Effect of the Zener-Hollomon parameter on the dynamic recrystallization kinetics of Mg–Zn–Zr–Yb magnesium alloy

Computational Materials Science 166 (2019) 221–229

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Effect of the Zener-Hollomon parameter on the dynamic recrystallization kinetics of Mg–Zn–Zr–Yb magnesium alloy

T

⁎

Lu Lia,b, , Yu Wanga, Hao Lia, Wei Jianga, Tao Wanga, Cun-Cai Zhanga, Fang Wanga, Hamid Garmestanib a b

School of Materials and Energy, Southwest University, Chongqing 400715, PR China School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, USA

A R T I C LE I N FO

A B S T R A C T

Keywords: Mg–Zn–Zr–Yb alloy DRX kinetics Zener-Hollomon parameter Iso-fraction contour map DRX sensitive rate

The isothermal compression tests for as-homogenized Mg–5.8 Zn–0.5 Zr–1.0 Yb magnesium alloy were performed on a Gleeble-1500 thermo-mechanical simulator at the temperatures of 250, 300, 350 and 400 °C and the strain rates of 0.001, 0.01, 0.1 and 1 s−1. The progress of dynamic recrystallization (DRX) was modeled by the modified Avrami type equation and the DRX kinetics was expressed as a function of the Zener-Hollomon (Z) parameter. The effects of the Z parameter on the microstructure evolution, the DRX fraction, the DRX rate, and the DRX sensitive rate were analyzed. An iso-fraction contour map was proposed to clarify the relations among the Z parameter, DRX fraction and deformation degree. Based on the spacing between the contours, the partial DRX stage was divided into three successive sub-stages and each of them was characterized by the mechanism underlying DRX evolution. Moreover, the ZK60 (Yb-free) alloy was adopted for comparative analysis to elucidate the effect of Yb addition on the DRX kinetics. The results showed that the Z parameter exerted a fundamental influence on the DRX evolution. Considering the robustness and efficiency of DRX, the deformation regime with the lnZ value less than 24 was preferable. Furthermore, the Yb addition can promote the onset of DRX but the progress should be retarded to some extent compared with the Yb-free counterpart. These conclusions were verified by the microstructure observations.

1. Introduction Rare-earth containing magnesium alloys are becoming attractive as high-performance structural materials for lightweight applications due to their satisfactory strength and ductility enhancement imparted by microstructural modifications after appropriate hot processing. Especially, in recent years, the studies on the thermal deformation behavior of Mg–Zn–Zr–Yb series alloys have considerably increased as they offer superior mechanical properties after hot processing. Yu [1] revealed that Yb had an obvious effect on refining microstructures of ZK60 alloys, and the extruded ZK60–2.0 Yb (wt%) alloy exhibited a tensile strength as high as ∼420 MPa with a low ductility of ∼3%. They claimed that the substantial strengthening in the as-extruded state was attributed to the fine microstructure induced by the incomplete dynamic recrystallization during extrusion. Moreover, there was a contrary tendency between the ultimate tensile stress (UTS) and the elongation to failure with the increase of Yb addition. Hence, a small amount of Yb was alloyed into the base alloy to improve the ductility [2]. In our preceding studies [3]. the thermal deformation behavior and ⁎

processing maps of as-homogenized ZK60–1.0 Yb (wt%) alloy were systematically investigated. The dynamic recrystallized (DRX) domain of 280–350 °C and 0.15–1 s−1 with an average DRXed grain size of ∼8 μm, is distinguished from other available DRX domains of ZK60 magnesium alloys with and without rare earth elements addition. It is apparent that DRX behavior acts an essential role in determining the formability, microstructure and mechanical properties of the studied alloys. Therefore, it is of importance to conduct an in-depth investigation on the DRX kinetics of Yb–containing magnesium alloys. Dynamic recrystallization (DRX) is one of the most important microstructural evolution mechanisms during thermal processing of metallic alloys. With respect to the hot deformation of magnesium alloys, DRX is readily to occur due to the nature of hexagonal close-packed (HCP) crystal structure and lower stacking fault energy (SFE) on the basal planes [4]. Grain refinement can be achieved by DRX, which is favorable to the improvement of workability and mechanical properties. Thus, it is of the essence to develop a comprehensive understanding of DRX kinetics to determine the optimal hot processing parameters and to obtain the desirable microstructure. To date,

Corresponding author at: School of Materials and Energy, Southwest University, Chongqing 400715, PR China. E-mail address: [email protected] (L. Li).

https://doi.org/10.1016/j.commatsci.2019.05.015 Received 7 February 2019; Received in revised form 25 April 2019; Accepted 6 May 2019 Available online 14 May 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.

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modeling of the DRX kinetics in magnesium alloys has been widely reported. Xu [5,6] established the DRX kinetics of as-cast AZ61 and AZ91D alloys by thermal deformation testing. Quan [7] studied the influence of deformation conditions on the DRX fraction of as-cast AZ80 magnesium alloy based on the DRX model. Lv [8,9] compared the DRX characteristics of Mg–2.0 Zn–0.3 Zr–0.2 Y and Mg–2.0 Zn–0.3 Zr–0.9 Y alloys by a series of isothermal upsetting experiments. These attempts mainly based on the stress-strain data under different deformation conditions to fit the Avrami type DRX model. However, the DRX kinetics and specific identification of processing parameter effects on the DRX behavior of Mg–Zn–Zr–Yb alloys have been scarcely revealed. In order to address these issues, the DRX kinetics model of ashomogenized Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy was established by true stress-strain curves through hot compression testing. The ZenerHollomon parameter, which is a temperature compensated strain rate factor, has been used to give a simplified description of the specific processing condition. The primary focus of this study is to develop a thorough understanding of the effect of the Z parameter on the DRX characteristics of the studied alloy based on the achieved DRX model and shed a light on the influence of Yb addition on the DRX kinetics compared with Yb-free counterpart. To the best of the authors’ knowledge, this is the first time a detailed investigation of the effect of the Z parameter on the DRX characteristics of the Mg–Zn–Zr–Yb series alloys and the analysis of Yb addition on dynamic recrystallization based on the kinetics model. These findings will offer insight into the accurate prediction and effective regulation of the DRX behavior in the newly developed Mg–Zn–Zr–Yb alloys during thermal deformation, which is quite essential for optimizing the hot-working processes to refine the grains, and thus for better developing high-performance Mg alloys with excellent microstructure and properties.

Fig. 1. True stress-strain curves of Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy at different deformation conditions.

to the stacking fault energy of the magnesium is comparatively low, the DRV caused by dislocation cross-slip and climb is insufficient to balance the WH. Hence, the flow stress rapidly increases. Once the accumulated dislocation density exceeds a critical strain the dynamic recrystallization occurs, which could reduce the WH rate by the annihilation of dislocation during the nucleation and growth of DRXed grains. After reaching the peak stress, the obvious dynamic softening is induced by dynamic recrystallization [12]. Finally, when the WH and dynamic softening (DRX and DRV) reach a dynamic balance, a steady-state flow behavior can be achieved [10,13]. As shown in Fig. 1, the numerical value lies behind each curve representing the corresponding value of the Z parameter. It is evident that the flow stress is highly influenced by deformation temperature, strain rate, and the Z parameter. The flow stress presents an ascending tendency with increasing strain rate as well as a decreasing trend with increasing deformation temperature, and thus an ascending tendency with increasing Z value. Moreover, the peak stress σp and the corresponding strain εp increase with increasing Z value from 21.4 (400 °C/ 0.001 s−1) to 36.4 (250 °C/1 s−1). In General, the ratio of the critical strain for the onset of DRX εc to the peak strain εp, (εc/εp =wc ), is a constant with the value between 0 and 1 for a given alloy deformed within a given temperature and strain rate range [14]. Therefore, εc also increases with rising Z value, which indicates that the DRX will be readily occurred under the deformation conditions with lower Z values. The main reason for this phenomenon is that higher deformation temperatures increase the driving force and the mobility of grain boundaries, while lower strain rates provide sufficient time for dislocation rearrangement and energy accumulation for DRX [15].

2. Experimental method The studied alloy with a nominal chemical composition of Mg–5.8 Zn–0.5 Zr–1.0 Yb (wt%) was melted in a crucible furnace using Mg–30% Zr, Mg–15% Yb, pure Zn, pure Mg and protected by a mixed atmosphere of SF6 (10 vol%) and CO2 (bal.). The as-cast ingots were heat-treated at 380 °C for 12 h and then machined into hot compression specimens of ϕ 10 mm × 15 mm. The detailed information about the starting microstructure, please refer to our preceding work[2]. Isothermal compression tests were performed on a Gleeble–1500 thermalmechanical simulator at the temperatures of 250, 300, 350 and 400 °C and the strain rates of 0.001, 0.01, 0.1 and 1 s−1. The initial height was reduced to a true strain of 0.69. After compression, the specimens were immediately cooled to room temperature to freeze the microstructure and then sectioned parallel to the compression axis for microstructure characterization. The microstructures were observed by Olympus optical microscopy (OM) and scanning electron microscopy (SEM, Sirion 200). More than 20 metallographic images with high magnification in the typical region were captured continuously at each sample to accurately measure the size of the fine DRX grains. And the average gain size was evaluated by using the linear intercept procedure according to ASTM E112 [10].

3.2. The kinetics of dynamic recrystallization 3.2.1. The models for peak strain and critical strain The work-hardening rate θ (θ = dσ /dε ) is widely used for the calculation of the critical strain εc , the peak stress σp , the saturated stress σsat , and the steady-state stress σss . The schematic of the relation between work-hardening rate and flow stress is shown in Fig. 2. A typical θ − σ curve is consisted with three segments, which reflects the distinct stages of microstructural evolution during the thermal deformation. As shown in Fig. 2, the first stage can be represented by an approximately linear segment, where the work hardening rate θ is positive but decreases steeply to the value of θ (σc ). The saturated stress σsat is readily achieved in this stage by making extrapolation (as illustrated in Fig. 2, go along the orange dotted line) to the value of θ = 0, which suggests a balance between the work hardening (WH) and dynamic recovery (DRV). At the second stage, from the critical stress σc to the peak stress

3. Results and discussion 3.1. Flow behavior and hot deformation mechanism The true stress-strain curves of Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy at designed temperatures and strain rates are shown in Fig. 1. It is obvious that the flow stress increases with increasing strain firstly and then decreases to a relatively steady state gradually, which indicative of the competing deformation mechanism of work hardening (WH), dynamic recovery (DRV), and dynamic recrystallization (DRX). At the incipient deformation stage, the continual increase and rapid accumulation of dislocation results in the obvious work hardening [11]. Meanwhile, due 222

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minimum point. Taking 250 °C and 300 °C for example, the relations are shown in Fig. 4(a) and (b), respectively. The point gives a good estimate of the critical stress and thus the corresponding critical strain can be obtained from the original flow curve [17]. The values of εc, εp, and εc/ εp at different deformation regimes are achieved and listed in Table 1. It is clear that the values of εc and εp increase with increasing strain rate and decreasing deformation temperature. Fig. 5(a) and Table 1 indicate the relation between εc and εp , indicating they follow a linear pattern, and it is summarized that the average value of εc/εp = 0.498. As can be used as an effective estimate for the onset of DRX, the ratio of εc/εp for different alloys have been widely reported in literature [18,19]. It is common that the critical strain εc accounts for about 80% of the peak strain σp , but it is noted that for the studied alloy, the ratio ranging from 0.446 to 0.553, which is smaller than that value of 0.8. It reveals that the DRX may occur at a smaller strain [20], which well agrees with the results below. The critical and peak strain are often represented as a function of the Zener-Hollomon parameter [21]. In our previous research [3], the formula of the Z parameter for the studied Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy has been determined. It can be described as follows:

Fig. 2. Schematic of the dependence of work-hardening rate on stress.

σp , the value of work-hardening rate is still positive but with a lower slope, and thus the curve decreases more slowly, which indicates that the thermal softening mainly induced by DRV and DRX becomes more and more predominant, then eventually exceeding WH. The third stage begins from the peak stress σp to the steady state stress σss , where the θ value is negative because the softening effect induced by DRX counteracts WH. So, another dynamic balance is achieved. The variations of strain hardening rate θ with flow stress σ of the studied Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy at evaluated conditions are illustrated in Fig. 3. It is obvious that these curves show good agreement with the above-mentioned features of the schematic. According to the calculation approach for the critical conditions proposed by Poliak and Jonas[16], the first derivative of strain hardening curve as a function of stress (i.e., −∂θ / ∂σ vs σ ) has a unique

158495 ⎞ Z = ε ̇ exp ⎛ ⎝ RT ⎠

(1) −1

where ε ̇ is the strain rate (s ); T is the deformation temperature (K); R is the gas constant (8.314 J/mol K). The average value of the activation energy for deformation, Q , for the studied alloy is obtained as 158.495 kJ/mol. Therefore, the specific processing condition in terms of strain rate and deformation temperature can be obtained by Eq. (1). Given a constant initial grain size, the critical and peak strains are usually expressed as a power-law function of the Z parameter, in the form of Eq. (2)

ε = AZ n

Fig. 3. The variation of strain hardening rate θ with flow stress σ at different deformation conditions: (a) 250 °C; (b) 300 °C; (c) 350 °C; (d) 400 °C. 223

(2)

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Fig. 4. The dependence of −∂θ / ∂σ on true stress σ at deformation conditions: (a) 250 °C and (b) 300 °C.

ε0.5 = 0.047Z 0.068

Table 1 The values of εc, εp , and εc, εp at different deformation regimes. Temperature/°C

ε /̇ s−1

εc

εp

εc/ εp

250

0.001 0.01 0.1 1

0.102 0.134 0.145 0.148

0.228 0.282 0.290 0.309

0.446 0.474 0.498 0.479

300

0.001 0.01 0.1 1

0.086 0.101 0.121 0.134

0.179 0.207 0.238 0.284

0.481 0.490 0.508 0.471

0.001 0.01 0.1 1

0.040 0.053 0.091 0.118

0.073 0.105 0.182 0.232

0.540 0.500 0.503 0.510

0.001 0.01 0.1 1

0.025 0.050 0.078 0.097

0.045 0.095 0.154 0.201

0.553 0.525 0.506 0.480

350

400

(7)

According to Eq. (6) and Eq. (7), kd and nd at different deformation regimes could be obtained by regression analysis. The mean values of the kd and nd can be calculated as kd = 1.342 and nd = 2.406, respectively. Thus, the DRX kinetics model of Mg–5.8 Zn–0.5 Zr–1.0 Yb alloy can be expressed as follows: 2.406

⎛ ε − εc ⎞ XDRX = 1 − exp ⎡ ⎢−1.342 ε0.5 ⎝ ⎠ ⎣ ⎜

= 1 − exp ⎡−1.342 ⎛ ⎢ ⎝ ⎣ ⎜

⎟

⎤ ⎥ ⎦ 2.406

ε − 0.003Z 0.115 ⎞ 0.047Z 0.068 ⎠ ⎟

⎤ ⎥ ⎦

3.2.3. DRX grain size As the dynamically recrystallized (DRXed) grain size under steady state can be determined by the Zener-Hollomon parameter [25], the relation between DRXed grain size and the Z parameter can be written as follows [26,27]:

dDRX = CZ β where A and n are material constants. The linear relations of lnεc − lnZ and lnεp − lnZ are presented in Fig. 5(b). With the estimation of the values of A and n, a similar type of equations are obtained for both peak and critical strains:

εp = 0.006Z 0.116

(3)

εc = 0.003Z 0.115

(4)

(9)

where C and β are material constants; dDRX is donated as the average grain size when fully DRX occurs which was obtained by metallographic measurement. The relation of lndDRX and lnZ is shown in Fig. 5(d). By linear regression fitting, the material constants are calculated as β = 0.064, C = 57.740. Therefore, the DRX grain size, dDRX as a function of the Z parameter can be written as Eq. (10). It is obvious that the higher Z value induced the finer dynamic recrystallized grains. In other words, the size of DRXed grains reduces with increasing strain rate and decreasing deformation temperature.

3.2.2. DRX fraction Generally, the dynamic recrystallization fraction, XDRX, at different deformation conditions can be estimated by the decrease of the flow stress, which is often written as [22]:

XDRX =

(8)

dDRX = 57.740Z 0.064

(10)

3.3. Microstructure evolution with the Zener-Hollomon parameter

σp − σ σp − σss

It is well known that the Zener-Hollomon parameter acts as a critical indicator of microstructure evolution under specific deformation condition. So the Z parameter and the value level were adopted to investigate the influence on the DRX behaviors. Fig. 6 depicts the evolution of compressed microstructure with different Z-value levels. When compressed at a relatively high Z value condition (as shown in Fig. 6(a) and (b)), corresponding to the deformation condition of 300 °C/1 s−1 and 300 °C/0.1 s−1, the alloy exhibits profuse twins and coarse original grains surrounded by fine DRXed grains, indicating a typical necklace structure. The deformation regime with a lower deformation temperature and higher strain rate proceeds the DRX process with a much higher nucleation rate and lower growth rate. The continuous growth of DRXed grains is limited because there is insufficient

(5)

In addition, the Kinetic model of DRX can be expressed by the modified Avrami equation as Eq. (6) [13,23,24]: n

ε − εc ⎞ d ⎞ XDRX = 1 − exp ⎜⎛−k d ⎛ ⎟ ⎝ ε0.5 ⎠ ⎠ ⎝ ⎜

⎟

(6)

where kd and nd are material constants. ε0.5 is the strain for 50% DRX fraction and can be determined from Eq. (5) by setting the XDRX value of 0.5. The variations of ε0.5 with different strain rates and deformation temperatures are listed in Table 2. Moreover, the lnε0.5–lnZ curve as demonstrated in Fig. 5(c) can be used to obtain the equation of ε0.5 as a function of the Z parameter. 224

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Fig. 5. The dependence of (a) critical strain on peak strain; And the influence of the Z parameter on (b) critical strain and peak strain, (c) the strain when achieves 50% DRX fraction, and (d) DRX grain size.

decreasing strain rate, the DRXed grains obtain sufficient time and energy to grow up freely, resulting in the highest average grain size and the DRX fraction. In addition, it is worth illustrating that the above-mentioned tendency is well consistent with the DRX kinetics curves in Section 3.4 and the underlying mechanism will be further discussed.

Table 2 The values of ε0.5 at different deformation conditions. ε /̇ s−1

0.001 0.01 0.1 1

T /°C 250

300

350

400

0.381 0.407 0.458 0.483

0.352 0.379 0.405 0.415

0.193 0.260 0.365 0.392

0.183 0.199 0.328 0.366

3.4. Effects of the Z parameters on DRX kinetics 3.4.1. Dependence of DRX fraction on the Z parameter Based on the established DRX kinetics model, the evolutions of the predicted DRX fractions with increasing strain (taking Z value as the index) are shown in Fig. 7(a), which all exhibit a typical S-shape. It is obvious that the DRX fraction increases with decreasing Z value at a fixed strain and gradually approaches the constant value of 1 with increasing strain, implying the alloy will be fully recrystallized undergoing sufficient deformation degree. Moreover, when deformed at a constant strain rate such as 1 s−1, as the bottom four curves shown in Fig. 7(a), it is seen that the increasing temperature promotes the Scurves steeper, which means that the recrystallization proceeds more rapidly. The higher temperatures can greatly promote the mobility of grain boundaries, resulting in more newborn nuclei to grow up faster. In contrast, when compressed at a fixed temperature for example 400 °C, as the top four curves shown in Fig. 7(a), it is evident that the deformation strain required for the same amount of DRX fraction increases with rising strain rate, which reveals that the DRX process is delayed. The lower strain rates could provide sufficient time and space for the DRX nuclei to grow up prior to the next round of recrystallized nuclei generated [28]. These results well agree with the metallographic observation in Fig. 6. Hence, when compressed at the deformation regimes with lower Z values, the alloy tends to achieve a higher DRX fraction with less deformation strain.

time for the original nuclei to grow up before the newborn nuclei formed at grain boundaries. Therefore, the smallest average size of DRXed grains is achieved despite the lowest DRX fraction of ∼70% observed. Moreover, lack of enough deformation degree further results in the characteristic of incomplete DRX at the deformation condition with a relatively high Z value. When compressed at a medium Z value condition (as shown in Fig. 6(c) and (d)), corresponding to the deformation condition of 350 °C/0.1 s−1 and 350 °C/0.01 s−1, many serrated and long striped grains can be observed, revealing further progress in DRX. With the rise of deformation temperature and the drop of strain rate, the mobility of grain boundary dislocations is improved due to the enhanced atomic diffusion and the longer deformation time resulting the growth rate of grains is comparable with the nucleation rate. Hence, the DRX fraction and the average size of DRXed grains are obviously increased. When the compression performed under a relatively low Z value condition (as shown in Fig. 6(e) and (f)), corresponding to the deformation condition of 400 °C/0.01 s−1 and 400 °C/0.001 s−1, an equiaxed and coarsened grain structure can be found. It implies that the state of fully DRX is achieved and the nucleation rate is lower than the growth rate of grains. With increasing deformation temperature and 225

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Fig. 6. Microstructure of specimens deformed at the strain of 0.6 at deformation regimes with different levels of Z values.

bottom four curves shown in Fig. 7(b), the DRX rate increases with decreasing Z value, indicating more grains would undergo DRX at the same increment of strain and lower Z values. In addition, when deformed at the same temperature, the effect is very analogous to the strain rate. Hence, at the deformation condition of 400 °C/0.001 s−1 corresponding to the lowest Z value in the studied deformation regimes, the alloy exhibits the highest DRX rate during whole DRX process.

3.4.2. Dependence of DRX rate on the Z parameter The DRX rate was defined to feature the change of the DRX fraction with the deformation degree and the expression gives:

υDRX =

∂XDRX ∂ε

(11)

Then taking the first partial derivative of Eq. (6) with respect to the deformation strain, the equation is changed as follows: nd − 1

n

υDRX =

∂XDRX k n ε − εc ⎞ d ⎤ ⎛ ε − εc ⎞ = d d ·exp ⎡−k d ⎛ · ⎢ ∂ε ε0.5 ⎝ ε0.5 ⎠ ⎥ ⎣ ⎦ ⎝ ε0.5 ⎠ ⎜

⎟

⎜

3.4.3. Relations among the Z parameter, DRX fraction and deformation degree It is well known that when the full recrystallization is achieved under a small deformation degree, the risk of damage accumulation, crack initiation, and even fracture induced by large strain may be effectively avoided. In addition, according to the aforementioned analysis, the alloy tends to achieve a higher DRX fraction when deformed at lower Z values at the same deformation degree. Therefore, it is essential to investigate the relations among the Z parameter, DRX fraction and deformation degree.

⎟

(12)

Fig. 7(b) shows the dependence of DRX rates on the DRX fraction throughout the whole DRX process (taking Z value as the index), which all present a typical ‘slow–rapid–slow’ characteristic. The υDRX increases rapidly at the initial deformation. After approaching the max value, the υDRX decreases gradually with more and more matrix transformed [29]. Moreover, the deformation condition (represented by the Z parameter) has a pronounced effect on the DRX rate. At the same strain rate, as the 226

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Fig. 7. The influence of the Z parameters on (a) the DRX fraction, and (b) the DRX rate.

spacing increases with rising Z values), which reveals the onset of DRX is sensitive to the deformation condition. When deformed at lower Z values corresponding to the lower strain rates and higher temperatures, the alloy is readily to trigger DRX due to the decreasing critical strain, resulting the spacing is narrower than those with higher Z values. This phenomenon is mainly attributed to the compound effects of the deformation temperature, the strain rate, and the local dislocation density. With decreasing deformation temperature, the dislocation slip and climb are impeded, resulting in an inherent difficulty in sub-grain formation and merger, which makes an increase in critical strain value and thus more strain required for the onset of DRX. On the other hand, the number of DRX nucleation sites, the DRXed grain growth rate, and the softening effect induced by DRX are also adversely impacted by increasing strain rate due to the insufficient time for dislocation moving, which results in the increase of the critical strain traded off by DRX softening and strain hardening. Moreover, the “adiabatic temperature rise effect” will further improve the dynamic recovery and reduce the local dislocation density, leading to a reduction of the deformation storage energy, which further hampers the occurrence of DRX by increasing the critical strain. The characteristic of the subsequent continuous transformation substage (10%–90%) is that the spacing of the contours decreases firstly then followed by a gradual increase with rising strain at a constant Z value, which is consistent with the feature of S-curves. With the increase of strain, DRX rate continually elevates and then gradually drops until fully DRX occurs, which leads to the variation of spacing in this stage fundamentally. It is worth noting that the contour spacings at the deformation conditions with higher Z values are much larger than that of lower ones, which reveals that the DRX is postponed with strain rate increasing or temperature decreasing. It may be attributed to the fact that when deformed at a higher strain rate, the density of dislocation induced in a very short time is much larger than that required for DRX, so the pining effect dominates by surplus dislocations on the movable boundaries [30]. Moreover, decreasing temperature results in lower diffusivity, which in general constrains grain boundaries to be mobile, thus delaying DRX effect. Hence, the deformation conditions with lower Z values are favorable to the structural preparation for the DRX progress in this stage. The last delayed finishing sub-stage with the XDRX ranging of 90–100% exhibits a much larger spacing, which implies that a larger amount of deformation is required for the alloy to achieve the fully recrystallized state. This may stem from the fact that the stored energy accumulated in the hardening, which is considered as the driving force for DRX, has been almost consumed by preceding DRX nucleation and

Fig. 8. The iso-fraction contour map of the parameter XDRX to depict the relations among the Z parameter, DRX fraction and deformation degree.

As shown in Fig. 8, the variation of DRX fraction with strain and the Z parameter can be plotted as an iso-fraction contour map of the parameter XDRX. It should be illustrated that a smoothing procedure was used to describe the overall tendency, thus some specific values in this map may slightly deviate with those in Fig. 7. The numbers against each contour represent the DRX fractions as percent. Three regions including the un-dynamically recrystallized (un-DRXed) region (colored by red shade), the partial DRX region (with two borders highlighted by thick lines), and the fully DRXed region (colored by green shade) suggest the successive stages of DRX with increasing XDRX from 0 to 100%. Additionally, the spacing between the contours reflects the difficult degree when 10% of DRX fraction increases. Therefore, a larger spacing means that at the same processing condition, more deformation degree is required. This difference is associated with the specific stage of DRX development in terms of the average DRXed grain size, dislocation density, and grain growth rate at that moment. In this study, the partial DRX stage (as shown in Fig. 8) was divided into three successive sub-stages according to the spacing of the contours to elucidate the DRX evolution. The first initial onset sub-stage with the XDRX ranging of 0–10% exhibits a narrow inverted triangle (i.e., the 227

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growth [31]. In addition, due to the primary softening effects induced by DRX and DRV, the dislocation density decreases extensively and thus the increment of strain energy is considerably reduced in this stage. In a word, the lack of energy for trigging DRX in the remaining unrecrystallized grains leads to the increased requirement for the deformation degree. But it is interesting to note that based on the present calculation, the full recrystallization (XDRX = 100%) can be obtained at the strain value of about 1 even deformed at a high lnZ value of 36, which suggests that the alloy exhibits a high feasibility of DRX as well as an overall good hot workability under the studied deformation conditions.

3.4.4. Dependence of ∂XDRX / ∂ ln Z on the Z parameter Generally, the industrial hot metal forming processes, such as hot forging, hot extrusion, and hot rolling are usually performed in regimes with varying Z values. The manipulation of keeping constant deformation temperature and strain rate is cost consuming because a number of factors can lead to the change in both of them, especially the deformation temperature. According to the discussion in Section 3.4.3, the Z parameter plays a fundamental role in the DRX fraction, which directly affects the DRX completion rate. Therefore, it is essential to understand the change of the DRX fraction with varying deformation conditions. The DRX sensitive rate (SRDRX) was defined and the expression gives:

SRDRX =

∂XDRX ∂ ln Z

Fig. 10. The dependences of εc and ε0.5 on the Z parameter of ZK60 [32] and ZK60–1.0 Yb alloys.

3.5. Comparing with Yb-free alloy on DRX kinetics To further elucidate the effect of Yb addition on the DRX kinetics, the ZK60 (Yb-free) alloy was adopted for comparative analysis. Based on the Avrami-type equation, the critical strain εc and ε0.5 of the Mg–5.78 Zn–0.76 Zr alloy was modeled by Qin [32], which can be described as Eq. (14) and (15), respectively.

(13)

The dependence of DRX sensitive rate, ∂XDRX / ∂ ln Z , on the deformation condition, lnZ, at different strains is shown in Fig. 9. It is interesting to find that at different deformation degrees, the DRX sensitive rates exhibit a similar varying tendency and simultaneously achieve their minimum values when lnZ ≈ 26. Exclusive of the strain ranging from 0 to 0.2 when the DRX may be just occurred with insufficient DRX fraction to be compared with, the change of the DRX sensitive rate with varying deformation conditions is significantly decreased with increasing deformation degree from 0.3 to 0.6. The steep variation of ∂XDRX / ∂ ln Z on both sides of the lnZ value equal to 26, indicating the high sensitivity of the DRX fraction with the change of the deformation regime, especially at the low deformation degree. Therefore, by comprehensive consideration of the robustness, the DRX rate, and the onset of DRX, it is recommended to perform the hot working at the deformation condition with the lnZ value less than 24 for the studied alloy.

22571 ⎞ εc = 1.34 × 10−4d 00.4695 ε·0.1511exp ⎛ ⎝ RT ⎠

(14)

11157 ⎞ ε0.5 = 5.25 × 10−3d 00.4428 ε·0.0466exp ⎛ ⎝ RT ⎠

(15)

Based on the established kinetics, the relations of lnεc–lnZ and lnε0.5–lnZ of the alloys with and without Yb addition are shown in Fig. 10. It is found that both the εc and ε0.5 values of Yb-containing alloy are slightly lower than that of the counterpart, especially the latter. With the increase of Z value, the value of critical strain εc is very close. These results indicate that the Yb addition can promote the onset of DRX but the progress should be retarded to some extent. The reduction of the εc value may be attributed to the decrease of the stacking fault energy (SFE) induced by Yb addition. According to the literature [33], the SFE of Yb is less than 10 ergs/cm2. As well known, the DRX behaviors are intimately related to SFE in metal. In the case of high SFE, cross-slip can easily occur due to the small-extended dislocation width, which results in DRV over DRX [34]. In contrast, in the case of low SFE, dislocation density increases during deformation because of the difficulty of cross-slip and thus the restriction of DRV, leading to the promotion of DRX or SRX [35]. In addition, another promotion factor may be associated with the particle-stimulated nucleation (PSN) mechanism during straining, which can promote DRX by producing local high-density dislocations. However, according to the literature [36–38], the particle with a diameter larger than 1 μm can be regarded as the ideal site for PSN. As the coarse particles are not so predominant in the micrographs, the effect of PSN should be further ascertained. Moreover, it is worth illustrating that the lower εc values of Yb-containing alloy at deformation regimes with lower Z values are due to the fact that the higher deformation temperature leads to an increase in mobility of grain boundaries, thus allowing DRX to occur at smaller strains [39]. On the other hand, the suppression effect on the DRX kinetics may be related to the extensive existence of fine precipitates (as shown in Fig. 6) before and during deformation [1,40]. With a high density of dispersed fine precipitates, dislocations induced by straining are pinned (Zener drag), which results in hindering dislocation rearranged to

Fig. 9. The influence of the Z parameter on the DRX sensitive rate with different strains. 228

Computational Materials Science 166 (2019) 221–229

L. Li, et al.

References

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4. Conclusion (1) The Zener-Hollomon parameter has a fundamental influence on the DRX kinetics. With decreasing Z value, the alloy is readily to trigger DRX due to the decrease of critical strain and tends to achieve a higher DRX rate and fraction with less deformation degree. These results are in good agreement with the metallographic observation with different levels of Z values. (2) The relations among the Z parameter, DRX fraction and deformation degree can be depicted by an iso-fraction contour map of the parameter XDRX. In view of DRX evolution, the partial DRX stage can be divided into three successive sub-stages: initial onset stage, continuous transformation stage and delayed finishing stage, according to the spacing of the contours. (3) Although the evolutions of DRX sensitive rates during straining exhibit a similar varying tendency at different deformation regimes, the change of the DRX sensitive rate with varying deformation conditions is significantly decreased with increasing deformation degree from 0.3 to 0.6. Considering the robustness and efficiency of DRX, hot processing with the lnZ value less than 24 is preferable for the studied alloy. (4) The addition of Yb plays an important role in adjusting the DRX kinetics of the Mg–Zn–Zr–Yb alloy by decreasing SFE and increasing the density of fine precipitates. Thus, Yb addition can promote the onset of DRX but the progress should be retarded to some extent compared with the Yb-free counterpart. Funding This work was supported by the National Natural Science Foundation of China (grant number 51605392). Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Declaration of Competing Interest The authors declare that there is no conflict of interest.

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