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ScienceDirect Materials Today: Proceedings 2S (2015) S440 – S452

Joint 3rd UK-China Steel Research Forum & 15th CMA-UK Conference on Materials Science and Engineering

Simulation and Experimental study of Recrystallization Kinetics of Nickel Based Single Crystal Superalloys Zhonglin Li, Qingyan Xu*, Baicheng Liu School of Materials Science and Engineering, Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, Tsinghua Unveristy, Beijing 100084, China

Abstract Hot compression tests on cylinders using nickel-based single crystal superalloy DD6 were conducted to introduce the stored energy for recrystallization. Annealing was carried out at different temperatures for different time to investigate the kinetics of recrystallization of DD6. The experimental results show that the recrystallization rate increases gradually with temperature. The stress, strain and stored energy distribution of single crystal superalloy were modeled based one macroscopic phenomenon-based elastic-plastic model, considering the orthotropic properties of SX superalloys. Recrystallization microstructure on the crosssection perpendicular to the cylinder axis was simulated using cellular automaton (CA) method. The model can predict experimental results. © 2014 The Authors. Published by Elsevier Ltd. © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Selection and Peer-review under responsibility of the Chinese Materials Association in the UK (CMA-UK). This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). responsibility of the Chinese Materials Association in the UK (CMA-UK). Selection and Peer-review access article under theunder CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Keywords: single crystal, superalloys, recrystalliztion, kinetics, simulation

1. Introduction Nowadays, single crystal superalloys have been used widely in the manufacture of turbine blades. The disadvantage is that the blades from single crystal superalloys become very more difficult to cast, and great care

* Corresponding author. Tel.: +86-10-62795482; fax: +86-10-62773637. E-mail address: [email protected]

2214-7853 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and Peer-review under responsibility of the Chinese Materials Association in the UK (CMA-UK). doi:10.1016/j.matpr.2015.05.060

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should be taken to prevent defects such as freckles[1], recrystallization[2] and so on. Recrystallization can introduces high-angle grain boundaries and degrades the creep[3-5] and fatigue[6] properties significantly. Recrystallization is intolerant in single crystal component, and can be ascribed to the plastic deformation during manufacturing process[7-8] which introduces high stored energy. Most of previous work on recrystallization of SX superalloys focuses on annealing conditions[2, 9-10], microstructural features[11] and mechanical properties. Xie et al.[12] investigated the orientational dependence of deformation and recrystallization in a Ni-based single crystal superalloy. Wang et al.[13] studied the effect of eutectics on the deformation and recrystallization behavior in SX superalloy CMSX-4. Their samples were deformed by indent. Till now, a lot of simulation on recrystallization has been conducted[14-18], but most about aluminium, magnesium and steel. Little work has been done on simulation of recrystallization of SX nickel-based superalloys. One simulation on recrystallization of SX superalloys was reported by Zambaldi et al.[19] in 2007. However, the kinetics used in their CA model and simulated results didn’t obviously conform to the reality. Besides, growth kinetics of recrystallization of SX superalloys has not been well researched and discussed. In this paper, a macroscopic phenomenon-based deformation model will be used to obtain the driving force for recrystallization. CA simulation of recrystallization will also be utilized to investigate the growth kinetics of recrystallization. 2. Experimental Procedure 2.1. Single-crystal Superalloy Materials The second generation single-crystal superalloy DD6 in as-cast condition was used for the experiments. The composition is given in Table 1. The material exhibits a two-phase microstructure, revealing coherent and cuboidal ¤ ’-precipitates are surrounded by γ-matrix. Dendritic morphology presents in as-cast microstructure of this material, fine and regular cubic gamma prime phase in dendritic core regions, coarse and irregular cubic gamma prime phase in interdendritic regions. Bulky eutectic structure appears in interdendritic regions. Table 1. Nominal composition of alloy DD6 Element wt.%

Ni Balance

Cr 4.3

Co 9

Mo 2

W 8

Ta 7.5

Re 2

Nb 0.5

AL 5.6

Hf 0.1

2.2. Hot Compression In this research, hot compression tests were chosen to obtain 10%~12% plastic strain, as a prerequisite for recrystallization in single crystal superalloys. Cylinder pieces of diameter 6 mm and length 10mm was cut using Electrical Discharge Machining (EDM) from as-cast test bars of SX DD6 of diameter 15 mm and length 300 mm. Compressive testing was carried out by Gleeble1500D (thermal physical simulator) at 980 ºC at a strain rate of 3×10-3 s-1. Great care was taken to ensure that only test pieces within 15° of along the axis were employed for the compressive testing. The test pieces were heated by 15 ºC/s and held for 1 minute before compression. 2.3. Solution Heat Treatment Every compression sample was cut into two same smaller cylinders using EDM, and tubed in silica glass under inert argon atmosphere to avoid oxidation. The samples were annealed at temperatures 1280 ºC, and cooling was done in air. The annealing times were 10min, 1h, 2h and 4h. 2.4. Electron Back Scatter Diffraction Observation In order to investigate the grain structure evolution during recrystallization, the middle section face of cylinder pieces was ground to a 2000# finish and then electropolished using perchloric acid (90%) and dehydrated alcohol (10%) to provide a deformation-free flat surface in order to collect high-quality Kikuchi pattern maps used for grain

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orientation measurement. The samples were examined by an Oxford detector within a MIRA3 LMH field emission gun scanning electron miroscope. EBSD data were collected using an accelerating voltage of 20 kV and a magnification of 100 times with a step size of 10 or 20 μm was used to achieve a good combination of accuracy and analysis time. The data were then further analyzed using the HKL CHANNEL5 suite of programs, assuming a FCC Ni-superalloy structure with a lattice parameter of 0.36 nm. 3. Macroscopic Phenomenon-based Deformation Model of SX Superalloys During last two decades, the crystal plasticity finite element method (CPFEM)[20-22] has gained a great popularity in describing the heterogeneous characteristics of deformation on mesoscale. However, it can hardly describe the stored-energy distribution in the entire blade scale for the limitation of its computing amount. Hence, for the present modeling, a macroscopic phenomenon-based deformation model considering orthotropic mechanical properties is utilized for SX superalloys DD6. The purpose of this model in this paper is to obtain the driving force for recrystallization. The elastic part of SX superalloys considers the orthotropic characteristics. For materials with cubic structure (BCC or FCC), whose three principal orientations (here denoted 1, 2, 3 respectively) are identical, elastic constitutive equation is determined by the generalized Hooke law, which can be expressed as follows:

^H ` > S @^V ` -1 ^V ` = > S @ ^H ` = >C @^H ` T ^H ` >H11H 22 H 33 J 12 J 23 J 31 @ T ^V ` >V 11V 22 V 33 W 12 W 23 W 31 @ > S @ >C @-1

0 0 0 · § 1 / E P / E P / E ¨ P / E 1 / E P / E ¸ 0 0 0 ¨ ¸ ¨ P / E P / E 1 / E 0 0 0 ¸ [S ] ¨ ¸ 0 0 1/ G 0 0 ¸ ¨ 0 ¨ 0 0 0 0 1/ G 0 ¸ ¨ ¸ 0 0 0 0 1/ G ¹ © 0

(1)

(2)

(3)

Where {V} and {H} are the stress and strain vector respectively. Sij and Cij denote the elastic compliance and stiffness constants, which measure the strain (or stress) necessary to maintain a given stress (or strain). E, ­ and G are Young’s modulus, poisson ratio and shear modulus respectively in three principal orientations , and . Since the matrix phase ¤and precipitate phase γ’ both exhibit FCC structure and have coherent interface, the nickel-based single-crystal superalloys can be approximately considered as orthotropic materials, whose three principal orientations are identical, though nickel-based single crystal superalloys are just acknowledged as engineering SX. There are only 3 independent constants in [S]. By the spatial geometry transformation, the Young’s modulus E’ and shear modulus G’ in a given crystallographic orientation can be obtained as follows:

1/ E '

S '11

S11  2 J ( S11  S12  S44 / 2)

(4)

Fig. 1 shows Young’s modulus in all orientations at 980 ϨC and 1070 ϨC, which confirms high anisotropy of SX superalloy DD6.

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Fig. 1 Orientational dependence of Young’s Modulus of SX superalloy DD6 at 980ϨC

For the anisotropic materials, the Hill yield criterion, a simple extension of the von Mises criterion, was usually employed. Hill’s potential function can be expressed in terms of rectangular Cartesian stress components as:

f V

F (V 22  V 33 )2  G(V 33  V 11 ) 2  H (V 11  V 22 ) 2  2 LV 232  2M V 312  2 NV 122

(5)

Where F, G, H, L, M and N are independent constants obtained by tensile or compression tests of the material in different orientations. If ƒ(V)>V0, the material will yield. V0 is the user-defined reference yield stress specified for the metal plasticity definition, usually using the yield strength in one principal orientation. In this article, V0=S, where S indicates the yield strength in one principal orientation. Since the mechanical properties are identical in the three principal orientations for face-centered cubic materials, ƒ(V) can be expressed as:

f V

1 2

(V 22  V 33 ) 2  (V 33  V 11 ) 2  (V 11  V 22 ) 2  2 K (V 232  V 312  V 122 )

(6)

K=L/F denotes anisotropic plastic parameter. The potential function will evolve into the form of von Mises criterion if K equals 3, meaning that the material is isotropic. Isotropic hardening criterion is employed, and the flow rule used in this modelling is as follows:

d H pl

dO

wf

dO

wV

f

b

Where O denotes plastic multiplier. From the definition ƒ(V) above, b can be expressed as:

(7)

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ªV 11  0.5(V 22  V 33 ) º «V  0.5(V  V ) » 11 33 « 22 » «V 33  0.5(V 11  V 22 ) » b=« » KV 12 « » « » KV 31 « » KV 23 «¬ »¼

(8)

In this paper, commercial available FEM software was used for the deformation calculation. 4. Microstructure Simulation of Recrystallization with CA Method 4.1. Recovery and Recrystallization of Single Crystal Superalloys In general, deformed metal or alloys will experience three main stages during heating: recovery, grain growth and grain coarsening, as shown in Fig. 1. The effect of recovery is strong for metals and alloys (such as Al) with high stacking fault energy (Al, 166 mJ/m2). The stacking fault energy of pure Ni is about 128 mJ/m2, while most research indicates that the value for single crystal nickel-based superalloys is below 20 mJ/m2. The strongly ordered phase gamma prime tends to soften on recovery. Thus, the recovery of SX superalloy is very weak, which is also demonstrated by some experiments. In the following simulation, recovery will be omitted[23].

Fig. 2 General change of deformed metal during heating

4.2. Prediction of Driving Force for Recrystallization In most previous work, the deformation stored energy was expressed in terms of dislocation density using a dislocation-based constitutive model. However, this model can hardly describe the SX superalloys. In this research, plastic dissipated energy given in the following form was used to calculate the driving force for recrystallization of single crystal superalloys.

dU pl

³

H pl

0

V  d H pl

(9)

Most of the work expended in deforming a metal is given out and only a very small amount (~1%) remains as energy stored in the material. Hence, 1% of plastic dissipated energy was used as the driving force for recrystallization.

P

0.01U pl

(10)

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4.3. Static Recrystallization Nucleation Model The nucleation in the CA model is based on more physical rules, e.g. can take into account the influence of physical temperature. The number of nuclei that can appear in the material volume per unit of time is controlled by the equation:

C0 ( P  P c ) exp(

N

Qa RT

)

(11)

Where Qa is the activation energy for nucleation which is independent of deformation conditions, R the universal gas constant, T the absolute temperature, C0 is the scaling parameter and Pc is the critical stored energy below which recrystallization will not occur. The activation energy for nucleation Qa and the parameter C0 in the above nucleation equation are not known precisely. The value of critical stored energy Pc was computed with the following equation[24]:

P

107 H c

c

2.2H c  1.1

J lagb

(12)

Where Hc is the critical plastic strain which is necessary to trigger nucleation and Jlagb is the low angle grain boundary energy. The probability of nucleation in the unit of time is computed as:

Znuc

NS N dt

(13)

Where SN is the volume in which the nucleus can appear and dt is the time step. 4.4. Grain Growth and Coarsening Model The movement of the recrystallization front is treated as a strain-induced boundary migration process, in which the difference in the strain energy between the deformed area and the recrystallized area provides the driving force. Thus, the velocity of the recrystallization front, V, moving into the deformed matrix can be expressed as follows:

V

MP

(14)

Where M is the grain boundary mobility and P is the net pressure (driving force) on the grain boundary. During this stage, the contribution of driving pressure arising from boundary curvature is neglected for its relative small amount. The mobility can be estimated by

M

D0b 2 kT

exp(

Qb RT

)

(15)

Where D0 is the diffusion constant, b the Burger’s vector, k the Boltzmann constant and Qb is the activation energy for grain boundary motion. The grain motion is controlled by the diffusion process from physical views. Grain coarsening is assumed to occur on the common boundaries of the recrystallized grains in a curvaturedriven grain growth after the recrystallized fronts impinge on each other. The driving force for boundary motion P can be expressed by

P

JN

(16)

Where J is the grain boundary energy and N is the grain-boundary curvature. In this present model, the grain boundary energy is assumed to be independent on the misorientation angle T between two neighboring grains. The grain boundary energy J can be calculated by the Read-Shockley equation:

J

§ T ·ª § T ·º J m ¨ ¸ «1  ln ¨ ¸ » © Tm ¹ ¬ © Tm ¹¼

(17)

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Where T is the grain-boundary misorientation, Jm the large angle grain boundary energy, and Tm the large angle grain boundary misorientation. The transformation matrix g of a grain with Euler angle (M1 ) M2) can be calculated as

g

sin M1 cos M2  cos M1 sin M2 sin ) sin M2 sin ) º ª cos M1 cos M2  sin M1 sin M2 cos ) «  cos M sin M  sin M cos M cos )  sin M sin M  cos M cos M cos ) cos M sin ) » 1 2 1 2 1 2 1 2 2 « » «¬ sin M1 sin )  cos M1 sin ) cos ) »¼

(18)

The transformation matrix 'g from grain A (gA) to grain B (gB) can be expressed as

'g

g B g A1 { g A g B1

Then, the misorientation T can be calculated by

T

min cos 1

^

(19)

`

tr (O432 'g )  1 2

(20)

Where O432 is the symmetry operator. The grain boundary curvature is theoretically defined as

1

N

r1



1 r2

(21)

Where r1 and r2 are the principle radii of the grain boundary segment. An equivalent model is adopted to calculate the grain boundary curvature, which can be written as

N

A Kink  Ni a

N 1

(22)

Where a is the cell size of the CA lattice; the coefficient A=1.28; the number of the first and second nearest neighborhoods N=24; Ni is the number of cells within the neighborhood belonging to grain I, and Kink=15 is the number of cells within the neighborhood belonging to grain I for a flat interface (N=0). 5. Cellular Automaton Algorithm Coupled with Deformation FEM Modelling In this paper, a deterministic CA model is utilized to simulate static recrystallization of SX nickel-based superalloys. A 2D square lattice (0.01mm length of one cell and 700h700 of cell number) is employed. A von Neumann’s neighbor rule, which consider the nearest four neighbors cells, is used. The state of each cell site is characterized by the following variables: the grain number variable representing the different grains, the stored energy variable representing the driving force, the order parameter variable indicating whether it belongs to recrystallized grain, deformed matrix or grain boundary, Euler angle variable to calculate the misorientation angle of neighboring grains and the fraction variable representing the recrystallized fraction. Some key parameters for this CA model are shown in Table 2. The essentials of this CA model are summarized, as follows: a) Obtaining the stored energy from FEM modelling results using above mapping algorithm; b) Applying nucleation model according to Eq. (11) to (13) if there exists the deformed matrix, and the stored energy of this cell is set to zero; c) Obtaining the mobility M of the grain boundary in Eq. (15); d) Determining the driving force: stored energy or grain-boundary energy via Eq. (17) using a Von Neumann neighborhood; e) Calculating the curvature of a boundary cell according to Eq. (22); f) Calculating the velocity of a boundary cell and determining the direction of the motion; g) Calculating the fraction of a recrystallization front cell or a boundary segment, as follows:

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f

Vx dt dx



Vy dt dy



VxVy dt 2

(23)

dxdy

Where Vx and Vy are the velocity in X and Y directions, dx and dy are the sizes of a cell along the X and Y axes, and dt is the length of the time-step; h) Reassigning the crystallographic orientation of a recrystallization front cell or a boundary cell, when the boundary migrates through it (ƒt1). Table 2. Key Parameters for CA model Parameter

Value

Unit

Activation Energy for Nucleation, Qa

295

kJ/mol

Activation Energy for Grain Growth, Qm

300

kJ/mol

Large Angle Boundary Energy, Jm

0.9

J/m2

Low Angle Boundary Energy, Jlagb

0.6

J/m2

Critical Plastic Strain for Recrystallization, Hc

0.02

Burgers vector, b

0.36

Diffusion Constant, D0

m2/s

-23

J/K

7.5e

Boltzman Constant, K

nm

-4

1.38e

Universal Gas Constant, R

8.3144

J/(mol*K)

Step Size

0.01

mm

6. Results and Discussion 6.1. Deformation Simulation The orientation [001] was chosen as compression axis. A one-eighth-cylindrical model was used due to the fourfold crystallographic symmetry and symmetry along the compression axis, shown in Fig. 3(a). Hexahedral element was chosen to conduct the FEM analysis, and 34154 nodes and 31200 elements were used. The distribution of dissipation plastic energy is nonuniform when considering the friction between compression head and test piece, and the distribution on the section close to the middle face is relatively uniform, as shown in Fig. 3(b) and Fig. 4. The section close to the middle face, whose energy distribution is relatively uniform, was chosen to provide stored energy and predict microstructure. As stated above, 1% of the dissipation plastic energy in FEM was transformed into CA square lattice results with the following map function, as shown in Fig. 4. The conversion result shows that the interpolated values are in good agreement with the theoretical values.

f ( x, y)

a0  a1 x  a2 y  a3 xy

(24)

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Fig. 3 The finite element mesh (a) and the simulated distribution (980 ºC, compression head moving ~0.5mm) of plastic dissipation energy (b) assuming the friction coefficient between head and test piece as 0.1

Fig. 4 Distribution of Plastic Dissipation Energy (a) and Stored Energy (b): map from FEM results to CA Square Lattice results (section with 0.5 mm to the middle face)

6.2. Effect of Annealing Time on Recrystallization Microstructure Fig. 5 and Fig. 6 show experimental and simulated microstructure evolution with different solution time respectively. These samples were deformed at 980 ºC and heated at 1280 ºC. Since current EBSD technique can’t distinguish the gamma phase from gamma prime phase due to their same crystal structure and similar lattice constants, colors in the IPF (inverse polar figure) map only represent crystal orientations instead of phases. Thus, gamma and gamma prime phases were treated as one phase for recrystallization simulation. If neglecting the twin grains effect, both experiments and simulation indicate that the grain size increases with time. Similarly, the simulation can predict the microstructure well. Another finding from the experiments is that during the early stage the grains tend to exhibit dendritic shape, just as illustrated in Fig. 5(a) and (b). One possible explanation is that the solidified dendrite affects the recrystallization. It’s easier for nucleation and grain boundary migration in the dendritic core region, and the eutectic region retards the motion of the grain boundary. This factor is neglected in this research, and should be dealt with in the future.

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Fig. 5 Experimental Microstructure Evolution with different solution time (a)10min; (b)1h; (c)2h; (d)4h (deformed at 980 ϨC and heated at 1280 ϨC)

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Fig. 6 Simulated Microstructure Evolution with different solution time (a)10min; (b)1h; (c)2h; (d)4h (deformed at 980 ϨC and heated at 1280 ϨC)

6.3. Kinetics of Recrystallization for SX Nickel-based Superalloys Since no experimental data for the activation energy Qb of the grain boundary at different temperatures are available in the literature, the activation energy Qb is estimated according to some literature. Zambaldi[19] set Qb 1290 kJ/mol to model recrystallization of a single-crystal nickel-base superalloy. In addition, it should be mentioned that Porter and Ralph[25] reported a high activation energy of 790 kJ/mol for recrystallization experiments on a warm-rolled polycrystalline Nimonic 115 alloy. However, these values are obviously high. As stated above, the grain motion process is controlled by the diffusion. Therefore, the activation energy for grain boundary motion can be estimated according to the diffusion values. For Ni-Al binary system, the activation energy for diffusion of Al in Ni is 284 kJ/mol, and activation energy for diffusion of other elements (Ti, Co, Ru, Re etc.) in nickel-based superalloys are from 230 to 315 kJ/mol[26]. In this work, the value of 300 kJ/mol is used. When examining the microstructure for the sample deformed at 980 ºC and heated at 1280 ºC for 10min, about 90% of deformed matrix has recrystallized. Fig. 7(a) shows the experimental local misorientation map, which can tell the difference between deformed matrix and recrystallized region. The simulation can predict the recrystallization fraction well, as confirmed in Fig. 7(b).

Fig. 7 Experimental (a) and simulated (b) recrystallized microstructure for the sample deformed at 980 ºC and heated at 1280 ºC for 10min (blue region standing for recrystallization, remaining for deformed matrix)

When considering the kinetics of recrystallization, JMAK model is used in this research. In general, the JMAK equation can be expressed as:

X

§ § t ·q · 1  exp ¨  ¨ ¸ ¸ © ©W ¹ ¹

(25)

Where X is the recrystallization fraction, W is a characteristic time for recrystallization (in short, recrystallization time X(W)=1-e-1), q is the Avrami exponent and t is the time. Simulation results can agree with this model well, and we can get the avrami exponent q 3.11 and the characteristic time W is 463.24 seconds from Fig. 8. However, the JMAK model assumes homogeneous nucleation and spatially and temporally constant growth rate. This assumption is obviously the violation of the reality that the as-cast microstructure is unhomogeneous. This will be considered in the future research.

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Fig. 8 Recrystallization kinetics for the samples deformed at 980 ºC and heated at 1280 ºC

7. Conclusion 1) 2) 3) 4)

A CA method coupled with a macroscopic phenomenon-based deformation model was proposed to predict the microstructure evolution during the recrystallization of one single crystal nickel-based superalloy. Dissipation plastic energy can be used to obtain the stored energy for recrystallization. Gamma and gamma prime phases can be treated as one phase for recrystallization simulation. CA method can simulate the recrystallization process from the view of grain size and predict the recrystallization fraction well. The grain boundary moving during recrystallization is controlled by the diffusion process. For nickel-based single crystal superalloys, the activation energy for grain boundary motion can reach as high as 300kJ/mol at the solution temperature of 1280 ºC.

Acknowledgements This research is funded by National Basic Research Program of China (No. 2011CB706801) and National Natural Science Foundation of China (No. 51171089). References [1] D. Ma, A. Bührig-Polaczek. The Geometrical Effect On Freckle Formation in the Directionally Solidified Superalloy Cmsx-4. Metall. Mater. Trans. A. 45 (2014) 1435-1444. [2] D.C. Cox, B. Roebuck, C. Rae, R.C. Reed. Recrystallisation of Single Crystal Superalloy Cmsx-4. Mater. Sci. Tech.-Lond. 19 (2003) 440446. [3] B. Zhang, C. Liu, X. Lu, C. Tao, T. Jiang. Effect of Surface Recrystallization On the Creep Rupture Property of a Single-Crystal Superalloy. Rare Metals. 29 (2010) 413-416. [4] J. Meng, T. Jin, X.F. Sun, Z.Q. Hu. Effect of Surface Recrystallization On the Creep Rupture Properties of a Nickel-Base Single Crystal Superalloy. Mat. Sci. Eng. A.-Struct. 527 (2010) 6119-6122. [5] G. Xie, L. Wang, J. Zhang, L.H. Lou. Influence of Recrystallization On the High-Temperature Properties of a Directionally Solidified NiBase Superalloy. Metall. Mater. Trans. A. 39A (2008) 206-210. [6] W. Schaef, M. Marx. A Numerical Description of Short Fatigue Cracks Interacting with Grain Boundaries. Acta Mater. 60 (2012) 2425̢ 2436. [7] C. Panwisawas, H. Mathur, J. Gebelin, D. Putman, C.M.F. Rae, R.C. Reed. Prediction of Recrystallization in Investment Cast Single-Crystal

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