Abstract
This paper performs a comprehensive investigation on the high cycle fatigue (HCF) life prediction of turbine blade with film cooling holes. The modified theory of critical distance (MTCD) method is proposed to estimate the fatigue life of the specimen considering the notch sensitivity coefficient and multiaxial stress effect. Then, two types of specimens were designed with regard to the single-hole and multihole conditions. Afterwards, the dangerous path and fatigue life of the two specimens were achieved implementing the MTCD method. Then, the experiments and failure analysis were carried out. The results show that the stress concentration and multiaxial stress resulting from the film cooling holes are the primary reasons that the cracks originated. Meanwhile, the dangerous path of the single-hole specimen is quite different from the multihole specimen due to the interhole interference. Finally, most of the calculated fatigue life is within the twice error band of the tested life.
1. Introduction
Nowadays, the increasing thrust demand has called for high turbine entry temperature (TET) and wide-range rotational speed of modern turbofan engine [1]. Consequently, the turbine blade works in a harsh environment and might experience fatigue failure. To be specific, the high temperature and high speed would generate very high stress concentration at the film cooling holes and some sharp edges, especially for the hollow blade. Besides, the notch effect cannot be ignored when the turbine blade employs many holes and fins. It would not only aggravate the stress concentration but also generate multiaxial stress [2] due to the existence of the holes. Meanwhile, the implementation of single-crystal materials on turbine blade also contributes to the failures since the crystal defects are inevitable [3]. Many studies show that the turbine blades are vulnerable to high cycle fatigue (HCF) failures [4–7]. Therefore, the prediction of fatigue life of turbine blade is crucial to guarantee a safety operation.
The theory of critical distance (TCD) method was first proposed by Neuber [8] and Peterson [9] and developed by Susmel and Taylor [10] and Taylor [11]. It provides a deep insight into the estimation of fatigue limit and evaluates the fatigue damage using the linear elastic stress field in the crack initiation region. By introducing the characteristic distance related to material properties and loading conditions [11], the notch sensitivity of the specimen is linked to local damage. It should be pointed out that the TCD method is applicable in both monotonous load and periodic load [12]. Furthermore, the point method (PM) and line method (LM) are most widely accepted whereas the area method (AM) and volume method (VM) are seldom used due to the complexity. It was reported by Taylor and Wang [13] that PM demonstrated the highest accuracy in predicting various forms of notched specimens. In recent years, it is often combined with the improved Wöhler curve method [14] to tackle the fretting fatigue issues [15–19]. Based on the multiaxial fatigue criterion and PM, Ronchei et al. [20] introduced the critical plane to evaluate the metal component life under fretting load. Another application of TCD method is the residual compressive stress assessment using the LM. It has been verified in high residual stress gradient cases [21, 22].
The TCD method is commonly accepted and implemented in the field of notch fatigue analysis. Susmel [23] reviewed the utilization of TCD method in the fatigue field and extended the usage of TCD method to torsional fatigue of notched specimens [24]. Besides, the accuracy of TCD theory in HCF mechanical model was investigated [25]. Liao et al. [26] summarized 8 life assessment procedures and verified the optimal one with experiments. Then, a systematical review was made focusing on the notch fatigue concerning the stress gradient effect [27]. Afterwards, the TCD method was compared with highly stressed volume (HSV) and weakest link theory (WLT) with regard to GH4169 specimen fatigue life prediction [28]. Jadallah et al. [29] made an improvement in the traditional TCD method and verified it by HCF tests. Different notch stress concentration factors were experimented with the adopted microstructure length of the material as a parameter combined with the gradient elasticity theory. Chen [30] implemented the LM to estimate the notch fatigue limit from both energy condition and crack propagation condition perspectives. Razavi et al. [31] investigated the fatigue lifetime of 3D-printed titanium alloy Ti6Al4V while the specimen and the notch were kept as the manufacturing condition.
One of the tough tasks remaining to be solved is to precisely predict the elastic-plastic stress and strain field resulting from the anisotropy of single-crystal alloy [32, 33]. Continuous studies on the constitutive of single-crystal alloy are still carried out around the world. Typically, it can be divided into phenomenological model and crystal slip model. The former is represented by Hill and Hill and Rice [34, 35] and mainly starts from the modified cubic symmetric anisotropy constitutive model. This model is based on macroscopic stress and strain and cannot reflect the true deformation of materials. Therefore, it is difficult to establish an accurate description of the crystal anisotropy and orientation sensitivity. The crystal plasticity theory based on the slip system was first developed by Taylor [36, 37]. This model reveals the relationship between shear stress and strain rate of slip system with time, which can better track the change of plastic anisotropy.
In this paper, a design process is proposed to achieve the turbine blade fatigue life with film cooling holes. The modified TCD is derived considering the notch sensitivity coefficient and multiaxial stress effect. Then, two types of specimens are designed with different holes. The fatigue life of the specimen is predicted with the proposed TCD method and the dangerous path is obtained. Afterwards, the experiment is carried out to verify the calculation results.
2. Methodology
2.1. Design Framework
In this scenario, the main purpose is to develop a modified TCD method to precisely predict the HCF life of the concerned component with film cooling holes. A systematic design framework is proposed to better demonstrate the work, as depicted in Figure 1. The first step is to introduce the film cooling hole notch parameter and multiaxial stress parameter to improve the accuracy of conventional TCD method. The impacts of film cooling holes and hole number are adopted in the component life prediction. Afterwards, the simulating specimen is designed to simulate the turbine blade with file cooling holes. It should be noted that both the single-hole and multihole specimens are considered for a fair comparison. The dangerous path is predicted according to the combination of finite element analysis and the modified TCD approach. It would provide a good opportunity to observe the stress concentration of the film cooling holes. Meanwhile, the dangerous path is estimated from the test and failure analysis results. The comparisons of the simulation and test would verify the accuracy and reliability of the proposed method. Finally, the predicted life is achieved and compared with the tested life, whereas the error band is obtained which would also validate the proposed method.
2.2. Modified TCD Method
2.2.1. Conventional TCD Method
Convectional TCD method is mainly used to calculate the fatigue life under multiaxial stress. The fundamental principle of fracture mechanism is that the fatigue crack originates from the maximum stress concentration location (MSCL) and propagates along the dangerous path until it ruptures. To be specific, when the stress decreases to a certain value along a dangerous path, the structure encounters failure. The determinant stress is defined as the equivalent stress that equals the fatigue limit of the material. It decides the distribution of maximum principal stress in the stress concentration area.
Tanaka and Taylor pointed out that the equations of PM and LM could be arranged as follows: where is the maximum principal stress, is the reference equivalent stress, is the equivalent stress, is the critical distance of line method, and is the critical distance of point method.
Regarding the HCF issues, the parameters in the above equations can be defined by fatigue stress amplitude and gradient. Figure 2 depicts the schematic diagram of PM and LM under elastic stress distribution. It indicates that the specimen would reach the fatigue limit when the distance away from the MSCL approaches the or at the dangerous path. For the conventional TCD method, the determination of critical distance and is highly dependent on the material properties and stress ratio.
(a)
(b)
In order to calculate the critical distance in a more convenient way, Susmel and Taylor [38] came up with the characteristic distance dependent on the critical stress intensity gradient and material fatigue limit , as shown in
For PM and LM, the relationship between characteristic distance and critical distance is described as follows:
Then, the fatigue life prediction model of the component is derived, as depicted in (4). It could be seen that the fatigue life is only related to , , and , where and are decided by the material property and stress ratio.
For the failure under static loading, the critical distance is defined in (5). The main difference between (2) and (5) is that the and are substituted with and , where represents the plane fracture toughness of the material.
2.2.2. Modified TCD Method
According to the smooth specimen curve, when the material is in the initial stage of the fatigue cycle, equals . Equation (5) could be rewritten as
However, when the material reaches the fatigue limit, that is, and substitute it into (5), then Equation (7) is obtained.
According to (6) and (7), the values of parameters and are obtained as follows:
It can be seen from (8) that and can be obtained once the and are determined. Afterwards, the relationship between critical distance and fatigue life is established. It could be implemented to predict the HCF life of the component.
Nevertheless, it is not accurate to predict the fatigue life of the notch specimen using the largest stress gradient or the hot point [24]. In this scenario, the notch sensitivity coefficient is adopted to include the effects of notches on the fatigue life. As presented in (9), notch factor is determined by the theoretical stress concentration factor and the fatigue strength ratio of smooth to notched specimens . Also, the multiaxial stress effect is also taken into consideration by introducing the factor , acting as the exponent. It should be noted that used to be setting as 1.
The variance of with temperature of the single-hole specimen is achieved and listed in Table 1. Meanwhile, the of the multihole case is listed at temperature 980°C.
Figure 3 presents the work flow chart of calculating the HCF life implementing the modified theory of critical distance (MTCD) method. The first step is to assume an estimated fatigue life , which needs to be as close as to the real life in order to reduce the iterations. It could be assumed according to the life of similar structure with equivalent stress level. After that, the maximum principal stress distribution along the dangerous path near the notch would be obtained using the PM approach. Consequently, the maximum principal stress at the location away from the notch tip would be achieved and denoted as , which is considered as the equivalent stress. Then, the is introduced into the iteration , and the loop continues until equals . This ensures that the impacts of film cooling hole and multiaxial stress are considered in the turbine blade life prediction.
2.2.3. Elastic-Plastic Model Based on Crystal Plasticity Theory
In this scenario, the elastic-plastic constitutive model based on crystal plasticity theory is utilized to calculate the dangerous path in finite element analysis. The constitutive model of crystal plastic model was developed by Taylor. The total deformation gradient is divided into two parts: elastic and plastic, as described in When the crystal is deformed and twisted, the lattice vector changes and the slip direction can be defined as
The normal vector of the slip face is listed as follows:
The velocity gradient is given by the standard equation:
can also be expressed as where is the deformation rate tensor and is the rotation tensor.
The plastic deformation consists of dislocations and slips: where is the slip rate of slip system and is the Schmidt factor of slip system , which can be defined as
Furthermore, the Cauchy stress tensor is represented by , and the weighed Cauchy stress tensor is expressed by . The relationship between the two factors is listed as follows:
The shearing stress is a component of the traction force along the slip direction, and it is related to the Cauchy stress through the Schmidt tensor.
The strain rate is expressed by a power function equation: where is the reference shear strain rate, is the strain rate sensitivity index, and is the reference shear stress which characterizes the current strain hardening state of the crystal and depends on the sum of the slip shear rate : where is cumulative slip strain:
In order to simplify the calculation, the strain hardening of the materials can be replaced by the evolution equation of :
In the equation, is a function of , which determines the hardening of slip system caused by the amount of slip shear in slip system , which can be obtained by the following formula: where is the hardening coefficient and is the hardening rate: where is the hardening modulus and and are the model parameters.
In this scenario, the tensile tests were carried out to obtain the model parameters of (24) [39, 40]. The parameters are listed in Table 2. Meanwhile, the data is transported to ABAQUS as user-defined subroutine.
3. Simulating Specimen Life Prediction
3.1. Simulating Specimen Design
As presented in Figure 4, the film cooling holes are employed to cool the turbine blade. Typically, there exists a high stress concentration at the film cooling holes area, which might pose HCF failure risk. In order to verify the proposed method, two types of specimens with film cooling holes are designed to simulate the turbine blade. The details of the specimen are presented in Figure 5. It can be seen clearly that the diameters of specimen with single hole and multihole are 0.4 mm, maintaining the same as the turbine blade film cooling holes. Meanwhile, two through-holes are observed at each specimen for the tensile cyclic loading. The thickness of the specimen is 2 mm while the radius of the transition area is 12 mm. It should be noted that the material remains nickel-based single-crystal superalloy DD6. To be specific, the tensile axis direction is [001] orientation while the thickness direction (the axial direction of the hole) is [010] orientation, and the width direction is [100] orientation. Besides, the orientation angle deviation is less than ±5°.
(a) Single-hole specimen
(b) Multihole specimen
3.2. Dangerous Path Prediction
Finite element analysis is performed to calculate the dangerous path using the model parameters generated in Table 2. Considering the temperature of the turbine blade film cooling hole reaches about 980°C, the data implemented in the analysis also chooses the 980°C model parameters. The maximum shear stress distribution of the single-hole and multihole specimens is depicted in Figures 6 and 7. An obvious observation is that the maximum shear stress of the single-hole specimen lies at the middle region of the single hole whereas the maximum stress occurs near the hole edges.
Figure 8 depicts the maximum shear stress along the dangerous path. The crack would originate from the maximum stress and propagate along the dangerous path. For the single-hole specimen, the maximum shear stress decreased rapidly along the dangerous path from 0 to 0.5 mm. Then, it demonstrates a stable trend until 2.3 mm. Nevertheless, for the multihole specimen, the maximum principal stress first increases and peaks at 0.35 mm. Afterwards, it goes up to the maximum stress value at the hole edge and presents a saddle-like shape curve. Figure 9 presents the calculated dangerous path of both the single-hole specimen and multihole specimens. The main difference is that the dangerous path of the single-hole specimen is normal to the tensile stress whereas it is the connecting line of the center hole and side hole.
(a) Single-hole specimen
(b) Multihole specimen
(a) Single-hole specimen
(b) Multihole specimen
3.3. Life Prediction
According to the maximum principal stress distribution and the calculated dangerous path, the MTCD method is utilized to predict the HCF life of the specimen. The parameters of critical distance under temperatures are obtained and listed in Table 3.
Equations (25)–(28) present the relationship between life cycles and notch factor and critical distance of the single-hole and multihole specimens. It should be noted that when TCD method is implemented to predict the fatigue life, the two parameters and in the equation are determined by the material properties and the stress ratio of HCF experiments. Considering that the hole drill process has little impact on the macroscopic tensile properties of the specimens, it is not included in the developed model and the HCF life prediction.
4. Experiment Verification
4.1. Failure Analysis
In this scenario, the fatigue test is carried out of the single-hole and multihole specimens under tensile stress loading. By implementing the stress-controlled loading mode, the axial tension-tension HCF experiments with a stress ratio of 0.1 are undertaken. The cyclic loading frequency is within 80-90 Hz. The specimens with single-film cooling holes are tested at 900°C, 980°C, and 1050°C whereas the cases with multi-film cooling holes are tested at 980°C.
The morphology of the single-hole fracture is depicted in Figure 10. The crack initiation region 1, propagation region 2, and final rupture region 3 are clearly observed in Figure 10(a). Meanwhile, two distinct areas with different colors are identified due to the oxidation effects. As expected, the crack initiation and propagation regions demonstrate dark gray appearance because of the longer time exposition to oxygen. On the contrary, the final fatigue rupture region is light gray. More importantly, the fracture morphology shows a trapezoid shape, and the hole diameter is not uniform along the thickness direction. Besides, the propagation area of the large hole section is much larger than that of the small hole, indicating that the large hole section propagates earlier than other parts.
(a) Fracture morphology
(b) Enlarged morphology of crack initiation region
(c) Enlarged morphology of crack propagation region
(d) Enlarged morphology of fatigue rupture region
A closer inspection on the microscopic appearance of the crack initiation region is depicted in Figure 10(b). The specimen structural integrity is damaged due to the existence of the film cooling holes. Consequently, the crack originated from the defects at the hole edge. Multisource cracks such as 4 and 5 in the figure are generated and propagated until rupture. Meanwhile, the enlarged morphology of the propagation region indicates that the cracks propagate in the normal direction to the tensile stress. Also, the crack propagation implies a typical river-like feature. Finally, the micromorphology of the final fatigue rupture region is presented in Figure 10(d). This area is rough, and stepped tearing characteristics are observed. The failure analysis of the single-hole specimen uncovers that the failure belongs to typical multisource fatigue rupture.
Similar to the single-hole specimen failure analysis results, the crack initiation region, propagation region, and final rupture region of the multihole specimen are presented in Figure 11(a). However, due to the interference effects of the multiholes, the stress distribution of the specimen is complex than the single-hole specimen. Consequently, there are two propagation regions 1 and 2 found in the morphology. Region 1 represents the crack propagating from the interhole whereas region 2 means the crack propagating from the hole edges. The enlarged morphology of region 1 is depicted in Figure 11(b). The crack originates from the faults of the inter hole area and belongs to multisource crack. On one hand, closer inspection shows that there are many radiated lines in the propagation region. The cracks gradually merged due to the stress and converged to a shell-shaped fracture. The color of this area is dark since it is exposed to the oxygen for long time. On the other hand, the cracks propagate along the dangerous path and incorporate the cracks initiating from the hole edge, as shown in Figure 11(c). The sharp drop in the cross-section area of the specimen results in the significant decrease in stress, and the hole around region 2 appears to be ruptured. As a result, the river-like and stepped tearing characteristics are observed. Finally, as shown in Figure 11(d), the fatigue rupture region 3 is in light gray color, indicating that oxidation effect did not occur. Moreover, the coupling effects of oxidation and fatigue stress make the recast layer falling off and becoming rough. It is especially the case in the middle hole of the specimen. The rupture of the multihole specimen is also multisource fatigue rupture.
(a) Fracture morphology
(b) Enlarged morphology of crack initiation region
(c) Enlarged morphology of crack propagation region
(d) Enlarged morphology of fatigue rupture region
4.2. Test Result Analysis
The test results of the single-hole and multihole specimens are fitted using the Basquin equation, as listed in
In the above equation, is the stress amplitude, is the fatigue strength coefficient of the material, and is the Basquin coefficient. The parameters of the Basquin equation are shown in Table 4.
The fitted Basquin equations are demonstrated in the following equations:
The fitted S-N curve implementing the tested data is shown in Figure 12. As expected, the cycle life of the specimen decreases with the increase in rupture stress. For the single-hole specimen, the rupture stress significantly dropped for the higher temperature case. For example, the rupture stress of 106 cycles would go down from 525 MPa to 390 MPa when temperature increases from 900°C to 1080°C. Another observation is that the cycle life of the multihole specimen is much lower than that of single-hole specimen under the same temperature.
4.3. Result Comparisons
Figure 13 shows the fracture specimen fracture morphology after HCF experiment under the condition of 980°C/400 MPa. The fracture surface of the specimen shows that for the single-hole specimen, the crack first occurs at the hole edge and then propagates along the [001] plane. The crack front is normal to the tensile axis and then ruptured instantly under tensile stress, as shown in Figure 13(a). For the multihole specimen, the crack propagates gradually along the path between the two holes on the [101] and [-101] surface and finally ruptures with another hole due to the interhole interference. In the unbroken area, the existence of interference fringes can be observed, which also proves that the crack propagated along the path between holes.
(a) Single-hole results
(b) Multihole results
The dangerous paths of the two types of specimens are depicted in Figure 14. It is achieved by the test results. Obviously, the dangerous path of the single-hole specimen is a straight line normal to the tensile axis along the circumference of the hole. On the contrary, the dangerous path of the multihole specimen was the line between the middle hole and the edge hole. It is generated due to the interference of the inner holes.
(a) Single-hole results
(b) Multihole results
The relationship between the predicted HCF life and the experimental HCF life is shown in Figure 15. It can be seen that most of the predicted life results implementing MTCD method locate within the twice error band, indicating that the proposed method is capable of estimating the HCF life of specimens with film cooling holes under high temperature conditions.
5. Conclusion
This paper performs a comprehensive investigation on the fatigue life prediction of the turbine blade with film cooling holes. The MTCD was proposed taking the notch effect into consideration. The main outcomes are as follows: (1)The MTCD method considers both the notch sensitivity and multiaxial stress effects by introducing two parameters. Then, it is implemented in the specimen fatigue life prediction iterations. Both the life of single-hole specimen and multihole specimen are predicted and verified by the tests. Comparison results show that most of the calculated fatigue life is within the twice error band of the tested life(2)Failure analysis shows that both the single-hole specimen and multihole specimen ruptured due to fatigue failure. The stress concentration and multiaxial stress resulting from the film cooling holes are the primary reasons that the cracks originated. The cracks then propagated along the dangerous path and resulted in failure(3)The dangerous path of the single-hole specimen is quite different from the multihole specimen. The crack first appears at the hole edge and propagates along the [001] plane. The crack front is normal to the tensile axis and then ruptured instantly under tensile stress. However, for the multihole specimen, the crack propagates gradually along the path between the two holes on the [101] and [-101] surface and finally ruptures with another hole due to the interhole interference
Nomenclature
Roman Symbols| : | Deformation rate tensor |
| : | Critical distance of line method |
| : | Critical distance of point method |
| : | Elastic modulus |
| : | Total deformation gradient |
| : | Elastic deformation gradient |
| : | Plastic deformation gradient |
| : | Shear modulus |
| : | Hardening modulus |
| : | Theoretical stress concentration factor |
| : | Plane fracture toughness of the material |
| : | Fatigue strength ratio of smooth to notched specimens |
| : | Critical stress intensity gradient |
| : | Characteristic distance |
| : | Critical distance |
| : | Critical distance under cyclic loading condition |
| : | Critical distance under static loading condition |
| : | Strain rate sensitivity |
| : | Lattice vector changes and the slip direction |
| : | Initial stage fatigue life |
| : | Fatigue life of static loading condition |
| : | Fatigue life |
| : | Normal vector |
| : | Schmidt factor of slip system |
| : | Notch sensitivity coefficient |
| : | Rotation tensor. |
| : | Correction parameter |
| : | Cauchy stress tensor |
| : | Maximum principal stress |
| : | Reference equivalent stress |
| : | Equivalent stress |
| : | Material fatigue limit |
| : | Ultimate stress |
| : | Yield stress |
| : | Cumulative slip strain |
| : | Reference shear strain rate |
| : | Reference shear stress |
| : | Hardening rate |
| : | Hardening coefficient |
| : | Multiaxial stress factor |
| : | Weighed Cauchy stress tensor |
| : | Correction stress |
| : | Poisson’s ratio. |
| AM: | Area method |
| HCF: | High cycle fatigue |
| HSV: | Highly stressed volume |
| LM: | Line method |
| MSCL: | Maximum stress concentration location |
| MTCD: | Modified theory of critical distance |
| PM: | Point method |
| TCD: | Theory of critical distance |
| TET: | Turbine entry temperature |
| VM: | Volume method. |
Data Availability
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to thank AECC Shenyang Engine Research Institute for the funding and support.