Low cycle fatigue behavior of single crystal superalloy with temperature gradient

European Journal of Mechanics A/Solids 29 (2010) 611–618

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Low cycle fatigue behavior of single crystal superalloy with temperature gradient N.X. Hou, Q.M. Yu, Z.X. Wen, Z.F. Yue* Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 April 2009 Accepted 31 December 2009 Available online 11 January 2010

In present study, low cycle fatigue model based on rate dependent constitutive formulation with kinematic hardening and fatigue damage rule is introduced to investigate the fatigue behavior of single crystal superalloys with temperature gradient. Low cycle fatigue tests with uniform temperature and with temperature gradient are carried out to investigate the influence of temperature gradient. The microstructure reveals that the slip deformation is a principal mechanism of low cycle fatigue with temperature gradient. Complex stress experiments of low cycle fatigue are carried out to verify the life prediction rule. Bauschinger effect, ratcheting effect and fatigue damage of single crystal superalloy at different temperatures are studied. Special attention is placed on the simulation of low cycle fatigue behavior with temperature gradient. The simulation results show that the ratcheting effect at high temperature is more remarkable than that at low temperature due to the influence of temperature gradient, which results in the difference of damage evolution between high temperature zone and low temperature zone. Comparison between experiments and simulations with temperature gradient proves that the predicted model based on the damage of the second cycle is reasonable. Crown Copyright Ó 2010 Published by Elsevier Masson SAS. All rights reserved.

Keywords: Low cycle fatigue Damage Temperature gradient Single crystal superalloy

1. Introduction Ni-base superalloy single crystals, served as cooled turbine blades, are usually subjected to cycle loading such as start up and shut down. Cooled blades made of single crystal are also subjected to very significant temperature gradient during service. One of major failure modes in current single crystal cooled blades is low cycle fatigue (LCF) caused by temperature gradient across the thickness wall. The life assessment of cooled blade under LCF is a major challenge for both scientific and industrial community. Therefore, it is important to know the LCF behavior of the single crystal superalloy with temperature gradient. The classical phenomenological approaches of assessing fatigue life relate the local stress–strain state to the number of load cycles to initiate a macroscopic crack. Fatigue theories are classified as stress-based, strain-based, energy-based, and fracture mechanics based. In the past work, Coffin (1954) and Manson (1965) have proposed the use of strain range to describe the life to LCF crack initiation. Several extensions to Manson–Coffin equations have been proposed taking into account the effects of mean stress, stress concentrations and environment (Chaboche, 1974; Warren and Weia, 2006; Nies1ony et al., 2008). For single crystal

* Corresponding author. Tel./fax: þ86 29 88431002. E-mail address: [email protected] (Z.F. Yue).

superalloys, extensive researches (Vasseur and Remy, 1994; Yue and Lu, 1998; Bhattacharyya et al., 1999; Yue et al., 2007; Levkovitch et al., 2006; Rahmani and Nategh, 2008; Hou et al., 2009) have been done to reveal the relationship between stress range, strain rate, strain range, and testing temperature on the cyclic stress response, deformation mode and fatigue life at high temperature. However, the LCF behavior of material under temperature gradient has been rarely reported. Hou et al. (2008) developed experimental system with high temperature gradient to carry out fatigue tests using alloyed steel (30CrMnSi) under temperature gradient condition. The results show that the temperature gradient has remarkable influences on the LCF life. To understand the LCF behaviors of single crystal superalloy under temperature gradient, many works such as experiments and simulations have to be carried out. In this work, the crystallographic constitutive model with kinematic hardening and LCF damage model with temperature gradient are introduced to study the cyclic stress response, deformation mode and fatigue life of single crystal superalloys. The LCF experiments with uniform temperature and with temperature gradient are carried out to verify the constitutive model and damage model. The scanning electron microscopy (SEM) is employed to investigate the deformation and failure mechanisms of LCF with temperature gradient. Furthermore, the LCF behaviors with uniform temperature and with temperature gradient are studied by the numerical simulations.

0997-7538/$ – see front matter Crown Copyright Ó 2010 Published by Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2009.12.008

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Fig. 1. Testing system with temperature gradient.

fatigue tests with different temperature conditions. In the present work only the technologically interesting [001] orientation of the specimen is considered. The cooled air flows through the inside of specimen, and the flux rate is controlled by the flowmeter, while the outside of the specimen is heated by the furnace, and the temperature is controlled by the temperature controller. During the fatigue tests with uniform temperature, the temperature of specimens is 900  C. During the fatigue tests with temperature gradient, the outer temperature of specimens is 900  C, the temperature of cooled air is 20  C, and the flow rate of cooled air is 1.73 g/s. For all the tests, triangular-wave type tensile stresses are applied. The stress ratio R is 0.1, and all tests are run with frequency of 0.33 Hz. The fatigue life with uniform temperature is 2727 cycles, while the fatigue life under temperature gradient is 6948 cycles. It can be concluded that there is remarkable influence of cooled air on fatigue life. Furthermore, the fatigue tests at 600  C, 750  C and 800  C are performed to obtain the model parameters of the single crystal superalloys.

2. Experiments 2.2. Metallurgical study 2.1. Low cycle fatigue tests with uniform temperature and with temperature gradient The thin-walled specimen and the testing system with temperature gradient (see in Fig. 1) are used to carry out low cycle

The scanning electron microscopy (SEM) is employed to investigate the deformation and failure mechanism with uniform temperature and with temperature gradient. Fig. 2 shows the fracture surface of the specimen at 750  C. Fig. 3 shows the fracture

Fig. 2. Fracture surface of the specimen at 750  C.

N.X. Hou et al. / European Journal of Mechanics A/Solids 29 (2010) 611–618

613

Fig. 3. Fracture surface of the specimen with temperature gradient.

surface of specimen with temperature gradient. In Fig. 2(a) and (b), the fracture surface at 750  C is made up of many skew surfaces. The tilt angle from the fracture surface to the tensile axis is about 45 . The main origin surface of crack is normal to the tensile axis (see in Fig. 2(c)). The crack starts at inner surface (seen in Fig. 2(b)

and (c)), and the specimen final fractures from the outer surface (see in Fig. 2(d)). Fig. 3 shows the fracture surface of the specimen with temperature gradient. Compared Fig. 3(a) and (b) with Fig. 2, the number of skew surfaces with temperature gradient is fewer than that at 750  C. The tilt angle from the skew plane to the tensile

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Table 1 Material constants of crystallographic constitutive model.

Table 3 Material constants of damage

Temperature ( C)

E (MPa)

n

G (MPa)

s0 (MPa)

g0a (MPa)

Temperature ( C)

nfat

25 760 850 900 980

131.50 105.50 91.46 91.83 80.50

0.344 0.377 0.383 0.386 0.390

155.07 115.43 105.85 97.21 85.60

382.6 376.0 357.2 280.0 240.0

366.9 408.0 410.5 324.9 275.0

600 750 800 900

0.61 0.65 0.8 0.48

axis is about 45 . In Fig. 3(c), the fatigue crack occurs in outer surface located in higher temperature zone. In Fig. 3(c) and (d), there is mixture zone between fatigue fracture zone and tensile fracture zone. The mixture zone is near the inner surface. The tensile fracture surface is characterized by cleavage planes which belong to {111} crystal plane (seen in Fig. 3(d)–(e)). It can be summarized that the slip deformation is a principal mechanism of LCF with temperature gradient.

Many works (Suresh, 1991; Yue et al., 2007) have shown that the LCF life of single crystal Ni-base superalloys correlates with value of the resolved shear stresses and shear strain rate on all slip systems. Crystallographic constitutive model with rate dependent constitutive formulation and kinematic hardening rule is adopted to study the LCF behavior of single crystal superalloys under temperature gradient. The shear strain rate on a slip system is assumed to be related to the resolved shear stress sðaÞ , which is defined as

sðaÞ ¼ P ðaÞ : s

(1)

through a power law relation

g_

¼

sðaÞ

g_ 0ðaÞ

!1=m (2)

g ðaÞ

where P ðaÞ is parameter, s is stress tensor, g ðaÞ is the slip system ðaÞ strength, m is the strain rate sensitivity exponent and g_ 0 is the reference shear rate. If the value of the strain rate sensitivity exponent m is chosen to be very low, the material response is effectively rate independent. The parameter P ðaÞ is defined as

P

ðaÞ

¼

1 2

ðaÞ ðaÞT

m

n

þn

g_ ðaÞ ¼

X b

    hab g_ ðbÞ 

(6)

where hab is the hardening coefficient, which can be obtained as

3. Constitutive model

ðaÞ

For the crystallographic model, the strain rate sensitivity expoðaÞ nent m ¼ 0.02 and the reference shear rate g_ 0 ¼ 0:001ðs1 Þ (note: this parameter can be changed to see the influence of the value on the calculation results). To simplify the program, the material strain hardening can be specified by the evolution of the function g ðaÞ , which is modeled by the following equation

ðaÞ

 T

ðaÞ

m

(3)

where nðaÞ and mðaÞ are the unit vectors normal to the slip plane and along the slip direction of the slip system a respectively. The functions g ðaÞ characterize the current strain hardened state of crystal. Here it is simply depended on the sum shear strain of the slip magnitudes g, i.e.

g ðaÞ ¼ gðaÞ ðgÞ

hab ¼ qab hb ðno sum on bÞ

(7)

qab is the matrix describing the latent hardening, and hb is a single hardening rate. Here (Kalidindi et al., 1992)

  ga b hb ¼ h0 1 

(8)

ss

where h0 is hardening modulus, ss and b are model parameters. The material parameters are

ss =s0 ¼ 1:5; h0 =s0 ¼ 1:2; b ¼ 1:3: For the low cycle fatigue of single crystal superalloys, the back stress X is described by the kinematic hardening rules. There are many kinds of kinematic hardening rules for the evolutionary equation of X. Based on the kinematic hardening rule introduced by Yaguchi et al. (2002), a new kinematic hardening rule is developed for the low cycle fatigue model. The evolutionary equation of the X is expressed as follows:

  o  n Q  m   Xj sgnðXÞ X_ ¼ C ag_ ðaÞ  ðX  YÞg_ ðaÞ   dexp RT  Y_ ¼ afYst sgnðXÞ þ YgjXjm

(10)

where C, a, d, Q, R, a, m are the material constants, T is temperature, the variable Y expresses the dynamic recovery term of the back stress, Yst denotes the saturation value of Y, the third term on the right-hand side in Eq. (9) expresses the thermal recovery term of the back stress, sgn()is the sign function, which is defined as sgnðxÞ ¼ 1 if x < 0 and sgnðxÞ ¼ 1 if x  0.

(4)

where

g ¼

+

X

(5)

8

a

Table 2 Material constants of kinematic hardening rule. C 6.0

1.6

No

jgðaÞ j

a 150.0

d 4.5  10

Q 7

85.0

R 8.31

05

a 1.6  10

7

m

Yst

4.275

5.0

(9)

A

Fig. 4. Geometry shape and dimensions of notch plate specimen.

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615

Fig. 7. FE model. Fig. 5. Finite element model of notch plate form specimen.

Therefore, the power law relation can be expressed as follow:

  g_ ðaÞ ¼ sgn sðaÞ  X g_ 0ðaÞ

#1 " sðaÞ  X m   g ðaÞ j

(11)

The damage model of LCF in this paper is based on the model presented by Tinga et al. (2009), which is based on the climb rate, effective resolved shear stress and slip rate. In the present work only the technologically interesting [001] orientation of the material is considered, which means that the cubic slip systems can be neglected. The fatigue damage DDfat i can be defined as:

"

Xjsa jmoct jg_ a j max max g_ oct soct a

DDfat ¼ i

#noct

  Q c RT

(12)

exp

where samax and g_ amax are the maximum values of resolved shear stress and slip rate at each cycle, soct is reference stress, g_ oct , moct and noct are determined by experiments. In order to reduce the number of variable, soct is expressed by the function of s0 , and moct is expressed by the function of t. The fatigue damage DDfat i at arbitrary temperature can be written as:

"

DDfat i

 12  X jsamax j t=100 jg_ amax j ¼ 2:5  s0 10 a¼1

#nfat

  Q c exp RT

(13)

where t is centigrade temperature, T is the absolute temperature, R and Qc are model parameters. Parameter nfat can be obtained from

the LCF experiments. The LCF life at arbitrary temperature can be predicted by above damage model. The above constitutive has been implemented into a user subroutine UMAT of FE software ABAQUS. 4. Parameter identification The proposed constitutive model has fourteen material constants, Young’s modulus E, Poisson’s ratio n, shear modulus G, s0 , ga0 , C, a, d, Q, R, a, m, Yst and nfat . The material constants were determined by the following method: (1) Assume that C, d, a, Yst and nfat were equal to zero, determine E, n, G, s0 and ga0 though the stress–strain curves at different temperatures. Material constants of crystallographic constitutive model at different temperatures are listed in Table 1. For single crystal superalloy, s0 and ga0 are very sensitive to the temperature. s0 and ga0 are determined at temperature T via:

s0 ¼ s0;T1 þ

s0;T2  s0;T1 T2  T1

ga0 ¼ ga0;T1 þ

ðT  T1 Þ

ga0;T2  ga0;T1 ðT  T1 Þ T2  T1

(14)

(2) Determine C, a, d, Q, R, a, m and Yst based on the stress–strain hysteresis hoop of experiments. s0 and ga0 were reduced to 95% due to the exist of the kinematic hardening rule. It is difficult to determine so many constants once time. The method of trial

10000

820°C Strain controlled 1000 8000

Stress(MPa)

Calculated Nf

500 6000

4000

0

-500 2000 -1000 0

0

2000

4000

6000

8000

Experimental Nf Fig. 6. Calculated life versus experimentally life.

10000

-0.015

-0.010

-0.005

0.000 Strain(%)

0.005

0.010

Fig. 8. Bauschinger effect with 3 ¼ 0.015 at 820  C.

0.015

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900

1000

900°C Stress controlled

820°C Stress controlled

800 700

Stress(MPa)

Stress(MPa)

800

600

400

600 500 400 300

200

200 100

0 0.0

0.2

0.4

0.6 0.8 Strain(%)

1.0

0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Strain(%) Fig. 9. Ratcheting effect at 820  C.

Fig. 11. Ratcheting effect at 900  C.

and error is applied to determine these constants. These material constants are listed as Table 2. (3) Determine nfat based on the LCF life and the samax , g_ amax obtained by finite element analysis. The nfat can be solved according the Eq. (13), and it is listed in Table 3.

5. Verification and validation study 5.1. Complex stress verify The flat specimen with notch is designed to verify the feasibility of the constitutive model and damage model introduced in this work. The Geometry shape and dimensions of notch plate specimen (all dimension are in mm) is shown in Fig. 4. The experimental temperature is 600  C, and all fatigue tests are run with frequency of 0.33 Hz. The experiments are carried out with stress control. Finite element model of specimen is shown in Fig. 5. The model is meshed with 3-deimension 8-node brick element C3D8. 10,250 elements are meshed for the model. The crystallographic orientation of the model is [001]–[010]–[100] in Axial x–y–z, shown in Fig. 5. The calculated fatigue life Nf is plotted versus the experimental fatigue life Nf in Fig. 6. It can be seen from Fig. 6 that the life Nf predicted by the proposed model is within a factor two comparing with the life determined by the experiment, which

0.000014

820°C Stress controlled

Fatigue damadge

0.000013

0.000012

proves that the model introduced in this work can be used to investigate the LCF life at uniform temperature. 5.2. Simulation of LCF behavior with uniform temperature The LCF behaviors controlled by stress and by strain are simulated by finite element method (FEM). The thin-walled specimen is meshed with 3-dimension 8-node brick element with total of 8640 elements. FE model is shown in Fig. 7. The crystallographic orientation of the cell model is [001]–[010]–[100]. Loading direction is along y direction, i.e. [001] crystallographic orientation. The Bauschinger effect and ratcheting effect at 820  C are shown in Figs. 8 and 9, respectively. It can be seen from Fig. 8 that there is Bauschinger effect under cyclic strain loading. The peak stress is increasing and trend to saturation with the increase of cycle times. It can be seen from Fig. 9 that there is ratcheting effect under cyclic stress loading. The ratcheting strain increases with the increase of number of cycles, and the rate of strain ratcheting approaches a steady state with the increasing number of stress cycles. The performance of material will reduce with the increase of cycle times. The ratcheting failure will take place when the total accumulated strain reaches a critical value. Fig. 10 shows the damage evolution at 820  C. It can be seen that the fatigue damage per cycle is decreasing with the increase of cycle times. Furthermore, the LCF behaviors controlled by stress at 900  C are investigated. Ratcheting effect and damage evolution at 900  C are shown in Figs. 11 and 12, respectively. It can be seen from Fig. 11 that ratcheting effect at 900  C is also obvious. It can be seen from Fig. 12 that damage rule at 900  C accords with the rule at 820  C. The LCF life of single crystal superalloys can be calculated by the damage on the first cycle or other cycle. In this work, the low cycle fatigue life with uniform temperature is calculated by the first cycle damage in order to simply calculation. 5.3. Simulation of LCF with temperature gradient

0.000011

0.000010

1

2

3

4

5

Cycles Fig. 10. Damage evolution at 820  C.

6

7

8

Numerical simulations with temperature gradient are carried out to investigate the LCF behavior of single crystal superalloys under temperature gradient condition. The fluid and heat transfer are solved with the CFD software of Fluent. The temperature distributions solved by CFD software are transferred to the FE model by the interpolation method. Temperature distributions of FE model and middle section of model are shown in Figs. 13 and 14, respectively. It is found that there is remarkable temperature gradient along the

N.X. Hou et al. / European Journal of Mechanics A/Solids 29 (2010) 611–618

617

0.00045

Damadge

0.00040

0.00035

0.00030

0.00025

0.00020 1

2

3

4

5

6

Cycles Fig. 14. Temperature distributions of middle section. Fig. 12. Damage evolution at 900  C.

radial direction of the hole. The temperature gradient at inlet is obvious than that at outlet. The temperature gradient at middle section is up to 54.1  C/mm. The cycle loading according to experiment is applied on the FE model. Node 67 which temperature is 900  C and node 103 which temperature is 818.8  C are used to study the deformation mechanics of LCF behavior with temperature gradient. Fig. 15 shows the ratcheting effects of node 67 and node 103 with temperature gradient. It can be seen from Fig. 15 that the ratcheting effect of node 67 is more remarkable than that of node 103. It means that material performance at high temperature zone reduces quickly than that at low temperature zone. Materials will failure from high temperature zone when the total accumulated strain reaches a critical value. Fig. 16 shows the fatigue damage distributions of middle section with temperature gradient. The damage at outer surface is great larger than that at inner surface, which proves that the fatigue crack initializes from the outer surface. The damage evolutions of node 67

Fig. 15. Ratcheting effects of node 67 and node 103 with temperature gradient.

Fig. 16. Fatigue damage distributions of middle section with temperature gradient.

Fig. 13. Temperature distributions of FE model.

and node 103 are shown in Fig. 17. The difference of fatigue damage between first cycle and second cycle is very obvious under temperature gradient (see Fig. 17), which is caused by the effect of temperature gradient. The fatigue life predicted by the first cycle is 4239 cycles, while the fatigue life predicted by the second cycle is 6640 cycles. In LCF test with temperature gradient, the fatigue life is 6948 cycles. It can be concluded that the fatigue life predicted by

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Acknowledgements

0.00045 0.00040

Damadge

The work was supported by the National Natural Science Foundation of China (GS1) (No. 50775183, 10802065), the Research Fund for the Doctoral Program of Higher Education (GS2) (20060699033, 200806991051). These supports are gratefully acknowledged.

Without cooled air flow With cooled air flow: node 67 With cooled air flow: node 103

0.00035 0.00030 0.00025 0.00020

References

0.00015 0.00010 0.00005 0.00000

1

2

3

4

5

6

Cycles Fig. 17. Damage evolution of node 103 and 67.

the second cycle is more accurate than the fatigue life predicted by first cycle. It is appropriate to predict the fatigue life with damage of the second cycle under temperature gradient condition. 6. Conclusions (1) Low cycle fatigue tests with uniform temperature and with temperature gradient are carried out to investigate the influence of temperature gradient. There is remarkable influence of temperature on fatigue life. The microstructure reveals that the slip deformation is a principal mechanism of LCF with temperature gradient. (2) The LCF model based on rate dependent constitutive formulation with kinematic hardening is suitable for investigating Bauschinger effect and ratcheting effect of single crystal superalloys at different temperatures. The damage model can be used to predict LCF life, which is verified by the complex stress tests. (3) The ratcheting effect at high temperature zone is more remarkable than that at low temperature zone with temperature gradient. The material performance reduces quickly at high temperature. The difference of fatigue damage between first cycle and second cycle is tremendous due to the influence of temperature gradient. It is appropriate to predict the fatigue life with damage of the second cycle under temperature gradient condition.

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