Anomalous behaviors of a single-crystal Nickel-base superalloy over a wide range of temperatures and strain rates

Accepted Manuscript

Anomalous behaviors of a single-crystal Nickel-base superalloy over a wide range of temperatures and strain rates Jianjun Wang , Wei-Guo Guo , Yu Su , Ping Zhou , Kangbo Yuan PII: DOI: Reference:

S0167-6636(15)00257-4 10.1016/j.mechmat.2015.11.015 MECMAT 2519

To appear in:

Mechanics of Materials

Received date: Revised date: Accepted date:

26 March 2015 18 October 2015 25 November 2015

Please cite this article as: Jianjun Wang , Wei-Guo Guo , Yu Su , Ping Zhou , Kangbo Yuan , Anomalous behaviors of a single-crystal Nickel-base superalloy over a wide range of temperatures and strain rates, Mechanics of Materials (2015), doi: 10.1016/j.mechmat.2015.11.015

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Highlights  The DD407 superalloy tested over a wide range of temperatures and strain rates. The anomalous peak of flow stress and tension/compression asymmetry observed.



Temperature-dependent rupture property of the superalloy investigated.



The strengthening and failure mechanisms discussed.

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A constitutive model developed to account for the experimental observations.

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

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Anomalous behaviors of a single-crystal Nickel-base superalloy over a wide range of temperatures and strain rates Jianjun Wanga, Wei-Guo Guo*, a, Yu Sub, Ping Zhoua, Kangbo Yuana a b

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

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Abstract: This study is concerned with the plastic behaviors of a newly developed single-crystal Nickel-base superalloy over a temperature range of 293 – 1,273 K and over a strain-rate range of 0.001 – 4,000 /s. Both static and dynamic loading test were conducted to study the true stress–true strain behaviors of the superalloy along the nominal orientation [001]. Anomalous high temperature peak of the flow stress and

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the property of tension/compression asymmetry were noticed in the tests. Investigation on the experimental results was carried out to study the strain-rate effect on the temperature-dependent anomalies of the flow stress. It was found that the peak of the flow stress shifts to a higher temperature as the applied strain rate increases. Subsequent study of the temperature-dependent rupture property of the material

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revealed that the path of the crack propagation is dependent on the temperature. The strengthening and failure mechanisms of the material were established based on the

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experimental observations. Finally, a constitutive model was developed to describe the temperature and strain-rate effects on the material’s yield behavior. The model is

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able to capture the anomalous temperature peak of the yield strength, which is dependent on the applied strain rate. It was demonstrated that the yield behavior of the

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superalloy can be adequately predicted over a wide range of temperatures and strain rates.

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Keywords: Nickel-base superalloy; Single crystal; Temperature; Strain rate; Yield strength; Constitutive model

_______________________________________ *Corresponding author: [email protected]

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1. Introduction The technological advances in aircraft engine design are highly depending on the development of aircraft materials. The challenging environment within the aircraft turbine engine requires high temperature strength and adequate resistance to degradation (Shenoy et al., 2008) in the material of turbine blades. Compared to polycrystalline metallic materials, the single-crystal Nickel-base superalloys

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(SCNBSs) possess superior mechanical properties and excellent resistance to degradation in a high-temperature environment (Zambaldi et al., 2007). As a result, SCNBSs have been widely employed as the material of turbine engine blades in aircraft applications.

SCNBSs are hardened by coherent ordered fcc Ni3Al (  ，-phase) particles

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dispersed in the disordered fcc matrix (  -phase). Anomalous phenomena can be observed during the plastic deformation of SCNBSs, e.g., the positive temperature dependence (Leverant et al., 1971; Nembach, 2006) and the property of tension/compression asymmetry (Miner et al., 1986). In general, based on the observed temperature effect on the quasi-static mechanical behaviors, the yield stress

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vs. temperature relation of SCNBSs can be divided into three regions, i.e., the low temperature region (below 873 K), the intermediate temperature region (873 – 1,023

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K), and the high temperature region (above 1,023 K). As the temperature increases, the yield stress of SCNBSs stays unaffected or slightly decreased in the low

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temperature region. In the intermediate temperature region, however, the yield stress increases anomalously with the increasing temperature. A rapid decrease of the yield

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stress is generally observed once the temperature further progresses into the high temperature region (Nembach et al., 2003). As a result, a peak is formed in the yield

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stress vs. temperature curve over the considered temperature range. The peak of the flow stress can shift to a higher temperature with increased strain rate (Leverant et al.,1971; Jenesen and Tien, 1985; Weaver et al., 1996; Cuniberti et al., 1997). This phenomenon is similar to the strain-rate effect of the third type strain aging (3rd SA) (Samuel at al., 2006; Guo and Gao, 2013; Su et al., 2013; Wang et al., 2015). Leverant et al. (1971) experimentally studied the strain-rate effect on the peak flow stress of Mar-M200 SCNBS. It was found that the peak occurs at the temperature of 880 K and at the strain rate of 5×10-6 /s. The peak temperature shifts to 1,019 K at the strain rate of 5×10-4 /s. Nembach (2006) proposed six forming mechanisms for the peak of the 3

ACCEPTED MANUSCRIPT yield strength or the critical resolved shear stress (CRSS) in Nickel-base superalloys. They are respectively the anomalous plastic behavior of the  ，-phase, the changes of the  ，-precipitate dispersion, the ternary phases, the dynamic strain aging effects, the lattice misfit of the  ，-precipitates and the dislocation line tension. Thus it turns to be quite a challenge to describe the thermo-mechanical constitutive behavior of the superalloy due to the existence of the anomalous peak of flow stress.

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Upon the deformation of SCNBSs, the course of microstructural transformation is dominated by several processes including the motion of dislocation in the matrix (Zhu et al., 2012), the forming of dislocation networks around  ， particles (Carrol et al., 2008), the cutting of  ， particles by dislocation (Sarosi et al., 2007), and the process of rafting (Reed et al., 2007). Generally, the process of the microstructure

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transformation in SCNBSs is dependent on the temperature and the stress/strain level. At relatively low temperature, most dislocations move into the matrix channel and there are hardly any in the  ，-precipitates. The decrease of the yield stress is governed by the solid solution strengthened  -matrix (Nembach, 2006). Milligan

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and Antolovich (1987) found that the decrease of the yield stress was enhanced beyond the thermally activated decrease due to the solid solution strengthening of

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 -matrix. They attributed the enhanced decrease of the yield stress to the decrease of the density of superlattice-intrinsic stacking faults in the  ，-precipitates and the

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decrease of the dislocation in the  -matrix. For the high temperature softening effects, two interpretations were considered by Nembach, et al. (2003; 2006). One

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interpretation is based on the coarsening and/or the dissolution of  ，-precipitates. The other interpretation is based on the thermal activation above Tp, which is the

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temperature corresponding to the peak of the yield strength or the critical resolved shear stress (CRSS). The process of thermal activation may be strong enough to help dislocations overcome the  ，-particles by local climb and/or cross-slip rather than by shearing. Thermal activation leads to the strain-rate sensitivity of the yield stress. As it has been shown, most of the earlier studies of SCNBSs are limited to quasi-static mechanical behaviors. As a matter of fact, while in service, the turbine blades get damaged not only by creep, fatigue, corrosion or oxidation, but also by foreign object impact (Mazur et al., 2005), which causes significant reduction in the

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ACCEPTED MANUSCRIPT blade’s lifetime. Therefore, investigations on the high strain-rate mechanical properties of SCNBSs are indispensible. In this study, the plastic behavior of a newly developed DD407 single-crystal Nickel-base superalloy was investigated with selected temperatures from 293 to 1,273 K and with selected strain rates from 0.001 to 4,000 /s. The temperature effect and the strain-rate effect were studied to understand the observed anomalies, including the anomalous peak of the flow stress and the property of tension/compression asymmetry.

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Discussion was given to explain the deformation and failure mechanisms over a wide range of temperatures. Finally, we established a constitutive model that takes into account the anomalous peak of the flow stress. 2. Material and experimental procedure

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The investigated material is selected to be a newly developed DD407 single-crystal Nickel-base superalloy. The material was provided by Central Iron and Steel Research Institute of China. The DD407 SCNBS was developed from DD3 superalloy that has been used in turbine blades of aircraft engine. The test samples of this material were all subjected to standard heat treatment. The chemical composition

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of the newly developed DD407 together with other SCNBSs is provided in Table 1. The quasi-static tensile strength vs. temperature curves of these SCNBSs are shown in

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Fig.1. Anomalous peak of stress can be observed for all these superalloys. The considered level of weight percent variation in the compositions, as shown in Table 1,

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has limited effect on the tensile strength for these SCNBSs. It is seen that the peak temperatures for all these SCNBSs are close to each other. The strengthening of the superalloy is believed to be largely due to the existence

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of the intermetallic  ，-phase (Ni3Al precipitate). The hardening of the superalloy is

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attributed to the interactions of the dislocations in the face-cetered cubic  -matrix with the precipitates. The elements Cr, Co, Mo and W are partitioned preferentially to Nickel-base  -matrix where they mainly serve as solid solution strengthening elements. Cr, which promotes the formation of a protective Cr2O3 oxide scale, also plays an important role for hot corrosion resistance. The elements Ti and Ta strengthen the  ， precipitates by being substituted for Al in Ni3Al. Al also plays a fundamental role in promoting the formation of a stable Al2O3 alumina surface scale which protects the alloy against further oxidation. Lopez and Hancock (1970) found that the anomalous peak in the flow stress vs. temperature curve can shift to lower 5

ACCEPTED MANUSCRIPT temperatures with the addition of Ti. Wee et al. (1980) argued that, since the anomalous increase in flow stress depends on the difference between APB energy on (001) and (111) planes, any element that strengthens such difference can enhance the anomalous increase in the resolved shear stress (CRSS) with temperature. Tancret (2012) reduced the fabrication cost by adding in iron and by avoiding the use of expensive alloying elements, such as Ta, Mo, Co, etc. In recent years, computational models were established for superior mechanical properties of multi-component

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Ni-based superalloys (Tancret, 2012; Jha, et al., 2015). Based on the genetic algorithms in the models, one is able to optimize the chemical composition of alloys. In other words, one could predict the complex interplay between the composition and mechanical properties through these models. As a result, novel alloys can be designed with the concerned properties of the material being simultaneously maximized.

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The SEM micrograph of this SCNBS is shown in Fig.2 with the scanning surface perpendicular to [001] orientation. It can be observed in the scanned image that the microstructure of the superalloy consists of submicron cubic  ， -precipitates dispersed in the  -phase matrix. The average diameter of the  ，-precipitates is

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around 0.6 μm, and the channel width of the matrix is around 0.1 μm. The volume fraction of the  ，-precipitates of DD407 superalloy is about 0.63.

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Table 1 Nominal composition of DD407 together with other superalloys（wt.%） Alloy

Cr

Co

Mo

W

Al

Ti

Ta

Re

Nb

Hf

V

Ni

8

5.5

2.25

5

6

2

3.5

_

_

_

_

Bal

9.5

5.2

3.63

5.36

5.8

2.24

_

_

_

_

_

Bal

10

15

3

_

5.5

4

_

_

_

_

1

Bal

SRR99 (MacLachlan and Knowles, 2001)

8

5

_

10

5.5

2.2

3

_

_

_

_

Bal

CMSX-4 (MacLachlan and Knowles, 2001)

6.5

9

0.6

6

5.6

1

6.5

3

_

0.1

_

Bal

2.2

3.3

0.4

5.5

5.8

0.2

8.3

6.2

0.1

0.03

_

Bal

DD3 (Luo et al., 2003)

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DD407 (in this study)

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RR2000 (MacLachlan and Knowles, 2001)

CMSX-10 (MacLachlan and Knowles, 2001)

Cylindrical specimens of the material were prepared with the nominal

dimensions of 4 mm in diameter and 4 mm in length, i.e.,  4mm  4 mm . The axis of symmetry of the cylindrical specimens is oriented along the [001] crystal orientation. To reduce the end friction during the quasi-static and dynamic loading experiments, both ends of the specimens were polished by waterproof silicon carbide paper (800 and 4,000 grit). The ends of the specimens were greased for 6

ACCEPTED MANUSCRIPT low-temperature tests (T<573 K), and the ends of the specimens were lubricated by molybdenum-powder lubricant for high temperature tests (T≥ 573 K). 1350 DD407

DD3

1050 900 750 600 450 300 150 0 200

DD407 DD3 (Luo et al., 2003) RR 2000 SRR 99 (MacLachlan and CMSX-4 Knowles, 2001) CMSX-10

400

600

800

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Tensile Strength (MPa)

1200

Peak temperature

1000

Temperature (K)

1200

1400

1600

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Fig.1 Quasi-static tensile strength of several SCNBSs as a function of temperature

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Fig. 2 The SEM micrograph of the DD407 single-crystal superalloy with the scanning surface perpendicular to [001] orientation

The quasi-static loading experiments with strain rate of 0.001 /s were first carried

out by using NPU’s servohydraulic testing machine. The test was conducted with selected temperatures from 293 to 1,273 K. A radiant-heating furnace was used to provide required high temperatures. The attained testing temperature was measured by a thermocouple arrangement. The temperature was maintained constant during the test with a fluctuation of ±3 °C. The deformation of the specimen was measured by a LVDT, which is mounted in the testing machine. Calibration was done by taking into 7

ACCEPTED MANUSCRIPT account the compliance of the loading frame. The dynamic loading experiments were conducted with the respective strain rates of 1000 /s and 4000 /s using NPU’s enhanced split Hopkinson bar system. The enhanced split Hopkinson bar technique was originally developed by Nemat-Nasser et al. (1997). The initial environment temperature for the tests was selected to be 293 K, 573 K, 873 K, 973 K, 1,073 K, 1,173 K and 1,273 K, respectively. A furnace was used to attain the required uniform temperature in the specimen during the

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high-temperature tests. However, temperature gradient may develop in the elastic bars during the high-temperature split Hopkinson bar tests. Such temperature gradient can cause unwanted change in the elastic constants and the mechanical impedance of the bars. As a result, the state of the stress wave propagation in the elastic bars can be affected. To prevent such influence, the incident and transmit bars of the split

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Hopkinson bar system are left outside the furnace with an environment temperature below 523 K. A synchronous bar-moving system is utilized to make sure that the incident and transmit bars are in full contact with the specimen right before the stress wave arrives at the far end of the incident bar. The setup of the NPU’s enhanced split

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Hopkinson bar system is schematically illustrated in Fig. 3. It was demonstrated that reliable experimental data can be attained if the cold (bars)-hot (specimen) contact

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time (CHCT) is no more than 50 ms (Nemat-Nasser and Isaacs, 1997; Lennon and Ramesh, 1998; Li et al., 2009). 1

3

4

5

6

7

8

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2

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9

11 12

10

13

14

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1.striking bar 2.incident bar 3.strain gauge 4.specimen 5.thermocouples 6.transmit bar 7.strain gauge 8.bar mover 9.gas gun 10.pulse shaper 11.temperature controller 12.sleeve 13. ultrahigh dynamic strainometer 14.transient wave recorder 15.computer Fig. 3 The schematic of the enhanced split Hopkinson bar system

3. Experimental results and analysis 3.1 Anomalous peak of the uniaxial compression flow stress Uniaxial compression tests were conducted on the DD407 superalloy along the 8

ACCEPTED MANUSCRIPT nominal [001] crystal orientation. The true stress – true strain curves were obtained over a wide range of temperatures from 293 to 1,273 K and over strain rate ranging from 0.001 to 4,000 /s. Considering the variation of the flow stress upon the temperature change, the true stress – true strain curves with the selected temperatures are plotted in three dimensional graphs, as shown in Figs. 4(a)-4(c). As seen in Fig. 4(a), the flow stress decreases rapidly once the temperature exceeds a critical value, Tp, which is around 1,073 K. In the cases where temperature is equal to 1,173 K and

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1,273 K, the specimens are sheared off at the strain of about 0.2. To further the understanding of the temperature effect on the flow stress, the flow stress vs. temperature relation is plotted for each applied strain rate in Figs. 5(a)-5(c). The flow stress is selected either corresponding to certain strain levels as indicated by the

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legends, or corresponding to the yield strength,  0.2 , at the residual strain level of 0.2 %. It should be noted that the temperature rise needs to be considered during the adiabatic process under high strain rate loading (Kapoor and Nemat-Nasser, 1998; Wang et al., 2015). As seen in Figs. 5(a)-5(c), for all selected strain rates, the flow stress is not very sensitive to the temperature within the low temperature region. Once

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the temperature exceeds a critical value, the flow stress starts to increase rapidly with the temperature. This critical temperature is found to be about 850 K for the strain rate

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of 0.001 /s, 900 K for 1,000 /s, and 980 K for 4,000 /s. For all the considered strain levels indicated by the legends, a maximum value of the flow stress is observed at the temperature of about 1020 K for the strain rate of 0.001 /s, 1,185 K for 1,000 /s and

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1,205 K for 4,000 /s. It is noticed that the flow stress will drop rapidly as the temperature further increases above the corresponding critical temperature.

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Fig. 6 shows the flow stress vs. temperature curve with various strain rates and

with the fixed strain level of 0.1. As seen in this figure, the temperature for the peak of

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the flow stress ( TP ) shifts to higher value with increased strain rate. To better understand the strain-rate effect on the flow stress – temperature relation, the values of TP for selected Nickel-base superalloys are plotted in Fig. 7 as a logarithmic function of the strain rate for comparison. Except for the data of DD407 superalloy, other data are obtained from reported experimental work (Leverant et al., 1971; Jensen and Tien, 1985) under quasi-static loading conditions. To maintain the consistency of the comparison, all measurements of TP in Fig. 7 are based on the yield strength  0.2 . It is evident that the value of TP increases with the strain rate for 9

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all selected Nickel-base superalloys.

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Fracture

（b）

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（a）

10

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1400 DD407, 0.001/s

1300

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1100 1000 900

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True Stress (MPa)

1200

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（c） Fig. 4 The true stress–true strain relation of DD407 SCNBS with selected temperature of 293K, 573K, 873K, 973K, 1,073 K, 1,173 K and 1,273 K, and with the strain rate of (a) 0.001 /s, (b) 1,000 /s, and (c) 4,000 /s.

800 700

 0.2

600

Strain:0.1

PT

500

Strain:0.2

400

400

600

800

1000

Temperature (K)

（a）

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CE

200

11

1200

1400

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1000 900 800

 0.2

700

Strain:0.05 Strain:0.1

600 200

400

600

800

1000

Temperature (K)

（b） 1400 DD407, 4000/s

1100 1000

 0.2

900

Strain:0.1 Strain:0.2

800 200

600

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400

1200

1400

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1200

M

True Stress (MPa)

1300

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True Stress (MPa)

1100

800

1000

1200

1400

Temperature (K)

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（c） Fig. 5 The flow stress as a function of the temperature with indicated strain rates of (a) 0.001 /s, (b) 1,000 /s, and (c) 4,000 /s. 1300

True Stress (MPa)

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DD407, Strain:0.1

1200

4000/s 1100 1000 1000/s

0.001/s

900 800 700 600 500 0

200

400

600

800

1000

1200

1400

Temperature (K)

Fig. 6 The strain-rate effect on the flow stress vs. temperature relation

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1200

1000

900

DD407 Mar-M200 (Leverant et al.,1971) UDIMET 115 (Jensen and Tien,1985)

800 -7

-5

-3

-1 log( )

1

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Tp (K)

1100

3

5

Fig. 7 Variation of TP with log( ) for selected Nickel-base superalloys

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3.2 Tension/Compression asymmetry

To compare with the compressive property of the DD407 superalloy, the tensile mechanical behavior along [001] crystallographic direction was experimentally studied. The temperature-dependent tensile/compressive flow stress is plotted in Fig. 8 with the strain level of 0.1 and with the strain rate of 0.001 /s. By comparison, it is

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noticed that a more remarkable peak exists in the tensile flow stress vs. temperature relation. The temperature effect on the flow stress exhibited a noticeable property of

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tension/compression asymmetry, which is more evident within the temperature range of the peak flow stress.

The mechanism of tension/compression asymmetry for SCNBSs was extensively

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studied (Lall et al., 1979; Miner et al., 1986; Yamashita and Kakehi, 2006; Wang et al., 2014). Tension/compression asymmetry in the flow stress of Nickel-base

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superalloys was explained in terms of the “core width effect” by Lall et al. (1979). They assumed that the width of core of the a/2[101] superpartial dislocation in  ，

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phase must be changed due to the effect of the components of the stress tensor when the superpartial dislocation undergoes cross-slip from {111} to {100}. Since these superpartial dislocations are most probably dissociated into two Shockley partial dislocations on {111} planes, the component of tensile stress tensor which tends to constrict the partial dislocations will promote cross-slip, whereas the component of compressive stress tensor which extends the partial dislocations will retard the cross-slip. Therefore, according to the “core width effect”, the tensile flow stress is greater than the compressive flow stress in [001] orientation. In other studies, such as 13

ACCEPTED MANUSCRIPT the work of Yamashita and Kakehi (2006), it is believed that tension/compression asymmetry is primarily due to the microtwin formation in

 ， phase for a

nickel-base superalloy, PWA1480. DD407, 0.001/s, Strain:0.1

1200 1000 800 600

Tension Compression

400 200

400

600

800

1000

1200

1400

1600

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Temperature (K)

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True Stress (MPa)

1400

Fig. 8 The temperature-dependent tensile/compressive flow stress with the strain level of 0.1 and with the strain rate of 0.001/s

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3.3 Tensile stress-rupture property

An effort is made in this part to study the tensile stress-rupture property and the

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corresponding mechanism. Schematics of the SCNBSs are outlined in Fig. 9 on different length scales from the macroscopic to the microscopic scale. Apparently, microscopic analysis is needed in order to understand the mechanism of the

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macroscopic fracture.

As illustrated in the rightmost schematic in Fig. 9, there could be multiple paths

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of the crack propagation in SCNBSs: the crack may propagate along the matrix channel or through the  ，-particle. In order to determine the path of the crack

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propagation, the temperature dependence of the critical resolved shear stress (CRSS) is plotted in Fig. 10 for the two-phase Nimonic 105, the single-phase  and the single-phase  ，, respectively. As observed, the CRSS of the single-phase  is close to that of the single-phase  ， in low-temperature range, whereas the CRSS of the single-phase  ， is much higher than that of the single-phase  in high-temperature range. Therefore, it is conjectured that the path of the crack propagation in the two-phase Nimonic 105 is mainly along the matrix channel at high temperature and

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ACCEPTED MANUSCRIPT may cross over the  ，-particles at low temperature. Inspired by such inference, microscopic and macroscopic analyses are performed to study the tensile rupture

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property of DD407 superalloys.

Fig. 9 Schematics showing the SCNBS on different scales 450 Single-phase γ Single-phase γ' Two-phase Nimonic 105

400 350

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τ0 (MPa)

300 250 200 150 100

0 0

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50 200

400

600

800

1000

1200

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Temperature (K)

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Fig. 10 Critical resolved (to the primary (111) [101] -slip system) shear stress measured in compression vs. temperature (by Nitz and Nembach, 1998)

We now consider the temperature-dependent tensile fracturing behaviors of the

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material. Fig. 11(a) shows the variation of the fracture strain and the corresponding fracture stress over a temperature range of 100 – 1,500 K for DD407 SCNBS and

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Ni3(Al,Ti) single-crystal. The test on the DD407 SCNBS was conducted with a strain rate of 0.001 /s and the test on the Ni3(Al,Ti) single-crystal with 0.0013 /s. As seen in the figure, the fracture strain first decreases and then increases with the increasing temperature for both materials. Thus a valley of the fracture strain is formed over the selected temperature range. The valley temperature of the DD407 superalloy is around 873 K.

Fig. 11(b) shows the temperature effect on the crack propagations in the

DD407 superalloy with a tensile loading rate of 0.001/s and with a selected temperature of 293, 873, 1,073 and 1,373 K, respectively. It is evident that the path of the crack propagation is temperature dependent. For the case of 293 K, necking did 15

ACCEPTED MANUSCRIPT not occur during the loading process. The crack surface is perpendicular to the loading direction. With the temperature of 873 K, shear fracture is observed. The path of the crack propagation is perpendicular to the loading direction close to the lateral surface of the sample. In the center region of the fractured sample, the crack surface and the loading direction form an angle of about 40o, which is the orientation of the twisted {111} slip plane. With the temperature of 1,073 K, the crack surface exhibits a serrated form with an angle of 45o or 135o to the loading direction. For the case of DD407

0.4

DD407, 0.001/s

1400

0.3

1200 1000

0.2

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800 600

400 100Ni3(Al.Ti)(Aoki 300 500and Izumi, 700 1979), 900 0.0013/s 1100 1300 600 Temperature (K) 400

0 100

300

M

200

500

700

900

1100

1300

0.1

0.0 1500 0.2

0.1

Fracture Strain

1600

Elongation

the loading direction. Fracture Stress (MPa) Fracture Stress (MPa)

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1,373 K, necking is observed close to the fracture surface, which is perpendicular to

0.0 1500

(a)

Fracture Stress Fracture Strain

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PT

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Temperature (K)

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（b） Fig. 11 The temperature-dependent tensile fracturing behaviors. (a) The variation of the fracture strain and the fracture stress with temperature for DD407 SCNBS and Ni3(Al,Ti) single-crystal. (b) The crack propagations of DD407 SCNBS with selected temperatures.

Earlier studies showed that carbides and micro-voids of matrix and eutectic

 /  ， phase are the main crack sources (Ding et al., 2003). With the temperature of 293 K, eutectic  /  ， phase acts as a defect in SCNBSs, and the crack initiation is along the〈001〉direction. As already shown in Fig. 10, the strength of  -matrix is comparable to that of  ，-phase at room temperature, whereas the strength of 16

ACCEPTED MANUSCRIPT  ，-phase is much higher in the high temperature region. Therefore, the crack can propagate across the  ，-phase at 293K. Figs. 12(a)-12(i) show the SEM micrographs of the tensile fracture surface of DD407 SCNBS with the temperature of 293, 873, 1,073 and 1,373 K and strain rate of 0.001 /s. The SEM observation at 293 K supports the consideration of the low-temperature crack propagation across the  ，-phase. As seen in the inset of Fig. 12(b), the  ， particles are pulled apart due to the tension

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loading. At 873 K, it is observed that the resolved shear stress leads to the crack initiation on the {111} plane. Carbides and micro-voids are the main sources of the crack initiation. Sub-fracture surfaces are noticed on the main surface of the crack, as shown in Fig. 12(c). As seen in the inset of Fig. 12(d) for the case of 873 K, some parts of the edges and corners of the  ，-phase particles are sheared off, and the cracks

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propagate mainly through the matrix. With the temperature of 1,073 K, it is noticed that the crack propagates alternatively along two groups of octahedral slip planes, which have the same resolved shear stress and strain. In another words, there are two possible paths for crack propagation. Once the crack propagates along one of the slip

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plane and encounters obstacles, the other path will be taken. As shown in Fig. 12(f) for the case of 1,073 K, the cracks propagate mainly through the matrix, and some

ED

 ，-phase particles are sheared off. With the temperature of 1,373 K, void aggregation in matrix leads to the fracture, and the fracture surface is perpendicular to the loading

AC

CE

PT

direction, as shown in Fig. 12(i).

（293 K）

17

ACCEPTED MANUSCRIPT

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CR IP T

（873 K）

PT

ED

M

（1,073K）

（1,373K）

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Fig. 12 The SEM micrographs of the tensile fracture surface of DD407 SCNBS with the temperature of (a)-(b) 293 K, (c)-(d) 873 K, (e)-(f) 1,073 K and (h)-(i) 1,373 K and strain rate of 0.001 /s

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4. Discussion

4.1 Strengthening and failure mechanisms According to the experimental results for DD407 superalloy and reported data for other Nickel-base superalloys (Jockweg and Nembach, 1995; Nitz and Nembach, 1998, 1999; Nembach and Nitz, 1999), it is affirmed that such  ，-strengthened two-phase Nickel-base superalloys possess anomalous plastic behaviors. The CRSS, yield stress and flow stress do not decrease monotonically as the temperature

18

ACCEPTED MANUSCRIPT increases. A peak of the flow stress is formed over the selected temperature range. The peak temperature, Tp, shifts to higher temperature with increased strain rate. A similar anomaly is also observed in the CRSS vs. temperature relation of the  ，-single-phase crystal, as shown in Fig. 10. The CRSS of the single-phase  ， remarkably increases with temperature and forms an anomalous peak of stress in the high-temperature region, and this anomaly does not exist in a single-phase  -matrix

CR IP T

material (Nitz and Nembach, 1998, 1999). Therefore, the peak of CRSS of the

 ，-strengthened superalloy is considered to be attributed to the CRSS of the single-phase L12-long range ordered  ，-phase (Nembach, 2006). In the followings, we employ  c ' ,  c and  c

to denote the CRSS of the  ， -crystal, the

 ，-strengthened superalloy and the single-phase  -matrix material, respectively.

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Copley and Kear (1967) related  c to the CRSSs of their two constituent phases:

 c and  c ’. Based on the assumption that the dislocations shear the  ， particles, a detailed dynamic model yields the following, as

 2b



S 1 '   K ( c   c ’) , br 2

(1)

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c 

where  represents the specific energy of antiphase boundaries in the  ，-phase, b

1 011 -Burgers vector, S is the dislocation line tension, r 2

ED

represents the length of the

PT

is the average radius of the  ， precipitates which are assumed to be spheres, and the parameter K` is a constant and is approximated as a unit. The first term on the right

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hand side of Eq. (1) is independent of temperature. The value of  c ’ in the bracket contributes to the anomalies of  c .

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The anomalous strength enhancement of  ，-phase is generally believed to be

caused by cross-slip. Aoki and Izumi (1979) concluded that this anomaly may be attributed to the cube cross-slip of the screw superdislocation, i.e. the screw superdislocation on the {111} slip planes tends to cross-slip onto the {100} plane to reduce the antiphase boundary (APB) energy. Since the probability of cube cross-slip increases with the temperature, the yield stress shows positive temperature dependence. On the other hand, at higher temperatures the {100}〈110〉slip takes the place of {111}〈110〉 since the CRSS for {111}〈110〉slip becomes much higher than 19

ACCEPTED MANUSCRIPT

the CRSS for {100}〈110〉slip. The yield stress begins to decrease with increasing temperature. As a result, an anomalous peak of stress is formed. The fracture of SCNBSs can be explained by the rapid decrease of the mobile dislocation density due to dislocation pinning. Such dislocation pinning mechanism is based on the cross-slip, which causes the positive temperature dependence of the flow stress. Based on the observations and analysis,  ，-phase is the main reason for the

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anomalies. However, the anomalies of  c are not as pronounced as that of the single

 ，-phase, as shown in Fig. 10. This is due to the unevenly distributed dislocations in the matrix phase and  ，-phase of the  ，-strengthened superalloys (Milligan and Antolovich, 1987). During the deformation, the generation and motion of dislocations are restricted in the  -matrix phase or piled up at the  /  ， interfaces. Few

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dislocations were observed within the  ，-phase since cutting off the  ，-phase by dislocations would require the formation of a high-energy antiphase boundary. Thus, it is not energetically preferable for a crack to cut through the  ，-phase. For material with cuboidal  ，-particles, the crack is able to propagate along the  -channels

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between the  ，-particles (Ott and Mughrabi, 1999). The plastic deformation in

 -channels is dominantly under the conditions of high temperatures (high strength of

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 ，-phase), wide channels (easier bowing of dislocations in the channels) and low

PT

strains (lack of strong dislocation pile-ups). 4.2 Constitutive model

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The hardening mechanism of SCNBSs is quite complex. Therefore it is a

challenge to develop a constitutive model, which is able to reproduce the anomalous

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peak of the flow stress over the considered temperature range. Theoretical models were proposed to obtain the strength of a superalloy by

considering the effects of precipitates and matrix flow stress (Koppnaal and Kuhlmann-Wilsdorf, 1964; Nembach and Chow, 1978; Reppich et al., 1982; Sinharoy et al., 2001; Milligan et al., 2004), as

 n   mn   pn ,

(2)

where  is the flow stress of the Nickel-base superalloy,  m is the flow stress of the matrix,  p indicates the flow stress due to the precipitation, and n is the 20

ACCEPTED MANUSCRIPT exponent varying between 1 and 2, depending on the hinder strength of precipitation for the mobile dislocation. Koppenaal and Kuhlmann-Wilsdorf (1964) and Foreman and Makin (1967) found that n=2 is of general validity. Reppich et al. (1982) assumed that solid solution hardening of  -matrix and  ，-particle hardening can be superimposed according to “3/2 law” (n=3/2). The value of n was found to be about 1.1 (Chanzer and Nembach, 1992; Sinharoy et al., 2001; Milligan et al., 2004), which is fairly close to 1. In our study, n is a material parameter, which is affected by the

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nature of the  /  ' microstructure, such as the volume fraction of the  ， phase. For simplification, the flow stress of the  ，-strengthened superalloys can be expressed as

  ( mn   pn )1/ n .

(3)

Earlier studies showed that the CRSS of the single-phase  -matrix exhibits no

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anomalies (Jockweg and Nembach, 1995): the state of  c is isotropic and  c / T is negative. In  -matrix, the resistances to dislocation motion mainly come from the Orowan resistance and solid solution resistance. Thus the flow stress of the matrix can be expressed as

(4)

M

m  o s ,

where  o and  s represent the Orowan resistance and the solid solution resistance,

ED

respectively.

Before the dislocations are forced through the matrix channels, the applied stress

PT

must be sufficiently high to overcome the local Orowan resistance of the channels,

CE

which is simply given by (Pollock and Argon, 1992)

o 

Gb , l

(5)

where G is the shear modulus, b is the Burgers vector, l is the channel dimension

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along the [110] direction, and  is a orientation constant based on the isotropy of the matrix. The solid solution resistance is controlled by thermal activation, as

 Q v  v0 ( s )m exp( ) , G kT

(6)

where k is the Boltzman constant, Q is the activation free energy for dislocation motion, v0 is a pre-exponential factor independent of both stress and temperature,

 s is the solid solution resistance to the dislocations, and v is the average velocity

21

ACCEPTED MANUSCRIPT of the mobile dislocation. The effective plastic strain rate can be related to the dislocation motion by Orowan’s equation as

   bm ，

(7)

where  is the plastic strain rate,  is the orientation factor, and  m is the density of the mobile dislocation. By solving Eqs. (6) and (7), one can obtain the expression of  s as

Q ) , mkT

(8)

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 s  A0 1/ m exp(

Gm where A0  . The solid solution resistance,  s , can be expressed as ( bm v0 )1/ m

 s  A 1/ m exp(

(9)

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where A  A0 /  .

Q )， mkT

Fig. 13 shows the schematic of the quantity of the activated octahedral slip and the cube slip as functions of temperature for SCNBSs. The quantity of the activated octahedral slip decreases with the increasing temperature, and the decrease is more

M

rapid once the temperature rises above Tp. The cube slip is activated once the temperature exceeds the activation temperature, Ta, as marked in Fig. 13. The quantity

ED

of the activated cube slip keeps rising with the increasing temperature. As a result, the octahedral slip system is the primary one during the process of dislocation glide with

PT

the temperature below Tp. Although the cube slip is activated when the temperature rises above Ta, the long-distance glide in the cube slip planes is still difficult. A

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segment of the leading part of dislocations leaves its primary octahedral slip plane and cross slips onto the cube slip plane. The cross slipped part of the dislocations gets stuck in the cube slip plane and hinders the further glide of the non-cross slipped part

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in its original octahedral slip plane. Kear-Wilsdorf (K-W) locks are thus formed (Kear and Wilsdorf, 1962; Caillard and Couret, 1996; VeyssiBre and Saada; 1996; Caillard et al., 1997). With the temperature higher than Tp, the cube slip system becomes dominant. The long distance glide in the slip plane is easier to take place due to its low APB energy, leading to the decrease of the quantity of the octahedral slip. The change from the octahedral slip to the cube slip can reduce the APB energy. The shaded area in Fig. 13 illustrates the possibility of cross-slip, which leads to the formation of K-W locks and shows normal temperature dependence (Aoki and Izumi, 22

ACCEPTED MANUSCRIPT 1979). A normal distribution of the possibility of the cross-slip with the temperature can be used to demonstrate the additional degree of resistance to the dislocation due to K-W locks, expressed as follows:

 T  T 2  p , p  p0 exp   2     

(10)

where Tp is the peak temperature at which the resistance of the K-W locks is the

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strongest, the parameter  is affected by the temperature range of the K-W locks, and the parameter p0 is affected by the peak of the strength of K-W locks. As seen

Activated temperature

Cube slip

Tp

Temperature (K)

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Ta

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Octahedral slip

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Quantity of Slip System

in Fig. 6, Tp ,  and p0 are functions of strain rate, which will be discussed later.

Fig. 13 The quantity of the octahedral slip and the cube slip as functions of the temperature

 p =p

CE

PT

The flow stress of the  ，-precipitate can be expressed as

 c ’ S

,

(11)

where  c ’ is the CRSS of  ，-precipitate and S indicates the Schmid factor, which is

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not equal to the theoretical value due to the deflection of actual orientation. The deflection of nominal orientation [001] makes the cube slip possible with temperatures above Tp. With temperatures below Tp, octahedral slip is predominant and S is a constant. When the temperature is above Tp, the cube slip is becoming prevalent with increasing temperature. For superalloys strengthened by high volume fraction of  ，-precipitates, the increment in CRSS due to the presence of the  ，-precipitates is formulated by (Milligan et al., 2004) 23

ACCEPTED MANUSCRIPT   G 'b 

1

1/2  c ’= [1.62w( )   f] , 2 b  r 

(12)

where G ' is the shear modulus, r represents the average diameter of  ， particles, f is the fraction of  ， particles, and the adjustable parameter, w, arises due to the elastic interaction between the super-dislocation pairs which are shearing the microstructure.

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The temperature dependent volume fraction, f, does not exhibit significant change with the varying temperature (Nembach et al., 2003; Foreman and Makin, 1967). Thus f is considered a constant in this study. The growth of the  ，-precipitates can be expressed as (Lifshitz and Slyozov, 1961; Nembach et al., 2003) r 3  r03   t ,

(13)

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where  is the growth constant, t is time, and r0 is the average radius at t=0. Because of the very short heating time, r is considered a constant during the material’s deformation. Therefore, the width of the matrix channel does not change with the varying temperature, and  o can also be regarded as a constant.

M

1   Gb  1/2 By setting B0  [1.62w( )   f ] , one can rewrite Eq. (12) as 2 b  r 

ED

  ’ B0 .

(14)

PT

Based on Eq. (14), Eq. (11) can be rewritten as

 T  T 2  p ,  p =Bp0 exp   2     

(15)

B0 . The value of B can be denoted by a constant B1 with the temperature S

CE

where B=

AC

below Tp. For temperature above Tp, B takes different values for the case of octahedral slip and the case of cube slip. For the purpose of simplicity, based on the temperature dependency of the quantity of the octahedral slip and the cube slip, we assume that the value of B varies linearly with temperature above Tp, as B  B1  B（ , where B2 2 T -Tp） is a constant. 4.3 Strain-rate effect on the anomalous peak of the flow stress As observed by Nembach et al. (2003), under the condition T< Tp, the yield strength is independent of the temperature and the strain rate, whereas the value of the 24

ACCEPTED MANUSCRIPT yield strength is dependent on the temperature and the strain rate for T> Tp. This result is in agreement with that obtained by Milligan and Antolovich (1987). However, all of the above results were obtained under quasi-static conditions. The experimental results in this study showed that the flow stress is sensitive to the strain rate (especially high strain rate) over the entire range of the selected temperatures, as shown in Fig. 6. As discussed in section 3, the peak of the flow stress shifts to the higher

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temperature with increasing strain rate. To quantitatively describe the strain-rate effect on the anomalous peak of the flow stress, Mills and Chrzan (1992) considered that the annihilation of K-W locks is thermally activated and used a strain rate law of the form to describe the relation between the annihilation temperature Tu and the strain rate:

where

Hu

is

the

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  C m exp( Hu / kTu ) , activation

enthalpy

for

unpinning

(16)

and

m

is

the

temperature-independent density of mobile dislocation. Similarly, the initiation of K-W locks and the strongest resistance of K-W locks against dislocation motion can be regarded as thermally activated. We use the same form to respectively describe the

M

relation between the corresponding temperatures ( Ti and Tp ) and the strain rate, as

ED

  C m exp( Hi / kTi )

and

(18)

PT

  C m exp( H p / kTp ) ,

(17)

where H i is the activation enthalpy for initiation and H p is the activation

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enthalpy for the strongest resistance. By taking the logarithmic function of Eqs. (16)-(18), one obtains the relations as

AC

follows:

Tu  Ti 

Du , F  ln 

Di , F  ln 

(19) (20)

and

Tp 

E , F  ln 

(21)

where the constants Du , Di , E and F satisfy the conditions Du  H u / k , Di  H i / k , 25

ACCEPTED MANUSCRIPT E  H p / k , and F  ln(C m ) , respectively.

The temperature range of the K-W locks can be expressed by

T  Tu  Ti 

D' , F  ln 

(22)

where D ' is a constant. Based on the 3 principle (99.7 percent of the values in the set are within three standard deviations of the set’s norm), the model parameter 

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is given by 2 D' 2 D 6 ,  T   6 F  ln  F  ln 

where D is constant.

(23)

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In Eq. (10) and (15), p0 represents the value of the peak flow stress, which decreases with increasing strain rate. A phenomenological expression is given for the relation between p0 and strain rate as

p0  1  p1 ln  , where p1 is a constant.

(24)

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In summary, the three parameters, Tp ,  and p0 , which characterize the

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anomalous peak of the flow stress can be expressed as follows

E D ,  , and p0  1  p1 ln  . F  ln  F  ln 

(25)

PT

Tp 

4.4 Summary of the constitutive model

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With the constitutive model developed for the yield stress of SCNBSs, the values of the constitutive parameters can be calibrated using the experimental data. The final

AC

constitutive relations with the determined parameters can be expressed as 1/1.2   ( m1.2   1.2 , p )

 m  770  6.5 105  1/0.56 exp(

(26)

70 ), T

(27)

and

 T  T 2  p ,  p =Bp0 exp   2      where Tp 

(28)

143200 20700 ,  and p0  1  0.059ln  . The value of B in Eq. 132  ln  132  ln  26

ACCEPTED MANUSCRIPT (28) is given by B  256 for T  Tp , and B  256  1.35（T -Tp） for T＞Tp . Based on the developed constitutive model, the predicted yield stress is plotted in Fig. 14 as a function of the temperature with selected strain rates. The corresponding experimental data are also plotted for comparison. It is evident that the proposed model is able to reproduce the strain-rate and temperature dependency of the yield

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stress as well as the anomalous peak of the yield stress.

1200 DD407 1100

900 1000/s

800

0.001/s

700 600

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Yield Stress (MPa)

4000/s 1000

Points: Experimental Results Solid Curves: Model Predictions

500 100

300

500

700

900

1100

1300

1500

Temperature (K)

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Results Fig.Experiment 14 The model predicted yield stress of DD407 SCNBS as a function of the temperature with Model Predictions

5. Conclusions

ED

selected strain rates. The experimental results are also plotted for comparison.

PT

In this work, a comprehensive experimental investigation was carried out over a wide range of temperatures (293-1,273 K) and strain rates (0.001-4,000 /s) to study

CE

the plastic behaviors of a newly developed DD407 superalloy along the [001] direction. The rupture properties and strain-rate effect of the anomalous peak of flow

AC

stress were studied. Based on the experimental results and the strengthening mechanism, we developed a constitutive model, which takes into account the effects of strain rate and temperature that features the anomalous peak of the flow stress. The following conclusions can be drawn from this study: (1)

The anomalous peak of the flow stress takes place at both low and high strain rates. The peak of the flow stress shifts to higher temperature with increased strain rate.

(2)

The property of tension/compression asymmetry exists and such asymmetry is more evident as the temperature is close to the peak temperature. In 27

ACCEPTED MANUSCRIPT comparison with the compression case, the peak of the flow stress is more remarkable in the tensile stress vs. temperature relation. (3)

The path of crack propagation is dependent on the temperature.

(4)

A constitutive model is developed to calculate the yield stress as a function of the temperature with selected strain rates. The model was shown to be able to capture the anomalous peak of the yield stress, which is dependent on the strain rate. The yield behavior of the superalloy along the [001] direction can

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be adequately predicted over a wide range of temperatures and strain rates. Acknowledgments

This research work was supported by the National Natural Science Foundation of China (No. 11372255) and the Doctorate Foundation of Northwestern Polytechnical

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University. References

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PT

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ED

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ACCEPTED MANUSCRIPT simulation of dislocation motion L12 alloys. Acta metal. mater. 40, 3051-3064. Miner, R.V., Gabb, T.P., Gayda, J., et al., 1986. Orientation and temperature dependence of some mechanical properties of the single-crystal nickel-base superalloy Rene N4:Part Ⅲ.Tension-Compression anisotropy. Metallurgical Transactions A 17A, 506-512. Nemat-Nasser, S., Isaacs, J.B., 1997. Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta-W alloy. Acta materialia 45, 907-919. Nembach, E., 2006. The high temperature peak of the yield strength of  -strengthened

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Nembach, E., Chow, C.K., 1978. Experimental investigation into the relation between 

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