Thermal gradient mechanical fatigue assessment of a nickel-based superalloy

International Journal of Fatigue 135 (2020) 105486

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Thermal gradient mechanical fatigue assessment of a nickel-based superalloy

T

Jingyu Suna, Huang Yuana,b, , Michael Vormwaldc ⁎

a

School of Aerospace Engineering, Tsinghua University, Beijing, China Institute for Aero Engines, Tsinghua University, Beijing, China c Institute of Steel Construction and Materials Mechanics, Technical University of Darmstadt, Germany b

ARTICLE INFO

ABSTRACT

Keywords: Thermal gradient mechanical fatigue (TGMF) Transient temperature analysis Phase angle Nickel-based superalloy Fatigue life prediction

In the present work, a radiation heating system was developed to simulate thermal gradient mechanical fatigue (TGMF) loads in turbines. The specimen is externally heated by radiation and internally cooled by compressed air. Experiments showed that the TGMF life of the nickel-based superalloy is significantly shorter than that of the thermo-mechanical and the isothermal fatigue, although the thermal stress is small. It was confirmed that the conventional fatigue models generated seriously deviations and could not catch effects of the thermal gradients. The modified TGMF model introduced a correction term of the temperature gradient effects and can describe the TGMF lifetime of Inconel 718 reasonably. The new model provides a uniform description of isothermal and complex thermo-mechanical fatigue.

1. Introduction The nickel-base superalloy Inconel 718 is widely used in the gas turbine engines [1]. In the past decades, isothermal fatigue (IF) tests were commonly used to predict the lifetime of the gas turbine components, which generally work under varying thermo-mechanical loading conditions. Numerous investigations on the elevated temperature LCF of the nickel-based superalloy were performed [2–5]. However, a number of studies showed that the thermo-mechanical fatigue (TMF) loads influenced the fatigue damage mechanisms significantly [6–10]. Thermo-mechanical fatigue means the mechanical loads and temperature are varying simultaneously. For different phases between the cyclic mechanical loading and temperature, the TMF lifetime can be significantly shorter or longer than the isothermal one [11]. Except varying temperature in turbine, many components possess significant temperature gradients, such as turbine blades, due to internal cooling. In the course of service, the turbine components are subjected to not only the alternating mechanical loads and varying temperature but also the temperature gradients. For internally cooled parts, the temperature gradients produce additional stresses, which are expressed as multiaxial compressive stresses on the hotter surface, while a multiaxial tensile load is displayed on the colder surface. Few works on the TGMF fatigue were published in the past decades. Brendel et al. [12] and Prasad et al. [13] investigated temperature gradients in TMF specimens and found that the temperature gradients ⁎

had a remarkable influence on the TMF lifetime. Baufeld et al. [14] and Bartsch et al. [15] carried out the thermal gradient mechanical fatigue testing for the single crystal superalloy CMSX-4 with a NiPtAl oxidation protection coating. More experimental observations found the temperature gradient had remarkable influence on fatigue life [16,17]. But the study only discussed the isothermal fatigue life, the thermo-mechanical loading conditions were not considered. Micro-structural changes, defects, phase evolution of the metal coatings and rafting of the / substrate morphology were investigated. They detected cracks at the inner specimen surfaces and substrate pores. However, the fatigue life of the TGMF was not estimated. To quantify effects of temperature gradients in thermo-mechanical fatigue and to establish a uniform fatigue life prediction model for complex thermo-mechanical fatigue, extensive TGMF testing is necessary, in combining with the investigation of constitutive modeling. In the present work, a radiation heating system is developed based on the DLR concept [15,14]. Based on extensive finite element computations the transient temperature distributions in thin-walled tubular specimens are optimized for thermal gradient mechanical fatigue tests. The numerical results are verified by experimental measurements and provide the basis for TGMF tests. LCF behavior of the nickel-based superalloy is experimentally tested under given IF, TMF and TGMF loading conditions. The systematically experimental results provide a database for assessing fatigue life models and for establishing new TGMF fatigue life concepts.

Corresponding author at: School of Aerospace Engineering, Tsinghua University, Beijing, China. E-mail address: [email protected] (H. Yuan).

https://doi.org/10.1016/j.ijfatigue.2020.105486 Received 4 September 2019; Received in revised form 24 December 2019; Accepted 15 January 2020 Available online 03 February 2020 0142-1123/ © 2020 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 135 (2020) 105486

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2. Experiments 2.1. Materials and specimens The nickel-based superalloy Inconel 718 investigated in the present paper was provided by ThyssenKrupp VDM GmbH, in the form of rods with the 20 mm diameter. The rods were solution-treated at 980° C for one and a half hours then cooled to room temperature in water. Then they were exposed for eight hours at 720°C, furnace cooled at 56° C/h to 621°C, where they were held for eight hours and forced air cooling to room temperature [18], as shown in Fig. 1. The chemical compositions of Inconel 718 are given in Table 1. The grain morphologies of the heattreated alloy are analyzed by electron backscatter diffraction (EBSD), as shown in Fig. 2. The material shows an isotropic grain distribution and the average grain size is 26.2 µ m. Two kinds of tubular specimens were used for IF, TMF and TGMF tests, as shown in Fig. 3. The geometry of the tubular specimen for IF and TMF is shown in Fig. 3(a), total length of 185 mm, an outer diameter of the gauge section of 10 mm and 35 mm in gauge length. The geometry of the thin-walled tubular specimen for TGMF is shown in Fig. 3(b), with 120 mm in the total length, a gauge length of 30 mm, an outer diameter of the gauge section of 8.5 mm and an inner diameter of 6.5 mm. Thus, the wall thickness of the tubular specimen in the gauge section is 1 mm. The dimensions of solid and tubular specimens are adapt to the dimension ratios suggested by the standards ASTM E606 [19] and E2207 [20], respectively.

Fig. 2. Microstructure of the investigated nickel-based superalloy Inconel 718 after solution treatment.

2.2. The TGMF testing system In the present work, a thermal gradient mechanical fatigue (TGMF) testing system was developed, as shown in Fig. 4. The fatigue specimen was defined as failed if the load reduction exceeded 25% of the total applied load. For TGMF tests, the mechanical load was conducted by a hydraulic servo fatigue test machine. The specimen was heated by a

Fig. 3. Dimensions of the specimen: (a) for LCF and TMF tests and (b) for TGMF tests.

radiation furnace. The thermal gradient is obtained by heating of the outer surface with the concentrating radiation and simultaneous internal cooling with pressurized air. The radiation furnace consists of 16 halogen lamps, and each lamp is related to a reflector. The geometry of the reflector is an elliptic cylinder. The lamp is located at one of the focus lines of the elliptic cylinder mirror, and the specimen is located at the common focus lines of all reflectors, as shown in Fig. 5. The light of the lamps is reflected by the mirrors and focus on the surface of the specimen. Radiant heating can heat metals, non-metals, composites, etc., and obtain complex temperature fields through multi-temperature zone control (Fig. 6). The total power of the radiation furnace is 6.4 kW and can heat the tubular specimens to melt within seconds. The TGMF testing system includes multiple subsystems such as mechanical loading, heating, air cooling, temperature measurement and so on. These subsystems work cooperatively to ensure the mechanical load and temperature over the specimen gauge section are simultaneously varied and independently controlled.

Fig. 1. The heat treatment process of Inconel 718 in accordance with AMS Standard 5663. Table 1 Chemical compositions of Inconel 718 (wt.%). Element

wt.%

Element

wt.%

C S Cr Ni Mn Si Mo Ti Nb Cu

0.02 <0.001 18.53 53.44 0.05 0.06 3.06 0.99 5.30 0.04

Fe P Al Pb Co B Ta Se Bi

17.71 0.007 0.56 0.0002 0.13 0.004 <0.01 <0.0003 <0.00003

2

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Fig. 6. Principle of the TGMF radiation furnace with temperature monitors.

thermocouples, the diameter of the thermocouple should be as small as possible. Therefore, the thermocouple with a diameter of 0.25 mm was used in the present tests. The temperature controller was used to control the temperature of the specimen. The controller accepts the temperature sensor (thermocouple or infrared thermometer) as input and compares the actual temperature with the desired temperature. It then provides an output signal to the power regulator to control the luminance of the lamps, i.e., the heat power of the specimen. The temperature controller of the radiation furnace is a proportional-integral-derivative (PID) controller. The values of the proportional, integral, and derivative gains of the PID controller can be determined by the PID tuning process. However, the emissivity of the specimen and the temperature range influence the PID gains of the temperature controller. Therefore, the PID gains should be tuned when the specimen surface is oxidized. The air cooling subsystem consists of two air compressors, an air dryer, metering valves, air pressure gauges, and cooling passages. The two air compressors were used to provide the cooling air of the specimen and the lamp holders, separately. Each air compressor has a power of 2.4 kW and provides the compressed air of 0.4–0.8 MPa. The compressed air with a pressure of 0.7 MPa is filtered through the air dryer and output to the metering valve. The air dryer is used for removing water vapor from the compressed air. During all of the TGMF tests, the pressure and the flow volume of the compressed air were kept as constant to 40 l/min. The metering valve was used to control the volume flow of the compressed air. The air cooling subsystem also provides compressed air for the lamp holders. The duration of the TGMF test is usually long, so the lamp holders have to be cooled to ensure that they can work properly. There are no relevant standards for TGMF testing. Since the test processes of the TGMF and TMF are very similar, the concepts in TGMF testing are defined as same as those of the TMF testing, such as:

Fig. 4. Experimental apparatus of TGMF test system.

Fig. 5. Schematic drawing of the furnace consisting of 16 mirror segments, where the 16 halogen lamps located at one of each focal line and the specimen at their common focal line.

The mechanical load is transmitted to the specimen through a fixture. As shown in Fig. 6, the cooling air can enter the internal of the tubular specimen through the fixture. In order to reduce the additional bending moments and the bending stress in the specimen, the fixture must provide a sufficiently high degree of coaxiality. Meanwhile, the frame of the MTS testing system provides an alignment device, and it can effectively reduce the additional bending moments in the specimen during the test. The total strain is measured by an extensometer with a gauge length of 10 mm and a measuring range of ±20%. The temperature of the specimen was measured by the thermocouples. In practice, the choice of the diameter of the thermocouple wire is significant. A thick thermocouple will cause a delay in the temperature measurement. It leads to temperature deviations between the thermocouple and the specimen, and it affects the response of the feedback control system. In order to improve the accuracy of temperature measurement and to increase the response speed of

• Thermal strain, , • Mechanical strain, , • Total strain, = + , / ratio, R = , • Loading • Phase angle of the thermal loading and mechanical loading, th

mech

tot

mech

th

mech,min

mech,max

T

.

The light reflected on the different gold-coated mirrors focuses on the specimen and induces temperature gradients along the specimen outer surface. The highest temperature occurs in the middle section of the specimen during the TGMF test. Unlike the induction heating, the axial temperature gradient of the specimen is difficult to be adjusted by modifying the radiation furnace. Therefore, the influence of the axial temperature gradients has to be considered with the help of finite element computations. 3

International Journal of Fatigue 135 (2020) 105486

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3. Results of TGMF tests Experimental conditions and results of thermo-mechanical fatigue tests are summarized in Table 2. The isothermal fatigue (IF) tests were performed at 650 °C, and the in-phase thermo-mechanical fatigue (TMF-IP) and the out-of-phase thermo-mechanical fatigue (TMF-OP) tests were carried out with a temperature range of 300 650 °C, as introduced in [11,21]. In order to compare with the TMF testing results, the TGMF tests with the same temperature range of 300 650 °C and two kinds of phase angles were carried out: 0° (IP) and 180° (OP). Each kind of TGMF tests was conducted with four or five different mechanical strain amplitudes. The load type TGMF-IP stands for the in-phase thermal gradient mechanical fatigue, and TGMF-OP means the out-ofphase thermal gradient mechanical fatigue. The nickel-based superalloy Inconel 718 used in [11,21] was investigated in the present work under the TGMF loading conditions. Therefore, it is reasonable to compare the fatigue performance each other. Because of the thermal gradients in radial and axial directions, the axial stress is not constant on the cross-section of the specimen. Therefore, the stress distribution of the specimen during the test has to be obtained with the help of finite element computations. In order to facilitate the comparison and analysis between the computational and experimental results, the nominal axial stress on the cross-section is used in the following figures, defined as

Fig. 7. Locations of the temperature monitors.

It was confirmed that the temperature fluctuates depending on time in a isothermal test is small ( 1.5 °C). In order to obtain the temperature distribution of the specimen during the TGMF testing, temperatures at the upper, center and lower positions in the 12 mm gauge section of the specimen were measured by three thermocouples wrapped around the specimen. Fig. 7 shows the locations of the thermocouples. In the present work, the temperature of the TGMF testing varies between 300 °C and 650 °C. The cycle period was predetermined as 240 s with a triangular shape temperature cycle, thus the heating and cooling rates were about 2.2 °C/s. Fig. 8(a) shows the transient temperature measurement during a TGMF cycle and Fig. 8(b) shows the temperature deviations between the upper/lower thermocouple and the temperature command. The maximum axial temperature difference during the TMF test is less than ± 30 °C. The deviations in the center of the specimen, where the fatigue failure occurs, are obviously smaller, less than ± 10 °C.

(1)

= F / A.

Fig. 9(a) and (c) show the stable hysteresis loops of TGMF tests under IP and OP loading conditions, respectively. Since there is a phase difference between the temperature cycle and the load cycle in the fatigue test, the tensile half cycle and the compressive half cycle have different cycle softening behaviors. For the TGMF-IP test, an evident cyclic softening was observed during the tensile half cycle, whereas for Table 2 Experimental conditions and results of thermal gradient mechanical fatigue tests. Test Type

± mech [%]

[s−1]

IF

1.00

1 × 10

3

1 × 10

3

1 × 10

3

1 × 10

3

1 × 10

3

1 × 10

3

0.80 0.70 0.60 0.50 0.45 0.40

TMF-IP

1.00

0.80 0.70 0.60

TMF-OP

1.00

0.80 0.70 0.65

TGMF-IP

0.80

0.70 0.60 0.50

0.425 TGMF-OP

Fig. 8. Varying temperatures in a tubular specimen with the gauge length of 15 mm, heated by the radiation furnace. (a) Temperature variations of three thermal elements during a loading cycle. (b) Temperature deviations in the axial direction.

0.80

0.60 0.50

0.425

4

mech

[°]

Nf [cycle]

–

231

–

592

T

6.4 × 10

– – – – –

3

326 1336 8449

15497

130585

0

58

0

248

2.22 × 10

4

1.78 × 10

4

1.56 × 10

4

1.33 × 10

4

0 180

209

180

429

0

176 1297

2.22 × 10

4

1.78 × 10

4

1.56 × 10

4

1.44 × 10

4

180

1.67 × 10

4

0

48

1.33 × 10

4

1.22 × 10

4

0

107

1.11 × 10

4

0.89 × 10

4

0

1.67 × 10

4

180

128

1.22 × 10

4

1.11 × 10

4

180

864

0.89 × 10

4

180

0 0

180 180

303 633

50

208

1066

375 3387

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Fig. 9. Experimental results of TGMF tests with the varying temperature between 300 °C and 650 °C. Noting that, the vertical axis axial stress is the nominal stress. (a) Half life stable hysteresis loops of TGMF-IP tests. (b) Peak and valley stresses of TGMF-IP tests. (c) Half life stable hysteresis loops of TGMF-OP tests. (d) Peak and valley stresses of TGMF-OP tests.

the TGMF-OP test, the alloy performed a significant cyclic softening during the compressive half cycle. Fig. 9(b) and (d) illustrate the peak, valley and mean values of the stress response in the TGMF-IP and TGMF-OP tests, respectively. Cyclic softening was observed during in in-phase and out-of-phase thermal gradient mechanical fatigue tests, but the phase difference between the temperature cycle and the load cycle leads to the different cycle softening behavior. In the temperature range of 300 650 °C, the higher is the temperature, the more obvious cyclic softening is the material. Therefore, under the TGMF-IP loading condition, the cyclic softening of the superalloy in the tensile half cycle is more evident, whereas the TGMF-OP load made the cyclic softening behavior more clear in the compressive half cycle. It is known that the mean stress cannot be ignored in low cycle fatigue. Especially for nickel-based superalloys with high strength and low toughness, the mean stress is an important factor affecting the fatigue life, as shown in Fig. 10. The mean stress response curve for the Inconel 718 alloy under the in-phase and reverse phase TGMF test conditions. It can be seen that for TGMF-IP, the mean stress is compressive stress, and the increase of the mechanical strain amplitude has few effects on the mean stress. For TGMF-IP test, the mean stress becomes compression stress since the lower temperature stress is compressive. When = 0.8% , the mean stress at the half-life cycle is about −30 MPa. For TGMF-OP test, the mean stress becomes tensile and increases with the mechanical strain amplitude. When = 0.8% , the mean stress at the half-life cycle is about 50 MPa. The asymmetry of cyclic softening in tension and compression results in variations of the mean stress in the fatigue tests towards the lower temperature loading direction. For different mechanical strain amplitudes, the mean stress evolution rate is similar, and the mean stress value increases with the number of cycles.

Fig. 10. The evolution of mean stresses under TMF-IP-TGMP and TMF-OPTGMP loading conditions.

The evolution of the mean stress in TGMF changes the strength of the alloy with temperature. Both in-phase and out-of-phase TGMF tests, the temperature cycles in the interval of 300 650 °C. The material strength varies with temperature. When the temperature is high, the strength of the alloy is low. It results in the stress asymmetry of the hysteresis loop. The mean stress develops towards the low-temperature half cycle. The reason is that, with increasing mechanical strain amplitude, the plastic deformation of the alloy at the high temperature half cycle increases, which aggravates the asymmetry of tensile and compressive stress. Additionally, it can be observed that the absolute value of the mean stresses increases gradually with cycle under TGMF loading conditions, which can be considered as a result of the more intense softening of the superalloy Inconel 718 during the higher temperature 5

International Journal of Fatigue 135 (2020) 105486

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figures the temperature cycles of the TMF and TGMF tests are the same, the only difference between the TMF and TGMF tests is the temperature gradient in the specimen. Therefore, the temperature gradient is considered to have a significant effect on the fatigue life under the thermomechanical loading condition. Comparing Fig. 12(a) with Fig. 12(b), the reduction of TGMF-IP life due to the temperature gradient is more evident than the reduction of TGMF-OP life. 4. Computational analysis of temperature gradients During the TGMF test, the temperature distribution on the outer surface of the specimen can be measured by thermocouples or an infrared thermography camera [22–24]. However, the temperature fields on the inner surface of the specimen are difficult to measure directly by monitors, since the monitors are affected by the internal cooling. Most of the published works use CFD/FEM to compute the temperature of the inner surface [25,23]. The accuracy of the computational result is dependent on the assumption of the boundary conditions. In this section, a new method combining experiment and calculation are introduced to determine the temperature fields of the specimen. The temperature fields of the specimen are controlled by the heat conduction, radiation and convection. The heat conduction of the specimen is only related with the heat conductivity (Table 3). The internal surface of the specimen was cooled with compressed air, as mentioned before. Because of the small internal diameter of 6.5 mm in the specimen, it is difficult to measure the temperature of the internal specimen surface during tests. A variety of measurement options was tried. The internal cooling air has a significant effect on the thermocouple temperature of the inner surface of the specimen. The thermocouple measurement is not the temperature of the inner surface, but the average temperature of the wall and the cooling air flow. Therefore, the calibrated finite element computation becomes an alternative way to obtain the temperature distribution of the specimen inner surface.

Fig. 11. Comparison of the fatigue lives under IF, TMF and TGMF loading conditions.

half cycle. In the isothermal fatigue test, the mean stress value is generally small, so the mean stress has negligible variations on the isothermal fatigue test. For fatigue tests under mechanical strain control, the fatigue life can generally be expressed as a function of mechanical strain amplitude. Fig. 11 shows the comparison of the fatigue lifetimes of IF, TMF and TGMF loading conditions, respectively. The scattering of the experimental data is dramatic, although the isothermal fatigue life (IF) was obtained from the material at the upper temperature 650 °C. All the TMF and TGMF results show much lower fatigue lives than the IF. This result confirms that the fatigue life assessment based on the isothermal fatigue is generally not conservative for the Inconel 718. The TGMF-IP life is much smaller than the TMF-IP life, as shown in Fig. 12a) and the TGMF-OP life is smaller than the TMF-OP life, as shown in Fig. 12(b). However, the difference is substantial. In the

4.1. FE-model The FE-model is shown in Fig. 13, because of the symmetry of the geometry and the external environment, only the gauge section of the specimen is discretized using axisymmetric elements. The gauge length is 12 mm, and the center cross-section (AB) of the specimen is taken as the symmetric plane for the FEM model. The outer surface of the specimen (BC) is defined as a temperature boundary condition. The temperature distribution of the outer surface was measured during the test. The inner surface of the specimen (AD) is considered as an enforced convection boundary. The convective heat transfer coefficient can be calculated by an empirical formula for turbulent flow in tubes. The density, the temperature and the volume flow of the cooling air were measured during the test. The upper surface (CD) of the model is considered as a thermal conduction boundary. For radiation calculations, the emissivity of the surface of the specimen is defined as 0.75. 4.2. Enforced convection in turbulent pipe flow The inner surface of the hollow specimen can be considered as a pipe. According to the formula of convection in turbulent pipe flow, the convective heat transfer coefficient of the inner surface of the specimen can be calculated. Nusselt number (Nu) is a dimensionless number, defined as the ratio of convective to conductive heat transfer across (normal to) the boundary. The Nusselt number is given as [26]

NuL =

hL , k

(2)

where h is the convective heat transfer coefficient of the fluid, L is the characteristic length, k is the thermal conductivity of the fluid. The Gnielinski’s correlation for turbulent flow in tubes [27] is given

Fig. 12. Comparison of fatigue lives of TMF and TGMF: (a) IP, (b) OP. 6

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Table 3 Heat conductivity of Inconel 718. Temperature [° C]

100

200

300

400

500

600

700

800

900

1000

Conductivity [W/(m· °C) ]

14.7

15.9

17.8

18.3

19.6

21.2

22.8

23.6

27.6

30.4

Table 4 Heat convection coefficients of the inner surface of the specimen. Quantity

Unit

Value

T p

K bar

288.15 6.5 7.724

kg/m3 l/min –

µ V Re Pr f NuD h

1.81E-05 40 55624 0.63 0.0205 106.50 423.10

W/(m2·K)

Sutherland’s constant C = 120 K. The density of dry air can be calculated using the ideal gas law, expressed as a function of temperature and pressure, as

=

Fig. 13. Schematic diagram of the finite element model.

(f /8)(ReD 1000)Pr , 1 + 12.7(f /8)1/2 (Pr2/3 1)

(3)

where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from the correlation developed by Petukhov [26],

f = (0.79ln(ReD)

1.64)

2,

(4)

4.3. Computational results

with

0.5 3000

Pr ReD

In [11,21], a constitutive model of the investigated superalloy Inconel 718 under multiaxial thermo-mechanical loading was suggested and implemented into the commercial FEM software ABAQUS. The model was verified under the isothermal as well as thermo-mechanical multiaxial loadings and it is used in the present finite element computations for TGMF testing. The major functions of the constitutive model is summarized as below:

(5)

2000, 5 × 106,

(6)

The Prandtl number (Pr) is a dimensionless number, defined as the ratio of momentum diffusivity to thermal diffusivity, defined as

Pr =

cp µ k

,

(7)

• Kinematic hardening:

where cp is specific heat, µ is dynamic viscosity and k is thermal conductivity. For air cp = 903.3 J/(kg·K) and k = 0.0258 W/(m·K) at 288.15 K. The Reynolds number (Re) is a dimensionless number, defined as the ratio of inertial forces to viscous forces within a fluid,

Re =

uL , µ

ak = r k

where is the density of the fluid, u is the velocity of the fluid with respect to the object, µ is the dynamic viscosity of the fluid, and L is a characteristic dimension. It is noted that the dynamic viscosity µ of an ideal gas is a function of the temperature. The Sutherland’s formula can be derived as follows

T +C T µ = µ0 0 T + C T0

,

2 3

k

H (f k )

(8)

3 2

(10)

where p is absolute pressure, T is absolute temperature, R is specific gas constant and equals 287.058 J/(kg·K) . According to the measurement results of the sensors, the average pressure and temperature of the cooling air are 6.5 bar and 288.15 K, respectively. Substituting the measurement values into Eqs. (2)–(10), the heat convection coefficients of the inner surface of the specimen is computed as 423.10W/(m2·K) . Table 4 lists all intermediate variable in the calculation process.

as

NuD =

p , RT

p

ak : rk

µk

p

ak p rk

µk p

(11)

ak rk

+ H (f k )

ak k r , rk

where (12)

r k = r0k + r k, and

r k = r ks [1

a1k e

b1k p

(1

a1k ) e

b2k p],

• Isotropic hardening:

(9)

where µ 0 is the reference viscosity at reference temperature T0 , C is Sutherland’s constant depending on the gaseous material. For air at T0 = 291.15 K the reference viscosity µ 0 = 1.827 × 10 5 Pa·s and

Y = Y0 + Y

np,

where Y0 denotes the yield stress for proportional loading and Y 7

(13)

(14) np

is

International Journal of Fatigue 135 (2020) 105486

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Fig. 14. Experimental and computational results under the thermal gradient mechanical loading conditions with varying temperature between 300 °C and 650 °C. The vertical axis the nominal normal stress. (a) The hysteresis loops of TGMF-IP tests. (b) Peak, valley and mean stress values of TGMF-IP tests. (c) The hysteresis loops of TGMF-OP tests. (d) Peak, valley and mean stress values of TGMF-OP tests.

the non-proportional hardening defined by

Y

np

=

p (Y nps

Y

np ) p ,

where Y nps is the saturated value of Y loading path.

tests, even if it is controlled within a small interval. The maximum temperature appears in the external symmetric surface. The local temperature affects the local strain variation and the higher temperature causes higher strain for the same stress. The deviations in Fig. 14 are resulted from temperature gradients. Under the TGMF-IP loading condition, the mechanical load and temperature simultaneously reach their respective maximum or minimum values. The maximum temperature occurs at the symmetric position of the outer surface of the specimen. Fig. 15(a) illustrates the temperature and axial stress distribution of the internal and external surface along the axial direction when the temperature reached its maximum value during the TGMF-IP cycle. The computational temperature in Fig. 15(a) is related to the experimental result in Fig. 8 at the moment of the 60th second within a thermo-mechanical loading cycle. The black squares in Fig. 15(a) are the temperature measured by the thermocouples. In fact, the measured temperature is not symmetrical with respect to the central cross-section of the specimen. However, for the convenience, the boundary conditions are simplified and the axial temperature difference is neglected. Since the axial temperature gradient is much smaller than the radial temperature gradient, the simplification has little effect on the fatigue life calculation results. The abscissa axis is the distance from the center section of the test specimen (y-axis). The temperature gradients induce an axial stress difference of ca. 20 MPa between the inner and outer surface of the specimen. The temperature and axial stress distribution of the internal and external surface along the radial direction are displayed in Fig. 15(b), when the temperature reached its maximum value during the TGMF-IP cycle. The abscissa axis is the distance from the center axis to the outer surface of the test specimen. The temperature gradients induce a radial stress deviation of ca. 20 MPa between the inner and outer surface of the specimen, i.e., ca. 2%. Additional plastic strain in the outer surface is

(15) np

under a non-proportional

In the above equations, k, µk , r0k, r ks, a1, b1, b2 , Y nps and p are temperature-dependent material parameters. The constitutive model was verified under the isothermal and thermo-mechanical multiaxial loadings and it was used in the present finite element computations for TGMF testing. The material parameters of Inconel 718 are introduced in [11]. Figs. 14(a) and (c) show that the experimental and computational stress–strain responses of the TGMF-IP and TGMF-OP tests, respectively. In the figures the solid symbols represent the experimental nominal strain and stress of the gauge section, and the solid lines represent the computational local strain and stress at point B in Fig. 13. Because of the thermal gradient, the cyclic plastic evolutions were asymmetric in the gauge section of the specimen in the TGMF tests. In the figures, additional tensile strains were observed from the computational results of the TGMF-IP tests, and additional compressive strains were found from the computational results of the TGMF-OP tests. Fig. 14(b) and (d) show the experimental and computational peak, valley and mean nominal stresses of the TGMF-IP and TGMF-OP tests, respectively. The comparison of the experimental and computational results reveals that the peak and valley stresses agree each other. The stress–strain loops show, however, significant deviations in both IP and OP results. In the figures experimental data were measured and represent the average values of the specimen, while the computational results were taken from the symmetric point of the specimen. The temperature analysis reveals that the temperature is not constant in all 8

International Journal of Fatigue 135 (2020) 105486

J. Sun, et al.

Fig. 15. Computational distributions of the temperature and normal stresses at the maximum temperature (650 °C) during the TGMF-IP test with the mechanical strain amplitude of 0.8%. (a) The axial distribution; (b) the radial distribution.

Fig. 16. Locations of crack initiation from the TGMF tests: (a) TGMF-IP 0.65%, (b) TGMF-OP 0.65%. Arrows denote the crack initiation locations.

much larger than the average strain determined in the TGMF tests. Fig. 14 shows that the maximum difference between the local strain and the average strain may exceed 30%. From such results, the TGMF life prediction cannot be made directly based on the stress and strain data obtained from experimental measurements. Obviously, the computational results do not contain the influence of the gravitation in heat convection. Due to thermal expansion in the air, both ends of the specimen are cooled differently, which is not included in the present analysis. However, the present computations give a reasonable estimate of the temperature distribution during the TGMF tests.

IP test, and the intergranular fracture plays a dominant role. Well-developed fatigue striations are observed in Fig. 17(c) and (d), which reveal transgranular crack growth is predominant during out-of-phase thermo-mechanical tests. The present fractographic analysis confirms that the TGMF failure mechanisms are similar to the high temperature fatigue. All specimens show fatigue crack nucleation occurs in the outer specimen surfaces, which implies that the TGMF could be assessed by the conventional or TMF fatigue failure models. In the present section, several known fatigue models are selected to evaluate the fatigue life of the nickel-based superalloy under both TMF and TMGF loading conditions. Combined with the critical plane concept the models are well developed and verified for isothermal fatigue.

5. Fatigue life assessment of TGMF 5.1. Fractography

5.2. Brown-Miller model

Analysis of fractography provides the locations of the crack initiation, the crack propagation and the direction of the fatigue striation. Typical fractographs in the region of crack initiation are shown in Fig. 16, and the arrows show the locations of the crack initiation. For both TGMF-IP and TGMF-OP loadings, crack initiation and subsequent failure have been identified as being initiated at the outer surface of the specimen. Fig. 17 shows the fractography of stable crack propagation. For the TGMF tests, the SEM investigations reveal that the dominant failure mechanism is changing with the phase angle of the thermal loading and mechanical loading, T . In Fig. 17(a) and (b), the fracture surfaces under TGMF-IP loading displays evident grain boundaries and slight fatigue striations. Consequently, it can be concluded that a mixture of transgranular and intergranular fracture mode is present in the TGMF-

Based on the critical plane concepts [28], Wang and Brown [29] proposed a fatigue model in the form of the equivalent shear strain amplitude, as

2

=

max

2

+S

n,

(16)

/2 is the equivalent shear strain range [29]. n represents the where normal strain acting on the plane with the maximum shear strain range max . The material dependent parameter S represents the influence of the normal strain on the crack propagation. The fatigue endurance is determined by 2 9

=A

f

E

(2Nf ) b + B f (2Nf ) c

(17)

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Fig. 17. Fractographs of fracture surfaces from the area of stable crack propagation.

with

5.4. Smith-Watson-Topper model

A=1+

e

+ (1

e)S

p

+ (1

p ) S.

The SWT model [32] can be extended to multiaxial fatigue assessment based on the principal strain range and the maximum stress on the plane of the principal strain range, that is,

and

B=1+

n,max

5.3. Fatemi-Socie model

max

=

2

n,max

1+k

,

(

where n,max is the maximum normal stress on the critical plane suffering from the maximum shear strain range max , and k is a material parameter. The sensitivity of the material to the normal stress is reflected in the ratio k/ y . The fatigue life model oriented to the shearbased damage is illustrated as

2

=

f

G

(2Nf ) b0 +

f

(2Nf ) c0.

k0 f

(

with

k0 =

(1 +

f

e) E

f

p) f

(2Nf ) b+ c .

+(

)=

4

f

2

E

(2Nf )2b

(21)

(2Nf )b + c .

f f

n)

+(

) max =

4 f2 (2Nf )2b0 G

(22)

(2Nf )b0 + c 0.

5.6. Chu-Conle-Bonnen model

(2Nf ) c0

(2Nf ) b + (1 +

f f

The critical plane is calculated from the virtual shear strain energy [33].

(20)

(2Nf )b0 + G

f f

n

+4

y

f

(2Nf ) 2b +

Above the critical plane is defined from maximizing the virtual tensile strain energy n n , termed as Liu I model. Note the model parameters in the present model differ from the others. The corresponding virtual shear strain energy model (Liu II model) is expressed as

(19)

(2Nf ) b

n )max

n

+4

Furthermore, McClaflin and Fatemi [31] proposed that the sensitivity parameter k varies with fatigue life and can be related to the tension and torsion property, that is,

k=

E

The virtual tensile strain energy based model suggested by Liu [33] can be written as

(18)

y

2

5.5. Liu’s virtual strain energy models

Fatemi and Socie [30] proposed that the normal strain term in Eq. (16) should be replaced by the normal stress. The equivalent shear strain amplitude is defined as

2

2

f

=

(2Nf )c

1.

An additional energy-based fatigue model suggested by Chu et al. [34] is expressed as 10

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J. Sun, et al.

n

+

2

+ 1.04

n

f f

= 1.02

2

max

f

2

E

(2Nf ) 2b

TN does not provide any information on the dispersion of fatigue life. The evaluation results are summarized in Fig. 19(a). As mentioned, all known models are not conservative and contain significant deviations in the mean life, besides the TMF and TGMF (To be introduced in the next section). The TMF model gives the best prediction in the mean life prediction. The present model gives a reasonable mean life prediction with a small amount of conservative. The second parameter describes the scattering of the data and is the mean square root error of the fatigue life, defined as

(23)

(2Nf )b + c .

Note the difference in the strain energy in comparing with the Liu’s models. The critical plane should provide the maximum strain energy in the whole loading range and in all directions. 5.7. TMF Model suggested by Sun and Yuan

[A ( =

n )max

n

4 f2 E

(2Nf

) 2b

+ B(

+4

f f

)] (2Nf

(

with

)b + c

t0+ tcyc t0

(t )exp

Q (t ) dt RT (t )

k

,

(25)

where (t ) is stress triaxiality, R is the ideal gas constant (8.31 × 10 3 kJ/mol·K), the time integral is from the start time t 0 of the cycle to the end time t 0 + t cyc, tcyc is the period of the tests, C and k are material constants. In Eq. (25), Q is defined as [35,36]

Q (t ) = Q0

0 n (t )

1

n (t )

2

,

ult

(26)

The Brown-Miller’s model, Fatemi–Socie’s model, Smith-WatsonTopper’s model, Chu-Conle-Bonnen’s model, Liu’s Energy model and TMF model (Eq. (24)) are tried to evaluate the thermal gradient mechanical fatigue. Fig. 18 summarizes the comparison between predicted and experimental fatigue life of the IF, TMF-IP, TMF-OP, TGMF-IP, and TGMF-OP tests. In assessments, the computational stresses and strains are used to reduce effects of inhomogeneous stress and strain distributions induced by the temperature gradients. To give a quantitative assessment of the fatigue models, two statistical parameters are used to evaluate the conformity of the predicted fatigue life, Np , and the experimental result, Nf [37–39,11]. The mean scatter of fatigue life is defined as

f (T , =

i=1

Nf, i , Np, i

4 f2 E

f (T ,

(30)

T )[A (

n )max

n

(2Nf ) 2b + 4

f f

+ B(

)]

(2Nf )b + c .

(

1

2 R

) (31)

T) = 1 + g

T , T n,max

Tmelt

(32)

where Tmelt is the melting point of the material, and T n,max is the temperature corresponding to the maximum normal stress, g is a model parameter in the unit of mm. Introducing f should account for the temperature gradient effects normalized by the maximum temperature. In fact, the fatigue models did not consider influences of thermal stresses, which is proportional to the temperature gradients. The linear term to the gradient modulus can be taken as the first approximation. For the nickel-based superalloy Inconel 718, Tmelt = 1300 °C. The model parameter g is determined by the optimization algorithm to minimize the life prediction mean-squared error, and g is determined as 3.47 mm in the present work. T is the modulus of the temperature gradient, as

with

log

.

The correction term f is introduced for considering temperature gradient, as

(27)

n

i=1

2

Nf, i Np, i

The previous section confirms that all known models do not give reasonable conservative predictions for the thermal gradient mechanical fatigue tests, since effects of the thermal gradients are too strong to be considered in the conventional fatigue models. In fact, the thermal stress is induced by the temperature gradient and cannot be represented by the applied mechanical stress, which is the major difference to the TMF. To give a reasonable prediction of the TGMF life, the temperature gradient has to be included in the life model in a suitable way. The TMF model [11] seems to give the best agreement to both TMF and TGMF tests, but not conservative, and will be extended for TGMF in the present work. Based on the TMF model, a correction term relying on the temperature gradient is to be included on the left-hand side of Eq. (24). The fatigue life model for TGMF is suggested as

5.8. Fatigue life prediction based on the known fatigue models

1 E¯ = n

log

6. Introduction of a new TGMF life model

where Q0 is the intrinsic activation energy, 0 represents the intrinsic activation volume, ult is the ultimate stress at the maximum temperature and n (t ) is the normal stress of the critical plane. As given in [36], the activation energy Q0 is set to 240 kJ/mol and the intrinsic activation volume 0 is 3.51 × 10 4 m3/mol . The ultimate strength ult of Inconel 718 at 650 °C was measured as 1305 MPa in monotonic tensile tests. The coefficients k, B and C are determined from minimizing the life prediction mean square error. Finally, k equals 1.2, B is 0.49, and C is set to 1.29 × 1012 s−1.

TN = 10 E

n

The mean square root error is applied to measure the differences between fatigue lives from predictions and experiments. TRMS = 1 is the minimum and represents the agreement between experimental points and predictions. A larger TRMS value represents the more dispersion between the experimental life and prediction. A quantitative assessment of fatigue life prediction results is illustrated in Fig. 19(b). The results in the figures illustrate that besides TGMF model to be introduced in the next section, the TMF model introduced in [11] provides the best result.

(24)

with A defined as

A= 1+C

1 n

ERMS =

)

2 1 R

(29)

TRMS = 10 ERMS,

Based on the Liu’s virtual tension strain energy model, Sun and Yuan [11] introduced a fatigue model for the multi-axial TMF life prediction, as

(28)

illustrates the comparison of the mean value of experimental and predicted lives. Above n is the total number of the compared results, and the subscript i indicates the ith result. TN = 1 denotes zero discrepancy between experimental mean value and model predictions. TN > 1 means the predicted fatigue lives are conservative, and TN < 1 means the predicted fatigue lives are non-conservative. The statistical parameter

T =

11

T x

2

+

T y

2

+

T z

2

.

(33)

International Journal of Fatigue 135 (2020) 105486

J. Sun, et al.

Fig. 18. Comparison between predicted fatigue life and experimental results of thermo-mechanical as well as thermal gradient mechanical fatigue tests. (a) BrownMiller model; (b) Fatemi–Socie model; (c) Smith-Watson-Topper model; (d) Chu-Conle-Bonnen model; (e) Liu Tension Energy model; (f) TMF model [11].

For T = 0, f (T , T ) = 1, the TGMF model Eq. (31) reduces to the TMF model [11]. Table 5 lists the parameters of the proposed TGMF model, where is the angle of the critical nom /2 is the nominal strain amplitude, plane, loc and loc are the local stress and strain at point B in Fig. 13. T n,max is the maximum normal stress in the cycle on the critical plane, T n,max , T / y and T / r are the temperature and temperature gradients

corresponding to the maximum normal stress location. Fig. 20 illustrates the fatigue life assessment results of the TGMF tests. Fig. 20(a) shows fatigue life curves for all fatigue tests and confirms the rather unique correlation between the fatigue damage parameter and life for both TMF and TGMF. Effects of temperature gradients are considered in the fatigue model properly. In Fig. 20(b), one can observe that deviations of the proposed model (Eq. (31)) for most of the tests are within 12

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J. Sun, et al.

Fig. 19. The quantitative evaluation of the TGMF life prediction results: (a) TN , (b) TRMS .

the scatter band with a factor of 2. Only two tests reach the five-time scatter band. In Fig. 19 the assessment of the present TGMF model is compared with known fatigue models. In the figure two statistical parameters TN (Eq. (27)) and TRMS (Eq. (29)) are used to evaluate the conformity of the predicted and experimental TGMF lives. The results in Fig. 19 illustrate that the present TGMF model generates the best mean value in comparing with all other models, while the present TGMF model possesses the least scattering. Combining the two parameters, the present TGMF model provides the best result in predicting the fatigue life for all experiments.

Fig. 20. Computational results of the TGMF tests, based on the present TGMF model. (a) Fatigue damage parameter versus experimental fatigue lifetime, (b) Comparison of the predicted fatigue lifetime and experimental fatigue lifetime.

fatigue loading conditions. Effects of temperature gradients in combing with the phase angle were studied. The results can be extended to different temperature regions. The following conclusions can be drawn from the present work:

• A new radiation heating system with automatic compensation is

developed and used for study TGMF in high temperature materials. The finite element computations reveal that the temperature gradient in a simple tubular specimen is not constant and varies in the whole specimen. Local temperature distribution can induce significant deviations in the strain distribution.

7. Conclusions The present paper experimentally investigated fatigue and its assessment for the nickel-based superalloy Inconel 718 under 300 650 °C thermo-mechanical and thermal gradient mechanical Table 5 Stress, strain, temperature and temperature gradient on the material plane. Load Type

nom /2

loc

T n,max

Nf

Np

[cycle]

[cycle]

0.092 0.092 0.092 0.092 0.092

1066 208 107 50 48

552 126 102 78 39

0.044 0.044 0.044 0.044

3387 864 375 128

11784 1637 697 26

T/ y

T/ r

[°C]

[°C/mm]

[°C/mm]

0.85 0.99 1.29 1.28 1.87

650 650 650 650 650

−11.6 −11.6 −11.6 −11.6 −11.6

0.78 1.02 1.42 2.00

300 300 300 300

−6.5 −6.5 −6.5 −6.5

loc

[%]

[° ]

[MPa]

[%]

TGMF-IP

0.425 0.50 0.60 0.70 0.80

0 0 0 0 0

1058.4 1298.3 1827.8 1758.6 1917.3

TGMF-OP

0.425 0.50 0.60 0.80

0 0 0 0

1562.6 1680.5 1834.4 1981.1

13

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J. Sun, et al.

• Experimental results show that the temperature gradients change

• • • •

fatigue performance of the nickel-based alloy significantly and cannot be identified to thermo-mechanical fatigue. The phase angle additionally affects the fatigue life assessment. The fractographic analysis reveals different failure mechanisms depending on TGMF loads. Conventional isothermal fatigue life models and TMF life models fail to account for TGMF influence and do not provide a satisfactory life prediction. Maximum deviations are larger than 10-times of the fatigue life. Effects of temperature gradients have to be considered in the fatigue model. Temperature gradients may introduce significant additional strains into the material. The mean stress in TMF tests varies with loading cycles and develops towards lower temperature loads. The variation of the mean stress makes it necessary to include characteristic features of the TGMF into the fatigue life model. A modified TGMF life model is proposed to consider influences from the temperature gradient, by introducing a correction term for the TMF loads. The modified TGMF model provides a significantly better agreement to the performed experiments and gives a uniform prediction for both TMF and TGMF fatigue. Further detailed TGMF experimental and computational investigations are necessary to understand the fatigue mechanisms, to quantify effects of temperature distributions and to establish a reliable life prediction model for mechanical parts under complex thermo-mechanical loading conditions.

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[25]

[26]

Acknowledgment

[27]

The present work is financed by the National Natural Science Foundation of China under the contract numbers 11572169 and 51775294.

[28] [29]

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