*Article* **High-Cycle Fatigue Behavior and Fatigue Strength Prediction of Differently Heat-Treated 35CrMo Steels**

**Mengqi Yang 1,†, Chong Gao 2,3,†, Jianchao Pang 2,\*, Shouxin Li <sup>2</sup> , Dejiang Hu <sup>1</sup> , Xiaowu Li <sup>3</sup> and Zhefeng Zhang 2,\***

	- Department of Material Physics and Chemistry, School of Materials Science and Engineering,

**Abstract:** In order to obtain the optimum fatigue performance, 35CrMo steel was processed by different heat treatment procedures. The microstructure, tensile properties, fatigue properties, and fatigue cracking mechanisms were compared and analyzed. The results show that fatigue strength and yield strength slowly increase at first and then rapidly decrease with the increase of tempering temperature, and both reach the maximum values at a tempering temperature of 200 ◦C. The yield strength affects the ratio of crack initiation site, fatigue strength coefficient, and fatigue strength exponent to a certain extent. Based on Basquin equation and fatigue crack initiation mechanism, a fatigue strength prediction method for 35CrMo steel was established.

**Keywords:** 35CrMo steel; high-cycle fatigue; damage mechanism; fatigue strength prediction; heat treatment

### **1. Introduction**

Chromium-molybdenum alloy steels (Cr-Mo steels) have been extensively applied in various industrial fields for their good mechanical properties, hydrogen resistance, and heat resistance. These fields include chemical industry, petrochemical industry, aviation industry, engineering vehicles, power industry, and many more [1,2]. The steels are mainly used to produce the parts of large equipment, such as safety valves, automobile clutches, pressure vessels [3], railway axles [4], gears [5,6], and bolts [7]. Most of these components are not only the independent parts of equipment, but are also subjected to cyclic loads. For instance, header bolts connect the engine's head cover with stay rings, and they are also subjected to pre-tightening loads and axial alternating loads from the head cover. Its reliability frequently determines the safe and stable operation of the engine subjected to complex loadings that can easily cause fatigue damage and may cause economic losses or even lead to major engineering accidents. In recent years, the fatigue research on Cr–Mo steels mainly focuses on the explorations of performance and mechanisms under extreme environments [3,8–12] or advanced technology [4,13,14]. However, there is little research on the prediction of fatigue strength for Cr–Mo steels. Therefore, the research on fatigue strength prediction of Cr–Mo steels cannot be ignored.

In addition, Cr–Mo steels can also be machined into components with different performance requirements, e.g., wear-resistant components with high hardness and high strength [15], mill liners with wear properties and impact toughness [16], shock-resisting tools with the superior combination of hardness and impact properties [17], bolts with high

**Citation:** Yang, M.; Gao, C.; Pang, J.; Li, S.; Hu, D.; Li, X.; Zhang, Z. High-Cycle Fatigue Behavior and Fatigue Strength Prediction of Differently Heat-Treated 35CrMo Steels. *Metals* **2022**, *12*, 688. https://doi.org/10.3390/ met12040688

Academic Editors: Antonio Mateo and Angelo Fernando Padilha

Received: 16 February 2022 Accepted: 24 March 2022 Published: 17 April 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

comprehensive mechanical properties [18], etc. Heat treatment is the main technique to achieve these properties by regulating the microstructures or surface chemical composition. For example, quenching can improve the hardness and wear resistance of steel; and the different tempering temperatures can obtain different strengths and toughness [19,20]. Therefore, in the process of designing heat treatment procedures of materials for the components, it is necessary to adjust and test their mechanical properties and fatigue performance. However, fatigue test is time and energy consuming, so it is important to predict fatigue strength from static mechanical properties.

The main methods of fatigue strength prediction are El Haddad et al.'s model and Murakami's √ *area* parameter model [21]. However, they have certain limitations, the former has no estimation method for 3-D inclusions; the latter believes that the same material has defects with the same size, and it has no effective estimate for internal and unknown size defects. Therefore, it is still necessary to explore the fatigue strength prediction method for engineering materials from the fatigue curve (S–N curve).

In the early 20th century, researchers found the linear relation between stress amplitude and life on log–log plots, and proposed a simple formula such that

$$
\sigma\_a = \sigma\_f'(2\mathbf{N}\_f)^b \tag{1}
$$

where *σ<sup>a</sup>* is the stress amplitude, *σ<sup>f</sup>* 0 is the fatigue strength coefficient, *b* is the fatigue strength exponent, and *N<sup>f</sup>* is the number of cycles to failure. The values of fatigue strength coefficient and fatigue strength exponent are the intercept and slope of the S–N curves, respectively, on log–log plots. Nowadays, this is the well-known Basquin equation, and it has become an important tool for determining the fatigue strength and design criterion of materials. In recent years, the characteristics of the S–N curve and Basquin equation have been studied by many investigators [22–24]. Some researchers have proposed formulas to estimate the values of *σ<sup>f</sup>* 0 and *b*, which are generally based on the inclusion size, hardness, and tensile strength [25–27]. However, the shape of S–N curve and the values of *σ<sup>f</sup>* 0 and *b* could be changed by many other factors, such as sample surface treatment, experimental environment, and loading type [22,28,29]. It is valuable to further explore the high cycle fatigue (HCF) strength prediction of Cr–Mo steels.

In this study, four heat-treatment procedures of 35CrMo (Chinese designation) steel were employed to investigate the microstructures, tensile and HCF behaviors, and the relations among them. The differences in the mechanical behaviors of variously heat-treated 35CrMo steels were also analyzed. According to the corresponding fracture mechanisms, a suitable formula of fatigue strength prediction for the Cr–Mo steel was established.

#### **2. Experimental Materials and Procedures**

The chemical composition of 35CrMo steel is shown in Table 1. To gain a wide range of strength, the as-received steel bars were heated at 860 ◦C for 30 min followed by the oil-quenching. Then, some of the steel bars were processed into specimens, and the rest of them were tempered at 200 ◦C, 400 ◦C, and 500 ◦C for 90 min, respectively, followed by air-cooling to room temperature. The four heat-treatment procedures are given in Table 2, and the corresponding specimens are named as Q, QT200, QT400, and QT500, respectively.

**Table 1.** Chemical composition of 35CrMo/%.


The dimensions of the tensile and fatigue specimens are shown in Figure 1. Tensile tests were conducted at a strain rate of 10−<sup>3</sup> s <sup>−</sup><sup>1</sup> by an Instron 5982 static testing machine (Instron Corporation, Boston, MA, USA). The HCF tests were conducted under symmetrical push-pull loading condition (R = −1) by using a GPS100 high-frequency fatigue tester (Sinotest Equipment Co., Ltd., Changchun, China) under room temperature in air. The HCF tests were proceeded at a resonance frequency of about 115 Hz. In this experiment, about 20 specimens were prepared for each heat-treatment condition. Tests were stopped when the specimen failed completely or achieved 10<sup>7</sup> cycles. The fatigue strength was determined using the staircase method in which five pairs of specimens were tested, namely, taking the average values of these stress levels. The S–N curves were fitted with the data of all failed specimens by the least square method, which means that half of the specimens could fail above the curves [30]. The fatigue strength coefficients and exponents were obtained by the same method. QT500 500 °C tempering for 90 min The dimensions of the tensile and fatigue specimens are shown in Figure 1. Tensile tests were conducted at a strain rate of 10−3 s−1 by an Instron 5982 static testing machine (Instron Corporation, Boston, MA, USA). The HCF tests were conducted under symmetrical push-pull loading condition (R = −1) by using a GPS100 high-frequency fatigue tester (Sinotest Equipment Co., Ltd., Changchun, China) under room temperature in air. The HCF tests were proceeded at a resonance frequency of about 115 Hz. In this experiment, about 20 specimens were prepared for each heat-treatment condition. Tests were stopped

**Samples Quenching Tempering** 

QT200 200 °C tempering for 90 min QT400 400 °C tempering for 90 min

Untempered

Preheating to 860 °C for 30 min and quenching in oil

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**Table 2.** Heat-treatment procedures of 35CrMo steel.

Q


**Table 2.** Heat-treatment procedures of 35CrMo steel. when the specimen failed completely or achieved 107 cycles. The fatigue strength was de-

**Figure 1.** Configurations and dimensions of specimens tested for tensile (**a**) and fatigue (**b**) properties. (Unit: mm). **Figure 1.** Configurations and dimensions of specimens tested for tensile (**a**) and fatigue (**b**) properties. (Unit: mm).

The microstructures of specimens with different heat-treatment procedures were examined by electron back scattered diffraction (EBSD, LEO Supra 35, Carl Zeiss AG, Oberkochen, Germany). The tensile and fatigue fracture surfaces of failed specimens were examined by scanning electron microscopy (SEM, JSM-6510, Japan Electronics Co., Ltd., To-The microstructures of specimens with different heat-treatment procedures were examined by electron back scattered diffraction (EBSD, LEO Supra 35, Carl Zeiss AG, Oberkochen, Germany). The tensile and fatigue fracture surfaces of failed specimens were examined by scanning electron microscopy (SEM, JSM-6510, Japan Electronics Co., Ltd., Tokyo, Japan).

#### kyo, Japan). **3. Results and Discussion**

### *3.1. Microstructure*

**3. Results and Discussion**  *3.1. Microstructure*  The EBSD microstructures of 35CrMo steel with four heat-treatment procedures are shown in Figure 2. It can be seen that Q specimen contains many lath martensites and some retained austenites. The microstructure of QT200 specimen consists of plate shaped The EBSD microstructures of 35CrMo steel with four heat-treatment procedures are shown in Figure 2. It can be seen that Q specimen contains many lath martensites and some retained austenites. The microstructure of QT200 specimen consists of plate shaped tempered martensites and some retained austenites. Both QT400 and QT500 specimens display the uniform microstructures of tempered troostite, as shown in Figure 2c,d.

tempered martensites and some retained austenites. Both QT400 and QT500 specimens

display the uniform microstructures of tempered troostite, as shown in Figure 2c,d.

**Figure 2.** EBSD microstructures for 35CrMo steel with four heat-treatment procedures. (**a**) Q, (**b**) QT200, (**c**) QT400, and (**d**) QT500. **Figure 2.** EBSD microstructures for 35CrMo steel with four heat-treatment procedures. (**a**) Q, (**b**) QT200, (**c**) QT400, and (**d**) QT500.

#### *3.2. Tensile Behaviors 3.2. Tensile Behaviors*

The tensile properties of 35CrMo with different heat-treatment procedures are provided in Figure 3. The tensile properties of 35CrMo steel at different tempering temperatures are listed in Table 3. As can be seen from Figure 3b, with the tempering temperature increasing, the tensile strength (*σb*) successively decreases; besides, the yield strength (*σy*) slowly increases at first and then decreases, which are in agreement with the cases of other steels [19,31]. It is observed that the percentage reduction of area (*Z*) and elongation after fracture (*A*) increase in different degrees with increasing tempering temperature as shown in Figure 3c. Figure 3d gives relations of the elongation after fracture and the percentage reduction of area versus the tensile strength of 35CrMo steel. As the tensile strength increases, the elongation after fracture and the percentage reduction of area decrease in varying degrees. This is consistent with the inverse relation between strength and ductility for lots of metals [19]. The tensile properties of 35CrMo with different heat-treatment procedures are provided in Figure 3. The tensile properties of 35CrMo steel at different tempering temperatures are listed in Table 3. As can be seen from Figure 3b, with the tempering temperature increasing, the tensile strength (*σb*) successively decreases; besides, the yield strength (*σy*) slowly increases at first and then decreases, which are in agreement with the cases of other steels [19,31]. It is observed that the percentage reduction of area (*Z*) and elongation after fracture (*A*) increase in different degrees with increasing tempering temperature as shown in Figure 3c. Figure 3d gives relations of the elongation after fracture and the percentage reduction of area versus the tensile strength of 35CrMo steel. As the tensile strength increases, the elongation after fracture and the percentage reduction of area decrease in varying degrees. This is consistent with the inverse relation between strength and ductility for lots of metals [19].


**Table 3.** Tensile properties for 35CrMo steel processed at different tempering temperatures. **Table 3.** Tensile properties for 35CrMo steel processed at different tempering temperatures.

The macroscopic fractographies of tensile specimens for 35CrMo steel are shown in Figure 4. It can be seen that the tensile specimens with different tempering temperatures have significant necking phenomena. With the increase of tempering temperature, the area ratio of fiber zone (the ratio of the fiber zone area to the fracture surface area) gradually decreases, and the area ratio of shear lip first increases and then decreases slightly. Q and QT200 specimens have no obvious radial pattern, as shown in Figure 4a,b. QT400 and QT500 specimens have radial zone, the area ratio of radial zone increases and radial pattern becomes pronounced with the increase of tempering temperature, as shown in Figure 4c,d.

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**Figure 3.** Tensile properties of 35CrMo steel. (**a**) Tensile engineering stress-strain curves; (**b**,**c**) the relation between strengths (tensile and yield strengths), percentages (elongation after fracture and percentage reduction of area), and tempering temperature; and (**d**) relations of percentages vs. ten-**Figure 3.** Tensile properties of 35CrMo steel. (**a**) Tensile engineering stress-strain curves; (**b**,**c**) the relation between strengths (tensile and yield strengths), percentages (elongation after fracture and percentage reduction of area), and tempering temperature; and (**d**) relations of percentages vs. tensile strength. *Metals* **2022**, *12*, x FOR PEER REVIEW 6 of 15

sile strength.

**Figure 4.** The macroscopic fractographies of tensile samples for 35CrMo steel processed at different tempering temperatures. (**a**) Untampered, (**b**) 200 °C, (**c**) 400 °C, and (**d**) 500 °C. **Figure 4.** The macroscopic fractographies of tensile samples for 35CrMo steel processed at different tempering temperatures. (**a**) Untampered, (**b**) 200 ◦C, (**c**) 400 ◦C, and (**d**) 500 ◦C.

Tensile fractographies in the fiber zone for 35CrMo steel are magnified in Figure 5. It can be seen that the fiber zones of these specimens are mainly composed of dimples with

voids in the fiber zone can be attributed to the transition of the stress states of the specimen from uniaxial to triaxial due to the necking of specimens. The plastic deformation at the axial center of the specimen is difficult to continue with the effect of triaxial stress, so that the stress concentration occurs at the inclusions or second-phase particles, where the voids eventually nucleate and grow. Consequently, the sizes of microcracks or voids are closely related to inclusions or second-phase particles. It can be noted from Figure 5 that the sizes of microcracks and voids increase with the increase of tempering temperature, and such a similar situation has also appeared in high-strength, high ductility steels [32]. It can be concluded that the strength and toughness affect the behaviors of inclusions or secondphase particles. This seems to be consistent with the effect of tensile loads on the behaviors of inclusion and second-phase particles at elevated temperature, which is due to the trans-

formation of tensile properties affected by high temperature [33].

Tensile fractographies in the fiber zone for 35CrMo steel are magnified in Figure 5. It can be seen that the fiber zones of these specimens are mainly composed of dimples with different sizes, implying the typical ductile fracture modes. Besides, few microcracks and some larger voids can also be seen from the figure. The formation of microcracks and voids in the fiber zone can be attributed to the transition of the stress states of the specimen from uniaxial to triaxial due to the necking of specimens. The plastic deformation at the axial center of the specimen is difficult to continue with the effect of triaxial stress, so that the stress concentration occurs at the inclusions or second-phase particles, where the voids eventually nucleate and grow. Consequently, the sizes of microcracks or voids are closely related to inclusions or second-phase particles. It can be noted from Figure 5 that the sizes of microcracks and voids increase with the increase of tempering temperature, and such a similar situation has also appeared in high-strength, high ductility steels [32]. It can be concluded that the strength and toughness affect the behaviors of inclusions or second-phase particles. This seems to be consistent with the effect of tensile loads on the behaviors of inclusion and second-phase particles at elevated temperature, which is due to the transformation of tensile properties affected by high temperature [33]. *Metals* **2022**, *12*, x FOR PEER REVIEW 7 of 15

**Figure 5.** Tensile fractographies in the fiber zone for 35CrMo steel processed at different tempering temperatures. (**a**) Untampered, (**b**) 200 °C, (**c**) 400 °C, and (**d**) 500 °C. **Figure 5.** Tensile fractographies in the fiber zone for 35CrMo steel processed at different tempering temperatures. (**a**) Untampered, (**b**) 200 ◦C, (**c**) 400 ◦C, and (**d**) 500 ◦C.

#### *3.3. High-Cycle Fatigue Behaviors 3.3. High-Cycle Fatigue Behaviors*

tempering temperatures.

The S–N curves of 35CrMo steel under different heat treatments are shown in Figure 6a. The fatigue properties of 35CrMo steel at different tempering temperatures are listed in Table 4. Obviously, QT200 specimens have the best fatigue resistance. The fatigue strengths (*σw*) increase first and then decrease with the increase of tensile strengths (Figure 6b), which were also found in many other materials [19,34]. The Basquin equations for The S–N curves of 35CrMo steel under different heat treatments are shown in Figure 6a. The fatigue properties of 35CrMo steel at different tempering temperatures are listed in Table 4. Obviously, QT200 specimens have the best fatigue resistance. The fatigue strengths (*σw*) increase first and then decrease with the increase of tensile strengths (Figure 6b), which were also found in many other materials [19,34]. The Basquin equations for these materials are as below (Equations (2)–(5)):

0.058 1718.57(2 )*-*

0.089 2261.03(2 )*-*

0.126 2539.02(2 )*-*

**Table 4.** Fatigue properties and S–N curves parameters for 35CrMo steel processed at different

**Sample** *σw***/MPa** *σf***'** *b*  Q 627 2040.42 −0.073 QT200 706 1718.57 −0.058 QT400 548 2261.03 −0.089 QT500 418 2539.02 −0.126

$$
\sigma\_a = 2040.42 \, (2N\_f)^{-0.073} \text{, for Q} \tag{2}
$$

*a f σ* = *N* , for QT200 (3)

*a f σ* = *N* , for QT400 (4)

*a f σ* = *N* , for QT500 (5)

$$
\sigma\_a = 1718.57(2N\_f)^{-0.058}, \text{ for QT200} \tag{3}
$$

$$
\sigma\_a = \text{2261.03}(2N\_f)^{-0.089}, \text{ for QT400} \tag{4}
$$

$$
\sigma\_{\mathfrak{a}} = 2539.02(2N\_f)^{-0.126}, \text{ for QT500} \tag{5}
$$

**Figure 6.** (**a**) Relations between stress amplitude and fatigue life (S–N curves), (**b**) the relation between tensile and fatigue strengths, (**c**) the relation between fatigue strength coefficient and tensile strength, and (**d**) the relation between fatigue strength exponent and tensile strength. **Figure 6.** (**a**) Relations between stress amplitude and fatigue life (S–N curves), (**b**) the relation between tensile and fatigue strengths, (**c**) the relation between fatigue strength coefficient and tensile strength, and (**d**) the relation between fatigue strength exponent and tensile strength.

In Equations (2)–(5), the obtained fatigue strength coefficient *σf*' and fatigue strength exponents *b* are reported for the considered cases. The relations of fatigue parameters (*σf*' **Table 4.** Fatigue properties and S–N curves parameters for 35CrMo steel processed at different tempering temperatures.


The fatigue strength coefficient and the fatigue strength exponent are mainly affected by strengthening mechanisms and damage mechanisms of materials respectively [27]. In order to understand the variation trends of fatigue strength coefficient and exponent for 35CrMo steel, it is necessary to study the fracture mechanism of failed specimens. The fatigue source regions of failed specimens with different heat-treatment procedures were observed by SEM. According to different crack initiation mechanisms, these specimens could be divided into five categories, as shown in Figure 7, such as (a) surface scratch; (b) surface inclusion; (c) subsurface inclusion, representing the inclusion whose distance from In Equations (2)–(5), the obtained fatigue strength coefficient *σ<sup>f</sup>* 0 and fatigue strength exponents *b* are reported for the considered cases. The relations of fatigue parameters (*σ<sup>f</sup>* 0 and *b*) vs. the tensile strengths are shown in Figure 6c,d. It can be seen that the increasing and decreasing trends of them are opposite and both curves have extreme values at data of QT200 specimens. This is inconsistent with the trend of steels for very high cycle fatigue (VHCF) [27]. Some researchers pointed out that HCF and VHCF behaviors are different for the same materials [22,35,36]. Therefore, it is essential to study the variations of the fatigue strength coefficient and exponent in a wide strength range from the perspective of HCF.

the surface is less than its own size in this paper; (d) inner inclusion, representing the inclusion whose distance from the surface is greater than its size; and (e) micro-facet comprising numerous small convex and concave, representing the trace of plastic deformation caused by non-inclusion crack [37,38]. For the convenience of statistics, some researchers have summarized the fatigue crack initiation sites into two types, namely, surface and inner [19]. Inner represents inner inclusion and micro-facet, and surface scratch, surface The fatigue strength coefficient and the fatigue strength exponent are mainly affected by strengthening mechanisms and damage mechanisms of materials respectively [27]. In order to understand the variation trends of fatigue strength coefficient and exponent for 35CrMo steel, it is necessary to study the fracture mechanism of failed specimens. The fatigue source regions of failed specimens with different heat-treatment procedures were observed by SEM. According to different crack initiation mechanisms, these specimens

inclusion, and subsurface inclusion are classified as surface, as shown in Figure 8.

HCF.

could be divided into five categories, as shown in Figure 7, such as (a) surface scratch; (b) surface inclusion; (c) subsurface inclusion, representing the inclusion whose distance from the surface is less than its own size in this paper; (d) inner inclusion, representing the inclusion whose distance from the surface is greater than its size; and (e) micro-facet comprising numerous small convex and concave, representing the trace of plastic deformation caused by non-inclusion crack [37,38]. For the convenience of statistics, some researchers have summarized the fatigue crack initiation sites into two types, namely, surface and inner [19]. Inner represents inner inclusion and micro-facet, and surface scratch, surface inclusion, and subsurface inclusion are classified as surface, as shown in Figure 8. *Metals* **2022**, *12*, x FOR PEER REVIEW 9 of 15 *Metals* **2022**, *12*, x FOR PEER REVIEW 9 of 15

**Figure 7.** Fatigue crack initiation morphologies. (**a**) Surface scratch, (**b**) surface inclusion, (**c**) subsurface inclusion, (**d**) inner inclusion, and (**e**,**f**) micro-facet. **Figure 7.** Fatigue crack initiation morphologies. (**a**) Surface scratch, (**b**) surface inclusion, (**c**) subsurface inclusion, (**d**) inner inclusion, and (**e**,**f**) micro-facet. **Figure 7.** Fatigue crack initiation morphologies. (**a**) Surface scratch, (**b**) surface inclusion, (**c**) subsurface inclusion, (**d**) inner inclusion, and (**e**,**f**) micro-facet.

The two types of failed specimens have been indicated in the S–N curves, as shown in Figure 9. In the figure, the circles represent the failed specimens with cracks initiated **Figure 8.** Schematic diagram of crack initiation site. **Figure 8.** Schematic diagram of crack initiation site.

on the surface and the solid circles represent the cracks initiated inside. It is found that the specimens with initiation of inner cracks are generally loaded at low-stress levels and have high fatigue life, which can be clearly seen in Figure 9a,b. The same situation has also been found by some other researchers [13,22,37]. Under high applied stress amplitude, the surface defects and processing defects are the obvious weak zones, since the plastic deformation preferentially occurs at surface due to lack of constrain. The locally accumulated plastic strain caused by high stress concentration at the surface defects and processing The two types of failed specimens have been indicated in the S–N curves, as shown in Figure 9. In the figure, the circles represent the failed specimens with cracks initiated on the surface and the solid circles represent the cracks initiated inside. It is found that the specimens with initiation of inner cracks are generally loaded at low-stress levels and have high fatigue life, which can be clearly seen in Figure 9a,b. The same situation has also been found by some other researchers [13,22,37]. Under high applied stress amplitude, the sur-The two types of failed specimens have been indicated in the S–N curves, as shown in Figure 9. In the figure, the circles represent the failed specimens with cracks initiated on the surface and the solid circles represent the cracks initiated inside. It is found that the specimens with initiation of inner cracks are generally loaded at low-stress levels and have high fatigue life, which can be clearly seen in Figure 9a,b. The same situation has also been found by some other researchers [13,22,37]. Under high applied stress

face defects and processing defects are the obvious weak zones, since the plastic deformation preferentially occurs at surface due to lack of constrain. The locally accumulated

applied, the locally accumulated plastic strain over those surface defects becomes weaker; at this time, some interior inclusions may have the potential to compete with those defects. Since the inner area of a cross section is generally much larger than the outer surface layer area, the probability for larger inclusions or harmful inclusions emerging in the inner area

defect will induce crack initiation. On the other hand, when the lower stress amplitude is applied, the locally accumulated plastic strain over those surface defects becomes weaker;

area, the probability for larger inclusions or harmful inclusions emerging in the inner area

amplitude, the surface defects and processing defects are the obvious weak zones, since the plastic deformation preferentially occurs at surface due to lack of constrain. The locally accumulated plastic strain caused by high stress concentration at the surface defects and processing defect will induce crack initiation. On the other hand, when the lower stress amplitude is applied, the locally accumulated plastic strain over those surface defects becomes weaker; at this time, some interior inclusions may have the potential to compete with those defects. Since the inner area of a cross section is generally much larger than the outer surface layer area, the probability for larger inclusions or harmful inclusions emerging in the inner area is definitely greater than that in the surface area. If so, the fatigue cracks may initiate from internal inclusions at the low stress amplitude. *Metals* **2022**, *12*, x FOR PEER REVIEW 10 of 15 is definitely greater than that in the surface area. If so, the fatigue cracks may initiate from internal inclusions at the low stress amplitude.

**Figure 9.** S–N curves for the specimens of Q (**a**), QT200 (**b**), QT400 (**c**) and QT500 (**d**). **Figure 9.** S–N curves for the specimens of Q (**a**), QT200 (**b**), QT400 (**c**) and QT500 (**d**).

From Figure 9, one can see that most of the failure samples for Q begin to fracture from the inside, and the number of such failed samples gradually decreases with the increase of tempering temperature of heat-treatment procedures. Until the tempering temperature reaches 500 °C, all the failed specimens begin to fracture on the surface. Figure 10a shows the relations between the ratios of surface/inner fatigue crack initiation sites (the ratios of the number of failures originating from the surface/inner to the total number of failures) and yield strengths. It can be seen that the ratio of surface initiation cracks decreases with the increase of yield strength. In other words, as the yield strength decreases, the trend of surface fatigue crack initiation increases. It is understood with lower yield strength, the severe locally accumulated plastic deformation will easily result in the surface defects as mentioned above. Furthermore, it can be roughly inferred from the figure that cracks will initiate from the surface for the specimens with yield strengths below 1200 MPa. The ratio of inner cracks will continue to increase when the yield strengths of the samples are higher than 1500 MPa. To sum up, it can be said that the yield strength affects the ratio of fatigue crack initiation site to a certain extent. Wang et al. [39] have concluded that the transition from surface to subsurface crack From Figure 9, one can see that most of the failure samples for Q begin to fracture from the inside, and the number of such failed samples gradually decreases with the increase of tempering temperature of heat-treatment procedures. Until the tempering temperature reaches 500 ◦C, all the failed specimens begin to fracture on the surface. Figure 10a shows the relations between the ratios of surface/inner fatigue crack initiation sites (the ratios of the number of failures originating from the surface/inner to the total number of failures) and yield strengths. It can be seen that the ratio of surface initiation cracks decreases with the increase of yield strength. In other words, as the yield strength decreases, the trend of surface fatigue crack initiation increases. It is understood with lower yield strength, the severe locally accumulated plastic deformation will easily result in the surface defects as mentioned above. Furthermore, it can be roughly inferred from the figure that cracks will initiate from the surface for the specimens with yield strengths below 1200 MPa. The ratio of inner cracks will continue to increase when the yield strengths of the samples are higher than 1500 MPa. To sum up, it can be said that the yield strength affects the ratio of fatigue crack initiation site to a certain extent.

initiation has a significant effect on the slope of S–N curve. As an extension, the intercepts and slopes of S–N curves (fatigue strength coefficient and exponent of Basquin equation) are related to fatigue crack initiation sites, as shown in Figure 10b. It can be seen that the fatigue strength coefficient decreases and the fatigue strength exponent increases with the increasing ratio of the inner crack site. Therefore, different ratios of crack initiation sites affect the fatigue strength coefficient and exponent of Basquin equation to a certain extent. The reason can be found from the distribution characteristics of different crack initiations Wang et al. [39] have concluded that the transition from surface to subsurface crack initiation has a significant effect on the slope of S–N curve. As an extension, the intercepts and slopes of S–N curves (fatigue strength coefficient and exponent of Basquin equation) are related to fatigue crack initiation sites, as shown in Figure 10b. It can be seen that the fatigue strength coefficient decreases and the fatigue strength exponent increases with the increasing ratio of the inner crack site. Therefore, different ratios of crack initiation sites

sites in Figure 9 and the relations in Figure 10b. Combined with the above conclusions

affect the fatigue strength coefficient and exponent of Basquin equation to a certain extent. The reason can be found from the distribution characteristics of different crack initiations sites in Figure 9 and the relations in Figure 10b. Combined with the above conclusions that the yield strength affects the ratio of fatigue crack initiation site and the cracking position affects the fatigue strength coefficient and exponent, it can be said that the fatigue strength coefficient and exponent are indirectly influenced by the yield strength. *Metals* **2022**, *12*, x FOR PEER REVIEW 11 of 15 that the yield strength affects the ratio of fatigue crack initiation site and the cracking position affects the fatigue strength coefficient and exponent, it can be said that the fatigue strength coefficient and exponent are indirectly influenced by the yield strength. *Metals* **2022**, *12*, x FOR PEER REVIEW 11 of 15 that the yield strength affects the ratio of fatigue crack initiation site and the cracking position affects the fatigue strength coefficient and exponent, it can be said that the fatigue strength coefficient and exponent are indirectly influenced by the yield strength.

**Figure 10.** (**a**) The relation between the ratios of fatigue crack initiation sites and yield strength, and (**b**) relations of the fatigue strength coefficient and exponent vs. ratio of inner crack site. **Figure 10.** (**a**) The relation between the ratios of fatigue crack initiation sites and yield strength, and (**b**) relations of the fatigue strength coefficient and exponent vs. ratio of inner crack site. **Figure 10.** (**a**) The relation between the ratios of fatigue crack initiation sites and yield strength, and (**b**) relations of the fatigue strength coefficient and exponent vs. ratio of inner crack site.

#### *3.4. Prediction of Fatigue Strength 3.4. Prediction of Fatigue Strength 3.4. Prediction of Fatigue Strength*

To predict fatigue strength by Basquin equation, some parameters are necessary to figure out. As shown in Figure 11a, the fatigue strength *σw* of a material can be determined by the fatigue strength coefficient, exponent, and the life of knee point *Nk* in the S–N curve. The knee point is the intersection of the curve fitted by the group method and the fatigue strength calculated by the staircase method. Obviously, the knee point is also a necessary parameter to predict fatigue strength. To predict fatigue strength by Basquin equation, some parameters are necessary to figure out. As shown in Figure 11a, the fatigue strength *σ<sup>w</sup>* of a material can be determined by the fatigue strength coefficient, exponent, and the life of knee point *N<sup>k</sup>* in the S–N curve. The knee point is the intersection of the curve fitted by the group method and the fatigue strength calculated by the staircase method. Obviously, the knee point is also a necessary parameter to predict fatigue strength. To predict fatigue strength by Basquin equation, some parameters are necessary to figure out. As shown in Figure 11a, the fatigue strength *σw* of a material can be determined by the fatigue strength coefficient, exponent, and the life of knee point *Nk* in the S–N curve. The knee point is the intersection of the curve fitted by the group method and the fatigue strength calculated by the staircase method. Obviously, the knee point is also a necessary parameter to predict fatigue strength.

**Figure 11.** The fatigue strength prediction model. (**a**) The schematic illustration of S–N curves, (**b**) the linear relation between *σ<sup>f</sup>* 0 and *σy*, (**c**) the linear relation between *b* and *σy*, and (**d**) the relation of lg(2*N<sup>k</sup>* ) and *σy*.

The logarithmic form of Basquin equation for S–N curves can be obtained,

$$\lg \sigma\_a = b \lg(2N\_f) + \lg \sigma'\_f \tag{6}$$

If *N<sup>k</sup>* is determined, the fatigue strength prediction equation can be written as

$$\lg \sigma\_w = b \lg(2N\_k) + \lg \sigma'\_f \tag{7}$$

Based on the above discussion, *σ<sup>f</sup>* 0 and *b* are linearly fitted with the yield strength, and the error bands are within the 10% and 5%, respectively, as shown in Figure 11b,c. In addition, the knee point is also fitted with the yield strength for the unification of variables and convenience of calculation. They have a quadratic relation with only 1% error band, as shown in Figure 11d. This is the relation between the intersection of the two lines and the yield strength, which has no practical significance. The fitting equations can be expressed in linear and quadratic equations as below, respectively,

$$
\sigma\_f' = m\sigma\_y + n\tag{8}
$$

$$b = \mathfrak{u}\sigma\_y + v\tag{9}$$

$$\lg(2N\_k) = \varkappa \sigma\_y^2 + y \sigma\_y + z \tag{10}$$

Substituting Equations (8)–(10) into Equation (7), a new relation can be obtained,

$$\log \sigma\_w = (u\sigma\_y + v)(\mathbf{x}\sigma\_y^2 + y\sigma\_y + z) + \lg(m\sigma\_y + n) \tag{11}$$

where, *m*, *n*, *u*, *v*, *x*, *y*, and *z* are the material constants, which can be obtained by data fitting.

For 35CrMo steel, the constants have been fitted and the fatigue strength prediction formula can be expressed as follows,

$$\begin{array}{l} \lg \sigma\_{w} = (2.193 \times 10^{-4} \sigma\_{y} - 0.382)(-1.987 \times 10^{-5} \sigma\_{y}^{-2} + 5.422 \times 10^{-2} \sigma\_{y} - 30.029) + \\\lg(-2.542 \sigma\_{y} + 5564.850) \end{array} \tag{12}$$

The results of fatigue strength prediction are shown in Figure 12, and it can be seen that the errors of this fatigue prediction equation are less than 10%.

The fatigue fracture morphologies and HCF properties of 35CrMo steel specimens

(1) With the increase of tempering temperature, martensite is gradually decomposed and the tensile strength decreases, but the yield strength and fatigue strength increase at first and then decrease. QT200 specimens have the best fatigue performance; (2) To some extent, the yield strength affects the ratio of crack initiation site for a specimen, and the crack initiation site affects the fatigue strength coefficient and fatigue strength exponent. Therefore, the yield strength affects the change of fatigue strength coefficient and fatigue strength exponent, and they have a linear relation for HCF

(3) A fatigue strength prediction method based on the damage mechanisms and Basquin equation was proposed. In this way, the values of fatigue strength coefficient, fatigue strength exponent, and knee point can be expressed by yield strength. This method can effectively predict the HCF strength of 35CrMo steel. The fatigue strength coefficient, fatigue strength exponent, and knee point are affected by many factors, and it is still necessary to further explore whether this method is suitable for other materi-

**Author Contributions:** Conceptualization, J.P. and Z.Z.; Funding acquisition, M.Y., J.P., D.H. and Z. Z.; Investigation, C.G.; Methodology, C.G., S.L., J.P., and X.L.; Project administration, M.Y., and D.H.; Resources, M.Y., and D.H.; Writing-original draft, C.G.; Writing - review & editing, J.P., S.L., X.L., and Z.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by project entitled "Research on Lifetime Prediction of Nonrotating Parts of Pump Turbine Unit Based on Rotor-Stator Interaction (RSI), Fluid-Structure Coupling and Fracture Mechanics" under Grant. No. 022200KK52180006, National Natural Science Foundation of China (NSFC) under Grant. No. 51871224, Natural Science Foundation of Liaoning

**Institutional Review Board Statement:** The study did not require ethical approval.

*Metals* **2022**, *12*, x FOR PEER REVIEW 13 of 15

**Figure 12.** The calculated vs. experimental values for fatigue strength. **Figure 12.** The calculated vs. experimental values for fatigue strength.

**4. Conclusions** 

tests of 35CrMo steel;

Province under Grant. No. 20180550880.

**Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable.

as below:

als.

### **4. Conclusions**

The fatigue fracture morphologies and HCF properties of 35CrMo steel specimens with different tensile strengths were studied. The main conclusions can be summarized as below:


**Author Contributions:** Conceptualization, J.P. and Z.Z.; Funding acquisition, M.Y., J.P., D.H. and Z.Z.; Investigation, C.G.; Methodology, C.G., S.L., J.P., and X.L.; Project administration, M.Y., and D.H.; Resources, M.Y., and D.H.; Writing-original draft, C.G.; Writing—review & editing, J.P., S.L., X.L., and Z.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by project entitled "Research on Lifetime Prediction of Nonrotating Parts of Pump Turbine Unit Based on Rotor-Stator Interaction (RSI), Fluid-Structure Coupling and Fracture Mechanics" under Grant. No. 022200KK52180006, National Natural Science Foundation of China (NSFC) under Grant. No. 51871224, Natural Science Foundation of Liaoning Province under Grant. No. 20180550880.

**Institutional Review Board Statement:** The study did not require ethical approval.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank Z. K. Zhao and H. Y. Zhang for their help of the fatigue experiment and SEM observations.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


**Zhenxian Zhang <sup>1</sup> , Zhongwen Li <sup>1</sup> , Han Wu 2,3 and Chengqi Sun 2,3,\***


**Abstract:** In this paper, the axial loading fatigue tests are at first conducted on specimens ofG20Mn5QT steel from axle box bodies in high-speed trains. Then, the size and shape effects on fatigue behavior are investigated. It is shown that the specimen size and shape have an influence on the fatigue performance of G20Mn5QT steel. The fatigue strength of the hourglass specimen is higher than that of the dogbone specimen due to its relatively smaller highly stressed region. Scanning electron microscope observation of the fracture surface and energy dispersive X-ray spectroscopy indicate that the specimen size and shape have no influence on the fatigue crack initiation mechanism. Fatigue cracks initiate from the surface or subsurface of the specimen, and some fracture surfaces present the characteristic of multi-site crack initiation. Most of the fatigue cracks initiate from the pore defects and alumina inclusions in the casting process, in which the pore defects are the main crack origins. The results also indicate that the probabilistic control volume method could be used for correlating the effects of specimen size and shape o the fatigue performance of G20Mn5QT steel for axle box bodies in high-speed trains.

**Keywords:** G20Mn5QT steel; crack initiation mechanism; fatigue strength; size effect; shape effect

### **1. Introduction**

The high-speed railway industry has developed rapidly in the past decade. Fatigue failure, as one of the main failure modes for engineering materials and components [1–5], is also a key mechanical problem for high-speed trains. Many studies concerning the fatigue problems in high-speed trains have been carried out [6–11]. For example, Lu et al. [12] studied the very-high-cycle fatigue behavior of an axle steel LZ50 under rotating bending fatigue loading and showed that LZ50 steel had the fatigue limit at 5 <sup>×</sup> <sup>10</sup>6~10<sup>9</sup> cycles. The fatigue fracture surface observation indicated that the fatigue crack initiated from the ferrite on the surface of the specimen. Chen et al. [13] investigated the high cycle and very-high-cycle fatigue performance of an axle steel EA4T, and found that there was still a conventional fatigue limit for EA4T steel. Beretta et al. [14] studied the corrosion fatigue behavior of an axle steel A1N exposed in rainwater, and the results showed that the rainwater significantly reduced the fatigue strength (>10<sup>6</sup> cycles) of the A1N steel. Wang et al. [15] analyzed the fatigue strength of the CRH2 motor bogie frame through simulation and online tests. Zhang et al. [16] studied the fatigue crack growth behavior in the gradient microstructure of the surface layer of S38C axle steel. The results indicated that the crack growth rate firstly decelerated and then accelerated with increasing the crack length in the gradient layer. Guagliano and Vergani [17] conducted experiments and numerical analysis on the sub-surface cracks in railway wheels. Gao et al. [18] studied the effect of artificial defects on the fatigue strength of an induction hardened S38C axle and showed that the influence of shallower impact damage (smaller than 200 µm) on fatigue

**Citation:** Zhang, Z.; Li, Z.; Wu, H.; Sun, C. Size and Shape Effects on Fatigue Behavior of G20Mn5QT Steel from Axle Box Bodies in High-Speed Trains. *Metals* **2022**, *12*, 652. https:// doi.org/10.3390/met12040652

Academic Editor: Denis Benasciutti

Received: 7 March 2022 Accepted: 9 April 2022 Published: 11 April 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

strength was negligible. Luke et al. [19] made conclusions on some important aspects and results related to the application of the fracture mechanics approach to the prediction of inspection intervals of railway axles under in-service conditions.

The specimen geometry is an important factor affecting fatigue properties [20–22]. The fatigue performance tends to decrease with the increase of specimen size [23,24]. The size and shape of actual components are usually different from the standard testing specimens. Therefore, studying the size and shape effects on the fatigue behavior of materials in key structures of high-speed trains has scientific significance and application value. Li et al. [25] studied the effects of specimen size and notch on the fatigue properties of an EA4T axle steel. The study indicated that, with the increase of specimen size, the fatigue strength of the dogbone specimen was considerably lower than that of the hourglass specimen under axial loading. Shen et al. [26] analyzed the effect of inclusion size on the fatigue strength of small specimens and railway axles, and showed that, due to the increase of risk volume, the critical stress of fatigue failure in axles induced by inclusion was about 50% of that in small specimens under rotating bending loading. Varfolomeev et al. [27] studied the effect of specimen shape on the fatigue crack growth rate of an EA4T axle steel and showed that the crack growth rate depended on the specimen shape and loading condition.

Axle box bodies are important components in high-speed trains, which are subject to cyclic loadings and might fail in service. However, there are few results available for the effects of specimen size and shape on the fatigue behavior of materials for axle box bodies. Therefore, revealing the size and shape effects on the fatigue behavior of materials for axle box bodies is of great importance. This paper studies G20Mn5QT steel from axle box bodies in high-speed trains. The axial loading fatigue tests are at first conducted on the specimens with different size and shape. Then, the fatigue failure mechanism of G20Mn5QT steel is studied based on the observation of the fracture surface by scanning electron microscope (SEM) and the analysis of the crack initiation region by energy dispersive X-ray spectroscopy (EDS). Finally, the size and shape effects on the fatigue performance are correlated by using the probabilistic control volume method for G20Mn5QT steel.

#### **2. Materials and Methods**

The material used is a G20Mn5QT steel cut from the new axle box bodies of a highspeed train. The chemical composition is 0.18 C, 0.34 Si, 1.20 Mn, 0.22 Ni, 0.065 Al, 0.03 Cr, 0.011 Cu, 0.017 P, and 0.009 S in weight percent (Fe balance). The axle box body was at first heated at 910 ± 10◦ for 3.5 h and oil quenched, and then it was tempered for 4 h at 640 ± 10◦ and cooled to below 300 ◦C with the furnace and then air cooled. The average tensile strength and yield strength of the material are 582 MPa and 399 MPa, respectively. The standard deviation is 0.58 for the tensile strength and 5.5 for the yield strength. They are obtained from three specimens by an MTS Landmark machine. The strain rate is <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> s −1 . The shape of the tension specimen is shown in Figure 1a. Fatigue tests were conducted on an MTS Landmark machine. The loading frequency is 1 Hz to 32 Hz and the stress ratio *R* is −1. Two kinds of specimens, the hourglass specimen and the dogbone specimen, are chosen for fatigue tests, as shown in Figure 1b,c, respectively. The elastic stress concentration factor *K<sup>t</sup>* is defined as the ratio of the maximum principal stress at the notch root to that of the cylindrical specimen with the same smallest cross section (i.e., nominal stress), and is obtained by using Abaqus 6.14 software. In the calculation, Young's modulus is *E* = 210 GPa and Poisson's ratio is ν = 0.3. All tests were carried out at room temperature in air. Before the fatigue test, the surface of the experimental section of the specimen was ground and polished. The surface roughness *R<sup>a</sup>* was less than 0.5 µm.

SEM is used to observe the fatigue fracture surface and analyze the crack initiation mechanism. EDS is conducted to determine the element composition in the typical crack initiation region.

*Metals* **2022**, *12*, x FOR PEER REVIEW 3 of 13

**Figure 1.** Shape and dimension of specimens (in mm). (**a**) Tensile specimen; (**b**) hourglass specimen, *Kt* = 1.05; (**c**) dogbone specimen. **Figure 1.** Shape and dimension of specimens (in mm). (**a**) Tensile specimen; (**b**) hourglass specimen, *K<sup>t</sup>* = 1.05; (**c**) dogbone specimen. mechanism. EDS is conducted to determine the element composition in the typical crack initiation region.

#### SEM is used to observe the fatigue fracture surface and analyze the crack initiation **3. Results 3. Results**

#### mechanism. EDS is conducted to determine the element composition in the typical crack *3.1. Measurement of Specimen Temperature 3.1. Measurement of Specimen Temperature*

initiation region. **3. Results** *3.1. Measurement of Specimen Temperature* A thermocouple was used to measure the surface temperature of the experimental section of several specimens during the fatigue tests, as shown in Figure 2a. The loading frequency was not increased until the measured temperature was stable after a number of fatigue cycles (e.g., 200, 1000, or 3000 cycles). Figure 2b presents the variation of surface temperature of the experimental section with the loading frequency under the normal stress amplitude of 300 MPa, 350 MPa and 380 MPa for the hourglass specimens. The temperature is the stable temperature at the frequency in the abscissa in Figure 2b. It is observed that, with the increase of the loading frequency, the temperature of the experimental section increases remarkably under the stress amplitude of 350 MPa and 380 MPa, A thermocouple was used to measure the surface temperature of the experimental section of several specimens during the fatigue tests, as shown in Figure 2a. The loading frequency was not increased until the measured temperature was stable after a number of fatigue cycles (e.g., 200, 1000, or 3000 cycles). Figure 2b presents the variation of surface temperature of the experimental section with the loading frequency under the normal stress amplitude of 300 MPa, 350 MPa and 380 MPa for the hourglass specimens. The temperature is the stable temperature at the frequency in the abscissa in Figure 2b. It is observed that, with the increase of the loading frequency, the temperature of the experimental section increases remarkably under the stress amplitude of 350 MPa and 380 MPa, whereas it increases slowly under the stress amplitude of 300 MPa. In order to eliminate the possible influence of the temperature increase on fatigue properties during fatigue tests, an appropriate frequency is adopted under different stress amplitudes according to the results in Figure 2b. A thermocouple was used to measure the surface temperature of the experimental section of several specimens during the fatigue tests, as shown in Figure 2a. The loading frequency was not increased until the measured temperature was stable after a number of fatigue cycles (e.g., 200, 1000, or 3000 cycles). Figure 2b presents the variation of surface temperature of the experimental section with the loading frequency under the normal stress amplitude of 300 MPa, 350 MPa and 380 MPa for the hourglass specimens. The temperature is the stable temperature at the frequency in the abscissa in Figure 2b. It is observed that, with the increase of the loading frequency, the temperature of the experimental section increases remarkably under the stress amplitude of 350 MPa and 380 MPa, whereas it increases slowly under the stress amplitude of 300 MPa. In order to eliminate the possible influence of the temperature increase on fatigue properties during fatigue tests, an appropriate frequency is adopted under different stress amplitudes according to the results in Figure 2b.

**Figure 2.** Measurement of surface temperature for experimental section of specimens. (**a**) Picture of the method for temperature measurement; (**b**) variation of temperature with loading frequency. **Figure 2.** Measurement of surface temperature for experimental section of specimens. (**a**) Picture of the method for temperature measurement; (**b**) variation of temperature with loading frequency.

#### *3.2. Stress-Life (S-N) Data*

**Figure 2.** Measurement of surface temperature for experimental section of specimens. (**a**) Picture of the method for temperature measurement; (**b**) variation of temperature with loading frequency. The S-N data of the tested specimens are plotted in Figure 3. Here, the local stress amplitude is used, i.e., the stress concentration is considered for the hourglass specimen.

For the dogbone specimen, the local stress amplitude is the nominal stress amplitude because the positions of the fatigue fracture surface are all at the parallel segment with the smallest section. The loading information of the specimens is listed in Table 1. It is seen from Figure 3 that, with the increase of fatigue life, the fatigue strength decreases for both the hourglass and dogbone specimens. Moreover, the results for a stress amplitude of 367.5 MPa for the hourglass specimen in Figure 3a indicate that the loading frequency has no influence on fatigue performance of the G20Mn5QT steel. Therefore, the effect of loading frequency on the fatigue behavior is not considered for the G20Mn5QT steel when the fatigue data are analyzed in this paper. For the dogbone specimen, the local stress amplitude is the nominal stress amplitude because the positions of the fatigue fracture surface are all at the parallel segment with the smallest section. The loading information of the specimens is listed in Table 1. It is seen from Figure 3 that, with the increase of fatigue life, the fatigue strength decreases for both the hourglass and dogbone specimens. Moreover, the results for a stress amplitude of 367.5 MPa for the hourglass specimen in Figure 3a indicate that the loading frequency has no influence on fatigue performance of the G20Mn5QT steel. Therefore, the effect of loading frequency on the fatigue behavior is not considered for the G20Mn5QT steel when the fatigue data are analyzed in this paper.

The S-N data of the tested specimens are plotted in Figure 3. Here, the local stress amplitude is used, i.e., the stress concentration is considered for the hourglass specimen.

*Metals* **2022**, *12*, x FOR PEER REVIEW 4 of 13

*3.2. Stress-Life (S-N) Data*

**Figure 3.** S-N data of the tested specimens, in which the arrows denote the unbroken specimens at the associated cycles. (**a**) Hourglass specimens; (**b**) dogbone specimens. **Figure 3.** S-N data of the tested specimens, in which the arrows denote the unbroken specimens at the associated cycles. (**a**) Hourglass specimens; (**b**) dogbone specimens.


**Table 1.** Loading information of the tested specimens. **Table 1.** Loading information of the tested specimens.


**Table 1.** *Cont.*

<sup>1</sup> Denotes that the specimen does not fail at the associated cycles.

#### *3.3. Crack Initiation Mechanism*

Figures 4 and 5 show the SEM images of the fracture surface of several hourglass specimens. It is seen that the fatigue cracks initiate from the specimen surface (Figures 4b and 5b) or the subsurface of the specimen (Figures 4d and 5d). Meanwhile, some specimens exhibit the characteristic of multi-site crack initiation on the fracture surface (Figure 5). *Metals* **2022**, *12*, x FOR PEER REVIEW 6 of 13

**Figure 4.** SEM images of the fracture surface for hourglass specimens with single-site crack initiation. (**a**) and (**b**): local stress amplitude *σa* = 315 MPa, *N* = 1.73 × 105; (**c**) and (**d**): local stress amplitude *σa* = 252 MPa, *N* = 1.37 × 105. **Figure 4.** SEM images of the fracture surface for hourglass specimens with single-site crack initiation. (**a**,**b**): local stress amplitude *<sup>σ</sup><sup>a</sup>* = 315 MPa, *<sup>N</sup>* = 1.73 <sup>×</sup> <sup>10</sup><sup>5</sup> ; (**c**,**d**): local stress amplitude *σa* = 252 MPa, *<sup>N</sup>* = 1.37 <sup>×</sup> <sup>10</sup><sup>5</sup> .

**Figure 5.** SEM images of the fracture surface for the hourglass specimen with multi-site crack initiation, local stress amplitude *σa* = 315 MPa, *N* = 7.96 × 104. (**a**): Fracture surface with low magnification;

SEM images of the fracture surface of several dogbone specimens are shown in Figures 6 and 7. Similar to hourglass specimens, the fatigue cracks initiate from the specimen

(**b**–**d**): close-ups of crack initiation regions A, B, and C in (**a**).

**Figure 4.** SEM images of the fracture surface for hourglass specimens with single-site crack initiation. (**a**) and (**b**): local stress amplitude *σa* = 315 MPa, *N* = 1.73 × 105; (**c**) and (**d**): local stress amplitude

*σa* = 252 MPa, *N* = 1.37 × 105.

**Figure 5.** SEM images of the fracture surface for the hourglass specimen with multi-site crack initiation, local stress amplitude *σa* = 315 MPa, *N* = 7.96 × 104. (**a**): Fracture surface with low magnification; (**b**–**d**): close-ups of crack initiation regions A, B, and C in (**a**). **Figure 5.** SEM images of the fracture surface for the hourglass specimen with multi-site crack initiation, local stress amplitude *<sup>σ</sup><sup>a</sup>* = 315 MPa, *<sup>N</sup>* = 7.96 <sup>×</sup> <sup>10</sup><sup>4</sup> . (**a**): Fracture surface with low magnification; (**b**–**d**): close-ups of crack initiation regions A, B, and C in (**a**).

SEM images of the fracture surface of several dogbone specimens are shown in Figures 6 and 7. Similar to hourglass specimens, the fatigue cracks initiate from the specimen SEM images of the fracture surface of several dogbone specimens are shown in Figures 6 and 7. Similar to hourglass specimens, the fatigue cracks initiate from the specimen surface (Figures 6b and 7b) or the subsurface of the specimen (Figure 6d), and some fracture surfaces present the multi-site crack initiation feature (Figure 7). *Metals* **2022**, *12*, x FOR PEER REVIEW 7 of 13 surface (Figures 6b and 7b) or the subsurface of the specimen (Figure 6d), and some fracture surfaces present the multi-site crack initiation feature (Figure 7).

**Figure 6.** SEM images of the fracture surface for dogbone specimens with single-site crack initiation. (**a**) and (**b**): *σa* = 220 MPa, *N* = 2.41 × 106; (**c**) and (**d**): *σa* = 350 MPa, *N* = 2.7 × 103. **Figure 6.** SEM images of the fracture surface for dogbone specimens with single-site crack initiation. (**a**,**b**): *<sup>σ</sup><sup>a</sup>* = 220 MPa, *<sup>N</sup>* = 2.41 <sup>×</sup> <sup>10</sup><sup>6</sup> ; (**c**,**d**): *<sup>σ</sup><sup>a</sup>* = 350 MPa, *<sup>N</sup>* = 2.7 <sup>×</sup> <sup>10</sup><sup>3</sup> .

**Figure 7.** SEM images of the fracture surface for the dogbone specimen with multi-site crack initiation, *σa* = 380 MPa, *N* = 2.38 × 103. (**a**): Fracture surface with low magnification; (**b**–**d**) Close-ups of

crack initiation regions A, B, and C in (**a**).

**Figure 6.** SEM images of the fracture surface for dogbone specimens with single-site crack initiation.

(**a**) and (**b**): *σa* = 220 MPa, *N* = 2.41 × 106; (**c**) and (**d**): *σa* = 350 MPa, *N* = 2.7 × 103.

surface (Figures 6b and 7b) or the subsurface of the specimen (Figure 6d), and some frac-

ture surfaces present the multi-site crack initiation feature (Figure 7).

**Figure 7.** SEM images of the fracture surface for the dogbone specimen with multi-site crack initiation, *σa* = 380 MPa, *N* = 2.38 × 103. (**a**): Fracture surface with low magnification; (**b**–**d**) Close-ups of crack initiation regions A, B, and C in (**a**). **Figure 7.** SEM images of the fracture surface for the dogbone specimen with multi-site crack initiation, *<sup>σ</sup><sup>a</sup>* = 380 MPa, *<sup>N</sup>* = 2.38 <sup>×</sup> <sup>10</sup><sup>3</sup> . (**a**): Fracture surface with low magnification; (**b**–**d**) Close-ups of crack initiation regions A, B, and C in (**a**). *Metals* **2022**, *12*, x FOR PEER REVIEW 8 of 13

The SEM observations show that most fatigue cracks initiate from pore defects (Figures 4d, 5c and 6b) or inclusions (Figures 5d and 6d) for both the hourglass specimen and the dogbone specimen, and pore defects are the main crack initiation origins. The specimen size and shape do not change the fatigue failure mechanism of G20Mn5QT steel. The EDS is further used to determine the composition of the inclusion in the crack initiation region. The accelerating voltage is 15 kV. Figure 8 shows the results for the location "+" in the crack initiation region by EDS. It indicates that the main composition of inclusions should be alumina. The SEM observations show that most fatigue cracks initiate from pore defects (Figures 4d,5c and 6b) or inclusions (Figures 5d and 6d) for both the hourglass specimen and the dogbone specimen, and pore defects are the main crack initiation origins. The specimen size and shape do not change the fatigue failure mechanism of G20Mn5QT steel. The EDS is further used to determine the composition of the inclusion in the crack initiation region. The accelerating voltage is 15 kV. Figure 8 shows the results for the location "+" in the crack initiation region by EDS. It indicates that the main composition of inclusions should be alumina.

**Figure 8.** Analysis of composition of the inclusion in crack initiation region in Figure 6d. (**a**) SEM image of the crack initiation region. The symbol "+" denotes the location analyzed by EDS; (**b**) result by EDS. **Figure 8.** Analysis of composition of the inclusion in crack initiation region in Figure 6d. (**a**) SEM image of the crack initiation region. The symbol "+" denotes the location analyzed by EDS; (**b**) result by EDS.

#### **4. Discussion 4. Discussion**

#### *4.1. Comparison of S-N Data 4.1. Comparison of S-N Data*

literature [23,25,26,28–32].

Figure 9 shows the comparison of the S-N data between hourglass specimens and dogbone specimens. It is seen from Figure 9a that the difference in the S-N data between the two kinds of specimens is not obvious in terms of nominal stress amplitude, while the fatigue life of the hourglass specimen is generally larger than that of the dogbone specimen for the same local stress amplitude, though the fatigue life data overlap at several Figure 9 shows the comparison of the S-N data between hourglass specimens and dogbone specimens. It is seen from Figure 9a that the difference in the S-N data between the two kinds of specimens is not obvious in terms of nominal stress amplitude, while the fatigue life of the hourglass specimen is generally larger than that of the dogbone specimen

low stress amplitudes. As is well-known, the scatter of the fatigue life data tends to be larger at the low stress amplitude (i.e., the long fatigue life). The overlap of the fatigue life

stressed regions of the different types of specimens. The hourglass specimens all fail at or very near the smallest section of the specimen, whereas the positions of the fatigue fracture surface are all located at the parallel segment with the smallest section for the dogbone specimens. The highly stressed region of the hourglass specimen is smaller than that of the dogbone specimen. From the viewpoint of the statistical distribution of microstructures or defects, the dogbone specimen has more possibility for defects or microstructural inhomogeneity that could induce the fatigue failure. This is the reason why the fatigue life of the hourglass specimen is higher than that of the dogbone specimen at the same local stress amplitude. The decrease of the fatigue performance due to the larger highlystressed region (or control volume) has also been shown for different types of steel in the

for the same local stress amplitude, though the fatigue life data overlap at several low stress amplitudes. As is well-known, the scatter of the fatigue life data tends to be larger at the low stress amplitude (i.e., the long fatigue life). The overlap of the fatigue life data at several low stress amplitudes might be due to the scatter and randomness of the fatigue life. This phenomenon could be explained by the differences among the highly stressed regions of the different types of specimens. The hourglass specimens all fail at or very near the smallest section of the specimen, whereas the positions of the fatigue fracture surface are all located at the parallel segment with the smallest section for the dogbone specimens. The highly stressed region of the hourglass specimen is smaller than that of the dogbone specimen. From the viewpoint of the statistical distribution of microstructures or defects, the dogbone specimen has more possibility for defects or microstructural inhomogeneity that could induce the fatigue failure. This is the reason why the fatigue life of the hourglass specimen is higher than that of the dogbone specimen at the same local stress amplitude. The decrease of the fatigue performance due to the larger highly-stressed region (or control volume) has also been shown for different types of steel in the literature [23,25,26,28–32]. *Metals* **2022**, *12*, x FOR PEER REVIEW 9 of 13

**Figure 9.** S-N data of specimens with different size and shape, in which the arrows denote the unbroken specimens at the associated cycles. (**a**) Nominal stress amplitude versus fatigue life; (**b**) local stress amplitude versus fatigue life. **Figure 9.** S-N data of specimens with different size and shape, in which the arrows denote the unbroken specimens at the associated cycles. (**a**) Nominal stress amplitude versus fatigue life; (**b**) local stress amplitude versus fatigue life.

#### Here, the probabilistic control volume method [25,28] is used to analyze the size and *4.2. Prediction of Size and Shape Effects*

respectively.

*4.2. Prediction of Size and Shape Effects*

shape effects on the fatigue performance of G20MnQT steel. This method considers that if the fatigue strength of specimens A and B can be regarded as the minimum value of many reference specimens with relatively small control volume under the same manufacturing process and heat treatment, and the fatigue strength of the reference specimen follows a Weibull distribution, the fatigue strength of specimens A and B with the same survival probability satisfies the following relation: − <sup>−</sup> <sup>=</sup> (1) Here, the probabilistic control volume method [25,28] is used to analyze the size and shape effects on the fatigue performance of G20MnQT steel. This method considers that if the fatigue strength of specimens A and B can be regarded as the minimum value of many reference specimens with relatively small control volume under the same manufacturing process and heat treatment, and the fatigue strength of the reference specimen follows a Weibull distribution, the fatigue strength of specimens A and B with the same survival probability satisfies the following relation:

$$\frac{\sigma\_A - \gamma}{\sigma\_B - \gamma} = \left(\frac{V\_A}{V\_B}\right)^{-\frac{1}{k}}\tag{1}$$

For the case of fatigue failure induced by the surface crack initiation, the following relation is used: − <sup>−</sup> <sup>=</sup> (2) where *σ<sup>A</sup>* and *σ<sup>B</sup>* denote the fatigue strength of specimens A and B, respectively; *V<sup>A</sup>* and *V<sup>B</sup>* denote the control volume, which is usually chosen as the region with no less than 90% of the maximum principal stress [28–32]; *k* > 0 and *γ* ≥ 0 are shape and location parameters, respectively.

where *SA* and *SB* denote the critical part of the specimen surface (i.e., control surface) with a certain thickness. For the case of fatigue failure induced by the surface crack initiation, the following relation is used:

$$\frac{\sigma\_A - \gamma}{\sigma\_B - \gamma} = \left(\frac{S\_A}{S\_B}\right)^{-\frac{1}{k}}\tag{2}$$

(4)

 = (3) where *S<sup>A</sup>* and *S<sup>B</sup>* denote the critical part of the specimen surface (i.e., control surface) with a certain thickness.

> =

From the consideration that the fatigue cracks initiate from the specimen surface or

the size and shape effects of the fatigue strength for the present G20Mn5QT steel. The control surface (the region where the principal stress is no less than 90% of the maximum

In particular, for the two-parameter Weibull distribution, the fatigue strength of specimens A and B at the same survival probabilities satisfies the following relation:

$$\frac{\sigma\_A}{\sigma\_B} = \left(\frac{V\_A}{V\_B}\right)^{-\frac{1}{\mathbb{E}}} \tag{3}$$

$$\frac{\sigma\_A}{\sigma\_B} = \left(\frac{S\_A}{S\_B}\right)^{-\frac{1}{k}}\tag{4}$$

From the consideration that the fatigue cracks initiate from the specimen surface or subsurface for all the hourglass and dogbone specimens, Equation (4) is used to analyze the size and shape effects of the fatigue strength for the present G20Mn5QT steel. The control surface (the region where the principal stress is no less than 90% of the maximum principal stress) is obtained by the finite element analysis. In the calculation, the linear elastic constitutive relation is used. The Young's modulus is *E* = 210 GPa and Poisson's ratio is ν = 0.3. At first, the maximum principal stress is calculated at a load of 100 N under the tensile stress, and then the region of the surface of the specimen where the principal stress is no less than 90% of the maximum principal stress is determined. The control surface of the hourglass and dogbone specimens are listed in Table 2. The parameters of the Weibull distribution of the fatigue strength are estimated by the method in the literature [25,28]. In this method, the bilinear model [25,28,33] is assumed for the S-N curve, i.e.,

$$\log\_{10} \sigma = \begin{cases} \begin{array}{c} a \log\_{10} N + A\_{\prime} & N < N\_{0} \\ B\_{\prime} & N \ge N\_{0} \end{array} \tag{5} \\ \tag{6}$$

where *a*, *A* and *B* are constants, and *N*<sup>0</sup> is the number of cycles at the knee point of the curve. Equation (5) can be written as

$$\log\_{10} \sigma = \begin{cases} a(\log\_{10} N - \log\_{10} N\_0) + B\_\prime & N < N\_0 \\ B\_\prime & N \ge N\_0 \end{cases} \tag{6}$$

For the specimens with the fatigue strength *σ<sup>k</sup>* and the associated fatigue life *N<sup>k</sup>* (*k* = 1, 2, . . . , *n*, and *n* is the number of specimens), the values of *a*, *B* and *N*<sup>0</sup> can be obtained by the minimum value of the following equation

$$\sum\_{N\_k < N\_0} \left[ \log\_{10} \sigma\_k - a \log\_{10} (N\_k / N\_0) - B \right]^2 + \sum\_{N\_k \ge N\_0} \left( \log\_{10} \sigma\_k - B \right)^2 \tag{7}$$

From Equation (6), the fatigue strength *σ<sup>k</sup>* at an arbitrary fatigue life *N<sup>k</sup>* can be transformed into the fatigue strength *σ* 0 *<sup>k</sup>* at a given fatigue life *N*<sup>0</sup> *k* , i.e.,

$$\log\_{10} \sigma'\_k = \begin{cases} a \log\_{10} \frac{N'\_k}{N\_k} + \log\_{10} \sigma\_{k'} \ N\_k < N\_0 \\\ a \log\_{10} \frac{N'\_k}{N\_0} + \log\_{10} \sigma\_{k'} \ N\_k \ge N\_0 \end{cases} \quad \text{for } N'\_k < N\_0 \tag{8}$$

or

$$\log\_{10} \sigma'\_k = \begin{cases} a \log\_{10} \frac{N\_0}{N\_k} + \log\_{10} \sigma\_{k'} & N\_k < N\_0 \\\log\_{10} \sigma\_{k'} & N\_k \ge N\_0 \end{cases} \quad \text{for } N'\_k \ge N\_0 \tag{9}$$

Then, the statistical analysis can be performed for the fatigue strength at different fatigue life and the probabilistic stress-life (P-S-N) curve is obtained.

Figure 10 shows the comparison between the predicted P-S-N curves and the experimental data for the hourglass specimen. It is seen that the predicted 50% survival probability curve is in the middle of the experimental data and almost all the experimental data are within the predicted 5% and 95% survival probability curves. This indicates that the predicted results accord well with the experimental data, namely that the method

in the literature [25,28] is reasonable for the estimation of the parameters of the Weibull distribution of fatigue strength.

**Table 2.** Control surface of specimens with different size and shape.

**Figure 10.** Comparison of predicted P-S-N curves with experimental data for hourglass specimens, in which the arrows denote the unbroken specimens at the associated cycles. **Figure 10.** Comparison of predicted P-S-N curves with experimental data for hourglass specimens, in which the arrows denote the unbroken specimens at the associated cycles. **Figure 10.** Comparison of predicted P-S-N curves with experimental data for hourglass specimens,

Figure 11 shows the comparison between the predicted P-S-N curves by the experimental data of the hourglass specimen and the experimental data for the dogbone specimen. It is seen that the predicted P-S-N curves are in agreement with the experimental data, indicating that the probabilistic control volume method is applicable for correlating the size and shape effects on the fatigue performance of G20Mn5QT steel. Figure 11 shows the comparison between the predicted P-S-N curves by the experimental data of the hourglass specimen and the experimental data for the dogbone specimen. It is seen that the predicted P-S-N curves are in agreement with the experimental data, indicating that the probabilistic control volume method is applicable for correlating the size and shape effects on the fatigue performance of G20Mn5QT steel. Figure 11 shows the comparison between the predicted P-S-N curves by the experimental data of the hourglass specimen and the experimental data for the dogbone specimen. It is seen that the predicted P-S-N curves are in agreement with the experimental data, indicating that the probabilistic control volume method is applicable for correlating the size and shape effects on the fatigue performance of G20Mn5QT steel.

in which the arrows denote the unbroken specimens at the associated cycles.

imens at the associated cycles. **Figure 11.** Comparison of predicted P-S-N curves with experimental data of dogbone specimens by using the experimental data of hourglass specimens, in which the arrows denote the unbroken specimens at the associated cycles. **Figure 11.** Comparison of predicted P-S-N curves with experimental data of dogbone specimens by using the experimental data of hourglass specimens, in which the arrows denote the unbroken specimens at the associated cycles.

using the experimental data of hourglass specimens, in which the arrows denote the unbroken spec-

#### **5. Conclusions**

In this paper, the size and shape effects on the fatigue behavior are investigated for G20Mn5QT steel of axle box bodies in high-speed trains. The main results are as follows:

The specimen size and shape have influence on the position of the fatigue fracture surface. For the hourglass specimen, it fails at or very near the smallest section of the specimen; whereas for the dogbone specimen, the positions of the fatigue fracture surface are all located at the parallel segment with the smallest section.

The specimen size and shape have no influence on the fatigue failure mechanism of G20Mn5QT steel under an axial loading fatigue test. The fatigue cracks initiate from the surface or the subsurface of the specimen, and some fatigue fracture surfaces exhibit the characteristic of multi-site crack initiation. Most of the fatigue cracks initiate from the pore defects and alumina inclusions in the casting process, and the pore defects are the main crack origins.

The specimen size and shape have an influence on the fatigue performance of G20Mn5QT steel. Due to the larger highly stressed region, the fatigue life of the hourglass specimen is generally higher than that of the dogbone specimen at the same local stress amplitude. The probabilistic control volume method is applicable to correlating the size and shape effects on the fatigue performance of G20Mn5QT steel.

The results are helpful in understanding the fatigue failure mechanism of G20Mn5QT steel and the size and shape effects on the fatigue behavior of metallic materials.

**Author Contributions:** Conceptualization, Z.Z., Z.L. and C.S.; investigation, Z.Z., Z.L. and C.S.; visualization, Z.L., H.W. and C.S.; writing—original draft preparation, Z.Z., Z.L., H.W. and C.S.; writing—review and editing, Z.Z., Z.L., H.W. and C.S.; supervision, C.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the National Key Research and Development Program of China (2017YFB0304600).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

