# Prediction of High-Cycle Fatigue Performance of 1Cr11Ni2W2MoV Stainless Steel Plate after Foreign Object Damage

Authors: Zhenhua Zhao; Lingfeng Wang; Chao Liu; Lulu Liu; Wei Chen
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

High-cycle fatigue (HCF) properties of 1Cr11Ni2W2MoV stainless steel impacted by a high-speed steel ball were studied by the foreign object damage (FOD) test and HCF test. The results show that the damage depthZ has the most obvious effect on the HCF limit of notched flat plate specimens, and the fatigue limit decreases with the increase of depth Z. The microcharacteristics of the FOD notch and HCF fracture of 1Cr11Ni2W2MoV stainless steel were observed by a scanning electron microscope (SEM). The results show that the microdamage features such as plastic deformation, loss of material, and microcracks promote the initiation and propagation of fatigue cracks, and the fatigue source area lies near the root of the notch. The Peterson formula and Worst Case Notch (WCN) mode were used to predict the HCF limit of flat plate specimens after FOD. The crack growth threshold was obtained by the crack growth test. The results show that the prediction results of both methods are conservative. For the notch with damage depth Z<1 mm, the prediction accuracy of the WCN model is higher (errorrange<30%). For the notch with damage depth Z>1 mm, the prediction results of both methods have large errors (>30%) with the WCN model being slightly more accurate.


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## Body

## 1. Introduction

During the service of aeroengine, it is unavoidable to draw nearby particles into the flow path of the engine, including birds, ice, and small hard objects. The impact of birds and ice belongs to soft impact, namely, bird strike [1] and ice impact [2], and the impact of small hard objects such as bolts, nuts, sand, and stones is called foreign object damage (FOD) [3]. FOD often occurs near the ground during the operation, forming a dent or notch at the leading edge of the blade [4]. FOD has become a key issue which must be considered in the design of aeroengine blades.The FOD on aeroengine blades is a complex transient dynamic problem, which involves the nonlinearity, contact, and large deformation of materials. Many scholars have paid more and more attention to FOD in recent years. The most intuitive way to study the FOD problem is to test the impact damage prefabricated. The most used method is the air cannon method [5]. Peters et al. [6] used an air cannon to launch steel balls to impact the surface of the Ti-6A1-4V plate and observed the bottom of the notch by a scanning electron microscope (SEM) and found obvious microcrack damage caused by high-speed impact. Farahani et al. [7] carried out ballistic impact tests by air cannon, launching steel foreign objects of different shapes on Udimet-500 nickel-based alloy (the first-stage rotor material of the Fiat gas turbine engine) with different impact velocities and incident angles. The result shows that there are many microcracks at the damage caused by cube foreign object incident at a 45° angle. Bache et al. [8] conducted FOD tests using ballistic systems. The Ti-6A1-4V plate samples were impacted with 2 mm diameter steel balls at different incident speeds. The results show that there are ductile indentations at the impact position under low energy impacts, while loss of material (LOM) was observed on the edge of damaged samples under high energy impacts.With the development of finite element analysis technology, the process of simulating the impact of foreign objects on the blade by finite element software has also become a common method in studying FOD. Oakley and Nowell [9] and Duó et al. [10] used software such as LS-DYNA, DYTRAN, ABAQUS, and material models such as Bammann and Johnson-Cook to perform dynamic simulation analysis for the foreign object impacting process. The damage morphology obtained by the simulation is in good agreement with the experimental results, which proves the applicability of simulation analysis on blade FOD. Chen and Hutchinson [11] used ABAQUS finite element software to simulate the process of steel ball striking the edge of the Ti-6A1-4V flat plate. After studying the residual stress distribution at the FOD location, it was found that a significant residual tension area is formed at the raised tip at the bottom of the notch due to lateral expansion and then resulted in fatigue cracks. Marandi et al. [12] used the finite element method and the experimental method to investigate the stress concentration factor at the notch root of the turbine blade leading edge. The results showed that the finite element calculation results agree well with the experimental data.Aeroengine blades are subjected to high-cycle fatigue (HCF) loads caused by vibration and short-time resonance during operation. Stress concentrations, residual stresses, and microscopic damage at the FOD location will accelerate the initiation of cracks under HCF, resulting in premature fatigue damage to engine blades in its designed cycles. That means FOD is an important factor limiting the high-cycle fatigue performance of engine blades [13]. Ruschau et al. [14] performed high-cycle fatigue tests on Ti-6Al-4V blades which were impacted by glass beads at different incident angles. Bache et al. [8] studied the effects of different foreign object sizes and shapes on the fatigue strength of the FOD flat specimen. Martinez et al. [15] performed high-cycle fatigue tests on FOD flat specimens with different radii at the leading edges and then studied the effect of different leading-edge radii on the high-cycle fatigue strength of FOD blades. Witek, a scholar at the Rzeszow University of Science and Technology in Poland, used a shaker to directly perform high-cycle vibration fatigue test research on real blades and achieved good results. He used ABAQUS software to calculate the stress distribution during blade vibration. The results showed that the blades would generate high stress loads due to vibration under the first-order mode, and there was a high stress gradient in the blade’s first-order nodal line cross-section [16]. Based on the research on high-cycle fatigue test or complete blade, Witek [17] prefabricated different shapes of notches on the leading edge of the PZL-10W engine compressor blade and conducted a high-cycle fatigue test to study the effect of the notch shape on the fatigue strength of the blade. When studying the cause of fatigue failure of a certain type of naval helicopter engine blade, Witek et al. [18] found that the corrosion pits at the blade leading edge caused by salt spray seriously endangered the HCF performance of blades, just like the defects at the blade leading edge caused by FOD. In order to improve the anti-FOD capability of engine blades, surface strengthening processes such as shot peening [19] and laser shock peening [20] have been performed to blades in recent years. The impact of these processes on the fatigue strength of FOD blades has also become focuses on FOD-related research.Structural fatigue problems usually occur during the engineering application of metal components, especially in high stress concentration positions such as notches and welded joints [21, 22]. Based on the idea of structural elements, Neuber [23] proposed an average stress model to predict notched fatigue strength and gave a Neuber formula to calculate notched fatigue coefficients Kf. Peterson [24] proposed a simpler Peterson formula based on Neuber’s research, which is widely used in engineering. Ruschau et al. [25] took the FOD damage made by an air cannon as a notch and predicted the HCF strength of the damaged TC4 specimen through the above-mentioned empirical formula of fatigue notch coefficient. It was found that the predicted values were higher than experimental results. Based on the research by Neuber and Peterson, Taylor [26] developed a method called the Theory of Critical Distance (TCD), which took the average stress of stresses within a certain area around the root cross-section of the notch as the effective stress to predict the fatigue limit. Hudak et al. [27] proposed a Worst Case Notch (WCN) model for HCF strength prediction for damaged parts. This model assumes that microcracks generate at the root of damage since the initial fatigue loading stage; thus, the HCF fatigue strength of damaged parts could be predicted by the boundary conditions of crack propagation. The boundary conditions of crack propagation can be used to estimate the HCF strength. Nowell et al. [28] applied a short crack arrest model which is similar to the WCN model to predict the HCF strength of three different-sized FOD notches. The prediction result is significantly lower than the experimental value. They believes that the main factor affecting the prediction accuracy is that the model does not consider the effect of compressive residual stress at the damage root on the crack propagation boundary. Chaves et al. [29] proposed a method of fatigue limit prediction through stress gradient and linear elastic analysis under remote loading based on the microstructural fracture mechanics model (Navarro and De los Rios [30]). The prediction model was applied to circle holes and V-notches under I-mode loading. The results show that the prediction error is within 12% and the accuracy is high.This paper focuses on the effect of FOD on the HCF limit of a 1Cr11Ni2W2MoV stainless steel flat plate specimen, as well as on the prediction method for the HCF limit. Through FOD and HCF tests, the influences of damage parameters on the notch fatigue limit were analysed, and the microcharacteristics of notch and fracture were observed. Both the average stress model and the WCN model were used to predict the fatigue limit of FOD notch, and the high accuracy range of the WCN model was given after the research.


## 2. Experimental Investigations

### 2.1. FOD Test

#### 2.1.1. Specimen Details

1Cr11Ni2W2MoV is a kind of multielement alloying martensitic stainless steel, which is used after quenching and tempering. The process plasticity and welding performance of 1Cr11Ni2W2MoV are great. It can be manufactured to die forgings with complex shapes and welded structures. It is suitable for manufacturing shaft, disk, blade, compression spring, and other components working under 550°C and in a high humidity environment. Material properties of 1Cr11Ni2W2MoV at room temperature are shown in Table1.Table 1
Material properties of 1Cr11Ni2W2MoV and GCr15 at room temperature.
MaterialDensityρ (kg/m3)Elastic modulusE (GPa)Poisson’s ratioμ1Cr11Ni2W2MoV78001980.301GCr1578502100.300A flat plate specimen is designed for the FOD test. The size of the specimen is shown in Figure1.Figure 1
Size of flat plate specimen.The foreign objects used in the FOD test are GCr15 steel balls with diameters of 2 mm, 3 mm, and 4 mm, respectively, which are fixed by a cylindrical nylon sabot. Material properties of GCr15 at room temperature are shown in Table1.


#### 2.1.2. Device Details

FOD tests on flat plate specimens were carried out by an air cannon system to obtain damages more consistent with features of that in real service. The test system consists of a gas cylinder, a gas storage tank, a loading mechanism, a pressure sensor, a target chamber, solenoid valves, and other components. The picture of the air cannon system is shown in Figure2.Figure 2
Air cannon system.After the FOD test, a HIROX KH-7700 three-dimensional stereomicroscope was used to observe the macrofeatures of the damage notch, whose minimum resolution is 1μm.


#### 2.1.3. Test Method

The impact speed was 290 m/s in the test, with an incident angle of 0°. The impact diagram is shown in Figure3.Figure 3
Impact diagram.


### 2.2. Fatigue Test

#### 2.2.1. Specimen Details

The HCF test specimens are divided into the smooth rectangular section specimen and notched specimen. The smooth rectangular section specimen designed according to GB3075-82 (a Chinese national standard for metal axial fatigue test) [31] is shown in Figure 4 and used to obtain the fatigue limit of 1Cr11Ni2W2MoV at stress ratio R=0.8 (working stress ratio of aeroengine at cruise state). The notched specimen is a flat plate sample after the FOD test in Section 2.1. In fact, the residual stress at the notch root caused by FOD will affect the stress ratio, which cannot be maintained at 0.8. However, considering the complexity of quantitative analysis on and the inconvenience of measurement on residual stress in engineering application, in-depth research on this point is not conducted in the paper. Thus, all tests are carried out with the stress ratio R=0.8.Figure 4
Designing parameters of smooth rectangular section specimen.


#### 2.2.2. Device Details

The HCF tests on both smooth rectangular section specimens and notched specimens were carried out by QBQ-100 a high-frequency fatigue testing machine.


#### 2.2.3. Test Method

The step-loading method (Maxwell and Nicholas [32]) was used to perform HCF tests to obtain the HCF limit of each smooth rectangular section specimen and notched specimen at the life of 107 cycles.The method is described as follows. If the specimen survived the initial step of loading, the stress level would be increased by 10% and the test repeated. This would continue until the specimen failed in less than 107 cycles. If we assume that the fatigue damage accumulates linearly in the last step of loading, the fatigue limit for the life of 107 cycles could be calculated by the following formula:
(1)σe=σpr+Nf107⋅σf−σpr,where σe is the fatigue limit at 107 cycles, σpr is the maximum stress level of the previous stress step, Nf is the number of cycles when the last stress step fails, and σf is the maximum stress level of the last stress step.The load spectrum of the notched specimen is shown in Figure5.Figure 5
Load spectrum for HCF test.


### 2.3. SEM Observation

The microcharacteristics of FOD notches and fatigue fractures of specimens were observed by JSM-7001 SEM. The main characteristics of FOD notches include loss of material (LOM), lamellar structure, and microcrack, and the main characteristics of fatigue fractures include the fatigue source area and crack growth area.The device parameters are as follows:(i)
Resolution: 1.4 nm(ii)
Acceleration voltage: 30 kV(iii)
Magnification: 40x~20000x


### 2.4. Crack Growth Test

#### 2.4.1. Specimen Details

The crack growth test specimen is a compact tensile sample. The shape and size of the sample are designed according to relevant requirements in GB/T6398 (a Chinese national standard for fatigue crack growth rate test) [33], and then, the sample is processed by wire cutting before the test. The designing size of the crack growth test specimen is shown in Figure 6. The crack growth direction is perpendicular to the rolling direction.Figure 6
Designing size of the crack growth test specimen.


#### 2.4.2. Device Details

The tests were carried out by the 809 axial/torsional test system (MTS), and the crack length was measured by a Crack Opening Displacement (COD) gauge, as shown in Figure7. Before the crack growth test, the precracking test [34, 35] was carried out and the length of precracking is 1.7 mm. The controlled stress ratio during the test is 0.8, the frequency of precracking is 8 Hz, and the frequency of crack growth is 10 Hz. When the crack length reaches about 36 mm, the test is terminated.Figure 7
809 axial/torsional test system and COD gauge.


#### 2.4.3. Test Method

TheK gradient method controlled by constant force was used in the test. During the test, the gradient value C of stress intensity factor K is constant, and the calculation method of stress intensity factor range ΔK is given by the following formula. The compliance method was used to measure the crack length:
(2)ΔK=ΔPBW×2+a1−α3/20.886+4.64α−13.32α2+14.72α3−5.6α4,where α=a/W, ΔP is the range of load, a is the calculated crack length, B is the thickness of the specimen, and W is the width of the specimen.During the steady-state crack growth stage, the relationship between the crack growth rateda/dN and stress intensity factor amplitude ΔK can be described by the Paris power function:
(3)dadN=CΔKn.Take logarithm operation on both sides of the above formula,(4)lgdadN=lgC+nlgΔK.Generally, at least 5 pairs ofda/dNi and corresponding ΔKi which satisfy 10−7mm/cycle≤da/dN≤10−6mm/cycle should be taken as the threshold value for fatigue crack calculation [35]. lgda/dN in formula (4) is taken as the independent variable for the matching of lgda/dN−lgΔK data points by the linear regression method. The fatigue crack growth threshold ΔKth is ΔK corresponding to da/dN=10−7mm/cycle according to the matching results.


## 2.1. FOD Test

### 2.1.1. Specimen Details

1Cr11Ni2W2MoV is a kind of multielement alloying martensitic stainless steel, which is used after quenching and tempering. The process plasticity and welding performance of 1Cr11Ni2W2MoV are great. It can be manufactured to die forgings with complex shapes and welded structures. It is suitable for manufacturing shaft, disk, blade, compression spring, and other components working under 550°C and in a high humidity environment. Material properties of 1Cr11Ni2W2MoV at room temperature are shown in Table1.Table 1
Material properties of 1Cr11Ni2W2MoV and GCr15 at room temperature.
MaterialDensityρ (kg/m3)Elastic modulusE (GPa)Poisson’s ratioμ1Cr11Ni2W2MoV78001980.301GCr1578502100.300A flat plate specimen is designed for the FOD test. The size of the specimen is shown in Figure1.Figure 1
Size of flat plate specimen.The foreign objects used in the FOD test are GCr15 steel balls with diameters of 2 mm, 3 mm, and 4 mm, respectively, which are fixed by a cylindrical nylon sabot. Material properties of GCr15 at room temperature are shown in Table1.


### 2.1.2. Device Details

FOD tests on flat plate specimens were carried out by an air cannon system to obtain damages more consistent with features of that in real service. The test system consists of a gas cylinder, a gas storage tank, a loading mechanism, a pressure sensor, a target chamber, solenoid valves, and other components. The picture of the air cannon system is shown in Figure2.Figure 2
Air cannon system.After the FOD test, a HIROX KH-7700 three-dimensional stereomicroscope was used to observe the macrofeatures of the damage notch, whose minimum resolution is 1μm.


### 2.1.3. Test Method

The impact speed was 290 m/s in the test, with an incident angle of 0°. The impact diagram is shown in Figure3.Figure 3
Impact diagram.


## 2.1.1. Specimen Details

1Cr11Ni2W2MoV is a kind of multielement alloying martensitic stainless steel, which is used after quenching and tempering. The process plasticity and welding performance of 1Cr11Ni2W2MoV are great. It can be manufactured to die forgings with complex shapes and welded structures. It is suitable for manufacturing shaft, disk, blade, compression spring, and other components working under 550°C and in a high humidity environment. Material properties of 1Cr11Ni2W2MoV at room temperature are shown in Table1.Table 1
Material properties of 1Cr11Ni2W2MoV and GCr15 at room temperature.
MaterialDensityρ (kg/m3)Elastic modulusE (GPa)Poisson’s ratioμ1Cr11Ni2W2MoV78001980.301GCr1578502100.300A flat plate specimen is designed for the FOD test. The size of the specimen is shown in Figure1.Figure 1
Size of flat plate specimen.The foreign objects used in the FOD test are GCr15 steel balls with diameters of 2 mm, 3 mm, and 4 mm, respectively, which are fixed by a cylindrical nylon sabot. Material properties of GCr15 at room temperature are shown in Table1.


## 2.1.2. Device Details

FOD tests on flat plate specimens were carried out by an air cannon system to obtain damages more consistent with features of that in real service. The test system consists of a gas cylinder, a gas storage tank, a loading mechanism, a pressure sensor, a target chamber, solenoid valves, and other components. The picture of the air cannon system is shown in Figure2.Figure 2
Air cannon system.After the FOD test, a HIROX KH-7700 three-dimensional stereomicroscope was used to observe the macrofeatures of the damage notch, whose minimum resolution is 1μm.


## 2.1.3. Test Method

The impact speed was 290 m/s in the test, with an incident angle of 0°. The impact diagram is shown in Figure3.Figure 3
Impact diagram.


## 2.2. Fatigue Test

### 2.2.1. Specimen Details

The HCF test specimens are divided into the smooth rectangular section specimen and notched specimen. The smooth rectangular section specimen designed according to GB3075-82 (a Chinese national standard for metal axial fatigue test) [31] is shown in Figure 4 and used to obtain the fatigue limit of 1Cr11Ni2W2MoV at stress ratio R=0.8 (working stress ratio of aeroengine at cruise state). The notched specimen is a flat plate sample after the FOD test in Section 2.1. In fact, the residual stress at the notch root caused by FOD will affect the stress ratio, which cannot be maintained at 0.8. However, considering the complexity of quantitative analysis on and the inconvenience of measurement on residual stress in engineering application, in-depth research on this point is not conducted in the paper. Thus, all tests are carried out with the stress ratio R=0.8.Figure 4
Designing parameters of smooth rectangular section specimen.


### 2.2.2. Device Details

The HCF tests on both smooth rectangular section specimens and notched specimens were carried out by QBQ-100 a high-frequency fatigue testing machine.


### 2.2.3. Test Method

The step-loading method (Maxwell and Nicholas [32]) was used to perform HCF tests to obtain the HCF limit of each smooth rectangular section specimen and notched specimen at the life of 107 cycles.The method is described as follows. If the specimen survived the initial step of loading, the stress level would be increased by 10% and the test repeated. This would continue until the specimen failed in less than 107 cycles. If we assume that the fatigue damage accumulates linearly in the last step of loading, the fatigue limit for the life of 107 cycles could be calculated by the following formula:
(1)σe=σpr+Nf107⋅σf−σpr,where σe is the fatigue limit at 107 cycles, σpr is the maximum stress level of the previous stress step, Nf is the number of cycles when the last stress step fails, and σf is the maximum stress level of the last stress step.The load spectrum of the notched specimen is shown in Figure5.Figure 5
Load spectrum for HCF test.


## 2.2.1. Specimen Details

The HCF test specimens are divided into the smooth rectangular section specimen and notched specimen. The smooth rectangular section specimen designed according to GB3075-82 (a Chinese national standard for metal axial fatigue test) [31] is shown in Figure 4 and used to obtain the fatigue limit of 1Cr11Ni2W2MoV at stress ratio R=0.8 (working stress ratio of aeroengine at cruise state). The notched specimen is a flat plate sample after the FOD test in Section 2.1. In fact, the residual stress at the notch root caused by FOD will affect the stress ratio, which cannot be maintained at 0.8. However, considering the complexity of quantitative analysis on and the inconvenience of measurement on residual stress in engineering application, in-depth research on this point is not conducted in the paper. Thus, all tests are carried out with the stress ratio R=0.8.Figure 4
Designing parameters of smooth rectangular section specimen.


## 2.2.2. Device Details

The HCF tests on both smooth rectangular section specimens and notched specimens were carried out by QBQ-100 a high-frequency fatigue testing machine.


## 2.2.3. Test Method

The step-loading method (Maxwell and Nicholas [32]) was used to perform HCF tests to obtain the HCF limit of each smooth rectangular section specimen and notched specimen at the life of 107 cycles.The method is described as follows. If the specimen survived the initial step of loading, the stress level would be increased by 10% and the test repeated. This would continue until the specimen failed in less than 107 cycles. If we assume that the fatigue damage accumulates linearly in the last step of loading, the fatigue limit for the life of 107 cycles could be calculated by the following formula:
(1)σe=σpr+Nf107⋅σf−σpr,where σe is the fatigue limit at 107 cycles, σpr is the maximum stress level of the previous stress step, Nf is the number of cycles when the last stress step fails, and σf is the maximum stress level of the last stress step.The load spectrum of the notched specimen is shown in Figure5.Figure 5
Load spectrum for HCF test.


## 2.3. SEM Observation

The microcharacteristics of FOD notches and fatigue fractures of specimens were observed by JSM-7001 SEM. The main characteristics of FOD notches include loss of material (LOM), lamellar structure, and microcrack, and the main characteristics of fatigue fractures include the fatigue source area and crack growth area.The device parameters are as follows:(i)
Resolution: 1.4 nm(ii)
Acceleration voltage: 30 kV(iii)
Magnification: 40x~20000x


## 2.4. Crack Growth Test

### 2.4.1. Specimen Details

The crack growth test specimen is a compact tensile sample. The shape and size of the sample are designed according to relevant requirements in GB/T6398 (a Chinese national standard for fatigue crack growth rate test) [33], and then, the sample is processed by wire cutting before the test. The designing size of the crack growth test specimen is shown in Figure 6. The crack growth direction is perpendicular to the rolling direction.Figure 6
Designing size of the crack growth test specimen.


### 2.4.2. Device Details

The tests were carried out by the 809 axial/torsional test system (MTS), and the crack length was measured by a Crack Opening Displacement (COD) gauge, as shown in Figure7. Before the crack growth test, the precracking test [34, 35] was carried out and the length of precracking is 1.7 mm. The controlled stress ratio during the test is 0.8, the frequency of precracking is 8 Hz, and the frequency of crack growth is 10 Hz. When the crack length reaches about 36 mm, the test is terminated.Figure 7
809 axial/torsional test system and COD gauge.


### 2.4.3. Test Method

TheK gradient method controlled by constant force was used in the test. During the test, the gradient value C of stress intensity factor K is constant, and the calculation method of stress intensity factor range ΔK is given by the following formula. The compliance method was used to measure the crack length:
(2)ΔK=ΔPBW×2+a1−α3/20.886+4.64α−13.32α2+14.72α3−5.6α4,where α=a/W, ΔP is the range of load, a is the calculated crack length, B is the thickness of the specimen, and W is the width of the specimen.During the steady-state crack growth stage, the relationship between the crack growth rateda/dN and stress intensity factor amplitude ΔK can be described by the Paris power function:
(3)dadN=CΔKn.Take logarithm operation on both sides of the above formula,(4)lgdadN=lgC+nlgΔK.Generally, at least 5 pairs ofda/dNi and corresponding ΔKi which satisfy 10−7mm/cycle≤da/dN≤10−6mm/cycle should be taken as the threshold value for fatigue crack calculation [35]. lgda/dN in formula (4) is taken as the independent variable for the matching of lgda/dN−lgΔK data points by the linear regression method. The fatigue crack growth threshold ΔKth is ΔK corresponding to da/dN=10−7mm/cycle according to the matching results.


## 2.4.1. Specimen Details

The crack growth test specimen is a compact tensile sample. The shape and size of the sample are designed according to relevant requirements in GB/T6398 (a Chinese national standard for fatigue crack growth rate test) [33], and then, the sample is processed by wire cutting before the test. The designing size of the crack growth test specimen is shown in Figure 6. The crack growth direction is perpendicular to the rolling direction.Figure 6
Designing size of the crack growth test specimen.


## 2.4.2. Device Details

The tests were carried out by the 809 axial/torsional test system (MTS), and the crack length was measured by a Crack Opening Displacement (COD) gauge, as shown in Figure7. Before the crack growth test, the precracking test [34, 35] was carried out and the length of precracking is 1.7 mm. The controlled stress ratio during the test is 0.8, the frequency of precracking is 8 Hz, and the frequency of crack growth is 10 Hz. When the crack length reaches about 36 mm, the test is terminated.Figure 7
809 axial/torsional test system and COD gauge.


## 2.4.3. Test Method

TheK gradient method controlled by constant force was used in the test. During the test, the gradient value C of stress intensity factor K is constant, and the calculation method of stress intensity factor range ΔK is given by the following formula. The compliance method was used to measure the crack length:
(2)ΔK=ΔPBW×2+a1−α3/20.886+4.64α−13.32α2+14.72α3−5.6α4,where α=a/W, ΔP is the range of load, a is the calculated crack length, B is the thickness of the specimen, and W is the width of the specimen.During the steady-state crack growth stage, the relationship between the crack growth rateda/dN and stress intensity factor amplitude ΔK can be described by the Paris power function:
(3)dadN=CΔKn.Take logarithm operation on both sides of the above formula,(4)lgdadN=lgC+nlgΔK.Generally, at least 5 pairs ofda/dNi and corresponding ΔKi which satisfy 10−7mm/cycle≤da/dN≤10−6mm/cycle should be taken as the threshold value for fatigue crack calculation [35]. lgda/dN in formula (4) is taken as the independent variable for the matching of lgda/dN−lgΔK data points by the linear regression method. The fatigue crack growth threshold ΔKth is ΔK corresponding to da/dN=10−7mm/cycle according to the matching results.


## 3. Results and Discussion

### 3.1. FOD Test Results

#### 3.1.1. FOD Size

FOD tests were conducted on 13 flat plate specimens. The maximum damage size along the length direction of the specimen is defined as the damage widthX, the maximum damage size along the thickness direction of the specimen is defined as the damage thickness Y, and the maximum damage size along the width direction of the specimen is defined as the damage depth Z. The definition of damage sizes is shown in Figure 8.Figure 8
Damage size definition: (a) 3D sketch of the notch; (b) damage widthX and damage thickness Y; (c) damage depth Z.
(a)(b)(c)The FOD test results of flat plate specimens are shown in Table2. The relationship between damage size and steel ball diameter is shown in Figure 9, where the straight line formed by blue stars represents the mean value of damage size. It can be found from Figure 9 that the width X, thickness Y, and depth Z of the damage notch all increase with the diameter of the steel ball. The damage width X and depth Z are relatively stable with the same diameter of the impact steel ball, while damage thickness Y shows great fluctuation due to the deviation of impacting position in the thickness direction.Table 2
HCF limit of notched specimens.
No.Diameter (mm)X (mm)Y (mm)Z (mm)σt (MPa)121.690.960.43615221.660.790.5776332.771.340.82667432.51.310.99658532.871.561.04637632.651.211.26610732.520.961.11761832.771.341.39599932.541.191.496001032.571.111.228141143.541.691.945071242.851.061.616071343.411.61.56567σt: HCF limit.Figure 9
The relationship curve between damage size and steel ball diameter: (a) damage widthX with steel ball diameter; (b) damage thickness Y with steel ball diameter; (c) damage depth Z with steel ball diameter.
(a)(b)(c)


#### 3.1.2. Microscopic Features of Notches

The damage notches on No. 7 samples were observed by SEM, and their microscopic characteristics are shown in Figure10. It can be seen from the figure that there is a relatively smooth extrusion area at the root of the notch formed by the high-speed impact of the steel ball, and there is obvious plastic deformation accompanied by loss of material (LOM) on the side of the notch. If further amplified by the notch root area, it can be found that there are lamellar structures forming at the side of the notch due to the shear stress in the impacting process, accompanying with microcracks. These microdamage characteristics provide favourable conditions for the initiation and propagation of fatigue cracks and reduce the fatigue performances of materials at the notch.Figure 10
The microscopic features of FOD notch on the No. 7 flat plate specimen.


### 3.2. Fatigue Test Results

#### 3.2.1. Fatigue Limit

The fatigue limit results of smooth rectangular section specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 3.Table 3
HCF limit of smooth rectangular section specimens.
No.σt (MPa)Mean value (MPa)B-1950.5949.83B-2949B-4950The fatigue limit results of 13 notched specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 2.The relationship between the HCF limit of FOD flat plate specimens and FOD parameters is shown in Figure10 and the parameters including damage width X, damage depth Z, and steel ball diameter.Figures11(a) and 11(b) show the relationship between the HCF limit and damage size, in which the red line is the fitting curve of the HCF limit. It is obvious that the HCF limit decreases with the increase of damage depth Z and damage width X. However, the dispersion of damage width X and HCF limit is larger than that of damage depth Z. It shows that the effect of notch depth on HCF is larger than that of notch width.Figure 11
The relationship between HCF limit and FOD parameters: (a) HCF limit with damage widthX; (b) HCF limit with damage depth Z; (c) HCF limit with steel ball diameter.
(a)(b)(c)The relationship between the HCF limit and steel ball diameter is shown in Figure11(c), in which the straight line formed by blue stars represents the mean value of the HCF limit. It can be seen that the HCF limit decreases with the increase of steel ball diameter, which is consistent with the relationship between damage depth Z and steel ball diameter shown in Figure 9(c). The HCF limit under the same steel ball diameter has certain dispersion, which is caused by the characteristics of the air cannon system.


#### 3.2.2. Fracture Microscopic Features

The fatigue fractures on No. 9 notched specimens were observed and analysed by SEM, as shown in Figure12. It can be seen from the figure that the fatigue fracture occurs at the root surface of the FOD notch. There are two fatigue source zones, Zones A and B in Figure 11(a), which are located on both sides of the thickness direction of the flat plate specimen and near the notch tip. The fatigue source zone usually occurs at the notch tip where the notch root surface meets the specimen surface. The crack initiation becomes flat and fine since it is torn open and extruded repeatedly by HCF loads. The fracture surface can be distinguished from the fatigue source area (dark) and the crack growth area (light). The darker colour of the fatigue source area is due to the long exposure time in the air and the oxidation of the material after the crack initiation. The red arrow in Figure 12(a) indicates the incident direction of the steel ball. Zone A and Zone B are the locations of the fatigue source area. Since the crack started at different heights during the initiation process, it would meet each other in the propagation process, and then, a river pattern came out converging to the fatigue source (Figure 12(b)). The adiabatic shear band (ASB) can be observed near the notch root surface in Zone C, caused by steel ball impact [36]. What is more, loss of material (LOM) and plastic deformation can be observed in the fatigue source area, indicating that FOD provides favourable conditions for the initiation of fatigue cracks. Under the action of the fatigue load, the fatigue crack gets rapid growth and finally causes the specimen fracture failure.Figure 12
The microscopic features of fatigue fracture on No. 9 flat plate specimen: (a) global view; (b) Zone A.
(a)(b)


### 3.3. Crack Growth Test Results

The crack growth rate measured by the crack growth test is shown in Figure13, from which the crack growth threshold of 1Cr11Ni2W2MoV can be calculated as 6.23 MPam when the stress ratio R=0.8.Figure 13
Crack growth rate of 1Cr11Ni2W2MoV (R=0.8).


### 3.4. Fatigue Limit Prediction

The stress concentration caused by the notch has a significant effect on the fatigue performance of the components. The theoretical stress concentration factorKt is based on the assumption that the material is elastic, and it only depends on the geometry of the component. However, in engineering practice, there is plastic deformation in varying degrees of the notch. Therefore, it is necessary to use the fatigue notch factor Kf to characterize the fatigue strength of notched components.The fatigue notch factorKf is defined as the ratio between the fatigue limit of the smooth rectangular section specimen and the fatigue limit of the notched specimen under the same conditions and cycles. The formula is as follows:
(5)Kf=σsmoothσnotched,where σsmooth=949.83 MPa, as shown in Table 3.


### 3.5. Prediction Model for FOD Specimen Fatigue Limit Based on Peterson Formula

On the basis of Neuber’s research [23], Peterson assumed that stress decreases linearly from the notch root inward. When the average stress within a certain distance from the notch root is greater than or equal to the fatigue limit of the smooth specimen, the fatigue failure will occur. The Peterson formula is as follows:
(6)Kf=1+Kt−11+ap/ρ,where ρ is the radius of the notch root, which is approximately equal to the radius of the steel ball in this paper. ap is the material constant under the influence of tensile stress value, which could be determined by the empirical formula ap=270/σb1.8. σb is tensile strength of 1Cr11Ni2W2MoV, which equals to 955 MPa.For sharp notches, the following equation can be used to calculateKt [37]:
(7)Ktρd=1+2dρfρd,where d is the depth of the notch, ρ is the radius of the notch root, and fρ/d is given by the following equation:
(8)fρd≅1+0.12211+ρ/d5/2.The theoretical stress concentration factorKt can be calculated according to formula (7), and the prediction fatigue notch factor Kf can be calculated according to formula (6). The fatigue limit of the notched specimen can be predicted according to formula (6). The calculation results and error analysis are listed in Table 4.Table 4
Fatigue limit predictions by Peterson formula and WCN model.
No.Diameter (mm)Z (mm)KtKfσP (MPa)ΔSeσW (MPa)σtest (MPa)ErrorPErrorW120.431.871.79530.76130.31651.54615-13.70%5.94%220.52.021.92494.58122.07610.37776-36.27%-21.34%330.822.112.04465.5098.12490.61667-30.21%-26.45%430.992.352.26420.0690.15450.74658-36.16%-31.50%531.042.422.33408.2988.16440.80637-35.90%-30.80%631.262.732.62363.2780.79403.95610-40.45%-33.78%731.112.522.42392.8485.59427.97761-48.38%-43.76%831.392.912.79340.9377.24386.19599-43.08%-35.53%931.493.052.92325.4874.81374.06600-45.75%-37.66%1031.222.672.56370.7281.99409.94814-54.46%-49.64%1141.943.002.90327.2566.22331.08507-35.45%-34.70%1241.612.652.57369.4572.19360.94607-39.13%-40.54%1341.562.602.52376.7873.25366.23567-33.55%-35.41%The error distribution on the fatigue limit of FOD flat plate specimens predicted by the Peterson formula is shown in Figure14, and the error is defined by the following formula. It can be seen from the figure that the prediction error lies within the range of (-55%, -30%), except one 13.7%. Prediction values for all FOD flat plate specimens are lower than test values, which is considered conservative:
(9)error=σprediction−σtestσtest.Figure 14
Prediction error distribution of Peterson formula.


### 3.6. Prediction Model for FOD Specimen Fatigue Limit Based on WCN Model

Microcracks and corresponding cracks which do not propagate along the stress gradient direction of the notch could be regarded as the WCN model [27]. The WCN model takes the nominal stress threshold as a function of elastic stress concentration factor Kt and crack size a. The nominal stress threshold decreases with the increase of Kt and notch sensitivity. However, the nominal stress threshold does not decrease with the increase of Kt when the predicted Kt (by the WCN model) exceeds the limit value. It is mainly caused by the generation of end crack and nonpropagating crack. For the notch with fixed depth and different root radii, the nominal stress threshold increases with radius, and there is a limit radius. When the radius exceeds the limit radius, the crack will not propagate.The aim of the WCN model is to quantify the correlation among notch depth, root radius, stress concentration factor, and other parameters, so as to solve the problem of the notch fatigue limit in a systematic way. Because of the great amount of microdamages in the notch caused by FOD and the fatigue cracks initiated at the microcrack or micronotch, the WCN model is selected to predict the fatigue limit of the notched flat plate specimen.Generally speaking, the stress threshold is a function of the FOD notch size, the loading mode, and the material parameters. In this paper, the crack propagation threshold of 1Cr11Ni2W2MoV is obtained by the crack growth test.The formula of the WCN model is(10)ΔSe=ΔKthFπa0+d,(11)a0=1πΔKthFΔσe2,where ΔSe is stress amplitude at the root of the notch, a0 is small crack parameters, ΔKth is the crack propagation threshold (obtained from the crack growth test result in Section 3.3), F is the finite-thickness correction function (if the crack depth is much smaller than its length, F is about 1.12, and most cases of FOD are those where cracks are small [37]), and Δσe is the fatigue limit of the smooth specimen at 107 cycles.Small crack parametersa0 can be calculated by formula (11), and the stress amplitude ΔSe can be calculated by formula (10). According to the relationship between stress amplitude ΔSe and stress ratio R, the prediction value of the HCF limit of the notch root can be calculated. The prediction results are shown in Table 4, and the error distribution graph is shown in Figure 15. It can be seen from the figure that the prediction error lies within the range of (-50%, 6%). Most prediction values of FOD flat plate specimens are lower than test values, which is considered conservative, just like the prediction results by the Peterson formula.Figure 15
Prediction error distribution of the WCN model.


### 3.7. Comparison

It can be seen from Figure16 and Table 4 that the fatigue limit predicted by the Peterson formula and WCN model is both conservative, and almost all prediction error of the Peterson formula is between (-55%, -30%), while that of the WCN model is between (-50%, 20%). For the notch with damage depth Z greater than 1 mm, the predictions by the Peterson formula and WCN model are similar, and the errors are all more than 30%. The predictions by the WCN model are slightly accurate. For the notch with damage depth Z less than 1 mm, the prediction accuracy of the WCN model is higher than that of the Peterson formula, and the error lies within 32%. The results show that the WCN model is more accurate for the HCF limit prediction for specimens with small damage depth (<1 mm).Figure 16
Prediction error distribution of WCN model and Peterson formula.The possible reasons for the prediction error are as follows. The Peterson formula is based on the point stress model. It is assumed in the Peterson formula that there is only uniaxial stress at the notch of the specimen, without considering the strengthening effect, three-dimensional stress concentration, loading conditions, and complex microdamage characteristics at the notch. Therefore, the prediction results are conservative. In the WCN model, when the notch size is small, the parameter notch depthd in formula (10) is accurate. However, when the notch size is large, the parameter is not accurate enough, so it is necessary to research the model expression more deeply.


## 3.1. FOD Test Results

### 3.1.1. FOD Size

FOD tests were conducted on 13 flat plate specimens. The maximum damage size along the length direction of the specimen is defined as the damage widthX, the maximum damage size along the thickness direction of the specimen is defined as the damage thickness Y, and the maximum damage size along the width direction of the specimen is defined as the damage depth Z. The definition of damage sizes is shown in Figure 8.Figure 8
Damage size definition: (a) 3D sketch of the notch; (b) damage widthX and damage thickness Y; (c) damage depth Z.
(a)(b)(c)The FOD test results of flat plate specimens are shown in Table2. The relationship between damage size and steel ball diameter is shown in Figure 9, where the straight line formed by blue stars represents the mean value of damage size. It can be found from Figure 9 that the width X, thickness Y, and depth Z of the damage notch all increase with the diameter of the steel ball. The damage width X and depth Z are relatively stable with the same diameter of the impact steel ball, while damage thickness Y shows great fluctuation due to the deviation of impacting position in the thickness direction.Table 2
HCF limit of notched specimens.
No.Diameter (mm)X (mm)Y (mm)Z (mm)σt (MPa)121.690.960.43615221.660.790.5776332.771.340.82667432.51.310.99658532.871.561.04637632.651.211.26610732.520.961.11761832.771.341.39599932.541.191.496001032.571.111.228141143.541.691.945071242.851.061.616071343.411.61.56567σt: HCF limit.Figure 9
The relationship curve between damage size and steel ball diameter: (a) damage widthX with steel ball diameter; (b) damage thickness Y with steel ball diameter; (c) damage depth Z with steel ball diameter.
(a)(b)(c)


### 3.1.2. Microscopic Features of Notches

The damage notches on No. 7 samples were observed by SEM, and their microscopic characteristics are shown in Figure10. It can be seen from the figure that there is a relatively smooth extrusion area at the root of the notch formed by the high-speed impact of the steel ball, and there is obvious plastic deformation accompanied by loss of material (LOM) on the side of the notch. If further amplified by the notch root area, it can be found that there are lamellar structures forming at the side of the notch due to the shear stress in the impacting process, accompanying with microcracks. These microdamage characteristics provide favourable conditions for the initiation and propagation of fatigue cracks and reduce the fatigue performances of materials at the notch.Figure 10
The microscopic features of FOD notch on the No. 7 flat plate specimen.


## 3.1.1. FOD Size

FOD tests were conducted on 13 flat plate specimens. The maximum damage size along the length direction of the specimen is defined as the damage widthX, the maximum damage size along the thickness direction of the specimen is defined as the damage thickness Y, and the maximum damage size along the width direction of the specimen is defined as the damage depth Z. The definition of damage sizes is shown in Figure 8.Figure 8
Damage size definition: (a) 3D sketch of the notch; (b) damage widthX and damage thickness Y; (c) damage depth Z.
(a)(b)(c)The FOD test results of flat plate specimens are shown in Table2. The relationship between damage size and steel ball diameter is shown in Figure 9, where the straight line formed by blue stars represents the mean value of damage size. It can be found from Figure 9 that the width X, thickness Y, and depth Z of the damage notch all increase with the diameter of the steel ball. The damage width X and depth Z are relatively stable with the same diameter of the impact steel ball, while damage thickness Y shows great fluctuation due to the deviation of impacting position in the thickness direction.Table 2
HCF limit of notched specimens.
No.Diameter (mm)X (mm)Y (mm)Z (mm)σt (MPa)121.690.960.43615221.660.790.5776332.771.340.82667432.51.310.99658532.871.561.04637632.651.211.26610732.520.961.11761832.771.341.39599932.541.191.496001032.571.111.228141143.541.691.945071242.851.061.616071343.411.61.56567σt: HCF limit.Figure 9
The relationship curve between damage size and steel ball diameter: (a) damage widthX with steel ball diameter; (b) damage thickness Y with steel ball diameter; (c) damage depth Z with steel ball diameter.
(a)(b)(c)


## 3.1.2. Microscopic Features of Notches

The damage notches on No. 7 samples were observed by SEM, and their microscopic characteristics are shown in Figure10. It can be seen from the figure that there is a relatively smooth extrusion area at the root of the notch formed by the high-speed impact of the steel ball, and there is obvious plastic deformation accompanied by loss of material (LOM) on the side of the notch. If further amplified by the notch root area, it can be found that there are lamellar structures forming at the side of the notch due to the shear stress in the impacting process, accompanying with microcracks. These microdamage characteristics provide favourable conditions for the initiation and propagation of fatigue cracks and reduce the fatigue performances of materials at the notch.Figure 10
The microscopic features of FOD notch on the No. 7 flat plate specimen.


## 3.2. Fatigue Test Results

### 3.2.1. Fatigue Limit

The fatigue limit results of smooth rectangular section specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 3.Table 3
HCF limit of smooth rectangular section specimens.
No.σt (MPa)Mean value (MPa)B-1950.5949.83B-2949B-4950The fatigue limit results of 13 notched specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 2.The relationship between the HCF limit of FOD flat plate specimens and FOD parameters is shown in Figure10 and the parameters including damage width X, damage depth Z, and steel ball diameter.Figures11(a) and 11(b) show the relationship between the HCF limit and damage size, in which the red line is the fitting curve of the HCF limit. It is obvious that the HCF limit decreases with the increase of damage depth Z and damage width X. However, the dispersion of damage width X and HCF limit is larger than that of damage depth Z. It shows that the effect of notch depth on HCF is larger than that of notch width.Figure 11
The relationship between HCF limit and FOD parameters: (a) HCF limit with damage widthX; (b) HCF limit with damage depth Z; (c) HCF limit with steel ball diameter.
(a)(b)(c)The relationship between the HCF limit and steel ball diameter is shown in Figure11(c), in which the straight line formed by blue stars represents the mean value of the HCF limit. It can be seen that the HCF limit decreases with the increase of steel ball diameter, which is consistent with the relationship between damage depth Z and steel ball diameter shown in Figure 9(c). The HCF limit under the same steel ball diameter has certain dispersion, which is caused by the characteristics of the air cannon system.


### 3.2.2. Fracture Microscopic Features

The fatigue fractures on No. 9 notched specimens were observed and analysed by SEM, as shown in Figure12. It can be seen from the figure that the fatigue fracture occurs at the root surface of the FOD notch. There are two fatigue source zones, Zones A and B in Figure 11(a), which are located on both sides of the thickness direction of the flat plate specimen and near the notch tip. The fatigue source zone usually occurs at the notch tip where the notch root surface meets the specimen surface. The crack initiation becomes flat and fine since it is torn open and extruded repeatedly by HCF loads. The fracture surface can be distinguished from the fatigue source area (dark) and the crack growth area (light). The darker colour of the fatigue source area is due to the long exposure time in the air and the oxidation of the material after the crack initiation. The red arrow in Figure 12(a) indicates the incident direction of the steel ball. Zone A and Zone B are the locations of the fatigue source area. Since the crack started at different heights during the initiation process, it would meet each other in the propagation process, and then, a river pattern came out converging to the fatigue source (Figure 12(b)). The adiabatic shear band (ASB) can be observed near the notch root surface in Zone C, caused by steel ball impact [36]. What is more, loss of material (LOM) and plastic deformation can be observed in the fatigue source area, indicating that FOD provides favourable conditions for the initiation of fatigue cracks. Under the action of the fatigue load, the fatigue crack gets rapid growth and finally causes the specimen fracture failure.Figure 12
The microscopic features of fatigue fracture on No. 9 flat plate specimen: (a) global view; (b) Zone A.
(a)(b)


## 3.2.1. Fatigue Limit

The fatigue limit results of smooth rectangular section specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 3.Table 3
HCF limit of smooth rectangular section specimens.
No.σt (MPa)Mean value (MPa)B-1950.5949.83B-2949B-4950The fatigue limit results of 13 notched specimens at the life of 107 cycles with stress ratio 0.8 are shown in Table 2.The relationship between the HCF limit of FOD flat plate specimens and FOD parameters is shown in Figure10 and the parameters including damage width X, damage depth Z, and steel ball diameter.Figures11(a) and 11(b) show the relationship between the HCF limit and damage size, in which the red line is the fitting curve of the HCF limit. It is obvious that the HCF limit decreases with the increase of damage depth Z and damage width X. However, the dispersion of damage width X and HCF limit is larger than that of damage depth Z. It shows that the effect of notch depth on HCF is larger than that of notch width.Figure 11
The relationship between HCF limit and FOD parameters: (a) HCF limit with damage widthX; (b) HCF limit with damage depth Z; (c) HCF limit with steel ball diameter.
(a)(b)(c)The relationship between the HCF limit and steel ball diameter is shown in Figure11(c), in which the straight line formed by blue stars represents the mean value of the HCF limit. It can be seen that the HCF limit decreases with the increase of steel ball diameter, which is consistent with the relationship between damage depth Z and steel ball diameter shown in Figure 9(c). The HCF limit under the same steel ball diameter has certain dispersion, which is caused by the characteristics of the air cannon system.


## 3.2.2. Fracture Microscopic Features

The fatigue fractures on No. 9 notched specimens were observed and analysed by SEM, as shown in Figure12. It can be seen from the figure that the fatigue fracture occurs at the root surface of the FOD notch. There are two fatigue source zones, Zones A and B in Figure 11(a), which are located on both sides of the thickness direction of the flat plate specimen and near the notch tip. The fatigue source zone usually occurs at the notch tip where the notch root surface meets the specimen surface. The crack initiation becomes flat and fine since it is torn open and extruded repeatedly by HCF loads. The fracture surface can be distinguished from the fatigue source area (dark) and the crack growth area (light). The darker colour of the fatigue source area is due to the long exposure time in the air and the oxidation of the material after the crack initiation. The red arrow in Figure 12(a) indicates the incident direction of the steel ball. Zone A and Zone B are the locations of the fatigue source area. Since the crack started at different heights during the initiation process, it would meet each other in the propagation process, and then, a river pattern came out converging to the fatigue source (Figure 12(b)). The adiabatic shear band (ASB) can be observed near the notch root surface in Zone C, caused by steel ball impact [36]. What is more, loss of material (LOM) and plastic deformation can be observed in the fatigue source area, indicating that FOD provides favourable conditions for the initiation of fatigue cracks. Under the action of the fatigue load, the fatigue crack gets rapid growth and finally causes the specimen fracture failure.Figure 12
The microscopic features of fatigue fracture on No. 9 flat plate specimen: (a) global view; (b) Zone A.
(a)(b)


## 3.3. Crack Growth Test Results

The crack growth rate measured by the crack growth test is shown in Figure13, from which the crack growth threshold of 1Cr11Ni2W2MoV can be calculated as 6.23 MPam when the stress ratio R=0.8.Figure 13
Crack growth rate of 1Cr11Ni2W2MoV (R=0.8).


## 3.4. Fatigue Limit Prediction

The stress concentration caused by the notch has a significant effect on the fatigue performance of the components. The theoretical stress concentration factorKt is based on the assumption that the material is elastic, and it only depends on the geometry of the component. However, in engineering practice, there is plastic deformation in varying degrees of the notch. Therefore, it is necessary to use the fatigue notch factor Kf to characterize the fatigue strength of notched components.The fatigue notch factorKf is defined as the ratio between the fatigue limit of the smooth rectangular section specimen and the fatigue limit of the notched specimen under the same conditions and cycles. The formula is as follows:
(5)Kf=σsmoothσnotched,where σsmooth=949.83 MPa, as shown in Table 3.


## 3.5. Prediction Model for FOD Specimen Fatigue Limit Based on Peterson Formula

On the basis of Neuber’s research [23], Peterson assumed that stress decreases linearly from the notch root inward. When the average stress within a certain distance from the notch root is greater than or equal to the fatigue limit of the smooth specimen, the fatigue failure will occur. The Peterson formula is as follows:
(6)Kf=1+Kt−11+ap/ρ,where ρ is the radius of the notch root, which is approximately equal to the radius of the steel ball in this paper. ap is the material constant under the influence of tensile stress value, which could be determined by the empirical formula ap=270/σb1.8. σb is tensile strength of 1Cr11Ni2W2MoV, which equals to 955 MPa.For sharp notches, the following equation can be used to calculateKt [37]:
(7)Ktρd=1+2dρfρd,where d is the depth of the notch, ρ is the radius of the notch root, and fρ/d is given by the following equation:
(8)fρd≅1+0.12211+ρ/d5/2.The theoretical stress concentration factorKt can be calculated according to formula (7), and the prediction fatigue notch factor Kf can be calculated according to formula (6). The fatigue limit of the notched specimen can be predicted according to formula (6). The calculation results and error analysis are listed in Table 4.Table 4
Fatigue limit predictions by Peterson formula and WCN model.
No.Diameter (mm)Z (mm)KtKfσP (MPa)ΔSeσW (MPa)σtest (MPa)ErrorPErrorW120.431.871.79530.76130.31651.54615-13.70%5.94%220.52.021.92494.58122.07610.37776-36.27%-21.34%330.822.112.04465.5098.12490.61667-30.21%-26.45%430.992.352.26420.0690.15450.74658-36.16%-31.50%531.042.422.33408.2988.16440.80637-35.90%-30.80%631.262.732.62363.2780.79403.95610-40.45%-33.78%731.112.522.42392.8485.59427.97761-48.38%-43.76%831.392.912.79340.9377.24386.19599-43.08%-35.53%931.493.052.92325.4874.81374.06600-45.75%-37.66%1031.222.672.56370.7281.99409.94814-54.46%-49.64%1141.943.002.90327.2566.22331.08507-35.45%-34.70%1241.612.652.57369.4572.19360.94607-39.13%-40.54%1341.562.602.52376.7873.25366.23567-33.55%-35.41%The error distribution on the fatigue limit of FOD flat plate specimens predicted by the Peterson formula is shown in Figure14, and the error is defined by the following formula. It can be seen from the figure that the prediction error lies within the range of (-55%, -30%), except one 13.7%. Prediction values for all FOD flat plate specimens are lower than test values, which is considered conservative:
(9)error=σprediction−σtestσtest.Figure 14
Prediction error distribution of Peterson formula.


## 3.6. Prediction Model for FOD Specimen Fatigue Limit Based on WCN Model

Microcracks and corresponding cracks which do not propagate along the stress gradient direction of the notch could be regarded as the WCN model [27]. The WCN model takes the nominal stress threshold as a function of elastic stress concentration factor Kt and crack size a. The nominal stress threshold decreases with the increase of Kt and notch sensitivity. However, the nominal stress threshold does not decrease with the increase of Kt when the predicted Kt (by the WCN model) exceeds the limit value. It is mainly caused by the generation of end crack and nonpropagating crack. For the notch with fixed depth and different root radii, the nominal stress threshold increases with radius, and there is a limit radius. When the radius exceeds the limit radius, the crack will not propagate.The aim of the WCN model is to quantify the correlation among notch depth, root radius, stress concentration factor, and other parameters, so as to solve the problem of the notch fatigue limit in a systematic way. Because of the great amount of microdamages in the notch caused by FOD and the fatigue cracks initiated at the microcrack or micronotch, the WCN model is selected to predict the fatigue limit of the notched flat plate specimen.Generally speaking, the stress threshold is a function of the FOD notch size, the loading mode, and the material parameters. In this paper, the crack propagation threshold of 1Cr11Ni2W2MoV is obtained by the crack growth test.The formula of the WCN model is(10)ΔSe=ΔKthFπa0+d,(11)a0=1πΔKthFΔσe2,where ΔSe is stress amplitude at the root of the notch, a0 is small crack parameters, ΔKth is the crack propagation threshold (obtained from the crack growth test result in Section 3.3), F is the finite-thickness correction function (if the crack depth is much smaller than its length, F is about 1.12, and most cases of FOD are those where cracks are small [37]), and Δσe is the fatigue limit of the smooth specimen at 107 cycles.Small crack parametersa0 can be calculated by formula (11), and the stress amplitude ΔSe can be calculated by formula (10). According to the relationship between stress amplitude ΔSe and stress ratio R, the prediction value of the HCF limit of the notch root can be calculated. The prediction results are shown in Table 4, and the error distribution graph is shown in Figure 15. It can be seen from the figure that the prediction error lies within the range of (-50%, 6%). Most prediction values of FOD flat plate specimens are lower than test values, which is considered conservative, just like the prediction results by the Peterson formula.Figure 15
Prediction error distribution of the WCN model.


## 3.7. Comparison

It can be seen from Figure16 and Table 4 that the fatigue limit predicted by the Peterson formula and WCN model is both conservative, and almost all prediction error of the Peterson formula is between (-55%, -30%), while that of the WCN model is between (-50%, 20%). For the notch with damage depth Z greater than 1 mm, the predictions by the Peterson formula and WCN model are similar, and the errors are all more than 30%. The predictions by the WCN model are slightly accurate. For the notch with damage depth Z less than 1 mm, the prediction accuracy of the WCN model is higher than that of the Peterson formula, and the error lies within 32%. The results show that the WCN model is more accurate for the HCF limit prediction for specimens with small damage depth (<1 mm).Figure 16
Prediction error distribution of WCN model and Peterson formula.The possible reasons for the prediction error are as follows. The Peterson formula is based on the point stress model. It is assumed in the Peterson formula that there is only uniaxial stress at the notch of the specimen, without considering the strengthening effect, three-dimensional stress concentration, loading conditions, and complex microdamage characteristics at the notch. Therefore, the prediction results are conservative. In the WCN model, when the notch size is small, the parameter notch depthd in formula (10) is accurate. However, when the notch size is large, the parameter is not accurate enough, so it is necessary to research the model expression more deeply.


## 4. Conclusions

FOD tests, HCF tests, and crack growth tests under stress ratioR=0.8 were carried out for 1Cr11Ni2W2MoV stainless steel specimens. The Peterson formula and WCN model were used to predict the HCF limit of notched specimens. The results obtained in this study allow us to draw the following conclusions:
(1)
FOD notch sizes (damage widthX, damage thickness Y, and damage depth Z) increase with the steel ball diameter, and the damage thickness Y shows great fluctuation(2)
The high-speed impact of the steel ball results in a smooth extrusion area at the notch root, accompanying plastic deformation, loss of material, and microcracks at the side of the notch. These microdamage characteristics promote the initiation and propagation of fatigue cracks. Therefore, fatigue fracture occurs at the root of the FOD notch, which displays a dark fatigue source area(3)
The HCF limit of FOD flat plate specimens was predicted by the Peterson formula and WCN model. The predictions by the Peterson formula are conservative, and the error lies within (-55%, -30%), except one 13.7%. The predictions by the WCN model are more accurate, especially for the damage depthZ<1 mm (errors less than 32%). The prediction errors of the notch fatigue limit for damage depth Z>1 mm are more than 30%, but most of them are still better than the predictions by the Peterson formulaZ: damage depth; Kt: calculation stress concentration factor; Kf: predicted notch fatigue factor; σP: HCF limit predicted by Peterson formula; ΔSe: stress amplitude at the root of notch; σW: HCF limit predicted by WCN model; σtest: HCF limit tested by HCF test; ErrorP: prediction error of Peterson formula; ErrorW: prediction error of WCN model.



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*Source: 8837564-2020-10-19.xml*