- Email: [email protected]
Modeling of flow stress for magnesium alloy during hot deformation
Materials Science and Engineering A 527 (2010) 2790–2797
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Modeling of flow stress for magnesium alloy during hot deformation Yin-Jiang Qin ∗ , Qing-Lin Pan, Yun-Bin He, Wen-Bin Li, Xiao-Yan Liu, Xi Fan School of Materials Science and Engineering, Central South University, Changsha, 410083, China
a r t i c l e
i n f o
Article history: Received 5 August 2009 Received in revised form 29 December 2009 Accepted 11 January 2010
Keywords: Flow stress Dynamic recrystallization Hot deformation Magnesium alloy ZK60
a b s t r a c t Based on the classical flow stress–dislocation density relation and kinetics of dynamic recrystallization (DRX), a model was developed to determine flow stress of magnesium alloy at hot deformation condition. The proposed model is capable of predicting the flow behavior of work hardening and dynamic recovery (DRV) region as well as the softening caused by DRX. To establish the model, the double-differentiation method was used to identify the critical strain for initiation of DRX, and the DRV parameter ˝ was evaluated from the work-hardening behavior prior to critical strain. The net softening attributable to DRX was defined as the difference between the DRV and experimental curves, and Avrami equation was employed to represent this softening behavior. The flow stress curves of ZK60 magnesium alloy predicted by the developed model are in good agreement with experimental results, which confirms the validity of the model. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Lightweight magnesium alloys have attracted significant interest in the last decade due to their potential applications for weight reduction in transportation vehicles. However, magnesium alloys have poor formability and limited ductility at room temperature ascribed to their hexagonal close-packed (hcp) crystal structure. As a result, to date, most of the magnesium products are fabricated by die casting process instead of employing plastic forming techniques such as rolling, forging, and extrusion. At elevated temperatures the workability of magnesium alloys substantially increases as additional slip systems, i.e., non-basal and c + a slip become sufficiently available by thermal activation [1]. Therefore, it is necessary to investigate the hot deformation behavior of magnesium alloys. The flow behavior of metals and alloys at hot deformation condition has a great importance for designer of metal forming processes because of its effective role on the required deformation energy as well as the kinetics of metallurgical transformation such as dynamic recovery (DRV) and dynamic recrystallization (DRX). However, the flow behavior is influenced by many factors such as strain, strain rate, deformation temperature, etc. and understanding their effects is a difficult task due to their complex nature. Recently, the fast development of computing techniques led to a wide application of finite element method (FEM) simulation to study materials forming processes [2–4]. Nevertheless, numerical simulation can be truly reliable only when a proper flow stress
∗ Corresponding author. Tel.: +86 0731 88830933. E-mail address: [email protected] (Y.-J. Qin).
model is available. A number of models were proposed to describe the flow stress of metals and alloys and extensive summary on different models was given in [5–7]. In the case of magnesium and its alloys, a few of previous investigations were performed to study the flow stress behavior. Frost and Ashby [8] reported that pure magnesium abided by the power law creep equation above 573 K. Galiyev et al. [9,10] found that magnesium alloy ZK60 complied with the creep equation. Based on an observation of a linear relationship between semi-log Zener-Hollomon parameter and proof stress, Takuda et al. [11,12] proposed a parametrical method to express the proof flow stress in the AZ31 and AZ91 tensile tests. Similar constitutive relationship for AZ31B magnesium alloy was determined and validated through comparison between simulated and real extrusion by Li et al. [13]. Sheng and Shivpuri [14] proposed an analytical method, which reflected temperature, strain and strain rate effect by introducing temperature-compensated strain rate, and the model was well applied on three published experimental data. Liu et al. [15] put forward a new model of flow stress characterizing DRX for magnesium alloy AZ31B. Owing to the relative low stacking fault energy (SFE) and the lack of easily activated slip systems, DRX plays an important role during hot deformation of magnesium alloy [16]. However, almost all the flow stress models aforementioned, except the one proposed by Liu et al. in 2008 are empirical and do not take into account the effect of DRX. Since the DRX is a thermally activated process, the recrystallized volume fraction can be regarded as a function of strain through Avrami equation. Furthermore, the descending of flow stress during hot deformation is mainly dominated by the recrystallized volume fraction. Based on this idea, the dislocation model developed by Bergstrom [17,18] together with Avrami-type
0921-5093/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.01.035
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2791
Fig. 1. True stress–strain curves for ZK60 Mg alloy at various temperatures with strain rates of (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 and (d) 1 s−1 .
recrystallization kinetics equation are used to describe the flow behavior of magnesium alloy during hot deformation in this study.
2. Experiments The material used in the present study was commercial grade magnesium alloy ZK60 (Mg–5.78%Zn–0.76%Zr). The alloy was fabricated by semi-continuous casting and homogenized at 673 K for 12 h prior to deformation. The initial grain size was about 100 m. Cylindrical specimens with 10 mm in diameter and 15 mm in height were machined from the homogenized materials for compression tests. Uniaxial compression tests were conducted in the temperature range of 473–673 K at intervals of 50 K and constant true strain rate ranging from 0.001 to 1 s−1 at intervals of an order of magnitude on a Gleeble-1500 thermal simulator up to 60% of height reduction. The Gleeble compression system works with a servo-hydraulic mechanism, which deforms the sample heated electrically by its Ohmic resistance at a constant strain rate. The temperature was controlled and measured by a thermocouple welded to the sample. Graphite foils and colloidal graphite were used as lubrication between the sample and the anvils. All the specimens were heated at 10 K/s up to deformation temperatures, held 3 min to homogenize the temperature in the sample, and then deformed and water quenched. The load-stroke data recorded for every compression tests were converted into true stress–true plastic strain curves using standard equations. The true stress–true strain curves of magnesium alloy ZK60 at different strain rates are shown in Fig. 1, according to which the characteristics of flow stress curves are represented as follows: • The overall level of flow stress curve increases with the decrease of deformation temperature and the increase of strain rate.
• In the initial stage of the deformation, hardening rate is higher than the softening rate and thus the stress increases abruptly, then the increasing rate is decreased due to the occurrence of DRV and DRX. When the hardening rate is equal to the softening rate the flow stress peak is reached. • After the stress peak, the softening induced by DRX exceeds the hardening and the stress drops steeply. The stress becomes steady when a new balance between softening and hardening is obtained. 3. Modeling of flow stress In the present approach, the experimental curves are considered to be the net result of the simultaneous operation of DRV and DRX in the manner typified by the schematic curves of Fig. 2. Here the uppermost curve ( drv ) is regarded as resulting from the operation of DRV alone, i.e., in the absence of DRX. Although not obtainable directly from experiment, it represents the assumed work-hardening behavior of the unrecrystallization regions, e.g., in the interiors of the grains as “necklacing” proceeds inwards. This can be derived from the work-hardening behavior prior to critical strain (εc ) for initiation of DRX, as will be demonstrated later. After the critical strain εc , it leads to more and more softening and is responsible for the difference between the drv curve and DRX curve ( drx ). At saturation, the value of the asymptotic stress is given by sat . On continued straining, sat represents the dislocation density in the most work-hardened grains therefore the driving force for the continuation of DRX. 3.1. Modeling of dynamic recovery During deformation, the dependence of the dislocation density on plastic strain ε is generally considered to be given by the
2792
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
often expressed in the following form [21]: ε0.5 = A0.5 d0m0.5 ε˙ n0.5 exp
Q
DRX
(7)
RT
where A0.5 , m0.5 and n0.5 are materials constants, d0 is the initial grain size (m), QDRX is the DRX activation energy (kJ/mol), R is the universal gas constant (8.314 J mol−1 K−1 ) and T is the absolute temperature (K). According to Fig. 2, it is obvious that the difference between the drv and drx curve can be represented as , which is the net softening directly attributable to DRX. The maximum value of is sat – ss , where ss is the steady state stress under DRX conditions. Then, the fractional softening due to DRX is expressed as X=
Fig. 2. Schematic diagram illustrating the work-hardening curve drv together with a typical experimental DRX flow stress curve.
following equation [19]: d = U − ˝ dε
(1)
Here, U represents the work hardening, which is a multiplication term and can be regarded as constant with respect to strain. The term ˝ is the contribution due to DRV through dislocation annihilation and rearrangement, and is often called the coefficient of DRV. By employing the initial condition = 0 at ε = 0, the differential Eq. (1) can be easily solved as = 0 exp(−˝ε) +
U ˝
[1 − exp(−˝ε)]
(2)
Using the classic relationship between stress and dislocation den√ sity = Gb [20], where ␥ is a materials constant, b is the distance between atoms in the slip direction and G is the shear modulus. Then the flow stress can be given by the following expression in terms of strain: 2 = 02 exp(−˝ε) + (Gb)
2
U ˝
[1 − exp(−˝ε)]
(3)
where 0 is the yield stress. When strain ε tends to infinity, i.e., in the steady state condition, the DRV saturation stress sat can be obtained from Eq. (3) as follows:
sat = Gb
U/˝
− drx = drv sat − ss sat − ss
(8)
by which the DRX volume fraction under different deformation conditions can be measured directly from the curves which are similar to Fig. 2. This method is particularly useful when the microstructure is not stable, so that metallographic measurements are difficult to carry out. Combining Eq. (8) with Eq. (6), the flow stress during DRX period under hot deformation can be given by the following expression:
drx = drv − (sar − ss ) 1 − exp −ˇd
ε − ε nd c
ε0.5
(ε > εc ) (9)
3.3. Determination of c , sat , 0 and ss In order to determine the critical strain εc for initiation of DRX, the double-differentiation method proposed by Poliak and Jonas [22] is employed in the present investigation. Firstly, the flow curves shown in Fig. 1 are fitted and smoothed with a seventh-order polynomial in the region of plastic deformation that encompasses the stress peak using the MatlabTM software. (In some problematic cases, a higher order polynomial is employed.) Then, the workis calculated and plotted against hardening rate, i.e., = d/dε|ε,T ˙ the true stress. The Poliak method defines εc as the point at which the second derivative of the work-hardening rate with respect to stress (∂2 /∂ 2 ) is zero, i.e., the inflection point of – curve. The saturation stress sat , in turn, is defined by the extrapolation of the – plot to = 0 (using only the linear portion of the curve relating to stress values just below critical stress c ). As can be seen, it is
(4)
Then the variation of flow stress in the work hardening and DRV region during hot deformation can be expressed as follows: 2 2 drv = [sat − (sat − 02 )exp(−˝ε)]
1/2
(ε < εc )
(5)
3.2. Modeling of dynamic recrystallization When DRX takes place, the simultaneous use of the above – relation and the dynamically recrystallized fraction makes it possible to predict the flow stress after the stress peak. In the present approach, a standard Avrami-type equation was used to describe the DRX kinetics and it can be expressed as [3]:
X = 1 − exp −ˇd
ε − ε nd c
ε0.5
(6)
where X is the recrystallized volume fraction, ˇd and nd are DRX parameters depending on chemical composition and hot deformation conditions. ε0.5 is the strain for 50% recrystallization, which is
Fig. 3. Flow stress dependence of the strain-hardening rate for ZK60 Mg alloy with strain rate of 0.1 s−1 .
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2793
A˝ , n˝ and m˝ are constants. By regression analysis, ˝ can be expressed as follows:
22895
˝ = 1.24 × 104 d0−0.5849 ε˙ −0.1919 exp −
RT
(15)
It should be mentioned that in order to investigate the effect of initial grain size d0 on ˝, εc and ε0.5 (the latter two will be discussed later), the flow stress curves of magnesium alloy ZK60 tested by Yang et al. [25] were employed. In Yang’s study, the initial grain size is 50 m while the value is 100 m in the present study. For comparison purposes, the calculated value of ˝ is also depicted in Fig. 5 which shows that Eq. (15) represents the dependence of ˝ on strain rate and temperature very well. 3.5. Dependences of 0 , sat and ss on p It is generally accepted to use the hyperbolic-sine meliorated Arrhenius equation to model the temperature and strain rate dependence of peak stress p as follows [6]:
Fig. 4. Plot of versus 2 employed to determine the slope k.
the work-hardening behavior prior to c that characterizes sat . As an example, the – plots for magnesium alloy ZK60 with strain rate of 0.1 s−1 and different temperatures is illustrated in Fig. 3, in which the critical stress c and saturation stress sat are indicated by arrows and symbol of diamond, respectively. In each case, a “yield stress” 0 is identified on the flow stress curve in terms of a 2% offset in the total strain and the steady state stress ss is obtained directly from the experimental flow stress curve.
To determine the value of ˝, the method described by Jonas et al. [23] is employed and it is outlined briefly here. By differentiating Eq. (5), it can be shown that: d −1/2 2 2 2 = 0.5[sat − (sat − 02 )exp(−˝ε)] ˝(sat − 02 )exp(−˝ε) dε (10) Multiplying d/dε by leads to (11)
Meanwhile, Eq. (5) can be expressed as 2 2 (sat − 02 )exp(−˝ε) = sat − 2
(12)
−Q
(16)
RT
where A, ␣ and n are materials constants, Q is the deformation activation energy. When the flow stress is low, Eq. (16) can be simplified as [26]:
ε˙ = A pn exp
−Q
(17)
RT
While at high stress level, Eq. (16) becomes ε˙ = A exp(ˇp ) exp
3.4. Determination of ˝
d 2 = 0.5[˝(sat − 02 )exp(−˝ε)] dε
ε˙ = A[sinh(˛p )]n exp
−Q
(18)
RT
in which ˇ = ˛n . According to Eqs. (17) and (18), the approximate value of n and ˇ are determined by linear regression of the plots of ln ε˙ − p and ln ε˙ − ln p at different temperatures as 7.88 and 0.08, respectively. And then the suitable value of ˛ is determined to be 0.01. The activation energy Q is an important physical parameter indicating the plastic deformability, and can be calculated as follows [27]:
∂ ln ε˙ Q =R ∂ ln[sinh(˛p )]
T
∂ ln[sinh(˛p )] ∂(1/T )
(19) ε˙
The first term on the right-hand side of Eq. (19) refers to the slope of linear fitting of ln ε˙ versus ln[sinh(˛ p )] at different deformation temperatures (Fig. 6), and the second term refers to the slope of
So that Eq. (11) can now be written as d 2 − 0.5˝ 2 = 0.5˝sat dε
(13)
According to Eq. (13): k=
d() = −0.5˝ d( 2 )
(14)
in which k is the slope of the tangent to – 2 plot. And then the value of ˝ can be determined as −2k. An example of determination k is shown schematically in Fig. 4. Note that the slope k is also derived from the experimental data obtained prior to c and εc . A typical plot of the strain rate and deformation temperature dependence of ˝ is presented in Fig. 5 for ZK60 magnesium alloy. It is clear from this graph that increasing the strain rate or decreasing the temperature results in a decrease in ˝. This is consistent with ˝ being a measure of the ease of DRV. To quantify the effects of processing parameters on ˝, Yoshie et al. [24] used an empirical equation of the form ˝ = A˝ d0n˝ ε˙ m˝ exp(−Q˝ /RT ), where d0 is the initial grain size, Q˝ is an apparent activation energy (kJ/mol), and
Fig. 5. Dependence of ˝ on temperature and strain rate.
2794
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
Fig. 6. Relationship between peak stress and strain rate.
Fig. 8. Relationship between flow stress and Zener-Hollomon parameter.
linear fitting of ln[sinh(˛ p )] versus 1/T at different strain rates (Fig. 7). Then the mean activation energy Q of ZK60 magnesium alloy during hot deformation is determined to be 153.372 kJ/mol. The combined effect of strain rate and temperature on deformation behavior can be characterized by Zener-Hollomon (Z) parameter, the physical meaning of which is the so-called temperature-compensated strain rate parameter. It can be defined as [28]:
p can be described by Eq. (21) in which ˛ = 0.01, A = 3.3648 × 1011 , n = 6.88 and Z = ε˙ exp(153317/RT ). For modeling purposes, it is useful to be able to express the derived and measured quantities 0 , sat and ss as ratios of p , as the latter is more readily measurable. The dependences of these quantities for flow curves of ZK60 magnesium alloy on p are illustrated in Fig. 9. Overall, the following expressions give a reasonably good fit to the data:
Z = ε˙ exp
Q RT
= A[sinh(˛p )]n
(20)
Then, the peak stress p , can be written as a function of Z parameter, considering the definition of the hyperbolic law [29]: p =
Z 1/n 1
˛
A
+
2/n Z A
1/2
+1
(21)
Taking the natural logarithm of Eq. (20): ln Z = ln A + n ln[sinh(˛p )]
(22)
According to Eq. (22), the stress exponent n and constant A are determined by linear regression of ln Z − ln[sinh(˛ p )]) plot (Fig. 8) as 6.88 and 3.3648 × 1011 , respectively. And then, the peak stress
Fig. 7. Relationship between peak stress and deformation temperature.
0 = 0.36p , sat = 1.20p and ss = 0.62p
3.6. Determination of the parameters for DRX kinetics Based on experimental data and literature [30–31], it can be found that the critical strain εc and ε0.5 are function of initial grain ˙ and both of size d0 , deformation temperature T and strain rate ε, them can be described by equations which have the form similar to Eq. (7). In order to determine the unknown constants in these equations, regression analysis are performed, and the results are listed as follows: ε0.5 = 5.25 × 10−3 d00.4428 ε˙ 0.0466 exp
11157 RT
Fig. 9. Dependences of 0 , sat and ss on p .
(23)
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2795
ε˙ = 1 s−1 . Because when the deformation is very quick, the test time is too short to allow for heat transfer, then the specimen temperature increases [32]. This makes the flow stress curve deviate from the ideal isothermal condition. It should be noticed that there are some other factors such as twining that influence the hot deformation behavior [33–35]. However, the deformation temperature in the present work is relatively high and the predominant factor during the deformation process is DRX. Thus, other factors are not taken into account. 4.2. Microstructure evolution
Fig. 10. Determination of the constants ˇd and nd in Avrami equation.
εc = 1.34 × 10−4 d00.4695 ε˙ 0.1511 exp
22571 RT
(24)
To determine the DRX kinetics, plots of ln[−ln(1 − X)] versus ln[(ε − ε0 )/ε0.5 are used and the parameters ˇd and nd can be easily obtained from the slope and intercept of the fitting line. As an example, the plot for condition of ε˙ = 1 s−1 and T = 523 K is illustrated in Fig. 10. Through analysis of the results at other conditions, the values of nd are found to be independent of ε˙ and T and to be about 1.6, while ˇd is a function of ε˙ and T and can be described as ˇd = 6.92 × 10−2 ε˙ 0.0704 exp
12070 RT
Microstructure evolution of ZK60 alloy compressed at 573 K and strain rate of 0.1 s−1 was shown in Fig. 13. During the initiation of hot deformation, the original grain boundaries became wavy and corrugated, as shown in Fig. 13(a) and (b). The DRX process occurred on the boundaries of original grains. The recrystallized grains grew greatly through the migration of grain boundaries during the hot deformation process, as shown in Fig. 13(c). Fig. 14(a) shows that the tangled dislocations focused on the original grain boundaries at ε = 0.1. As a result, the recrystallized grains were firstly formed on the original grain boundaries. Such a process of new grains formation may be similar to that in conventional DRX in cubic metals, i.e., the bulging out of part of the serrated grain boundaries and the migration of grain boundaries for growth of DRX grains [36]. As further deformation was carried out, a large amount of DRX grains were observed, as shown in Fig. 13(d) and Fig. 14(b). Nevertheless, as a result of work hardening, some dislocations still remained in these DRX grains, and new DRX grains were formed in these
(25)
4. Results and discussion 4.1. Modeling results 4.1.1. The kinetics of DRX The strain dependence of dynamic recrystallized volume fraction at different temperatures and strain rates are illustrated in Fig. 11(a and b). It is shown clearly that the kinetics of DRX is strongly depended on deformation temperature and strain rate. The DRX is thermally activated so that the X increases with temperature. Also, the rate of DRX increases with decreasing strain rate, which provide longer time for dislocation annihilation and rearrangement. Since the DRX is a continuous process of deformation, nucleation of grains and subsequent migration of grain boundaries, X increases with the strain increased. As the strain increases, X reaches to a constant value, 100%, at which the flow stress reaches to steady state. 4.1.2. Prediction of flow curves In order to verify the developed flow stress model, comparisons between the experimental and predicted results are carried out. Fig. 12(a–d) shows the predicted and experimental flow stress curves of ZK60 magnesium alloy at strain rate of 0.001, 0.01, 0.1 and 1 s−1 , respectively. It can be easily found that the proposed flow stress model gives an accurate and precise estimate of the flow stress for ZK60 magnesium alloy under most deformation conditions, except the case with ε˙ = 1 s−1 and T = 473 K. The exception is probably arose by the specimen crack which leads to stress relaxation. So the proposed model can be used to numerically analyze the hot deformation processes. Additionally, it should be mentioned that the thermal softening result in a part of the flow stress reduction when the strain rate is relative high, for example
Fig. 11. Predicted results of the relation between recrystallized volume fraction and strain: (a) ε˙ = 0.01 s−1 and various temperatures and (b) 573 K and various strain rates.
2796
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
Fig. 12. Comparison between predicted and experimental flow stress curves of ZK60 Mg alloy under strain rate of (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 and (d) 1 s−1 .
Fig. 13. Microstructure evolution of ZK60 alloy compressed at a strain rate of 0.1 s−1 under 573 K: (a) ε = 0.1; (b) ε = 0.3; (c) ε = 0.5; (d) ε = 0.9.
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2797
Fig. 14. TEM images of ZK60 alloy deformed to various strains at T = 573 K and ε˙ = 0.1 s−1 : (a) ε = 0.1; (b) ε = 0.9.
grains. It means that repetitive DRX occurred and grain refinement as well as flow softening were attained through DRX process at ε = 0.9. Above analysis is consistent with the results reported by Yu et al. [37] and Wang et al. [38]. 5. Conclusion In this study, the hot compression tests were carried out to investigate the flow behavior of magnesium alloy ZK60. Based on the experimental results, a model was developed to predict the flow stress curve of magnesium alloy at hot deformation condition. The proposed model is capable of predicting the flow behavior of work hardening and DRV region as well as the softening caused by DRX. To establish the model, the double-differentiation method was used to determine the critical strain for initiation of DRX, and the DRV parameter ˝ was evaluated from the work-hardening behavior prior to critical strain. The net softening attributable to DRX was defined as the difference between the DRV and experimental curves, and Avrami equation was employed to represent this softening behavior. The flow stress curves of ZK60 magnesium alloy predicted by the developed model are in good agreement with experimental results, which confirms the validity of the proposed model. References [1] T. Al-Samman, G. Gottstein, Materials Science and Engineering A 490 (2008) 411–420. [2] Y.C. Lin, M.-S. Chen, J. Zhong, Computational Materials Science 43 (2008) 1117–1122. [3] Y.-C. Lin, M.-S. Chen, Journal of Materials Processing Technology 209 (2009) 4578–4583. [4] Y.C. Lin, M.-S. Chen, J. Zhong, Materials & Design 30 (2009) 908–913. [5] Z. Gronostajski, Journal of Materials Processing Technology 106 (2000) 40–44. [6] H.J. McQueen, N.D. Ryan, Materials Science and Engineering A 322 (2002) 43–63. [7] H.J. McQueen, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 33 (2002) 345–362.
[8] H.J. Frost, M.F. Ashby, Deformation Mechanisum Maps, Pergamon Press, Oxford, 1982. [9] A. Galiyev, R. Kaibyshev, G. Gottstein, Acta Materialia 49 (2001) 1199–1207. [10] A. Galiyev, O. Sitdikov, R. Kaibyshev, Materials Transactions 44 (2003) 426–435. [11] H. Takuda, H. Fujimoto, N. Hatta, Journal of Materials Processing Technology 80–81 (1998) 513–516. [12] H. Takuda, T. Morishita, T. Kinoshita, N. Shirakawa, Journal of Materials Processing Technology 164–165 (2005) 1258–1262. [13] L. Li, J. Zhou, J. Duszczyk, Journal of Materials Processing Technology 172 (2006) 372–380. [14] Z.Q. Sheng, R. Shivpuri, Materials Science and Engineering A 419 (2006) 202–208. [15] J. Liu, Z. Cui, C. Li, Computational Materials Science 41 (2008) 375–382. [16] S.E. Ion, F.J. Humphreys, S.H. White, Acta Metallurgica 30 (1982) 1909–1919. [17] Y. Bergstrom, Materials Science and Engineering 5 (1970) 193–200. [18] W. Roberts, Y. Bergstrom, Acta Metallurgica 21 (1973) 457–469. [19] Y. Estrin, H. Mecking, Acta Metallurgica 32 (1984) 57–70. [20] H. Mecking, U.F. Kocks, Acta Metallurgica 29 (1981) 1865–1875. [21] Y.-S. Jang, D.-C. Ko, B.-M. Kim, Journal of Materials Processing Technology 101 (2000) 85–94. [22] E.I. Poliak, J.J. Jonas, Acta Materialia 44 (1996) 127–136. [23] J.J. Jonas, X. Quelennec, L. Jiang, E. Martin, Acta Materialia 57 (2009) 2748–2756. [24] A. Yoshie, H. Morikawa, Y. Onoe, K. Itoh, Transactions of the Iron and Steel Institute of Japan 27 (1987) 425–431. [25] Y.Q. Yang, B.C. Li, Z.M. Zhang, Materials Science and Engineering A 499 (2009) 238–241. [26] H.E. Hu, L. Zhen, L. Yang, W.Z. Shao, B.Y. Zhang, Materials Science and Engineering A 488 (2008) 64–71. [27] K. Yu, W. Li, J. Zhao, Z. Ma, R. Wang, Scripta Materialia 48 (2003) 1319–1323. [28] A. Mwembela, E.B. Konopleva, H.J. McQueen, Scripta Materialia 37 (1997) 1789–1795. [29] X. He, Z. Yu, X. Lai, Computational Materials Science 44 (2008) 760–764. [30] L. Zhang, W. Yang, Z. Sun, Journal of University of Science and Technology Beijing: Mineral Metallurgy Materials (English Edition) 14 (2007) 130–135. [31] S.-L. Liao, L.-W. Zhang, C.-X. Yue, J.-B. Pei, H.-J. Gao, Journal of Central South University of Technology (English Edition) 15 (2008) 575–580. [32] F.A. Slooff, J. Zhou, J. Duszczyk, L. Katgerman, Scripta Materialia 57 (2007) 759–762. [33] J. Jiang, A. Godfrey, W. Liu, Q. Liu, Scripta Materialia 58 (2008) 122–125. [34] M.R. Barnett, Materials Science and Engineering A 464 (2007) 1–7. [35] M.R. Barnett, Materials Science and Engineering A 464 (2007) 8–16. [36] A.M. Wusatowska, H. Miura, T. Sakai, Materials Science and Engineering A 323 (2002) 177–186. [37] K. Yu, X.Y. Wang, S.T. Rui, R.C. Wang, Journal of Central South University of Technology 15 (2008) 434–437. [38] S.R. Wang, S.B. Kang, J.H. Cho, Journal of Materials Science 44 (2009) 5475–5484.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Modeling of flow stress for magnesium alloy during hot deformation Yin-Jiang Qin ∗ , Qing-Lin Pan, Yun-Bin He, Wen-Bin Li, Xiao-Yan Liu, Xi Fan School of Materials Science and Engineering, Central South University, Changsha, 410083, China
a r t i c l e
i n f o
Article history: Received 5 August 2009 Received in revised form 29 December 2009 Accepted 11 January 2010
Keywords: Flow stress Dynamic recrystallization Hot deformation Magnesium alloy ZK60
a b s t r a c t Based on the classical flow stress–dislocation density relation and kinetics of dynamic recrystallization (DRX), a model was developed to determine flow stress of magnesium alloy at hot deformation condition. The proposed model is capable of predicting the flow behavior of work hardening and dynamic recovery (DRV) region as well as the softening caused by DRX. To establish the model, the double-differentiation method was used to identify the critical strain for initiation of DRX, and the DRV parameter ˝ was evaluated from the work-hardening behavior prior to critical strain. The net softening attributable to DRX was defined as the difference between the DRV and experimental curves, and Avrami equation was employed to represent this softening behavior. The flow stress curves of ZK60 magnesium alloy predicted by the developed model are in good agreement with experimental results, which confirms the validity of the model. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved.
1. Introduction Lightweight magnesium alloys have attracted significant interest in the last decade due to their potential applications for weight reduction in transportation vehicles. However, magnesium alloys have poor formability and limited ductility at room temperature ascribed to their hexagonal close-packed (hcp) crystal structure. As a result, to date, most of the magnesium products are fabricated by die casting process instead of employing plastic forming techniques such as rolling, forging, and extrusion. At elevated temperatures the workability of magnesium alloys substantially increases as additional slip systems, i.e., non-basal and c + a slip become sufficiently available by thermal activation [1]. Therefore, it is necessary to investigate the hot deformation behavior of magnesium alloys. The flow behavior of metals and alloys at hot deformation condition has a great importance for designer of metal forming processes because of its effective role on the required deformation energy as well as the kinetics of metallurgical transformation such as dynamic recovery (DRV) and dynamic recrystallization (DRX). However, the flow behavior is influenced by many factors such as strain, strain rate, deformation temperature, etc. and understanding their effects is a difficult task due to their complex nature. Recently, the fast development of computing techniques led to a wide application of finite element method (FEM) simulation to study materials forming processes [2–4]. Nevertheless, numerical simulation can be truly reliable only when a proper flow stress
∗ Corresponding author. Tel.: +86 0731 88830933. E-mail address: [email protected] (Y.-J. Qin).
model is available. A number of models were proposed to describe the flow stress of metals and alloys and extensive summary on different models was given in [5–7]. In the case of magnesium and its alloys, a few of previous investigations were performed to study the flow stress behavior. Frost and Ashby [8] reported that pure magnesium abided by the power law creep equation above 573 K. Galiyev et al. [9,10] found that magnesium alloy ZK60 complied with the creep equation. Based on an observation of a linear relationship between semi-log Zener-Hollomon parameter and proof stress, Takuda et al. [11,12] proposed a parametrical method to express the proof flow stress in the AZ31 and AZ91 tensile tests. Similar constitutive relationship for AZ31B magnesium alloy was determined and validated through comparison between simulated and real extrusion by Li et al. [13]. Sheng and Shivpuri [14] proposed an analytical method, which reflected temperature, strain and strain rate effect by introducing temperature-compensated strain rate, and the model was well applied on three published experimental data. Liu et al. [15] put forward a new model of flow stress characterizing DRX for magnesium alloy AZ31B. Owing to the relative low stacking fault energy (SFE) and the lack of easily activated slip systems, DRX plays an important role during hot deformation of magnesium alloy [16]. However, almost all the flow stress models aforementioned, except the one proposed by Liu et al. in 2008 are empirical and do not take into account the effect of DRX. Since the DRX is a thermally activated process, the recrystallized volume fraction can be regarded as a function of strain through Avrami equation. Furthermore, the descending of flow stress during hot deformation is mainly dominated by the recrystallized volume fraction. Based on this idea, the dislocation model developed by Bergstrom [17,18] together with Avrami-type
0921-5093/$ – see front matter. Crown Copyright © 2010 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2010.01.035
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2791
Fig. 1. True stress–strain curves for ZK60 Mg alloy at various temperatures with strain rates of (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 and (d) 1 s−1 .
recrystallization kinetics equation are used to describe the flow behavior of magnesium alloy during hot deformation in this study.
2. Experiments The material used in the present study was commercial grade magnesium alloy ZK60 (Mg–5.78%Zn–0.76%Zr). The alloy was fabricated by semi-continuous casting and homogenized at 673 K for 12 h prior to deformation. The initial grain size was about 100 m. Cylindrical specimens with 10 mm in diameter and 15 mm in height were machined from the homogenized materials for compression tests. Uniaxial compression tests were conducted in the temperature range of 473–673 K at intervals of 50 K and constant true strain rate ranging from 0.001 to 1 s−1 at intervals of an order of magnitude on a Gleeble-1500 thermal simulator up to 60% of height reduction. The Gleeble compression system works with a servo-hydraulic mechanism, which deforms the sample heated electrically by its Ohmic resistance at a constant strain rate. The temperature was controlled and measured by a thermocouple welded to the sample. Graphite foils and colloidal graphite were used as lubrication between the sample and the anvils. All the specimens were heated at 10 K/s up to deformation temperatures, held 3 min to homogenize the temperature in the sample, and then deformed and water quenched. The load-stroke data recorded for every compression tests were converted into true stress–true plastic strain curves using standard equations. The true stress–true strain curves of magnesium alloy ZK60 at different strain rates are shown in Fig. 1, according to which the characteristics of flow stress curves are represented as follows: • The overall level of flow stress curve increases with the decrease of deformation temperature and the increase of strain rate.
• In the initial stage of the deformation, hardening rate is higher than the softening rate and thus the stress increases abruptly, then the increasing rate is decreased due to the occurrence of DRV and DRX. When the hardening rate is equal to the softening rate the flow stress peak is reached. • After the stress peak, the softening induced by DRX exceeds the hardening and the stress drops steeply. The stress becomes steady when a new balance between softening and hardening is obtained. 3. Modeling of flow stress In the present approach, the experimental curves are considered to be the net result of the simultaneous operation of DRV and DRX in the manner typified by the schematic curves of Fig. 2. Here the uppermost curve ( drv ) is regarded as resulting from the operation of DRV alone, i.e., in the absence of DRX. Although not obtainable directly from experiment, it represents the assumed work-hardening behavior of the unrecrystallization regions, e.g., in the interiors of the grains as “necklacing” proceeds inwards. This can be derived from the work-hardening behavior prior to critical strain (εc ) for initiation of DRX, as will be demonstrated later. After the critical strain εc , it leads to more and more softening and is responsible for the difference between the drv curve and DRX curve ( drx ). At saturation, the value of the asymptotic stress is given by sat . On continued straining, sat represents the dislocation density in the most work-hardened grains therefore the driving force for the continuation of DRX. 3.1. Modeling of dynamic recovery During deformation, the dependence of the dislocation density on plastic strain ε is generally considered to be given by the
2792
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
often expressed in the following form [21]: ε0.5 = A0.5 d0m0.5 ε˙ n0.5 exp
Q
DRX
(7)
RT
where A0.5 , m0.5 and n0.5 are materials constants, d0 is the initial grain size (m), QDRX is the DRX activation energy (kJ/mol), R is the universal gas constant (8.314 J mol−1 K−1 ) and T is the absolute temperature (K). According to Fig. 2, it is obvious that the difference between the drv and drx curve can be represented as , which is the net softening directly attributable to DRX. The maximum value of is sat – ss , where ss is the steady state stress under DRX conditions. Then, the fractional softening due to DRX is expressed as X=
Fig. 2. Schematic diagram illustrating the work-hardening curve drv together with a typical experimental DRX flow stress curve.
following equation [19]: d = U − ˝ dε
(1)
Here, U represents the work hardening, which is a multiplication term and can be regarded as constant with respect to strain. The term ˝ is the contribution due to DRV through dislocation annihilation and rearrangement, and is often called the coefficient of DRV. By employing the initial condition = 0 at ε = 0, the differential Eq. (1) can be easily solved as = 0 exp(−˝ε) +
U ˝
[1 − exp(−˝ε)]
(2)
Using the classic relationship between stress and dislocation den√ sity = Gb [20], where ␥ is a materials constant, b is the distance between atoms in the slip direction and G is the shear modulus. Then the flow stress can be given by the following expression in terms of strain: 2 = 02 exp(−˝ε) + (Gb)
2
U ˝
[1 − exp(−˝ε)]
(3)
where 0 is the yield stress. When strain ε tends to infinity, i.e., in the steady state condition, the DRV saturation stress sat can be obtained from Eq. (3) as follows:
sat = Gb
U/˝
− drx = drv sat − ss sat − ss
(8)
by which the DRX volume fraction under different deformation conditions can be measured directly from the curves which are similar to Fig. 2. This method is particularly useful when the microstructure is not stable, so that metallographic measurements are difficult to carry out. Combining Eq. (8) with Eq. (6), the flow stress during DRX period under hot deformation can be given by the following expression:
drx = drv − (sar − ss ) 1 − exp −ˇd
ε − ε nd c
ε0.5
(ε > εc ) (9)
3.3. Determination of c , sat , 0 and ss In order to determine the critical strain εc for initiation of DRX, the double-differentiation method proposed by Poliak and Jonas [22] is employed in the present investigation. Firstly, the flow curves shown in Fig. 1 are fitted and smoothed with a seventh-order polynomial in the region of plastic deformation that encompasses the stress peak using the MatlabTM software. (In some problematic cases, a higher order polynomial is employed.) Then, the workis calculated and plotted against hardening rate, i.e., = d/dε|ε,T ˙ the true stress. The Poliak method defines εc as the point at which the second derivative of the work-hardening rate with respect to stress (∂2 /∂ 2 ) is zero, i.e., the inflection point of – curve. The saturation stress sat , in turn, is defined by the extrapolation of the – plot to = 0 (using only the linear portion of the curve relating to stress values just below critical stress c ). As can be seen, it is
(4)
Then the variation of flow stress in the work hardening and DRV region during hot deformation can be expressed as follows: 2 2 drv = [sat − (sat − 02 )exp(−˝ε)]
1/2
(ε < εc )
(5)
3.2. Modeling of dynamic recrystallization When DRX takes place, the simultaneous use of the above – relation and the dynamically recrystallized fraction makes it possible to predict the flow stress after the stress peak. In the present approach, a standard Avrami-type equation was used to describe the DRX kinetics and it can be expressed as [3]:
X = 1 − exp −ˇd
ε − ε nd c
ε0.5
(6)
where X is the recrystallized volume fraction, ˇd and nd are DRX parameters depending on chemical composition and hot deformation conditions. ε0.5 is the strain for 50% recrystallization, which is
Fig. 3. Flow stress dependence of the strain-hardening rate for ZK60 Mg alloy with strain rate of 0.1 s−1 .
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2793
A˝ , n˝ and m˝ are constants. By regression analysis, ˝ can be expressed as follows:
22895
˝ = 1.24 × 104 d0−0.5849 ε˙ −0.1919 exp −
RT
(15)
It should be mentioned that in order to investigate the effect of initial grain size d0 on ˝, εc and ε0.5 (the latter two will be discussed later), the flow stress curves of magnesium alloy ZK60 tested by Yang et al. [25] were employed. In Yang’s study, the initial grain size is 50 m while the value is 100 m in the present study. For comparison purposes, the calculated value of ˝ is also depicted in Fig. 5 which shows that Eq. (15) represents the dependence of ˝ on strain rate and temperature very well. 3.5. Dependences of 0 , sat and ss on p It is generally accepted to use the hyperbolic-sine meliorated Arrhenius equation to model the temperature and strain rate dependence of peak stress p as follows [6]:
Fig. 4. Plot of versus 2 employed to determine the slope k.
the work-hardening behavior prior to c that characterizes sat . As an example, the – plots for magnesium alloy ZK60 with strain rate of 0.1 s−1 and different temperatures is illustrated in Fig. 3, in which the critical stress c and saturation stress sat are indicated by arrows and symbol of diamond, respectively. In each case, a “yield stress” 0 is identified on the flow stress curve in terms of a 2% offset in the total strain and the steady state stress ss is obtained directly from the experimental flow stress curve.
To determine the value of ˝, the method described by Jonas et al. [23] is employed and it is outlined briefly here. By differentiating Eq. (5), it can be shown that: d −1/2 2 2 2 = 0.5[sat − (sat − 02 )exp(−˝ε)] ˝(sat − 02 )exp(−˝ε) dε (10) Multiplying d/dε by leads to (11)
Meanwhile, Eq. (5) can be expressed as 2 2 (sat − 02 )exp(−˝ε) = sat − 2
(12)
−Q
(16)
RT
where A, ␣ and n are materials constants, Q is the deformation activation energy. When the flow stress is low, Eq. (16) can be simplified as [26]:
ε˙ = A pn exp
−Q
(17)
RT
While at high stress level, Eq. (16) becomes ε˙ = A exp(ˇp ) exp
3.4. Determination of ˝
d 2 = 0.5[˝(sat − 02 )exp(−˝ε)] dε
ε˙ = A[sinh(˛p )]n exp
−Q
(18)
RT
in which ˇ = ˛n . According to Eqs. (17) and (18), the approximate value of n and ˇ are determined by linear regression of the plots of ln ε˙ − p and ln ε˙ − ln p at different temperatures as 7.88 and 0.08, respectively. And then the suitable value of ˛ is determined to be 0.01. The activation energy Q is an important physical parameter indicating the plastic deformability, and can be calculated as follows [27]:
∂ ln ε˙ Q =R ∂ ln[sinh(˛p )]
T
∂ ln[sinh(˛p )] ∂(1/T )
(19) ε˙
The first term on the right-hand side of Eq. (19) refers to the slope of linear fitting of ln ε˙ versus ln[sinh(˛ p )] at different deformation temperatures (Fig. 6), and the second term refers to the slope of
So that Eq. (11) can now be written as d 2 − 0.5˝ 2 = 0.5˝sat dε
(13)
According to Eq. (13): k=
d() = −0.5˝ d( 2 )
(14)
in which k is the slope of the tangent to – 2 plot. And then the value of ˝ can be determined as −2k. An example of determination k is shown schematically in Fig. 4. Note that the slope k is also derived from the experimental data obtained prior to c and εc . A typical plot of the strain rate and deformation temperature dependence of ˝ is presented in Fig. 5 for ZK60 magnesium alloy. It is clear from this graph that increasing the strain rate or decreasing the temperature results in a decrease in ˝. This is consistent with ˝ being a measure of the ease of DRV. To quantify the effects of processing parameters on ˝, Yoshie et al. [24] used an empirical equation of the form ˝ = A˝ d0n˝ ε˙ m˝ exp(−Q˝ /RT ), where d0 is the initial grain size, Q˝ is an apparent activation energy (kJ/mol), and
Fig. 5. Dependence of ˝ on temperature and strain rate.
2794
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
Fig. 6. Relationship between peak stress and strain rate.
Fig. 8. Relationship between flow stress and Zener-Hollomon parameter.
linear fitting of ln[sinh(˛ p )] versus 1/T at different strain rates (Fig. 7). Then the mean activation energy Q of ZK60 magnesium alloy during hot deformation is determined to be 153.372 kJ/mol. The combined effect of strain rate and temperature on deformation behavior can be characterized by Zener-Hollomon (Z) parameter, the physical meaning of which is the so-called temperature-compensated strain rate parameter. It can be defined as [28]:
p can be described by Eq. (21) in which ˛ = 0.01, A = 3.3648 × 1011 , n = 6.88 and Z = ε˙ exp(153317/RT ). For modeling purposes, it is useful to be able to express the derived and measured quantities 0 , sat and ss as ratios of p , as the latter is more readily measurable. The dependences of these quantities for flow curves of ZK60 magnesium alloy on p are illustrated in Fig. 9. Overall, the following expressions give a reasonably good fit to the data:
Z = ε˙ exp
Q RT
= A[sinh(˛p )]n
(20)
Then, the peak stress p , can be written as a function of Z parameter, considering the definition of the hyperbolic law [29]: p =
Z 1/n 1
˛
A
+
2/n Z A
1/2
+1
(21)
Taking the natural logarithm of Eq. (20): ln Z = ln A + n ln[sinh(˛p )]
(22)
According to Eq. (22), the stress exponent n and constant A are determined by linear regression of ln Z − ln[sinh(˛ p )]) plot (Fig. 8) as 6.88 and 3.3648 × 1011 , respectively. And then, the peak stress
Fig. 7. Relationship between peak stress and deformation temperature.
0 = 0.36p , sat = 1.20p and ss = 0.62p
3.6. Determination of the parameters for DRX kinetics Based on experimental data and literature [30–31], it can be found that the critical strain εc and ε0.5 are function of initial grain ˙ and both of size d0 , deformation temperature T and strain rate ε, them can be described by equations which have the form similar to Eq. (7). In order to determine the unknown constants in these equations, regression analysis are performed, and the results are listed as follows: ε0.5 = 5.25 × 10−3 d00.4428 ε˙ 0.0466 exp
11157 RT
Fig. 9. Dependences of 0 , sat and ss on p .
(23)
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2795
ε˙ = 1 s−1 . Because when the deformation is very quick, the test time is too short to allow for heat transfer, then the specimen temperature increases [32]. This makes the flow stress curve deviate from the ideal isothermal condition. It should be noticed that there are some other factors such as twining that influence the hot deformation behavior [33–35]. However, the deformation temperature in the present work is relatively high and the predominant factor during the deformation process is DRX. Thus, other factors are not taken into account. 4.2. Microstructure evolution
Fig. 10. Determination of the constants ˇd and nd in Avrami equation.
εc = 1.34 × 10−4 d00.4695 ε˙ 0.1511 exp
22571 RT
(24)
To determine the DRX kinetics, plots of ln[−ln(1 − X)] versus ln[(ε − ε0 )/ε0.5 are used and the parameters ˇd and nd can be easily obtained from the slope and intercept of the fitting line. As an example, the plot for condition of ε˙ = 1 s−1 and T = 523 K is illustrated in Fig. 10. Through analysis of the results at other conditions, the values of nd are found to be independent of ε˙ and T and to be about 1.6, while ˇd is a function of ε˙ and T and can be described as ˇd = 6.92 × 10−2 ε˙ 0.0704 exp
12070 RT
Microstructure evolution of ZK60 alloy compressed at 573 K and strain rate of 0.1 s−1 was shown in Fig. 13. During the initiation of hot deformation, the original grain boundaries became wavy and corrugated, as shown in Fig. 13(a) and (b). The DRX process occurred on the boundaries of original grains. The recrystallized grains grew greatly through the migration of grain boundaries during the hot deformation process, as shown in Fig. 13(c). Fig. 14(a) shows that the tangled dislocations focused on the original grain boundaries at ε = 0.1. As a result, the recrystallized grains were firstly formed on the original grain boundaries. Such a process of new grains formation may be similar to that in conventional DRX in cubic metals, i.e., the bulging out of part of the serrated grain boundaries and the migration of grain boundaries for growth of DRX grains [36]. As further deformation was carried out, a large amount of DRX grains were observed, as shown in Fig. 13(d) and Fig. 14(b). Nevertheless, as a result of work hardening, some dislocations still remained in these DRX grains, and new DRX grains were formed in these
(25)
4. Results and discussion 4.1. Modeling results 4.1.1. The kinetics of DRX The strain dependence of dynamic recrystallized volume fraction at different temperatures and strain rates are illustrated in Fig. 11(a and b). It is shown clearly that the kinetics of DRX is strongly depended on deformation temperature and strain rate. The DRX is thermally activated so that the X increases with temperature. Also, the rate of DRX increases with decreasing strain rate, which provide longer time for dislocation annihilation and rearrangement. Since the DRX is a continuous process of deformation, nucleation of grains and subsequent migration of grain boundaries, X increases with the strain increased. As the strain increases, X reaches to a constant value, 100%, at which the flow stress reaches to steady state. 4.1.2. Prediction of flow curves In order to verify the developed flow stress model, comparisons between the experimental and predicted results are carried out. Fig. 12(a–d) shows the predicted and experimental flow stress curves of ZK60 magnesium alloy at strain rate of 0.001, 0.01, 0.1 and 1 s−1 , respectively. It can be easily found that the proposed flow stress model gives an accurate and precise estimate of the flow stress for ZK60 magnesium alloy under most deformation conditions, except the case with ε˙ = 1 s−1 and T = 473 K. The exception is probably arose by the specimen crack which leads to stress relaxation. So the proposed model can be used to numerically analyze the hot deformation processes. Additionally, it should be mentioned that the thermal softening result in a part of the flow stress reduction when the strain rate is relative high, for example
Fig. 11. Predicted results of the relation between recrystallized volume fraction and strain: (a) ε˙ = 0.01 s−1 and various temperatures and (b) 573 K and various strain rates.
2796
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
Fig. 12. Comparison between predicted and experimental flow stress curves of ZK60 Mg alloy under strain rate of (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 and (d) 1 s−1 .
Fig. 13. Microstructure evolution of ZK60 alloy compressed at a strain rate of 0.1 s−1 under 573 K: (a) ε = 0.1; (b) ε = 0.3; (c) ε = 0.5; (d) ε = 0.9.
Y.-J. Qin et al. / Materials Science and Engineering A 527 (2010) 2790–2797
2797
Fig. 14. TEM images of ZK60 alloy deformed to various strains at T = 573 K and ε˙ = 0.1 s−1 : (a) ε = 0.1; (b) ε = 0.9.
grains. It means that repetitive DRX occurred and grain refinement as well as flow softening were attained through DRX process at ε = 0.9. Above analysis is consistent with the results reported by Yu et al. [37] and Wang et al. [38]. 5. Conclusion In this study, the hot compression tests were carried out to investigate the flow behavior of magnesium alloy ZK60. Based on the experimental results, a model was developed to predict the flow stress curve of magnesium alloy at hot deformation condition. The proposed model is capable of predicting the flow behavior of work hardening and DRV region as well as the softening caused by DRX. To establish the model, the double-differentiation method was used to determine the critical strain for initiation of DRX, and the DRV parameter ˝ was evaluated from the work-hardening behavior prior to critical strain. The net softening attributable to DRX was defined as the difference between the DRV and experimental curves, and Avrami equation was employed to represent this softening behavior. The flow stress curves of ZK60 magnesium alloy predicted by the developed model are in good agreement with experimental results, which confirms the validity of the proposed model. References [1] T. Al-Samman, G. Gottstein, Materials Science and Engineering A 490 (2008) 411–420. [2] Y.C. Lin, M.-S. Chen, J. Zhong, Computational Materials Science 43 (2008) 1117–1122. [3] Y.-C. Lin, M.-S. Chen, Journal of Materials Processing Technology 209 (2009) 4578–4583. [4] Y.C. Lin, M.-S. Chen, J. Zhong, Materials & Design 30 (2009) 908–913. [5] Z. Gronostajski, Journal of Materials Processing Technology 106 (2000) 40–44. [6] H.J. McQueen, N.D. Ryan, Materials Science and Engineering A 322 (2002) 43–63. [7] H.J. McQueen, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science 33 (2002) 345–362.
[8] H.J. Frost, M.F. Ashby, Deformation Mechanisum Maps, Pergamon Press, Oxford, 1982. [9] A. Galiyev, R. Kaibyshev, G. Gottstein, Acta Materialia 49 (2001) 1199–1207. [10] A. Galiyev, O. Sitdikov, R. Kaibyshev, Materials Transactions 44 (2003) 426–435. [11] H. Takuda, H. Fujimoto, N. Hatta, Journal of Materials Processing Technology 80–81 (1998) 513–516. [12] H. Takuda, T. Morishita, T. Kinoshita, N. Shirakawa, Journal of Materials Processing Technology 164–165 (2005) 1258–1262. [13] L. Li, J. Zhou, J. Duszczyk, Journal of Materials Processing Technology 172 (2006) 372–380. [14] Z.Q. Sheng, R. Shivpuri, Materials Science and Engineering A 419 (2006) 202–208. [15] J. Liu, Z. Cui, C. Li, Computational Materials Science 41 (2008) 375–382. [16] S.E. Ion, F.J. Humphreys, S.H. White, Acta Metallurgica 30 (1982) 1909–1919. [17] Y. Bergstrom, Materials Science and Engineering 5 (1970) 193–200. [18] W. Roberts, Y. Bergstrom, Acta Metallurgica 21 (1973) 457–469. [19] Y. Estrin, H. Mecking, Acta Metallurgica 32 (1984) 57–70. [20] H. Mecking, U.F. Kocks, Acta Metallurgica 29 (1981) 1865–1875. [21] Y.-S. Jang, D.-C. Ko, B.-M. Kim, Journal of Materials Processing Technology 101 (2000) 85–94. [22] E.I. Poliak, J.J. Jonas, Acta Materialia 44 (1996) 127–136. [23] J.J. Jonas, X. Quelennec, L. Jiang, E. Martin, Acta Materialia 57 (2009) 2748–2756. [24] A. Yoshie, H. Morikawa, Y. Onoe, K. Itoh, Transactions of the Iron and Steel Institute of Japan 27 (1987) 425–431. [25] Y.Q. Yang, B.C. Li, Z.M. Zhang, Materials Science and Engineering A 499 (2009) 238–241. [26] H.E. Hu, L. Zhen, L. Yang, W.Z. Shao, B.Y. Zhang, Materials Science and Engineering A 488 (2008) 64–71. [27] K. Yu, W. Li, J. Zhao, Z. Ma, R. Wang, Scripta Materialia 48 (2003) 1319–1323. [28] A. Mwembela, E.B. Konopleva, H.J. McQueen, Scripta Materialia 37 (1997) 1789–1795. [29] X. He, Z. Yu, X. Lai, Computational Materials Science 44 (2008) 760–764. [30] L. Zhang, W. Yang, Z. Sun, Journal of University of Science and Technology Beijing: Mineral Metallurgy Materials (English Edition) 14 (2007) 130–135. [31] S.-L. Liao, L.-W. Zhang, C.-X. Yue, J.-B. Pei, H.-J. Gao, Journal of Central South University of Technology (English Edition) 15 (2008) 575–580. [32] F.A. Slooff, J. Zhou, J. Duszczyk, L. Katgerman, Scripta Materialia 57 (2007) 759–762. [33] J. Jiang, A. Godfrey, W. Liu, Q. Liu, Scripta Materialia 58 (2008) 122–125. [34] M.R. Barnett, Materials Science and Engineering A 464 (2007) 1–7. [35] M.R. Barnett, Materials Science and Engineering A 464 (2007) 8–16. [36] A.M. Wusatowska, H. Miura, T. Sakai, Materials Science and Engineering A 323 (2002) 177–186. [37] K. Yu, X.Y. Wang, S.T. Rui, R.C. Wang, Journal of Central South University of Technology 15 (2008) 434–437. [38] S.R. Wang, S.B. Kang, J.H. Cho, Journal of Materials Science 44 (2009) 5475–5484.