Fatigue Behaviour and Life Prediction of YSZ Thermal Barrier Coatings at Elevated Temperature under Cyclic Loads

Abstract

The concentration of interfacial normal stress at the free edges of thermal barrier coatings (TBCs) can result in coating spallation. Fatigue cracking is one of the main reasons for creating free edges under complex loads. It is crucial to investigate the fatigue cracking of coatings under cyclic loads to assess potential coating failure. To address this issue, a novel model was proposed to predict the fatigue life of the YSZ topcoat under stress parallel to the interface. Firstly, this study conducted uniaxial and tensile-torsional fatigue tests at elevated temperatures on specimens with atmospheric plasma-sprayed TBCs. The test results revealed that fatigue cracks appeared in the topcoat under cyclic loads, but these cracks did not propagate into the bondcoat or substrate immediately. The number of cycles before the topcoat cracked was found to be associated with the magnitude of the cyclic load. Secondly, this study analyzed the test conditions using the finite element method. Simulations indicated that the crack direction in the topcoat under complex loading conditions was aligned with the first principal stress direction. Finally, the fatigue life prediction model of the topcoat was established based on experiments and simulations. The predicted results fell within a fourfold scatter band.

1. Introduction

With the development of gas turbines to achieve better performance, the operating temperature requirements of turbine blades are increasing. The adoption of film-cooling technology and the preparation of thermal barrier coatings on the superalloy substrate are widely used for turbine blades to address these increased operating temperatures [1,2].

The widely-used thermal barrier coatings consist of a ceramic (7–8 wt% Yttria Stabilized Zirconia, i.e., YSZ) topcoat, and an MCrAlY bondcoat, which can be prepared by the atmospheric plasma spraying method (APS), the electron-beam physical vapor deposition method (EB-PVD), or the high-velocity oxygen-fuel method (HVOF) [3,4]. Multi-layer structures are prone to cracking and spalling under complex loads. The large-scale spallation of thermal barrier coatings can lead to the failure of cooling systems, which ultimately leads to turbine blade failures [5]. Therefore, evaluating the life of thermal barrier coatings is important for analyzing the life of turbines in gas turbine engines.

The primary task in analyzing the durability of coatings is to understand the underlying causes of coating failure. Coating failure mechanisms are complex [6]. The most prevalent failure mode is delamination due to strain mismatch [7]. This strain disparity between the coating and the substrate creates internal stresses within the coating. Significant shear and peel stresses are present at the edges of the coating or crack sites perpendicular to the coating interface. These interfacial stresses often serve as the initial trigger for delamination in multilayer structures under strain mismatch conditions [8,9,10], as shown in Figure 1. Several factors contribute to coating cracking, such as thermal cycling, which induces cracking due to cyclic thermal stresses [11]. Additionally, at elevated temperatures, plastic deformation and creep strain of the substrate may lead to coating cracking [12]. Moreover, cyclic mechanical stresses can also be responsible for coating cracks [13]. Nonetheless, there is limited research on the fatigue failure process of coatings on substrates under high temperatures, and there is also a scarcity of studies regarding the quantitative relationship between the magnitude of cyclic loads and the fatigue life of coatings.

In contemporary practice, prevalent fatigue life models encompass stress-based formulations (e.g., the Basquin model [14]), strain-based representations (e.g., the Manson–Coffin model [15]), and energy-based approaches [16]. Utilizing stress as a damage parameter to estimate fatigue life presents several advantages. First, it eliminates the necessity to differentiate between thermal strain and mechanical strain, making it especially convenient for thermal barrier coatings which subjected to thermal cycles. Second, by analyzing structural stress, fatigue life predictions can be made for areas experiencing asymmetric cyclic loads and significant stress gradients. This could be accomplished by introducing a parameter that is related to the local stress ratio and the local stress gradient. Consequently, stress-based fatigue life models are widely used in the fatigue life analysis of engineering structures. Typical stress-based fatigue life models include the Wöhler model [17] and the Basquin model [14].

Due to the complex shapes of actual coated structures, additional factors such as the effects of stress concentration and multi-axial loadings must be considered while predicting the fatigue life. Stress concentrations can occur in the coating near the cooling holes on the leading edges of turbine blades, which might greatly affect the fatigue strength of components [18]. Apart from the structural characteristics that cause stress concentration, the coating process introduces pores that also contribute to stress concentration within the material [19,20]. The stress concentrations induced by the pores in the materials can have nonnegligible effects on the mechanical properties of the structure [21]. Therefore, using life prediction models established for standard specimens to predict the fatigue life of structures with stress gradients is often overly conservative [22,23]. Under multi-axial loading conditions, the stress-strain response of materials becomes significantly more complex, affecting both the initiation and propagation paths of cracks. Then, a single fatigue life prediction model may not provide accurate results. Therefore, it is essential to modify fatigue analysis methods that account for the effects of multi-axial loads [24,25].

The work presented here intends to analyze the fatigue fracture of APS thermal barrier coatings under cyclic loads parallel to the coating interface and to establish a mathematical model to predict the fatigue life of the coatings. A series of fatigue tests were conducted on the superalloy specimens covered with thermal barrier coatings. The experimental process was monitored and parts of the specimens were investigated by the scanning electron microscope after tests. Finite element calculations were also performed to study the formation of the fatigue cracks. Based on the analysis of the above investigation, the fatigue life prediction model was proposed. Then the prediction results obtained from the proposed model were compared to the experimental results to validate the model’s accuracy.

2. Specimen Preparation and Experimental Procedure

2.1. Specimen Preparation

The substrate material of the specimens used for fatigue tests was directionally solidified alloy DZ411. The specimens were manufactured from 16 mm diameter bars. The nominal chemical composition of DZ411 alloy is listed in

Table 1. Chemical composition of DZ411 (wt%).

Element	C	Cr	Co	Ni	W	Mo	Al	Ti	Ta	B
Content (%)	0.07∼0.12	13.5∼14.3	9.0∼10.0	rest	3.5∼4.1	1.3∼1.7	2.8∼3.4	4.6∼5.2	2.5∼3.1	0.007∼0.02
Element	Si	P	S	Pb	Bi	As	Sn	Sb
Content (%, ≤)	0.2	0.005	0.01	0.0005	0.0001	0.005	0.002	0.001

. The standard heat treatment regime of castings was as follows: solution treatment 1225 °C ± 10 °C, 2 h, air cooling; single aging, 1120 °C ± 10 °C, 2 h, air cooling; secondary aging, 850 °C ± 10 °C, 24 h, air cooling.

The specimens included uniaxial fatigue bars, uniaxial fatigue tubes with a radial hole, and one multiaxial fatigue tube with a radial hole. The diameter of the effective section of uniaxial fatigue bars was 6 mm. The length of the effective section was 12 mm. The outer diameter of the effective section of uniaxial fatigue tubes was 10 mm, and the inner diameter was 8 mm. Thus the thickness of the tube wall was 1 mm. The length of the effective section was 20 mm. The outer diameter of the effective section of multiaxial fatigue tubes was 12 mm, and the inner diameter was 10 mm. Thus the thickness of the tube wall was 1 mm. The length of the effective section was 30 mm. The radial holes in the tubes were set to simulate the cooling holes in the leading edge of the turbine blade. The diameter of the radial hole was 1 mm or 2 mm. The thermal barrier coatings covered the gauge section and the transition arc section, extending to the threaded position connecting to the fixture, while the multi-axis test specimen was sprayed up to the clamping section.

The bondcoat was prepared using the HVOF method, and consisted of MCrAlY powder with the following weight percentage composition: 10.5% Al, 18.6% Cr, 32.6% Co, 37.2% Ni, and 0.4% Y. The ceramic topcoat was made of 8% yttria-stabilized zirconia (8YSZ) and was applied using the Atmospheric Plasma Spraying (APS) method, with the weight percentage composition of 24.5% O, 67.5% Zr, and 8% Y. For coated bars, a 400 $\mathsf{\mu }$m thick topcoat and a 250 $\mathsf{\mu }$m thick bondcoat were prepared, while for thin-walled tubes, a 200 $\mathsf{\mu }$m thick topcoat and a 125 $\mathsf{\mu }$m thick bondcoat were applied. Geometries and images of different specimens are shown in Figure 2, Figure 3 and Figure 4. The regions with fill patterns are the position where the thermal barrier coatings cover.

2.2. Test Equipment

The fatigue tests at elevated temperatures were performed using the Instron 8852 servo-hydraulic system and an electric furnace with a capacity of 1000 °C. The fatigue testing process was recorded by a wide-range crack measuring instrument through the observation window of the furnace. This system allows for the capture of coating cracks at 100× magnification. Test equipment is shown in Figure 5a and the example of images obtained by the cracking measuring instrument is shown in Figure 5b.

For the uniaxial fatigue tests, specimens were connected to the fixtures via threaded ends, and an axial load was applied to the coated specimens through a hydraulic servo control system. In the tensile-torsional multiaxial fatigue tests, both axial force and torque were simultaneously transmitted to the specimens through the fixtures. The tests were load-controlled. The loading waveforms were in sinusoidal form at a frequency of 1 Hz.

3. Fatigue Experimental Procedure and Results

3.1. Uniaxial Fatigue Test of Rods

This study primarily focuses on the thermal barrier coatings to turbine blades, with a particular emphasis on their performance under high temperatures and centrifugal loads. The fatigue tests in this section were carried out at elevated temperatures of 800 °C and 870 °C, similar to the temperature in partial areas of turbine blades used in heavy gas turbines. The uniaxial fatigue tests with a stress ratio of 0.05 were conducted to simulate the loading and unloading processes of centrifugal loads. The load-controlled uniaxial fatigue tests were conducted on the DZ411 rods with TBCs. The stress ratio was 0.05. The detailed test parameters are listed in

Table 2. Uniaxial fatigue test results of rods with thermal barrier coatings.

Number	Temperature (°C)	Max Load (kN)	Life of Topcoat (Cycle)
FAB-800-1040-1	800	29.41	1360
FAB-800-1020-1		28.84	1400
FAB-800-1000-1		28.27	1600
FAB-800-980-1		27.71	3600
FAB-800-980-2			800
FAB-800-960-1		27.14	1700
FAB-800-920-1		26.01	2100
FAB-800-800-1		22.62	5600
FAB-870-840-1	870	23.75	900
FAB-870-840-2			1
FAB-870-820-1		23.18	1050
FAB-870-800-1		22.62	1440
FAB-870-770-1		21.77	840
FAB-870-770-2			2700
FAB-870-760-1		21.49	1700
FAB-870-730-1		20.64	4700
FAB-870-720-1		20.36	2100
FAB-870-660-1		18.66	5900
FAB-870-600-1		16.97	15,000

. The type of the uniaxial fatigue tests was defined as FAB in number rules, and the reference stress was defined as dividing the axial load by DZ411’s sectional area. Repeated tests under the same load were numbered consecutively. The number rule is the FAB-temperature-reference stress-number of the repeated test. At 100× magnification, the number of cycles in which cracks on the topcoat can be observed by the naked eye was taken as the fatigue life of the coating. The results are listed in Table 2 and are plotted in Figure 6.

The cracks opened and closed more evidently while the cyclic loading acted on the specimen. A single circumferential crack usually appeared first, and then the number of cracks increased as the number of cycles increased. Take specimens FAB-800-1020-1 and FAB-870-760-1 as examples, the images of specimens obtained during the experimental procedure are shown in Figure 7 and Figure 8.

Before the cross-sectional cut, the specimen was cold-mounted in epoxy resin to prevent the additional damage to the coating. The cylindrical specimen was cut along its axis with a water knife. The prepared sample is shown in Figure 9a. The scanning electron microscope (SEM) was performed on the sample and the image is shown in Figure 9b. In the SEM image, cracks were observed in the ceramic layer, extending from the top surface down to the interface with the bondcoat. However, these cracks did not propagate into the bondcoat. The interface between the topcoat and the bondcoat remained connected.

After the substrate fractured, as illustrated in Figure 10, the bondcoat was separated from the substrate near the fracture.

A high porosity is a typical characteristic of APS ceramic topcoat, and such microstructural features significantly reduce the fatigue resistance of the ceramic layer. Moreover, the distribution of pores in the ceramic layer is irregular and difficult to predict, leading to considerable variability in the results of tensile fatigue tests. For example, the evident crack can be observed in the specimen FAB-870-840-2 during the initial stage of the loading procedure, as shown in Figure 11.

3.2. Uniaxial Fatigue Test of Tubes

Fatigue tests were also conducted on coated DZ411 tubes each with a single radial hole (shown in Figure 3) under similar uniaxial test conditions. The radial holes were designed to simulate the cooling holes on the turbine blades. Two levels of loads were selected at two temperatures for specimens with radial holes of different diameters, as detailed in

Table 3. Uniaxial fatigue test results of tubes with thermal barrier coatings.

Number	Temperature (°C)	Max Load (kN)	Diameters of Radial Hole (mm)	Life of Topcoat (Cycle)
FAT-800-810-H1-1	800	22.90	1	100
FAT-800-810-H2-1			2	1
FAT-800-630-H1-1		17.81	1	300
FAT-800-630-H2-1			2	60
FAT-870-600-H1-1	870	17.36	1	270
FAT-870-600-H2-1			2	90
FAT-870-450-H1-1		12.72	1	1380
FAT-870-450-H2-1			2	1300

. The specimens were designated as ‘FAT’. Throughout the tests, the experimental process at elevated temperatures was monitored through the observation window of a high-temperature furnace, by a wide-range crack measurement instrument. The time at which visible cracks in the coating were first detected was recorded as the coating’s fatigue life. The test data results are presented in Table 3.

During the experiment, it was observed that cracks in the topcoat consistently appeared at the edge of the radial hole. The crack direction was oriented at a 90° or −90° relative to the axial direction of the cylindrical tube. Define the sign of angles: a clockwise direction is considered positive, while a counterclockwise direction is considered negative. Taking specimens FAT-800-630-H2-1 and FAT-870-600-H1-1 as examples, as illustrated in Figure 12 and Figure 13. The coated tube which was tested until the substrate fractured, is shown in Figure 14. The coatings on the tube were relatively thin, and no significant delamination between the substrate and the bondcoat was observed after fracturing.

3.3. Multiaxial Fatigue Test of Tubes

To investigate the failure process of the coating around blade holes under complex loading conditions, a DZ411 tube with a radial hole was designed as shown in Figure 4. An axial-torsional cyclic fatigue test was conducted at a controlled load of 800 °C.

In the experiment, axial force and torque were controlled. The load ratio was −1. The frequency was 1 Hz. The maximum axial load and the torque were 22.98 kN and 84.2 N/m, respectively. The loading waveform was a sinusoidal wave and the loading path followed a circular trajectory with a 90° phase difference, as shown in the load spectrum (Figure 15). The post-fracture specimen is depicted in Figure 16, where no significant delamination between the substrate and the bondcoat was observed. After 30 cycles, cracks appeared in the topcoat around the hole. Unlike under uniaxial loading conditions, the direction of crack propagation was not at 90° or −90° but at −70° and 110°, as depicted in Figure 17.

4. Finite Element Analysis

4.1. Computation Method

The results of uniaxial fatigue tests on coated circular rods show that the predominant failure mode of the coating is the occurrence of circumferential cracks in the ceramic layer with increasing cycles. Therefore, attention needs to be focused on the axial stress within the ceramic layer under axial loading. Firstly, it is necessary to calculate the axial stress within the topcoat corresponding to different axial loads. The material parameters used for the calculations are presented in

Table 4. The material parameters of YSZ, Al₂O₃, MCrAlY and DZ411 [26,27,28,29].

	Property	Temperature (°C)	Elastic Modulus (GPa)	Poisson’s Ratio	Thermal Expansion Coefficient (10−6K−1)	Density (kg·m−3)	Specific Heat Capacity (J·kg−1·K−1)
Material
YSZ (TC) [26,27]		20	17.5	0.2	blank	5200	437
		220	16.3
		420	15.2
		620	14.0
		820	12.9
		1020	11.7
Al2O3 (TGO) [28]		22	386.0	0.257	6.0	3978	857
		566	347.6		8.0
		1149	312.1		8.9
MCrAlY (BC) [28]		22	137.9	0.27	15.16	7320	501
		566	124.0		15.37
		1149	93.8		17.48
DZ411 (sub) [29]		25	129.9	0.3	11.9	8344	469
		100	128.0		11.9		474
		200	126.0		12.4		482
		300	123.0		12.6		491
		400	118.0		12.9		501
		500	114.0		13.2		511
		600	110.0		13.6		522
		700	106.0		14.0		534
		800	101.0		14.5		547
		900	95.0		15.0		561
		1000	86.0		15.6		575

and

Table 5. The thermal expansion coefficient of 8YSZ prepared by APS.

Temperature (°C)	50	100	150	200	250	300	350	400	450	500	550
Themal Expansion Coefficient (${\mathbf{10}}^{-\mathbf{6}}{\mathbf{K}}^{-\mathbf{1}}$)	12.16	11.81	11.41	12.08	11.88	11.53	11.66	11.68	11.66	11.66	11.66
Temperature (°C)	600	650	700	750	800	850	900	950	1000	1050	1100
Themal Expansion Coefficient (${\mathbf{10}}^{-\mathbf{6}}{\mathbf{K}}^{-\mathbf{1}}$)	11.43	11.51	11.46	11.41	11.41	11.33	11.28	11.21	11.18	10.96	10.66

, including elastic modulus, Poisson’s ratio, density, thermal expansion coefficient, and specific heat capacity. In the calculations, the anisotropy of the coating and substrate was disregarded, and the materials were treated as isotropic.

The thermal stress in the coating-substrate structure is considerable, greatly influencing the stress experienced by the coating. The methodology used to calculate thermal strain is outlined in reference [10].

$${\epsilon }_{\mathrm{T}}={\alpha }_{\mathrm{r}}\left(T\right)(T-T_{\mathrm{r}})$$

(1)

Namely, the thermal strain ${\epsilon }_{\mathrm{T}}$ at temperature T is the product of the relative thermal expansion coefficient and the temperature difference. The temperature difference is defined as the difference between the current temperature T and the reference temperature $T_{\mathrm{r}}$.

The reference temperature in the thermal expansion coefficient measurement experiment is denoted as $T_0$. The thermal expansion coefficient at temperature T is denoted as ${\alpha }_0\left(T\right)$. When the reference temperature is set to $T_{\mathrm{r}}$, the relative thermal expansion coefficient at temperature T could be calculated with Equation (2).

$${\alpha }_{\mathrm{r}}\left(T\right)={\alpha }_0\left(T\right)+{\displaystyle \frac{T_{\mathrm{r}}-T_0}{T-T_{\mathrm{r}}}}\left({\alpha }_0\left(T\right)-{\alpha }_0\left(T_{\mathrm{r}}\right)\right)$$

(2)

Next, consider the effects of the preparation process on the reference temperature. The temperature is $T_{\mathrm{c}}$ before the powder is sprayed, and the substrate is preheated to $T_{\mathrm{s}}$. Then, the equilibrium temperature expression for the coating-substrate system is expressed as Equation (3).

$$T_{\mathrm{b}}={\displaystyle \frac{{\rho }_{\mathrm{C}}{h}_{\mathrm{C}}{c}_{\mathrm{C}}{T}_{\mathrm{c}}+\left({\rho }_{\mathrm{B}}{h}_{\mathrm{B}}{c}_{\mathrm{B}}+{\rho }_{\mathrm{S}}{h}_{\mathrm{S}}{c}_{\mathrm{S}}\right)T_{\mathrm{s}}}{{\rho }_{\mathrm{C}}{h}_{\mathrm{C}}{c}_{\mathrm{C}}+{\rho }_{\mathrm{B}}{h}_{\mathrm{B}}{c}_{\mathrm{B}}+{\rho }_{\mathrm{S}}{h}_{\mathrm{S}}{c}_{\mathrm{S}}}}$$

(3)

where $\rho$, h, c are the density, thickness and specific heat capacity, respectively. The subscript C, B and S denote the topcoat, the bondcoat and the substrate, respectively. Assuming that the multilayered material is stress-free at the equilibrium temperature $T_{\mathrm{b}}$, the reference temperature $T_{\mathrm{r}}$ can be obtained, i.e.,

$${\epsilon }_{\mathrm{T},\mathrm{C}}\left(T_{\mathrm{b}}\right)={\epsilon }_{\mathrm{T},\mathrm{B}}\left(T_{\mathrm{b}}\right)={\epsilon }_{\mathrm{T},\mathrm{S}}\left(T_{\mathrm{b}}\right)$$

(4)

The stress in the topcoat can be calculated using the finite element method. The element type used in the finite element calculations was Solid185. The interfaces between different materials share the nodes. A symmetric model was created to reduce the number of the mesh. The nodes on the side surface at one end were coupled along the axial direction, and a virtual node $N_{\mathrm{v}}$ was placed 5 mm away from the side surface as the master node. The axial force $F_{\mathrm{z}}$ was applied to the virtual node using the local stiffening method, as depicted in Figure 18.

4.2. Analysis of Computational Results

The calculated axial stress in the topcoat and the number of cycles until the topcoat cracks obtained from experiments, are plotted in Figure 19. The horizontal axis represents the number of cycles until the topcoat cracks, while the vertical axis represents the max axial stress in the topcoat. Both axes use a logarithmic scale. Combining that the load ratio is 0.05, the axial stress amplitude in the topcoat during one cycle can also be calculated. The stress amplitude and the number of cycles until the topcoat cracks are plotted in Figure 20.

The temperature difference between the two experimental conditions was 70 °C. It is assumed that the temperature difference has almost no influence on fatigue performance. The primary distinction lies in the thermal stress. Therefore, the stress ratio $R_{\sigma }$ varies. The stress ratio can be determined using the stress amplitude ${\sigma }_{\mathrm{a}}$ and the maximum stress value ${\sigma }_{\max }$. The relationship between the stress ratio and the fatigue life is shown in Figure 21.

The failure in the coated tubes with a radial hole primarily occurred at the hole edge. The circumferential stress at the hole edge caused cracking in the topcoat. Therefore, the focus in this section is the circumferential stress. The outer radius of the DZ411 tube was 5 mm, and the inner radius was 4 mm. The thickness of the bondcoat was 125 $\mathsf{\mu }$m, and the thickness of the topcoat was 200 $\mathsf{\mu }$m. An axial force of 17.36 kN was applied to the coated tube under 870 °C. The calculation domain length was 20 mm, which was equal to the gauge length of the specimen. Two models were considered, with radial hole diameters of 1 mm and 2 mm, labeled as H1 and H2, respectively.

The distribution of circumferential stress in the topcoat near the hole edge in the two models is shown in Figure 22. The axial direction of the radial hole is $z_1$ axis, and the circumferential stress is along the circumferential direction of the radial hole. Additionally, the circumferential stress in the topcoat near the hole edge at 870 °C without applying an axial force was calculated, as shown in Figure 23. The maximum circumferential stress near the hole edge occurred in the direction perpendicular to the axial direction of the tube whether with or without the axial load, i.e., at 90° or −90° around the hole edge. The difference lies in the variation of the maximum stress locations. When an axial load was applied, the point of maximum circumferential stress was located between the upper and lower surfaces of the topcoat at the hole edge. As the hole diameter increased, the net cross-sectional area of the tube decreased, increasing the circumferential stress. Without axial load, only thermal stress existed. The maximum circumferential stress at the hole edge was located at the outer surface of the topcoat. As the hole diameter increased, the circumferential stress at the hole edge slightly decreased.

Subsequently, the stress near the hole edge under multi-axial loading at 800 °C was calculated. Figure 24 shows the variation in the first principal stress distribution in the topcoat near the radial hole during the loading process. Throughout one cycle, both the location and value of the maximum first principal stress point change continuously. In subfigure ➀ in Figure 24, the distribution corresponds to the uniaxial tensile state. The clockwise direction is positive. The location of the maximum value point is at −90°. The changes in location and value of the maximum first principal stress point near the hole edge in the topcoat during the loading process are depicted in Figure 25.

Considering that the structure and the load are symmetry, to avoid the step changes in the position of the maximum first principal stress point caused by numerical calculations, the position of the maximum first principal stress point was considered within ±90° from the 0° position.

The direction of the first principal stress at the maximum stress point is consistent with the tangential direction at the hole edge at that location. The direction of the first principal stress at the maximum stress point was consistent with the tangential direction of the hole edge at the maximum stress point. During one loading cycle, the maximum first principal stress point was located at approximately −68° (as indicated by the red circle in Figure 25), which is almost the same as the actual cracking position of the specimen (−70°, as indicated by the white line in Figure 17). The variation of the first principal stress at the point with the red circle is also plotted in Figure 25.

5. Proposal of Fatigue Life Model

The opening direction of fatigue cracks in the topcoat aligns with the direction of the first principal stress [30]. Therefore, a fatigue life model using the first principal stress as the damage parameter was proposed. Initially, a fatigue life model for YSZ, considering the influence of the stress ratio, was established using fatigue test data from smooth specimens. The form of the fatigue life model based on the equivalent stress ${\sigma }_{\mathrm{eq}}$ is as follows,

$$\lg \left(N_{\mathrm{f}}\right)=A_1+A_2\lg ({\sigma }_{\mathrm{eq}}-A_4)$$

(5)

where $N_{\mathrm{f}}$ means fatigue life, i.e., the cycle number when the topcoat cracks. The equivalent stress ${\sigma }_{\mathrm{eq}}$ is related to the stress ratio $R_{\sigma }$. The expression of the equivalent stress is,

$${\sigma }_{\mathrm{eq}}={\sigma }_{\max }{(1-R_{\sigma })}^{A_3}$$

(6)

where $A_i$ is the fitting parameter.

The calculated stress ratio of the test data was concentrated between 0.3 and 0.45. Thus, the parameter $A_3$ related to the stress ratio was set to 0.5. During the fitting process, the parameter $A_3$ was not included in the least squares regression analysis. The next step was setting an initial value for $A_4$. Let

$$A_4={\displaystyle \frac{\min \left({\sigma }_{\mathrm{eq}}\right)}{2}}$$

(7)

the ${\sigma }_{\mathrm{eq}}$ can be obtained from Equation (6). Then, the linear least squares regression analysis on Equation (5) was performed. The fitting results for parameters $A_1$ and $A_2$ were then used as initial values for the nonlinear regression analysis of Equation (5) while $A_4$ was uncertain.

Following this procedure, the test data obtained at 870 °C (excluding the data of the specimen FAB-870-840-2 which cracked immediately upon loading) were used for obtaining parameters by fitting based on Equations (5) and (6). This yielded the parameters in the fatigue life model of the topcoat under stress parallel to the interface, as listed in

Table 6. The fitting results of parameters in the fatigue life model of the topcoat under stress parallel to the interface.

Parameter	A1	A2	A3	A4
Value	4.7926	−1.1892	0.5	70.8073

.

After completing the parameter fitting, the Durbin–Watson test was used to evaluate the validity of the proposed fatigue life model [31,32]. The test statistic $D_{\mathrm{m}}$ is given.

$$D_{\mathrm{m}}={\displaystyle \frac{{\sum }_{i=2}^n{({\delta }_{\mathrm{SR},i}-{\delta }_{\mathrm{SR},i-1})}^2}{{\sum }_{i=1}^n{\delta }_{\mathrm{SR},i}^2}}$$

(8)

where n refers to the number of the data points and ${\delta }_{\mathrm{SR},i}$ represents the standardized residual of the i-th data point sorted by increasing equivalent stress. The expression for ${\delta }_{\mathrm{SR},i}$ is as follows:

$${\delta }_{\mathrm{SR},i}={\displaystyle \frac{\left|\lg N_{\mathrm{f},\mathrm{e}}-\lg N_{\mathrm{f},\mathrm{p}}\right|}{{\delta }_{\mathrm{SD}}}}$$

(9)

${\delta }_{\mathrm{SR},i}$ is the ratio of the absolute deviation of the logarithmic predicted fatigue life from the logarithmic experimental fatigue life to the standard deviation. The expression for the standard deviation ${\delta }_{\mathrm{SD}}$ is as follows:

$${\delta }_{\mathrm{SD}}=\sqrt{\sum _{i=1}^n{\displaystyle \frac{{[\lg N_{\mathrm{f},\mathrm{e}}-\lg N_{\mathrm{f},\mathrm{p}}]}^2}{n-k_n}}}$$

(10)

where $k_n$ denotes the number of fitting parameters. Since parameter $A_3$ is not included in the fitting process, in this case, $k_n=3$. Substituting the data, the value of the statistic $D_{\mathrm{m}}$ is 0.8190. If $D_{\mathrm{m}}<2-4.73/n^{0.555}$, the model would be deemed unsuitable. In this case, the right-hand side of the inequality yields 0.6822, indicating that the model can describe the stress-fatigue life relationship of the YSZ topcoat undergoing stress parallel to the interface.

Using the load data at 800 °C, calculations based on the obtained material model were performed, and a comparison of experimental and predicted fatigue life is shown in Figure 26.

Eighty percent of the predicted lifetimes fell within the twofold scatter band, and one predicted lifetime fell outside the threefold scatter band. However, the one outside the threefold scatter band was lower than the actual experimental lifetime, indicating a conservative prediction. Most of the predicted lifetimes at 800 °C were conservative. It is likely due to the temperature being lower than the temperature corresponding to the data used for fitting the model parameters.

Under complex loading conditions, the effects of stress gradients and multiaxial loading on the coating’s life must also be considered [33,34]. Firstly, the stress distribution of circumferential stress around the hole edge in the coated tube with a radial hole was analyzed. The circumferential stresses for axial loading of 17.36 kN at 870 °C are provided as an example. The extraction path of the stress is shown in Figure 27, starting from the position where the max stress was located, taking along the circumferential direction of the tube. The circumferential stress-to-distance from the hole edge curve and the normalized circumferential stress-to-distance from the hole edge curve are shown in Figure 28a and Figure 28b, respectively. The normalized stress ${\sigma }_{\mathrm{H}}/{\sigma }_{\mathrm{H}},\max$ represents the ratio of the stress at different points along the path to the maximum stress at the hole edge. The normalized distance $x/r_{\mathrm{H},\max }$ represents the ratio of the distance from different points along the path to the hole edge to the notch characteristic parameter. In this case, the characteristic parameter is the hole radius.

According to reference [34], the stress gradient impact factor Y is defined as the reciprocal of twice the area enclosed by the normalized stress curve and the horizontal axis over the interval [0, 0.5], i.e.,

$$Y={\displaystyle \frac{1}{2S_{0.5}}}$$

(11)

where, $S_{0.5}$ represents the integral area of the partial normalized curve. The stress gradient impact factor was 1.04 when the hole diameter was 1 mm, and the factor was 1.43 when the hole diameter was 2 mm. The hole diameter has an insignificant effect on the stress gradient impact factor. The stress-fatigue life model considering stress gradients is the same as Equation (5), but the expression of effective stress must include the stress gradient impact factor. The specific expression is as follows.

$${\sigma }_{\mathrm{eq}}=Y^{-m}{\sigma }_{\max }{(1-R_{\sigma })}^{A_3}$$

(12)

where the stress gradient index factor m is related to the maximum stress. Considering the dimensionless nature of m, the relationship between m and ${\sigma }_{\max }$ is expressed as follows.

$$m=A_m{\left({\displaystyle \frac{{\sigma }_{\max }}{{\sigma }_{\mathrm{b}}}}\right)}^{B_m}$$

(13)

where ${\sigma }_{\mathrm{b}}$ is the ultimate strength of the YSZ coatings, taken as 130 MPa in this case. Parameters $A_m$ and $B_m$ are fitting parameters, with fitting results $A_m=6.286$ and $B_m=-3.508$.

Finite element analysis was performed on a coated tube with a radial hole under uniaxial fatigue testing conditions, yielding the maximum stress and stress ratio listed in

Table 7. The max circumferential stresses at the hole edge, and the stress ratio, as well as the corresponding life prediction results.

Temperature (°C)	Hole Diameter (mm)	Max Load (kN)	Max Stress (MPa)	Stress Ratio	Predicted Life (Cycle)	Test Life (Cycle)
800	1	22.90	288.44	0.201	153	100
800	1	17.81	233.37	0.237	317	300
870	1	17.36	240.51	0.277	297	270
870	1	12.72	190.16	0.338	2018	1380
800	2	22.90	310.61	0.169	126	1
800	2	17.81	249.90	0.198	238	60
870	2	17.36	255.47	0.232	230	90
870	2	12.72	199.63	0.283	1015	1300

. The predicted life and experimental life are also listed in Table 7, with their comparison shown in Figure 29.

At 800 °C, the tube with a radial hole whose diameter was 2 mm under an axial load of 22.90 kN exhibited significant coating cracking at the hole edge during the initial cycle. The circumferential stress at the hole edge for this tube was the highest among the eight test conditions, which correlated with its earliest failure during testing. Prediction results obtained from the proposed model indicate that the life of the tube with a radial hole whose diameter was 2 mm was shorter, consistent with the experimental findings. The remaining predictions are fairly accurate, with a dispersion band within a factor of two.

According to the conclusion in Section 4, the direction of the first principal stress is the same as the actual cracking position. Thus, under complicated loading conditions, the first principal stress is suitable as a damage parameter. Under multiaxial conditions, considering the effect of non-proportional loading on life, the equivalent stress for life estimation is expressed as follows.

$${\sigma }_{\mathrm{eq}}=(1+{\alpha }_{\mathrm{m}}{\Phi }_{\mathrm{m}})Y^{-m}{\sigma }_{\max }{(1-R_{\sigma })}^{A_3}$$

(14)

where ${\alpha }_{\mathrm{m}}$ represents the material’s non-proportional strengthening coefficient under multiaxial loading, reflecting the material’s sensitivity to additional non-proportional strengthening, and was taken as 0.5 for life estimation. ${\Phi }_{\mathrm{m}}$ denotes the load non-proportionality, reflecting the impact of loading history and path on material reinforcement. In this case, it was a circular path loading with a phase difference of 90°, and the load non-proportionality was taken as 1. During calculations, the parameter was taken as the stress at the point of maximum principal stress value during loading, which was 246.91 MPa. According to the cyclic stress shown in Figure 15, the stress ratio was 0.03. The principal stresses along the radial path from the point of maximum principal stress were extracted for normalization to obtain the stress gradient impact factor, which was 1.45. The predicted life was calculated to be 104 cycles, approximately 3.5 times the experimental life, as shown in Figure 29.

Combining the life prediction results of tubes with holes under uniaxial conditions, the process outlined in this section shows that the life predictions under high-stress conditions align with the experimental results. Despite our efforts to emulate actual turbine blade loading conditions, it should be noted that certain factors, such as temperature gradients, creep effects, oxidation effects and thermal-mechanical cycles, were not fully considered in this study. However, the computational methods and models proposed in this study aim to provide methods for these scenarios. Additionally, further investigation may be necessary to fully comprehend the fatigue performance of thermal barrier coatings under extremely high temperatures (such as temperatures exceeding 1000 °C).

6. Conclusions

This study outlines uniaxial and tensile-torsion fatigue tests carried out at 800 °C and 870 °C on high-temperature alloy rods and tubes coated with thermal barrier coatings. The tests aimed to determine the number of cycles before coating cracked under different loads, along with the location and orientation of the cracks. A fatigue life model was proposed for coating cracks under cyclic stress parallel to the coating interface based on the observed experimental phenomena. Combining the finite element analysis, the model predicted the fatigue life of the coating under experimental loads, and comparison results confirmed the model’s accuracy. The main conclusions are as follows:

(1)

The study revealed that under cyclic stress, the ceramic topcoat on the high-temperature alloy might experience fatigue cracking before the substrate fractures. Specimens that did not fracture were embedded in epoxy resin and sectioned axially. The SEM images of the sectioned specimen revealed that cracks in the ceramic topcoat extended from the topcoat to the interface between the bondcoat and topcoat, but did not penetrate the bondcoat. Additionally, there were no interfacial cracks found in the bondcoat and substrate.

(2)

During axial tensile fatigue tests, cracks in the topcoat initially appeared perpendicular to the direction of the load. When subjected to tensile-torsion loads with a 90° phase difference, the cracks on the edges of the radial hole in the tube were oriented at −70° and 110° relative to the edges of the hole. Calculations revealed that the location of the maximum principal stress corresponded to the location of the initial crack, and the direction of the principal stress aligned with the direction of the crack opening.

(3)

A fatigue life model was proposed, utilizing the maximum principal stress as the damage parameter, taking into account the stress gradient and the impact of multiaxial loads on fatigue life. This model was used to predict the cracking life of ceramic topcoat thin-walled tubes with radial holes under loads parallel to the coating interface. Excluding two data points where cracking occurred immediately upon loading, the life prediction results fell within a four-fold scatter band, with 77% of the prediction results falling within a two-fold scatter band.