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compression asymmetry of [001] single-crystal nickel-based superalloy DD6 during low cycle fatigue
Materials Science & Engineering A 593 (2014) 31–37
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Tension/compression asymmetry of [001] single-crystal nickel-based superalloy DD6 during low cycle fatigue B.Z. Wang, D.S. Liu, Z.X. Wen, Z.F. Yue n Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710072, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2013 Received in revised form 24 August 2013 Accepted 3 September 2013 Available online 12 September 2013
In order to study the tension/compression asymmetry of (001) single-crystal nickel-based superalloy DD6 during low cycle fatigue, fully reversed cycle fatigue tests were conducted at 760 1C and 980 1C, with strain rates of 10 3 and 5 10 3 s 1, respectively. The total strain ranges were varied from 7 0.5% to 71.2%. The DD6 single crystals show that tensile stress (T) is higher than compressive stress (C) at the initial cycles in the above conditions. Under the strain rate 10 3 s 1at strain ranges 7 0.7%/760 1C and 70.55%/980 1C or under the strain rate 5 10 3 s 1at 7 0.7%/980 1C, the tension–compression asymmetry of C 4T was observed. According to the above experiments, an advanced rate-dependent constitutive model based on the crystal plasticity theory was proposed. The effects of elastic deformation (for T 4C at the initial cycles of LCF), dislocation accumulation in the matrix channel (for kinematic hardening) and the shearing of precipitates by Superlattice Intrinsic Stacking Faults (SISF) (for kinematic softening) were taken into account. Numerical simulations were conducted using the proposed constitutive equations, such as monotonic tension, compression and cyclic deformation. The present model is shown to be successful in simulating the inelastic behavior of DD6 single crystals under low cycle fatigue. & 2013 Elsevier B.V. All rights reserved.
Keywords: Single crystal Superalloy Tension/compression asymmetry Low cycle fatigue Constitutive modeling
1. Introduction Single-crystal (SC) nickel-based superalloys have been widely used in the hot parts of jet engines because of their excellent mechanical properties at high temperatures. The microstructure of these superalloys consists of a γ matrix phase and a large volume fraction of γ′ Ni3Al precipitate phase. The superior high temperature behavior of SC superalloys is attributed to the above twophase microstructure. Meanwhile, the two-phase composite microstructure increases the difficulty in reliably predicting the mechanical characteristics. In the last few decades, numerous studies have been carried out to simulate the mechanical behavior of superalloys. In these studies, the crystal plasticity finite-element method has been applied to solve a broad variety of crystal mechanical problems and has shown great flexibility with respect to various constitutive formulations for plastic flow and hardening at the elementary shear system level. For SC nickel-based superalloys, initially, it was treated as a homogenous material [1–5]. The influence of microscopic two-phase structures and the dislocations on the deformation mechanism were neglected. The above constitutive laws are empirical viscoplastic formulations and could characterize the macroscopic properties of SC superalloys well.
n
Corresponding author. Tel./fax: þ 86 29 88431002. E-mail address: [email protected] (Z.F. Yue).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.09.013
However, owing to the anomalous mechanical behavior of SC superalloys, the evolution of microstructures must be introduced into the constitutive framework [6–12]. Because these physicsbased multiscale models take into account numerous microstructure details, such as dimensions and properties of two phases, considerable computational efforts are needed, which are not feasible in engineering practice. Tension/compression asymmetry has been observed in SC nickel-based superalloys under monotonic loading and cyclic loading [13–19]. This behavior depends on the sense of the stress condition and crystallographic orientation. For orientations close to [001], the tensile flow stress (T) is greater than the compressive flow stress (C). In the operating condition of turbine blade, besides the centrifugal tensile stress, the contact areas between blade and disc are subjected to considerable compressive stress. Therefore, tension/compression asymmetry must be considered in the strength design. During LCF, the cyclic deformation behavior shows kinematic hardening or kinematic softening depending on the temperature, strain rate and strain range [20–25]. Roughly speaking, the kinematic behaviors occur with respect to the dislocation structures [26–28]. The deformation starts in the γ matrix. At the small strain range, the dislocation dipoles fill the matrix channels homogeneously and form the dislocation networks. The internal stresses produced by the above dislocations deposited on the phase boundaries are accompanied by kinematic hardening. As the strain
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B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
Table 1 Chemical composition of DD6. C
Cr
Ni
Co
W
Mo
Al
Zr
Sb
Bi
0.001–0.04
3.8–4.8
Balance
8.5–9.5
7.0–9.0
1.5–2.5
5.2–6.2
r 0.10
r 0.001
r 0.00005
Mn
Si
P
S
Cu
Mg
Ag
Pb
Sn
As
r0.015
r 0.20
r0.018
r 0.004
r 0.10
r 0.003
r 0.0005
r 0.0005
r 0.001
r 0.001
range increases, the γ′ Ni3Al phase precipitates are sheared by the superdislocation, which may decompose to form the partial plus superlattice intrinsic stacking faults (SISF) that relate to the kinematic softening; this may cause a change of tension/compression asymmetry behavior, C 4T near (001). The purpose of this paper is to investigate the influence of the fatigue test condition on cyclic hardening behavior. The LCF tests of (001)-oriented DD6 SC nickel-based superalloy were performed at 700 1C and 980 1C. A homogenous model, which also considered the microstructure of dislocations, was introduced in this paper to simulate the cyclic deformation behavior of SC superalloys. This model mainly regards the tension/compression asymmetry, cyclic hardening and softening behavior during the low cycle fatigue (LCF). According to the test results, the constitutive model was examined in detail for describing LCF behavior of SC superalloy.
2. Fatigue asymmetry behavior of single-crystal nickel-based superalloy 2.1. Experimental procedure SC nickel-based superalloy DD6 was casted into solid round bars of diameter 15 mm with a longitudinal length of 200 mm. The bars were subjected to heat treatment under general conditions: 1290 1C 1 hþ 1300 1C 2 hþ 1315 1C 4 h/AC þ1120 1C 4 h/ AC þ870 1C 32 h/AC. The chemical composition of the material and the mechanical properties at high temperatures are summarized in Tables 1 and 2, respectively. The specimens were machined from the above cast bars, with axes parallel to [001] orientation. The diameter of all specimens was 5 mm and the parallel length was 25 mm. The monotonic tension tests and cyclic deformation tests were performed using an Instron 8801 hydraulic servo testing machine with a gauge length of 12.5 mm. The tests were conducted under strain control and the strain ratio R¼ 1. 2.2. Cyclic deformation behavior As shown in Table 3, two sets of LCF tests were conducted at 760 1C and 980 1C. For one set the strain rate was 5 10 3 s 1 and for the other set the strain rate was 10 3 s 1. The SC superalloys exhibit a cyclic hardening or softening behavior due to the temperature, strain rate and strain range. Cyclic hardening/softening curves are shown in Fig. 1. As can be seen, both the sets of results show that the initial maximal tension stress is higher than the initial maximal compression stress (T 4C). At 760 1C, all specimens (strain range from 70.7% to 71.2%) exhibit a cyclic hardening in tension and a cyclic softening under high strain rates 5 10 3 s 1, which means that the tension stress is always greater than the compression stress during the whole test. We call this cyclic deformation behavior normal hardening in this paper. Under the low strain rates, 10 3 s 1, when the strain range is 70.65% the cyclic deformation behavior is similar to that under strain rate 5 10 3 s 1. However, when the strain range is 70.7%, the specimen exhibits cyclic softening in tension and cyclic
Table 2 Mechanical properties at 760 1C and 980 1C. Temperature (1C)
Young's modulus (GPa)
Yield stress (MPa)
Elongation (%)
Reduction of area (%)
760 980
101 85
935 880
8.0 27.0
12.0 34.0
Table 3 Test parameters and results for low cycle fatigue tests. Orient T (1C) Strain rate (s 1) Strain range (%) Rupture life t/c Asymmetry 001 001 001 001 001 001 001 001 001 001 001
760 760 760 760 760 980 980 980 980 980 980
10 3 10 3 5 10 3 5 10 3 5 10 3 10 3 10 3 5 10 3 5 10 3 5 10 3 5 10 3
70.65 70.7 70.7 70.9 71.2 70.5 70.55 70.5 70.6 70.7 71.0
6200 5200 12,450 850 10 5750 5000 8000 2850 900 300
t 4c t 4c-c 4t t 4c t 4c t 4c t 4c t 4c-c 4t t 4c t 4c t 4c-c 4t t 4c-c 4t
hardening in compression. The tension/compression asymmetry changes from initial T4 C into C 4T. We call this abnormal hardening in this paper. At 980 1C, under both the sets of strain rates, the cyclic deformation shows normal hardening at the small strain range, but abnormal hardening at the large strain range. The hysteresis loops at the first cycle and after stabilization are plotted, as shown in Fig. 2. The results also show that under the strain rate 5 10 3 s 1, at the strain range 70.7%, the SC superalloy shows normal hardening at 760 1C, but abnormal hardening at 980 1C. When the temperature is 760 1C and strain range is 70.7%, the materials show normal hardening under 5 10 3 s 1but abnormal hardening under 10 3 s 1. Therefore, the strain range, strain rate and temperature have great influences on cyclic deformation behavior. 2.3. Analysis of the cyclic deformation behavior mechanism The initial tension/compression asymmetry is generally explained in terms of the “core width effect”: the a/2〈110〉 screw dislocation is initially dissociated into two Shockley partials, a/6 ̄ ̄2〉, on the octahedral plane. For [001]-oriented 〈2̄11〉 and a/6〈11 single crystal nickel-based superalloy, tensile stress tends to constrict Shockley partials, which decrease the resistance to the cross slip from the original octahedral plane to the cube plane and above cross-slip generates K-W locks, which pins the dislocation. In compression, the Shockley partials will be extended. Therefore, the tensile stress is higher than the compressive stress. However, at low strain levels, K-W locks are not found and deformation
B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
600
33
1000
Stress amplitude / MPa
Stress amplitude / MPa
550
-0.55%
500
0.5% 450
0.55% -0.5%
400
800
1.0%
0.7%
600
0.6% 0.5%
400
350
Tensile stress Compressive stress
980°C / 10-3 300 0.0
0.2
0.4
0.6
0.8
Tensile stress Compressive stress
980°C / 5×10-3 200 0.0
1.0
0.2
0.4
N / Nf 900
800
1.0
1.2%
0.6%
Stress amplitude / MPa
Stress amplitude / MPa
0.8
1200
-0.7%
0.7%
700
-0.6% 600
500
Tensile stress Compressive stress
760°C / 10-3 400 0.0
0.6
N / Nf
0.2
0.4
0.6
0.8
1.0
1000
0.9% 800
0.7% 600
400
200 0.0
Tensile stress Compressive stress
760°C / 5×10-3 0.2
N / Nf
0.4
0.6
0.8
1.0
N / Nf Fig. 1. The experimental cyclic responses of peak stress.
occurs primarily by the moving of a/2〈110〉 dislocations between and around the precipitates [18,25,29]. The dislocations are predominantly edge in character. In the next section, a new tension/ compression model based on the above mechanism will be introduced. As shown in Fig. 1, normal hardening, which includes hardening in tension and softening in compression, usually happens at small strain range. In this condition, the dislocation structures of specimens fatigue tested to failure are almost exclusively in the γ phase and form the dislocation networks at the phase interfaces. During the fatigue test, the dislocation network structures become gradually stable. The mobile dislocations will interact with the stable dislocation network structures and the mobile ability will be affected. This behavior is similar to the kinematic hardening of general metals. On the contrary, abnormal hardening includes softening in tension and hardening in compression. This cyclic deformation behavior is attributed to the evolution of the dislocation microstructures. For the situation of abnormal hardening (C 4T), as the strain level increases, the SSFs traversing the entire γ′ precipitates can be observed [23,24,26]. The a/3〈112̄ 〉 partially enters the γ' phase and creates an SSF, whereas the a/6〈12̄1〉 lies at the interface, pinned by the high-energy APB that would be created if it entered the precipitate. The cyclic response of C 4T is believed to be associated with the above (111)〈112̄ 〉 slip producing SSFs in the precipitate. For all of the experiments, the initial several cycles of fatigue experiments show T 4C. When the strain range, strain rate and temperature are satisfied, as the cyclic number increases, the deformation by SSF shearing of the γ′ phase becomes a predominant mechanism, which is related to abnormal hardening.
3. Simulations 3.1. Model framework A physically based homogeneous single crystal viscoplasticity constitutive framework is adopted in this paper. The kinematics of crystal plasticity are employed. The deformation gradient F is decomposed into F ¼ Fe FP , where Fe is the elastic deformation gradient, which represents the deformation of lattice and rigidbody rotations; FP is the plastic deformation gradient generated by the shear stress in slip direction. The velocity gradient is L ¼ F_ U F 1 ¼ Le þ Lp , where L ¼ F_ UF 1 ¼ Le þ Lp and the plastic velocity gradient Lp is related to the slip rate γ_ ðαÞ on the slip system α: N
Lp ¼ F e U F_ U ðF p Þ 1 U ðF e Þ 1 ¼ ∑ γ_ ðαÞ mðαÞ nðαÞ p
α¼1
ð1Þ
where unit vectors mðαÞ and nðαÞ describe the slip direction and the normal to the slip plane of the slip system α, respectively. The above formulas of crystal deformation kinetics give a relationship between slip shear deformation and macro deformation. The homogenous constitutive model is as follows: τ ðαÞ m ðαÞ γ_ ðαÞ ¼ γ_ ðαÞ 0 ðαÞ sgn½τ τ crit
ð2Þ
In this rate-dependent model, the slip rate on a slip system is related to the resolved shear stress τ ðαÞ ¼ τ:PðαÞ ; γ_ ðαÞ 0 is a reference strain rate; and τ ðαÞ crit is the critical resolved shear stress [30,31].
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B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
the edge dislocations. Actually, this mechanism is supported by the observed dislocation structures, which primarily consist of long straight screw dislocations after plastic deformation. Peierls's result is as follows: 2G 4πζ τPN ¼ exp ð4Þ 1υ b
800
Cycle 1 Cycle 1000
600
Stress / MPa
400 200 0 -200 -400 -600 -800 -0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
Strain 1000
2
εen ¼ l εe11 þ m2 εe22 þ n2 εe33 þ mnγ e23 þnlγ e13 þ lmγ e12
Cycle 1 Cycle 100
800
400
Stress / MPa
ð5Þ
where l, m and n are direction cosines. We assume that the critical shear stress is
600
τCRSS ¼ τ0 þ τP ðTÞ
200
ð6Þ
where τ0 is the athermal component and τP ðTÞ is the thermal component of yield stress. The thermal component τP ðTÞ is controlled by the Peierls mechanism and the effect of elastic strain in the tension/compression deformation on the Peierls barrier is considered: !! 4πh 1 þ εeðhÞ τP ðTÞ ¼ τ0P exp 1 ð7Þ ð1 υÞb 1 þεðbÞ e
0 -200 -400 -600 -800 -1000 -0.012
where ζ ¼ h=2ð1 υÞ; h is the distance between glide planes; b is the Burgers vector and υ is Poisson's ratio. Nabarro's result has a different coefficient and the exponential factor is expð 8πζ=bÞ. During elastic deformation, the distance between glide planes and the distance between atoms along the slip direction are not fixed. In microscopic view, the elastic deformation derives from the change of distance between atoms. The difference of elastic deformation in tension and compression will result in the asymmetry of yield stress. For elastic strain εen in arbitrary direction in a cartesian coordinate system
-0.008
-0.004
0.000
0.004
0.008
0.012
Strain Fig. 2. Stress–strain hysteresis loop at the first cycle and stabilized cycle. (a) 760 1C/ 5 10 3/ 7 0.7%, (b) 980 1C/5 10 3/ 7 1.0%.
The evolution of τ ðαÞ crit presents the strain hardening of a slip system α N β _ τ_ ðαÞ crit ¼ ∑ hαβ γ β¼1
ð3Þ
where hαβ is the slip hardening modulus related to the density and configuration of dislocations. This yield criterion agrees with the Schmid rule and the tendency of tension/compression is symmetry. 3.2. Tension/compression asymmetry The purpose of this work is to predict tension/compression asymmetry evolution during cyclic loading. As can be seen in Table 3, all of the results show tension/compression asymmetry, even at very small strain amplitude in which the micro-plastic strains are believed to be carried by a small density of mobile edge dislocations. In Österle and Bettge et al.'s work, there are no K-W locks superlattice and no stacking faults (SSF) observed at the beginning of deformation (plastic strain r0.2%) at 650 1C, 750 1C and 850 1C [18]. KW-locks are not appropriate to demonstrate the asymmetry of initial yield stress of tension and compression. In this section, we will present a modified hardening rule to simulate the non-Schmid effects of nickel-base SC. Peierls and Nabarro proposed a model of calculating critical stress based on a cubic lattice with elastic isotropy, known as the P–N model. It is commonly accepted that at plastic strains lower than about 0.2%, deformation is produced via the rapid motion of
where εðiÞ e is the elastic strain in the ðiÞ direction. The Burgers vectorpofffiffiffi dislocations on octahedral slip {111}〈110〉 has magnitude b ¼ a= 2 and the pffiffiffi separation between the two octahedral slip planes is h ¼ a= 3. The parameter a is the crystallographic lattice constant of FCC. At small strain amplitude 70.7%/760 1C, the misfit stress lowered the applied effective stress and the test results also αÞ show tension/compression asymmetry. The misfit stress τðmisf is it αÞ expressed as τðmisf ¼ E U δ with E being the elastic modulus and δ it
being the misfit of the matrix and precipitate. The misfit can be relaxed by plastic deformation. When the misfit is accommodated by plastic slip, the misfit stress would vanish. For 760 1C/ 70.7%, a αÞ value of τðmisf ¼ 145 MPa is calculated, with E¼ 109 MPa and it
δ¼ 1.33 10 3.
3.3. Kinematic hardening and softening For fatigue kinematic hardening, a two parameter model is employed to model the cyclic response. This model has been employed in predicting the cyclic saturation of several SC nickelbased superalloys. In this model, a state variable ωðαÞ is introduced to represent the slip back stress, which respects the accumulation in matrix channels. Kinematic hardening rule evolves according to the following relationship: _ ¼ λð_γ ðαÞ ω1 j_γ ðαÞ jωðαÞ Þ ω
ð8Þ
where λ and ω1 are material constants. The second term in the bracket expresses the dynamic recovery property of the back stress. For abnormal hardening, the kinematic softening effect can be incorporated by adding the softening stress, which is related to the
B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
Table 4 Constants/parameters in the constitutive model.
35
1000 800
Parameters
Symbol Value
Units
Inverse rate sensitivity exponent Athermal shear stress Back stress coefficients Initial dislocation density
m τ0 λ, ω1
0.02 10 100, 48 1.5 108
Dimensionless MPa MPa cm 2
20 1 0. 5, 0.45 1.25 10 5, 20
Dimensionless Dimensionless nm Dimensionless
Test data cycle 5 Model predictions
600
Softening stress constant Softening stress exponent Dislocation spacing Dislocation density evolution constants
Stress / MPa
ρðαÞ 0 ksof t n L1, L2 K1, K2
400 200 0 -200 -400 -600 -800 -1000 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010
1200
Strain
C 1100 1000 800
T 900
T
400
[011]
800
[001]
Experiment Simulation
700
600 0.002
Test data cycle 30 Model predictions
600
C Stress / MPa
Stress / MPa
1000
200 0 -200 -400
0.004
0.006
0.008
0.010
0.012
0.014
-600
Strain
-800
Fig. 3. Tension/compression asymmetry under monotonic loading at 700 1C and strain rate 10 3 s 1 for [001] and [011] orientations.
-1000 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010
Strain Fig. 5. Predicted stress–strain hysteresis loop against experimental data at 980 1C under strain rate 5 10 3 s 1. Results are shown for the cycles 5 (a) and 30 (b).
800 600
Test data cylce 5 Model Predicitions
Stress / MPa
400
Therefore, the evolution of ρðαÞ is expressed as
200
ρ_ ðαÞ ¼ 0
ð10Þ
where L1 is the dislocation mean free path and L2 represents the critical annihilation length. K1 and K2 are material constants. The effect of tension/compression asymmetry and hardening can be included in our model by replacing the τðαÞ in Eq. (2)
-200 -400 -600 -800 -0.008
1 K1 K 2 L2 ρðαÞ γ_ ðαÞ b L1
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
ð αÞ τ ωα þ τðαÞ þ τðαÞ m misf it sof t γ_ ðαÞ ¼ γ_ ðαÞ sgn½τðαÞ ωα 0 τðαÞ
ð11Þ
crit
Strain Fig. 4. Predicted stress–strain hysteresis loop against experimental data at 760 1C under strain rate 5 10 3 s 1.
The above discussed constitutive equations have been implemented in a user defined material subroutine in software ABAQUS. The determined values of material constants are listed in Table 4.
dislocation density evolution. αÞ τðsof ¼ 7 ksof t t
ρðαÞ ρðαÞ 0
!n ð9Þ
where ρðαÞ is the density of a/6〈12̄1〉 dislocations that lie at the phase interface and pinned by the high-energy APB. The 7 selection depends on the sense of loading direction. According to the αÞ test results, τðsof is related to the strain rate and strain range. t
4. Simulation results The model introduced in the previous sections has been used to simulate the mechanical properties of the SC nickel-based superalloy DD6. The simulated tension/compression and fatigue results are compared with experimental results.
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B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
tensile stress is also higher than the compressive stress. As the cycle number increases, the shearing of γ′ Ni3Al precipitate phase by SSFs becomes the predominant deformation mechanism as mentioned in Section 2.3. The peak value of tensile stress gradually decreases and that of the compressive stress gradually increases. Our model could simulate the abnormal hardening well. The difference between the simulation and the test is related to the decrease of tensile elastic module after about 100 cycles. The evolution of dislocation density ρðαÞ is shown in Fig. 7.
1000
Stress amplitude / MPa
900
C 800
T 700
600
5. Conclusions 500
Test data Model predictions 400
0
50
100
150
200
250
Number of cycles Fig. 6. Predicted results of cyclic responses of peak stress against experimental data at 980 1C under strain rate 5 10 3 s 1.
3.00E+012
/m
-2
2.50E+012
Dislocation density
2.00E+012
1.50E+012
1.00E+012
5.00E+011
0.00E+000 0
50
100
150
200
250
Number of cycles Fig. 7. The calculated dislocation density as a function of fatigue cycles at 980 1C under strain rate 5 10 3 s 1.
4.1. Simulation of tension/compression asymmetry The present model is used to simulate the monotonic tests at 700 1C under the strain rate 10 3 s 1. For monotonic loading, the cyclic hardening and softening parameters, ωα and τðαÞ , are equal sof t to zero. The results are shown in Fig. 3. As can be seen, there is obviously asymmetry of tension/compression in [001] and [011], especially for the [011] orientation about 1097 MPa in compression and 928 MPa in tension. 4.2. Simulation of cyclic hardening Fig. 4 shows the tension/compression asymmetry of the first cycle in fatigue test at 760 1C and 70.7%. In this condition, the plastic strain is very small, about 0.05%, but it also shows apparent tension/compression asymmetry, 705 MPa in tension and 675 MPa in compression. Because nearly no K-W lock forms as mentioned in Section 2.3, the reason for the asymmetry in this condition should be the effect of the elastic strain during deformation. During the reverse cycle, the back stress ωα is not yet zero; the parameters of normal hardening are shown in Table 4. For cyclic loading at 980 1C and 71.0% under strain rate 5 10 3 s 1, the simulation results are shown in Figs. 5 and 6. In this condition, the peak value curve shows apparent abnormal hardening. As can be seen, during the initial several cycles, the
At 760 1C and 980 1C under the strain rates 5 10 3 and 10 3 s 1, the LCF tests of DD6 single crystals with (001) orientation show that the tensile stress is higher than the compressive stress during the initial cycles. As the number of cycles increases, the asymmetry of tension/compression will be reversed according to the loading conditions. At 760 1C, under 5 10 3 s 1, T 4C during the whole life of fatigue, but under 10 3 s 1 at strain range larger than 70.7%, C 4T was observed. At 980 1C, under 5 10 3 s 1at the strain range larger than 7 0.7% and under 10 3 at the strain range larger than 7 0.55%, C4 T as the cyclic number increases. The different cyclic responses are related to the dislocation microstructures in the materials. A developed constitutive model for SC superalloy DD6 has been implemented in the commercial finite-element software ABAQUS as a user material subroutine to simulate the above cyclic responses. The formulations governing the mechanical response have been calibrated against experimental data. First, the tension/ compression asymmetry is modeled by incorporating P-N stress into constitutive relations. The effect of elastic deformations parallel to the slip direction and perpendicular to the slip plane on the critical resolved shear stresses is introduced. Second, normal kinematic hardening, which relates to the dislocation accumulation in the matrix channel at small strain ranges, is simulated by an empirical equation. Dislocation density-based equations for the deformation mechanism of precipitate phase shearing by SSF are proposed. Abnormal kinematic hardening, softening in tension and hardening in compression, is attributed to the above dislocation density evolution. Ultimately, the of tension/compression asymmetry behavior of SC nickel based superalloy during LCF was investigated at 760 1C and 980 1C. A viscous constitutive formulation, which could accurately represent the cyclic response of low cycle fatigue of SC nickel-based superalloy, was developed.
Acknowledgments The work was supported by the National Natural Science Foundation of China (51210008 and 51175424) and the Basic Research foundation of Northwestern Polytechnical University (JC201239). These supports are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Tension/compression asymmetry of [001] single-crystal nickel-based superalloy DD6 during low cycle fatigue B.Z. Wang, D.S. Liu, Z.X. Wen, Z.F. Yue n Department of Engineering Mechanics, Northwestern Polytechnical University, Xi'an 710072, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 July 2013 Received in revised form 24 August 2013 Accepted 3 September 2013 Available online 12 September 2013
In order to study the tension/compression asymmetry of (001) single-crystal nickel-based superalloy DD6 during low cycle fatigue, fully reversed cycle fatigue tests were conducted at 760 1C and 980 1C, with strain rates of 10 3 and 5 10 3 s 1, respectively. The total strain ranges were varied from 7 0.5% to 71.2%. The DD6 single crystals show that tensile stress (T) is higher than compressive stress (C) at the initial cycles in the above conditions. Under the strain rate 10 3 s 1at strain ranges 7 0.7%/760 1C and 70.55%/980 1C or under the strain rate 5 10 3 s 1at 7 0.7%/980 1C, the tension–compression asymmetry of C 4T was observed. According to the above experiments, an advanced rate-dependent constitutive model based on the crystal plasticity theory was proposed. The effects of elastic deformation (for T 4C at the initial cycles of LCF), dislocation accumulation in the matrix channel (for kinematic hardening) and the shearing of precipitates by Superlattice Intrinsic Stacking Faults (SISF) (for kinematic softening) were taken into account. Numerical simulations were conducted using the proposed constitutive equations, such as monotonic tension, compression and cyclic deformation. The present model is shown to be successful in simulating the inelastic behavior of DD6 single crystals under low cycle fatigue. & 2013 Elsevier B.V. All rights reserved.
Keywords: Single crystal Superalloy Tension/compression asymmetry Low cycle fatigue Constitutive modeling
1. Introduction Single-crystal (SC) nickel-based superalloys have been widely used in the hot parts of jet engines because of their excellent mechanical properties at high temperatures. The microstructure of these superalloys consists of a γ matrix phase and a large volume fraction of γ′ Ni3Al precipitate phase. The superior high temperature behavior of SC superalloys is attributed to the above twophase microstructure. Meanwhile, the two-phase composite microstructure increases the difficulty in reliably predicting the mechanical characteristics. In the last few decades, numerous studies have been carried out to simulate the mechanical behavior of superalloys. In these studies, the crystal plasticity finite-element method has been applied to solve a broad variety of crystal mechanical problems and has shown great flexibility with respect to various constitutive formulations for plastic flow and hardening at the elementary shear system level. For SC nickel-based superalloys, initially, it was treated as a homogenous material [1–5]. The influence of microscopic two-phase structures and the dislocations on the deformation mechanism were neglected. The above constitutive laws are empirical viscoplastic formulations and could characterize the macroscopic properties of SC superalloys well.
n
Corresponding author. Tel./fax: þ 86 29 88431002. E-mail address: [email protected] (Z.F. Yue).
0921-5093/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msea.2013.09.013
However, owing to the anomalous mechanical behavior of SC superalloys, the evolution of microstructures must be introduced into the constitutive framework [6–12]. Because these physicsbased multiscale models take into account numerous microstructure details, such as dimensions and properties of two phases, considerable computational efforts are needed, which are not feasible in engineering practice. Tension/compression asymmetry has been observed in SC nickel-based superalloys under monotonic loading and cyclic loading [13–19]. This behavior depends on the sense of the stress condition and crystallographic orientation. For orientations close to [001], the tensile flow stress (T) is greater than the compressive flow stress (C). In the operating condition of turbine blade, besides the centrifugal tensile stress, the contact areas between blade and disc are subjected to considerable compressive stress. Therefore, tension/compression asymmetry must be considered in the strength design. During LCF, the cyclic deformation behavior shows kinematic hardening or kinematic softening depending on the temperature, strain rate and strain range [20–25]. Roughly speaking, the kinematic behaviors occur with respect to the dislocation structures [26–28]. The deformation starts in the γ matrix. At the small strain range, the dislocation dipoles fill the matrix channels homogeneously and form the dislocation networks. The internal stresses produced by the above dislocations deposited on the phase boundaries are accompanied by kinematic hardening. As the strain
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B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
Table 1 Chemical composition of DD6. C
Cr
Ni
Co
W
Mo
Al
Zr
Sb
Bi
0.001–0.04
3.8–4.8
Balance
8.5–9.5
7.0–9.0
1.5–2.5
5.2–6.2
r 0.10
r 0.001
r 0.00005
Mn
Si
P
S
Cu
Mg
Ag
Pb
Sn
As
r0.015
r 0.20
r0.018
r 0.004
r 0.10
r 0.003
r 0.0005
r 0.0005
r 0.001
r 0.001
range increases, the γ′ Ni3Al phase precipitates are sheared by the superdislocation, which may decompose to form the partial plus superlattice intrinsic stacking faults (SISF) that relate to the kinematic softening; this may cause a change of tension/compression asymmetry behavior, C 4T near (001). The purpose of this paper is to investigate the influence of the fatigue test condition on cyclic hardening behavior. The LCF tests of (001)-oriented DD6 SC nickel-based superalloy were performed at 700 1C and 980 1C. A homogenous model, which also considered the microstructure of dislocations, was introduced in this paper to simulate the cyclic deformation behavior of SC superalloys. This model mainly regards the tension/compression asymmetry, cyclic hardening and softening behavior during the low cycle fatigue (LCF). According to the test results, the constitutive model was examined in detail for describing LCF behavior of SC superalloy.
2. Fatigue asymmetry behavior of single-crystal nickel-based superalloy 2.1. Experimental procedure SC nickel-based superalloy DD6 was casted into solid round bars of diameter 15 mm with a longitudinal length of 200 mm. The bars were subjected to heat treatment under general conditions: 1290 1C 1 hþ 1300 1C 2 hþ 1315 1C 4 h/AC þ1120 1C 4 h/ AC þ870 1C 32 h/AC. The chemical composition of the material and the mechanical properties at high temperatures are summarized in Tables 1 and 2, respectively. The specimens were machined from the above cast bars, with axes parallel to [001] orientation. The diameter of all specimens was 5 mm and the parallel length was 25 mm. The monotonic tension tests and cyclic deformation tests were performed using an Instron 8801 hydraulic servo testing machine with a gauge length of 12.5 mm. The tests were conducted under strain control and the strain ratio R¼ 1. 2.2. Cyclic deformation behavior As shown in Table 3, two sets of LCF tests were conducted at 760 1C and 980 1C. For one set the strain rate was 5 10 3 s 1 and for the other set the strain rate was 10 3 s 1. The SC superalloys exhibit a cyclic hardening or softening behavior due to the temperature, strain rate and strain range. Cyclic hardening/softening curves are shown in Fig. 1. As can be seen, both the sets of results show that the initial maximal tension stress is higher than the initial maximal compression stress (T 4C). At 760 1C, all specimens (strain range from 70.7% to 71.2%) exhibit a cyclic hardening in tension and a cyclic softening under high strain rates 5 10 3 s 1, which means that the tension stress is always greater than the compression stress during the whole test. We call this cyclic deformation behavior normal hardening in this paper. Under the low strain rates, 10 3 s 1, when the strain range is 70.65% the cyclic deformation behavior is similar to that under strain rate 5 10 3 s 1. However, when the strain range is 70.7%, the specimen exhibits cyclic softening in tension and cyclic
Table 2 Mechanical properties at 760 1C and 980 1C. Temperature (1C)
Young's modulus (GPa)
Yield stress (MPa)
Elongation (%)
Reduction of area (%)
760 980
101 85
935 880
8.0 27.0
12.0 34.0
Table 3 Test parameters and results for low cycle fatigue tests. Orient T (1C) Strain rate (s 1) Strain range (%) Rupture life t/c Asymmetry 001 001 001 001 001 001 001 001 001 001 001
760 760 760 760 760 980 980 980 980 980 980
10 3 10 3 5 10 3 5 10 3 5 10 3 10 3 10 3 5 10 3 5 10 3 5 10 3 5 10 3
70.65 70.7 70.7 70.9 71.2 70.5 70.55 70.5 70.6 70.7 71.0
6200 5200 12,450 850 10 5750 5000 8000 2850 900 300
t 4c t 4c-c 4t t 4c t 4c t 4c t 4c t 4c-c 4t t 4c t 4c t 4c-c 4t t 4c-c 4t
hardening in compression. The tension/compression asymmetry changes from initial T4 C into C 4T. We call this abnormal hardening in this paper. At 980 1C, under both the sets of strain rates, the cyclic deformation shows normal hardening at the small strain range, but abnormal hardening at the large strain range. The hysteresis loops at the first cycle and after stabilization are plotted, as shown in Fig. 2. The results also show that under the strain rate 5 10 3 s 1, at the strain range 70.7%, the SC superalloy shows normal hardening at 760 1C, but abnormal hardening at 980 1C. When the temperature is 760 1C and strain range is 70.7%, the materials show normal hardening under 5 10 3 s 1but abnormal hardening under 10 3 s 1. Therefore, the strain range, strain rate and temperature have great influences on cyclic deformation behavior. 2.3. Analysis of the cyclic deformation behavior mechanism The initial tension/compression asymmetry is generally explained in terms of the “core width effect”: the a/2〈110〉 screw dislocation is initially dissociated into two Shockley partials, a/6 ̄ ̄2〉, on the octahedral plane. For [001]-oriented 〈2̄11〉 and a/6〈11 single crystal nickel-based superalloy, tensile stress tends to constrict Shockley partials, which decrease the resistance to the cross slip from the original octahedral plane to the cube plane and above cross-slip generates K-W locks, which pins the dislocation. In compression, the Shockley partials will be extended. Therefore, the tensile stress is higher than the compressive stress. However, at low strain levels, K-W locks are not found and deformation
B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
600
33
1000
Stress amplitude / MPa
Stress amplitude / MPa
550
-0.55%
500
0.5% 450
0.55% -0.5%
400
800
1.0%
0.7%
600
0.6% 0.5%
400
350
Tensile stress Compressive stress
980°C / 10-3 300 0.0
0.2
0.4
0.6
0.8
Tensile stress Compressive stress
980°C / 5×10-3 200 0.0
1.0
0.2
0.4
N / Nf 900
800
1.0
1.2%
0.6%
Stress amplitude / MPa
Stress amplitude / MPa
0.8
1200
-0.7%
0.7%
700
-0.6% 600
500
Tensile stress Compressive stress
760°C / 10-3 400 0.0
0.6
N / Nf
0.2
0.4
0.6
0.8
1.0
1000
0.9% 800
0.7% 600
400
200 0.0
Tensile stress Compressive stress
760°C / 5×10-3 0.2
N / Nf
0.4
0.6
0.8
1.0
N / Nf Fig. 1. The experimental cyclic responses of peak stress.
occurs primarily by the moving of a/2〈110〉 dislocations between and around the precipitates [18,25,29]. The dislocations are predominantly edge in character. In the next section, a new tension/ compression model based on the above mechanism will be introduced. As shown in Fig. 1, normal hardening, which includes hardening in tension and softening in compression, usually happens at small strain range. In this condition, the dislocation structures of specimens fatigue tested to failure are almost exclusively in the γ phase and form the dislocation networks at the phase interfaces. During the fatigue test, the dislocation network structures become gradually stable. The mobile dislocations will interact with the stable dislocation network structures and the mobile ability will be affected. This behavior is similar to the kinematic hardening of general metals. On the contrary, abnormal hardening includes softening in tension and hardening in compression. This cyclic deformation behavior is attributed to the evolution of the dislocation microstructures. For the situation of abnormal hardening (C 4T), as the strain level increases, the SSFs traversing the entire γ′ precipitates can be observed [23,24,26]. The a/3〈112̄ 〉 partially enters the γ' phase and creates an SSF, whereas the a/6〈12̄1〉 lies at the interface, pinned by the high-energy APB that would be created if it entered the precipitate. The cyclic response of C 4T is believed to be associated with the above (111)〈112̄ 〉 slip producing SSFs in the precipitate. For all of the experiments, the initial several cycles of fatigue experiments show T 4C. When the strain range, strain rate and temperature are satisfied, as the cyclic number increases, the deformation by SSF shearing of the γ′ phase becomes a predominant mechanism, which is related to abnormal hardening.
3. Simulations 3.1. Model framework A physically based homogeneous single crystal viscoplasticity constitutive framework is adopted in this paper. The kinematics of crystal plasticity are employed. The deformation gradient F is decomposed into F ¼ Fe FP , where Fe is the elastic deformation gradient, which represents the deformation of lattice and rigidbody rotations; FP is the plastic deformation gradient generated by the shear stress in slip direction. The velocity gradient is L ¼ F_ U F 1 ¼ Le þ Lp , where L ¼ F_ UF 1 ¼ Le þ Lp and the plastic velocity gradient Lp is related to the slip rate γ_ ðαÞ on the slip system α: N
Lp ¼ F e U F_ U ðF p Þ 1 U ðF e Þ 1 ¼ ∑ γ_ ðαÞ mðαÞ nðαÞ p
α¼1
ð1Þ
where unit vectors mðαÞ and nðαÞ describe the slip direction and the normal to the slip plane of the slip system α, respectively. The above formulas of crystal deformation kinetics give a relationship between slip shear deformation and macro deformation. The homogenous constitutive model is as follows: τ ðαÞ m ðαÞ γ_ ðαÞ ¼ γ_ ðαÞ 0 ðαÞ sgn½τ τ crit
ð2Þ
In this rate-dependent model, the slip rate on a slip system is related to the resolved shear stress τ ðαÞ ¼ τ:PðαÞ ; γ_ ðαÞ 0 is a reference strain rate; and τ ðαÞ crit is the critical resolved shear stress [30,31].
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B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
the edge dislocations. Actually, this mechanism is supported by the observed dislocation structures, which primarily consist of long straight screw dislocations after plastic deformation. Peierls's result is as follows: 2G 4πζ τPN ¼ exp ð4Þ 1υ b
800
Cycle 1 Cycle 1000
600
Stress / MPa
400 200 0 -200 -400 -600 -800 -0.008
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
Strain 1000
2
εen ¼ l εe11 þ m2 εe22 þ n2 εe33 þ mnγ e23 þnlγ e13 þ lmγ e12
Cycle 1 Cycle 100
800
400
Stress / MPa
ð5Þ
where l, m and n are direction cosines. We assume that the critical shear stress is
600
τCRSS ¼ τ0 þ τP ðTÞ
200
ð6Þ
where τ0 is the athermal component and τP ðTÞ is the thermal component of yield stress. The thermal component τP ðTÞ is controlled by the Peierls mechanism and the effect of elastic strain in the tension/compression deformation on the Peierls barrier is considered: !! 4πh 1 þ εeðhÞ τP ðTÞ ¼ τ0P exp 1 ð7Þ ð1 υÞb 1 þεðbÞ e
0 -200 -400 -600 -800 -1000 -0.012
where ζ ¼ h=2ð1 υÞ; h is the distance between glide planes; b is the Burgers vector and υ is Poisson's ratio. Nabarro's result has a different coefficient and the exponential factor is expð 8πζ=bÞ. During elastic deformation, the distance between glide planes and the distance between atoms along the slip direction are not fixed. In microscopic view, the elastic deformation derives from the change of distance between atoms. The difference of elastic deformation in tension and compression will result in the asymmetry of yield stress. For elastic strain εen in arbitrary direction in a cartesian coordinate system
-0.008
-0.004
0.000
0.004
0.008
0.012
Strain Fig. 2. Stress–strain hysteresis loop at the first cycle and stabilized cycle. (a) 760 1C/ 5 10 3/ 7 0.7%, (b) 980 1C/5 10 3/ 7 1.0%.
The evolution of τ ðαÞ crit presents the strain hardening of a slip system α N β _ τ_ ðαÞ crit ¼ ∑ hαβ γ β¼1
ð3Þ
where hαβ is the slip hardening modulus related to the density and configuration of dislocations. This yield criterion agrees with the Schmid rule and the tendency of tension/compression is symmetry. 3.2. Tension/compression asymmetry The purpose of this work is to predict tension/compression asymmetry evolution during cyclic loading. As can be seen in Table 3, all of the results show tension/compression asymmetry, even at very small strain amplitude in which the micro-plastic strains are believed to be carried by a small density of mobile edge dislocations. In Österle and Bettge et al.'s work, there are no K-W locks superlattice and no stacking faults (SSF) observed at the beginning of deformation (plastic strain r0.2%) at 650 1C, 750 1C and 850 1C [18]. KW-locks are not appropriate to demonstrate the asymmetry of initial yield stress of tension and compression. In this section, we will present a modified hardening rule to simulate the non-Schmid effects of nickel-base SC. Peierls and Nabarro proposed a model of calculating critical stress based on a cubic lattice with elastic isotropy, known as the P–N model. It is commonly accepted that at plastic strains lower than about 0.2%, deformation is produced via the rapid motion of
where εðiÞ e is the elastic strain in the ðiÞ direction. The Burgers vectorpofffiffiffi dislocations on octahedral slip {111}〈110〉 has magnitude b ¼ a= 2 and the pffiffiffi separation between the two octahedral slip planes is h ¼ a= 3. The parameter a is the crystallographic lattice constant of FCC. At small strain amplitude 70.7%/760 1C, the misfit stress lowered the applied effective stress and the test results also αÞ show tension/compression asymmetry. The misfit stress τðmisf is it αÞ expressed as τðmisf ¼ E U δ with E being the elastic modulus and δ it
being the misfit of the matrix and precipitate. The misfit can be relaxed by plastic deformation. When the misfit is accommodated by plastic slip, the misfit stress would vanish. For 760 1C/ 70.7%, a αÞ value of τðmisf ¼ 145 MPa is calculated, with E¼ 109 MPa and it
δ¼ 1.33 10 3.
3.3. Kinematic hardening and softening For fatigue kinematic hardening, a two parameter model is employed to model the cyclic response. This model has been employed in predicting the cyclic saturation of several SC nickelbased superalloys. In this model, a state variable ωðαÞ is introduced to represent the slip back stress, which respects the accumulation in matrix channels. Kinematic hardening rule evolves according to the following relationship: _ ¼ λð_γ ðαÞ ω1 j_γ ðαÞ jωðαÞ Þ ω
ð8Þ
where λ and ω1 are material constants. The second term in the bracket expresses the dynamic recovery property of the back stress. For abnormal hardening, the kinematic softening effect can be incorporated by adding the softening stress, which is related to the
B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
Table 4 Constants/parameters in the constitutive model.
35
1000 800
Parameters
Symbol Value
Units
Inverse rate sensitivity exponent Athermal shear stress Back stress coefficients Initial dislocation density
m τ0 λ, ω1
0.02 10 100, 48 1.5 108
Dimensionless MPa MPa cm 2
20 1 0. 5, 0.45 1.25 10 5, 20
Dimensionless Dimensionless nm Dimensionless
Test data cycle 5 Model predictions
600
Softening stress constant Softening stress exponent Dislocation spacing Dislocation density evolution constants
Stress / MPa
ρðαÞ 0 ksof t n L1, L2 K1, K2
400 200 0 -200 -400 -600 -800 -1000 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010
1200
Strain
C 1100 1000 800
T 900
T
400
[011]
800
[001]
Experiment Simulation
700
600 0.002
Test data cycle 30 Model predictions
600
C Stress / MPa
Stress / MPa
1000
200 0 -200 -400
0.004
0.006
0.008
0.010
0.012
0.014
-600
Strain
-800
Fig. 3. Tension/compression asymmetry under monotonic loading at 700 1C and strain rate 10 3 s 1 for [001] and [011] orientations.
-1000 -0.010 -0.008 -0.006 -0.004 -0.002 0.000 0.002 0.004 0.006 0.008 0.010
Strain Fig. 5. Predicted stress–strain hysteresis loop against experimental data at 980 1C under strain rate 5 10 3 s 1. Results are shown for the cycles 5 (a) and 30 (b).
800 600
Test data cylce 5 Model Predicitions
Stress / MPa
400
Therefore, the evolution of ρðαÞ is expressed as
200
ρ_ ðαÞ ¼ 0
ð10Þ
where L1 is the dislocation mean free path and L2 represents the critical annihilation length. K1 and K2 are material constants. The effect of tension/compression asymmetry and hardening can be included in our model by replacing the τðαÞ in Eq. (2)
-200 -400 -600 -800 -0.008
1 K1 K 2 L2 ρðαÞ γ_ ðαÞ b L1
-0.006
-0.004
-0.002
0.000
0.002
0.004
0.006
0.008
ð αÞ τ ωα þ τðαÞ þ τðαÞ m misf it sof t γ_ ðαÞ ¼ γ_ ðαÞ sgn½τðαÞ ωα 0 τðαÞ
ð11Þ
crit
Strain Fig. 4. Predicted stress–strain hysteresis loop against experimental data at 760 1C under strain rate 5 10 3 s 1.
The above discussed constitutive equations have been implemented in a user defined material subroutine in software ABAQUS. The determined values of material constants are listed in Table 4.
dislocation density evolution. αÞ τðsof ¼ 7 ksof t t
ρðαÞ ρðαÞ 0
!n ð9Þ
where ρðαÞ is the density of a/6〈12̄1〉 dislocations that lie at the phase interface and pinned by the high-energy APB. The 7 selection depends on the sense of loading direction. According to the αÞ test results, τðsof is related to the strain rate and strain range. t
4. Simulation results The model introduced in the previous sections has been used to simulate the mechanical properties of the SC nickel-based superalloy DD6. The simulated tension/compression and fatigue results are compared with experimental results.
36
B.Z. Wang et al. / Materials Science & Engineering A 593 (2014) 31–37
tensile stress is also higher than the compressive stress. As the cycle number increases, the shearing of γ′ Ni3Al precipitate phase by SSFs becomes the predominant deformation mechanism as mentioned in Section 2.3. The peak value of tensile stress gradually decreases and that of the compressive stress gradually increases. Our model could simulate the abnormal hardening well. The difference between the simulation and the test is related to the decrease of tensile elastic module after about 100 cycles. The evolution of dislocation density ρðαÞ is shown in Fig. 7.
1000
Stress amplitude / MPa
900
C 800
T 700
600
5. Conclusions 500
Test data Model predictions 400
0
50
100
150
200
250
Number of cycles Fig. 6. Predicted results of cyclic responses of peak stress against experimental data at 980 1C under strain rate 5 10 3 s 1.
3.00E+012
/m
-2
2.50E+012
Dislocation density
2.00E+012
1.50E+012
1.00E+012
5.00E+011
0.00E+000 0
50
100
150
200
250
Number of cycles Fig. 7. The calculated dislocation density as a function of fatigue cycles at 980 1C under strain rate 5 10 3 s 1.
4.1. Simulation of tension/compression asymmetry The present model is used to simulate the monotonic tests at 700 1C under the strain rate 10 3 s 1. For monotonic loading, the cyclic hardening and softening parameters, ωα and τðαÞ , are equal sof t to zero. The results are shown in Fig. 3. As can be seen, there is obviously asymmetry of tension/compression in [001] and [011], especially for the [011] orientation about 1097 MPa in compression and 928 MPa in tension. 4.2. Simulation of cyclic hardening Fig. 4 shows the tension/compression asymmetry of the first cycle in fatigue test at 760 1C and 70.7%. In this condition, the plastic strain is very small, about 0.05%, but it also shows apparent tension/compression asymmetry, 705 MPa in tension and 675 MPa in compression. Because nearly no K-W lock forms as mentioned in Section 2.3, the reason for the asymmetry in this condition should be the effect of the elastic strain during deformation. During the reverse cycle, the back stress ωα is not yet zero; the parameters of normal hardening are shown in Table 4. For cyclic loading at 980 1C and 71.0% under strain rate 5 10 3 s 1, the simulation results are shown in Figs. 5 and 6. In this condition, the peak value curve shows apparent abnormal hardening. As can be seen, during the initial several cycles, the
At 760 1C and 980 1C under the strain rates 5 10 3 and 10 3 s 1, the LCF tests of DD6 single crystals with (001) orientation show that the tensile stress is higher than the compressive stress during the initial cycles. As the number of cycles increases, the asymmetry of tension/compression will be reversed according to the loading conditions. At 760 1C, under 5 10 3 s 1, T 4C during the whole life of fatigue, but under 10 3 s 1 at strain range larger than 70.7%, C 4T was observed. At 980 1C, under 5 10 3 s 1at the strain range larger than 7 0.7% and under 10 3 at the strain range larger than 7 0.55%, C4 T as the cyclic number increases. The different cyclic responses are related to the dislocation microstructures in the materials. A developed constitutive model for SC superalloy DD6 has been implemented in the commercial finite-element software ABAQUS as a user material subroutine to simulate the above cyclic responses. The formulations governing the mechanical response have been calibrated against experimental data. First, the tension/ compression asymmetry is modeled by incorporating P-N stress into constitutive relations. The effect of elastic deformations parallel to the slip direction and perpendicular to the slip plane on the critical resolved shear stresses is introduced. Second, normal kinematic hardening, which relates to the dislocation accumulation in the matrix channel at small strain ranges, is simulated by an empirical equation. Dislocation density-based equations for the deformation mechanism of precipitate phase shearing by SSF are proposed. Abnormal kinematic hardening, softening in tension and hardening in compression, is attributed to the above dislocation density evolution. Ultimately, the of tension/compression asymmetry behavior of SC nickel based superalloy during LCF was investigated at 760 1C and 980 1C. A viscous constitutive formulation, which could accurately represent the cyclic response of low cycle fatigue of SC nickel-based superalloy, was developed.
Acknowledgments The work was supported by the National Natural Science Foundation of China (51210008 and 51175424) and the Basic Research foundation of Northwestern Polytechnical University (JC201239). These supports are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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