A new kinetics model of dynamic recrystallization for magnesium alloy AZ31B

Materials Science and Engineering A 529 (2011) 300–310

Contents lists available at SciVerse ScienceDirect

Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

A new kinetics model of dynamic recrystallization for magnesium alloy AZ31B J. Liu a,∗ , Z. Cui a , L. Ruan b a b

National Die & Mold CAD Engineering Research Center, Shanghai Jiao Tong University, Shanghai, China Department of Mechanical Engineering, Kumamoto University, Kumamoto, Japan

a r t i c l e

i n f o

Article history: Received 13 February 2011 Received in revised form 1 August 2011 Accepted 9 September 2011 Available online 16 September 2011 Keywords: Dynamic recrystallization (DRX) Kinetics model Strain softening Magnesium alloy AZ31B

a b s t r a c t The classical kinetics models of dynamic recrystallization (DRX) in the form of Avrami function describe the development of DRX process to a large extent; however, because of the characteristics of exponent function, the conventional models cannot exactly exhibit the development speed of DRX process. Based on this analysis, a new kinetics model of DRX was proposed, which represents the ‘slow–rapid–slow’ property of DRX development. According to the new model, the development process of DRX can be divided into three phases: slow-beginning phase, rapid-increasing phase and slow-rising-to-balance phase. Because the turning point between the second phase and the third one corresponds to the inflexion from the faster velocity of DRX development to the slower one, the strain at this moment can be considered as the most appropriate and economic strain that guarantees fine grains and saves energy consumption. Take a typical metal characterized by DRX magnesium alloy AZ31B for instance, the Gleeble-1500 thermomechanical simulation compression tests were conducted together with microscopic examination, according to which the model parameters were determined. Statistics shows that the experimental results are in good agreement with the predicted values, which validates the accuracy of the new kinetics model. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The objective of metal forming is not only to form desired shapes and dimensions, but to obtain superior mechanical properties as well. The in-depth study on microstructural evolutions of hot deformation will be helpful to determine the optimal process parameters of hot working. Dynamic recrystallization (DRX) is considered as one of the most important microstructural evolution mechanisms, which is beneficial to obtain fine metallurgical structures, eliminate defects and improve mechanical properties of products. With the development of computer technologies and numerical simulations, it is crucial to model the microstructural evolutions and predict the microstructural changes. In the last three decades, many researchers have proposed some microstructural evolution models suitable for different materials. Although the DRX kinetics models proposed by researchers have some differences in parameters and forms, they are all based on the Avrami function. By analyzing several typical and highly influential models, a new DRX kinetics model, which is able to more reasonably demonstrate the velocity of DRX was built. The new DRX kinetics model is in accord with the usual way that DRX develops, and has fewer parameters. Through the new model the most

∗ Corresponding author. Tel.: +86 21 62813430; fax: +86 21 62827605. E-mail addresses: [email protected], [email protected] (J. Liu). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.09.032

favorite and economic strain corresponding to fine and uniform grains is able to be obtained at a given temperature and strain rate, which is significant to design and optimize hot deformation processes.

2. Reviews and discussions Sellars and co-workers [1,2] conducted pioneering research on microstructural evolution on the basis of a great quantity of Gleeble thermomechanical tests, and a series of models which cover almost all the probable physical metallurgical phenomena related to DRX, static recrystallization (SRX) and grain growth were put forward. The DRX kinetics model proposed by Sellars based on Avrami function is as follows: XDRX = 1 − exp



−0.693

 t  t0.5

(1)

where XDRX is the dynamically recrystallized volume fraction, t0.5 is the time for 50% DRX, t is the time for XDRX to occur. In Sellars’ model, the using of the constant 0.693 is crucial and very skillful, which make all the parameters have obvious physical meanings. Later, some researchers such as McQueen et al. [3] published DRX kinetics models where the dynamically recrystallized volume fraction are also formulated in the form of time function.

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

Yada et al. [4] built the following DRX kinetics model of plain C–Mn steel in the form of strain function:

⎧ ⎪ ⎨

 XDRX = 1 − exp

 ε − ε 2

−0.693

⎪ ⎩ ε0.5 = 1.144 × 10−3 d0 0.28 ε˙ 0.05 exp



 6420 

(2)

T

where εc is the critical strain, ε0.5 is the strain for 50% DRX, d0 is the initial grain size, T is the deformation temperature, ε˙ is the strain rate. Although the different variables, in comparison to those in Sellars’ model, are used in Yada’s model, i.e., Sellars’ model is the function of time and Yada’s model is the function of strain, both of them are same in nature. It is because that the function of time is consistent with the function of strain when the stain rate is a constant. So, Yada’s model can be applied to unsteady numerical simulations during hot deformations as the result of the using of the strain form. Thus, many kinds of commercial FEM softwares, such as Marc, Superform and Deform employ Yada’s model to predict microstructural evolution. Different from Sellars’ model, the physical meaning of the variable of ε0.5 in Yada model is not obvious, which does not precisely represent the strain for 50% DRX any more, but a statistic. So, it is difficult to determine the value of ε0.5 . Apart from this, the difference between Sellars’ model and Yada’s model is that the exponent used in Sellars’ model is 1, but in Yada’s model 2. In recent years Kim and co-workers [5,6], developed the following DRX kinetics model in the form of modified Avrami equation based on the thermomechanical simulation tests, the equation is as follows:

  ⎧ m ⎪ ⎨ XDRX = 1 − exp − ε −∗εc ε  Z −0.08 ⎪ ⎩ m = 1.12

(3)

A

where ε* is the strain for maximum softening rate, Z is the temperature-compensated strain rate, i.e. the Zener–Hollomon parameter, A is a material constant. In Kim’s model, the strain for maximum softening rate—ε* is used instead of the strain for 50% DRX ε0.5 of Yada’s model. According to the classic theory, the development process of DRX is similar to that of phase transition. The velocity of DRX begins from zero and slowly increases, and after a latent period the velocity of DRX starts to rapidly increase, and after it reaches the maximum the velocity of DRX gradually decreases, in the end the velocity of DRX is close to zero. Since ε* represents the strain for maximum softening rate, i.e., the strain for maximum velocity of DRX, ε* in Kim’s model physically coincides with ε0.5 (the strain for 50% DRX) in Yada’s model. The constant 0.693, however, is not used in Kim’s model and the component mˇı is described as the function of Z through regression analysis. Compared with Sellars’ model, the parameters in Kim’s model do not have the obvious physical meanings, but just statistic meanings. Kopp and co-workers [7,8] put forward the following DRX kinetics model in the form of strain function:

XDRX = 1 − exp



−kd

 ε − ε  c

ε0.5 = k1 Z n1

ε0.5 − εc

Laasraoui and Jonas [9], Serajzadeh and Taheri [10] made great efforts to build the DRX kinetics models suitable for different kinds of steel, the equations can be generalized as the following form:

c

ε0.5

(4)

where εc is the critical strain, ε0.5 is the strain for 50% DRX, kd , k1 and n1 are material constants. Kopp did not assign the favorite value to kd . By analyzing it is able to be found that the variables in Kopp’s model have obvious physical meanings when kd is equal to 0.693. Similar with Sellars’ model, the exponent in Kopp’s model equals 1.

301

XDRX = 1 − exp



−kd

ε − εc εp

nd

(5)

where εp is the peak strain, εc is the critical strain, kd and nd are material constants. In Eq. (5) ε0.5 is substituted by εp in contrast to Yada model. As far as kd and nd was concerned, some researchers think that they are the function of Zener–Hollomon parameter or the function of ε˙ and T, but other researchers think that they are constants under different deformation conditions. Recently, Eq. (5) is often used as the general form of the DRX kinetics model. By analysis, it can be found that all the above typical kinetics models of DRX are based on Avrami function and the differences among them are the choices of the constants and the exponents and the using of ε0.5 , ε* and εp in the models. Synthetically considering Sellars’ model, Kopp’s model, Yada’s model and the general form Eq. (5), the following modified Kopp’s model is used to discuss the classical DRX kinetics models based on Avrami form. XDRX = 1 − exp



 ε − ε nd  c

−0.693

ε0.5 − εc

(6)

where εc is the critical strain, ε0.5 is the strain for 50% DRX, and nd is a material constant. The using of 0.693 in Eq. (6) makes all the variables in model have obvious physical meanings. As the values of the exponent nd used in Sellars’ model, Yada’s model and Eq. (5) are different; the influence of different nd values on the DRX kinetics models based on Avrami form is able to be investigated by analyzing the modified Kopp’s models with different nd values. So, as a typical example of the classical DRX kinetics models, the modified Kopp’s model Eq. (6) is discussed in the next section. 3. A new kinetics model of DRX The characteristics of DRX are as follows: the dynamically recrystallized volume fraction equals zero when the strain is smaller than the critical strain, and the maximum of the dynamically recrystallized volume fraction equals 1; once the strain exceeds the critical strain, the dynamically recrystallized volume fraction first slowly increases, and then rapidly increases, at last slowly increases. Based on these characteristics of DRX process and the feature of the limit of exponent function, the following new kinetics model of DRX was proposed. XDRX = [1 + kv

(1−(ε−εc /ε0.5 −εc )) −1

]

(7)

where εc is the critical strain; ε0.5 is the strain for 50% DRX; kv is a constant related to the velocity of DRX, which is mainly decided by the initial grain size and the stacking-fault energy. Take εc of 0.2 and ε0.5 of 0.5 for example, the curves of DRX kinetics based on the modified Kopp’s model at different values of nd by Eq. (6) are shown in Fig. 1(a). Take εc of 0.2 for example, the curves of DRX kinetics based on the new kinetics model by Eq. (7) at different values of kv and ε0.5 are shown in Fig. 1(b). It can be found from Fig. 1(a) that when the strain exceeds the critical strain εc , the kinetics curves of DRX based on the modified Kopp’s model exhibit different rising trends with increasing strain at different values of nd . Moreover, the dynamically recrystallized volume fraction is greater than zero when the strain is less than the critical strain εc and the value of nd is equal to even numbers 2 and 4; the dynamically recrystallized volume fraction is less than zero when the strain is less than the critical strain εc and the value of nd is equal to odd numbers 1 and 3. It is inconsistent with the rule of DRX process and it is caused by the features of exponential function. Fig. 1(b) shows that the new kinetics model of DRX is consistent with the rule of DRX. By analyzing Eq. (7), it is able to be

302

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

Fig. 1. Schematic of kinetics curves based on different DRX kinetics model (a) based on modified Kopp’s model (b) based on new kinetics model.

found that once the model parameter kv is appropriately chosen as a larger value, the value of the dynamically recrystallized volume fraction—XDRX tends to zero when ε is less than εc , XDRX tends to 1 when ε is more than εc and XDRX identically equals 0.5 when ε is equal to ε0.5 . The velocity function of DRX is proposed for further analyzing the process of DRX. It is referred to as the variation of the dynamically recrystallized volume fraction with strain, namely, the first partial derivative of the dynamically recrystallized volume fraction to strain, the form is as follows:

vDRX =

∂XDRX ∂ε

(8)

where vDRX is the velocity of DRX process, XDRX is the dynamically recrystallized volume fraction, ε is the equivalent strain. The DRX velocity function vDRX in Eq. (8) physically reflects how fast DRX happens. In general, the velocity function of a physical quantity can be written as its first derivative with respect to time. As to the kinetics model of DRX, the first derivative of the dynamically recrystallized volume fraction with respect to strain is same to its first derivative with respect to time when the strain rate is a constant. So, the velocity function of DRX using the differential coefficient divided for stain is employed. By plotting the velocity curves based on the different models, the disadvantages of the modified Kopp’s model and the advantages of the new kinetics model of DRX are explained below. Differentiating Eq. (6) with respect to strain, the velocity function of DRX based on the modified Kopp’s model can be written as

follows:

vDRX =

∂XDRX 0.693nd exp = ε0.5 − εc ∂ε



 ε − ε nd −1 c

·

 ε − ε nd  c

−0.693

ε0.5 − εc (9)

ε0.5 − εc

Differentiating Eq. (7) with respect to strain, the velocity function of DRX based on the new kinetics model can be written as follows:

vDRX =

∂XDRX ln kv (1−(ε−εc /ε0.5 −εc )) = kv ε0.5 − εc ∂ε ×(1 + kv

(1−(ε−εc /ε0.5 −εc )) −2

)

(10)

Take εc of 0.2 and ε0.5 of 0.5 for example, the velocity curves based on the modified Kopp’s model at different values of nd by Eq. (9) are shown in Fig. 2(a). Take εc of 0.2 for example, the velocity curves based on the new kinetics model of DRX at different values of kv and ε0.5 by Eq. (10) are shown in Fig. 2(b). From Fig. 2(b), it can be seen that the velocity curves based on the new kinetics model of DRX present a single peak. First, the velocity of DRX begins with zero and slowly increases, and then rapidly increases, reaches a maximum and rapidly decreases, finally slowly decreases and gradually tends to zero, which is in accord with the classic theory, and the velocity curves of DRX is similar to the curves when the exponent nd is greater than or equal to 3 in Fig. 2(a). In Sellars’ and Kopp’s models, the exponent is used as 1. When the exponent nd is equal to 1, Fig. 1(a) shows that the curve of DRX kinetics is characterized by convex increasing tendency and

Fig. 2. Schematic of velocity curves based on different DRX kinetics model (a) based on modified Kopp’s model (b) based on new kinetics model.

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

Fig. 2(a) shows that the velocity curve of DRX exhibits monotone decreasing tendency, which is inconsistent with the developing process of DRX. In Yada’ model, the exponent is used as 2. When the exponent nd is equal to 2, Figs. 1 and 2(a) show that the curves are basically in agreement with the kinetics law of DRX. In the beginning, however, the velocity of DRX increases quickly until it reaches the maximum, after which it decreases more slowly. It can also be seen that the velocity of DRX has begun to decrease before the strain for 50% DRX. When the exponent nd is greater than 2, it can be found from Fig. 2(a) that the increasing speed of the velocity of DRX is similar to it decreasing speed and the peak velocity of DRX is approximately equal to ε0.5 . By further analyzing the new model, three key strains during the developing process of DRX during are investigated. Construct the first differential of the velocity function of DRX Eq. (10) and let the first differential be zero, the stagnation point of this function will be obtained. Because the kinetics curve of DRX is a monotonic increasing one, this point is the extreme point of velocity function. ∂vDRX ∂(∂XDRX /∂ε) = =0 ∂ε ∂ε

(11)

Substitute Eq. (10) with Eq. (11) and simplify: ∂vDRX = ∂ε

 ln k 2 v ε0.5 − εc ×(1 + kv

kv

(1−(ε−εc /ε0.5 −εc ))

(1−(ε−εc /ε0.5 −εc )) −3

)

(1 − kv

(1−(ε−εc /ε0.5 −εc ))

=0

)

303

mean grain size is slowly refined. For this reason, in order to obtain optimal microstructure, save energy consumption and reduce production cost, ε3 is the most economic strain which can guarantee fine grains and better mechanical properties. Compared with the kinetics model of DRX in the form of Avrami equation, the new kinetics model has following special features: 1) Due to the characteristics of exponential function, the curves of DRX kinetics model based on Avrami equation are not equal to zero at the strain less than the critical strain. The new model, however, the dynamically recrystallized volume fraction tends to zero when the strain is less than the critical strain. 2) In this new model, ε0.5 has clear physical meaning, which exactly represents the strain corresponding to the dynamically recrystallized volume fraction of 50%. So, the calculation is convenient by parameter regression. 3) The parameter kv reflects the developing speed of DRX process and the extent of rapid-increasing phase of DRX process. 4) Compared with the usual form Eq. (5), fewer parameters need be determined by regression analysis in this new model. There is the only parameter kv to be determined, apart from the parameters εc and ε0.5 . 5) The most economic strain ε3 that guarantees fine grains can be obtained at a given temperature and strain rate, which is important to determine processes in industrial production.

(12) 4. Material and experimental results

The solution of Eq. (12) is ε = ε0.5 . It is clear that the value of Eq. (10) is the maximum at the strain of ε0.5 , whose physical meaning is that the increase rate of the dynamically recrystallized volume fraction is the quickest at this point. Then, construct the second differential of the velocity function of DRX and let the second differential be zero, the turning point of Eq. (10) will be obtained. ∂2 vDRX = ∂ε2

 ln k 3 v ε0.5 − εc −4kv

kv

(1−(ε−εc /ε0.5 −εc ))

(1−(ε−εc /ε0.5 −εc ))

(kv

+ 1)(1 + kv

2(1−(ε−εc /ε0.5 −εc ))

(1−(ε−εc /ε0.5 −εc )) −4

)

=0 (13)

The solution of Eq. (13) is √ kv =2± 3 (1−(ε−εc /ε0.5 −εc ))

(14)

By simplifying, the final solution is ε = ε0.5 ∓

ε0.5 − εc 0.76 ln kv

(15)

The strains in Eq. (15) are corresponding to two turning points of the velocity function of DRX. Considering the strain corresponding to the peak of the velocity curve of DRX, there are three key strains, which is rewritten by ε1 = ε0.5 −

ε0.5 − εc 0.76 ln kv

Take magnesium alloy AZ31B (Mg–3%Al–1%Zn) for example, the diameter of the sample is 10 mm and the length is 15 mm. The isothermal compression tests were performed on the thermomechanical simulator Gleeble-1500 at the temperatures ranging from 523 K to 673 K and at the strain rates 0.001 s−1 , 0.01 s−1 , 0.1 s−1 and 1 s−1 . The maximum true strain is 1.0. The stress–strain curves obtained from the tests are shown in Fig. 3. According to Fig. 3, the characteristics of AZ31B stress–strain curves are represented as follows: 1) In the initial stage of the forming process, the stress abruptly increases to a peak due to the dominance of work hardening. 2) When the strain rate increases at the constant temperature, or the temperature decreases at the constant strain rate, the overall level of the flow curve enhances correspondingly due to the growing work hardening. 3) When deformation exceeds the peak strain, the flow stress decreases at a rate, which reduces with increasing stain as softening caused by DRX overtakes hardening caused by work hardening. 4) The flow stress shows steady state region due to the equilibrium of work softening and work hardening.

ε2 = ε0.5

ε3 = ε0.5 +

ε0.5 − εc 0.76 ln kv

(16)

According to the three key strains, the developing process of DRX can be divided into three phases: slow-beginning phase {0, ε1 }; rapid-increasing phase {ε1 , ε3 }; and slow-rising-to-balance phase {ε3 , 1}. It can be seen from Eq. (16) that ε2 is situated in the middle of rapid-increasing phase, and at that moment the velocity of DRX process is maximum. Of the three key strains, ε3 is of great significance to optimize industrial processes. When the strain is less than ε3 , the developing process of DRX has rapid velocity, i.e., the mean grain size is rapidly refined; when the strain is greater than ε3 , the developing process of DRX has slow velocity, i.e., the

5. Determination of parameters and verification Bergstrom [11] pointed out that for the material characterized by DRX, the flow stress can be written by  = p − XDRX (p − ss )

(17)

where XDRX is the dynamically recrystallized volume fraction;  p and  ss are the peak stress and the steady-state stress, respectively. XDRX can be calculated according to the flow stress curves by converting Eq. (17) into the following form: XDRX =

 − p ss − p

(18)

By Eq. (7), the following form can be obtained. 1 − XDRX (1−(ε−εc /ε0.5 −εc )) = kv XDRX

(19)

304

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

a

b

200

180

180

160

160

140

True stress, σ (MPa)

True stress, σ (MPa)

200

0

250 C

120

0

300 C 0

350 C

100

0

400 C

80 60 40

0

350 C

120

0

400 C

100 80 60 40 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain,

ε

True strain,

d

200 0

160

0

350 C

140

ε

200 180

0

300 C

True stress, σ (MPa)

250 C

180

True stress, σ (MPa)

0

300 C

140

20

20

c

0

250 C

0

400 C

120 100 80 60 40

160 140 120 100 80 60 40

0

250 C

0

300 C

0

350 C

20

20

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

True strain,

0

400 C

ε

True strain,

ε

Fig. 3. The stress–strain curves of magnesium alloy AZ31B (a) ε˙ = 0.001 s−1 , (b) ε˙ = 0.01 s−1 , (c) ε˙ = 0.1 s−1 , and (d) ε˙ = 1 s−1 .

Take log of both sides of Eq. (19): ln

1 − X

DRX

XDRX





= ln kv 1 −

ε − εc ε0.5 − εc

 (20)

The value of kv can be obtained after the relation between ln ((1 − XDRX )/XDRX ) and 1 − (ε − εc )/(ε0.5 − εc ) is determined. In order to determine the relationship between ln ((1 − XDRX )/XDRX ) and 1 − (ε − εc )/(ε0.5 − εc ), the parameters εc and ε0.5 should be regressed in advance. As to the material characterized by strain softening at elevated temperatures, the schematic of the relation between workhardening rate  ( = d/dε) and  is shown in Fig. 4. It can be seen that the flow stress corresponds to the peak stress  p and the steady state stress  ss when  equals 0. Kim and co-workers [5,6] thought that the – curve can be divided into three segments. The first segment begins from the initial stress  0 to the critical stress of DRX  c , where the work-hardening rate  is positive and the curve has greater slope and decreases quickly due to the interaction between work hardening and DRV. The second segment begins from the critical stress  c to the peak stress  p , where the work-hardening rate  is still positive but has lower slope and decreases slowly. It is because that the softening caused by DRX do not counteract the work hardening. So, the critical strain can be determined according the – curve approximately. The third segment begins from the peak stress  p to the steady state stress  ss , where the work-hardening rate  is negative since DRX is the predominant microstructure evolution mechanism. The flow stress becomes steady state when DRX and work hardening reach equilibrium.

The critical strain for initiation of DRX is considered to be difficult to be determined by metallography, since large numbers of samples before and after the critical strain require observing and analyzing and microstructural changes during cooling from the elevated temperature to the room temperature can influence the deformed structure. Generally, the relation between the critical strain and the peak strain is written by, εc = kεp

(21)

θ

σc

σ ss

σp

σs

Fig. 4. Schematic of typical – curve.

σ

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310 Table 1 The values of coefficient k used in the references. Authors

Material

Coefficient

Fernández et al. [12] Imbert and McQueen [13] Anelli [14] Cabrera et al. [15] Karjalainen et al. [16] Barnett et al. [17] Kim et al. [5] Kim and Yoo [6] Sun and Hawbolt [18]

Nb and Nb–Ti microalloyed steels A2 and M2 tool steels C–Mn and eutectoid steels Steels Nb microalloyed Steels Steels Microalloyed medium carbon steel AISI 304 stainless steel 0.8333

0.70–0.83 0.6–0.7 0.85 0.8 0.6–0.9 0.8 0.6184 0.60–0.7 0.8333

where k is a coefficient. The values of k used in references are various for different materials as listed in Table 1, from which it can be seen that the values of k range from 0.6 to 0.9. Recently, Policak and Jonas [19], Najafizadeh and Jonas [20] pointed that the onset of DRX was corresponding to the inflection point of the curves of the strain hardening rate –, the relation between  and  can be represented by a third order equation (before the peak stress) and the critical strain can be obtained by setting the second derivative of this equation to zero. However, the value of critical strain for magnesium alloy AZ31B is approximate 0.4 by the approach proposed by Policak and Jonas, which is less than the normal range 0.6–0.9. The – curves at different temperatures and strain rates are shown in Fig. 5. It can be seen that the relation between  and  is inconsistent with the third order equation, which is probably the reason causing the computation error. Under the inspiration of the approach put forward by Policak and Jonas [19], Najafizadeh and Jonas [20], the critical strain was approximately determined according to the relation between  and . From Fig. 5, it can be seen that before the peak stress, i.e. the strain hardening rate  is equal to zero, the value of  appears to be

305

a decent trend. Take the stress–strain curve at the temperature of 573 K and the strain rate of 0.1 s−1 as an example, the relationship between  and ε, the relationship between  and , the relationship between d/d and  and the relationship between d2 /d 2 and  are shown in Fig. 6(a)–(d), respectively. The stain hardening rate  ( = d/dε), d/d and d2 /d 2 are obtained by second-order central difference scheme. The – curve at the temperature of 573 K and the strain rate of 0.1 s−1 shows a turning point which is related to the onset of working softening by DRX. In order to determine the position of the turning point, d/d– curve and d2 /d 2 – curve are plotted and it can be found that when the strain exceeds the specific value (marked with a little red microscopic “o”), the value of d2 /d 2 shows a sharp decline. According to a closest estimate, the ratio of the critical strain to the peak strain is equal to 0.81. So, Eq. (21) can be rewritten as follows: εc = 0.81 · εp

(22)

In general, the value of εp can be written as the function of the Z parameter, which is referred to as temperature-compensated strain rate i.e. the Zener–Hollomon parameter and the formula is as follows: Z = ε˙ exp

Q  RT

(23)

where ε˙ is the strain rate, Q is the effective activation energy for deformation, R is the gas constant, T is the temperature. Sellars and Whiteman [2] pointed out that the stress can be considered to be dependent on the temperature and strain rate and modeled using creep equation: Z = A(sin h(˛))

n

(24)

where A, ˛ and n are the undertermined parameters. McQueen et al. [3] have pointed out that for metal with DRX,  can be referred to

Fig. 5. Schematic of the relationship between  and  for magnesium alloy AZ31B (a) ε˙ = 0.001 s−1 , (b) ε˙ = 0.01 s−1 , (c) ε˙ = 0.1 s−1 , and (d) ε˙ = 1 s−1 .

306

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

Fig. 6. Schematic of the relationship between strain hardening rate and stress at the temperature of 573 K and the strain rate of 0.1 s−1 for magnesium alloy AZ31B (a)  versus ε (b)  versus  (c) d/d versus  (d) d2 /d 2 versus .

a 5.5

b

160

σp (MPa)

5.0

120

lnσp

4.5 4.0

o

250 C o 300 C o 350 C o 400 C

3.5 3.0 -8

-7

-6

-5

-4

.-3

lnε

-2

-1

0

d

1.0 0.5 0.0

0.001/s 0.01/s 0.1/s 1/s

-0.5 -1.0 -1.5 1.4

1.5

1.6

1.7

-3

1.8

1/T (10 /K)

1.9

80

0 -8

1

1.5

o

250 C o 300 C o 350 C o 400 C

40

lnsinh(ασp)

lnsinh(ασp)

c

200

-7

-6

-5

-4

.-3

lnε

-2

-1

1

1.5 1.0 0.5 0.0 o

250 C o 300 C o 350 C o 400 C

-0.5 -1.0

2.0

0

-1.5 -8

-7

-6

-5

-4

-3

.

lnε

-2

-1

0

1

˙ (b) Schematic of  p versus ln ε, ˙ (c) schematic of Fig. 7. Schematic of the dependence of the peak stress on temperature and strain rate: (a) Schematic of ln  p versus ln ε, ln (sin h(˛ p )) versus 1/T, (d) schematic of ln (sin h(˛ p )) versus ln ε˙ [21].

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

0.0

307

0.0 Experiment data

-0.5

-0.5

Linear fit

-1.0 -1.0 lnε0.5

lnεp

-1.5 -2.0 -2.5

-1.5 -2.0

-3.0

Experiment data

-2.5 -3.5

Linear fit

-3.0

-4.0 5

10

15

20

25

30

35

40

45

0

50

the peak stress  p or the start of steady state stress  m and the only difference between the Q’s values is the value of Q in terms of  p is higher than the value of Q in terms of  m . As the value of the peak stress  p is easier to be obtained from the Gleeble tests than  m , the peak stress  p is used in the present study. The values of A and n can be obtained from the plot of ln Z versus ln (sin h(˛ p )). But the values of ˛ and Q must be determined before the values of A and n are fixed. The approximate value of ˛ is determined as follows.

20

30

40

50

lnZ

lnZ Fig. 8. Schematic of the dependence of εp on Zener–Hollomon parameter.

10

Fig. 9. Schematic of the dependence of ε0.5 on Zener–Hollomon parameter.

The hot working favored the power law at the high stress level and the equation is presented as follows: 

Z ≈ A1  n

(25)

The favorite equation is the exponent law at the low stress level, which is given by Z ≈ A2 exp (ˇ)

Fig. 10. Schematic of the relationship between  and ε for magnesium alloy AZ31B (a) ε˙ = 0.001 s−1 , (b) ε˙ = 0.01 s−1 , (c) ε˙ = 0.1 s−1 , (d) ε˙ = 1 s−1 .

(26)

308

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

Strain for max softening rate

1.0

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

ε0.5

0.6

0.8

1.0

Fig. 11. Schematic of the comparison between the strain for maximum work softening rate and ε0.5 .

Fig. 12. Schematic of ln ((1 − XDRX )/XDRX ) versus 1 − (ε − εc )/(ε0.5 − εc ).

 p plot at the high stress level. The value of Q is calculated in the following equation: Assuming the value of n equals n, the approximate value of n can be taken as the slope of the plot of ln ε˙ versus ln  at the low stress level. It can be seen by comparing Eq. (24) with Eq. (26) that A is equal A2 /2n and the value of ˛ is approximately equal to ˇ/n. The value of A2 and ˇ can be determined according to the ln ε˙ versus

 Q =R

∂(ln (sin h(˛))) ∂(1/T )

 · ε˙

∂ ln ε˙ ∂(ln (sin h(˛)))

(27) T

on the right-hand side of the above formula the first term represents the slope of the ln (sin h(˛ p )) versus 1/T plot and the

Fig. 13. Schematic of the kinetics curves of DRX of magnesium alloy AZ31B (a) ε˙ = 0.001 s−1 , (b) ε˙ = 0.01 s−1 , (c) ε˙ = 0.1 s−1 , and (d) ε˙ = 1 s−1 .

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

309

The relationship of εp and the Z parameter is obtained according to Fig. 8 and the simplified form is as follows: εp = 0.0075 × Z 0.1029

(28)

By Eqs. (22) and (28), the critical strain εc is obtained. εc = 0.0061 × Z 0.1029

(29)

The strain for 50% DRX ε0.5 can be obtained by Eq. (16). The relationship of ε0.5 and the Z parameter is obtained according to Fig. 9 and the simplified form is as follows: ε0.5 = 0.0426 × Z 0.0781

Fig. 14. The initial grain of magnesium alloy AZ31B.

second term represents the reciprocal value of inclination of the ln (sin h(˛ p )) versus ln ε˙ plot. The approximate value of n is 8.6625 according to Fig. 7(a), and that of ˇ is 0.0978 according to Fig. 7(b). So, the suitable value of ˛ is 0.0113. The average value of slope of the ln (sin h(˛ p )) versus 1/T is approximately 3.0875 × 103 according Fig. 7(c). The reciprocal value of inclination of the ln (sin h(˛ p )) versus ln ε˙ is approximately 6.1837 according to Fig. 7(d). So, the value of Q equals 158.7323 kJ/mol and the corresponding Z parameter can be obtained.

(30)

In order to further discuss the physical meaning of ε0.5 , the strain corresponding to the maximum softening rate is obtained by plotting the –ε curves at different temperatures and strain rates, which is marked with a red microscopic “o”, as shown in Fig. 10. The comparison between the strain for maximum work softening rate from Fig. 8 and ε0.5 calculated according to Eq. (30) is shown in Fig. 11. There is no significant difference between them, even under some deformation conditions, they are almost equal. So, it can be considered to be reasonable that the velocity of DRX reaches the maximum or the work softening rate reaches the maximum when the strain is ε0.5 in the new kinetics model of DRX. Finally, according to Eq. (20) the relation between ln ((1 − XDRX )/XDRX ) and 1 − (ε − εc )/(ε0.5 − εc ) is fitted as shown in Fig. 12 and the parameter kv is obtained, whose value is 150. So, the new kinetics model of DRX is as follows: XDRX = (1 + 150(1−(ε−εc /ε0.5 −εc )) )−1

(31)

By Eqs. (16), (29) and (30), the most economic strain ε3 is obtain, ε3 = 0.0529 × Z 0.0765

Fig. 15. Micrographs of magnesium alloy AZ31B at 300 ◦ C at the strain rate of 0.1 s−1 with the strain of (a) 0.15, (b) 0.3, (c) 0.5, and (d) 0.7.

(32)

310

J. Liu et al. / Materials Science and Engineering A 529 (2011) 300–310

6. Conclusions

Volume fraction of DRX, Xdrx

1.0 0.9

The following conclusions can be drawn:

0.8 0.7 0.6 0.5 0.4 0.3 0.2

Predicted by model Experiment data

0.1 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

True strain, ε Fig. 16. Comparison of the predicted value by the model with the experimental data.

The DRX kinetics curves of magnesium alloy AZ31B at different temperatures and strain rates are shown in Fig. 13 according to Eqs. (29)–(31). It can be seen from Fig. 13 that the dynamically recrystallized volume fraction increases with increasing strain. When the strain is constant, the dynamically recrystallized volume fraction is higher at the higher deformation temperature or lower strain rate. In order to validate the accuracy of the new model, the experiments by manual measuring method in quantitative metallography were conducted. The initial grains of magnesium alloy AZ31B are shown in Fig. 14 and the mean grain size is 22 ␮m. The micrographs of magnesium alloy AZ31B at the temperature of 300 ◦ C, at the strain rate of 0.1 s−1 and at different strains are shown in Fig. 15. It can be found that from Fig. 15: at the strain of 0.15 DRX occurs only in a very small proportion of grains, at the strain of 0.3 DRX occurs in a little more part of grains, at the strain of 0.5 DRX occurs in the most part of grains and at the strain of 0.7 DRX occurs in almost all the grains. In conclusion, the recrystallized grains increase and the mean grain size decreases with the increase of strain. The comparison of the predicted dynamically recrystallized volume fraction according the new kinetics model with the experimental results is shown in Fig. 16, which demonstrates that the calculated values basically are in accordance with the experimental ones. Consequently, the new kinetics model is in agreement with the development law of DRX process (slowly–rapidly–slowly), and includes few parameters, which have clear physical meaning and are easy to determine, moreover, coincides with the microscopic examination.

1) The new kinetics model of DRX reflects the ‘slow–rapid–slow’ property of DRX development. According to the new model, the development process of DRX can be divided into three phases: slow-beginning phase, rapid-increasing phase and slow-risingto-balance phase. The strain at the turning moment between the second phase and the third one can be considered as the most appropriate and economic strain that guarantees fine grains and saves energy consumption. 2) This kinetics model of DRX has fewer parameters—kv except εc and ε0.5 compared to the conventional kinetics models, the parameters of which include kd and nd except εp and εc or ε0.5 and εc . 3) This kinetics model of DRX is in good agreement with microscopic observation. The results manifest that the proposed model is valid and accurate. Acknowledgements This work was supported by the National Basic Research Program of China under Grant No. 50905110. References [1] C.M. Sellars, Materials Science and Technology 6 (1990) 1072–1081. [2] C.M. Sellars, J.A. Whiteman, Metal Science 13 (3–4) (1979) 187–194. [3] H.J. McQueen, S. Yue, N.D. Ryan, et al., Journal of Materials Processing Technology 53 (1995) 293–310. [4] H. Yada, In: G.E. Ruddle and A.F. Crawley, (Eds.), Proc. Int. Symp. Accelerated Cooling of Rolled Steel, Conf. of Metallurgists, CIM, Winnipeg, MB, Canada, Aug. 24–26, 1987, Pergamon Press, Canada, pp. 105–120. [5] S.-I. Kim, Y. Lee, D.-L. Lee, et al., Materials Science and Engineering A 355 (2003) 384–393. [6] S.-I. Kim, Y.-C. Yoo, Materials Science and Engineering A 311 (2001) 108–113. [7] R. Kopp, M.L. Cao, M.M.d. Souza, Proceedings of the Second International Conference on Technology of Plasticity, 1987, pp. 1129–1134. [8] R. Kopp, K. Karnhausen, M.M.d. Souza, Scandinavian Journal of Metallurgy 20 (1991) 351–363. [9] A. Laasraoui, J.J. Jonas, Metallurgical Transactions A 22A (7) (1991) 151–160. [10] S. Serajzadeh, A.K. Taheri, Mechanics Research Communications 30 (2003) 87–93. [11] Y. Bergstrom, Materials Science and Engineering (5) (1969/1970) 193–200. [12] A.I. Fernández, P. Uranga, B. López, J.M. Rodriguez-Ibabe, Materials Science and Engineering A361 (2003) 367–376. [13] C.A.C. Imbert, H.J. McQueen, Materials Science and Engineering A313 (2001) 104–116. [14] E. Anelli, ISIJ International 32 (3) (1992) 440–449. [15] J.M. Cabrera, J. Ponce a, J.M. Prado, Journal of Materials Processing Technology 143–144 (2003) 403–409. [16] L.P. Karjalainen, T.M. Maccagno, J.J. Jonas, ISIJ International 35 (12) (1995) 1523–1531. [17] M.R. Barnett, G.L. Kelly, P.D. Hodgson, Scripta Materialia 43 (2000) 365–369. [18] W.P. Sun, E.B. Hawbolt, ISIJ International 37 (10) (1997) 1000–1009. [19] E.I. Poliak, J.J. Jonas, ISIJ International 43 (5) (2003) 684–691. [20] A. Najafizadeh, J.J. Jonas, ISIJ International 46 (11) (2006) 1679–1684. [21] J. Liu, Z. Cui, C. Li, Computational Materials Science (SCI/EI) 41 (3) (2008) 375–382.