density. Consequently, a modification on the dislocation distribution is expected to have an immediate effect on the dynamic/static recrystallization kinetics. In agreement with this, when the strain is reversed at a strain lower than that required for the onset of dynamic recrystallization, dynamic recrystallization is delayed [18]. Concerning static recrystallization, it is generally accepted that the reversal modifies its kinetics [19] and,
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2003.09.018
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
as a general trend, a strain reversal implies a retardation for the onset of the recrystallization, as compared to the effect of the same total strain applied in a monotonic test. However, the results obtained by different authors present some contradictions. For aluminum, the reversal from tension to compression was observed to cause a retardation in the recrystallization kinetics, being a function of the total absolute strain [5]. For microalloyed steels, the reversal of the strain from tension to compression was observed to always accelerate the recrystallization, although, when this last is expressed in terms of the total absolute strain, a delay is also observed. [13,20]. The present authors [21] found, using torsion tests, a net retardation of the static recrystallization when the strain is partially reversed, but just the opposite effect when it is fully reversed and, consequently, that the total absolute strain is not an appropriate variable to express the recrystallization kinetics [22]. New experiments on microalloyed steels by the present authors and also by Karjalainen and Somani [23] confirm such behavior. In the present work, the effect of different amounts of reversal on the recrystallization and strain-induced precipitation are analyzed in more depth and commented upon in terms of the available models. 2. Experimental procedure A microalloyed steel with a composition shown in Table 1 was used to perform multipass torsion tests at a strain rate of 0.1 s1 , in order to determine the static softening kinetics following different strain paths. The testing conditions are summarized in Table 2. The deformation sequences were of the following type: • Two-pass tests: One pass to a given strain followed by a certain holding time at the deformation temperature and a second pass to analyze the effect of the holding. • Three-pass tests: One pass to a given strain ðef Þ, some reversal of the strain ðer Þ by twisting in the reverse followed by a holding time at the deformation temperature and a third pass also in the reverse to analyze the effect of the holding. The static fractional softening was determined from the stress–strain curves by the 2% offset method as described elsewhere [24]. Some samples were water quenched after a certain holding time in order to perform microstructural observations at the sub-surface on a longitudinal section parallel to the torsion axis, located at about 0:9R (R ¼ specimen radius).
Table 2 Testing conditions Reheating Deformation Tests temperature (C) temperature (C)
Strain path ðeÞ
1250
0.12 0.15 0.20 0.25 0.2; )0.06 0.2; )0.10 0.2; )0.15 0.2; )0.20 0.25 2 0.29; )2 0.25; )2 0.25; )0.1 0.25; )0.25
1100
Monotonic
Strain reversal
1200
1050
Monotonic Strain reversal
Carbon extraction replicas were prepared from the torsion specimens, using conventional methods in order to investigate the strain-induced precipitation by transmission electron microscopy (TEM). The observations were carried out in a Philips CM12 STEM equipment fitted with an EDS system from EDAX. 3. Results The graph in Fig. 1 shows examples of the stress strain curves for torsion test involving a strain reversal. For simplicity, the absolute stress and strains have been considered. Just after the reversal, the stress rises to reach a stress level lower than the pre-stress, followed by a ‘‘yield stress plateau’’. After the plateau, the stress starts increasing again due to strain hardening. It can be 90
Monotonic 80 70
Stress (MPa)
334
60
Curves after the reversal
50 40 30
1050ºC
20
εf =0.25
10
0.1s
εf =0.29
-1
0 0
0.2
0.4
0.6
0.8
1
1.2
Strain
Fig. 1. Flow curves for tests involving a reversal of the strain. Effect of the applied pre-strain, ef , on the transient after the reversal.
Table 1 Steel composition (wt%) C
Si
Mn
P
S
Al
Nb
V
N
Mo
Cu
Cr
Ni
.15
.30
1.42
.012
.002
.037
.033
.011
.007
.003
.012
0.02
0.03
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
70
Stress (MPa)
60
0.25 1050ºC 0.25 0.2 1100ºC 0.15 0.12
0.9 0.8
Fractional softening
observed in the figure, that the behavior of the material after the reversal depends on the applied pre-strain. For a reversal applied before the onset of the dynamic recrystallization, the shape of the curve after the Bauschinger strain is the same than that corresponding to a monotonic test, if the origin of this last is properly shifted. After the plateau, the strain hardening activates again and the flow stress rises to the peak, followed by the softening stage produced by dynamic recrystallization. However, when the pre-strain before the reversal enters the dynamic recrystallization range, very small strain hardening is observed after the plateau and the stress goes directly to the level corresponding to the steady state. This result has been employed to make sure that the applied pre-strains were, for all the cases, outside the range of dynamic recrystallization and, consequently, the softening during the holding time is not affected by this mechanism. This ensures that the recrystallization, taking place during the holding, is static for all the tests. The graph in Fig. 2, shows an example of the flow curve obtained for a three-pass test. The last curve after the holding time is used to determine the fractional softening as explained in the experimental procedure. The fractional softening curves are shown in Fig. 3, for straining under monotonic conditions (two-pass tests). It is not easy to quantify metallographically the recrystallized fraction in the present steel. However, the micrographs in Fig. 4, show the microstructure at the shoulder and at the gauge length sub-surface of a specimen deformed at 1100 C to a strain 0.2 and a subsequent 35 s holding time, followed by water quenching. The grain refinement observed in the gauge length region, as compared to the initial grain size (that at the shoulders), clearly indicates the occurrence of static recrystallization for this condition, that has softened a 40% during the holding time, see Fig. 3.
335
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1
10
100
1000
10000
Fig. 3. Static softening curves obtained after monotonic straining.
Fig. 4. Austenite microstructure on a longitudinal section parallel to the torsion axis after deformation at 1100 C to a 0.2 strain followed by a holding time of 35 s, (a) at the shoulder and (b) at the gauge length.
Reverse
50
Reverse 40
Forward
30
1100 ºC
20
-1
0.1 s 10 0 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 2. Example of a multipass torsion curves obtained for a three-pass test schedule. A 50 s holding time was applied between the last pass and the previous one, both passes performed without any reversal of the twist.
The curves in Fig. 3 show the normal trend: increasing the pre-strain shortens the holding time required to onset the static recrystallization. However, independently of the applied condition within the studied range, a general characteristic of the different curves is that they stagnate at a certain recrystallization level. The level of occurrence of the stagnation depends on the applied pre-strain. The consequence of this stagnation is that full softening of the material is not reached within the time interval used in the present work. At short times, before the stagnation, a Johnson– Mehl–Avrami–Kolmogorov(JMAK) type equation
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X ¼ 1 exp 0:693
t
n
300
ð1Þ
t0:5
Monotonic 1100 ºC Strain reversal Monotonic Strain reversal 1050 ºC
250
with X, the fraction softened, t, the holding time and t0:5 , the time required for 50% softening, can be used to fit the experimental data. An exponent of n ¼ 1, gives a good fit at low recrystallized fractions for the most of the experimental curves, as can be seen in Fig. 3 (hatched lines). Some deviations are observed for the lowest applied strains in which case an exponent higher that 1 seems to give a better fit. However, for these cases, the stagnation happens for short recrystallized fractions and the range over which the fitting can be done is quite narrow. The time, ts , has been recorded at the stage the experimental data deviate from the predictions of the JMAK equation. Deformation sequences involving different amounts of reversal (three-pass tests) were performed in order to investigate the effect of the reversal on the recrystallization kinetics. The obtained softening curves are shown in Fig. 5. As for the monotonic tests, Eq. (1) with n ¼ 1 has been used to fit the experimental points, before the softening stagnates. The constant value of the time exponent for the JMAK equation indicates that the strain
200
t0.5 (s)
t0.5 ∝ ε
-3.4
Reversion
150
100
50
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 6. Time for the 50% softening, t0:5; as a function of the total absolute strain.
1
Fractional softening
0.9
0.2
0.8
0.2,-0.06
0.7
0.2,-0.1 0.2,-0.15
0.6
0.2,-0.2
0.5 0.4 0.3 0.2
1100 ºC
0.1 0 1
10
100
1000
Holding time (s) 1 ε = + 0.25
0.9
ε = + 0.25, -0 0.1
Fractional softening
0.8
ε = + 0.25; -0 0.25
0.7 0.6 0.5 0.4 0.3 0.2 1050 ºC 0.1 0 1
10
100
1000
10000
Fig. 5. Static softening curves as a function of the holding time following different strain paths.
Fig. 7. TEM micrographs obtained on carbon extraction replicas. (a) and (b) Ti–Nb-rich particles image and microanalysis and (c) cluster of Nb(C,N) precipitates. Sample deformed at 1100 C to a strain of +0.12, followed by a holding of 150 s and subsequently water quenched to room temperature.
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
path acts on the incubation time but (excepting the softening stagnation) do not affect the rate at which recrystallization progresses. It is clear from experiment that the reversal of the strain shortens the time required for the onset of static recrystallization. However, for a partial reversal of the strain of 0:1 (end of the strain hardening stagnation after the reversal), the recrystallization kinetics is significantly delayed. This is true for the two testing temperatures used in the present work. An intermediate behavior between these two extremes is observed for the rest of the reversed strains. This is more clearly illustrated in Fig. 6 in which the time required to reach the 50% softening has been plotted as a function of the total absolute strain. The values of t0:5 have been deduced from Eq. (1) and correspond to extrapolated values for the case the softening stagnation happens below this fraction. The TEM micrographs in Fig. 7, show the precipitates present in a sample deformed at 1100 C to e ¼ 0:12 and held for 150 s at the deformation temperature, corresponding, as can be seen in Fig. 3, to the plateau in the softening curve. Two types of precipitates are observed: relatively coarse (20–60 nm) Ti–Nb-rich particles and fine Nb(C,N) precipitates. The Ti does not appear in the composition but may be present in the steel as a residual element. The sizes of the Nb(C,N) are lower than about 10 nm and appear forming clusters of particles, as the one shown in the TEM micrograph in Fig. 7(c).
4. Discussion The effect of strain reversal on the flow curve is clearly apparent in Fig. 1 and follows the general trend observed for this steel and analyzed elsewhere [14] that can be summarized as follows. Just after the reversal, the stress rises to reach a stress level lower than the prestress, followed by a ‘‘yield stress plateau’’. This is a relatively common behavior after the reversal, and has been reported for a relatively broad set of materials and deformation conditions [11,12,14–16]. The strain hardening stagnation is generally attributed to the partial dissolution of the dislocation substructure [10,17]. As reported elsewhere [14], the final shape of the curve after the reversal depends strongly on the applied pre-strain. The curves in Fig. 1, clearly illustrate this effect. For strains lower than that required for the onset of dynamic recrystallization during the forward deformation, strain hardening activates after the plateau and the flow curve recovers the shape corresponding to a monotonic tests. Dynamic recrystallization occurring during pre-straining in the forward, even before the effect of the first dynamically recrystallized grains can be detected on the flow curve [22], significantly modifies the stress–strain curve after the reversal.
337
The 2% offset method has been applied to deduce the fractional softening, for different strain paths. Some authors claim [25] that recovery has a large contribution to the softening taking place after hot working in Nbmicroalloyed steels. However, Fernandez et al. [24] compared the softening data obtained, by the use of the mentioned 2% offset method, from double-hit torsion tests performed on samples with initial coarse austenite grain sizes, to the fraction recrystallized determined by metallography at the specimen subsurface. The results they obtained indicate that recovery at 1100 C and 1 s1 in Nb-containing steel is responsible for not more than about 10% of the observed static softening. For the present steel, it is not straightforward, under the conditions used, to determine the recrystallized fraction metallographically. Recrystallization manifests by the microstructure grain refinement taking place at the deformed zone, Fig. 4. The obtained results at low recrystallized fractions can be fitted, by Eq. (1) with an exponent of n ¼ 1, see Fig. 3. This value of n is commonly found for the static recrystallization after hot working in steels [26,27]. However, as the holding time increases, the experimental points deviate from the trend defined by the above equation and the fractional softening stagnates due probably to the stop of recrystallization caused by the pinning produced by strain-induced precipitation of Nb carbonitrides, Nb(C,N), [28,29]. The temperatures at which the stagnation is observed in the present work are higher than those that can be determined from current physically based models [25,30] for precipitation on dislocations, but the TEM observations clearly show the presence of fine Nb carbonitrides in the samples, Fig. 7(c). The supersaturation of Nb in the present steel can be estimated, according to the solubility product for Nb(C,N) in [30], to be about 5 at 1050 C and 3.4 at 1100 C. The empirical model developed by Abad et al. [31] predicts the start of the strain-induced precipitation of the Nb carbonitrides in the present steel, for an approximate holding time of 100 s at 1050 C, which is in agreement with the observed start of the stagnation, Fig. 3, at this temperature. However, this same model predicts a precipitation time close to 800 s for the samples deformed at 1100 C to a pre-strain of 0.25. This time is significantly higher than the value of ts deduced from the experimental curves. The TEM observations carried out on carbon extraction replicas, Fig. 7, show Ti–Nb-rich particles too large to interact with recrystallization, but Nb(C,N) precipitates sizes within the range of those expected when strain-induced precipitation leads to the stagnation of the softening [32]. Different contributions can explain the apparent discrepancy between the prediction of the models and the experimental results. The tendency of Nb to segregate during solidification is well
338
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known and the presence in the sample of some Nb enriched regions could contribute to explain the observed behavior. The presence in the steel of some residual Ti can also have some contribution. Some recovery, concurrent with recrystallization, probably contributes to the recrystallization stagnation for long times by lowering the stored energy of the non recrystallized regions and help understanding why, even for long holding times, the progress of the recrystallization remains inhibited. The softening curves in Fig. 5 clearly show that the recrystallization kinetics depends on the applied strain path. Other authors have also reported such an effect for different materials [5,19,20]. As mentioned before, the total absolute strain is often used to investigate the effect of the strain path on the recrystallization kinetics, which is not applicable to the present results. Lindh et al. [33] found for copper after cold deformation (compressiontension tests), that the redundant deformation is 65% as effective as permanent straining in terms of its influence on recrystallization. The results in the present work converge to this same type of relation for large reverse strains but the relation is clearly not applicable for strains within the transient after the reversal. For small reverse strains, a more complex behavior is observed, as shown in Fig. 6. Increasing the reversal, the t0:5 increases to reach a maximum and decreases afterwards. The maximum recrystallization time is reached at a reverse strain close to that required for the onset of strain hardening after the plateau, er ffi 0:1 for both conditions. After this, the softening times decrease with increasing the reverse strain. As will be discussed in the following, this can be related to the partial dissolution of the dislocation structure. The deformation dislocation substructure provides the driving pressure, FR , for recrystallization that can be expressed as 1 FR ¼ lb2 q: 2
ð2Þ
where l is the shear modulus, b the Burgers vector and q the dislocation density driving the migration of the recrystallization front into the deformed areas. The driving pressure can be assumed to decrease with time due to the early consumption during recrystallization of the regions with the highest stored energy, the variations in the mobility of the migrating front due to the segregation of the solutes and/or to recovery processes in the non recrystallized areas. A quite general formula [34–36] relating the decay with time of the driving pressure for recrystallization can be written in the form u t FR ðtÞ ¼ FR0 1 þ : ð3Þ s1 With s1 a relaxation time and 0 < u < 1 and FR0 being expressed by Eq. (2). Assuming site saturation, the
fraction recrystallized can be expressed, in absence of interaction with precipitation, in the way [25] " Z t 3 # mðtÞFR ðtÞdt ð4Þ Xv ¼ 1 exp NRex 0
with NRex the number of nuclei per unit volume. Assuming the temperature dependent mobility, m, remains constant with time, which means that the solutes remain constant during the process of precipitation-free recrystallization, this equation can be rewritten as " u 3 # Z t t 0 : ð5Þ Xv ¼ 1 exp NRex mFR 1þ dt si 0 To make, after integration, the time dependence in Eq. (4) consistent with the one found experimentally (n ¼ 1) when applying Eq. (1) to the present steel: u ¼ 1 ðn=3Þ ¼ ð2=3Þ. For s1 t, the integration leads to s 3 t 3ð1uÞ t 1 0 3 ¼ NRex ðmFR Þ : ð6Þ 0:693 t0:5 s1 1u For site saturated nucleation at grain boundaries, Nrex 2 can be expressed by kSv ðFR0 Þ , where k is a geometric factor and Sv the austenite specific grain boundary area before recrystallization starts [25]. Assuming the driving pressure for nucleation and for growth are approximately the same during the early stages of recrystallization and substituting FR0 by its value expressed by Eq. (2), the effective initial dislocation density, q0 can be related to t0:5 as " #1=5 0:693 32 1 q0 ¼ : ð7Þ 5 27 kms1 ðlb2 Þ Sv t0:5 In this equation, q0 and Sv depend on the strain and strain path; q0 through the storage or partial elimination of dislocations and Sv through the change of the shape of the austenite grains produced by the forward or reverse deformation. Based on the above equation, the following expression: q0ðeÞ Svðe¼0:2Þ t0:5ðe¼0:2Þ 1=5 ¼ ð8Þ q0ðe¼0:2Þ SvðeÞ t0:5ðeÞ allows making an estimation of the relative change in the effective dislocation density involved in the recrystallization process for different strain paths. The estimation is referred to a particular forward deformation (e ¼ 0:2) and is done as a function of the recrystallization times and Sv . This last has been determined [37], assuming the full reversibility of the shape of the grains during the reversal [38]. The obtained results are shown in Fig. 8, as a function of the total absolute strain. The effect of the transient after the reversal is clearly recognizable. According to the obtained results, a maximum decrease in the effective dislocation density for
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341 1.4
ρo (at ε)/ρo(at ε =0.2)
1.2
Monotonic
1
0.8
0.6 Reversion 0.4 Estimated from t0.5 ; Eq.(8) 0.2
Estimated from σ; Eq. (10)
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.
.45
Total absolute strain
Fig. 8. Relative dislocation density for recrystallization at different strain paths, as compared to that corresponding to a monotonic prestrain of 0.2. Estimations made on the basis of Eqs. (8) and (10).
recrystallization can be estimated as being about 18% and happens for a reverse strain close to 0.1 (total absolute strain ¼ 0.3). This diminution is in close agreement with the one measured by TEM during the transient after the reversal in high purity Al by Hasegawa et al. [10] (17% decrease of the total dislocation density). However, it is significantly lower than the one obtained in Al–2Mg by Zhu and Sellars [5] (30–40% decrease of the internal dislocation density). Eq. (4) is based on a series of assumptions and oversimplifications that can put into question its validity. The driving pressure for recrystallization expressed by Eq. (2) is broadly used but is a simplified version of the one taking also into account subgrains [39]. The term s1 in Eq. (3) is not strictly speaking independent of the dislocation density [35] or the subgrain size [34]. However, the main concerns relate to the applicability of such an approach for tests involving a reversal of the strain. The experimental data in [5] have been used to investigate the relation between the experimentally measured internal dislocation density, qi , and the recrystallization time for different strain paths involving or not a reversal of the strain. It is found that these data agree reasonably well with a general relationship of the form 0:21 1 qi / : ð9Þ Sv t0:5 The exponent is quite close to the one deduced in the present work through a theoretical approach, Eq. (7). On the other hand, the total dislocation density, q, can be related to the stress according to the following expression: 2 r q¼ ð10Þ Malb with M the Taylor factor and a a constant. This equation is known to lose validity during the transients
339
involving sudden changes in strain rate or temperature [40], but it is used here to estimate the change in the equivalent dislocation density for different strain paths. Comparison has been made, as previously, with the value obtained for a monotonic pre-strain of 0.2. The obtained results are also shown in Fig. 8. It can be seen that these are in relatively close agreement with those deduced by using Eq. (8) and that the most important deviations are observed during the transient for reverse strains within the plateau. The transient in the flow curve is expected to be due to the superposition of two contributions: a softening due to the dissolution/dissociation of the dislocation substructure [4,5,10], and a strain hardening due to the dislocation storage as a result of the rebuilding compatible with the new deformation conditions [41]. The transient after the reversal can be associated to the micro-localization of plastic flow and the formation of microbands parallel to the plane of the newly most active slip system [17]. These microbands cut through the ‘‘old’’ dislocation sheets and produce channels for easy transport of mobile dislocations. This leads to the progressive destruction of the pre-existing structure. Additionally, the reversal of the strain involves the remobilization of dislocations during the transient before they are annihilated by recovery or trapped by the structure having the new polarity. The release of dislocations contributes to sustain the imposed strain rate [42] without the need to produce new ones (no strain hardening). However, after a certain strain, in reverse mode, the prior substructure is expected to have no longer the ability to release enough dislocations to compensate for the dislocation storage in the newly formed substructure and the strain hardening activates again. The recrystallization kinetics relates to the stored energy and more precisely to the way in which this last is distributed in the material. The strain range over which the progressive dissolution of the prior deformation structure gains over the progressive build up of the new one (plateau in the flow curve) is expected to delay nucleation and consequently retard the recrystallization. The maximum mean substructure ‘‘dilution’’ is expected to happen at the end of the plateau (longest recrystallization times), Fig. 6. For higher reverse strains, the ‘‘old substructure’’ is no longer able to relax enough dislocations and the strain hardening activates again as a result of the dislocation accumulation in the ‘‘new substructure’’ and the recrystallization times decrease with the increase of the (reverse) strain, as happens for monotonic tests. During the above part of the discussion no interaction between recrystallization and precipitation has been considered. However, the stagnation of the recrystallization caused by the strain-induced precipitation [32] of Nb(C,N), Fig. 7(c), can be observed for all the applied
340
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conditions. The time for the start of the stagnation, ts , has been plotted in Fig. 9 as a function of t0:5 and a potential dependence of the form m ts ¼ At0:5
ð11Þ
can be used to fit the experimental set of data, for each deformation temperature. This relation agrees with the usual equations relating the recrystallization and the strain-induced precipitation times [26,27,43–46] to the deformation conditions and the initial grain size, D0 : t0:5 ¼ A1 Da0 eb e_c f1 ðT Þ;
ð12Þ
ts ¼ A2 ed e_e f2 ðT Þ:
ð13Þ
For the present steel, the values of b ffi 3:4 (Fig. 6) and d ffi 1:5 were found to fit reasonably to the experimental results for monotonic tests [22]. The value for b is in the range of those reported by other authors [26,44] and d is slightly higher than the value (d ¼ 1) generally accepted [43]. However, the above-mentioned values lead to a b=d quotient similar to the value of m deduced from the fit of the Eq. (11) to the experimental data, see Fig. 9. This shows the reliability of the above equations. However, the main consequence of the data plot in Fig. 9 is the lack of effect of the strain path on the relation between ts and t0:5 . The recrystallization stagnation is a consequence of the competition between the driving pressure for recrystallization, FR , and the pinning pressure, FPin , caused by the strain-induced precipitation [32] of Nb(C,N) precipitates like those shown in Fig. 7(c). Different authors have developed physically based models to describe the precipitation kinetics [30] and the interaction between this and the softening mechanisms [25]. Most of these models take into account homogeneous precipitation on dislocations. Assuming the Zener equation correctly describes the pressure exerted by a particle distribution characterized by a volume fraction, fv , and a particle mean radius, r, the following equation can be
m = 0.67
180 160
m = 0.47 140
ts (s)
120 100 80 60
1100 ºC 1050 ºC 1100 ºC 1050 ºC
40 20
Monotonic Strain reversal
0 0
50
100
150
200
250
Fig. 9. Relation between ts and the time required to reach a 50% softening.
used to describe the net driving pressure for recrystallization: 1 3cfv DF ¼ lb2 q 2 r
ð14Þ
with c the interfacial energy. This equation replaces Eq. (2) when recrystallization interacts with precipitation. To note that the values of ts deduced from the curves are not the time for the start of the precipitation, but actually the time at which the net driving pressure for recrystallization cancels out. By using the formulation proposed by other authors [25], together with fv ¼ 43 pr3 N (N ¼ number of straininduced particles per unit volume), Eq. (14) can be rewritten as 1 DF ¼ lb2 q 4cpr2 Ntot ½1 expðHtÞ; 2
ð15Þ
where Ntot is the number of total precipitate nucleation density sites and H ¼ Zb expðDGn=kT Þ with Z the Zeldovich factor, b the atomic impingement rate [47], DGn the free energy for nucleation and k the Boltzmann constant. For nucleation on dislocations, Ntot has been expressed as 0.5q1:5 [30]. The driving pressure for recrystallization and the pinning pressure depend on the local dislocation density. Consequently, a variation of the local dislocation density due to a reversal of the strain will affect at the same time the local driving and the pinning pressures, which could explain the observed relation between t0:5 and ts being independent of the strain path.
5. Conclusions • The strain path has a clear effect on the static recrystallization kinetics after hot working. The total absolute strain is not the variable to be used to describe the recrystallization kinetics because small reverse strains produce a retardation of the recrystallization. • The longest recrystallization times are observed at the end of the ‘‘yield stress plateau’’ which is consistent with the highest ‘‘old structure’’ dissolution produced by the reversal taking place at this stage (onset of the strain hardening after the plateau). • Full softening is not reached for any one of the strain paths/temperatures selected. The recrystallization stagnation is due to the strain-induced precipitation of Nb carbonitrides. • The results have been discussed in terms of the available recrystallization and strain-induced precipitation models. The application of these models to describe the material behavior during the holding after a reversal of the strain is probably a rough approach from the point of view of the complex dislocation interactions taking place. However, within the frame of
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
the present knowledge on the subject, reasonable results are obtained, allowing to predict the recrystallization kinetics trend. • The times ts correspond to the instant at which the driving pressure for recrystallization cancels out with the pinning pressure exerted by the strain-induced Nb(C,N) precipitates. Both ts and t0:5 relate to the local dislocation density. The relation between ts and t0:5 has been observed to be independent of the strain path.
Acknowledgements The authors would like to acknowledge the CICYT (Spain) for the financial support of this work. The authors are greatly indebted to M. Diaz-Fuentes for his assistance with the TEM work.
References [1] Urcola JJ, Sellars CM. Acta Metall 1987;35:2635. [2] Baxter GJ, Furu T, Zhu Q, Whiteman JA, Sellars CM. Acta Mater 1999;47:2367. [3] Furu T, Shercliff HR, Baxter GJ, Sellars CM. Acta Mater 1999;47:2377. [4] Zhu Q, Sellars CM. In: Hutchinson B et al., editors. Proc Conf on Thermomechanical Processing in Theory Modelling and Practice. Sweden: ASM; 1996. p. 193. [5] Zhu Q, Sellars CM. In: McNelley TR, editor. Proc 3rd Int Conf on Recrystallization and Related Phenomena. Monterey, USA: Monterey Institute of Advanced Studies; 1997. p. 195. [6] Davenport SB, Higginson RL. J Mater Process Technol 2000;98:267. [7] Davenport SB, Higginson RL, Sellars CM. Philos Trans R Soc Lond A 1999;357:1645. [8] McLaren AJ, Sellars CM. In: Ryan ND, Brown AJ, McQueen HJ, editors. Proc Int Symp on Strip Casting, Hot and Cold Working of Stainless Steels. Quebec: Canadian Institute of Mining, Metallurgy and Petroleum; 1993. p. 107. [9] Rollet AD, Lowe T, Kocks UK, Stout MG. In: Kallend JS, Gottstein, G, editors. Proc Eight Int Conf on Textures of Material (ICOTOM 8), 1998, p. 473. [10] Hasegawa T, Yakou T, Karashima S. Mater Sci Eng 1975;20:267. [11] Stout MG, Rollet AD. Metall Trans A 1990;21A:3201. [12] Takur A, Vecchio KS, Nemat-Nasser S. Metall Mater Trans A 1996;27A:1739. [13] Karjalainen LP, Somani MC, Peura P, Porter DA. In: Proc Conf on Thermomechanical Processing of Steels. London: IOM; 2000. p. 130. [14] Bartolome R, Astiazaran JI, Iza-Mendia A, Gutierrez I. In: Proc Conf on Thermomechanichal Processing of Steels. London: IOM; 2000. p. 221. [15] Pinheiro IP, Barbosa R, Cetlin PR. In: Yue S, Essadiqui E, editors. Proc on Thermomechanical Processing of Steel J.J. Jonas Symposium, COM 2000, Met Soc, Ottawa, Canada, 2000, p. 221. [16] Zhu Q, Sellars CM. In: Beynon JH et al., editors. Proc 2nd Int Conf on Modelling of Metal Rolling Processes. London: The Institute of Materials; 1996. p. 158.
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Study of the strain reversal effect on the recrystallization and strain-induced precipitation in a Nb-microalloyed steel D. Jorge-Badiola, I. Gutierrez
*
Centro de Estudios e Investigaciones T ecnicas de Gipuzkoa and University of Navarra, TECNUN, P Manuel de Lardizabal, 15, 20018 Donostia-San Sebasti an, Spain Received 17 July 2003; received in revised form 17 September 2003; accepted 18 September 2003
Abstract Efforts to produce more accurate through-process models for hot working need to take into account variations in the deformation conditions. In the present work, the effect of the reversal of the strain for a Nb-microalloyed steel is discussed. It is shown that, depending on the magnitude of the reversal, the static softening kinetics can be accelerated or delayed. The same is true for the case of the strain-induced Nb-carbonitride precipitation, leading to an interaction between both processes and the deformation path. Current models of recrystallization and precipitation have been used to analyze the obtained results. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hot working; Strain path; Recrystallization; Strain-induced precipitation
1. Introduction In industrial hot working local strain reversal, together with strain rate and temperature changes, characteristic of each particular forming process, take place during deformation passes. Efforts have been carried out to investigate the material behavior during transients following instantaneous changes of conditions [1–7]. For the case of hot working, it has been demonstrated that when combining the outputs obtained from a finite element model with empirical equations of static recrystallization, important discrepancies can be found between the predictions of the model and the true behavior of the material [8]. This has been attributed to the way in which the final equivalent strains are computed from the incremental strains. These differences can be considered negligible for most of the cases in materials undergoing fast static recrystallization after deformation. However, when the recrystallization kinetics becomes slower, due to the material and/or to a decrease
*
Corresponding author. Tel.: +34-943-21-28-00; fax: +34-943-21-30-
76. E-mail address: [email protected] (I. Gutierrez).
on the deformation temperature, the effect can be non-negligible. When reversing the strain during deformation, a decrease in stress is generally observed to take place which is very often accompanied by a transient decrease of the work hardening rate [9], leading to the formation of a ‘‘yield stress plateau’’ extending over a relatively large strain interval [4,10–15]. This transient is generally attributed to the partial dissolution, during the first stages after the reversal [10], of the substructure present before changing the direction of deformation. This has been reported to affect the geometrically necessary dislocations [6,16] but to activate recovery processes affecting all the dislocations [17]. Both the static and the dynamic recrystallization depend on the amount and distribution of the stored energy, which is directly related to the local dislocation density. Consequently, a modification on the dislocation distribution is expected to have an immediate effect on the dynamic/static recrystallization kinetics. In agreement with this, when the strain is reversed at a strain lower than that required for the onset of dynamic recrystallization, dynamic recrystallization is delayed [18]. Concerning static recrystallization, it is generally accepted that the reversal modifies its kinetics [19] and,
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2003.09.018
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
as a general trend, a strain reversal implies a retardation for the onset of the recrystallization, as compared to the effect of the same total strain applied in a monotonic test. However, the results obtained by different authors present some contradictions. For aluminum, the reversal from tension to compression was observed to cause a retardation in the recrystallization kinetics, being a function of the total absolute strain [5]. For microalloyed steels, the reversal of the strain from tension to compression was observed to always accelerate the recrystallization, although, when this last is expressed in terms of the total absolute strain, a delay is also observed. [13,20]. The present authors [21] found, using torsion tests, a net retardation of the static recrystallization when the strain is partially reversed, but just the opposite effect when it is fully reversed and, consequently, that the total absolute strain is not an appropriate variable to express the recrystallization kinetics [22]. New experiments on microalloyed steels by the present authors and also by Karjalainen and Somani [23] confirm such behavior. In the present work, the effect of different amounts of reversal on the recrystallization and strain-induced precipitation are analyzed in more depth and commented upon in terms of the available models. 2. Experimental procedure A microalloyed steel with a composition shown in Table 1 was used to perform multipass torsion tests at a strain rate of 0.1 s1 , in order to determine the static softening kinetics following different strain paths. The testing conditions are summarized in Table 2. The deformation sequences were of the following type: • Two-pass tests: One pass to a given strain followed by a certain holding time at the deformation temperature and a second pass to analyze the effect of the holding. • Three-pass tests: One pass to a given strain ðef Þ, some reversal of the strain ðer Þ by twisting in the reverse followed by a holding time at the deformation temperature and a third pass also in the reverse to analyze the effect of the holding. The static fractional softening was determined from the stress–strain curves by the 2% offset method as described elsewhere [24]. Some samples were water quenched after a certain holding time in order to perform microstructural observations at the sub-surface on a longitudinal section parallel to the torsion axis, located at about 0:9R (R ¼ specimen radius).
Table 2 Testing conditions Reheating Deformation Tests temperature (C) temperature (C)
Strain path ðeÞ
1250
0.12 0.15 0.20 0.25 0.2; )0.06 0.2; )0.10 0.2; )0.15 0.2; )0.20 0.25 2 0.29; )2 0.25; )2 0.25; )0.1 0.25; )0.25
1100
Monotonic
Strain reversal
1200
1050
Monotonic Strain reversal
Carbon extraction replicas were prepared from the torsion specimens, using conventional methods in order to investigate the strain-induced precipitation by transmission electron microscopy (TEM). The observations were carried out in a Philips CM12 STEM equipment fitted with an EDS system from EDAX. 3. Results The graph in Fig. 1 shows examples of the stress strain curves for torsion test involving a strain reversal. For simplicity, the absolute stress and strains have been considered. Just after the reversal, the stress rises to reach a stress level lower than the pre-stress, followed by a ‘‘yield stress plateau’’. After the plateau, the stress starts increasing again due to strain hardening. It can be 90
Monotonic 80 70
Stress (MPa)
334
60
Curves after the reversal
50 40 30
1050ºC
20
εf =0.25
10
0.1s
εf =0.29
-1
0 0
0.2
0.4
0.6
0.8
1
1.2
Strain
Fig. 1. Flow curves for tests involving a reversal of the strain. Effect of the applied pre-strain, ef , on the transient after the reversal.
Table 1 Steel composition (wt%) C
Si
Mn
P
S
Al
Nb
V
N
Mo
Cu
Cr
Ni
.15
.30
1.42
.012
.002
.037
.033
.011
.007
.003
.012
0.02
0.03
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
70
Stress (MPa)
60
0.25 1050ºC 0.25 0.2 1100ºC 0.15 0.12
0.9 0.8
Fractional softening
observed in the figure, that the behavior of the material after the reversal depends on the applied pre-strain. For a reversal applied before the onset of the dynamic recrystallization, the shape of the curve after the Bauschinger strain is the same than that corresponding to a monotonic test, if the origin of this last is properly shifted. After the plateau, the strain hardening activates again and the flow stress rises to the peak, followed by the softening stage produced by dynamic recrystallization. However, when the pre-strain before the reversal enters the dynamic recrystallization range, very small strain hardening is observed after the plateau and the stress goes directly to the level corresponding to the steady state. This result has been employed to make sure that the applied pre-strains were, for all the cases, outside the range of dynamic recrystallization and, consequently, the softening during the holding time is not affected by this mechanism. This ensures that the recrystallization, taking place during the holding, is static for all the tests. The graph in Fig. 2, shows an example of the flow curve obtained for a three-pass test. The last curve after the holding time is used to determine the fractional softening as explained in the experimental procedure. The fractional softening curves are shown in Fig. 3, for straining under monotonic conditions (two-pass tests). It is not easy to quantify metallographically the recrystallized fraction in the present steel. However, the micrographs in Fig. 4, show the microstructure at the shoulder and at the gauge length sub-surface of a specimen deformed at 1100 C to a strain 0.2 and a subsequent 35 s holding time, followed by water quenching. The grain refinement observed in the gauge length region, as compared to the initial grain size (that at the shoulders), clearly indicates the occurrence of static recrystallization for this condition, that has softened a 40% during the holding time, see Fig. 3.
335
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1
10
100
1000
10000
Fig. 3. Static softening curves obtained after monotonic straining.
Fig. 4. Austenite microstructure on a longitudinal section parallel to the torsion axis after deformation at 1100 C to a 0.2 strain followed by a holding time of 35 s, (a) at the shoulder and (b) at the gauge length.
Reverse
50
Reverse 40
Forward
30
1100 ºC
20
-1
0.1 s 10 0 0
0.2
0.4
0.6
0.8
1
1.2
Fig. 2. Example of a multipass torsion curves obtained for a three-pass test schedule. A 50 s holding time was applied between the last pass and the previous one, both passes performed without any reversal of the twist.
The curves in Fig. 3 show the normal trend: increasing the pre-strain shortens the holding time required to onset the static recrystallization. However, independently of the applied condition within the studied range, a general characteristic of the different curves is that they stagnate at a certain recrystallization level. The level of occurrence of the stagnation depends on the applied pre-strain. The consequence of this stagnation is that full softening of the material is not reached within the time interval used in the present work. At short times, before the stagnation, a Johnson– Mehl–Avrami–Kolmogorov(JMAK) type equation
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X ¼ 1 exp 0:693
t
n
300
ð1Þ
t0:5
Monotonic 1100 ºC Strain reversal Monotonic Strain reversal 1050 ºC
250
with X, the fraction softened, t, the holding time and t0:5 , the time required for 50% softening, can be used to fit the experimental data. An exponent of n ¼ 1, gives a good fit at low recrystallized fractions for the most of the experimental curves, as can be seen in Fig. 3 (hatched lines). Some deviations are observed for the lowest applied strains in which case an exponent higher that 1 seems to give a better fit. However, for these cases, the stagnation happens for short recrystallized fractions and the range over which the fitting can be done is quite narrow. The time, ts , has been recorded at the stage the experimental data deviate from the predictions of the JMAK equation. Deformation sequences involving different amounts of reversal (three-pass tests) were performed in order to investigate the effect of the reversal on the recrystallization kinetics. The obtained softening curves are shown in Fig. 5. As for the monotonic tests, Eq. (1) with n ¼ 1 has been used to fit the experimental points, before the softening stagnates. The constant value of the time exponent for the JMAK equation indicates that the strain
200
t0.5 (s)
t0.5 ∝ ε
-3.4
Reversion
150
100
50
0 0
0.1
0.2
0.3
0.4
0.5
0.6
Fig. 6. Time for the 50% softening, t0:5; as a function of the total absolute strain.
1
Fractional softening
0.9
0.2
0.8
0.2,-0.06
0.7
0.2,-0.1 0.2,-0.15
0.6
0.2,-0.2
0.5 0.4 0.3 0.2
1100 ºC
0.1 0 1
10
100
1000
Holding time (s) 1 ε = + 0.25
0.9
ε = + 0.25, -0 0.1
Fractional softening
0.8
ε = + 0.25; -0 0.25
0.7 0.6 0.5 0.4 0.3 0.2 1050 ºC 0.1 0 1
10
100
1000
10000
Fig. 5. Static softening curves as a function of the holding time following different strain paths.
Fig. 7. TEM micrographs obtained on carbon extraction replicas. (a) and (b) Ti–Nb-rich particles image and microanalysis and (c) cluster of Nb(C,N) precipitates. Sample deformed at 1100 C to a strain of +0.12, followed by a holding of 150 s and subsequently water quenched to room temperature.
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path acts on the incubation time but (excepting the softening stagnation) do not affect the rate at which recrystallization progresses. It is clear from experiment that the reversal of the strain shortens the time required for the onset of static recrystallization. However, for a partial reversal of the strain of 0:1 (end of the strain hardening stagnation after the reversal), the recrystallization kinetics is significantly delayed. This is true for the two testing temperatures used in the present work. An intermediate behavior between these two extremes is observed for the rest of the reversed strains. This is more clearly illustrated in Fig. 6 in which the time required to reach the 50% softening has been plotted as a function of the total absolute strain. The values of t0:5 have been deduced from Eq. (1) and correspond to extrapolated values for the case the softening stagnation happens below this fraction. The TEM micrographs in Fig. 7, show the precipitates present in a sample deformed at 1100 C to e ¼ 0:12 and held for 150 s at the deformation temperature, corresponding, as can be seen in Fig. 3, to the plateau in the softening curve. Two types of precipitates are observed: relatively coarse (20–60 nm) Ti–Nb-rich particles and fine Nb(C,N) precipitates. The Ti does not appear in the composition but may be present in the steel as a residual element. The sizes of the Nb(C,N) are lower than about 10 nm and appear forming clusters of particles, as the one shown in the TEM micrograph in Fig. 7(c).
4. Discussion The effect of strain reversal on the flow curve is clearly apparent in Fig. 1 and follows the general trend observed for this steel and analyzed elsewhere [14] that can be summarized as follows. Just after the reversal, the stress rises to reach a stress level lower than the prestress, followed by a ‘‘yield stress plateau’’. This is a relatively common behavior after the reversal, and has been reported for a relatively broad set of materials and deformation conditions [11,12,14–16]. The strain hardening stagnation is generally attributed to the partial dissolution of the dislocation substructure [10,17]. As reported elsewhere [14], the final shape of the curve after the reversal depends strongly on the applied pre-strain. The curves in Fig. 1, clearly illustrate this effect. For strains lower than that required for the onset of dynamic recrystallization during the forward deformation, strain hardening activates after the plateau and the flow curve recovers the shape corresponding to a monotonic tests. Dynamic recrystallization occurring during pre-straining in the forward, even before the effect of the first dynamically recrystallized grains can be detected on the flow curve [22], significantly modifies the stress–strain curve after the reversal.
337
The 2% offset method has been applied to deduce the fractional softening, for different strain paths. Some authors claim [25] that recovery has a large contribution to the softening taking place after hot working in Nbmicroalloyed steels. However, Fernandez et al. [24] compared the softening data obtained, by the use of the mentioned 2% offset method, from double-hit torsion tests performed on samples with initial coarse austenite grain sizes, to the fraction recrystallized determined by metallography at the specimen subsurface. The results they obtained indicate that recovery at 1100 C and 1 s1 in Nb-containing steel is responsible for not more than about 10% of the observed static softening. For the present steel, it is not straightforward, under the conditions used, to determine the recrystallized fraction metallographically. Recrystallization manifests by the microstructure grain refinement taking place at the deformed zone, Fig. 4. The obtained results at low recrystallized fractions can be fitted, by Eq. (1) with an exponent of n ¼ 1, see Fig. 3. This value of n is commonly found for the static recrystallization after hot working in steels [26,27]. However, as the holding time increases, the experimental points deviate from the trend defined by the above equation and the fractional softening stagnates due probably to the stop of recrystallization caused by the pinning produced by strain-induced precipitation of Nb carbonitrides, Nb(C,N), [28,29]. The temperatures at which the stagnation is observed in the present work are higher than those that can be determined from current physically based models [25,30] for precipitation on dislocations, but the TEM observations clearly show the presence of fine Nb carbonitrides in the samples, Fig. 7(c). The supersaturation of Nb in the present steel can be estimated, according to the solubility product for Nb(C,N) in [30], to be about 5 at 1050 C and 3.4 at 1100 C. The empirical model developed by Abad et al. [31] predicts the start of the strain-induced precipitation of the Nb carbonitrides in the present steel, for an approximate holding time of 100 s at 1050 C, which is in agreement with the observed start of the stagnation, Fig. 3, at this temperature. However, this same model predicts a precipitation time close to 800 s for the samples deformed at 1100 C to a pre-strain of 0.25. This time is significantly higher than the value of ts deduced from the experimental curves. The TEM observations carried out on carbon extraction replicas, Fig. 7, show Ti–Nb-rich particles too large to interact with recrystallization, but Nb(C,N) precipitates sizes within the range of those expected when strain-induced precipitation leads to the stagnation of the softening [32]. Different contributions can explain the apparent discrepancy between the prediction of the models and the experimental results. The tendency of Nb to segregate during solidification is well
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known and the presence in the sample of some Nb enriched regions could contribute to explain the observed behavior. The presence in the steel of some residual Ti can also have some contribution. Some recovery, concurrent with recrystallization, probably contributes to the recrystallization stagnation for long times by lowering the stored energy of the non recrystallized regions and help understanding why, even for long holding times, the progress of the recrystallization remains inhibited. The softening curves in Fig. 5 clearly show that the recrystallization kinetics depends on the applied strain path. Other authors have also reported such an effect for different materials [5,19,20]. As mentioned before, the total absolute strain is often used to investigate the effect of the strain path on the recrystallization kinetics, which is not applicable to the present results. Lindh et al. [33] found for copper after cold deformation (compressiontension tests), that the redundant deformation is 65% as effective as permanent straining in terms of its influence on recrystallization. The results in the present work converge to this same type of relation for large reverse strains but the relation is clearly not applicable for strains within the transient after the reversal. For small reverse strains, a more complex behavior is observed, as shown in Fig. 6. Increasing the reversal, the t0:5 increases to reach a maximum and decreases afterwards. The maximum recrystallization time is reached at a reverse strain close to that required for the onset of strain hardening after the plateau, er ffi 0:1 for both conditions. After this, the softening times decrease with increasing the reverse strain. As will be discussed in the following, this can be related to the partial dissolution of the dislocation structure. The deformation dislocation substructure provides the driving pressure, FR , for recrystallization that can be expressed as 1 FR ¼ lb2 q: 2
ð2Þ
where l is the shear modulus, b the Burgers vector and q the dislocation density driving the migration of the recrystallization front into the deformed areas. The driving pressure can be assumed to decrease with time due to the early consumption during recrystallization of the regions with the highest stored energy, the variations in the mobility of the migrating front due to the segregation of the solutes and/or to recovery processes in the non recrystallized areas. A quite general formula [34–36] relating the decay with time of the driving pressure for recrystallization can be written in the form u t FR ðtÞ ¼ FR0 1 þ : ð3Þ s1 With s1 a relaxation time and 0 < u < 1 and FR0 being expressed by Eq. (2). Assuming site saturation, the
fraction recrystallized can be expressed, in absence of interaction with precipitation, in the way [25] " Z t 3 # mðtÞFR ðtÞdt ð4Þ Xv ¼ 1 exp NRex 0
with NRex the number of nuclei per unit volume. Assuming the temperature dependent mobility, m, remains constant with time, which means that the solutes remain constant during the process of precipitation-free recrystallization, this equation can be rewritten as " u 3 # Z t t 0 : ð5Þ Xv ¼ 1 exp NRex mFR 1þ dt si 0 To make, after integration, the time dependence in Eq. (4) consistent with the one found experimentally (n ¼ 1) when applying Eq. (1) to the present steel: u ¼ 1 ðn=3Þ ¼ ð2=3Þ. For s1 t, the integration leads to s 3 t 3ð1uÞ t 1 0 3 ¼ NRex ðmFR Þ : ð6Þ 0:693 t0:5 s1 1u For site saturated nucleation at grain boundaries, Nrex 2 can be expressed by kSv ðFR0 Þ , where k is a geometric factor and Sv the austenite specific grain boundary area before recrystallization starts [25]. Assuming the driving pressure for nucleation and for growth are approximately the same during the early stages of recrystallization and substituting FR0 by its value expressed by Eq. (2), the effective initial dislocation density, q0 can be related to t0:5 as " #1=5 0:693 32 1 q0 ¼ : ð7Þ 5 27 kms1 ðlb2 Þ Sv t0:5 In this equation, q0 and Sv depend on the strain and strain path; q0 through the storage or partial elimination of dislocations and Sv through the change of the shape of the austenite grains produced by the forward or reverse deformation. Based on the above equation, the following expression: q0ðeÞ Svðe¼0:2Þ t0:5ðe¼0:2Þ 1=5 ¼ ð8Þ q0ðe¼0:2Þ SvðeÞ t0:5ðeÞ allows making an estimation of the relative change in the effective dislocation density involved in the recrystallization process for different strain paths. The estimation is referred to a particular forward deformation (e ¼ 0:2) and is done as a function of the recrystallization times and Sv . This last has been determined [37], assuming the full reversibility of the shape of the grains during the reversal [38]. The obtained results are shown in Fig. 8, as a function of the total absolute strain. The effect of the transient after the reversal is clearly recognizable. According to the obtained results, a maximum decrease in the effective dislocation density for
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341 1.4
ρo (at ε)/ρo(at ε =0.2)
1.2
Monotonic
1
0.8
0.6 Reversion 0.4 Estimated from t0.5 ; Eq.(8) 0.2
Estimated from σ; Eq. (10)
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.
.45
Total absolute strain
Fig. 8. Relative dislocation density for recrystallization at different strain paths, as compared to that corresponding to a monotonic prestrain of 0.2. Estimations made on the basis of Eqs. (8) and (10).
recrystallization can be estimated as being about 18% and happens for a reverse strain close to 0.1 (total absolute strain ¼ 0.3). This diminution is in close agreement with the one measured by TEM during the transient after the reversal in high purity Al by Hasegawa et al. [10] (17% decrease of the total dislocation density). However, it is significantly lower than the one obtained in Al–2Mg by Zhu and Sellars [5] (30–40% decrease of the internal dislocation density). Eq. (4) is based on a series of assumptions and oversimplifications that can put into question its validity. The driving pressure for recrystallization expressed by Eq. (2) is broadly used but is a simplified version of the one taking also into account subgrains [39]. The term s1 in Eq. (3) is not strictly speaking independent of the dislocation density [35] or the subgrain size [34]. However, the main concerns relate to the applicability of such an approach for tests involving a reversal of the strain. The experimental data in [5] have been used to investigate the relation between the experimentally measured internal dislocation density, qi , and the recrystallization time for different strain paths involving or not a reversal of the strain. It is found that these data agree reasonably well with a general relationship of the form 0:21 1 qi / : ð9Þ Sv t0:5 The exponent is quite close to the one deduced in the present work through a theoretical approach, Eq. (7). On the other hand, the total dislocation density, q, can be related to the stress according to the following expression: 2 r q¼ ð10Þ Malb with M the Taylor factor and a a constant. This equation is known to lose validity during the transients
339
involving sudden changes in strain rate or temperature [40], but it is used here to estimate the change in the equivalent dislocation density for different strain paths. Comparison has been made, as previously, with the value obtained for a monotonic pre-strain of 0.2. The obtained results are also shown in Fig. 8. It can be seen that these are in relatively close agreement with those deduced by using Eq. (8) and that the most important deviations are observed during the transient for reverse strains within the plateau. The transient in the flow curve is expected to be due to the superposition of two contributions: a softening due to the dissolution/dissociation of the dislocation substructure [4,5,10], and a strain hardening due to the dislocation storage as a result of the rebuilding compatible with the new deformation conditions [41]. The transient after the reversal can be associated to the micro-localization of plastic flow and the formation of microbands parallel to the plane of the newly most active slip system [17]. These microbands cut through the ‘‘old’’ dislocation sheets and produce channels for easy transport of mobile dislocations. This leads to the progressive destruction of the pre-existing structure. Additionally, the reversal of the strain involves the remobilization of dislocations during the transient before they are annihilated by recovery or trapped by the structure having the new polarity. The release of dislocations contributes to sustain the imposed strain rate [42] without the need to produce new ones (no strain hardening). However, after a certain strain, in reverse mode, the prior substructure is expected to have no longer the ability to release enough dislocations to compensate for the dislocation storage in the newly formed substructure and the strain hardening activates again. The recrystallization kinetics relates to the stored energy and more precisely to the way in which this last is distributed in the material. The strain range over which the progressive dissolution of the prior deformation structure gains over the progressive build up of the new one (plateau in the flow curve) is expected to delay nucleation and consequently retard the recrystallization. The maximum mean substructure ‘‘dilution’’ is expected to happen at the end of the plateau (longest recrystallization times), Fig. 6. For higher reverse strains, the ‘‘old substructure’’ is no longer able to relax enough dislocations and the strain hardening activates again as a result of the dislocation accumulation in the ‘‘new substructure’’ and the recrystallization times decrease with the increase of the (reverse) strain, as happens for monotonic tests. During the above part of the discussion no interaction between recrystallization and precipitation has been considered. However, the stagnation of the recrystallization caused by the strain-induced precipitation [32] of Nb(C,N), Fig. 7(c), can be observed for all the applied
340
D. Jorge-Badiola, I. Gutierrez / Acta Materialia 52 (2004) 333–341
conditions. The time for the start of the stagnation, ts , has been plotted in Fig. 9 as a function of t0:5 and a potential dependence of the form m ts ¼ At0:5
ð11Þ
can be used to fit the experimental set of data, for each deformation temperature. This relation agrees with the usual equations relating the recrystallization and the strain-induced precipitation times [26,27,43–46] to the deformation conditions and the initial grain size, D0 : t0:5 ¼ A1 Da0 eb e_c f1 ðT Þ;
ð12Þ
ts ¼ A2 ed e_e f2 ðT Þ:
ð13Þ
For the present steel, the values of b ffi 3:4 (Fig. 6) and d ffi 1:5 were found to fit reasonably to the experimental results for monotonic tests [22]. The value for b is in the range of those reported by other authors [26,44] and d is slightly higher than the value (d ¼ 1) generally accepted [43]. However, the above-mentioned values lead to a b=d quotient similar to the value of m deduced from the fit of the Eq. (11) to the experimental data, see Fig. 9. This shows the reliability of the above equations. However, the main consequence of the data plot in Fig. 9 is the lack of effect of the strain path on the relation between ts and t0:5 . The recrystallization stagnation is a consequence of the competition between the driving pressure for recrystallization, FR , and the pinning pressure, FPin , caused by the strain-induced precipitation [32] of Nb(C,N) precipitates like those shown in Fig. 7(c). Different authors have developed physically based models to describe the precipitation kinetics [30] and the interaction between this and the softening mechanisms [25]. Most of these models take into account homogeneous precipitation on dislocations. Assuming the Zener equation correctly describes the pressure exerted by a particle distribution characterized by a volume fraction, fv , and a particle mean radius, r, the following equation can be
m = 0.67
180 160
m = 0.47 140
ts (s)
120 100 80 60
1100 ºC 1050 ºC 1100 ºC 1050 ºC
40 20
Monotonic Strain reversal
0 0
50
100
150
200
250
Fig. 9. Relation between ts and the time required to reach a 50% softening.
used to describe the net driving pressure for recrystallization: 1 3cfv DF ¼ lb2 q 2 r
ð14Þ
with c the interfacial energy. This equation replaces Eq. (2) when recrystallization interacts with precipitation. To note that the values of ts deduced from the curves are not the time for the start of the precipitation, but actually the time at which the net driving pressure for recrystallization cancels out. By using the formulation proposed by other authors [25], together with fv ¼ 43 pr3 N (N ¼ number of straininduced particles per unit volume), Eq. (14) can be rewritten as 1 DF ¼ lb2 q 4cpr2 Ntot ½1 expðHtÞ; 2
ð15Þ
where Ntot is the number of total precipitate nucleation density sites and H ¼ Zb expðDGn=kT Þ with Z the Zeldovich factor, b the atomic impingement rate [47], DGn the free energy for nucleation and k the Boltzmann constant. For nucleation on dislocations, Ntot has been expressed as 0.5q1:5 [30]. The driving pressure for recrystallization and the pinning pressure depend on the local dislocation density. Consequently, a variation of the local dislocation density due to a reversal of the strain will affect at the same time the local driving and the pinning pressures, which could explain the observed relation between t0:5 and ts being independent of the strain path.
5. Conclusions • The strain path has a clear effect on the static recrystallization kinetics after hot working. The total absolute strain is not the variable to be used to describe the recrystallization kinetics because small reverse strains produce a retardation of the recrystallization. • The longest recrystallization times are observed at the end of the ‘‘yield stress plateau’’ which is consistent with the highest ‘‘old structure’’ dissolution produced by the reversal taking place at this stage (onset of the strain hardening after the plateau). • Full softening is not reached for any one of the strain paths/temperatures selected. The recrystallization stagnation is due to the strain-induced precipitation of Nb carbonitrides. • The results have been discussed in terms of the available recrystallization and strain-induced precipitation models. The application of these models to describe the material behavior during the holding after a reversal of the strain is probably a rough approach from the point of view of the complex dislocation interactions taking place. However, within the frame of
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the present knowledge on the subject, reasonable results are obtained, allowing to predict the recrystallization kinetics trend. • The times ts correspond to the instant at which the driving pressure for recrystallization cancels out with the pinning pressure exerted by the strain-induced Nb(C,N) precipitates. Both ts and t0:5 relate to the local dislocation density. The relation between ts and t0:5 has been observed to be independent of the strain path.
Acknowledgements The authors would like to acknowledge the CICYT (Spain) for the financial support of this work. The authors are greatly indebted to M. Diaz-Fuentes for his assistance with the TEM work.
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