A physically-based constitutive model for a typical nickel-based superalloy

Computational Materials Science 83 (2014) 282–289

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

A physically-based constitutive model for a typical nickel-based superalloy Y.C. Lin ⇑, Xiao-Min Chen, Dong-Xu Wen, Ming-Song Chen School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China

a r t i c l e

i n f o

Article history: Received 28 July 2013 Received in revised form 25 October 2013 Accepted 5 November 2013

Keywords: Hot deformation Nickel-based superalloy Dynamical recovery Dynamic recrystallization

a b s t r a c t Due to their excellent properties, nickel-based superalloys are extensively used in critical parts of modern aero engine and gas turbine. The hot deformation behaviors of a typical nickel-based superalloy are investigated by hot compression tests with strain rate of (0.001–1) s1 and forming temperature of (920–1040) °C. Results show that the flow stress is sensitive to the forming temperature and strain rate. With the increase of forming temperature or the decrease of strain rate, the flow stress decreases significantly. Under the high forming temperature and low strain rate, the flow stress–strain curves show the obvious dynamic recrystallization. Based on the stress–dislocation relation and kinetics of dynamic recrystallization, a two-stage constitutive model is developed to predict the flow stress of the studied nickel-based superalloy. Comparisons between the predicted and measured flow stress indicate that the established physically-based constitutive model can accurately characterize the hot deformation behaviors for the studied nickel-based superalloy. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Generally, material flow behaviors during hot forming processes (such as rolling, forging, and extrusion) are often complex [1]. It is well known that the work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX) often occur in the metals and alloys with low stacking fault energy during the hot deformation [2,3]. For the multi-pass hot forming process, the static and metadynamic recrystallizations [4–7] also occur. The hardening and softening mechanisms are both significantly affected by the thermo-mechanical parameters, such as forming temperature, deformation degree, and strain rate. On the one hand, a given combination of thermo-mechanical parameters determines the final microstructures and properties of the products. On the other hand, microstructural changes in alloys during the hot-forming in turn affect the flow behaviors, and hence influence the forming process. The constitutive relations are often used to describe the plastic flow properties of metals and alloys in a form that can be used in the computer code to simulate the thermo-mechanical response of mechanical parts under the prevailing loading conditions [1,8,9]. In recent years, many constitutive models have been developed or improved to describe the flow behaviors of metals or alloys. Lin and Chen [1] presented a critical review on some experimental ⇑ Corresponding author at: School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China. Tel.: +86 013469071208. E-mail addresses: [email protected], [email protected] (Y.C. Lin). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.11.003

results and constitutive descriptions for metals and alloys under hot working in recent years, and the constitutive models are divided into three categories, including the phenomenological [9–35], physically-based [36–38] and artificial neural network models [39–45], to introduce their developments, prediction capabilities, and application scopes. One of the most commonly-used phenomenological constitutive models is the Arrhenius hyperbolic-sine equation. Lin et al. [10] proposed a modified Arrhenius model to characterize the hot deformation behavior of 42CrMo steel by the compensation of strain and strain rate. Also, similar modified Arrhenius models were developed to predict the flow behaviors of 2124-T851 aluminum alloy [11], 800H superalloy [12], modified 9Cr–1Mo (P91) steel [13], Ni–20.0Cr–2.5Ti–1.5Nb– 1.0Al superalloy [14], Inconel 600 superalloy [15], as-extruded 7075 aluminum alloy [16], TC4-DT alloy [17], Mg–Zn–Cu–Zr magnesium alloy [18], as-cast 21Cr economical duplex stainless steel [19], CP–Ti alloy [20], AISI 321 austenitic stainless steel [21], Al–3Cu–0.5Sc alloy [22], GCr15 steel [23], as-cast Ti60 titanium alloy [24], cast A356 aluminum alloy [25], and Ni–42.5Ti–7.5Cu alloy [26]. Considering the coupled effects of strain, strain rate and forming temperature on the material flow behaviors of Al–Zn–Mg–Cu and Al–Cu–Mg alloys, Lin et al. [27–30] proposed new phenomenological constitutive models to describe the thermo-viscoplastic responses of Al–Zn–Mg–Cu and Al–Cu–Mg alloys under hot working condition. In their proposed models, the material constants are presented as functions of strain rate, forming temperature and strain. Amongst the empirical and

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

semi-empirical models, Johnson–Cook model [1] was successfully used to predict the hot deformation behaviors of a variety of materials. Also, some modified Johnson–Cook model were established to predict hot deformation behaviors of 42CrMo steel [31], boron steel sheet [32], titanium matrix composites [33], boron steel B1500HS [34], and 20CrMo alloy steel [35]. Based on the classical stress–dislocation relation and the kinetics of dynamic recrystallization, the constitutive equations were established to describe the flow stress during the working hardening-dynamic recovery and dynamical recrystallization periods for 42CrMo steel [36], 7050 aluminum alloy [37], and N08028 alloy [38]. In addition, neural network models were developed to predict the flow stresses of 42CrMo steel [39], A356 aluminum alloy [40], as-cast 904L austenitic [41], Al/Mg based nanocomposite [42], glass fiber reinforced polymers [43], as-cast Ti–6Al–2Zr–1Mo–1V alloy [44], and 28CrMnMoV steel [45]. Due to their comprehensive strength and toughness, excellent mechanical properties and high resistance to oxidation and corrosion, nickel-based superalloys are extensively used in critical parts of modern aero engine and gas turbine [46,47]. Generally, the nickel-based superalloy components are made through the complex thermo-mechanical processes, and their properties are significantly sensitive to the forming temperature, strain rate, as well as strain [48,49]. Therefore, in order to achieve the excellent properties of nickel-based superalloy parts, it is significant to investigate the hot deformation behaviors and optimize the thermomechanical processing parameters. Due to the limited forming temperature range and complex microstructural evolution, the hot deformation characteristics and microstructural evolution of the nickel-based superalloys were carried out by many researchers [46–53]. Ning et al. [46] investigated the hot deformation behavior of GH4169 superalloy associated with stick d phase dissolution during the isothermal compression process, and found that the d phase has great effects on the DRX and high-temperature flow behavior of GH4169 superalloy. Wang et al. [49] investigated the hot deformation of X-750 nickel-based superalloy, and established the processing maps for the studied material. Ning et al. [50] studied the hot deformation behavior of the post-cogging FGH4096 superalloy with fine equiaxed microstructure, and developed a phenomenological constitutive model to characterize the dependence of steady flow stress on the forming temperature and strain rate. Wang et al. [51,52] studied the hot deformation behaviors of superalloy 718 with d phase, and confirmed that the nucleation mechanisms of DRX in superalloy 718 are strongly dependent on the Zener–Hollomon parameter. Wu et al. [53] studied the hot compressive deformation behavior of a new hot isostatically pressed Ni–Cr–Co based powder metallurgy superalloy. Obviously, previous studies [1] mainly developed or improved the phenomenal models to describe the flow behavior of nickel-based superalloys. Although these modes can give an accurate and precise estimate of the flow stress, the clear physical meanings are still absent. Therefore, the physically-based constitutive models for nickel-based superalloys should be further studied. In this study, the hot deformation behaviors of a typical nickelbased superalloy are investigated by isothermal compression tests with the wide ranges of forming temperature and strain rate. Based on the experimental results, a physically-based model considering dynamic recovery and dynamic recrystallization mechanisms is developed to describe the relationship between the flow stress, strain rate, and forming temperature.

283

1.00Ti–0.59Al–0.01Co–0.03C–(bal.) Fe. Cylindrical specimens with a diameter of 8 mm and a height of 12 mm were machined from the wrought billet. All the specimens were solution treated at 1040 °C for 0.75 h, then followed by the water quenching. Fig. 1 shows the microstructure of the studied superalloy before the hot deformation. It is found that the microstructure is composed of fine equiaxed grains with a mean size of 75 lm. Hot compression tests were conducted on Gleeble-3500 thermo mechanical simulator under the forming temperatures of 920, 950, 980, 1010, and 1040 °C. The strain rates were selected as 0.001, 0.01, 0.1, and 1 s1, and the height reduction was 70%. Prior to the hot compression, all specimens were heated at a rate of 10 °C/s and soaked for 300 s at the forming temperature. Tantalum foil with the thickness of 0.1 mm was used between the specimen and dies to reduce the friction. The stress–strain data were automatically recorded by the testing system during the hot compression. In order to study the effects of the forming processing on the microstructure, the specimens were immediately quenched by water after hot compression. Then, the deformed specimens were sliced along the compression axis section for microstructure analysis. After polished mechanically and etched in a solution consisting of HCl(100 mL) + CH3CH2OH(100 mL) + CuCl2(5 g) at room temperature for 3–5 min, the exposed surfaces were observed by optical microscope (OM).

3. Results and discussion 3.1. Flow characteristics of the studied superalloy Due to the unavoidable interfacial friction between the specimen and dies, the deformation is inhomogeneous, and the deformed specimens reveal the barrel-type shape [41]. Generally, the calculated flow stress–strain curves overestimate the actual ones. Therefore, the effect of friction on the flow stress should be considered to acquire the accurate true stress–strain curves of the studied superalloy. In present study, the detailed method to correct the flow stress can be found in Refs. [54,55]. Fig. 2 shows the true stress–strain curves considering the effects of friction on the hot deformation behavior of the studied superalloy. Obviously, the flow stress is significantly influenced by the forming temperature and strain rate. From Fig. 2a, it can be found that the flow stress increases with decreasing the forming temperature under given strain rate. This is because the number of slip systems is limited, and the process of softening is not obvious under relatively low forming temperatures. Therefore, the work hardening mechanisms, such as dislocation intersections and pileups, lead to the increasingly high stress for the continuous deformation. While the rates for the vacancy diffusion, cross-slip of screw dislocations

2. Materials and experiments The chemical compositions (wt.%) of the studied nickel-based superalloy are as follows: 52.82Ni–18.96Cr–5.23Nb–3.01Mo–

Fig. 1. Microstructure of the studied superalloy before the hot deformation.

284

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

650

Strain rate: 0.001s-1

200 920OC

150

950OC

100

980OC 1010OC O

50

1040 C

(a)

0

0.0

0.2

0.4

0.6

0.8

True Strain

True Stress (MPa)

True Stress (MPa)

250

Forming temperature: 920OC 1s

520

-1

-1

0.1s

390

-1

0.01s

260 -1

0.001s

130 0 0.0

(b) 0.2

0.4

0.6

0.8

True Strain

Fig. 2. True stress–strain curves for the studied superalloy under the tested conditions. (a) Strain rate e_ ¼ 0:001 s1; (b) forming temperature T = 920 °C.

and climb of edge dislocations increase with increasing the forming temperature. Thus, the dynamic recovery becomes easy to take place and overcome the work hardening caused by the dislocation pile-up and tangling [56]. In addition, it is beneficial for the mobility of grain boundaries under the high forming temperature. Therefore, the nucleation and growth of dynamically recrystallized grains are accelerated (shown in Fig. 3), which decreases the flow stress [57]. As shown in Fig. 2a, it is evident that the flow stresses under high forming temperatures are much lower than those under low forming temperatures. Besides, it is obvious that the flow stress increases with the increase of the strain rate (shown in Fig. 2b). The main reason for this phenomenon is that lower strain rates can provide longer time for the nucleation and growth of dynamically recrystallized grains, as well as the dislocation motions (including slip, cross-slip and climb) [58,59]. So, the flow stress under low strain rate is lower than those under high strain rate. According to the dynamic softening mechanisms during hot deformation, the flow stress–strain curves can be divided into two types, i.e., dynamic recovery and dynamic recrystallization types. As shown in Fig. 4, the curves marked with ‘a’ is corresponding to the dynamic recovery type, which assumes the dynamic recovery is the main softening mechanism. At the beginning of deformation, the flow stress rapidly increases to the peak stress (rp) due to the work hardening. When a balance between the work hardening and dynamic recovery is achieved, a saturation flow stress (rsat) appears. While for the true stress–strain curve with dynamic recrystallization feature (marked with ‘b’ in Fig. 4), the flow stress drops gradually when the deformation degree exceeds the peak stress (ep), and then a steady flow stress (rss) appears.

Fig. 4. Sketch map of flow stress–strain curves. (Symbols ‘a’ and ‘b’ show the dynamic recovery type and dynamic recrystallization type, respectively.).

Generally, the true stress–strain curve with dynamic recrystallization characteristics can be divided into three stages: stage I (work hardening stage), stage II (softening stage) and stage III (steady stage). The stage I is the shortest and characterized by an sharp increase of flow stress. It is attributed to the fact that the generation and multiplication of dislocation occur rapidly, leading to the high work hardening rate. Meanwhile, the dynamic recovery caused by the dislocation climbing, sliding, and cross-slip is too weak to overcome the work hardening effect. In stage II, the dynamic recrystallization takes place, resulting in the decreased work hardening rate with the increase of strain. When the dynamic softening (DRV and DRX) rate is equal to the work hardening rate, the

Fig. 3. Typical dynamically recrystallization grains of the deformed superalloy under the strain rate of 0.01 s1 and deformation temperatures of: (a) 920 °C; (b) 1040 °C.

285

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

flow stress achieves a peak value. Then, the flow stress gradually decreases due to DRX. The stage II has the longest duration. When a new balance between the dynamic softening and work hardening is reached, the flow stress maintains at a fairly constant level regardless of the increased strain (in stage III). For the studied superalloy, the flow stress–strain curves show three obvious stages under low strain rates and high forming temperatures; while the steady stage is not obvious under the high strain rate and low forming temperatures, as shown in Fig. 2.

can be determined as the horizontal intercept of the tangent line of h  r plot through the inflection point. It is demonstrated that the saturation flow stress (rsat) can be expressed as the function of peak stress rp, which can be easy to be measured [38]. For the studied superalloy, the dependences of rsat on the peak stress (rp) is depicted in Fig. 6. It indicates that there is good linear relationship between rsat and rp, and the relationship can be expressed as,

3.2. Constitutive modeling of flow stress

In general, the combined effects of forming temperature and strain rate on the flow stress of metals and alloys can be characterized by the Zener–Hollomon parameter. The expression of Zener– Hollomon parameter (Z) is given as [62],

3.2.1. Modeling the work-hardening and dynamic recovery The evolution of the dislocation density with strain is generally controlled by the competition between the dislocation storage and annihilation, and the dependence of dislocation density on strain can be expressed by [36],

dq ¼ U  Xq de

ð1Þ

where dq/de is the increasing rate of dislocation density with stain; U represents the work hardening, which is a multiplication, and it can be regard as a constant respect to strain. Xq represents the dynamic recovery due to the dislocation annihilation and rearrangement, and X is the coefficient of dynamic recovery [60]. Integrating Eq. (1) gives,

q¼

U

X

 

U

X

  q0 eXe

ð2Þ

where q0 is the initial dislocation density. It is known that the effective stress can be negligible compared with the internal stress at high temperatures [36]. Thus, the applied stress can be directly estipffiffiffiffi mated by the square root of the dislocation density, r ¼ alb q; where a is the material constant; l is the shear modulus; b is the distance between atoms in the slip direction [61]. Substituting this expression into Eq. (2), the flow stress during work-hardening and dynamic recovery period can be represented as,



r ¼ r2sat þ ðr20  r2sat ÞeXe Þ

0:5

ð3Þ

where r is the flow stress.pThe rsat and the yield ffiffiffiffiffiffiffiffiffiffisaturationpstress ffiffiffiffiffiffi stress r0 are equal to alb U=X and alb q0 ; respectively. It can be found in Eq. (3) that three parameters (rsat, r0 and X) need to be determined. The saturation flow stress (rsat) is usually determined from the relationship between the work-hardening rate (h = dr/de) and flow stress (r), as shown in Fig. 5. In other words, the inflection point of h  r curve is firstly obtained (indicated by rectangle symbols). Then, the saturation flow stress (rsat)

rsat ¼ 11:30 þ 1:07rp

Z ¼ e_ expðQ =RTÞ

ð5Þ

where e_ is the strain rate (s1). r is the flow stress (MPa). R is the universal gas constant (8.31 J mol1 K1). T is the absolute temperature (K). Q is the activation energy (J mol1). In order to obtain the value of Zener–Hollomon parameter, the activation energy (Q) should be determined firstly. The detailed procedures to calculate the value of Q can be seen in authors’ previous publication [63]. The activation energy of the studied superalloy is estimated as 4.74  105 J mol1, which is close to the result (4.43  105 J mol1) of Wang et al. [52]. The yield stress r0 at variant forming temperatures and strain rates can be directly obtained from the flow stress–strain curves. Fig. 7 illustrates the relationship between the yield stress (r0) and Zener–Hollomon parameter (Z). Obviously, there is a good linear relationship between the yield stress (r0) and Zener–Hollomon parameter (Z). Then, the yield stress (r0) can be expressed as a function of Zener–Hollomon parameter,

r0 ¼ 14:86 ln Z  478:07

ð6Þ

According to Eq. (3), the coefficient of dynamic recovery (X) can be calculated by,

Xe ¼ ln



r2sat  r20 r2sat  r2

 ð7Þ

Based on the flow stress–strain before critical strain, the values of X can be determined for all the tested conditions. Fig. 8 shows the relationship between X and Zener–Hollomon parameter. It can be found that the dynamic coefficient increases with the decrease of Zener–Hollomon parameter, and X can be expressed as an exponential function of Zener–Hollomon parameter,

X ¼ 2344:39Z 0:12208

ð8Þ

600

2400

0.001s-1

920 oC

500

0.01s-1

2000 1600

σsat (MPa)

0.1s-1

θ (MPa)

ð4Þ

1s-1

1200

400 300

800

200

400

100 σsat

0 0

130

260

σsat

390

σsat

520

σsat

650

σ (MPa) Fig. 5. Relationship between the work hardening rate and stress under the forming temperature of 920 °C and different strain rates.

0

0

100

200

300

400

500

600

σp (MPa) Fig. 6. Relationship between rsat and rp. (Symbols for the experimental result, the solid line for the fitting line.).

286

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

200

stress due to the dynamic recrystallization. Substituting Eqs. (11) into (10), the flow stress during the dynamic recrystallization period can be obtained by,

150

r ¼ rrec  ðrsat  rss Þ 1  exp K d

100

As shown in Eq. (12), there are five unknown parameters including ec, ep, rss, Kd and nd. The peak strain can be easily determined by the measured flow stress curves. Fig. 9 shows the relationship between ep and Zener–Hollomon parameter (Z). So, it can be found that the peak strain (ep) can be represented as a function of Zener–Hollomon parameter (Z),

250

σ0 (MPa)



50 0 36

39

42

45

48

ln Z



e  ec ep

nd 

ðe P ec Þ

ep ¼ 0:00188Z 0:11741

Fig. 7. Relationship between the yield stress (r0) and Zener–Hollomon parameter. (Symbols for the experimental results, the solid line for the fitting line.).

ð12Þ

ð13Þ

-0.4 -0.8

ln εp

3.5

3.0

ln Ω



2.5

-1.2 -1.6 -2.0

2.0

-2.4

36

39

42

45

48

ln Z

1.5

36

39

42

45

48

Fig. 9. Relationship between ep and Zener–Hollomon parameter. (Symbols for the experimental results, the solid line for the fitting line.).

ln Z Fig. 8. Relationship between X and Zener–Hollomon parameter. (Symbols for the experimental results, the solid line for the fitting line.).

500

Therefore, the flow stress constitutive equations during the work hardening-dynamic recovery period of the studied superalloy can be summarized as,

r0 ¼ 14:86 ln Z  478:07 > > > > X ¼ 2344:39Z 0:12208 > > : Z ¼ e_ expð4:74  105 =RTÞ

ð9Þ

σss (MPa)

8 > r ¼ ½r2sat þ ðr20  r2sat ÞeXe 0:5 > > > > > < rsat ¼ 11:30 þ 1:07rp

400 300 200 100 0 0

3.2.2. Modeling the dynamic recrystallization Once the deformation degree is larger than the critical strain ec, the nucleation and growth of dynamic recrystallization grains will take place near the grain boundaries, twin boundaries, as well as the deformation bands. Especially, under high forming temperatures and low strain rates, the occurrence of dynamic recrystallization becomes more and more obvious. Generally, the volume fraction of dynamic recrystallization (XD) can be expressed as [36],

ep

rrec  r XD ¼ ðe P e c Þ rsat  rss

300

400

500

600

Fig. 10. Relationship between rss and rp. (Symbols for the experimental results, the solid line for the fitting line.).

0.0 -0.5

ð10Þ

where ec and ep are the critical strain and peak strain, respectively. Kd and nd are material constants. Meanwhile, the expression of XD can also be represented as [36,61],

200

σp (MPa)

ln Κd

    e  ec nd ðe P ec Þ X D ¼ 1  exp K d

100

-1.0 -1.5 -2.0

ð11Þ

where rrec is the flow stress when the dynamic recovery is the main softening mechanism, and it can be calculated by Eq. (3). r is the flow stress. rsat is the saturation stress due to the balance between the work hardening and the dynamic recovery. rss is the steady

-2.5 36

39

42

45

48

ln Z Fig. 11. Relationship between Kd and Zener–Hollomon parameter. (Symbols for the experimental results, the solid line for the fitting line.).

287

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

Measured Predicted

600

O

Temperature: 920 C -1

600

1s

450

0.1s

300

0.01s

-1

-1

-1

0.001s

150

True Stress (MPa)

True Stress (MPa)

750

(a)

0 0.0

0.4

0.6

-1

300

-1

0.01s

200 100

0.8

-1

0.001s

(b) 0.0

0.2

500

O

Temperature: 980 C -1

440

1s

330 -1

0.1s

220

-1

0.01s

110

-1

0.0

0.4

0.6

Measured Predicted

O

-1

1s

-1

0.1s

200 -1

0.01s

100

0.8

-1

(d)

0.001s

0.0

0.2

True Strain

0.4

0.6

0.8

True Strain

400

True Stress (MPa)

0.8

Temperature: 1010 C

300

0

0.2

0.6

400

0.001s

(c)

0

0.4

True Strain

True Stress (MPa)

True Stress (MPa)

Measured Predicted

-1

1s

0.1s

True Strain 550

O

Temperature: 950 C

400

0

0.2

Measured Predicted

500

Measured Predicted

O

Temperature: 1040 C -1

1s

300

-1

0.1s

200

-1

0.01s

100

-1

0.001s

(e)

0

0.0

0.2

0.4

0.6

0.8

True Strain Fig. 12. Comparisons between the predicted and measured flow stress curves of the studied superalloy under the temperatures of: (a) 920 °C; (b) 950 °C; (c) 980 °C; (d) 1010 °C; (3) 1040 °C.

Then, the critical strain can be estimated as [64],

Fig. 10 shows the dependences of the steady stress (rss) on the peak stress (rp). By linear fitting method, the following linear equation can be obtained to describe the relationship between rss and rp,

Then, substituting the measured flow stress data (after critical strain) into Eq. (12), the parameters Kd and nd can be easily obtained for all the tested conditions. The values of nd are calculated in range of 1.25–2.42, and the average value can be evaluated as 1.85. The relationship between Kd and Zener–Hollomon parameter is illustrated in Fig. 11. It is notable that the dependence of Kd on the Zener–Hollomon parameter can be characterized as,

rss ¼ 0:74925rp

K d ¼ 0:00033Z 0:16103

ec ¼ 0:85ep ¼ 0:00160Z

0:11741

ð14Þ

ð15Þ

Therefore, the constitutive equations during dynamic recrystallization period ðe P ec Þ of the studied superalloy can be expressed as,

700

Predicted flow stress (MPa)

ð16Þ

R=0.987

600 500 400 300 200 100 0 0

100

200

300

400

500

600

700

Measured flow stress (MPa) Fig. 13. Correlation between the measured and predicted flow stress.

n h  nd io 8 eec > r ¼ r  ð r  r Þ 1  exp K > rec sat ss d ep > > > > > > rrec ¼ ½r2sat þ ðr20  r2sat ÞeXe 0:5 > > > > > rsat ¼ 11:30 þ 1:07rp > > > > r0 ¼ 14:86 ln Z  478:07 > > > > < X ¼ 2344:39Z 0:12208 rss ¼ 0:74925rp > > > > > ep ¼ 0:00188Z 0:11741 > > > > > ec ¼ 0:00160Z 0:11741 > > > > > K d ¼ 0:00033Z 0:16103 > > > > > nd ¼ 1:85 > : Z ¼ e_ expð4:74  105 =RTÞ

ð17Þ

288

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289

3.2.3. Verification of the developed constitutive model In order to verify the developed constitutive model, the comparisons between the measured and predicted results are carried out. Fig. 12 shows the measured and predicted flow stress curves of the studied superalloy under all the tested conditions. It indicates that the predicted flow stresses well agree with the measured ones. However, for the work hardening-dynamic recovery period, there is a relatively large deviation when the forming temperatures are 920 °C and 950 °C, and the strain rate is 1 s1. The main reason for this phenomenon is that the predicted dynamic recovery rate is higher than the measured one. It is well known that the superalloy is generally kind of low stacking fault energy alloy. In materials with low stacking fault energy, the reduced mobility of dislocations lowers the rate of recovery, and high local gradients of dislocation density can induce the large grain boundary migration rate [65]. Especially under high strain rate (such as 1 s1), there is no enough time for the transformation of subgrain into grain. In addition, when the forming temperature is below or equal to the completely dissolution temperature of the nano-sized precipitate phase c00 (about 950 °C), the nano-sized precipitate phase can still strengthens the studied superalloy, which leads to the excessive deformation resistance [66]. Thus, the actual dynamic recovery rate is low, resulting in the deviation between the measured and predicted flow stress. In order to further evaluate the prediction accuracy of the developed constitutive model, the scatter map of the predicted and measured flow stresses for all the tested conditions is shown in Fig. 13. The correlation coefficient (R) and the average absolute relative error (AARE) between the predicted and measured flow stress are employed. The AARE is calculated through a term by term comparison of the relative error and therefore is an unbiased statistical parameter for determining the predictability of the equation. The correlation coefficient (R) provides information on the strength of linear relationship between the experimental and the predicted values. They can be expressed as,

PN i¼1 ðX i  XÞðY i  YÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN PN 2 2 i¼1 ðX i  XÞ i¼1 ðY i  YÞ AARE ¼

1 XN

Y i  X i

 100% i¼1 N Xi

ð18Þ

ð19Þ

where Xi and Yi are the measured and predicted flow stress, respectively. X and Y are the mean values of Xi and Yi, respectively. N is the number of data used in this investigation. The calculated correlation coefficient (R) is 0.987, which indicates that there is a good correlation between the predicted and measured data. Meanwhile, the average absolute relative error (AARE) is only 2.93%, which implies the good prediction capability of the developed model. Compared to other work, the average absolute relative error (AARE) is 4.83% when Yu [67] modeled high-temperature tensile deformation behavior of AZ31B magnesium alloy considering strain effects. Therefore, the developed constitutive equation can be used to describe the flow behavior of the studied superalloy under the elevated forming temperature. 4. Conclusions The hot compressive deformation behaviors of a typical superalloy are investigated. Based on the true stress–strain data, a physically-based constitutive model is developed to characterize the flow behavior of the studied superalloy. The proposed model can well describe the flow stress during the work hardening-dynamic recovery and dynamic recrystallization periods. The good agreements between the measured and predicted results confirm that

the developed model can give an accurate estimation of the hot deformation behavior of the studied superalloy over wide ranges of forming temperature and strain rate. However, under the low forming temperature and high strain rate, the flow stress shows a relatively low dynamic recovery rate in the early deformation stage, leading to a relatively large deviation between the measured and predicted flow stress. It is attributed to the low stacking fault energy feature of the studied superalloy, as well as the effect of nano-sized precipitate phase c00 on the hot deformation behavior. Acknowledgements This work was supported by National Natural Science Foundation of China (Grant No. 51375502), 973 program (No. 2013CB035801), Sheng-hua Yu-ying Program of Central South University, and the Fundamental Research Funds for the Central Universities of Central South University (2013zzts031), China. References [1] Y.C. Lin, X.M. Chen, Mater. Des. 32 (2011) 1733–1759. [2] M.S. Chen, Y.C. Lin, X.S. Ma, Mater. Sci. Eng. A 556 (2012) 260–266. [3] A. Momeni, K. Dehghani, G.R. Ebrahimi, J. Alloys Compd. 509 (2011) 9387– 9393. [4] S.M. Seyed, S. Serajzadeh, Comp. Mater. Sci. 49 (2010) 773–781. [5] Y.C. Lin, M.S. Chen, J. Zhong, Comp. Mater. Sci. 44 (2008) 316–321. [6] Y.C. Lin, M.S. Chen, J. Zhong, J. Mater. Process. Technol. 209 (2009) 2477–2482. [7] J. Liu, Y.G. Liu, H. Lin, M.Q. Li, Mater. Sci. Eng. A 565 (2013) 126–131. [8] S. Mandal, A.K. Bhaduri, V.S. Sarma, Metall. Mater. Trans. A 43 (2012) 2056– 2068. [9] G.Z. Quan, A. Mao, G.C. Luo, J.T. Liang, D.S. Wu, J. Zhou, Mater. Des. 52 (2013) 98–107. [10] Y.C. Lin, M.S. Chen, J. Zhong, Comput. Mater. Sci. 42 (2008) 470–477. [11] Y.C. Lin, Y.C. Xia, X.M. Chen, M.S. Chen, Comput. Mater. Sci. 50 (2010) 227–233. [12] C.Y. Sun, J.R. Liu, R. Li, Q.D. Zhang, Acta Met. Sinica 47 (2011) 191–196. [13] D. Samantaray, C. Phaniraj, S. Mandal, A.K. Bhaduri, Mater. Sci. Eng. A 528 (2011) 1071–1077. [14] Y.Q. Ning, Z.K. Yao, Mater. Sci. Eng. A 551 (2012) 7–12. [15] H.Y. Wu, F.J. Zhu, S.C. Wang, W.R. Wang, C.C. Wang, C.H. Chiu, J. Mater. Sci. 47 (2012) 3971–3981. [16] G.Z. Quan, G.S. Li, Y. Wang, W.Q. Lv, C.T. Yu, J. Zhou, Mater. Res. 16 (2013) 19– 27. [17] X.N. Peng, H.Z. Guo, Z.F. Shi, C. Qin, Z.L. Zhao, Mater. Des. 50 (2013) 198–206. [18] H. Yu, H.S. Yu, Y.M. Kim, B.S. You, G.H. Min, Trans. Nonferr. Met. Soc. China 23 (2013) 756–764. [19] D.N. Zou, K. Wu, Y. Han, W. Zhang, B. Cheng, G.J. Qiao, Mater. Des. 51 (2013) 975–982. [20] J. Peng, C.Y. Zhou, Q. Dai, X.H. He, Mater. Des. 50 (2013) 968–976. [21] M. Haj, H. Mansouri, R. Vafaei, G.R. Ebrahimi, A. Kanani, Int. J. Min. Met. Mater. 20 (2013) 529–534. [22] G.F. Xu, X.Y. Peng, X.P. Liang, X. Li, Z.M. Yin, Trans. Nonferr. Met. Soc. China 23 (2013) 1549–1555. [23] F. Yin, L. Hua, H.J. Mao, X.H. Han, Mater. Des. 43 (2013) 393–401. [24] W.W. Peng, W.D. Zeng, Q.J. Wang, H.Q. Yu, Mater. Des. 51 (2013) 95–104. [25] N. Haghdadi, A. Zarei-Hanzaki, H.R. Abedi, Mater. Sci. Eng. A 535 (2012) 252– 257. [26] A. Etaati, K. Dehghani, Mater. Chem. Phys. 140 (2013) 208–215. [27] Y.C. Lin, L.T. Li, Y.Q. Jiang, Exp. Mech. 52 (2012) 993–1002. [28] Y.C. Lin, Q.F. Li, Y.C. Xia, L.T. Li, Mater. Sci. Eng. A 534 (2012) 654–662. [29] Y.C. Lin, L.T. Li, Y.X. Fu, Y.Q. Jiang, J. Mater. Sci. 47 (2012) 1306–1318. [30] Y.C. Lin, Y. Ding, M.S. Chen, J. Deng, Mater. Des. 52 (2013) 118–127. [31] Y.C. Lin, X.M. Chen, Comput. Mater. Sci. 49 (2010) 628–633. [32] D.T. Nguyen, T.L. Banh, D.W. Jung, S.H. Yang, Y.S. Kim, High Temp. Mater. Proc. 31 (2012) 37–45. [33] W.D. Song, J.G. Ning, X.N. Mao, Mater. Sci. Eng. A 576 (2013) 280–289. [34] H.P. Li, L.F. He, G.Q. Zhao, L. Zhang, Mater. Sci. Eng. A 580 (2013) 330–348. [35] A. He, G.L. Xie, H.L. Zhang, X.T. Wang, Mater. Des. 52 (2013) 677–685. [36] Y.C. Lin, M.S. Chen, J. Zhong, Mech. Res. Commun. 35 (2008) 142–150. [37] B. Wu, M.Q. Li, D.W. Ma, Mater. Sci. Eng. A 542 (2012) 79–87. [38] L. Wang, F. Liu, Q. Zuo, C.F. Chen, Mater. Des. 47 (2013) 737–745. [39] Y.C. Lin, J. Zhang, J. Zhong, Comput. Mater. Sci. 43 (2008) 752–758. [40] N. Haghdadi, A. Zarei-Hanzaki, A.R. Khalesian, H.R. Abedi, Mater. Des. 49 (2013) 386–391. [41] Y. Han, G. Qiao, J. Sun, D. Zou, Comput. Mater. Sci. 67 (2013) 93–103. [42] V. Senthilkumar, A. Balaji, D. Arulkirubakaran, Trans. Nonferr. Met. Soc. China 23 (2013) 1737–1750. [43] H. Fazilat, M. Ghatarband, S. Mazinani, Z.A. Asadi, M.E. Shiri, M.R. Kalaee, Comput. Mater. Sci. 58 (2012) 31–37. [44] G.Z. Quan, W.Q. Lv, Y.P. Mao, Y.W. Zhang, J. Zhou, Mater. Des. 50 (2013) 51–61. [45] H.Y. Li, D.Y. Wei, Y.Y. Li, X.F. Wang, Mater. Des. 35 (2012) 557–562.

Y.C. Lin et al. / Computational Materials Science 83 (2014) 282–289 [46] Y.Q. Ning, M.W. Fu, X. Chen, Mater. Sci. Eng. A 540 (2012) 164–173. [47] K. Wu, G.Q. Liu, B.F. Hu, C.Y. Wang, Y.W. Zhang, Y. Tao, J.T. Liu, Mater. Sci. Eng. A 528 (2011) 4620–4629. [48] S.L. Guo, D.F. Li, H.J. Pen, Q.M. Guo, J. Hu, J. Nucl. Mater. 410 (2011) 52–58. [49] Y. Wang, Q.L. Pan, Y.F. Song, C. Li, Z.F. Li, Mater. Des. 51 (2013) 154–160. [50] Y.Q. Ning, Z.K. Yao, Y.Y. Lei, H.Z. Guo, M.W. Fu, Mater. Charact. 62 (2011) 887– 893. [51] Y. Wang, W.Z. Shao, L. Zhen, B.Y. Zhang, Mater. Sci. Eng. A 528 (2011) 3218– 3227. [52] Y. Wang, W.Z. Shao, L. Zhen, L. Yang, X.M. Zhang, Mater. Sci. Eng. A 497 (2008) 479–486. [53] K. Wu, G.Q. Liu, B.F. Hu, F. Li, Y.W. Zhang, Y. Tao, J.T. Liu, Mater. Des. 32 (2011) 1872–1879. [54] J.L. Uvira, J.J. Jones, F. Fereshteh-Saniee, F. Fatehi-Sichani, J. Mater. Process. Technol. 177 (2006) 478–482. [55] R. Ebrahimi, A. Najafizadeh, J. Mater. Process. Technol. 152 (2004) 136–143.

289

[56] F. Chen, Z. Cui, S. Chen, Mater. Sci. Eng. A 528 (2011) 5073–5080. [57] G.Z. Quan, G.S. Li, T. Chen, Y.X. Wang, Y.W. Zhang, J. Zhou, Mater. Sci. Eng. A 528 (2011) 4643–4651. [58] Y.C. Lin, M.S. Chen, J. Zhong, Mater. Lett. 62 (2008) 2132–2135. [59] A. Momeni, K. Dehghani, Met. Mater. Int. 16 (2010) 843–849. [60] A. Laasraoui, J.J. Jonas, Metal. Trans. A 22 (1991) 1545–1558. [61] R.W.K. Honeycombe, Edward Arnold, London, 1981. [62] C. Zener, J.H. Hollomon, J. Appl. Phys. 15 (1944) 22–32. [63] Y.C. Lin, M.S. Chen, J. Zhong, J. Mater. Process. Technol. 205 (2008) 308–315. [64] C.M. Sellars, J.A. Whiteman, Metal Sci. 13 (1979) 187–195. [65] F. Montheillet, J. Thomas, Metall. Mater. High Struct. Effic. 146 (2004) 357– 368. [66] J. Jaeger, D. Solas, T. Baudin, O. Fandeur, J. Schmitt, C. Rey, 12th Int. Symp Superalloys. 2012, pp. 663-672. [67] D.H. Yu, Mater. Des. 51 (2013) 323–330.