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Computational method for the analysis of erosion-induced stress and damage in thermal barrier coatings
Surface & Coatings Technology xxx (xxxx) xxxx
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Computational method for the analysis of erosion-induced stress and damage in thermal barrier coatings Jianan Songa, Hongyu Qia,b, Shaolin Lia,b,∗, Duoqi Shia,b, Xiaoguang Yanga,b a b
Department of Energy and Power Engineering, Beihang University, Beijing, 100191, China Beijing Key Laboratory of Aero-Engine Structure and Strength, Beijing, 100191, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal barrier coatings Erosion FEM Crack initiation Load waveform
In this study, we study the effects of erosion on the distribution of stress and cracking behaviors in thermal barrier coatings (TBCs) using the finite element method. The distribution of stress and failure processes of models that include effects of erosion were different from those of non-erosion models. Erosion increased the average temperature in thermal growth oxidation layers and caused interfacial cracks to appear earlier. Moreover, a high erosion rate can accelerate stress growth in TBCs and induce greater interfacial damage during the first few cycles. For TBCs subjected to various thermal cycle fatigue loading waveforms, the difference between the maximum stresses for erosion-included and non-erosion models increases with dwelling time at elevated temperature. Therefore, the effect of erosion was deemed worth consideration in future lifetime prediction studies on TBCs.
1. Introduction Thermal barrier coating systems (TBCs) are applied to the hot components of aero-engines (e.g., turbine blades and discs) to protect substrates from elevated temperatures. The application of TBCs can increase the turbine inlet temperature of a gas turbine engine by approximately 160 °C [1,2]. However, the failure of TBCs can expose a substrate to elevated-temperature gaseous environments, leading to the early failure of turbine blades and significantly threatening flight safety. Therefore, based on the increasing use of TBCs, research on TBCs failure processes is essential for TBCs design to ensure the safety of hot components and aero-engines [3]. TBCs failure can be attributed to many different factors, such as foreign object damage, thermal growth oxidation (TGO) growth stress, phase transformation calcium-magnesium-aluminum-silicate attacks, and sintering [4–12], etc. According to previous researches, the failure of TBCs on turbine blades during service can be attributed to two major issues: oxidation of the bond coat (BC) layers [13–15], and erosion of projectiles attacked in the gas stream [16–19]. Oxidation induces additional stress in TBCs, eventually leading to cracking and spallation. The velocity of turbine blades can reach up to 300 m/s, meaning small, hard particles will erode the surfaces of TBCs in the combustion gases [16]. Furthermore, erosion decreases the thickness of the top-coat (TC) layers [20], which accelerates the expansion of TGO layers. Previous experimental researches on TBCs erosion provided erosion ∗
mechanism and rage of erosion rate in turbomachines, ect. Hamed et al. [21] found that the erosion rate of turbine blades is mainly affected by rotational speed and air-flow conditions, and the blade geometry, TBCs material and characteristics of erosion particle also have couple effect of blades’ erosion. The studies of Tabakoff et al. [22] and Nicholls et al. [23] indicated that the erosion rate of TBCs, both the electron beamphysical deposited (EB-PVD) and plasma sprayed (PS) TBCs, increased linearly with the particles impact velocity. Many researchers investigated the relationship of TBCs erosion and particles motion through via numerical integration [24–26]. Hammed et al. [25] developed method to predict the erosion life of turbine blade based on the erosion experimental results in the laboratory. However, there has been little research on the combined effects of erosion and TGO. In above studies, the failure processes of the TBCs were studied using uniform or gradient temperatures, which neglect the erosion effects of TC layers. Therefore, the main purpose of this study was to analyze the combined effects of TBCs erosion and TGO growth using the finite element method (FEM). First, the influence of erosion on thermal mismatch stresses was investigated. Next, this combined effect on stress distributions and cracking behaviors were analyzed through the user subroutines. Finally, the differences between the erosion-included and non-erosion model were analyzed with different dwelling times and loading waveforms. Furthermore, because erosion rate was changed on the different locations of turbine blade [27], we also analyzed the influence of different TBCs erosion rate.
Corresponding author. Department of Energy and Power Engineering, Beihang University, Beijing, 100191, China. E-mail address: [email protected] (S. Li).
https://doi.org/10.1016/j.surfcoat.2019.125089 Received 17 July 2019; Received in revised form 20 September 2019; Accepted 22 October 2019 0257-8972/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Jianan Song, et al., Surface & Coatings Technology, https://doi.org/10.1016/j.surfcoat.2019.125089
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Erosion direction
Equation constrain
Swelling elements
y
Cohesive elements x
Fig. 1. Geometry and meshing of FE models.
4.3 × 10-5 mm/s, according to the particle impact velocities, impingement angles and operating temperatures of erosion environment. Therefore, we set the thickness reduction rate as 3.8 × 10-6, 1.9 × 10-5 and 4.3 × 10-5 mm/s, respectively, in Section 3.3, to investigate the influence of erosion rate on stress distributions and interfacial cracking behaviors in the TBCs. However, in Section 3.1, 3.2 and 3.4, the reduction rate was set as 1.9 × 10-5 mm/s. It is worth mentioning that, the erosion option was only activated in the y-direction during the high temperature process. Moreover, considering the width of the FE models was less than the average diameter of erosion particle [31–33], the erosion induced TC layer reduction was assumed to be uniform. Therefore, we only discuss the effects of erosion on the temperature fields in TBCs that can influence TGO growth rates and failure processes of the TBCs.
2. FEM modeling In the present work, considering the effects of erosion, stress distributions and interface cracking behaviors were analyzed using the FEM. The influence of erosion rate and thermal load waveforms was also analyzed. In consideration of the efficiency of the FEM models, thermal cycles were halted after most interface elements failed. 2.1. Geometric and FEM mesh model Generally, TBCs are composed of three layers – a TC layer, TGO layer, and BC layer, each of which has different mechanical behaviors. Furthermore, in this work, an idealized cosine curve was introduced to simulate the irregular interfaces of TBCs. Because interface roughness varies between manufacturers [28,29], we used average geometrical values in the FEM models, namely 2.55 μm for amplitude (A) and 21.5 μm for wavelength (λ), as shown in Fig. 1. To reduce out-of-plane stress, generalized plane strain elements (CPE8G) were adopted in the FEM models. Because element size could significantly influence the stress distributions near interfaces, to improve the accuracy of the FEM models, the element size was set to be less than 0.1 μm. Cohesive elements were employed to analyze the cracking behaviors of the TC-TGO interface. The thickness of the cohesive elements was set to 0.002 μm, which should be negligible in an FE model. Moreover, to facilitate to analysis, the direction along the thickness of the TBCs was defined as the y direction, and the axial direction of the substrate was defined as the x direction, as plotted in Fig. 1.
2.3. TGO growth model The growth stress caused by TGO growth is one of the main reasons for TBCs failure [14]. The erosion would decrease the thickness of TC layers, which reduced the thermal insulation performance of TC layers [33]. Moreover, the reduction of thermal insulation also increased the operating temperature of TGO layers. Because TGO growth is significantly affected by temperature, in this study, Tamman's law was adopted in our models to describe TGO growth behaviors, as follows,
h = A·exp(−
Q n1 )·t T
(1)
where t denotes the dwelling time at high-temperature. Based on the previous research [35–37], the coefficients A and Q were set to 2165.96 μm/h0.36 and 8140.16 °C, respectively. The exponent n1 has a value of 0.36. The oxidation kinetics curves of TGO layers at different temperatures can be found in Ref. [36]. The “SWELLING” subroutine was employed in the ABAQUS software to simulate the process of TGO growth. The growth was only activated in the thickness direction. Therefore, the “SWELLING” option was defined as follows:
2.2. TC erosion model In this work, the erosion of the TC layers was conducted using a “UMESHMOTION” subroutine. Adaptive meshes were employed in the TC layers to avoid mesh distortion during the erosion process. According to the experimental results [30], the erosion-induced thickness reduction rate of TC layer was changed from 3.8 × 10-6 mm/s to 2
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Table 1 Temperature dependence of thermal expansion coefficient α, Young's modulus E, and Poisson ratio υ for the different layers. T (°C)
TBC
TGO
28 200 300 400 500 650 760 850 980 1050 1100
BC
α ( × 10 )
E (GPa)
υ
α ( × 10 )
E (GPa)
υ
α ( × 10-6)
E (GPa)
υ
10.02 9.78 10.02 10.34 10.29
65 56 58 54 53 44 40 31 29 23 18
0.1 0.1 0.1 0.1 0.1 0.11 0.11 0.11 0.12 0.12 0.12
8.2 8.4 8.7 9 9.3 9.6
400 390 380 355 325 320
0.23 0.23 0.24 0.25 0.25 0.25
20.5 24.3 34.2
198 189 162 151 143 134 129 124 119
0.31 0.32 0.32 0.33 0.33 0.34 0.34 0.35 0.37
-6
1 έoxidation = rii έ cr 3
-6
Table 3 Temperature dependence of Young's modulus E for the substrate.
(2)
where the vector i = 1, 2, 3 denotes the local coordinate system and r is the coefficient of TGO growth. As mentioned above, because the growth option was only activated in the thickness direction of TGO, in this work, the coefficients were set to r22 = 3 and r11 = r33 = 0. According to Ref [38], the mechanical behavior of an initial TGO layer is different from that of a newly generated TGO layer. Therefore, in the FE models, the elastic modulus and Poisson's ratio of the newly generated TGO layer were set to 300 GPa and 0.28, respectively [38]. Furthermore, the creep constant B of a newly generated TGO layer was set to B = 1.02 × 10-6 [39].
0.9 1.0 1.0 -
10 7.794 6.029 5.074 4.412 4.412 4
20.5 24.3 34.2
9.45 11.15 15.35 20.20 24.55 30.90
1100
E(GPa) ν
131.5 0.344
107.5 0.371
107 0.374
98 0.383
80.5 0.390
69.5 0.399
67.5 0.413
-8
1.8 × (10 -10 7.3 × 10-10 6.54 × 10-19 2.20 × 10-21 1.84 × 10-7 2.15 × 10-8 4.85 × 10-36 2.25 × 10-9
-10
)
n
T(°C)
1 1 4.57 2.99 1.55 2.45 1 3
1000 1000 ≤600 700 800 ≥850 10 1200
As mentioned above, crack initiation and propagation on the TCTGO interface is one of the main reasons for TBCs failure. Therefore, in this work, we introduced cohesive elements (COH2D4) on the TC-TGO interface to investigate the effect of erosion on crack initiation and propagation behaviors. The adaptive mesh has employed in the TC layers, to solve the issue of mesh distortion caused by the erosion process. Furthermore, the cracks in the TC layers were neglected considering the convergence of the FEM models. The constitutive model of cohesive elements was based on continuum damage mechanics. Specifically, the whole model was regarded as a continuous structure before the interfacial cracks initiated. The mechanical behaviors of cohesive elements were controlled by traction-separation law. The damage evolution was defined as follows,
Table 2 Temperature dependence of heat conductivity coefficient K (W/m·°C) for the different layers.
20 200 400 600 800 1000 1100
1070
2.6. Interfacial cracking behaviors
Each thermal cycle is divided into three steps. In the first step, the surface temperature of the TC layer increases from 25 °C to 1070 °C and the inside wall of the substrate is heated from 25 °C to 950 °C over 5 s. Next, the temperature maintained for 670 s. Finally, the FEM models are cooled to 25 °C over 55 s. The creep behavior of the material is only activated during the dwelling step.
SUB
980
The displacement boundary condition of the FEM model is shown in Fig. 1. During our calculations, the y-direction displacement of the bottom side of the model was fixed. A constraint equation was employed to combine the x-direction displacement of the right sides, which can reduce the boundary effects of the model.
2.5. Boundary and thermal loading condition
BC
850
Sub [53].
Here, B and n2 denote temperature-dependent empirical coefficients, as listed in Table 4 [51–53].
TGO
700
TBC [51,52] TGO [51,52] BC [51,52]
(3)
TBC
650
B(s-1MPa-n)
In our FEM models, TBCs are treated as elastic materials and combined temperature-displacement analysis steps are employed to simulate heterogeneous temperature fields in TBCs during the service. The mechanical and heat-transfer properties of each layer are listed in Tables 1–3. Previous studies have shown that the creep properties of TBCs influence stress distributions significantly at high temperatures [40]. Therefore, the Norton creep power law is applied to the TBCs in our models, and is defined as follows:
T (°C)
25
Table 4 Creep data for the TBCs.
2.4. Materials parameters
έ cr = Bσ n2
T(°C)
0, D=
δf (δ − δ0) δ (δf − δ0)
1,
δ ≤ δ0 ,
δ 0 ≤ δ ≤ δf δf ≤ δ
(4)
where δ, δ0, and δf represent the opening displacement, critical displacement, and fracture displacement, respectively. The critical tensile stress and shear stress were set to 200 MPa and 100 MPa, respectively 3
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Fig. 2. Average temperature in TGO layer of erosion-included and non-erosion models.
layers significantly. Moreover, the differences in temperature fields increase with the number of cycles. After 20 thermal cycles, the TCTGO interface is undamaged. These results also indicate that thermal mismatch stress is insufficient for interfacial crack initiation and propagation in TBCs, but the effects of thermal growth stress should be considered.
[41]. The stiffness of cohesive elements was set to 200 GPa/mm, and the fracture energy of mode-I (Gn) and mode-II cracks (Gt) was set to 20 and 60 J/m2, respectively [42,43].
3. Results and discussion 3.1. Influence of erosion on stress distributions in the TC
3.2. Influence of erosion on TGO growth and cracking behaviors As discussed above, the erosion of TCs changes the temperature field in the TBCs, which has an influence on thermal mismatch stress in the TBCs and growth rate of the TGO layer. Firstly, we discussed the effect of erosion on the average temperature of the TGO layer and thermal mismatch stress distributions in the TC. The TGO growth option was disabled in those FEM models. The average temperatures of TGO for TBCs with and without erosion process are plotted in Fig. 2. Because erosion decreases the thickness of the TC layer, which reduced the insulation of TBCs, the difference between TGO average temperatures increases with the number of thermal cycles. After 19 thermal cycles, the average temperature in the TGO layer increased from 1010 °C to 1028 °C for erosion-included model while the TGO temperature of non-erosion model was 1010 °C. This change in temperature can potentially affect both the creep behaviors of TBCs and the growth rates of the TGO layers. The stress distributions of different FEM models after 20 thermal cycles are shown in Fig. 3. For the TBCs without an erosion process, the maximum tensile stress S22 and shear stress S12 reach 31.30 MPa and 24.20 MPa, respectively. The maximum tensile stress and shear stress both appear at off-peak locations in the TC-TGO interface. However, the stress amplitudes increase slightly for the FEM models with erosion process, based on inelastic deformation in different temperature fields. Therefore, erosion only has a small influence on thermal mismatch stresses in TBCs but can change the temperature amplitudes in TGO
Our previous thermal-displacement analysis indicated that erosion can influence the temperature fields of TGO layers, which increases the TGO growth rate in the TBCs. Moreover, compared to the FEM models without erosion process, the difference in temperature increased with the number of thermal cycles and dwell time at high temperature. Therefore, we designed a temperature-dependent TGO-growth model to investigate the effect of erosion on TGO growth behaviors. After 19 thermal cycles, most interfacial elements failed and calculation was halted. As shown in Fig. 4, the maximum tensile stress (S22) for TBCs without erosion is 133.07 MPa and the maximum shear stress (S12) is 120.95 MPa. However, for the erosion-included FE model, the maximum tensile stress (S22) of the TC layer increased by 8.2%. The erosion of TC layers also cause an increase in shear stress of about 9.22 MPa higher than the non-erosion models. When comparing these two different FEM models, we noted that erosion only increases the stress amplitude and does not change the stress distribution in TBCs. The maximum tensile and shear stress still appeared in the middle and off-peak locations on the TC-TGO interface. The results indicate that the mode-ΙΙ cracks were more likely to initiate in the off-peak location of the TC-TGO interface and the mode-Ι cracks were more likely to initiate in the middle interfacial location and propagate along the x-direction. Finally, the transverse and interfacial cracks converged, leading to spallation of the TC layers. 4
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Fig. 3. Thermal mismatch stress distributions of erosion-included (a), (b) and non-erosion models (c), (d).
Because erosion can induce an increasing temperature field in TGO layers, the TGO growth behaviors of the erosion-included models are different from those of the non-erosion models. Specifically, as the number of thermal cycles increases, the erosion-included models exhibit higher temperatures in the TGO layers, which induced an
increasing TGO growth rate and larger stress amplitude in the TC layer of the TBCs. According to the FEM results, the interfacial cracks in TBCs were initiated in the off-peak locations. Thus, the stress evolution in this area was investigated. Increased temperature would induce a higher TGO growth rate. Therefore, the difference between these two kinds of
Fig. 4. Stress distributions of the erosion-included (a), (b) and non-erosion (c), (d) models. 5
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130
with erosion without erosion
S22 in the off-peak location / MPa
120 110 100 90 80 70 60 50 40 30 20 10 0 -10 0
2000
4000
6000
8000
10000
12000
14000
Time / s
S12 in the off-peak location / MPa
Fig. 5. Evolution of tensile stress S22 for the erosion-included and non-erosion models. 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130
with erosion without erosion
0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 6. Evolution of shear stress S12 for erosion-included and non-erosion models.
3.3. Influence of erosion rate and TGO growth rate on cracking behaviors
models became larger with the thermal fatigue test conducted. After one thermal cycle, the tensile stress S22 in the off-peak location of the TC-TGO interface was 31.14 MPa for the erosion-included model and 28.47 MPa for the non-erosion model, as plotted in Fig. 5. After 18 thermal cycles, the tensile stress for the erosion model reached 130.14 MPa, which is approximately 30% greater than the value for the non-erosion model. However, erosion had little effect on shear stress evolution. As shown in Fig. 6, the erosion only increased about 6 MPa of the shear stress in the off-peak location. The evolution of tensile and shear stress also indicated that, for the erosion-included TBCs, the cracks in the off-peak location would change from model-I to mixed model-I and II as the number of thermal cycles increased. Interfacial cracks initiate at the off-peak location and the corresponding damage of interfacial elements is shown in Fig. 7. For the models without erosion, the interfacial damage increased to 0.71 after 10 thermal cycles and reach to 0.97 after 20 cycles. For the erosioninduced models, interfacial damage appeared after 8 cycles and increased to 1 after 15 cycles. Therefore, erosion has a greater effect on TBCs with a longer service at the elevated temperature, and causes cracks to appear earlier. Furthermore, increasing stress may induce earlier failure of the TBCs. A lifetime estimation of the TBCs without considering erosion effects are overly optimistic. Therefore, these factors should be considered in future lifetime prediction studies.
During TBCs service on turbine blades, the erosion rate changes based on the angle and speed of airflow [44,45]. According to the experimental studies of Wellman [46], the erosion rate varies from 3.8 × 10-6 mm/s to 4.3 × 10-5 mm/s when the particle impact velocities, impingement angles particle size, the material of TC layers and operating temperatures changed. In this work, we investigated the influence of erosion rate on stress distributions and interfacial cracking behaviors in the TBCs. It is worth mentioning that the rate of decrease of the TC layer thickness is assumed to be uniform, and cracks induced from being struck by foreign objects are neglected. Furthermore, when the erosion rate increased to 4.3 × 10-5 mm/s, the interface failed after 18 thermal cycles. Thus, each calculation was halted after 18 cycles. A higher erosion rate increases the average temperatures in TGO layers and accelerates stress growth in TC layers. As shown in Fig. 8, after 18 thermal cycles, the maximum tensile stress (S22) increases to 131.63 MPa for the TBCs with the lower erosion rate (3.8 × 10-6 mm/ s), while for the higher erosion rate (4.3 × 10-5 mm/s), the residual tensile stress increases to 142.36 MPa. Furthermore, the difference in tensile stress increases with number of TCF cycles. As plotted in Fig. 9, after 18 thermal cycles, the tensile stress for the high-erosion-rate TBCs is about 8.3% greater than that for the low-erosion-rate condition. Nevertheless, the increasing erosion rate does not change the location 6
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1.0
Interfacial damage D
0.8
0.6
0.4
0.2
with erosion without erosion 0.0 0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 7. Evolution of interfacial damage for erosion-included and non-erosion models.
Fig. 8. Tensile stress distributions for models with different erosion rates, (a) for 3.8 × 10-6 mm/s, (b) for 1.9 × 10-5 mm/s and (c) for4.3 × 10-5 mm/s, after 18 TCF cycles.
3.8×10-6 μm/s 1.9×10-5 μm/s 4.3×10-5 μm/s
150
Maximum S22 in the off-peak location
140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 9. Tensile stress evolutions of models with different erosion rate.
maximum shear stress of the high erosion rate model was about 16% higher than of low erosion rate model. Moreover, a faster erosion rate increases the accumulation of interfacial damage during the first few steps. As plotted in Fig. 12, the interfacial damage appeared after 7 cycles. For TBCs with the low erosion rate, the damage reached 0.11 during the cooling step. This value increases to 0.78 for the high-erosion-rate TBCs. According to Ref. [47], interfacial strength varies between different
of maximum tensile stress. For both models, the maximum tensile stress appears in the middle of the TC-TGO interface. The shear stress (S12) distributions of TBCs with different erosion models are shown in Fig. 10. An increase of erosion rate did not change the location of the maximum shear stress, which remain located at the off-peak location in the TC-TGO interface. Furthermore, as plotted in Fig. 11, after one thermal cycle, there is no difference in maximum shear stress between the models. However, after 18 cycles, the 7
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Fig. 10. Shear stress distributions of models with different erosion rate, (a) for 3.8 × 10-6 mm/s, (b) for 1.9 × 10-5 mm/s and (c) for4.3 × 10-5 mm/s, after 18 cycles.
3.8×10-6 mm/s 1.9×10-5 mm/s 4.3×10-5 mm/s
Maximum S12 in the off-peak location
40 20 0 -20 -40 -60 -80 -100 -120 -140
0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 11. Shear stress evolutions of models with different erosion rate. 1.0
Interfacial damage D
0.8
0.6
0.4
3.8×10-6 mm/s 1.9×10-5 mm/s 4.3×10-5 mm/s
0.2
0.0 0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 12. Interfacial damage evolutions of models with different erosion rate.
3.4. Influence of thermal loading waveforms
manufacturers. Therefore, at a higher erosion rate (such as that at the leading edge of a turbine blade), TBCs with low interfacial strength may fail during the first few cycles. Moreover, a greater erosion rate accelerates the stress growth, which would has a greater effect on the lifetime of long-time-service TBCs.
We discussed the importance of the combined effects of TGO growth and erosion at elevated temperatures in the sections above. Here, we investigate the influence of thermal loading waveforms on stress distributions and the failure processes of TBCs. The dwelling times were set to 1 s, 670 s and 1200 s, respectively. The TBCs failed after 18 cycles with 670 s dwelling, and 6 thermal cycles with 1200s dwelling.
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Fig. 13. Tensile stress distributions of models with different dwelling times, (a), (b) and (c) for erosion-included models and (e), (f) and (d) for non-erosion models.
Fig. 14. Shear stress distributions of models with different dwelling times, (a), (b) and (c) for erosion-included models and (e), (f) and (d) for non-erosion models.
Fig. 15. It is worth mentioning that the maximum shear stress of TBCs with 1 s dwelling process was located in the TC layer, as plotted in Fig. 14(a) and (d), but when the dwelling time increases to 670 s, the maximum shear stress appears in the TC-TGO interface, as shown in Fig. 14(b) and (e). This transform of maximum shear stress location indicates that loading waveforms can influence the final failure modes of TBCs. Therefore, according to the FEM results, erosion has a greater influence on TBCs with longer dwelling times at elevated temperatures. This indicates that erosion-included stress growth deserves greater consideration for TBCs with longer dwelling processes to obtain more
However, for TBCs with only 1 s of dwelling process, there was no interfacial damage after 20 thermal cycles, and then the calculation was halted. As plotted in Fig. 13 and Fig. 14, after 20 thermal cycles, the differences in tensile and shear stress is 0.25 and 0.11 MPa, respectively. When the dwelling time increases to 670 s, the tensile stress difference between erosion-included and non-erosion models is 11.02 MPa and the difference for shear stress is 9.22 MPa. Moreover, the difference increased further when the dwelling time reached 1200 s (11.99 MPa for tensile stress and 10.51 MPa for shear stress). The evolution of stress difference between the erosion and non-erosion models is plotted in
9
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Difference on stress of each models / MPa
J. Song, et al.
12 11 10 9 8 7 6 5 4 3 2
tensile sterss shear stress
1 0 0
200
400
600
800
1000
1200
1400
Dwelling time of each cycle / s Fig. 15. Difference on stress of erosion-induced and non-erosion models.
accurate lifetime estimates. Furthermore, TCF loading waveforms can influence the stress distribution and final failure modes of the TBCs. According to the FEM results, cracks are more likely to initiate in the TC layer for TBCs with short dwelling times, while for long-dwelling loading waveforms, cracks initiate and propagate in the TC-TGO interface. This phenomenon has been observed in the previous research [48–50].
(06) (2016) 477–517. [4] Y.C. Zhou, Q.X. Liu, L. Yang, D.J. Wu, W.G. Mao, Failure mechanisms and life prediction of thermal barrier coatings, Chin. J. Solid Mech. 31 (5) (2010) 504–531. [5] R.G. Wellman, J.R. Nicholls, Erosion, corrosion and erosion–corrosion of EB PVD thermal barrier coatings, Tribol. Int. 41 (7) (2008) 657–662. [6] H. Bhatnagar, S. Ghosh, M.E. Walter, Parametric studies of failure mechanisms in elastic EB-PVD thermal barrier coatings using FEM, Int. J. Solids Struct. 43 (14–15) (2006) 4384–4406. [7] E.A.G. Shillington, D.R. Clarke, Spalling failure of a thermal barrier coating associated with aluminum depletion in the bond-coat, Acta Mater. 47 (1999) 1297–1305. [8] H. Dong, G. Yang, C. Li, et al., Effect of TGO thickness on thermal cyclic lifetime and failure mode of plasma-sprayed TBCs, J. Am. Ceram. Soc. 97 (4) (2014) 1226–1232. [9] W. Zhu, L. Yang, J.W. Guo, et al., Numerical study on interaction of surface cracking and interfacial delamination in thermal barrier coatings under tension, Appl. Surf. Sci. 315 (315) (2014) 292–298. [10] J. Jiang, W. Wang, X. Zhao, et al., Numerical analyses of the residual stress and top coat cracking behavior in thermal barrier coatings under cyclic thermal loading, Eng. Fract. Mech. 196 (2018) 191–205. [11] J. Su, Z. Ji, Z. Han, et al., Study of crack initiation and propagation in TBCs by experiments and extended fnite element method, Appl. Mech. Mater. 331 (2013) 129–132. [12] C.J. Li, H. Dong, H. Ding, et al., The correlation of the TBC lifetimes in burner cycling test with thermal gradient and furnace isothermal cycling test by TGO effects, J. Therm. Spray Technol. 26 (3) (2017) 378–387. [13] A.M. Karlsson, J.W. Hutchinson, A.G. Evans, A fundamental model of cyclic instabilities in thermal barrier systems, J. Mech. Phys. Solids 50 (2002) 1565. [14] D.R. Mumm, A.G. Evans, I.T. Spitsberg, Characterization of a cyclic displacement instability for a thermally grown oxide in a thermal barrier system, Acta Mater. 49 (2001) 1793. [15] P.K. Wright, A.G. Evans, Mechanisms governing the performance of thermal barrier coatings, Curr. Opin. Solid State Mater. Sci. 4 (3) (1999) 255–265. [16] X. Chen, R. Wang, N. Yao, A.G. Evans, J.W. Hutchinson, R.W. Bruce, Foreign object damage in a thermal barrier system: mechanisms and simulations, Mater. Sci. Eng. A 352 (2003) 221. [17] J.R. Nicholls, M.J. Deakin, D.S. Rickerby, A comparison between the erosion behavior of thermal spray and electron beam physical vapour deposition thermal barrier coating, Wear 23 (1999) 352. [18] R.G. Wellman, J.R. Nicholls, A mechanism for the erosion of EB-PVD TBCs, Wear 242 (2000) 89. [19] N.A. Fleck, T. Zisis, The erosion of EB-PVD thermal barrier coatings: the competition between mechanisms, Wear 268 (11) (2010) 1214–1224. [20] D. Shin, A. Hamed, Experimental Investigation and Parametric DOE Appraisal of Thermal Barrier Coating High-Temperature Erosion, Aiaa Science & Technology Forum & Exposition, 2016. [21] A. Hamed, W.C. Tabakoff, R.V. Wenglarz, et al., Erosion and deposition in turbomachinery, J. Propuls. Power 22 (2) (2006) 350–360. [22] W. Tabakoff, Investigation of coatings at high temperature for use in turbomachinery, Surf. Coat. Technol. 39 (1–3) (1989) 97–115. [23] J.R.,M.J. Nicholls, et al., A comparison between the erosion behaviour of thermal spray and electron beam physical vapour deposition thermal barrier coatings, Wear 233–235 (none) (1999) 352–361. [24] M.F. Hussein, W. Tabakoff, Three dimensional dynamic characteristics of solid particles suspended by polluted air flow in a turbine stage, 11th Aerospace Sciences Meeting, AIAA, 1972. [25] A. Hamed, W. Tabakoff, R.B. Rivir, et al., Turbine blade surface deterioration by erosion, J. Turbomach. Trans. ASME 127 (3) (2005) 445–452.
4. Conclusion In this work, we performed FEM analysis of TBCs with and without erosion. The influence of erosion and load waveform on stress distributions and cracking behaviors in the TBCs was discussed. The FEM results indicated that erosion increases stress amplitudes and induces early failure of TBCs, which is worth considering in future research. The main conclusions of this study are as follows: Erosion has little influence on residual thermal mismatch stresses in TBCs. However, because erosion decreases the heat insulation performance of TBCs, the temperature in the TBCs increases, which accelerates the growth rate of the TGO layer. Furthermore, changes in TGO growth can increase the stress amplitude and cause cracks to appears earlier. The difference between the stress amplitude of erosion-included and non-erosion model increased with the numbers of TCF loading. Moreover, a higher erosion rate has a more significant influence on stress distributions in TBCs and induces the initiation of interfacial cracks in the first few cycles. For TBCs with long-dwelling thermal cycles, erosion increases the maximum stress and should not be neglected as a research object when TBCs are applied to turbine blades. Acknowledgments The authors would like to thank the Fundamental Research Funds for the Central Universities (No. YWF-19-BJ-J-338) and National Natural Science Foundation of China (No. 51571010) the for the support given to this research. References [1] W. Beele, G. Marijnissen, A. Van Lieshout, et al., The evolution of thermal barrier coatings-status and upcoming solutions for today's key issues, Surf. Coat. Technol. (1999) 61–67. [2] H.D. Steffens, R. Kaczmarek, Thermal barrier coating for engine, Weld. World 11/ 12 (1990) 224–230. [3] Tiejun Wang, Xueling Fan, Yongle Sun etc, Research progress on stress and crack in the thermal barrier coating system of heavy duty gas turbine, Acta Mech. Solida Sin.
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[26] J. Di, S. Wang, L. Zhang, et al., Study on the erosion characteristics of boride coatings by finite element analysis, Surf. Coat. Technol. (2018) 115–124. [27] P. Walsh, J. Quets, R. Tucker, Coatings for the protection of turbine blades from erosion, J. Eng. Gas Turbines Power 117 (1995) 152–155. [28] D. Liu, M. Seraffon, P.E. Flewitt, et al., Effect of substrate curvature on residual stresses and failure modes of an air plasma sprayed thermal barrier coating system, J. Eur. Ceram. Soc. 33 (15) (2013) 3345–3357. [29] D. Zhang, S. Gong, H. Xu, et al., Effect of bond coat surface roughness on the thermal cyclic behavior of thermal barrier coatings, Surf. Coat. Technol. 201 (3) (2006) 649–653. [30] R.G. Wellman, J.R. Nicholls, A review of the erosion of thermal barrier coatings, J. Phys. D 40 (16) (2007). [31] A.G. Evans, N.A. Fleck, S. Faulhaber, et al., Scaling laws governing the erosion and impact resistance of thermal barrier coatings[J], Wear 260 (7) (2006) 886–894. [32] L. Wang, D. Li, J. Yang, et al., Modeling of thermal properties and failure of thermal barrier coatings with the use of finite element methods: a review[J], J. Eur. Ceram. Soc. 36 (6) (2016) 1313–1331. [33] S. Hassani, M. Bielawski, W. Beres, L. Martinu, M. Balazinski, J. Klemberg-Sapieha, Predictive tools for the design of erosion resistant coatings, Surf. Coat. Technol. 203 (2008) 204–210. [35] L.Y. Lim, S.A. Meguid, Temperature Dependent Dynamic Growth of Thermally Grown Oxide in Thermal Barrier Coatings, Materials & Design, 2019. [36] S. Ahmadian, C. Thistle, E.H. Jordan, Experimental and finite element study of an air plasma sprayed thermal barrier coating under fixed cycle duration at various temperatures, J. Am. Ceram. Soc. 96 (2013) 3210–3217. [37] B. Li, X. Fan, K. Zhou, et al., Effect of oxide growth on the stress development in double-ceramic-layer thermal barrier coatings, Ceram. Int. 43 (17) (2017) 14763–14774. [38] Dae-Jin Kim, Evaluation of the degradation of plasma sprayed thermal barrier coatings using nano-indentation, J. Nanosci. Nanotechnol. 9 (12) (2009). [39] J. Schwarzer, D. LöHe, O. VöHringer, Influence of the TGO creep behavior on delamination stress development in thermal barrier coating systems, Mater. Sci. Eng. A (2004) 692–695. [40] D. Pan, M.W. Chen, P.K. Wright, et al., Evolution of a diffusion aluminide bond coat
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for thermal barrier coatings during thermal cycling, Acta Mater. 51 (8) (2003) 2205–2217. J. Aktaa, K. Sfar, D. Munz, et al., Assessment of TBC systems failure mechanisms using a fracture mechanics approach, Acta Mater. 53 (16) (2005) 4399–4413. M. Białas, Finite element analysis of stress distribution in thermal barrier coatings, Surf. Coat. Technol. 202 (24) (2008) 6002–6010. C. Bumgardner, B. Croom, X. Li, High-temperature delamination mechanisms of thermal barrier coatings: in-situ, digital image correlation and fnite element analyses, Acta Mater. 128 (2017) 54–63. R. Darolia, Thermal barrier coatings technology: critical review, progress update, remaining challenges and prospects, Int. Mater. Rev. (58) (2013) 315–348. Y.Q. Xiao, L. Yang, Y.C. Zhou, et al., Dominant parameters affecting the reliability of TBCs on a gas turbine blade during erosion by a particle-laden hot gas stream, Wear 390 (2017) S0043164816306408. R.G. Wellman, M.J. Deakin, J.R. Nicholls, The effect of TBC morphology on the erosion rate of EB PVD TBCs, Wear 258 (1) (2005) 349–356. Z.H. Xu, Y. Yang, P. Huang, et al., Determination of interfacial properties of thermal barrier coatings by shear test and inverse finite element method, Acta Mater. 58 (18) (2010) 5972–5979. K. Kokini, T.R. Hornack, Transient thermal load effects on coatings bonded to cylindrical substrates and containing circumferential cracks, J. Eng. Mater. Technol. 110 (1) (1988) 35. D. Zhu, R.A. Miller, Investigation of thermal high cycle and low cycle fatigue mechanisms of thick thermal barrier coatings, Mater. Sci. Eng. A 245 (2) (1998) 212–223. G. Pulci, J. Tirillò, F. Marra, et al., High temperature oxidation of MCrAlY coatings modified by Al 2 O 3, PVD overlay, Surf. Coat. Technol. 268 (2015) 198–204. J. Rösler, M. Bäker, K. Aufzug, A parametric study of the stress state of thermal barrier coatings. Part I: creep relaxation, Acta Mater. 52 (2004) 4809–4817. M. Ranjbar-Far, J. Absi, G. Mariaux, et al., Simulation of the effect of material properties and interface roughness on the stress distribution in thermal barrier coatings using finite element method, Mater. Des. 31 (2010) 772–781. Editorial Board of Chinese Aeronautical Materials, Handbook of Chinese Aeronautical Materials, China Standard Press, Beijing, 2002, pp. 260–278.
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Surface & Coatings Technology journal homepage: www.elsevier.com/locate/surfcoat
Computational method for the analysis of erosion-induced stress and damage in thermal barrier coatings Jianan Songa, Hongyu Qia,b, Shaolin Lia,b,∗, Duoqi Shia,b, Xiaoguang Yanga,b a b
Department of Energy and Power Engineering, Beihang University, Beijing, 100191, China Beijing Key Laboratory of Aero-Engine Structure and Strength, Beijing, 100191, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Thermal barrier coatings Erosion FEM Crack initiation Load waveform
In this study, we study the effects of erosion on the distribution of stress and cracking behaviors in thermal barrier coatings (TBCs) using the finite element method. The distribution of stress and failure processes of models that include effects of erosion were different from those of non-erosion models. Erosion increased the average temperature in thermal growth oxidation layers and caused interfacial cracks to appear earlier. Moreover, a high erosion rate can accelerate stress growth in TBCs and induce greater interfacial damage during the first few cycles. For TBCs subjected to various thermal cycle fatigue loading waveforms, the difference between the maximum stresses for erosion-included and non-erosion models increases with dwelling time at elevated temperature. Therefore, the effect of erosion was deemed worth consideration in future lifetime prediction studies on TBCs.
1. Introduction Thermal barrier coating systems (TBCs) are applied to the hot components of aero-engines (e.g., turbine blades and discs) to protect substrates from elevated temperatures. The application of TBCs can increase the turbine inlet temperature of a gas turbine engine by approximately 160 °C [1,2]. However, the failure of TBCs can expose a substrate to elevated-temperature gaseous environments, leading to the early failure of turbine blades and significantly threatening flight safety. Therefore, based on the increasing use of TBCs, research on TBCs failure processes is essential for TBCs design to ensure the safety of hot components and aero-engines [3]. TBCs failure can be attributed to many different factors, such as foreign object damage, thermal growth oxidation (TGO) growth stress, phase transformation calcium-magnesium-aluminum-silicate attacks, and sintering [4–12], etc. According to previous researches, the failure of TBCs on turbine blades during service can be attributed to two major issues: oxidation of the bond coat (BC) layers [13–15], and erosion of projectiles attacked in the gas stream [16–19]. Oxidation induces additional stress in TBCs, eventually leading to cracking and spallation. The velocity of turbine blades can reach up to 300 m/s, meaning small, hard particles will erode the surfaces of TBCs in the combustion gases [16]. Furthermore, erosion decreases the thickness of the top-coat (TC) layers [20], which accelerates the expansion of TGO layers. Previous experimental researches on TBCs erosion provided erosion ∗
mechanism and rage of erosion rate in turbomachines, ect. Hamed et al. [21] found that the erosion rate of turbine blades is mainly affected by rotational speed and air-flow conditions, and the blade geometry, TBCs material and characteristics of erosion particle also have couple effect of blades’ erosion. The studies of Tabakoff et al. [22] and Nicholls et al. [23] indicated that the erosion rate of TBCs, both the electron beamphysical deposited (EB-PVD) and plasma sprayed (PS) TBCs, increased linearly with the particles impact velocity. Many researchers investigated the relationship of TBCs erosion and particles motion through via numerical integration [24–26]. Hammed et al. [25] developed method to predict the erosion life of turbine blade based on the erosion experimental results in the laboratory. However, there has been little research on the combined effects of erosion and TGO. In above studies, the failure processes of the TBCs were studied using uniform or gradient temperatures, which neglect the erosion effects of TC layers. Therefore, the main purpose of this study was to analyze the combined effects of TBCs erosion and TGO growth using the finite element method (FEM). First, the influence of erosion on thermal mismatch stresses was investigated. Next, this combined effect on stress distributions and cracking behaviors were analyzed through the user subroutines. Finally, the differences between the erosion-included and non-erosion model were analyzed with different dwelling times and loading waveforms. Furthermore, because erosion rate was changed on the different locations of turbine blade [27], we also analyzed the influence of different TBCs erosion rate.
Corresponding author. Department of Energy and Power Engineering, Beihang University, Beijing, 100191, China. E-mail address: [email protected] (S. Li).
https://doi.org/10.1016/j.surfcoat.2019.125089 Received 17 July 2019; Received in revised form 20 September 2019; Accepted 22 October 2019 0257-8972/ © 2019 Elsevier B.V. All rights reserved.
Please cite this article as: Jianan Song, et al., Surface & Coatings Technology, https://doi.org/10.1016/j.surfcoat.2019.125089
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Erosion direction
Equation constrain
Swelling elements
y
Cohesive elements x
Fig. 1. Geometry and meshing of FE models.
4.3 × 10-5 mm/s, according to the particle impact velocities, impingement angles and operating temperatures of erosion environment. Therefore, we set the thickness reduction rate as 3.8 × 10-6, 1.9 × 10-5 and 4.3 × 10-5 mm/s, respectively, in Section 3.3, to investigate the influence of erosion rate on stress distributions and interfacial cracking behaviors in the TBCs. However, in Section 3.1, 3.2 and 3.4, the reduction rate was set as 1.9 × 10-5 mm/s. It is worth mentioning that, the erosion option was only activated in the y-direction during the high temperature process. Moreover, considering the width of the FE models was less than the average diameter of erosion particle [31–33], the erosion induced TC layer reduction was assumed to be uniform. Therefore, we only discuss the effects of erosion on the temperature fields in TBCs that can influence TGO growth rates and failure processes of the TBCs.
2. FEM modeling In the present work, considering the effects of erosion, stress distributions and interface cracking behaviors were analyzed using the FEM. The influence of erosion rate and thermal load waveforms was also analyzed. In consideration of the efficiency of the FEM models, thermal cycles were halted after most interface elements failed. 2.1. Geometric and FEM mesh model Generally, TBCs are composed of three layers – a TC layer, TGO layer, and BC layer, each of which has different mechanical behaviors. Furthermore, in this work, an idealized cosine curve was introduced to simulate the irregular interfaces of TBCs. Because interface roughness varies between manufacturers [28,29], we used average geometrical values in the FEM models, namely 2.55 μm for amplitude (A) and 21.5 μm for wavelength (λ), as shown in Fig. 1. To reduce out-of-plane stress, generalized plane strain elements (CPE8G) were adopted in the FEM models. Because element size could significantly influence the stress distributions near interfaces, to improve the accuracy of the FEM models, the element size was set to be less than 0.1 μm. Cohesive elements were employed to analyze the cracking behaviors of the TC-TGO interface. The thickness of the cohesive elements was set to 0.002 μm, which should be negligible in an FE model. Moreover, to facilitate to analysis, the direction along the thickness of the TBCs was defined as the y direction, and the axial direction of the substrate was defined as the x direction, as plotted in Fig. 1.
2.3. TGO growth model The growth stress caused by TGO growth is one of the main reasons for TBCs failure [14]. The erosion would decrease the thickness of TC layers, which reduced the thermal insulation performance of TC layers [33]. Moreover, the reduction of thermal insulation also increased the operating temperature of TGO layers. Because TGO growth is significantly affected by temperature, in this study, Tamman's law was adopted in our models to describe TGO growth behaviors, as follows,
h = A·exp(−
Q n1 )·t T
(1)
where t denotes the dwelling time at high-temperature. Based on the previous research [35–37], the coefficients A and Q were set to 2165.96 μm/h0.36 and 8140.16 °C, respectively. The exponent n1 has a value of 0.36. The oxidation kinetics curves of TGO layers at different temperatures can be found in Ref. [36]. The “SWELLING” subroutine was employed in the ABAQUS software to simulate the process of TGO growth. The growth was only activated in the thickness direction. Therefore, the “SWELLING” option was defined as follows:
2.2. TC erosion model In this work, the erosion of the TC layers was conducted using a “UMESHMOTION” subroutine. Adaptive meshes were employed in the TC layers to avoid mesh distortion during the erosion process. According to the experimental results [30], the erosion-induced thickness reduction rate of TC layer was changed from 3.8 × 10-6 mm/s to 2
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Table 1 Temperature dependence of thermal expansion coefficient α, Young's modulus E, and Poisson ratio υ for the different layers. T (°C)
TBC
TGO
28 200 300 400 500 650 760 850 980 1050 1100
BC
α ( × 10 )
E (GPa)
υ
α ( × 10 )
E (GPa)
υ
α ( × 10-6)
E (GPa)
υ
10.02 9.78 10.02 10.34 10.29
65 56 58 54 53 44 40 31 29 23 18
0.1 0.1 0.1 0.1 0.1 0.11 0.11 0.11 0.12 0.12 0.12
8.2 8.4 8.7 9 9.3 9.6
400 390 380 355 325 320
0.23 0.23 0.24 0.25 0.25 0.25
20.5 24.3 34.2
198 189 162 151 143 134 129 124 119
0.31 0.32 0.32 0.33 0.33 0.34 0.34 0.35 0.37
-6
1 έoxidation = rii έ cr 3
-6
Table 3 Temperature dependence of Young's modulus E for the substrate.
(2)
where the vector i = 1, 2, 3 denotes the local coordinate system and r is the coefficient of TGO growth. As mentioned above, because the growth option was only activated in the thickness direction of TGO, in this work, the coefficients were set to r22 = 3 and r11 = r33 = 0. According to Ref [38], the mechanical behavior of an initial TGO layer is different from that of a newly generated TGO layer. Therefore, in the FE models, the elastic modulus and Poisson's ratio of the newly generated TGO layer were set to 300 GPa and 0.28, respectively [38]. Furthermore, the creep constant B of a newly generated TGO layer was set to B = 1.02 × 10-6 [39].
0.9 1.0 1.0 -
10 7.794 6.029 5.074 4.412 4.412 4
20.5 24.3 34.2
9.45 11.15 15.35 20.20 24.55 30.90
1100
E(GPa) ν
131.5 0.344
107.5 0.371
107 0.374
98 0.383
80.5 0.390
69.5 0.399
67.5 0.413
-8
1.8 × (10 -10 7.3 × 10-10 6.54 × 10-19 2.20 × 10-21 1.84 × 10-7 2.15 × 10-8 4.85 × 10-36 2.25 × 10-9
-10
)
n
T(°C)
1 1 4.57 2.99 1.55 2.45 1 3
1000 1000 ≤600 700 800 ≥850 10 1200
As mentioned above, crack initiation and propagation on the TCTGO interface is one of the main reasons for TBCs failure. Therefore, in this work, we introduced cohesive elements (COH2D4) on the TC-TGO interface to investigate the effect of erosion on crack initiation and propagation behaviors. The adaptive mesh has employed in the TC layers, to solve the issue of mesh distortion caused by the erosion process. Furthermore, the cracks in the TC layers were neglected considering the convergence of the FEM models. The constitutive model of cohesive elements was based on continuum damage mechanics. Specifically, the whole model was regarded as a continuous structure before the interfacial cracks initiated. The mechanical behaviors of cohesive elements were controlled by traction-separation law. The damage evolution was defined as follows,
Table 2 Temperature dependence of heat conductivity coefficient K (W/m·°C) for the different layers.
20 200 400 600 800 1000 1100
1070
2.6. Interfacial cracking behaviors
Each thermal cycle is divided into three steps. In the first step, the surface temperature of the TC layer increases from 25 °C to 1070 °C and the inside wall of the substrate is heated from 25 °C to 950 °C over 5 s. Next, the temperature maintained for 670 s. Finally, the FEM models are cooled to 25 °C over 55 s. The creep behavior of the material is only activated during the dwelling step.
SUB
980
The displacement boundary condition of the FEM model is shown in Fig. 1. During our calculations, the y-direction displacement of the bottom side of the model was fixed. A constraint equation was employed to combine the x-direction displacement of the right sides, which can reduce the boundary effects of the model.
2.5. Boundary and thermal loading condition
BC
850
Sub [53].
Here, B and n2 denote temperature-dependent empirical coefficients, as listed in Table 4 [51–53].
TGO
700
TBC [51,52] TGO [51,52] BC [51,52]
(3)
TBC
650
B(s-1MPa-n)
In our FEM models, TBCs are treated as elastic materials and combined temperature-displacement analysis steps are employed to simulate heterogeneous temperature fields in TBCs during the service. The mechanical and heat-transfer properties of each layer are listed in Tables 1–3. Previous studies have shown that the creep properties of TBCs influence stress distributions significantly at high temperatures [40]. Therefore, the Norton creep power law is applied to the TBCs in our models, and is defined as follows:
T (°C)
25
Table 4 Creep data for the TBCs.
2.4. Materials parameters
έ cr = Bσ n2
T(°C)
0, D=
δf (δ − δ0) δ (δf − δ0)
1,
δ ≤ δ0 ,
δ 0 ≤ δ ≤ δf δf ≤ δ
(4)
where δ, δ0, and δf represent the opening displacement, critical displacement, and fracture displacement, respectively. The critical tensile stress and shear stress were set to 200 MPa and 100 MPa, respectively 3
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Fig. 2. Average temperature in TGO layer of erosion-included and non-erosion models.
layers significantly. Moreover, the differences in temperature fields increase with the number of cycles. After 20 thermal cycles, the TCTGO interface is undamaged. These results also indicate that thermal mismatch stress is insufficient for interfacial crack initiation and propagation in TBCs, but the effects of thermal growth stress should be considered.
[41]. The stiffness of cohesive elements was set to 200 GPa/mm, and the fracture energy of mode-I (Gn) and mode-II cracks (Gt) was set to 20 and 60 J/m2, respectively [42,43].
3. Results and discussion 3.1. Influence of erosion on stress distributions in the TC
3.2. Influence of erosion on TGO growth and cracking behaviors As discussed above, the erosion of TCs changes the temperature field in the TBCs, which has an influence on thermal mismatch stress in the TBCs and growth rate of the TGO layer. Firstly, we discussed the effect of erosion on the average temperature of the TGO layer and thermal mismatch stress distributions in the TC. The TGO growth option was disabled in those FEM models. The average temperatures of TGO for TBCs with and without erosion process are plotted in Fig. 2. Because erosion decreases the thickness of the TC layer, which reduced the insulation of TBCs, the difference between TGO average temperatures increases with the number of thermal cycles. After 19 thermal cycles, the average temperature in the TGO layer increased from 1010 °C to 1028 °C for erosion-included model while the TGO temperature of non-erosion model was 1010 °C. This change in temperature can potentially affect both the creep behaviors of TBCs and the growth rates of the TGO layers. The stress distributions of different FEM models after 20 thermal cycles are shown in Fig. 3. For the TBCs without an erosion process, the maximum tensile stress S22 and shear stress S12 reach 31.30 MPa and 24.20 MPa, respectively. The maximum tensile stress and shear stress both appear at off-peak locations in the TC-TGO interface. However, the stress amplitudes increase slightly for the FEM models with erosion process, based on inelastic deformation in different temperature fields. Therefore, erosion only has a small influence on thermal mismatch stresses in TBCs but can change the temperature amplitudes in TGO
Our previous thermal-displacement analysis indicated that erosion can influence the temperature fields of TGO layers, which increases the TGO growth rate in the TBCs. Moreover, compared to the FEM models without erosion process, the difference in temperature increased with the number of thermal cycles and dwell time at high temperature. Therefore, we designed a temperature-dependent TGO-growth model to investigate the effect of erosion on TGO growth behaviors. After 19 thermal cycles, most interfacial elements failed and calculation was halted. As shown in Fig. 4, the maximum tensile stress (S22) for TBCs without erosion is 133.07 MPa and the maximum shear stress (S12) is 120.95 MPa. However, for the erosion-included FE model, the maximum tensile stress (S22) of the TC layer increased by 8.2%. The erosion of TC layers also cause an increase in shear stress of about 9.22 MPa higher than the non-erosion models. When comparing these two different FEM models, we noted that erosion only increases the stress amplitude and does not change the stress distribution in TBCs. The maximum tensile and shear stress still appeared in the middle and off-peak locations on the TC-TGO interface. The results indicate that the mode-ΙΙ cracks were more likely to initiate in the off-peak location of the TC-TGO interface and the mode-Ι cracks were more likely to initiate in the middle interfacial location and propagate along the x-direction. Finally, the transverse and interfacial cracks converged, leading to spallation of the TC layers. 4
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Fig. 3. Thermal mismatch stress distributions of erosion-included (a), (b) and non-erosion models (c), (d).
Because erosion can induce an increasing temperature field in TGO layers, the TGO growth behaviors of the erosion-included models are different from those of the non-erosion models. Specifically, as the number of thermal cycles increases, the erosion-included models exhibit higher temperatures in the TGO layers, which induced an
increasing TGO growth rate and larger stress amplitude in the TC layer of the TBCs. According to the FEM results, the interfacial cracks in TBCs were initiated in the off-peak locations. Thus, the stress evolution in this area was investigated. Increased temperature would induce a higher TGO growth rate. Therefore, the difference between these two kinds of
Fig. 4. Stress distributions of the erosion-included (a), (b) and non-erosion (c), (d) models. 5
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130
with erosion without erosion
S22 in the off-peak location / MPa
120 110 100 90 80 70 60 50 40 30 20 10 0 -10 0
2000
4000
6000
8000
10000
12000
14000
Time / s
S12 in the off-peak location / MPa
Fig. 5. Evolution of tensile stress S22 for the erosion-included and non-erosion models. 30 20 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 -130
with erosion without erosion
0
2000
4000
6000
8000
10000
12000
14000
Time / s Fig. 6. Evolution of shear stress S12 for erosion-included and non-erosion models.
3.3. Influence of erosion rate and TGO growth rate on cracking behaviors
models became larger with the thermal fatigue test conducted. After one thermal cycle, the tensile stress S22 in the off-peak location of the TC-TGO interface was 31.14 MPa for the erosion-included model and 28.47 MPa for the non-erosion model, as plotted in Fig. 5. After 18 thermal cycles, the tensile stress for the erosion model reached 130.14 MPa, which is approximately 30% greater than the value for the non-erosion model. However, erosion had little effect on shear stress evolution. As shown in Fig. 6, the erosion only increased about 6 MPa of the shear stress in the off-peak location. The evolution of tensile and shear stress also indicated that, for the erosion-included TBCs, the cracks in the off-peak location would change from model-I to mixed model-I and II as the number of thermal cycles increased. Interfacial cracks initiate at the off-peak location and the corresponding damage of interfacial elements is shown in Fig. 7. For the models without erosion, the interfacial damage increased to 0.71 after 10 thermal cycles and reach to 0.97 after 20 cycles. For the erosioninduced models, interfacial damage appeared after 8 cycles and increased to 1 after 15 cycles. Therefore, erosion has a greater effect on TBCs with a longer service at the elevated temperature, and causes cracks to appear earlier. Furthermore, increasing stress may induce earlier failure of the TBCs. A lifetime estimation of the TBCs without considering erosion effects are overly optimistic. Therefore, these factors should be considered in future lifetime prediction studies.
During TBCs service on turbine blades, the erosion rate changes based on the angle and speed of airflow [44,45]. According to the experimental studies of Wellman [46], the erosion rate varies from 3.8 × 10-6 mm/s to 4.3 × 10-5 mm/s when the particle impact velocities, impingement angles particle size, the material of TC layers and operating temperatures changed. In this work, we investigated the influence of erosion rate on stress distributions and interfacial cracking behaviors in the TBCs. It is worth mentioning that the rate of decrease of the TC layer thickness is assumed to be uniform, and cracks induced from being struck by foreign objects are neglected. Furthermore, when the erosion rate increased to 4.3 × 10-5 mm/s, the interface failed after 18 thermal cycles. Thus, each calculation was halted after 18 cycles. A higher erosion rate increases the average temperatures in TGO layers and accelerates stress growth in TC layers. As shown in Fig. 8, after 18 thermal cycles, the maximum tensile stress (S22) increases to 131.63 MPa for the TBCs with the lower erosion rate (3.8 × 10-6 mm/ s), while for the higher erosion rate (4.3 × 10-5 mm/s), the residual tensile stress increases to 142.36 MPa. Furthermore, the difference in tensile stress increases with number of TCF cycles. As plotted in Fig. 9, after 18 thermal cycles, the tensile stress for the high-erosion-rate TBCs is about 8.3% greater than that for the low-erosion-rate condition. Nevertheless, the increasing erosion rate does not change the location 6
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Fig. 8. Tensile stress distributions for models with different erosion rates, (a) for 3.8 × 10-6 mm/s, (b) for 1.9 × 10-5 mm/s and (c) for4.3 × 10-5 mm/s, after 18 TCF cycles.
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maximum shear stress of the high erosion rate model was about 16% higher than of low erosion rate model. Moreover, a faster erosion rate increases the accumulation of interfacial damage during the first few steps. As plotted in Fig. 12, the interfacial damage appeared after 7 cycles. For TBCs with the low erosion rate, the damage reached 0.11 during the cooling step. This value increases to 0.78 for the high-erosion-rate TBCs. According to Ref. [47], interfacial strength varies between different
of maximum tensile stress. For both models, the maximum tensile stress appears in the middle of the TC-TGO interface. The shear stress (S12) distributions of TBCs with different erosion models are shown in Fig. 10. An increase of erosion rate did not change the location of the maximum shear stress, which remain located at the off-peak location in the TC-TGO interface. Furthermore, as plotted in Fig. 11, after one thermal cycle, there is no difference in maximum shear stress between the models. However, after 18 cycles, the 7
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Fig. 10. Shear stress distributions of models with different erosion rate, (a) for 3.8 × 10-6 mm/s, (b) for 1.9 × 10-5 mm/s and (c) for4.3 × 10-5 mm/s, after 18 cycles.
3.8×10-6 mm/s 1.9×10-5 mm/s 4.3×10-5 mm/s
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Time / s Fig. 12. Interfacial damage evolutions of models with different erosion rate.
3.4. Influence of thermal loading waveforms
manufacturers. Therefore, at a higher erosion rate (such as that at the leading edge of a turbine blade), TBCs with low interfacial strength may fail during the first few cycles. Moreover, a greater erosion rate accelerates the stress growth, which would has a greater effect on the lifetime of long-time-service TBCs.
We discussed the importance of the combined effects of TGO growth and erosion at elevated temperatures in the sections above. Here, we investigate the influence of thermal loading waveforms on stress distributions and the failure processes of TBCs. The dwelling times were set to 1 s, 670 s and 1200 s, respectively. The TBCs failed after 18 cycles with 670 s dwelling, and 6 thermal cycles with 1200s dwelling.
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Fig. 13. Tensile stress distributions of models with different dwelling times, (a), (b) and (c) for erosion-included models and (e), (f) and (d) for non-erosion models.
Fig. 14. Shear stress distributions of models with different dwelling times, (a), (b) and (c) for erosion-included models and (e), (f) and (d) for non-erosion models.
Fig. 15. It is worth mentioning that the maximum shear stress of TBCs with 1 s dwelling process was located in the TC layer, as plotted in Fig. 14(a) and (d), but when the dwelling time increases to 670 s, the maximum shear stress appears in the TC-TGO interface, as shown in Fig. 14(b) and (e). This transform of maximum shear stress location indicates that loading waveforms can influence the final failure modes of TBCs. Therefore, according to the FEM results, erosion has a greater influence on TBCs with longer dwelling times at elevated temperatures. This indicates that erosion-included stress growth deserves greater consideration for TBCs with longer dwelling processes to obtain more
However, for TBCs with only 1 s of dwelling process, there was no interfacial damage after 20 thermal cycles, and then the calculation was halted. As plotted in Fig. 13 and Fig. 14, after 20 thermal cycles, the differences in tensile and shear stress is 0.25 and 0.11 MPa, respectively. When the dwelling time increases to 670 s, the tensile stress difference between erosion-included and non-erosion models is 11.02 MPa and the difference for shear stress is 9.22 MPa. Moreover, the difference increased further when the dwelling time reached 1200 s (11.99 MPa for tensile stress and 10.51 MPa for shear stress). The evolution of stress difference between the erosion and non-erosion models is plotted in
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accurate lifetime estimates. Furthermore, TCF loading waveforms can influence the stress distribution and final failure modes of the TBCs. According to the FEM results, cracks are more likely to initiate in the TC layer for TBCs with short dwelling times, while for long-dwelling loading waveforms, cracks initiate and propagate in the TC-TGO interface. This phenomenon has been observed in the previous research [48–50].
(06) (2016) 477–517. [4] Y.C. Zhou, Q.X. Liu, L. Yang, D.J. Wu, W.G. Mao, Failure mechanisms and life prediction of thermal barrier coatings, Chin. J. Solid Mech. 31 (5) (2010) 504–531. [5] R.G. Wellman, J.R. Nicholls, Erosion, corrosion and erosion–corrosion of EB PVD thermal barrier coatings, Tribol. Int. 41 (7) (2008) 657–662. [6] H. Bhatnagar, S. Ghosh, M.E. Walter, Parametric studies of failure mechanisms in elastic EB-PVD thermal barrier coatings using FEM, Int. J. Solids Struct. 43 (14–15) (2006) 4384–4406. [7] E.A.G. Shillington, D.R. Clarke, Spalling failure of a thermal barrier coating associated with aluminum depletion in the bond-coat, Acta Mater. 47 (1999) 1297–1305. [8] H. Dong, G. Yang, C. Li, et al., Effect of TGO thickness on thermal cyclic lifetime and failure mode of plasma-sprayed TBCs, J. Am. Ceram. Soc. 97 (4) (2014) 1226–1232. [9] W. Zhu, L. Yang, J.W. Guo, et al., Numerical study on interaction of surface cracking and interfacial delamination in thermal barrier coatings under tension, Appl. Surf. Sci. 315 (315) (2014) 292–298. [10] J. Jiang, W. Wang, X. Zhao, et al., Numerical analyses of the residual stress and top coat cracking behavior in thermal barrier coatings under cyclic thermal loading, Eng. Fract. Mech. 196 (2018) 191–205. [11] J. Su, Z. Ji, Z. Han, et al., Study of crack initiation and propagation in TBCs by experiments and extended fnite element method, Appl. Mech. Mater. 331 (2013) 129–132. [12] C.J. Li, H. Dong, H. Ding, et al., The correlation of the TBC lifetimes in burner cycling test with thermal gradient and furnace isothermal cycling test by TGO effects, J. Therm. Spray Technol. 26 (3) (2017) 378–387. [13] A.M. Karlsson, J.W. Hutchinson, A.G. Evans, A fundamental model of cyclic instabilities in thermal barrier systems, J. Mech. Phys. Solids 50 (2002) 1565. [14] D.R. Mumm, A.G. Evans, I.T. Spitsberg, Characterization of a cyclic displacement instability for a thermally grown oxide in a thermal barrier system, Acta Mater. 49 (2001) 1793. [15] P.K. Wright, A.G. Evans, Mechanisms governing the performance of thermal barrier coatings, Curr. Opin. Solid State Mater. Sci. 4 (3) (1999) 255–265. [16] X. Chen, R. Wang, N. Yao, A.G. Evans, J.W. Hutchinson, R.W. Bruce, Foreign object damage in a thermal barrier system: mechanisms and simulations, Mater. Sci. Eng. A 352 (2003) 221. [17] J.R. Nicholls, M.J. Deakin, D.S. Rickerby, A comparison between the erosion behavior of thermal spray and electron beam physical vapour deposition thermal barrier coating, Wear 23 (1999) 352. [18] R.G. Wellman, J.R. Nicholls, A mechanism for the erosion of EB-PVD TBCs, Wear 242 (2000) 89. [19] N.A. Fleck, T. Zisis, The erosion of EB-PVD thermal barrier coatings: the competition between mechanisms, Wear 268 (11) (2010) 1214–1224. [20] D. Shin, A. Hamed, Experimental Investigation and Parametric DOE Appraisal of Thermal Barrier Coating High-Temperature Erosion, Aiaa Science & Technology Forum & Exposition, 2016. [21] A. Hamed, W.C. Tabakoff, R.V. Wenglarz, et al., Erosion and deposition in turbomachinery, J. Propuls. Power 22 (2) (2006) 350–360. [22] W. Tabakoff, Investigation of coatings at high temperature for use in turbomachinery, Surf. Coat. Technol. 39 (1–3) (1989) 97–115. [23] J.R.,M.J. Nicholls, et al., A comparison between the erosion behaviour of thermal spray and electron beam physical vapour deposition thermal barrier coatings, Wear 233–235 (none) (1999) 352–361. [24] M.F. Hussein, W. Tabakoff, Three dimensional dynamic characteristics of solid particles suspended by polluted air flow in a turbine stage, 11th Aerospace Sciences Meeting, AIAA, 1972. [25] A. Hamed, W. Tabakoff, R.B. Rivir, et al., Turbine blade surface deterioration by erosion, J. Turbomach. Trans. ASME 127 (3) (2005) 445–452.
4. Conclusion In this work, we performed FEM analysis of TBCs with and without erosion. The influence of erosion and load waveform on stress distributions and cracking behaviors in the TBCs was discussed. The FEM results indicated that erosion increases stress amplitudes and induces early failure of TBCs, which is worth considering in future research. The main conclusions of this study are as follows: Erosion has little influence on residual thermal mismatch stresses in TBCs. However, because erosion decreases the heat insulation performance of TBCs, the temperature in the TBCs increases, which accelerates the growth rate of the TGO layer. Furthermore, changes in TGO growth can increase the stress amplitude and cause cracks to appears earlier. The difference between the stress amplitude of erosion-included and non-erosion model increased with the numbers of TCF loading. Moreover, a higher erosion rate has a more significant influence on stress distributions in TBCs and induces the initiation of interfacial cracks in the first few cycles. For TBCs with long-dwelling thermal cycles, erosion increases the maximum stress and should not be neglected as a research object when TBCs are applied to turbine blades. Acknowledgments The authors would like to thank the Fundamental Research Funds for the Central Universities (No. YWF-19-BJ-J-338) and National Natural Science Foundation of China (No. 51571010) the for the support given to this research. References [1] W. Beele, G. Marijnissen, A. Van Lieshout, et al., The evolution of thermal barrier coatings-status and upcoming solutions for today's key issues, Surf. Coat. Technol. (1999) 61–67. [2] H.D. Steffens, R. Kaczmarek, Thermal barrier coating for engine, Weld. World 11/ 12 (1990) 224–230. [3] Tiejun Wang, Xueling Fan, Yongle Sun etc, Research progress on stress and crack in the thermal barrier coating system of heavy duty gas turbine, Acta Mech. Solida Sin.
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