A new method to establish dynamic recrystallization kinetics model of a typical solution-treated Ni-based superalloy

Computational Materials Science 122 (2016) 150–158

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A new method to establish dynamic recrystallization kinetics model of a typical solution-treated Ni-based superalloy Ming-Song Chen a,b,⇑, Y.C. Lin a,b,c,⇑, Kuo-Kuo Li a,b, Ying Zhou a a

School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China c Light Alloy Research Institute of Central South University, Changsha 410083, China b

a r t i c l e

i n f o

Article history: Received 14 February 2016 Received in revised form 8 May 2016 Accepted 15 May 2016

Keywords: Dynamic recrystallization DRX kinetics model DRX volume fraction Ni-based superalloy Microstructure

a b s t r a c t The dynamic recrystallization (DRX) behavior of a typical solution-treated Ni-based superalloy is investigated by hot compressive tests and metallographic observations. The DRX volume fractions of deformed specimens are evaluated based on optical micrographs. Results show that the DRX volume fraction and grain size increase with the increase of deformation temperature or the decrease of strain rate. It is found that the method to calculate the DRX volume fraction based on true stress–strain curves is not suitable for the studied superalloy. i.e., the relationships between the DRX volume fraction and strain are hard to obtain. It results in the lack of adequate experimental data to establish the DRX kinetics model by the conventional method. Therefore, a new method, in which only the DRX volume fractions in the center part of deformed specimens need to be employed, is proposed. Considering the friction-induced inhomogeneous deformation in specimens, the values of strain and strain rate in the center part of deformed specimens are obtained by finite element simulation. The material constants of DRX kinetics model are determined by the least square method. An agreement between the predicted and experimental results shows that the established model can well describe the DRX behavior of the studied superalloy. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction Microstructural evolution of metals or alloys is often very complex during hot deformation [1–3]. Complex microstructural evolution mechanisms, including work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX), often occur in the metals or alloys with low stacking fault energy [4–6]. Besides, the static [7–11] and metadynamic recrystallizations [12,13] also take place in the multi-pass hot deformation process. It is well known that DRX is one of the most important microstructural evolution mechanisms, which can refine initial coarse grains and improve mechanical properties [14–16]. Therefore, understanding the DRX behaviors of metals and alloys is very significant. DRX kinetics model, which describes the relationships between the DRX volume fraction and strain, strain rate and deformation temperature, is essential to control the microstructures of forgings. In recent decades, some efforts have been made on the DRX ⇑ Corresponding authors at: School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China. E-mail addresses: [email protected] (M.-S. Chen), [email protected] (Y.C. Lin). http://dx.doi.org/10.1016/j.commatsci.2016.05.016 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.

behaviors and kinetics models of various metals and alloys [17–47]. Beladi and Hodgson [17] investigated the effect of carbon content on the DRX kinetics of Nb-steels, and found that the DRX depends strongly on thermomechanical parameters, as well as the chemical compositions. Cui et al. [18] studied the DRX behavior of TiAl alloy, and found that the DRX of c grains is the main softening mechanism. Mirzadeh et al. [19] investigated the DRX behavior of a 304 H austenitic stainless steel by the electron backscattered diffraction (EBSD) technique, and found that the recrystallized fraction can be determined from the grain average misorientation distribution based on the threshold value of 1.55°. The influence of initial microstructure on discontinuous dynamic recrystallization (DDRX) of high purity and ultra-high purity austenitic stainless steels was investigated by Wahabi et al. [20]. They concluded that larger initial grain sizes can promote a delay in the DDRX onset in the two alloys. Chen et al. [21] investigated the DRX behavior of 42CrMo steel, and established the DRX kinetics model, in which the effects of initial grain size on the DRX behavior were considered. Based on true stress–strain curves, Lv et al. [22] established the DRX kinetics model of Mg–2.0Zn–0.3Zr alloy. By isothermal compressive tests and metallographic observations, Liu et al. [23] investigated the DRX behavior of 300 M steel, and the Avrami equation was established to determine the DRX kinetics. Mirzadeh

M.-S. Chen et al. / Computational Materials Science 122 (2016) 150–158

and Najafizadeh [24] studied the DRX softening behavior of a 17–4 PH stainless steel, and found that the Avrami kinetics is suitable for extrapolation of flow curves to higher strains. Jiang et al. [25] investigated the substructure and twin boundary evolution of alloy 617B during DRX, and found that the evolution of substructure and twin boundaries have a significant effect on DRX process. Ning et al. [26,27] studied the DRX behavior and microstructural evolution of the hot isostatic pressed FGH4096 superalloy. Chen et al. [28] presented a two-dimensional cellular automata (CA) approach to predict the microstructural evolution of 316LN austenitic stainless steel during DRX. In addition, the DRX behaviors and kinetics models of 410 martensitic stainless steel [29], titanium-modified austenitic stainless steel [30,31], SCM435 steel [32], 304stainless steel [33,34], X70 pipeline steel [35], 38MnVS6 Steel [36], Cu–0.4 Mg alloy [37], magnesium alloys [38–40], Mg–Y–Nd–Zr alloy [41], Mg–Li–Al–Nd duplex alloy [42], Mg–Gd–Y–Zr alloy [43] and Ti55511 titanium alloy [44] were studied. In the previous reports, the Avrami equation [45] was often used to describe the relationships between the DRX volume fraction and strain at given deformation temperature and strain rate. ec (the critical strain for the onset of DRX) and e0:5 (the strain for 50% DRX), which are the functions of deformation temperature and strain rate [17–47], are the characteristic parameters of Avrami equation. The true stress– strain curves were often used to determine the values of ec and e0:5 at different deformation conditions [46,44,47]. Then, based on obtained ec and e0:5 , the material constants of DRX kinetics model were determined by the linear fitting method. The Ni-based superalloys, typical precipitation strengthened alloys with ultrahigh alloying degree, are widely applied in aerospace and energy industries [48,49]. Over the last decades, some investigations on hot deformation behaviors of Ni-based superalloys have been carried out [48–62]. Based on the true stress–strain data, Lin et al. [49,50], Liu et al. [51], Etaati et al. [52], Yu et al. [53] and Zuo et al. [54] developed the flow stress constitutive equations of typical Ni-based superalloys. Wen et al. [55], Zhang et al. [56] and Zhang et al. [57] established the processing maps to gain the optimum hot deformation domains of typical Ni-based superalloys. Reyes et al. [58] established the grain size model of a Ni-based superalloy using CA algorithm. Kaoumi and Hrutkay [59] found that microstructural evolution of 617 superalloy is significantly influenced by temperature and strain rate under tensile load. Zhang et al. [1,2] studied the effects of strain rate on microstructural evolution of a Ni-based superalloy during hot deformation, and found that the mechanisms for the nucleation of DDRX and CDRX are closely related to strain rate. Chen et al. [60] investigated the DRX behavior of a typical Ni-based superalloy by hot compressive tests, and the segmented models were established to describe the DRX kinetics. Liu et al. [61] studied the DRX kinetics and microstructural evolution by a CA model. Lin et al. [62] investigated the microstructural evolution of hot deformed Ni-based superalloy, and found that DDRX is the dominant nucleation mechanism. In the paper, the DRX behavior of a typical solution-treated Ni-based superalloy is investigated by hot compressive tests and metallographic observations. A new method is proposed to establish DRX kinetics model. In the new method, the frictioninduced inhomogeneous deformation of specimens is considered. The strain and strain rate in the center of deformed specimens are obtained by finite element simulation. The DRX volume fractions are evaluated based on optical micrographs. The material constants of DRX kinetics model are determined based on the DRX volume fractions and strain, strain rate and deformation temperature in the center part of deformed specimens. In addition, for the studied Ni-based superalloy, the feasibility to evaluate the DRX volume fraction by analyzing the true stress–train curves is discussed.

151

2. Material and experiments 2.1. Experimental material The material used in the study is a typical Ni-based superalloy with the composition (wt%) of 52.82Ni–18.96Cr–5.23Nb–3.01M o–1.00Ti–0.59Al–0.01Co–0.03C–(bal.) Fe. Cylindrical specimens were machined with a size of /10 mm
12 mm from a forged billet. All specimens were heated to 1040 °C and hold for 45 min in heat treatment furnace, then quenched by water. 2.2. Hot compressive experiments Hot compressive experiments were done on Gleeble-3500 thermo-mechanical simulator. Five different deformation temperatures (920, 950, 980, 1010 and 1040 °C) and four strain rates (0.001, 0.01, 0.05 and 0.1 s1) were used in the hot compressive experiments, and the deformation degrees for all specimens are 60%. The tantalum foils with thickness of 0.1 mm were used in order to minimize the friction between the specimen and dies. Before deformation, each specimen was heated to the deformation temperature at a heating rate of 10 °C/s, and then held for 300 s to eliminate the thermal gradient. The true stress–strain data were recorded by the testing system. All deformed specimens were immediately quenched by water. 2.3. Metallography experiments In order to observe the microstructures, the deformed specimens were cut along the compressive axis section. Then, the compressive axis sections were mechanically polished and etched in the solution of HCI (100 ml) + CH3CH2OH (100 ml) + CuCl2 (5 g) at room temperature for 3–5 min. The microstructure was observed by a Leica optical microscope (OM). The initial optical microstructure of the solution-treated specimens is shown in Fig. 1. From Fig. 1, it is found that the d phase is absolutely dissolved. 3. Results and discussion 3.1. True stress–true strain curves Fig. 2 shows the typical true stress–strain curves of the studied superalloy under tested conditions. All the curves are corrected to reduce the effects of frictions using the methods in Ref. [63]. It is obvious that the flow stress is sensitive to the deformation temperature and strain rate. It can be found that the flow stress increases with the increase of strain rate or the decrease of deformation temperature. This is because the critical shear stress of dislocation

Fig. 1. Initial optical microstructure of specimens after solution treatment.

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Fig. 2. Typical true stress–true strain curves of the studied superalloy at: (a) T = 980 °C; (b) e_ ¼ 0:1 s1 .

motion increases with the increased strain rate or the decreased deformation temperature [64]. In additional, the occurrence of DRX is not easy at high strain rate or low deformation temperature. From Fig. 2, it can also be observed that the flow stress firstly increases rapidly to a peak, and then decreases monotonically with the increase of strain. It is attributed to the synthetical effects of WH, DRV and DRX mechanisms. At the beginning of deformation, the rapid generation and multiplication of dislocation result in high work hardening rate [65]. So, the flow stress rapidly increases. Then, the DRV caused by dislocation climbing and sliding occurs, which makes the increase rate of flow stress slow. Once the critical strain (ec ) for initiating DRX is reached, the DRX occurs and dynamic softening is strengthened. Therefore, the increase rate of stress gradually reduces to zero, and the flow stress achieves a peak value. Afterwards, the flow stress gradually decreases until a balance between the WH, DRV and DRX is reached.

3.2. Effects of deformation parameters on DRX In this section, a new method to calculate DRX volume fractions is firstly introduced. Fig. 3(a) shows the typical optical micrograph of the deformed specimen. It can be observed that the abundant small DRX grains appear within initial large grains, especially around grain boundaries. Generally, the DRX volume fraction (X drx ) can be defined by:

X drx ¼

Adrx At

ð1Þ

in which Adrx is the area of DRX grains. At is total area of optical micrograph, which is also equal to the sum of Adrx and Ainitial . Ainitial is the area of initial grains which are not replaced by DRX grains.

Therefore, for a given optical micrograph, X drx can be evaluated by Eq. (1). However, it is hard to directly measure Adrx via the optical micrograph. Nevertheless, it is well known that the number of pixel is evenly distributed for a photo. Thus, if the number of pixel of the DRX grains area and optical micrograph are known, X drx can be evaluated by:

X drx ¼

Ndrx Ntotal

ð2Þ

in which Ndrx is the number of pixel of DRX grains area, and N total is the total number of pixel of optical micrograph. Also,

X drx ¼ 1 

Ninitial Ntotal

ð3Þ

in which N initial is the number of pixel of initial grains which are not replaced by DRX grains. The values of Ninitial and N total can be evaluated using the Photoshop software (or another image analysis software), as shown in Fig. 3(b). Therefore, for a given optical micrograph, the value of X drx can be evaluated by Eq. (3). For obtaining the DRX volume fractions in the center of deformed specimens, three different optical micrographs in the central area were used. 3.2.1. Effects of deformation temperature on DRX behavior Fig. 4 shows the optical micrographs of the central part of deformed specimens at different deformation temperatures when the strain rate is 0.01 s1. It can be found that the deformation temperature greatly affects the DRX behavior. When the deformation temperatures are 920 °C and 950 °C, the initial large grains are squashed during hot deformation. A lot of small DRX grains appear within initial grains, especially around grain boundaries. With the increase of deformation temperature, the DRX grains obviously become large, and DRX volume fraction markedly increases. When the deformation temperature is increased to 1040 °C, the DRX volume fraction almost approaches to 100%.

Fig. 3. Sketch maps for evaluating of X drx using the Photoshop software: (a) the initial optical micrograph; (b) the optical micrograph which is dealt with in the Photoshop software.

M.-S. Chen et al. / Computational Materials Science 122 (2016) 150–158

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Fig. 4. Deformed optical micrograph at the strain rate of 0.01 s1 and deformation temperatures of: (a) 920 °C; (b) 950 °C; (c) 980 °C; (d) 1010 °C; (e) 1040 °C (the central part of deformed specimens).

The DRX volume fractions at the central part of deformed specimens are evaluated by the method introduced in Section 3.2.1, as shown in Fig. 5. It is obvious that the high deformation temperature accelerates the DRX behavior. The reason is that the mobility of grain boundaries increases with the increase of deformation temperature. Therefore, the small DRX grains can rapidly grow, and replace initial large grains at the relatively high temperatures. 3.2.2. Effects of strain rate on DRX behavior Figs. 6 and 7 show the effects of strain rate on the DRX behavior at the deformation temperature of 980 °C. It can be observed that the DRX volume fraction decreases with the increase of strain rate. The DRX grains become smaller and smaller with the increase of strain rate. It indicates that the strain rate is one of major factors that affect the DRX behavior. The reasons mainly include two aspects. On the one hand, the low strain rate reduces the critical dislocation density for the onset of DRX, and thus promotes the nucleation of DRX. On the other hand, the low strain rate provides long time for the growth of small DRX grains.

Fig. 5. Effects of deformation temperature on the DRX volume fraction at the strain rate of 0.01 s1 (the central part of deformed specimens).

4. DRX kinetics model 4.1. Typical DRX kinetics model In general, the following equation is widely applied to describe the DRX kinetics [21,23,45,66,67].

8 h
nd i eec > > < X drx ¼ 1  exp 0:693 e0:5 ec e0:5 ¼ a1 e_ m1 expðQ 1 =TÞ > > : ec ¼ a2 e0:5

ð4Þ

where X drx is the DRX volume fraction, e is the strain; ec is the critical strain for the onset of DRX, e0:5 is the strain for 50% DRX, T is the deformation temperature (unit: K), a1 ; a2 ; m1 ; nd and Q 1 are the material constants. ec and e0:5 can be expressed as the functions of Zener–Hollomon parameter [60]. 4.2. Conventional method to establish DRX kinetics model based on true stress–strain curves The method to determine the values of material constants of DRX kinetics model can be obtained in many Refs. [4–6,17,21]. For the conventional method, the most important step to establish DRX kinetics model is to obtain the X drx –e curves. If the X drx  e curves are obtained at the studied deformation temperatures and strain rates, the material constants of Eq. (5) can be easily determined by the linear fitting method. In general, the volume fraction of DRX can be determined by true stress–strain curves [21,40,67,68]. Fig. 8 shows typical true stress–strain curve for dynamic recrystallization behavior. The true stress–strain curve can be divided into three stages, I, II and III. In stage I, only WH occurs. The flow stress is linearly increased with the increase of strain. In stage II, the WH and DRV occurs. The change of flow stress is influenced only by WH and DRV until the onset of DRX. If only WH and DRV occur during hot deformation, the true stress–strain curve will change along the dotted line

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Fig. 6. Deformed optical micrograph at the deformation temperature of 980 °C and strain rates of: (a) 0.001 s1; (b) 0.01 s1; (c) 0.05 s1; (d) 0.1 s1 (The central part of deformed specimens).

shown in Fig. 8. In stage III, the softening induced by DRX will decrease the flow stress of material until a balance between WH, DRV and DRX is achieved. For the conventional method, the volume fraction of DRX is evaluated by [21,67,68]:

X drx ðeÞ ¼

Fig. 7. Effects of strain rate on the DRX fraction at the deformation temperature of 980 °C (The central part of specimens).

rdrv ðeÞ  rðeÞ rsat  rss

ð5Þ

where X drx ðeÞ is the DRX volume fraction at the strain of e, rdrv ðeÞ is the flow stress at the strain of e if only WH and DRV occur in hot deformation, rðeÞ is the flow tress at the strain of e obtained by experiments, rsat is the saturation stress when a balance is achieved between the WH and DRV, rss is the steady stress under the balance of WH, DRV and DRX. The above method is suitable for some metals and alloys, such as 42CrMo steel [21] and Plain carbon steel 0.19C–0.20Si–0.40Mn [68]. However, there is a doubt whether it suits for the studied superalloy with ultrahigh alloying degree. This is because an important condition which makes Eq. (5) valid is that the dislocation motion is only inhibited by other dislocations. Therefore, the flow stress is only determined by the average dislocation density which can be decreased by DRX. i.e., the flow stress ‘at absolute zero’ is set equal to [69,70]

r^ ¼ a^ lbq1=2

ð6Þ

where q is the average dislocation density, l is an appropriate shear modulus, b is the magnitude of the Burgers vector of dislocations, a is a constant of order unity which partly depends on the strength of the dislocation/dislocation interaction. Because the effective obstacle strength can be decreased by thermal activation, the flow stress at a finite temperature (T) and strain rate (e_ ) becomes

r ¼ sðe_ ; TÞa^ lb q1=2 Fig. 8. Typical true stress–strain curve for dynamic recrystallization behavior (The solid line represent the true stress–strain curves obtained by experiment, the dotted line represents rdrv –e curve if only WH and DRV occur in hot deformation).

ð7Þ

where sðe_ ; TÞ is a function of T and e_ , and is equal to 1 as T = 0 K. In most materials, there are other contributions to the flow resistance, such as lattice resistance, grain size and solution hardening. Especially for the studied superalloy, the solution hardening

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takes a very important role in the impediment to dislocation motion. For these cases, the flow stress can be expressed as [69]:

r ¼ sðe_ ; TÞa^ lbq1=2 þ r0 ðe_ ; TÞ

ð8Þ

where r0 ðe_ ; TÞ is the addition stress caused by other contributions. Generally, some other contributions to the flow stress obey more complex, nonlinear superposition rules [69]. Thus, Eq. (5) may not adapt to evaluate the X drx for the studied superalloy. Therefore, it is necessary to verify whether Eq. (5) is suitable for the studied superalloy by applying the following method. Firstly, the values of rss can be evaluated by the true stress–strain curves, and X drx obtained by optical micrographs using Eq. (5). Then, determine whether the evaluated values of rss agree with the actual ones. Take the cases deformed at the temperature of 980 °C and all the tested strain rates as examples. According to Eq. (5), it can be known that rss can be evaluated by:

rss ¼ rsat 

rdrv;0:92  r0:92 X drx;0:92

ð9Þ

in which, r0:92 , rdrv;0:92 and X drx;0:92 are the values of r, rdrv and X drx at the strain of 0.92 (The strain for all deformed specimens is 0.92). The values of r0:92 can be directly obtained by true stress–strain curves. While the values of rsat and rdrv;0:92 cannot be directly obtained by true stress–strain curves. The method to obtain the values of rsat and rdrv ð0:92Þ are as followings. If the average dislocation density (q) is determined by the WH and DRV, the evolution of q can be expressed by KM model [69,70].

dq=de ¼ k1 q1=2  k2 q

Fig. 9. Comparisons between rdrv and true stress of the studied superalloy under tested conditions. (The solid lines represent the real true stress–strain curves; the dotted lines represent rdrv –e curves).

Table 1 The values of

rsat , C, k2 , X drx;c , rdrv;0:92 , r0:92 and rss; max when T = 980 °C.

e_ (s1)

k2

C

rsat

0.001 0.05 0.01 0.1

125.51 167.49 242.90 317.13

22.71 78.56 263.90 367.13

40.72 62.96 77.85 51.89

rdrv;0:92

X drx;c

r0:92

rss; max

77.08 69.69 27.54 21.52

98.442 120.36 201.44 250.01

90.39 99.86 92.35 5.23

Unit: MPa 125.51 167.49 242.90 317.13

ð10Þ

The production term, k1 q1=2 , is related to the athermal storage of moving dislocations which become immobilized after having traveled a distance proportional to the average spacing between the dislocations, q1=2 . The second term, k2 q; is associated with DRV which is assumed to follow the first order kinetics, i.e. to be linear in q:k2 is called as DRV coefficient. Combining Eqs. (7) and (10), the relationship between the hardening rate (h ¼ dr=de) and the stress (r) can be obtained as,

h ¼ dr=de ¼

1 1 sðe_ ; TÞalbk1  k2 r 2 2

ð11Þ

Letting

hI ¼

1 sðe_ ; TÞalbk1 2

ð12Þ Fig. 10. Finite element model for the simulation of hot compressive process.

Eq. (11) can be revised as:

1 h ¼ dr=de ¼ hI  k2 r 2

ð13Þ

If only WH and DRV occur during hot deformation, the relationships between rdrv and e can be obtained by integrating Eq. (13).

rdrv ¼ rsat  Cexpð0:5k2 eÞ

ð14Þ

in which C is the coefficient related to the critical strain of DRV. For a given deformation temperature and strain rate, the values of rsat ; C and k2 can be obtained by the non-linear fitting to the true stress–strain data of WH and DRV stage (stage II, shown in Fig. 8). By the non-linear fitting method(shown in Fig. 9), the values of rsat , C and k2 at the deformation temperature of 980 °C and strain rates of 0.001, 0.05, 0.01 and 0.1 s1 can be evaluated, as listed in Table 1. Based on the values of rsat , C and k2 , the value of rdrv;0:92 can be evaluated by Eq. (14). The values of X drx in the center part of deformed specimens (X drx;c ) were evaluated by the methods introduced in Section 3.2.1, as listed in Table 1. Due to the uneven deformation, the strain at in the center part of deformed specimens is

larger than the average strain (0.92), as shown in Fig. 10. Thus, the value of X drx;c is also larger than that X drx;0:92 . So,

rss ¼ rsat 

rdrv;0:92  r0:92 X drx;0:92

6 rsat 

rdrv;0:92  r0:92 X drx;c

¼ rss; max

ð15Þ

Based on the values of rsat , r0:92 , rdrv;0:92 and X drx;c , the values of rss; max can be evaluated by Eq. (15), as listed in Table 1. From Table 1, it can be found that the values of rss; max evaluated by Eq. (15) don’t conform to the actual. Especially, the evaluated value of rss; max for the strain rate of 0.1 s1 is only 5.23 MPa. It is greatly smaller than that for the strain rate of 0.001 s1, which is abnormal. Moreover, the evaluated values of rss; max for the strain rates of 0.001, 0.05 and 0.01 s1 are very close, which is also impossible. It indicates that Eq. (5) is not suitable for the studied Ni-based superalloy. i.e., the volume fraction of DRX cannot be evaluated by the true stress–strain curves. Therefore, the X drx  e curves cannot be obtained based on the true stress–strain curves. So, it is infeasible to establish the accurate DRX kinetics model by the conventional method due to lack of X drx  e curves.

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Table 2 Experimental and predicted X drx in the center of specimens under tested conditions. Tested conditions in the center part of specimens T (°C)

e_ (s1)

e

920 920 920 920 950 950 950 950 980 980 980 980 1010 1010 1010 1010 1040 1040 1040 1040

0.1207 0.0630 0.0134 0.0016 0.1272 0.0668 0.0129 0.0015 0.1207 0.0630 0.0140 0.0015 0.1293 0.0685 0.0129 0.0016 0.1239 0.0652 0.0132 0.0015

1.11 1.16 1.23 1.46 1.17 1.23 1.19 1.39 1.11 1.16 1.29 1.39 1.19 1.26 1.19 1.46 1.14 1.20 1.21 1.42

Experimental X drx;c (%)

Predicted X drx;c (%)

Absolute error (%)

1.59 2.52 17.23 19.82 12.28 14.54 22.54 56.77 23.16 38.43 65.63 86.45 34.87 53.59 75.54 96.54 58.75 68.45 91.66 100.00

1.96 3.86 10.74 35.33 9.40 14.42 24.21 57.88 20.20 27.90 51.37 82.16 43.49 55.43 70.27 96.96 63.78 75.13 89.98 99.72

0.37 1.34 6.49 15.51 2.88 0.12 1.67 1.11 2.96 10.53 14.26 4.29 8.62 1.84 5.27 0.42 5.03 6.68 1.68 0.28

4.3. A new method to establish the DRX kinetics model In order to establish the DRX kinetics model, a new method is proposed in the section. In the new method, the value of volume fractions of DRX in the center part (X drx;c ) at different deformation temperatures and strain rates are obtained by the method introduced in Section 3.2.1, as shown in Table 2. The friction-induced inhomogeneous deformation of specimens is considered. The strains and average strain rates in the center of deformed specimens are obtained by finite element simulation. The material constants of DRX kinetics model are determined by the least square method.

4.3.1. Finite element model to obtain the effective strain and strain rate Fig. 10(a) shows the established finite element (FE) model by DEFORM-3D software. The tetrahedral element was used in meshed specimen, and the number of elements is 20,000. During the simulation, the specimen will be remeshed if the mesh becomes unusable (negative Jacobian). The tabular data format is applied to define the flow stress of the studied superalloy. The true stress–strain data obtained by hot compressive tests are loaded

into the material models. The boundary conditions of FE models are the same as those of hot compressive tests. In order to verify the FE models, comparisons between the experimental and simulated maximum diameters (dmax ) of deformed specimens were done, as shown in Fig. 11(a). It can be found that the simulated dmax well agree with the experimental ones. It indicates that the established the FE models can accurately simulate the hot compressive process of specimens. Fig. 11(b) shows that the typical distribution of effective strain in the compressive axis section of deformed specimens. It can be found that the effective strain in the center part is greatly larger than the average strain (0.92). The values of effective strain and the average strain rate at the center part of deformed specimens under the tested conditions were obtained, as listed in Table 2. 4.3.2. Determination of material constants For the conventional method, the values of material constants a1 ; a2 ; m1 ; nd and Q 1 are determined by the linear fitting method [21]. It needs enough experimental data. For example, the values of e0:5 at different deformation temperatures and strain rates should be firstly obtained to determine the values of a1 , m1 and Q 1 . However, the values of e0:5 cannot be directly obtained by analyzing true stress–strain curves for the studied superalloy. Therefore, a new non-linear least squares fitting method is proposed in order to obtain the values of a1 ; a2 ; m1 ; nd and Q 1 . In the proposed method, the values of material constants a1 ; a2 ; m1 ; nd and Q 1 make the following function minimized.

Eða1 ; a2 ; m1 ; Q 1 ; nd Þ ¼

n X 

exp X pre drx i  X drx i

2

ð16Þ

i¼1

in which E is the sum of squared errors between the experimental are experimenand predicted X drx under the tested conditions, X exp drx i tal volume fractions of DRX under the tested conditions, as shown in Table 2. X pre drx i are the predicted volume fractions of DRX obtained by Eq. (4). The subscript i denotes the different tested conditions (deformation temperature and strain rates). n is the sum of tested conditions, and selected as is 20, as listed in Table 2. Based on the experimental data shown in Table 2, the values of a1 , a2 ; m1 ; nd and Q 1 can be evaluated as 6:43937
106 , 0.2315, 0.14281982, 1.8 and 16038.638, respectively. Thus, the DRX kinetics model for studied superalloy can be expressed as:

 8
1:8  > e ec > X ¼ 1  exp 0:693 > drx e0:5 ec <

> e0:5 ¼ 6:43937
106 e_ 0:14281982 expð16038:638=TÞ > > : ec ¼ 0:2315e0:5

ð17Þ

Fig. 11. The simulated results: (a) comparisons between the experimental and simulated dmax ; (b) the typical distribution of effective strain in the compressive axis section of deformed specimens (half of the specimen).

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Fig. 12. Comparisons between the measured and predicted X drx;c under all the tested conditions.

In order to verify the established DRX kinetics model, comparisons between the experimental and predicted X drx;c are done, as shown in Fig 12. It shows that the predicted X drx well agree with the experimental ones. The absolute errors between the experimental and predicted X drx are evaluated, as shown in Table 2. It can be found the absolute errors are smaller than 7% for most of cases. Moreover, the maximum absolute error is only 15.51%. It indicates the established DRX kinetics model can well predict the DRX behavior of the studied superalloy.

5. Conclusions The DRX behavior of a typical solution-treated Ni-based superalloy is investigated by hot compressive tests and metallographic observations. Some important results can be summarized below. (1) The effects of deformation parameters on DRX behavior are significant. The DRX volume fraction and grain size increase with the increase of deformation temperature or the decrease of strain rate. (2) It is found that the conventional method to calculate the DRX volume fraction based on true stress–strain curves is not suitable for the studied superalloy due to its ultrahigh alloying degree. Therefore, the DRX volume fractions of deformed specimens are evaluated based on optical micrographs by Photoshop software. (3) A new method, in which only the DRX volume fractions in the center part of deformed specimens need to be employed, is proposed. Considering the friction-induced inhomogeneous deformation, the values of strain and strain rate in the center of deformed specimens are obtained by finite element simulation. The material constants of DRX kinetics model are determined by the least square method. An agreement between the experimental and predicted results shows that the established model can well describe the DRX behavior of the studied superalloy.

Acknowledgements This work was supported by National Natural Science Foundation of China (No. 51305466, 51375502), National Key Basic Research Program (Grant No. 2013CB035801), State key laboratory of High Performance Complex Manufacturing (No. zzyjkt2014-01), the Open-End Fund for the Valuable and Precision Instruments of

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