Creep residual life prediction of a nickel-based single crystal superalloy based on microstructure evolution

Accepted Manuscript Creep residual life prediction of a nickel-based single crystal superalloy based on microstructure evolution Chengjiang Zhang, Weibing Hu, Zhixun Wen, Wenwei Tong, Yamin Zhang, Zhufeng Yue, Pengfei He PII:

S0921-5093(19)30445-9

DOI:

https://doi.org/10.1016/j.msea.2019.03.132

Reference:

MSA 37749

To appear in:

Materials Science & Engineering A

Received Date: 18 September 2018 Revised Date:

30 March 2019

Accepted Date: 31 March 2019

Please cite this article as: C. Zhang, W. Hu, Z. Wen, W. Tong, Y. Zhang, Z. Yue, P. He, Creep residual life prediction of a nickel-based single crystal superalloy based on microstructure evolution, Materials Science & Engineering A (2019), doi: https://doi.org/10.1016/j.msea.2019.03.132. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Manuscript:

Creep residual life prediction of a nickel-based single crystal superalloy based on microstructure evolution Chengjiang Zhanga,b, Weibing Hub, Zhixun Wenc*,Wenwei Tongc, Yamin Zhangc, Zhufeng Yuec,

a

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Pengfei Hea*

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, P. R. China

School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, P. R.

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b

China c

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School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, P. R. China

*

Corresponding author

Email address: Zhixun Wen

[email protected]

[email protected]

Pengfei He

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Abstract: Microstructure evolution occurs throughout the high temperature creep process inside nickel-based single crystal superalloys. In this study, creep tests of a nickel-based single crystal superalloy in the [001] orientation at 980°C and 1100°C were performed. The results indicated that creep failure occurred due to the formation of micropores and deterioration. Scanning electron

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microscopy (SEM) observations showed that the material degradation during creep was primarily reflected in the coarsening of the matrix channel and the precipitation of the TCP phase. Meanwhile,

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transmission electron microscopy (TEM) observations indicated that the dislocation morphology of the [001] orientation is consistent with the characteristics of the octahedral slip system. To obtain a quantifying explanation of such microscopic phenomena, a material constitutive model and creep damage model considering the Orowan effect and the dislocation effect were established based on the crystal plasticity theory. The model parameters were fitted according to the experimental results. Based on the quantitative description of the microstructure evolution during the creep process, the residual life prediction model of a single crystal superalloy was established for guidance in engineering applications. Keywords: residual life; single crystal; crystal plasticity theory; microstructure evolution. 1

ACCEPTED MANUSCRIPT 1. Introduction The excellent high temperature mechanical properties of a nickel-based single crystal superalloy are primarily derived from its γ/γ' phase microstructure. The microstructure is formed by a face-centred cubic (FCC) γ' precipitate phase (approximately 70% of the volume fraction) located

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uniformly in the γ matrix phase [1]. At temperatures greater than 900°C, the microstructure evolution of a single crystal superalloy expresses rafting/topological inversion behaviour in the loaded state [2], and the critical strain for rafting is 0.12%-0.16%. The γ' precipitate phase gradually roughens into a layered plate structure from the initial cubic shape and determines the macroscopic mechanical

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properties of a single crystal superalloy in a high temperature service environment [3]. Pearson et al. [4] proposed that rafting is a hardening process to enhance the creep behaviour of single crystal alloys in

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the [001] orientation by inhibiting the dislocation climb of the γ' precipitate phase, greatly increasing the resistance to creep deformation. However, studies [5, 6] suggested that the directional coarsening of the γ' precipitate phase during the creep process is a softening behaviour, which weakens the mechanical properties of the single crystal superalloy at high temperatures. The oriented coarsening behaviour of the γ' precipitate phase is governed by the lattice

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mismatch, elastic constant and element content. The difference in the atomic chemical potential of the lattice mismatch is considered the driving force of atomic diffusion. Some studies suggested that creep resistance is also stronger under a larger mismatch [7]. In addition, the rafting structure folds [8-10]

. Zhou et al. [11] found that the increase in the

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due to the unevenness of the lattice mismatch

difference between the elastic modulus of the strengthening phase and the matrix phase increases the stress gradient near the phase interface and that it increases the driving force of rafting in the initial [12]

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stage of creep. Tian et al.

conducted a statistical analysis of alloy composition using TEM and

found that the increase in the Re element content can reduce the migration of the Ni and Al elements, inhibiting the exchange of elements between two phases, and thus increase the critical stress and temperature of rafting. However, research on the fourth generation of nickel-based single crystal superalloys shows that the alloy composition does not affect the driving force of rafting [13]. At high temperatures, external stress can significantly affect the aspect ratio and phase interface dislocation density of the γ' precipitate phase at the γ/γ' interface

[14, 15]

. The stress causes a difference in the

atomic chemical potential at the interface, which provides a driving force for diffusion of the γ' phase. Moreover, the completion time of the rafting decreases with increasing load [2, 16-23]. Huron and Shui 2

ACCEPTED MANUSCRIPT found that increasing temperature significantly reduces the critical stress of atomic diffusion and greatly increases the oriented coarsening rate of the γ' precipitate phase [24, 25]. The crystal plastic model of the rafting process was established by Desmorat et al.

[26]

to

describe microstructure evolution by γ channel width evolution coupled with the Kelvin method, [27]

described the

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which can effectively simulate the directional coarsening. Similarly, Fan et al.

isotropic material directional coarsening using the square root of the volume fraction of the γ' precipitate phase based on the Ostwald model, thus, the Orowan stress can be used in the crystal plasticity model to describe the effect of rafting on creep deformation. The model can predict the

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direction of rafting under complex load conditions and the channel width during rafting. 2. Material and methods

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The material studied is a second generation nickel-based superalloy DD6 provided by the Institute of Aeronautical Materials. Single crystal bars were prepared under the standard heat treatment, which is 1315°C×6 h/AC +1130°C×4 h/AC + 870°C×32 h/AC (AC means air-cooling). The orientation deviation of the material was controlled within 5°C, and the chemical composition of the material is listed in Table 1.

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The microstructure of the original (001) crystal plane after electrochemical etching is shown in Fig. 1. The cubic block particles are the γ' strengthening phase, the particle size is 0.3-0.6 µm (measured by the ImageJ software), and the main component is Ni3Al. The sizes of strengthening

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phases inside interdendrites are close to 0.6 µm, and inside dendrites are close to 0.3 µm. The atomic arrangement is face-centred cubic (FCC), with a volume fraction of 67% in the alloy (obtained by

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etching the matrix phase with electrochemical corrosion), uniform distribution and neat alignment.

Fig. 1. Two-phase structure of the single crystal DD6 original material 3

ACCEPTED MANUSCRIPT Table 1 Chemical composition of the single crystal superalloy (in wt.%). Cr

Co

W

Al

Ta

Re

Mo

Hf

Ni

4.0

8.0

7.0

6.0

7.0

3.0

2.0

0.4

Bal.

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Creep tests were conducted at 980°C and 1100°C, with I-shaped samples adopted. The size of the specimen is shown in Fig. 2. The SEM and TEM observations are operated at the gauge lengths in the middle of the I-shaped samples. The samples for the creep residual life test were taken from

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the serviced materials, and the axial direction of the samples is the [001] orientation.

3. Results

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3.1. Rafting

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Fig. 2. Size and physical drawings of the I-shaped sample and serviced blade

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Fig. 3. Microstructure evolution of the [001] orientation at 980°C /250 MPa

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(a) 20 h, (b) 40 h, (c) 60 h, (d) 100 h, (e) 150 h, (f) 200 h

Fig. 4. Microstructure evolution of the [001] oriented (001) crystal plane at 980°C /250 MPa (a) 5 h, (b) 100 h

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Fig. 5. Width evolution of matrix channels at the [001] orientation at 980°C

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Microstructural observations at different creep stages indicate that the microstructure undergoes significant evolution at 980°C, as the initial cubic γ' phase is gradually transformed into a plate-like structure in the vertical stretching direction. As observed from Fig. 3, in the initial stage of rafting, the γ' phases in the horizontal direction are gradually connected to each other, and the matrix channels between them are collapsed into dots. After 60 h of creep, the entire strengthening phase

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formed a complete N-type rafting structure along the direction of the vertical tensile stress axis. The (010) crystal plane and the (100) crystal plane have approximately the same length as that of the γ' phase, showing a rugged appearance. Moreover, the γ' phase of the (001) crystal plane gradually evolved from a cubic shape to a mesh shape (Fig. 4).

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At the same time, the channel width of the matrix phases becomes wider and the change rate of the channel is faster at the initial stage of rafting. With the completion of rafting and the beginning of

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topological inversion, the change rate of the matrix channel width decreases significantly (Fig. 5). Especially in the topological inversion stage, the width of the channel of the matrix phase remains basically unchanged, whereas the long matrix phases are broken from the middle and eventually become the thick and short matrix phases. The statistics show that the degree of roughening of the matrix channel is proportional to the 0.5th power of the creep time. By comparing the rafting process under different stresses (Fig. 5), it is clearly observed that the creep life decreases and the rafting process accelerate with increasing external stresses. The velocity of matrix phases broadening increases significantly, maintaining a fast and then slow rate of evolution. 6

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Fig. 6. Microstructure evolution of the [001] orientation at 1100°C /140 MPa (a) 20 h, (b) 40 h, (c) 60 h, (d) 80 h, (e) 100 h, (f) 150 h

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Fig. 7. Width evolution of matrix channels at the [001] orientation at 1100°C

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As observed from Fig. 6, by comparing the microstructure evolutions of the [001] orientation rafted at 1100°C and 980°C, it is found that at both temperatures, the cubic γ' phases gradually transform into the plate-like raft structure perpendicular to the tensile direction. In the case of a similar creep life, the rafting rate and coarsening rate are obviously faster at 1100°C than that of at 980°C. For example, after 20 h of rafting, the γ' phase at 980°C has not been completely joined, but

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the γ' phase at 1100°C has formed a complete plate rafting form. These results show that the higher the temperature is, the more intense the atomic diffusion between the strengthening phase and the matrix will be. It is observed from Fig. 7 that at 1100°C, the greater the stress is, the faster the matrix phase channel widens.

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3.2. Dislocation motion

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Fig. 8. Dislocation distribution of the [001] orientation after creep at 980°C /250 MPa for 20 h

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The double beam TEM observations were performed in the (110) crystal ribbon axis and the (111) operation vector, as well as the geometric model and dislocation diagram are shown in Fig.8.

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Under 980°C /250 MPa, the [001] orientation γ' phase has not completely roughened into the plate structure at 20 h during the creep process. The creep deformation of the material is dominated by dislocations, and the dislocation network at the γ/γ' interface is derived from the reaction between interface dislocations. At this time, the dislocation motion is primarily conducted in the form of slip and climb, and it cannot shear the strengthening phase. The matrix channel in the direction of

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stretching shows a dislocation of the bow, which indicates that the dislocation needs to overcome the obstruction beyond the strengthening phase and is called the Orowan bypass mechanism. Meanwhile, the dislocation network appears in the matrix channels and present 45° in the direction of stretching. Interface dislocations on different slip surfaces can react as follows: (1)

a / 2[101] − a / 2[1 10] → a / 2[011]

(2)

a / 2 < 1 10 > {111} ,

the slip surfaces are (111), (-11-1), (1-1-1),

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a / 2[101] − a / 2[0 11] → a / 2[110]

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Moreover, the active slip system is

and (-1-11), and the slip directions are [0-11], [011], [10-1], [101], [1-10], and [110] of the octahedral slip system. When the rafting process of the γ' phase is completed, the dislocation network tends to be stable. Zhang et al.

[13]

found that the hindrance of the dislocation network to dislocation motion

increases with the dislocation density in the dislocation network. Therefore, the obstacle mechanism of the Orowan mechanism and dislocation network should be considered in the creep constitutive model and creep damage model.

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Fig. 9. Dislocation distribution of the [001] orientation after creep at 1100°C /140 MPa for 20 h

The [001] orientation sample was observed in the [110] ribbon axis and the (200) operation

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vector. Under 1100°C/140 MPa, the evolution of the microstructure is primarily dominated by rafting, as shown in Fig.9. The most common consideration for creep damage is the rapid coarsening of the two-phase microstructure due to high temperatures. Two features of early deformation are directional

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coarsening of the matrix channel and interface dislocation nets1 at the two-phase interface. The interface dislocation network is generated by a/2<1-10> {111} creeping dislocations in the matrix

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phase. 4. Discussion

4.1. Creep constitutive model and damage model The FE model applied in this paper is based on the crystal plasticity theory proposed by Hill and Rice [28, 29], and it is assumed that the creep shear strain rate γ& (α ) of the α slip system follows the common creep rule:

γ& (α ) = A(τ α ) n

(3)

where A and n are creep parameters, and τ α is the resolved shear stress of the α slip system. The 10

ACCEPTED MANUSCRIPT creep parameters stay the same in each slip system belonging to a particular slip system group, while changing from group to group. The resolved shear stress of the α slip system can be expressed as

τ α = σ : P (α ) σ

is the stress tensor under the crystal axis. P(α ) is the orientation factor, defined as: P (α ) =

(

T 1 (α ) (α )T m n + n (α )m (α ) 2

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where

(4)

)

(5)

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where m(α ) indicates the slip direction of the starting slip system, and n(α ) represents the unit normal vector of the slip surface in the slip system.

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The elastic strain rate ε& e can be obtained from the macroscopic stress rate

σ&

, and ε& c is

obtained from the resolved shear stress of the slip system:

ε& = ε& e + ε& c

(6)

The elastic strain rate ε& e follows Hooke's law and can be obtained from the knowledge of

orientation factor, namely:

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elastic mechanics. ε& c is obtained by multiplying the shear strain rate of the slip system by the

(7)

N

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ε&ijc = ∑ γ& (α )P (α ) α =1

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Decomposing the creep strain gives:

(ε ) = (ε ) ij c

Oct1 ij c

+ (ε ij )c

Oct 2

+ (ε ij )c

Cub

(8)

The three terms on the right-hand side of equation (8) correspond to the creep strains of the octahedral slip system, the dodecahedral slip system and the hexahedral slip system, respectively. Assuming that the elastic part ε& e and the inelastic part ε& c of the macroscopic strain rate do not interact with each other, the σ& caused by creep deformation can be expressed as:

σ& = Ce : ε& where Ce is an anisotropic elastic tensor. 11

(9)

ACCEPTED MANUSCRIPT Nickel-based single crystal superalloys have a special two-phase microstructure. The strengthening of the γ' phase to creep is primarily due to its hindrance to dislocation motion. When dislocations interact with the γ' phase, they are drawn at the γ/γ' interface, as indicated by the white arrows in Fig. 10. When the material undergoes plastic deformation during creep, dislocations need

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to cross the strengthening phase to overcome the obstruction. This behaviour is known as the Orowan bypass mechanism. The force required to bypass the strengthening phase is the threshold stress of creep and is called Orowan stress, which is denoted as τ or . Kakehi et al.

[30-32]

considered

the γ' phase), namely:

τ or = λ

Gb

κ

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this slip resistance to be inversely proportional to the width of the matrix channel (the space between

(10)

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where κ is the width of the matrix channel, G is the shear modulus, b is the modulus of the Burgers

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vector, and λ is the material constant, which varies with creep conditions.

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(a)

(b)

Fig. 10. The dislocation in the matrix channel lunges past the γ' phase (a) TEM image, (b) schematic diagram

According to the SEM analysis, the roughening of the matrix channel with the creep time is roughly parabolic and is expressed as:

κ = κ 0 + c1 t

(11)

where κ 0 is the initial base channel width, which is 0.05 µm for the DD6 single crystal alloy. c1 is a parameter that characterizes the matrix channel roughening rates are positively correlated with creep temperature and stress, which can be roughly calculated as c1 = 12

σ 2.7T 14.2 . t is the creep time. 5.4 × 1050

ACCEPTED MANUSCRIPT Moreover, the barrier stress generated by the dislocation network formed in the γ phase also hinders creep deformation. Literature

[33-37]

studies of high temperature materials including

nickel-based single crystal superalloys indicate that the strengthening effect of dislocations is proportional to the square root of the dislocation density, namely: (12)

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(α ) (α ) τ net = c2Gb ρ110

(α ) where c2 is a constant, b is the modulus of the Burgers vector, G is the shear modulus, and ρ110 is

(α ) (α ) (α ) ρ&110 = (k1 ρ110 − k2 ρ110 ) γ&M(α )

(13)

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the a / 2 < 110 > -dislocation density in the γ matrix phase, namely:

where k1 and k2 are the material constants characterizing the dislocation stress hardening,

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respectively and are taken as 2×109 and 25[38].

Whether a single crystal material undergoes long-term creep rupture or short-term tensile damage depends on the relationship between the load stress and yield stress. When the yield stress is exceeded, the γ' phase of the single crystal is sheared and the material is momentarily pulled off. The

surface

[39]

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critical stress is called the critical shear stress τ c when the yield stress is decomposed into each slip . The following equation is used here as a creep (long-term) failure criterion and a tensile

(short-term) failure criterion:

(14)

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(α ) R (α ) = τ (α ) − τ net − τ or

1. When R (α ) ≤ 0 , the decomposition shear stress fails to reach the critical value that enables

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dislocations to enter the γ phase, therefore no slip is activated in the matrix, then γ& (α ) =0 ; (α ) 2. When 0 < R (α ) < τ c − τ net − τ or , creep deformation is observed. Based on the continuous

damage model proposed by Kachanov and Ravbotnov [40, 41], and the damage evolution rate proposed by Yeh et al.[42], a creep damage model dominated by the shear stress and the shear strain rate was established: (α ) 1  (α ) (α )  R & & γ = γ 0  (α ) ×  1 − ω (α )  τ

13

n

(15)

ACCEPTED MANUSCRIPT ω&

(α )

= ω& 0

χ

R (α )

βτ c

(α )

γ& α γ&s

(16)

φ

( )

where χ and φ are temperature dependent parameters. β is a constant, taken here as 2.5. ω is (α )

the damage to the material. ω =1 represents the damage value when the material breaks. γ&s

is the

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steady-state creep rate, generally taken as 10. γ& (0α ) and ω& 0 are the initial creep rate and the initial damage rate, respectively. γ& (0α ) is a function of temperature and stress, namely:

(17)

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γ&0(α ) = A ⋅ exp ( −Q RT ) ⋅ (τ α ) n

In Eq. (17), T is the absolute temperature and R is the gas constant. Q is the activation energy,

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taken as Qoct = 6.97 E − 19 J / atom for the octahedral slip system.

(α ) − τ or , i.e., τ (α ) ≥ τ c , the external load is greater than the yield strength of 3. When R (α ) ≥ τ c − τ net

the material, and the structure undergoes instantaneous tensile failure. In this situation, the dislocation will be cutting the γ' phase, and the creep constitutive equation and the creep damage equation will no longer apply.

model:

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By integrating Eq. (16) from 0 to 1, the total creep time is calculated using the life prediction

tf =

1

N

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ω& ∑ α =1

0

R

(α )

βτ c

χ

(18)

(1 + n ⋅ φ )

where N is the number of the slip surface in a slip system, taken as 12 in the octahedral slip system

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(Oct1, <110> [111]) and the dodecahedral slip system (Oct2, <112> [111]), and taken as 6 in the hexahedral slip system (Cub, <110> [100]). Based on the crystal plasticity theory, the relationship between the shear stress τ and the macroscopic stress σ decomposed to each slip surface is expressed by the Schmid equation:

τ = S fσ

(19)

Here, S f is the Schmid factor. Therefore, the critical shear stress τ c of a slip system in each orientation can be determine from σ 0.2 . 14

ACCEPTED MANUSCRIPT The dislocations in the initial stage of creep are observed using transmission electron microscopy and indicate that the slip system of the [001] orientation is obtained as an octahedral slip system. Eq. (18) indicates that the creep life is proportional to the (- χ ) square of R (α ) , and R (α ) is positively correlated with the creep load σ . Therefore, the creep curve under different stresses can

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be used to find χ , adapting the creep model to all stresses at a specific temperature, as shown in Fig. 11 and Fig. 12. The specific elastic parameters and creep parameters used in the constitutive model and damage model are shown in Table 2 and Table 3. Table 2

Orientations

Elastic Modulus /GPa

980

[001]

80.5

1100

[001]

Poisson's ratio

Shear modulus /GPa

0.39

91

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Temperature/°C

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Elastic parameters of the nickel-based single crystal superalloy DD6.

67.5

Table 3

0.4

70

Creep parameters of the nickel-based single crystal superalloy DD6. Orientations

980

[001]

1100

[001]

A

n

ω& 0

χ

φ

τ c /MPa

Octahedron

1.2×10-20

7.5

12

5.2

0.4

277.6

Octahedron

1.5×10-18

7.5

38

5.6

0.4

157.2

Slip systems

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Temperature /°C

Fig. 11. Test curve and simulations at the [001] orientation under 980 °C 15

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Fig. 12. Test curve and simulations at the [001] orientation under 1100 °C

4.2. Creep residual life model

After serving for a period of time at high temperature, the rafting of the single crystal microstructure will lead to the degradation of mechanical properties, which is primarily reflected in the reduction of Orowan resistance to slip (Eq. (10) shows that the coarsening of the matrix channel

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during the rafting process decreases the Orowan stress). Based on the Orowan mechanism, the reduction of τ or will result in a decrease in the critical shear stress

τc

of the material. The

elastoplastic constitutive model of the nickel-based single crystal superalloy can be expressed by a

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viscoplastic expression:

where

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γ& (α )

τ or

 τ (α ) (α )  = γ&0  (α )  g

1/ m

   

  τ (α )  1 − exp  −  τ or  

n

   sgn τ (α )  

( )

(20)

is derived from Eq. (10), and g(α ) is the slip resistance, namely: g& (α ) = ∑ hαβ γ& ( β )

(21)

(α ≠ β ) is the latent hardening modulus and

hαα

β

hαβ is the slip hardening modulus ( hαβ

(not summed) is the self-hardening modulus). Here, the self-hardening modulus expression without considering the Bauschinger effect [28, 29] is adopted: hαα = h (γ ) = h0 sec h 2 16

h0γ τ s −τ 0

(22)

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h0 is the initial hardening modulus; τ0 is the initial yield stress, equal to the initial value of

the slip resistance g(α ) ;

is the saturated shear stress; and γ is the Taylor cumulative shear

τs

strain:

The latent hardening modulus is:

hαβ = qh(γ )

(23)

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t (α ) dt γ = ∑ ∫ γ& α0

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where q is a constant. The values of the parameters used in the critical shear stress

Table 4

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model are shown in Table 4.

τc

(24)

evolution

Constitutive model parameters of the nickel-based single crystal superalloy[38]. Model parameter

Symbol

Value

Unit

Reference slip rate

γ&0

1.5×10-10

s-1

m

0.085

/

7

/

98

MPa

h0

20

GPa

τ

s

110

MPa

q

1.3

/

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Rate sensitivity exponent slip resistance Rate sensitivity exponent Orowan threshold

，τ

g 0(α )

Critical shear stress

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Initial hardening modulus Saturation shear stress

n

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Latent hardening constant

0

With the coarsening of the matrix channel, the yield strength of the single crystal material shows a significant decrease. The uniaxial tensile curves along the [001] orientation of the DD6 single crystal superalloy for different matrix channel widths at 1100°C are shown in Fig. 13.

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Fig. 13. Tensile experiment of the DD6 single crystal alloy for different matrix channel widths at the [001]

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orientation under 1100°C

The uniaxial tensile curves of the DD6 single crystal superalloys with different matrix channel widths are basically the same, especially in terms of the slope of the elastic stage (i.e., the elastic modulus) and the ultimate deformation. The yield stress decreases as the initial matrix channel width

κ

increases, but the magnitude of the decrease decreases as the channel width increases. When

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κ > 0.2µm , the yield strength σ 0.2 tends to be constant. Substituting the channel width into the above constitutive model, the variation law of the yield strength of the DD6 single crystal at 1100°C

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is obtained, as shown in Fig. 14.

Fig. 14. Effect of the matrix channel size on the yield strength of the nickel-based single crystal superalloy at 1100°C

Through the elastoplastic constitutive model of the nickel-based single crystal superalloy under 18

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constitutive equation and the creep damage equation established above, the damage of single crystal creep is dominated by the shear stress and the shear strain rate. Therefore, the inclusion of pores and the TCP phase will cause a local stress concentration in the creep process of the single crystal material, thereby shortening the creep residual life. The above influencing factors are summarized as

γ& For the original material, let

(α )

= γ&

(α ) 0

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before creep, and Eq. (8) evolves to:

 R (α )  1  (α ) × (α )  1 − ω0 − ω  τ

n

(25)

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ω0

materials with initial damage

ω0 =0 ; for the material that has been in service for a period of time,

ω 0 = ω 1 + ω 2 + ω 3 , where the terms to the right of the equal sign represent the initial damage

caused by the micropores, the TCP phase, and the coarsening of the matrix channel.

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Most creep ruptures are caused by microcracks emitted by micropores. Liu[43] observed evolutions in the porosity of materials at each stages of creep at 1100°C /130 MPa, as shown in Fig. 15. The thick black line is the strain curve during the creep process, which is obtained by the creep

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test machine. The positions indicated by the arrows represent the porosity of the material when the strain reaches this value. Similarly, the porosity is obtained by the ImageJ software. The reslut

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indicates that only a few micropores are present in the initial stage of creep (Fig. 15 (a)) and that they are mostly formed during the casting process. With creep time increase, under the action of stress and high temperature, the dislocations appearing at the γ/γ' two-phase interface cause vacancies and new micropores are generated in large quantities, causing the size and number of micropores to significantly increase (Figs. 15(b-d)). In the last stage of creep, radial microcracks are formed around the micropores and extend along a plane perpendicular to the stress axis (cleavage plane) (Figs. 15(e-f)), which eventually lead to fracture of the samples. The increase in micropores reduces the effective area of the material, and the stress concentration around the micropores tends to cause the slip. The greater the stress is, the greater the 19

ACCEPTED MANUSCRIPT damage caused by the micropores is, and the greater η1 is. When the stress changes from 0 to 500 MPa, the value η1 varies from 0 to 0.2 (if the material has a stress gradient, the stress of the dangerous section is considered). When the volume fraction of the micropores is ϕ x , the damage

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ω1 can be defined as:

ω1 =η（ （ / ϕxmax -ϕx0） 1 ϕx -ϕx0）

(26)

where ϕ x 0 is the porosity of the original material and ϕ xmax is the porosity at the time of the

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fracture. As shown in Fig. 15, ϕ x 0 is 0.00055 for a single crystal material, and ϕ xmax is 0.018.

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Fig. 15. Evolution of micropores during creep

Due to the diffusion and segregation of insoluble elements such as Re and W at high [44]

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temperatures, the TCP phase occurs simultaneously with the initiation of micropores

. The TCP

phase consists of the µ phase with the Re element and the σ phase with the W element. As shown in Fig. 16(a)(b), at 1100°C /140 MPa, the TCP phase primarily follows the {111} crystal plane along the slip system (111) [-1-12], (-1-11) [112], (-11-1) [1-1-2] and (1-1-1) [-11 -2]. Since the TCP phase has a large elastic modulus, it is more brittle and more susceptible to fracture than the surrounding matrix (Fig. 16(c)), causing microcracks to occur around the TCP phase, as shown in Fig. 16(d). The cracks grow and expand along the slip system (-1-11) [112] and (111) [-1-12].

20

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Fig. 16. Evolution of the TCP phase during creep test of the [001] orientation at 1100°C /140 MPa (a) 100 h, (b) 110 h, (c) 150 h, (d) 187 h

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The appearance of the TCP phase interrupts the rafting structure generated during the creep process, and the microcracks that are generated reduce the effective area. Therefore, it can be assumed that the damage is proportional to the TCP phase volume fraction. When the stress changes

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from 0 to 500 MPa, the value of η2 varies from 0 to 0.1 (if the material has a stress gradient, the stress of the dangerous section is considered). If the TCP phase volume fraction is ϕ y , then the

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damage ω2 can be defined as:

ω2 =η2ϕy / ϕymax

(27)

ϕ ymax is the TCP phase volume fraction at the break, which increases with the Re element content and is approximately 0.01 for DD6. Under the action of temperature and stress, the roughening of the matrix channel not only reduces the Orowan stress and the yield strength of the material but also causes the initial damage. The variation law of the matrix channel (Eq. (11)) is taken as the abscissa, and the damage evolution law (Eq. (16)) is used as the ordinate to establish the curve. The functional relationship can be fitted at 980°C /250 21

ACCEPTED MANUSCRIPT MPa and at 1100°C /140 MPa (Fig. 17): (28)

（b）

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（a）

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y =y0 + A1× exp(( x − x0 ) / t1 )

Fig. 17. Correspondence between the matrix channel size and creep damage (a) 980°C /250 MPa, (b)1100°C /140 MPa

Fig. 17 indicates that even though the creep conditions have changed, the corresponding relationship of the damage increasing with the width of the base channel basically remains

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unchanged. By averaging the parameters under two fitting conditions, the evolution of the damage with the coarseness of the matrix channel is obtained:

ω3 =η3 (0.00563 + 0.00772exp((κ − 0.05346) / 0.04845))

(29)

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η1 ，η2 and η3 in Eqs. (26), (27) and (29) represent the weights of the micropores, the TCP

η3 = 1 −η1 −η2 .

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phase and the rafting contribution to the total damage, respectively, and 4.3. Model verification

After measuring the width of the matrix channel κ , the value of the yield strength τ c can be obtained by Eq. (20). On the other hand, after measuring the width of the matrix channel κ , as well as the volume fraction of the micropores ϕ xmax and TCP phase ϕ ymax of the material after service, the value of initial damage ω0 can be obtained by Eq. (26), Eq. (27) and Eq. (29). By substituting into the creep damage equation Eq. (25) based on crystal plasticity theory, the creep residual life of materials under different temperatures and stress conditions can be obtained. 22

ACCEPTED MANUSCRIPT To verify the creep residual life model, five samples under different service conditions were subjected to the creep residual life test and finite element simulation prediction, as shown in Fig. 18. We got the width of the matrix channel κ , as well as the volume fraction of the micropores ϕ xmax and TCP phase ϕ ymax by scanning electron microscope (SEM), then carried out creep test on each

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samples (solid lines in Fig. 18). Simultaneously, substitute the microstructure parameters into the

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life prediction model to obtain the simulated creep curve (dotted lines in Fig. 18).

Fig. 18. Residual life prediction

For the material used in test 1,

κ =0.12µ m ，ϕ x =0.003 ，ϕ y =0 , and the test condition is 1100°C

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/140 MPa; for the material used in test 2, κ =0.16µ m ，ϕ x =0.005 ，ϕ y =0.001 , and the test condition

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is 1100°C /140 MPa; for the material used in test 3, κ =0.19µ m ， ϕ x =0.005 ， ϕ y =0 , and the test condition is 980°C /250 MPa; for the material used in test 4, κ =0.22µ m ，ϕ x =0.01 ，ϕ y =0.002 , and the test condition is 980°C /250 MPa; and for the material used in test 5, κ =0.22µ m ， ϕ x =0.005 ，

ϕ y =0.001 , and the test condition is 1100°C /140 MPa. The simulation results of the creep residual life model fit well with the test results.

5. Conclusions Creep tests and finite element simulations were conducted at 980°C and 1100°C. The creep constitutive model and creep damage equation at different temperatures were established based on 23

ACCEPTED MANUSCRIPT the crystal plasticity theory. In addition, based on microstructure evolution analysis, a residual life model of single crystal materials was established. The main conclusions are as follows: 1. The quantitative analysis of matrix channel coarsening, TCP phase growth and micropore growth during the creep process is performed via SEM observations. It is found that the coarsening

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of the matrix channel is proportional to the 0.5th power of the creep time. Moreover, the volume fraction of the micropores and the TCP phase are rapidly increased, which deteriorates the mechanical properties of the nickel-based single crystal superalloy.

2. Based on the observation results of TEM and crystal plasticity theory, a creep constitutive

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model and creep damage model were established, attributing the creep damage in the [001] orientation to the activity of the octahedral slip system. This model can calculate the creep rate and

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damage rate under a stress gradient and can describe the rapid tensile failure behaviour of materials when the loading stress approaches or exceeds the critical shear stress. 3. The material deterioration and micropore growth of the single crystal material after service are considered in the creep residual life model. The Orowan resistance decrease caused by matrix channel coarsening and the stress concentration caused by micropores and the TCP phase are

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considered in deriving the evolution of yield strength and the initial damage. The creep residual life

equation.

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Acknowledgements

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of the superalloy can be obtained by substituting these influencing factors into the creep damage

We are grateful for the financial support provided by the National Natural Science Foundation of China (51875461, 51875462), the Basic Research Plan of Natural Science in Shaanxi Province - key projects (2018JZ5002), the Fundamental Research Funds for the Central Universities (3102018gxc029) and the Innovation Capacity Support Plan in Shaanxi Provincial (2018KJXX-007).

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