**An Ultra-High Frequency Vibration-Based Fatigue Test and Its Comparative Study of a Titanium Alloy in the VHCF Regime**

#### **Wei Xu 1,\*, Yanguang Zhao 2, Xin Chen 1, Bin Zhong 1, Huichen Yu 1, Yuhuai He <sup>1</sup> and Chunhu Tao <sup>1</sup>**


Received: 1 October 2020; Accepted: 22 October 2020; Published: 24 October 2020

**Abstract:** This paper proposes an ultra-high frequency (UHF) fatigue test of a titanium alloy TA11 based on electrodynamic shaker in order to develop a feasible testing method in the VHCF regime. Firstly, a type of UHF fatigue specimen is designed to make its actual testing frequency reach as high as 1756 Hz. Then the influences of the loading frequency and loading types on the testing results are considered separately, and a series of comparative fatigue tests are hence conducted. The results show the testing data from the present UHF fatigue specimen agree well with those from the conventional vibration fatigue specimen with the loading frequency of 240 Hz. Furthermore, the present UHF testing data show good consistency with those from the axial-loading fatigue and rotating bending fatigue tests. But the obtained fatigue life from ultrasonic fatigue test with the loading frequency of 20 kHz is significantly higher than all other fatigue test results. Thus the proposed ultra-high frequency vibration-based fatigue test shows a balance of high efficiency and similarity with the conventional testing results.

**Keywords:** vibration-based fatigue; ultra-high frequency; very high cycle fatigue; fatigue test; titanium alloy

#### **1. Introduction**

Aviation equipments, such as aeroplanes and aeroengines always undergo cyclic stress during service time, thus fatigue damage has been a major concern in the researches of aeronautical materials and structures. In recent decades, the academic and engineering communities have gradually realized that fatigue fracture of materials can occur after 10<sup>7</sup> cycles or even 108cycles. Especially for aviation equipment, the failure forms of many structural components belong to very high cycle fatigue (VHCF) regime. As a result, VHCF has gradually been paid an increasing number of attentions in the design of the aviation equipments [1], with the higher requirements of service life and reliability.

Fatigue testing is an essential aspect in fatigue researches. Due to the ultra-high failure cycles, improving the loading efficiency is very crucial in the VHCF testing. Several testing equipments have been used for the testing of VHCF, such as rotating-bending fatigue tester, servo-hydraulic fatigue tester, electromagnetic resonance tester and ultrasonic fatigue tester. The first three types of the testing approaches are usually regarded as conventional fatigue testers, which have the general

loading frequency range within 10 to 100 Hz. Thus it would be extremely time-consuming using the conventional fatigue testers in the VHCF regime [2]. In contrast, the development of the ultrasonic fatigue tester has greatly improved the loading efficiency of the VHCF testing since the actual loading frequency reaches up to 20 kHz [3]. Benefit from the huge progress in fatigue testing efficiency, the researches in VHCF of titanium alloys and other metals have been widely conducted and some interesting fatigue testing data and failure mechanism have been obtained [4–6].

However, the significant increase in loading frequency would remarkably influence the fatigue strength and life of some materials in the VHCF regime, which have been reported by many researchers [2,4,7] Also it is controversial whether the fatigue failure mechanism under the conditions of ultrasonic fatigue testing is similar to those of conventional high-frequency fatigue testing [8,9]. Meanwhile, some investigations have supported that the results from the ultrasonic fatigue tester were close to those from the conventional fatigue testers [2]. Anyway, it can be stated that the application of the ultrasonic fatigue testing in the VHCF regime is still in controversy. Furthermore, a widely-accepted testing standard of the ultrasonic fatigue has not been proposed, while the currently existing testing standards (e.g., ISO and ASTM standards) are only applicable for the conventional fatigue tests.

For aviation engineering, VHCF issues are generally introduced by the vibration of the moving components such as blades and vanes. When subjected to fatigue loadings during the working condition, these components would always experience bending or twist loads at high frequencies [10,11]. For one thing, the actual working frequency of these aviation components cannot reach as high as the loading frequency (i.e., 20 kHz) of the ultrasonic fatigue testing. For another thing, axial-loading fatigue data does not provide a sufficient representation of HCF or VHCF behaviors of the vibrating components [12]. These situations are not adequately simulated by axial-loading fatigue tests. In short, both of the conventional axial loading fatigue testing and the ultrasonic fatigue testing data are insufficient for the design of the vibrating components in the VHCF regime.

Accordingly, vibration-based fatigue tests have been developed and carried out to obtain more meaningful data for the vibrating components. Also it is a proper testing approach to study the bending fatigue properties within the reduced experimental period since the testing frequency is much larger than the conventional axial-loading fatigue tests [13]. In other words, the vibration-based fatigue test is a sort of speeding-up fatigue test. Vibration-based bending fatigue tests have been usually carried out by electrodynamic shakers. It is widely known that the response amplitude of a specimen reaches its maximum value when the specimen vibrates under resonance condition for same excitation amplitude [14]. Thus the specimens are usually excited in a high frequency resonant mode for the purpose of reducing the power-consuming of the testing system [15,16]. This can be supported by several vibration fatigue studies of different materials [17,18], most of which have been carried out in the regime of conventional fatigue cycle though. And few work of vibration fatigue has been reported in the VHCF regime of materials.

This research proposes an experimental method for ultra-high frequency fatigue of materials using an electrodynamic shaker. For a type of titanium alloy commonly used in aero engines, an ultra-high frequency (UHF) fatigue specimen is independently designed and the fatigue experiments in the HCF and VHCF regimes are conducted. Furthermore, the influences of the loading frequency and loading types on the testing results of the fatigue life are considered separately, and a series of comparative fatigue tests are hence conducted, with the testing results compared with the present UHF results finally.

#### **2. UHF Fatigue Specimen Design**

The fatigue testing method used in the present study is actually based on resonance. Thus the loading frequency of the present fatigue testing system is very close to the resonance frequency of the specimen. It is widely known that the number of the vibration mode of a continuous system is infinite, and each vibration mode has the corresponding natural frequency. For the moving components in power engineering, such as blades, the first-order bending mode is the most common form in actual service. In other aspect, the first-order vibration mode is easy to be conducted in fatigue tests compared

to the higher-order vibration modes. Thus only the first-order longitudinal vibration mode has been merely considered in both the conventional electromagnetic loading and ultrasonic loading. Similarly, the first-order bending vibration mode is considered in the present UHF specimen. And the natural angular frequency ω in the first-order mode is the key parameter to be concerned during the design.

$$
\omega = f(l, \rho, E, \mu) \tag{1}
$$

where ρ, *E* and μ denote density, Young's modulus and Poisson's ratio, respectively. By dimensional analysis, a derived equation with a dimensionless form can be obtained by transferring Equation (1), as shown in Equation (2):

$$\frac{\frac{\partial l}{\partial \mu}}{\sqrt{\frac{E}{\rho}}} = f(\mu) \tag{2}$$

For two specimens with the same shape, material and boundary condition, the only difference is the characteristic length or size *l*, thus the proportional relation expressed by Equation (3) can be obtained. <sup>ω</sup><sup>1</sup>

$$\frac{\alpha\_1}{\alpha\_2} = \frac{l\_2}{l\_1} \tag{3}$$

Accordingly, any higher natural frequency of a fatigue specimen could be achieved by reducing the size proportionally in theory. However, the clamping reliability and the fatigue dangerous zone (i.e., working section of specimen in which the fatigue failure is most likely to occur) should be simultaneously considered, thus a novel UHF fatigue specimen cannot be determined by simply reducing the size proportionally of the existing vibration-based fatigue specimen. In the present study, an iterative method was adopted to obtain the geometry of UHF fatigue specimen, with the flow chart shown in Figure 1. Here, two basic goals should be mentioned: One of them is that the first-order bending natural frequency or resonance frequency *f* of UHF fatigue specimen should fall within the range 1600 Hz < *f* < 2000 Hz, which ensure the testing period of 10<sup>9</sup> cycles is less than one week. Of course, the resonance frequency beyond 2000 Hz is also not welcome in order to avoid possible frequency effect. Furthermore, the present frequency range is close to that of the aeroengine blade with the present titanium alloy in order to make the present fatigue testing results more valuable for the blade. The other basic goal is that the maximum stress σmax in the fatigue dangerous zone should be significantly larger than the mean value σ<sup>m</sup> in the same zone. And a relation σmax ≥ 1.5σ<sup>m</sup> is adopted in the present study.

Accordingly, finite element method (FEM) was employed to determine the geometry of UHF fatigue specimen. A series of FEM models with different geometries was established by a commercially available FEM code ABAQUS (v6.14, Dassault Systemes, Providence, RI, USA). Generally, the maximum stress levels for HCF and VHCF tests are far less than the yield strength of specimens. Thus the present specimen is modeled as a linear-elastic solid, with the fundamental material properties listed in Table 1. Only the first bending vibration mode was considered in the study and the natural frequencies can be obtained by the Lanczos eigensolver integrated in ABAQUS. And the stress distribution on the surface of the specimen during the vibration can be also obtained.

**Table 1.** Material parameters of TA11titanium alloy [19].


**Figure 1.** Iterative method to design ultra-high frequency specimen.

After the FEM calculation, a design of UHF specimen was finally determined, with the geometry shown in Figure 2. The two holes in the right side are used to install bolts to mount the specimen on testing system, while the three small holes in the left side are used to adjust the natural frequency and stress distribution of the specimen. The first-order mode bending natural frequency by the FEM calculation is 1775 Hz, which just meets the aforementioned frequency requirement. And the surface normalized axial stress contour *S*<sup>11</sup> is shown in Figure 3a. Noting the area with the two mounting holes would be totally clamped by the fixture, thus clamping area can be replaced by the boundary condition of fixed support, and the two holes are not necessarily included in the FEM model.

**Figure 2.** Geometry of the present ultra-high frequency (UHF) fatigue specimen (unit: mm).

In order to validate the FEM result, a stress measurement based on strain gauge was used to obtain the stress values along the central line of the area with arc segment. Benefit from the FEM calculation, a continuous stress distribution curve along the same central path can be obtained and shown in Figure 3b, with *O* and *E* representing the origin and end points on the central path, respectively. The comparison of the stress distributions from the FEM and strain gauge can be found in Figure 3b, which shows very good consistency. It should be noted that the maximum stress point is located not exactly at the midpoint, but slight near to the clamping end (normalized location = 0.41353), which is

resulted from the bending deformation of the specimen. Furthermore, the mean stress σ<sup>m</sup> along the central path can be obtained by the Equation (4):

$$
\sigma\_{\mathbf{m}} = \frac{1}{l} \int\_{\text{path}} \sigma\_{11} \mathbf{d}l \tag{4}
$$

where *l* denotes the length of the path. The normalized value ofσ<sup>m</sup> can be hence obtained, with the value of 0.56. Noting the normalized maximum stress σmax is equal to 1, thus the aforementioned relation σmax ≥ 1.5σ<sup>m</sup> can be satisfied.

**Figure 3.** FEM calculation and validation of the present UHF specimen: (**a**) Surface normalized axial stress contour; (**b**) comparison of the stress distributions from FEM and strain gauge (*O* and *E* represent the origin and end points on the central path of the specimen).

Furthermore, it should be pointed out the optimal geometry of the present UHF fatigue specimen is dependent on the material and geometrical parameters. And the geometry of the present UHF specimen is proposed based on the present titanium alloy. Although this geometry is not exactly applicable to other materials, it is still important reference geometry for the similar fatigue tests of other materials.

#### **3. Material and Experimental Details**

#### *3.1. Experimental Material*

A near-alpha titanium alloy TA11 alloy equivalent to Ti-8Al-1Mo-1V was used in the present fatigue study. It has been usually used in advanced turbine engines as low pressure compressor blades and due to its excellent damping capacity, low density, high Young's modulus, and fine welding and anti-oxidation performance [20]. The chemical composition of TA11 used in the present study is list in Table 2.


**Table 2.** Chemical composition of TA11 titanium alloy (mass fraction/%).

#### *3.2. UHF Fatigue Testing Setup*

A vibration-based bending fatigue test was subsequently conducted using the present UHF specimens shown in Figure 2. All the specimens were cut from the same batch of raw material in order to make the results reliable. The test was conducted on a vibration-based fatigue testing system, of which the major body is an electrodynamic shaker (ES-10D-240 Electrodynamic Shaker System) located in a soundproof room. The maximum loading capacity of the shaker is 10 kN and the frequency range is 5 to 3000 Hz, which meets the mode-I bending vibration testing requirement of the present specimens. Similar to conventional fully-reversed bending fatigue tests, the ratio between the maximum and minimum stress in the present test is equal to −1.

In order to clamp the specimen firmly, a specified fixture was designed and manufactured. As same as illustrated in Figure 4, one end was clamped firmly and the other end was free, which is shown in Figure 4a. An accelerometer was used to monitor the shaker input load and a laser vibrometer was located above the free end to monitor the amplitude. The excitation direction was vertical to the specimen, with the excitation force having a sine waveform. Simultaneously, a small-size strain gauge was mounted longitudinally on the surface of the specimen, exactly locating at the maximum stress location, as shown in Figure 4a. It should be mentioned that a small-size strain gauge (sensitive pattern area: 1 <sup>×</sup> 1 mm2) was adopted since both of the specimen and the fatigue dangerous zone are small.

**Figure 4.** Vibration-based fatigue experiment for the UHF specimen: (**a**) Testing equipment (**b**) Sketch of the experimental system.

In this test, the amplitude of the free edge was a main object feedback, by which the excitation frequency could be automatically adjusted. The specimens were expected to be tested in the resonance condition and the excitation frequency could be adjusted automatically to keep the vibration amplitude stable [16]. Consequently, the amplitude could be steadily controlled. Benefiting from the automatic self-adjusting, the experimental system could run by the means of so-called closed-loop control, which is sketched in Figure 4b.

#### *3.3. Resonance Frequency and Stress Calibration*

It is widely known that the response amplitude of a specimen can reach its maximum value when the specimen vibrates under resonance condition for the same excitation load. Therefore, the vibration-based fatigue tests are always expected to be carried out under the resonance condition for the purpose of reducing the power-consuming of the experimental system. Accordingly, the resonance frequency should be identified before the vibration-based fatigue test. In order to obtain the vibration characteristics, the excitations with a series of increasing frequency was imposed upon the specimen, and the excitation frequency-response curves can be hence obtained and the resonant frequency of the specimen can be further determined, with the value of around 1756 Hz, which is obtained by the frequency sweeping testing shown in Figure 5. Noting that specimen is clamped by the designed fixture in the test, the resonant frequency obtained from the test is actually that of the combination of the specimen and the fixture, and it would be less than the natural frequency (i.e., 1775 Hz) of the single specimen, which is obtained by the computation presented in Section 2.

**Figure 5.** Determination of the resonance frequency of the UHF specimen by frequency sweeping.

During the vibration-based fatigue test process, the resonant frequency was monitored to determine the failure moment of the specimen. In this study, the excitation frequency was preset using the same value of the obtained resonant frequency, which is a stable value (i.e., around 1756 Hz) during the fatigue testing process. As the crack propagates in the fatigue dangerous zone, the resonant frequency decreases gradually to a critical value. When the resonant frequency drop rate reaches to a critical value with the value of 1%, the fatigue test will be terminated and the specimen is considered to be failure.

Since the strain gauge would fail soon after several cycles in the fatigue test, the stress-control of vibration-based fatigue tests has been always achieved by controlling the amplitude. Thus a calibration relation between the measured strain and the amplitude should be determined prior to the fatigue testing. The clamped specimen is similar to a normal slender cantilever beam and the transverse stress can be ignored. Accordingly, three typical values of the amplitude were selected and the peak-valley values of the strain along the 1-direction were measured during the vibration testing. A linear calibration relation between the measured strain (peak-valley value εP−V) and double-amplitude 2*a* of the present UHF specimen can be obtained, shown in Figure 6. For a specific amplitude *a*, the value of σ<sup>1</sup> can be gained by the strain gauge measure together with the stress-strain relation σ<sup>1</sup> = 0.5 · *E*εP−V. Thus the present stress-control vibration-based fatigue test is actually the realized by control the amplitude, and various stress levels can be realized by varying amplitude *a*.

**Figure 6.** Calibration relation between the measured strain and displacement amplitude of the UHF specimen.

#### *3.4. Fatigue Tests for Comparison*

In order to verify and compare the present UHF fatigue testing results, some comparative tests have been conducted from two aspects:

On one hand, the effect of the loading frequency on the testing results should be verified. Some previous studies for VHCF tests have also considered this issue [7,9], But the comparative studies have been always performed among different types of fatigue tests, such as the comparison between the rotating bending fatigue test and ultrasonic fatigue test. Although the loading frequencies of the tests are definitely different, the loading condition also influences the testing results. Accordingly, the effect of the loading frequency can best be considered separately. In the present study, a conventional vibration fatigue (CVF) specimen shown in Figure 7 was used to explore the influence of the loading frequency. The two holes in the right side are used for mounting bolts to fix the specimen on the testing system. The minimum width of the fatigue dangerous zone is 10 mm. The geometry of the conventional vibration fatigue specimen is taken from a Chinese testing standard HB 5277, which is widely used in the field of vibration-based fatigue testing for the aeroengine blade materials in China. After a similar frequency-response test mentioned by Section 3.3, the actual loading frequency close to the resonance frequency is obtained, with the value of approximately 240 Hz.

**Figure 7.** Geometry of the conventional vibration fatigue (CVF) specimen (unit: mm).

One the other hand, the effect of the fatigue loading types should be considered. Some other types of fatigue tests were also conducted, which includes conventional axial loading (CA) fatigue test, rotating bending (RB) fatigue test and ultrasonic axial loading (UA) fatigue test. All these tests have been widely carried out in fatigue community, thus the comparison with them is helpful to verify the applicability of the present testing method in VHCF testing. Here, the CA loading fatigue test and RB fatigue test were carried out in an electromagnetic resonant fatigue testing system and a rotating bending fatigue tester, respectively. And the UA fatigue test was carried out in a commercial ultrasonic fatigue test machine (USF-2000, Shimadzu, Japan). All the fatigue tests were conducted in room temperature, with the stress ratio *R* equal to −1. The loading frequencies for the CA, RB and UA fatigue tests were 120 Hz, 83.3 Hz and 20 kHz, respectively. Considering the ultra-high loading frequency (i.e., 20 kHz) in the UA fatigue testing, a compressive dry air cooling system was used to cool the UA specimen during the testing in order to ensure the specimen temperature is maintained at room temperature. Furthermore, an infrared thermometer was used to monitor the surface temperature of the specimen during the UA fatigue test.

The geometry of the specimens for comparison is shown in Figure 8. All the specimens have hourglass-type shape. It should be pointed out all the specimens for comparison were machined from the same batch of raw materials with the present UHF specimen, in order to make the testing results more comparable.

**Figure 8.** Geometry of the specimens used in fatigue tests for comparison (unit: mm): (**a**) Conventional axial loading (CA) specimen; (**b**) rotating bending (RB) specimen and (**c**) ultrasonic axial loading (UA) specimen.

#### **4. Results and Discussion**

#### *4.1. Vibration-Based Fatigue Testing Results*

Figure 9 shows the obtained *S-N* data and the relevant fitting curves for the present vibration-based fatigue tests, which involves the UHF and CVF specimens, with the actual loading frequencies of about 1756 Hz and 240 Hz, respectively. Several stress levels have been selected in the present UHF fatigue testing, covering the stress range from 400 MPa to 540 MPa. And the maximum failure cycle can reach up close to 109. In order to compare the results between the two types of vibration-based fatigue tests, the same stress levels have been also considered in the test for CVF specimens. Noting the actual loading frequency of CVF specimens is about 240 Hz, the fatigue testing in the VHCF regime (i.e., >10<sup>7</sup> cycles) has not been considered due to its high time-consuming. Finally, there are 20 and 17 valid data obtained in the UHF and CVF tests, respectively, which are given in Table 3. Considering the significant scatter of the obtained fatigue lives by the tests, several specimens were tested at the same stress level in order to make the results statistically reliable.

**Figure 9.** *S-N* data and the fitting curves for the present vibration-based fatigue tests involving the UHF and CVF specimens.


**Table 3.** *S-N* testing data from the UHF specimens and CVF specimens.

The *S-N* curves can be obtained by a regression calculation with a three-parameter *S-N* model, which is named as Stromeyer model [21], expressed by:

$$\log \text{N}\_{\text{f}} = a - b \lg(\sigma\_{\text{max}} - \text{S}\_0) \tag{5}$$

where *N*<sup>f</sup> and σmax denote the failure cycle and stress level (or maximum cyclic stress). And *a*, *b* and *S*<sup>0</sup> are the fitting parameters. Noting σmax would approach to *S*<sup>0</sup> when *N*<sup>f</sup> approaches to infinite, thus *S*<sup>0</sup> can be regarded as a fatigue limit from a mathematical point of view. Consequently, the *S*-*N* curves from Stromeyer model would exhibit obvious curvature characteristics and the fitting model has been widely used in the fatigue community. The values of *a*, *b* and *S*0for UHF specimens are 10.7, 2.50 and 395, respectively, while those for CVF specimens are 8.05, 1.32 and 421, respectively.

In addition, typical surface crack morphology when UHF specimen fails are shown in Figure 10. It can be found an obvious continuous crack locating approximately at normalized location of 0.38 initiated at *O* point shown in Figure 3b, which is close to the maximum stress point determined in Section 2, not at the narrowest width of the specimen. And the crack propagation direction is basically perpendicular to the axial stress direction (i.e., 1-direction). Thus it is reasonable to determine the strain measurement location as mentioned in Section 3.2.

**Figure 10.** Typical crack morphology when UHF specimen fails.

#### *4.2. E*ff*ect of the Loading Frequency*

As shown in Figure 9, the *S*-*N* curves between the UHF and CVF specimens are close to each other in the same life cycle regime. Both of the UHF and CVF specimens are sheet specimens used in the vibration-based testing with the same type of fatigue loads. The major difference between them is merely the loading frequency, which does not influence the *S*-*N* curves clearly as shown in Figure 9. Considering the two cases share the same fatigue stress levels and several testing data have been obtained at those fatigue stress levels, further comparison and analysis can be conducted in order to explore the effect of loading frequency further.

In the present vibration-based fatigue testing, three fatigue stress levels were considered with the values 440 MPa, 480 MPa and 540 MPa. Figure 11 shows the comparison of fatigue lives from the UHF and CVF specimens at these fatigue stress levels. It can be found the fatigue lives for the two types of specimens are close to each other at these stress levels. Strictly speaking, the fatigue lives for the CVF specimen are slightly shorter than those for the UHF specimen, which is clear at the stress level of 440 MPa. One reason leading to the tiny discrepancy is probably the influence of the specimen size. It can be found there is a significant difference in the size of the two types of specimens, despite their shapes are similar. It has been previously pointed out a smaller size fatigue specimen would have a longer fatigue life due to the smaller risky volume and less likelihood of containing defects [22]. Thus the present tiny discrepancy in fatigue lives between the two specimens can be mainly attributed to specimen size instead of the loading frequency.

In addition, the error bars shown in Figure 11 to reflect the dispersion of result data are worth mentioning. The length of the error is generally increased as the fatigue stress level is decreased. It suggests the dispersion of the fatigue lives is more significant for the lower stress level, which has been widely verified by the previous HCF and VHCF researches [23,24]. It should be noted the length of error bars for UHF specimen is significantly shorter than those for CVF at the low fatigue stress levels (i.e., 440 and 480 MPa), which suggests the results for UHF specimens are probably less dispersive than the CVF specimens. Although there is no solid reason to explain it, it is at least concluded the data stability by the present UHF specimens is not inferior to the conventional specimens with the lower frequency.

**Figure 11.** Comparison of fatigue life results from the UHF and CVF specimens for the same fatigue stress levels.

#### *4.3. E*ff*ect of the Testing Types*

Figure 12 shows the comparison of the results of the present UHF specimens and other types of fatigue specimens in consideration. The *S-N* curves can be also obtained by Equation (5), with the values of the fitting parameters given in Table 4.

**Figure 12.** Comparison of the results of ultra high frequency (UHF) specimen and other types of fatigue specimens.

**Table 4.** Fitting parameters of Equation (5) for the comparative tests.


In general, the *S*-*N* data based on the present UHF specimens get close to those based on the conventional HCF methods, including the CA and RB specimens. In contrast, the *S-N* curve from the UA specimens has significant discrepancy compared with those from all other testing methods. It can be found that the fatigue life data obtained by the UA test are much longer than other types of tests at the same stress level.

Generally, the factors influencing the fatigue testing results includes the temperature rise of specimen, material uniformity and testing method. Noting the cooling system was simultaneously used during the UA testing process, the surface temperature was kept at the range of 12~15◦C obtained by the infrared thermometer. Thus the temperature rise of the UA specimens can be neglected. In addition, all the fatigue specimens in this study were obtained based on same batch of Ti-alloy, thus it can be inferred the discrepancy of the results shown in Figure 12 is caused by the testing method.

For fatigue testing method, an important factor is the specimen size, which can be usually considered to explain the discrepancy of the testing data. It has been usually widely thought that the fatigue specimen with smaller size would have longer fatigue life [22,25]. The specimen in the UA testing usually has a small risky volume which means the specimen volume subjected to a stress amplitude larger than the 90% of its maximum value [26]. Some previous studies have attributed the size effect of fatigue specimens to the influence of the risky volume [25,26]. However, another important factor should be noted is the loading frequency. Since both the loading frequency and the risky volume could influence the fatigue life of specimens, the explanation from the risky volume is feasible when the loading-frequencies of the fatigue tests are close to each other. For example, for the present two conventional comparative fatigue tests involving the CA and RB specimens, their loading frequencies are both close to 100 Hz. Accordingly, the discrepancy between their corresponding *S*-*N* curves could be explained by the risky volume theory, which means the risky volume of RB specimen is smaller than that of CA, resulting in the fatigue life of RB specimen is longer. It should be pointed out the risky volume of the UA specimen is actually small, but still larger than that of the present UHF and RB specimens. Note the fatigue life obtained by the UA test is much longer than other types of tests, the discrepancy between their results cannot be explained by the risky volume theory.

Instead, another important factor in the testing method is the loading frequency, which could be the major factor to cause the discrepancy of the testing results. It has been found that the fatigue lives of some materials have been proved to be almost unaffected by the loading frequency, while the situation of some other materials are opposite [9]. In general, the materials with an obvious strain rate-related effect are more susceptible to loading frequency [7], thus it can be inferred the present titanium alloy is a material with obvious strain rate-related effect and the UA testing method is probably not suitable for its VHCF testing. In contrast, although the frequency of the present UHF fatigue method is also high (1756 Hz) compared with the conventional testing methods, the obtained fatigue lives by the present UHF method are not clearly influenced by the high frequency.

#### *4.4. Discussion*

Although the testing results from the present UHF specimens is generally close to those from CA and RB specimens, especially in the long life regime, the discrepancy between them deserves further explanation from the perspective of the failure criterion. It should be paid attention that the present vibration-based UHF test has a different failure criterion compared with that of the aforementioned conventional fatigue tests. Actually, the failure criterion of the most conventional fatigue tests is the separation of specimen. However, the present UHF test adopts the failure criterion that the resonance frequency of specimens drops by 1% of the initial value. The critical value of resonance frequency used as the failure criterion was determined based on the previous vibration-based fatigue tests [27]. It has been widely known that the life of the crack initiation and the growth of the micro-structurally small crack accounts for a large proportion of the total life in long life regime [28]. For the low stress cases in present UHF test, once the crack initiation occurs, the specimen would fail very soon since the loading frequency is very high. Thus the critical frequency for the failure criterion can be feasible. However, for the high stress cases in the present UHF test, the proportion of crack initiation life in the total life decreases while the proportion of the crack propagation life increases. Consequently, the specimen would not fail very soon after the crack initiation and the growth of the micro-structurally small crack.

Here, it should be clarified the reason why the specimen separation is not applicable for the failure criterion of the present UHF test. Firstly, the present vibration-based fatigue specimen has been usually clamped at only one side, with the other side free. If the testing continues until the specimen is separated, the free side is likely to impact and damage the surrounding devices and personnel when the separation occurs, because the loading frequency is very high. Secondly, the stress control of the testing is achieved by controlling the vibration amplitude of the specimen. As the fatigue test continues, the damage evolution would continue so that the resonance frequency would drop simultaneously. But the vibration amplitude should be maintained during the whole testing period to satisfy the testing stress condition, which needs a close-loop control method. However, in the period before the specimen's final separation, the resonance frequency drops dramatically as the damage evolution develops sharply. Thus it is so difficult to maintain the vibration amplitude at a constant in the final period. Considering the final period approaching to the separation of specimen is usually short, thus a critical frequency drop has been usually adopted as the failure criterion, instead of the specimen separation.

However, the failure criterion that the resonance frequency of specimens drops by 1% is just an empirical one, which has been proved feasible in some conventional vibration-based fatigue tests. Thus it has been adopted as a recommended failure criterion in the Chinese vibration-based fatigue testing standard HB 5277, which is also followed in the present study. However, the failure criterion of the frequency drop by 1% may be not the optimal one for the present non-standard UHF specimen especially in the higher fatigue stress cases. Thus it is necessary to figure out the optimal failure criterion for the present UHF specimen in the future, which helps to further improve the accuracy of the present testing results.

#### **5. Conclusions**

In summary, this paper proposes an ultra-high frequency (UHF) fatigue test of a titanium alloy TA11 based on electrodynamic shaker in order to develop a feasible testing method in the VHCF regime. Firstly, a type of UHF fatigue specimen is designed to make its actual testing frequency reach as high as 1756 Hz. Then the influences of the loading frequency and loading types on the testing results of the fatigue life are considered separately, and a series of comparative fatigue tests are hence conducted. The results show the testing data from the present UHF fatigue specimen agree well with those from the conventional vibration fatigue specimen with the loading frequency of 240 Hz. Furthermore, the present UHF testing data show good consistency with those from the axial-loading fatigue and rotating bending fatigue tests. But the fatigue life obtained from the ultrasonic fatigue test is significantly higher than all other fatigue testing results. Thus the proposed ultra-high frequency vibration-based fatigue test will have a good application prospect in the VHCF testing due to its balance of high efficiency and similarity with the conventional testing results.

**Author Contributions:** Conceptualization, W.X.; Funding acquisition, Y.H. and C.T.; Investigation, W.X.; Methodology, B.Z.; Project administration, H.Y.; Software, B.Z.; Supervision, H.Y., and Y.H.; Validation, X.C; Visualization, Y.Z.; Writing—original draft, W.X. and X.C.; Writing—review & editing, Y.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (91860112), the National Key Research and Development Program of China (2017YFB0702004), the Materials Special Project (JPPT-KF2008-6-1) and the Open Project from State Key Laboratory of Structural Analysis of Industrial Equipment of DLUT (GZ18116).

**Acknowledgments:** W.X. would like to acknowledge the advice of Sun Chengqi from Institute of Mechanics, Chinese Academy of Sciences.

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

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28. Przybyla, C.P.; Musinski, W.D.; Castelluccio, G.M.; McDowell, D.L. Microstructure-sensitive HCF and VHCF simulations. *Int. J. Fatigue* **2013**, *57*, 9–27. [CrossRef]

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## *Article* **Hydrogen Assisted Fracture of 30MnB5 High Strength Steel: A Case Study**

#### **Garikoitz Artola 1,\* and Javier Aldazabal <sup>2</sup>**


Received: 6 November 2020; Accepted: 26 November 2020; Published: 30 November 2020

**Abstract:** When steel components fail in service due to the intervention of hydrogen assisted cracking, discussion of the root cause arises. The failure is frequently blamed on component design, working conditions, the manufacturing process, or the raw material. This work studies the influence of quench and tempering and hot-dip galvanizing on the hydrogen embrittlement behavior of a high strength steel. Slow strain rate tensile testing has been employed to assess this influence. Two sets of specimens have been tested, both in air and immersed in synthetic seawater, at three process steps: in the delivery condition of the raw material, after heat treatment and after heat treatment plus hot-dip galvanizing. One of the specimen sets has been tested without further manipulation and the other set has been tested after applying a hydrogen effusion treatment. The outcome, for this case study, is that fracture risk issues only arise due to hydrogen re-embrittlement in wet service.

**Keywords:** hydrogen re-embrittlement; environmentally assisted cracking; galvanic protection; high strength steel

#### **1. Introduction**

High strength steels offer multiple design and cost advantages. Their most publicized application is in automotive components, where their use is being promoted by progressively more restrictive CO2 emission control policies. Components manufactured in these steels favor a reduction in emissions and an improved gas mileage thanks to their lightweight design. Other sectors such as oil & gas are also prone to take advantage of high strength steels, such as in jack-ups and mooring chains for offshore platforms [1]. In this case, weight reduction is relevant to optimize the cost of keeping the structures floating and moored at their intended position.

The fastener market also benefits from high strength steels in terms of cost competitiveness [2]. Employing class 10.9 instead of class 8.8 bolts not only allows a reduction in the number of elements employed in a joint thanks to the 20% higher strength of the 10.9 class; it also means a bolt diameter reduction that is accompanied by a flange size reduction and a reduced installation time. Figure 1 shows a scheme in terms of cost.

Despite the fact that high strength steels are attractive cost-wise, they suffer some drawbacks, such as their increased risk of showing Hydrogen Embrittlement (HE) issues. HE causes a deterioration of mechanical properties which is often related to corrosion processes [3,4] and HE affected components fracture under applied stresses which are well below their design specifications. Bolts are a representative example of components which are concerned by this HE susceptibility, as recognized by the existence of fastener-specific standards to account for HE control in the final product [5].

**Figure 1.** Scheme of the cost advantages that the use of high strength steels implies in bolted joints [2].

When the hydrogen is incorporated into the steels, during the manufacturing process of the component or during service, is referred as internal or external hydrogen, respectively [6].

Internal hydrogen can be incorporated at different steps of the manufacturing route [7], starting upstream in the molten metal [8]. Hydrogen intake is minimized at this stage by applying vacuum degassing techniques on the melt. The importance of this degassing is such that it has become compulsory in some industries [9]. In the case of cast steel parts, the hydrogen left in the melt is expelled from the metal into micro-shrinkage and gas porosity voids during solidifications. The hydrogen trapped in this way recombines into gaseous H2 in the pores and it is difficult to dissociate it again into atomic hydrogen for removal. In the case of wrought products, hot forging and/or hot rolling aid in the closing the porosity due to solidification and help the removal of hydrogen excess from the steel by the combination of deformation and temperature. The higher the rolling reduction, the lower the hydrogen content in the final material.

After casting and/or forging processes, there are two known sources of internal hydrogen: the intake during heat treatment [10,11], especially when involving austenitizing processes, as hydrogen solubility increases with temperature and in the presence of austenite (Figure 2); and absorption from an electrolyte [12], like acidic media [13] from cleaning and pickling processes or from electrolytic coating processes as in zinc plating [14].

**Figure 2.** Solubility of H in iron at P-1 bar (made from [15]).

External hydrogen can be incorporated into the steel when the steel is working under high H2 partial pressures [16] or under acidic conditions, among which sour service with H2S stands out and has led to the existence of a very specific regulation [17]. A more common source of external hydrogen is the intake from an electrolyte both when impressed current cathodic protection systems and sacrificial coatings are used (e.g., zinc plating or hot dip galvanizing) [18,19]. These processes involving external hydrogen leading to what is known as Environmentally Assisted Cracking (EAC).

Figure 3 shows free corrosion and how galvanic protection processes lead to EAC, in the case of steels working in wet conditions. In free corrosion, iron dissolves in the water as Fe2<sup>+</sup> ion donating two free electrons. These electrons recombine involving H+, OH<sup>−</sup> and H2O. When there is an oxygen excess Fe2O3 forms on the steel surface. The reduction of H<sup>+</sup> to atomic H does not occur and EAC is avoided. For zinc coated steel in wet service with a discontinuity that exposes the steel substrate, the galvanic potential between the zinc anode and the steel cathodic zone is negative enough to allow hydrogen ion reduction, and drives a H<sup>+</sup> current to the steel surface, where atomic hydrogen can undergo the reaction of the H<sup>+</sup> ion with an electron to yield adsorbed molecular H on the steel surface. From there, the hydrogen enters the steel by diffusion.

**Figure 3.** Hydrogen involving reactions in naked (**left**) and zinc coated (**right**) steel in contact with water.

The hydrogen then distributes inside the steel in form of diffusible and trapped hydrogen, the difference being the ability of the hydrogen atoms to move or not across the microstructure [20–22]. The fraction of diffusible hydrogen from the total hydrogen is known to affect EAC, as diffusion allows H atoms to accumulate in the maximum stress triaxiality sites of the material. Thus, hydrogen trap control has become a key resource to develop EAC resistant steels as demonstrated by Fielding [23] and Yamasaki [24]. With a similar approach, as diffusible hydrogen excess tends to effuse from the material when temperature is raised at oven temperatures, oven degassing treatments are industrially used to assess if steel integrity has been compromised by hydrogen [9,25], and this approach is used in this work. More specifically, the Slow Strain Rate Tensile testing (SSRT) results of specimens that have been oven dehydrogenated, compared to non-dehydrogenated, have been used to determine whether internal or external hydrogen was the cause of an in-field failure of a set of galvanized bolts.

The bolt in Figure 4 shows the actual HE case that motivates this study. It is a hot dip galvanized M52 class 10.9 bolt that was installed in an outdoors structure in a coastal onshore location in Northern Spain. The structure consisted of 90 bolts from the same batch and the fractured bolt is part of a set of seven bolts that suffered delayed fracture after a week in service at the same site. All the bolts were tightened with a dynamometric wrench to the desired fastening torque. The torque was calculated to reach the design clamping force of the joint, employing the measured friction coefficient between the bolt and the nut greased threads. The bolt-nut batch had passed regular quality control checks and no deviation was found in the fractured products; they were free of forging defects, decarburization, liquid metal embrittlement or quench cracking.

**Figure 4.** M52 class 10.9 bolt that failed in service due to delayed fracture.

When the failure surface of the cracked bolts was inspected, a trans-granular brittle cleavage pattern was observed, which is shown in Figure 5a, while the expected fractographic texture for the bolt corresponds to a ductile failure pattern like that shown in Figure 5b.

**Figure 5.** (**a**) Brittle cleavage texture in the crack surface of the M50 class 10.9 bolt shown in Figure 1; (**b**) ductile collapse texturein the tensile specimens extracted from the shaft of the same bolt; (**c**)microstructure of the bolt in the center of the shaft diameter.

This type of delayed fracture has been reported in hot dip galvanized bolts with sections fully immersed in water [26], EAC being the root cause of failure. The failed bolts of concern in this case study, though, were not immersed in water. Thus, this work is aimed at answering the following question: was the delayed fracture due to HE from internal hydrogen introduced by the manufacturing process or was it due to EAC from external hydrogen taken up in a scenario of water drop condensation on coating discontinuities at the bolt surface?

With this question in mind, a SSRT based study was performed with the same material supply of fractured bolts. Tests on dehydrogenated and non-dehydrogenated samples were performed at different stages of the industrial production process (delivery condition, heat treatment and galvanizing) and, for the galvanized specimens were tested, both immersed in seawater and in dry conditions.

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

The material employed for the study is steel grade 30MnB5 according to UNE-EN 10083-3 [27] in bright bar format according to UNE-EN 10263-4 [28]. The material was received with dimensions of φ48 mm × 450 mm length (Figure 6) and in the usual condition of raw material for the manufacture of bolts. This material was sourced from the same mill as the fractured M52 bolts that motivated this study.

**Figure 6.** 30MnB5 bars employed for the study.

The chemical composition of the material was checked during incoming inspection. Table 1 shows the result of the analyses performed by spark spectrometry, with an expanded measurement uncertainty (K = 2) of ±0.03% for all the elements of the table. It is noted that the material meets the specifications set for alloyed quench and tempering steels under the designation 30MnB5, except for a slight excess in Mn content. The observed addition of Cr is intended to improve hardenability and is in accordance with the limits imposed by the current regulations [27], which allow the 30MnB5 nomenclature to be maintained.


**Table 1.** Chemical composition check (by weight%) of 30MnB5 bars.

\* Results of the chemical analysis performed on the bars by spark emission spectroscopy. \*\* Additions up to 2% Cr permissible by standard [27] for improved hardenability.

The material also meets the chemical composition requirements established for Class 8.8 to 10.9 [29] fasteners. It should be remembered that the bolt classes are determined mainly by the resistance levels as shown in Table 2.


**Table 2.** Nominal values of mechanical properties for high-strength bolt classes [29].

For the study of internal hydrogen uptake during the production process, 27 specimens were prepared and divided into three sets of specimens, one for each stage of production (Figure 7):


The quench and tempering sequence that was applied is reported below, and it reproduces that of the fractured M52 bolts that motivated this work (same furnaces and processing conditions):


Non-acidic surface conditioning (cleaning, etching and rinsing) and hot dip galvanizing was performed in an industrial production facility.

**Figure 7.** Materials for the study of internal hydrogenation in the manufacturing process for class 10.9 bolts.

Three specimens from each set were submitted to SSRT in air at room temperature without applying a dehydrogenating treatment, while the other three were dehydrogenated in an oven at 250 ◦C for 2 h before being tested under the same conditions. This baking process at 250 ◦C is considered sufficient to express a drop in the diffusible hydrogen from the material affected by delayed fracture [9], if any. A loss of ductility of non-conditioned samples relative to dehydrogenated samples would indicate the presence of diffusible hydrogen, incorporated into the material during the production process.

For the heat treated and galvanized specimens, three further specimens were dehydrogenated to perform SSRT immersed in seawater, to assess the severity of the EAC produced by the concurrence of the zinc coating and water. All tests were carried out at room temperature (23 ◦C). Table 3 summarizes the tests that were performed, indicating the identification code that will be used to refer to each material and testing condition in the following.


**Table 3.** Summary of the tensile testing battery.

As is shown in Table 3, one specimen per set was tested under conventional tensile testing conditions [30] to verify that the material performance was as expected and to accept the material condition for further testing. The results of these tests are presented in Table 4. It is noted that the bars in delivery condition could be used directly to manufacture Class 8.8 bolts. After heat treatment, the specimens met the mechanical requirements for class 10.9 bolts as expected. It is important to stress that the heat treatments were carried out in all cases on specimens already machined and not on bars, so that the detection of surface effects caused by the process was not hidden by further machining and it was verified that the control of the protective atmosphere of the furnace had been correct (absence of carburized or decarburized layer).

**Table 4.** Verification of the mechanical properties of 30MnB5 samples. The tolerances correspond to the expanded measurement uncertainty of each result (K = 2).


The media used in submerged specimen tests was a seawater substitute according to ASTM D1141 [31]. All electrolytes used for the experimental part of this work were prepared in accordance with this standard in the heavy metal version and with the composition indicated in Table 5. The distilled water used as the base of the mixture met in all preparations the level of purity required by ASTM D1193 [32], Type II. The pH of the electrolyte was adjusted in all cases to 8.2 ± 0.1 by additions of NaOH 0.1N after the compound mixing process was complete.


**Table 5.** Chemical composition of synthetic seawater employed in the tests.

The SSRT method employed for the assessment of process hydrogenations and zinc coating induced EAC is explained in ASTM G129-00 [33]. This method consists of a test analogous to that of uniaxial tensile testing, but at a very low strain rate. Using a reduced deformation rate (e.g., 10−<sup>5</sup> s<sup>−</sup>1) is a key element in the case of studying external embrittlement processes as in this work, as it allows hydrogen distribution to evolve in the microstructure. In conventional tensile test speeds, are at least an order of magnitude faster [30] than in slow strain rate tests, and hydrogen absorption kinetics and diffusion kinetics are not able to modify material behavior and therefore do not allow hydrogen embrittlement to be assessed. It is important to remark that SSRT results are comparative in nature, i.e., they are used to determine whether one material behaves better than another in the same embrittling conditions, or whether the same material has different susceptibility to HE in two different hydrogenation environments. For this reason, the SSRT results are usually provided as a ratio between the values obtained for the condition of interest and a control condition against which it is compared.

The SSRT were performed on a Zwick Roell Model 1475 universal machine (Zwick Roell, Ulm, Germany) with a maximum load capacity of 100 kN. Circular cross-sectional specimens of φ10 mm were used according to UNE EN-ISO 6892-1:2017 (Figure 8).

**Figure 8.** Manufacturing sketches of the specimens used in tests at low deformation speed. Dimensions in mm.

The 30MnB5 steel specimens were obtained from bars of φ48 mm as shown in Figure 9. The heat treatments and galvanizing were applied directly to the specimens. In this way, the skin of the tested material reproduces the working condition of interest, like that of the commercial bolts whose failure motivated this work.

The tests were carried out in all cases with a crosshair feed of 0.03 mm/min, which corresponds to a strain rate of 10−<sup>5</sup> s<sup>−</sup>1.

**Figure 9.** Specimen extraction sketch of bolt steel specimens. Dimensions in mm.

#### **3. Results and Discussion**

The average results of the SSRT tests targeted to assessing the HE at different stages of the manufacturing process are shown in Table 6. These tests were performed in air. It should be noted that these averages are calculated on three specimens and, to avoid generating a sense of false accuracy, experimental deviations corresponding to the 95% confidence interval on the average have been included using the expression (1).

$$IC95 = \pm \sqrt{(\mathcal{U}(K=2))^2 + \left(\frac{t\_{\mathcal{K}=0.05,2} \cdot S\_{\mathcal{V}}}{\sqrt{3}}\right)^2} \tag{1}$$

where *U* (*K* = 2) is the expanded uncertainty of each individual measure, *t*<sup>∝</sup> = 0.05, 2 is the value of Student's t for a significance value of 0.05% in a distribution of two tails and two degrees of freedom and *Sv* is the standard deviation.


**Table 6.** Average values of mechanical properties obtained in SSRT at different stages of the production process of class 10.9 bolts with 30MnB5 steel.

To evaluate if a step in the process had caused any significant change in HE, the results in Table 6 were analyzed by hypothesis tests which were performed for the difference in the averages of normal distributions. The hypothesis tests concerned all the four SSRT outputs considered in this work, Rp0.2, Rm, E and RA with a significance level of α = 0.01. The conditions that were chosen for the analyses are the following:


• QT-ND versus GA-ND and QT-DH versus GA-DH: to assess if galvanizing modified the mechanical behavior from the heat-treated condition.

There is no statistically significant difference between the average values of the experimental data in any of the comparisons stated above. Thus, no effect of the production process can be associated with HE.

Figure 10 has been elaborated to help in visualizing this fact. It plots the average values in Table 6 with their standard deviation as the parameters of Normal probability distribution functions. Starting with the values of strength, the results have been divided first by class and then by condition. Figure 10a shows, that for the raw material in delivery condition, Rm distributions overlap by a high percentage, while Rp0.2 coincide in about 50% of their dispersion. Accounting for the reduced number of repeats, this 50% overlapping has been enough to discard the existence of statistically significant differences between the averages of Rp0.2 for DC-ND and DC-DH.

The plots in Figure 10b represent the averages for QT and GA specimen sets. These curves are similar to those of DC specimens in the sense that the curves overlap at a high percentage. This fact removes the possibility of affirming that the difference observed between the means is significant, even for the most disperse values such as Rm on GA-DH when working with three repeats.

**Figure 10.** Probability density plot assuming Normal distribution of average strengths and their uncertainties in Table 6 as mean and standard deviation values. (**a**) Plot for the DC Class 8.8 specimens. (**b**) Plot for the QT and GA Class 10.9 specimens.

The same plotting approach for E and RA is shown in Figure 11. For these two properties, which are highly affected by HE and EAC, the curve overlapping is even higher than that for the strengths. This confirms that no major mechanical property decay due to production process-related internal hydrogen HE was produced in the original M52 fractured bolts that motivated this work. Major decays would have been detected even with three repeats.

Regarding the SSRT tests in seawater, targeted to assess the weight of EAC in the failure of the fractured M52 bolts, Table 7 gathers the obtained average results. In this case, the hypothesis tests were again performed in terms of the existence of a difference in the average of the results. The following results were compared:


**Figure 11.** Probability density plot assuming Normal distribution of average elongations and area reduction and their uncertainties in Table 6 as mean and standard deviation values. (**a**) Plot for the DC Class 8.8 specimens. (**b**) Plot for the QT and GA Class 10.9 specimens.

**Table 7.** Average values of the mechanical properties obtained in water submerged SSRT with 30MnB5 steel treated to class 10.9 bolt strength, both for naked and hot-dip galvanized specimens.


The outcome of the statistical analysis indicates that the effect of the immersion is not significant in terms of strength for either of the conditions; neither naked (QT), nor galvanized (GA). Figure 12 reflects this fact, as the gaussian curves overlap clearly for QT (Figure 12a) and slightly less so for GA (Figure 12b) specimens. The most doubtful result is observed for Rm in GA-DH and GA-SW, but again about 50% of the curve overlaps in the worst case and three repeats do not allow the conclusion that this is a significant variation.

**Figure 12.** Probability density plot assuming Normal distribution of average strengths and their uncertainties in Table 7 as mean and standard deviation values. (**a**) Plot for the QT specimens. (**b**) Plot for GA specimens.

When the hypothesis tests are performed on the difference of averages of the ductility related properties, the scenario changes, not for the elongation, E, but for the reduction in area, RA. The outcome of the statistical analysis is that the drop in the average RA is significant for the galvanized specimens tested in submerged condition, with a level of significance of α = 0.01. Figure 13 plots this: the RA curves of the graph in Figure 13a show that the Gaussian for GA-DH and the Gaussian for GA-SW do not touch each other. Consequently, it is statistically sound to affirm that an embrittlement process has

been developed in the GA-SW SSRT specimens and their differences in both E and RA can be considered as an actual effect of EAC and not a statistical artifact. This combination of hot dip galvanizing Zn coatings and water as the critical factor for the failure of bolts in the field has been reported by other case studies [26,34].

**Figure 13.** Probability density plot assuming a Normal distribution of average strengths and their uncertainties in Table 7 as mean and standard deviation values. (**a**)Plot for the QT specimens. (**b**) Plot for GA specimens.

Once the presence of EAC is confirmed, it is possible to assess the embrittlement susceptibility of the GA-SW condition, employing the embrittlement ratios as indicated in the standard testing method ASTM G129. The embrittlement ratios measured in this case for the average E and RA values in GA-SW condition using GA-DH as control condition are 85% and 72% respectively (Table 8).

**Table 8.** Embrittlement susceptibility ratios for the GA-SW condition.


The EAC was also confirmed by visual inspection of the fractured SSRT specimens (Figure 14). Only GA-SW specimens showed a very severe cracking pattern all along the necking section. The cracks caused by EAC appear remarked in white, due to material deposition from the electrolyte in the areas of the substrate exposed by the appearance of the cracks themselves. Similar EAC cracking patterns had been observed in previous work [18] for steels with the same strength level as the GA-DH specimens, when a cathodic protection potential was applied to seawater immersed SSRT.

**Figure 14.** Pictures of the necking areas of representative SSRT specimens showing no EAC except for GA-SW.

#### **4. Conclusions**

Tensile testing specimens were manufactured following exactly the same raw material grade and source, and the same industrial manufacturing route (same suppliers and process parameters), as a set of class 10.9 hot-dip galvanized M52 bolts that had suffered a delayed fracture. The intention was to elucidate whether the fracture was HE affected by internal hydrogen uptake of the bolts during their manufacture, or was an EAC issue affected by external hydrogen uptake of the bolts due to the water condensation on a discontinuity of the zinc coating. The results have made clear that the industrial production process does not promote the presence of internal HE in the studied steps: the supply condition of the raw material, the heat treatment, and the hot-dip galvanizing. The specimens processed according to the production route did not require subsequent dehydrogenation processes. Though this does not ensure that there cannot be a mistake in the supply chain, it at least shows the right capability and points to external hydrogen as the principal process involved in the failure of the M52 bolts.

Regarding EAC, it has been observed that mere immersion in seawater does not produce any embrittlement effect on the naked material, while the presence of a galvanic coating in this medium causes a significant susceptibility to EAC. In this particular case study of bolt delayed fracture, the evidence points to a combination of poor zinc coating condition (no matter whether from the galvanizing or due to scratching during poor handling) and the presence of condensation from seawater droplets or aerosol in the installation site (coastal Northern Spain) as the origin of the problem. As the EAC did not affect the whole installed bolt set, it is likely that some of the bolts were unscratched, as the whole batch was installed under the same ambient humidity. Though hydrogen induced fracture-preventing practices are usual in bolt industry, small details such as poor handling can still lead to premature failures. Even in the absence of coating defects, a sufficient coating thickness must be guaranteed, as thin zinc layers may eventually dissolve and expose the EAC susceptible steel substrate.

**Author Contributions:** Conceptualization, J.A. and G.A.; methodology, J.A. and G.A.; validation, G.A. and J.A.; formal analysis, G.A. and J.A.; writing—original draft preparation, G.A.; writing—review and editing, G.A. and J.A.; supervision, J.A. Both authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

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

#### **References**


34. Álvarez, J.A.; Lacalle, R.; Arroyo, B.; Cicero, S.; Gutiérrez-Solana, F. Failure analysis of high strength galvanized bolts used in steel towers. *Metals* **2016**, *6*, 163. [CrossRef]

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## *Article* **The Evaluation of Front Shapes of Through-the-Thickness Fatigue Cracks**

**Behnam Zakavi 1,\*, Andrei Kotousov <sup>1</sup> and Ricardo Branco <sup>2</sup>**


**Abstract:** Fatigue failure of structural components due to cyclic loading is a major concern for engineers. Although metal fatigue is a relatively old subject, current methods for the evaluation of fatigue crack growth and fatigue lifetime have several limitations. In general, these methods largely disregard the actual shape of the crack front by introducing various simplifications, namely shape constraints. Therefore, more research is required to develop new approaches to correctly understand the underlying mechanisms associated with the fatigue crack growth. This paper presents new tools to evaluate the crack front shape of through-the-thickness cracks propagating in plates under quasi-steady-state conditions. A numerical approach incorporating simplified phenomenological models of plasticity-induced crack closure was developed and validated against experimental results. The predicted crack front shapes and crack closure values were, in general, in agreement with those found in the experimental observations.

**Keywords:** crack front shape; structural plates; through-the-thickness crack; steady-state loading conditions; small-scale yielding

#### Branco, R. The Evaluation of Front Shapes of Through-the-Thickness Fatigue Cracks. *Metals* **2021**, *11*, 403. https://doi.org/10.3390/met11030403

Academic Editor: Dariusz Rozumek

**Citation:** Zakavi, B.; Kotousov, A.;

Received: 1 February 2021 Accepted: 21 February 2021 Published: 1 March 2021

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#### **1. Introduction**

The evaluation of fatigue life and failure conditions of structural components is of permanent and primary interest for engineers. Over the past five decades, significant progress has been made toward the development of more appropriate fatigue crack growth models and life assessment procedures. Significant research effort has been directed to the study of the fatigue crack closure phenomenon, which was first introduced by Elber [1] to explain the experimentally observed features of fatigue crack growth in aluminium alloys. The number of publications grew rapidly since this pioneering study, and continues to grow. It is now commonly accepted that the contributions of various crack closure mechanisms, specifically plasticity-induced crack closure, roughness-induced crack closure, and oxideinduced closure, are significant, and these mechanisms are capable of explaining many fatigue crack growth phenomena, e.g., the influence of thickness on crack growth rates, retardation effects associated with overloads, or higher propagation rates of small cracks in comparison with long cracks [2].

It is well-established that for relatively long cracks propagating in a non-aggressive environment, the plasticity-induced crack closure dominates over the roughness-induced crack and oxide-induced closures. The plasticity-induced crack closure models rely on far fewer assumptions than the two other closure mechanisms. The first theoretical model was developed by Budianski and Hutchinson [3] based on the two-dimensional Dugdale stripyield model [4]. The theoretical results demonstrated that opening stress intensity factor is surprisingly high, and increases with an increase in the *R* ratio. All early crack closure models for plate components utilised both plane strain and plane stress simplifications, although real cracks are inherently three-dimensional (3D). To examine the thickness effect

on crack propagation rates, empirical constraint factors were often used, demonstrating a stronger correlation with experimental results. With the advance of numerical methods and the increase in computational power, it became possible to eliminate these simplifications and study more realistic geometries, as well as various 3D effects [5,6].

In 3D problems, the order of the singularity at the intersection of the crack front with the free surface depends on Poisson's ratio and the intersection angle. From energy considerations, it follows that fatigue cracks have to preserve the 1/ <sup>√</sup>*<sup>r</sup>* singularity. Therefore, the fatigue crack has to intersect the free surface at a critical angle, βcr, which is a function of Poisson's ratio. Several experimental studies have reported that, at least, the Mode I fatigue crack front is shaped to ensure the square root singular behaviour along the entire crack front. However, it seems that the effect of 3D corner singularity is not very significant in the presence of a sufficiently large crack front process zone [7]. This is because the 3D corner singularity effect is a point effect, and is very much localised. Therefore, it might be negated by the plasticity and damage formation near the surface. For example, in an experimental study of steel circular bars subjected to bending and torsion, the experimental intersection angles were found to be very different from the theoretically predicted critical angles [8].

Considerably less effort has been directed toward the study of the effects of the 3D corner singularity and elasto-plastic constraints on plasticity-induced crack closure. Generally, the direct 3D elasto-plastic simulations of fatigue crack growth demand much greater computational resources [9]. These simulations have many issues associated with the validation of the numerical solution and the accuracy of the obtained results. A number of factors affect the accuracy, which are difficult to control: the mesh refinement, the type of finite element, the crack advance scheme (which usually consists of releasing nodes ahead of the crack front), contact conditions, and the local criterion of crack front opening. Branco et al. [10] recently provided an exhaustive review concerning these aspects. The overall conclusion was that the direct numerical approaches are capable of describing the shape evaluation of fatigue cracks. However, the application of these approaches to particular problems can be quite cumbersome. Each problem needs a large effort to calibrate the solution and verify the results. These efforts are usually focused on the reduction in the number of finite elements, the number of simulations required in the analysis, or, eventually, the computation time, which cannot be considered to be of practical relevance [11].

In the present paper, a simplified procedure for the evaluation of the fatigue crack front shapes of through-the-thickness cracks propagating under the cyclic loading conditions is presented. The procedure is based on simplified methods for the evaluation of the plasticity-induced crack closure effect, namely the equivalent-thickness method introduced by Yu and Guo [12,13], as well as the analytical model developed by Kotousov et al. [14,15]. The outcomes of the simulation are compared with available experimental results obtained at the same propagation conditions for validation purposes. The paper is organised as follows: Section 2 addresses the method used to evaluate the crack front shape, as well as the models introduced to evaluate the crack closure along the crack front. Section 3 describes the finite element model developed to calculate the stress intensity factors along the crack front. Section 4 compares the predicted crack front shapes with those obtained experimentally for different materials and propagation conditions. The paper ends with some concluding remarks.

#### **2. Crack Shape Simulation and Crack Closure Models**

The main idea behind the evaluation of the steady-state shape of a fatigue crack front proposed in this paper is to select a curve from a parametric family that minimises the deviation of the fatigue driving force along the crack front. In other words, we first specified a possible parametric set of curves in the crack plane (e.g., parabolic, hyperbolic, or elliptical shapes) and then evaluated the local fatigue driving force using a finite element model, along with simplified plasticity-induced crack closure models. In this study, the

local fatigue driving force was defined by the effective stress intensity factor range, Δ*K*eff, given by the formula:

$$
\Delta K\_{\rm eff} = \mathcal{U} \cdot \Delta K = \mathcal{U} \cdot (K\_{\rm max} - K\_{\rm min}) \tag{1}
$$

where *U* is the normalised load ratio parameter, or normalised effective stress intensity factor, which is often used to describe the effects of loading and geometry on crack closure, and Δ*K* is the traditional linear-elastic stress intensity factor range [16] defined by the maximum and minimum values of the stress intensity factor experienced for a given load cycle. The load ratio is therefore given by *R* = *K*min/*K*max. In the case of 3D problems, this normalised load ratio is not a constant, but rather a function of the position along the crack front, *U* = *U*(*z*). Thus, the local crack growth rate is a function of the effective stress intensity factor range, i.e.,

$$
\Delta K\_{\rm eff}(z) = K\_{\rm max}(z) - K\_{\rm op}(z) = lI(z)\Delta K(z) \tag{2}
$$

where *K*op(*z*) is the local opening load stress intensity factor, which corresponds to the minimum load at which the crack faces, at point *z*, which are fully separated.

A number of sophisticated finite-element (FE) models were developed to evaluate *U*(*z*) for different geometries and loading conditions. However, as discussed above, these models have many limitations, and are quite difficult to apply in fatigue calculations. Below, we consider two simplified methods for the evaluation of the normalised load ratio, the equivalent-thickness model introduced by Yu and Guo [13], and the analytical model proposed by Kotousov et al. [14,15], which are addressed in Sections 2.1 and 2.2, respectively. These methods will be further incorporated into the 3D linear elastic finite element simulations to evaluate the shape of the through-the-thickness cracks. This evaluation will be performed via the corner singularity method [17], which is briefly presented in Section 2.3.

#### *2.1. Equivalent-Thickness Model*

For through-the-thickness cracked plates, She et al. [17] proposed defining the equivalent thickness based on a numerical analysis of the 3D distribution of the out-of-plane stresses and constraint factor, *T*z, which is defined as:

$$T\_{\mathbf{z}} = \frac{\sigma\_z}{\sigma\_x + \sigma\_y} \tag{3}$$

where σ*x*, σ*y*, and σ*<sup>z</sup>* are the normal stresses. This method is illustrated in Figure 1. The equivalent thickness, 2*h*eq, for point P on the crack front is identified as the plate thickness, which leads to the same distribution of *T*z at the mid-plane.

**Figure 1.** Schematic illustration of the equivalent-thickness method in the through-the-thickness cracks: (**a**) original straight through-the-thickness cracked geometry; (**b**) final straight-through-the-thickness cracked geometry with equivalent thickness.

An empirical equation was suggested to evaluate the equivalent-thickness as follows:

$$\frac{h\_{\rm eq}}{h} = 1 - \left(\frac{z}{h}\right)^2\tag{4}$$

where *z* is the distance from the mid-plane and *h* is the half-thickness of the plate. The normalised load ratio parameter in this method can be calculated as follows:

$$
\hat{\omega}L = \frac{\sqrt[3]{\kappa}}{1 - R} \tag{5}
$$

where κ is a function of the *R* ratio:

$$\kappa = \frac{\left(1 - R^2\right)^2 \left(1 + 10.34R^2\right)}{\left(1 + 1.67R^{1.61} + \frac{1}{0.15\pi^2\alpha\_\theta}\right)^{4.6}}\tag{6}$$

and α<sup>g</sup> is a global constraint factor, αg, defined by the formula:

$$\alpha\_{\mathbb{R}} = \frac{1+t}{1-2\mathbf{v}+t} \tag{7}$$

where ν is the Poisson's ratio and *t* is given by:

$$t = 0.2088 \sqrt{\frac{r\_0}{h\_{\rm eq}}} + 1.5046 \frac{r\_0}{h\_{\rm eq}} \tag{8}$$

with:

$$r\_0 = \frac{\pi}{16} \left(\frac{K\_{\text{max}}}{\sigma\_0}\right)^2 \tag{9}$$

where σ<sup>0</sup> is the flow stress. These empirical equations were extended to the corner, and surface cracks and were extensively validated using 3D finite element analyses.

#### *2.2. Analytical Model for the Evaluation of Crack Closure*

Another method for the evaluation of local plasticity-induced closure is based on a simplified 3D analytical model. In accordance with this model, the parameter *U* for Mode I loading under small-scale yielding conditions can be approximated from the following expression:

$$\mathcal{U}(\mathbb{R}, \mathfrak{v}) = a(\mathfrak{v}) + b(\mathfrak{v})\mathbb{R} + c(\mathfrak{v})\mathbb{R}^2 \tag{10}$$

where the fitting functions *a*, *b* and *c* can be written in the form:

$$\begin{array}{l} a(\eta) = 0.446 + 0.266 \cdot \mathbf{e}^{-0.41\eta} \\ b(\eta) = 0.373 + 0.354 \cdot \mathbf{e}^{-0.235\eta} \\ c(\eta) = 0.2 - 0.667 \cdot \mathbf{e}^{-0.515\eta} \end{array} \tag{11}$$

where η = *K*max/(*h* <sup>√</sup>σf) is a dimensionless parameter.

The above equations were obtained within the first-order plate theory based on the Budiansky–Hutchinson crack closure model [3,15]. The results, which correspond to the classical two-dimensional theories (or plane stress state, or plane strain state), can be obtained as limiting cases of very thin and very thick plates, i.e., when η → ∞ or η → 0, respectively. The details of the derivation of these equations can be found in the original paper by Codrington and Kotousov [14].

#### *2.3. Corner Singularity Method*

In this study, the evaluation of the steady-state front in the through-the-thickness cracks was carried out using the corner singularity method. First, we approximated the shape of the crack front by a two-parameter elliptical curve, which can be described as:

$$\mathbf{x} = b\sqrt{1 - \frac{z^2}{a^2}} \ -h \le z \le h \tag{12}$$

where *a* and *b* are the major and minor semi-axes of an ellipse, respectively, as shown in Figure 2.

**Figure 2.** Elliptical-arc crack front shape for geometrical parameters crack propagation.

The crack front tends to intersect the free plate surface at the critical angle, βc, when the plasticity effects are small. The critical angle is a function of the Poisson's ratio and the type of loading. We found that the critical intersection angle can be approximated by the following formula [18]:

$$
\tan \beta\_{\mathbf{c}} = \frac{\mathbf{v} - \mathbf{2}}{\mathbf{v}} \tag{13}
$$

where ν is the Poisson's ratio. Typically, when the size of the plastic zone is greater than 1% of the plate thickness, the stress state near the vertex location is not controlled by the elastic singularity. In these cases, the plasticity effects become more important, and together with the vertex singularity effect, lead to greater critical angles for elastic-plastic materials. To find *b*, we need to make sure that:

$$\frac{\partial \mathbf{x}}{\partial z}|\_{z=\pm h} = -\frac{bh}{a\sqrt{a^2 - h^2}} = \frac{\mathbf{v}}{\mathbf{v} - 2} \tag{14}$$

where *b* is defined by:

$$b = \frac{a\mathbf{v}}{(2-\mathbf{v})} \sqrt{\frac{a^2}{h^2} - 1} \tag{15}$$

Substituting Equation (15) into Equation (12), we obtain:

$$\mathbf{x}(z) = \frac{a\mathbf{v}}{(2-\mathbf{v})} \sqrt{\frac{a^2}{h^2} - 1} \cdot \sqrt{a^2 - z^2} \quad -h \le z \le h \tag{16}$$

This equation meets the condition that the crack front intersects with the free surface at the critical angle given by Equation (13), and represents a parametric curve with one single parameter, *a*. Further, the steady-state condition of the crack propagation requires that the projection of the effective stress intensity factor along the crack propagation direction (*x*-direction, Figure 1) is constant for all points along the crack front. This condition cannot be satisfied exactly with any multi-parametric equation describing the possible crack front shapes. However, the shape that minimises the difference of the effective stress intensity factor along the crack front can be considered as the best approximation of the actual fatigue crack front shape.

#### **3. Numerical Approach**

This section describes the numerical model developed in this research to determine the stress intensity factor ranges along the crack front. The stress intensity factor ranges, along with the crack closure models described in the previous section, enabled the computation of the local fatigue driving force, which was used to obtain a steady-state crack front shape. The steady-state crack front shape was selected as the one producing the minimum deviation of Δ*K*eff along the crack front. This evaluation needs to be completed for each curve from the parametric set.

To reduce the computational overhead, we developed a simplified geometry by introducing adequate boundary conditions, capable of describing 3D effects near the crack front. Section 3.1 describes the details of the numerical modelling, and Section 3.2 addresses the boundary conditions considered in this paper. The last section, Section 3.3, is devoted to the validation of the stress intensity factor values obtained with the proposed approach.

#### *3.1. Finite Element Model Description*

The typical finite element geometry, developed here to study a through-the-thickness crack in an elastic plate, is shown in Figure 3. As can be seen, the rectangular cross-section geometries were modelled to evaluate the stress and displacement fields near the crack tip. The size of the finite element models is sufficient to avoid the effect of the finite boundaries on the stress state. By taking advantage of the symmetry conditions (i.e., XY symmetry, XZ symmetry, and YZ symmetry), only one-eighth of the crack problem was modelled. The height of the FE models taken was approximately ten times larger than the plate thicknesses. In accordance with the previous studies, this is sufficient to accurately describe the 3D effects near the crack front [19,20].

**Figure 3.** Finite element mesh: (**a**) assembled model; (**b**) detail of the crack front; (**c**) detail of the spider web pattern.

The FE models corresponding to different values of *a* (Figure 1) were meshed with linear 8-node hexahedral elements of type C3D8R. A reasonably uniform element grid with a structured mesh was considered. A denser mesh, with a spider-web pattern (Figure 3c) was used near the crack front, where the stress gradients were expected to be maximum (Figure 3b), consisting of 5 concentric rings centred at the crack tip with a radial discretisation of 10◦ (Figure 3c). Thirty nodes along the plate half-thickness (Figure 3b) were used to define the crack front shape. The specimen was subjected to uniaxial loading applied at the bottom surface (i.e., at the XZ-plane with a Y-coordinate equal to *H*/2). The assembled mode is exhibited in Figure 3a. Further details about the modelling approach can be found in papers published by the present authors [19,21].

The numerical simulations were carried out using Abaqus/CAE 2020 (© Dassault Systèmes, 2019), assuming a homogeneous, isotropic, and linear-elastic behaviour. The mechanical properties inserted into Abaqus/CAE 2020 to perform the numerical simulations were the Young's modulus and the Poisson's ratio of the tested materials (Table 1). The displacement field far from the crack tip was calculated in accordance with the William's solution using MATLAB R2020b, and the obtained results were applied for the boundary conditions. The 3D solutions of the J-integral were used to calculate the stress intensity factor near the crack front. One layer of elements surrounding the crack front was used to calculate the first contour integral. The additional layer of elements was used to compute the subsequent contours. The different contour solutions were approximately coincident after eight contours. The results from averaging contours five through eight was considered. A similar strategy, either in terms of mesh framework or simulation analysis, was carried out for all geometries and crack configurations studied in the present paper.

**Table 1.** Mechanical properties of the selected materials.


#### *3.2. Boundary Conditions*

The plane-stress displacements far from the crack tip were calculated in accordance with William's solution [22]:

$$u\_{\mathbf{x}}(r,\theta) = \left(\frac{r}{2\pi}\right)^{1/2} \frac{(1+\theta)}{E} [\mathbb{K}\_{\mathbf{I}}^{\infty} f\_{\mathbf{x}}^{\mathbf{I}}(\theta)] \tag{17}$$

$$
\mu\_{\rm Y}(r,\theta) = \left(\frac{r}{2\pi}\right)^{1/2} \frac{(1+\nu)}{E} [\mathcal{K}\_{\rm I}^{\infty} f\_{\rm Y}^{\rm I}(\theta)] \tag{18}
$$

Being:

$$f\_{\mathbf{x}}^{1}(\theta) = \cos \frac{\theta}{2} \left( k - 1 + 2 \sin^{2} \frac{\theta}{2} \right) \tag{19}$$

$$f\_{\mathcal{Y}}^{1}(\theta) = \sin\frac{\theta}{2} \left( k + 1 + 2\cos^{2}\frac{\theta}{2} \right) \tag{20}$$

where *r* is the distance from the crack tip, θ is the angle measured from the symmetry line, *K*<sup>∞</sup> <sup>I</sup> is the remotely applied Mode I stress intensity factor, and *k* is Kolosov's constant for plane stress and plane strain conditions. The plane stress *k* value was considered in the boundary conditions, i.e.,

$$k = \frac{3 - \mathbf{v}}{1 + \mathbf{v}}\tag{21}$$

where ν is the Poisson's ratio. Bakker [23] showed that a cracked plate under plane stress undergoes a change to plane strain behaviour near the crack tip. He proved that the radial position, where the plane stress to plane strain transition takes place, strongly depends on the position in the thickness direction. The degree of plane strain is essentially zero at distances from the tip greater than five times the thickness, even in the middle plane of the plate [24].

#### *3.3. Validation Study*

The numerical results obtained for the maximum stress intensity factor are presented in Figure 4 as a function of the thickness for a Poisson's ratio of 0.3. The classical results for both the plane stress state and the plane strain state are also given in Figure 4. It is evident from Figure 4 that the stress intensity factor changes with the thickness of the plate until the thickness exceeds a critical value. In this particular problem, the results showed that the critical thickness is 25 mm. Once the thickness exceeds the critical dimension, the stress field in the vertex singularity region has a negligible impact on the behavior of the whole structure. The stress intensity factor becomes relatively constant in the sufficiently thick plate, and is equal to the value for plane strain conditions.

**Figure 4.** The effect of the thickness on the maximum stress intensity factor.

#### **4. Crack Front Shape Evaluation and Comparison with Experimental Studies**

The proposed method for the evaluation of the steady-state crack front shapes was compared against two independent experimental studies. The specimen geometries used in the experimental tests are exhibited in Figure 5, and were made of 6082-T6 aluminium alloy and polymethyl methacrylate (PMMA), separately. The main mechanical properties of both materials are listed in Table 1. The former (Figure 5a) consisted of a standard middle-crack tension specimen with a thickness of 3 mm [11,25]. The tests were conducted under constant-amplitude axial loading using a stress ratio equal to 0.25. Figure 6a shows an example of the typical fracture surfaces obtained in the tests. Fatigue cracks grew over a sufficiently large distance from the initial notch to ensure the quasi-steady-state conditions of propagation. The beach-marking technique was applied to mark the crack front at the fracture surface.

**Figure 5.** Specimen geometries used in the crack front shape evaluation: (**a**) 6082-T6 aluminium alloy and (**b**) polymethyl methacrylate (PMMA). All dimensions are in mm.

**Figure 6.** The crack front shapes observed in the experiments for the: (**a**) 6082-T6 aluminium alloy reprinted with permission from ref. [11], copyright 2021 Elsevier and (**b**) polymethyl methacrylate reprinted with permission from ref. [26], copyright 2021 Elsevier. Propagation direction is from left to right in case (**a**) and from bottom to top in case (**b**). All dimensions are in millimetres.

Regarding the latter (Figure 5b), the specimen geometry was made of polymethyl methacrylate. It had a rectangular cross-section (Figure 5b), with a thickness of 40 mm [26,27], and an initial straight notch at the middle of the specimen. The tests were conducted under four-point bending loading conditions using a stress ratio equal to 0. The crack front shape was evaluated in situ using a high-resolution digital camera. As in the previous case, fatigue cracks propagated over a sufficiently large distance from the initial notch to ensure the quasi-steady state conditions of propagation. An example of the crack front shapes observed in the experiments is exhibited in Figure 6b.

Figure 7a,b displays a comparison of the experimental crack front shapes and those obtained with the proposed methods for the 6082-T6 aluminium alloy and PMMA, respectively. Overall, the results showed that the equivalent-thickness method provides a satisfactory approximation for the fatigue crack propagation under small yielding conditions. Moreover, the experimental results confirmed that the angle at which the crack front intersects the free surface is greater than the proposed empirical equations in the sufficiently plastic materials. We think that the careful combination of the hyperbolic and elliptical functions might provide accurate crack front shape estimation in the presence of residual stresses or large crack closure effects. The good agreement demonstrated in the previous analysis confirmed the possibility of the accurate evaluation of stress intensity factors using the proposed approach in materials controlled by 3D corner singularity effects.

**Figure 7.** A comparison between the predicated crack shapes and experimental data for the specimens composed of: (**a**) 6082-T6 aluminium alloy and (**b**) polymethyl methacrylate.

This methodology can also be applied to conduct parametric studies associated with the main variables affecting the fatigue crack growth of through-the-thickness cracks. A subject that can be analysed with the developed approach is the effect of the stress ratio on crack closure values. Figure 8 plots the ratio of the opening stress intensity factor (*K*o) to the maximum stress intensity factor (*K*max) along the crack front for both materials. As shown, the plane stress curve represents the upper limit, while the plane strain curve represents the lower limit. The values of *K*o/*K*max are between two limiting cases, and decrease with an increase in the stress ratio. In addition, at lower stress ratios, the differences between the maximum and minimum values of *K*o/*K*max are higher for PMMA and tend to be closer for the aluminium alloy.

Figure 9 plots the variation in the *K*o/*K*max ratio at the crack surface obtained from the presented 3D FE simulations against previously published relationships based on experimental tests that incorporated plasticity-induced crack closure. Notably, the results of the presented procedure agree well with the outcomes of the experimental and theoretical studies reported in the literature [1,16,27–29]. The variation between the presented method and published data decreases with an increase in the *R* ratio, as the size of the reverse plasticity zone (or monotonic plastic zone) becomes smaller in the fatigue crack growth rates. These results provide further support to and validation of the numerical technique outlined in this paper.

**Figure 8.** The ratio of the opening stress intensity factor to the maximum stress intensity factor as a function of the *R* ratio along the crack front: (**a**) 6082-T6 aluminium alloy; (**b**) polymethyl methacrylate.

**Figure 9.** The ratio of the of the opening stress intensity factor to the maximum stress intensity factor as a function of the *R* ratio along the crack front and past published functions: (**a**) 6082-T6 aluminium alloy; (**b**) polymethyl methacrylate.

#### **5. Conclusions**

In this paper, new numerical modelling tools capable of simulating the crack shape development of through-the-thickness fatigue cracks in finite plates were presented. The proposed approaches assume a pre-defined crack front shape, and include plasticityinduced crack closure. The methodology was successfully tested for cracked rectangular cross-section geometries when subjected to Mode I loading. The following conclusions can be drawn:

1. The maximum stress intensity factor becomes relatively constant in the sufficiently thick plates and is equal to the value obtained for plane strain state conditions. The plane strain fatigue models (2D) may lead to inaccurate predictions when applied to the analysis of fatigue crack growth of thin structural plates;


The comparison with experimental results is encouraging, and demonstrates the validity of the underlying assumptions: (1) the crack front shape intersects the free plate surface at the critical angle; ad (2) the local stress intensity factor can be considered as the fatigue crack driving force, which leads to the formation of the crack front shape under high cycling loading. The above assumptions might not be correct in the case of large plastic effects near the crack tip. In this case, the plasticity-induced crack closure, which is significantly different along the crack front, will be the one of the most influential factors affecting the crack front shape.

Future work will be directed to the application of the proposed methodology to more complex problems in terms of geometry, loading scenario, and crack shape configuration. Lastly, the simplicity and speed of calculation of the proposed approach, compared to the current numerical solutions used for the same purpose, make it quite attractive for simulating the fatigue crack growth, in both practical applications and parametric studies.

**Author Contributions:** Conceptualization, B.Z. and A.K.; methodology, B.Z. and A.K.; software, B.Z; validation, B.Z. and R.B.; formal analysis, B.Z. and R.B.; investigation, B.Z. and R.B.; data curation, B.Z. and R.B.; writing—original draft preparation, B.Z.; writing—review and editing, B.Z., R.B. and A.K.; visualization, B.Z., A.K. and R.B.; supervision, A.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was sponsored by FEDER funds through the program COMPETE (Programa Operacional Factores de Competitividade) and by national funds through FCT (Fundação para a Ciência e a Tecnologia) under the project UIDB/00285/202.

**Data Availability Statement:** The data presented in this study are available from the corresponding author, upon reasonable request. The data are not publicly available due to ethical restrictions.

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

#### **References**


## *Article* **Analysis of the Deceleration Methods of Fatigue Crack Growth Rates under Mode I Loading Type in Pearlitic Rail Steel**

**Grzegorz Lesiuk 1,\*, Hryhoriy Nykyforchyn 2, Olha Zvirko 2, Rafał Mech 1, Bartosz Babiarczuk 1, Szymon Duda 1, Joao Maria De Arrabida Farelo 1,3 and Jose A.F.O. Correia <sup>4</sup>**


**Abstract:** The paper presents a comparison of the results of the fatigue crack growth rate for raw rail steel, steel reinforced with composite material—CFRP—and also in the case of counteracting crack growth using the stop-hole technique, as well as with an application of an "anti-crack growth fluid". All specimens were tested using constant load amplitude methods with a maximum loading of Fmax = 8 kN and stress ratio R = σmin/σmax = 0.1 in order to analyze the efficiency of different strategies of fatigue crack growth rate deceleration. It has been shown that the fatigue crack grows fastest in the case of the raw material and slowest in the case of "anti-crack growth fluid" application. Additionally, the study on fatigue fracture surfaces using light and scanning electron (SEM) microscopy to analyze the crack growth mechanism was carried out. As a result of fluid activity, the fatigue crack closure occurred and significantly decreased crack driving force and finally resulted in fatigue crack growth decrease.

**Keywords:** pearlitic steel; CFRP patches; crack retardation; fatigue crack growth; failure analysis

#### **1. Introduction**

It is commonly known that each industry, e.g., automotive, marine, building, rail, etc., requests specific parameters and types of different materials, including steel. There are many examples of product orientation in terms of loads, strength, or wear resistance, e.g., in the railway industry. It is expected to have high strength and high wear resistance (rails and wheels of vehicles moving on them). However, much higher focus will be placed on welding properties and corrosion resistance in the marine industry. Pearlitic steel is one of the standard steels which is usually used in places where high loads and wear are expected. It is feasible for this material to work in conditions where it might be exposed to high loads, mainly where high loads occur in a small area. These types of loads cause a local accumulation of stresses, which often exceed the yield point's value. Unfortunately, preventing plastic deformation is not easy or, in many cases, it is simply impossible to avoid. In connection with this, more and more research is carried out on the effects of these deformations, namely cracks and their propagation. The ability to predict the places of crack formation and their propagation paths allows for a sufficiently quick reaction before a tragic catastrophe occurs (break or simply the destruction of the element). The development of fracture mechanics and methods of predicting and determining the lifetime of components under the influence of fatigue undoubtedly contributes to the improvement of safety and reduction of operating costs [1].

Safety, reduction of costs, and changing technologies require optimizing different types of materials in each industry. To optimize and find the proper material for a specific

**Citation:** Lesiuk, G.;

Nykyforchyn, H.; Zvirko, O.; Mech, R.; Babiarczuk, B.; Duda, S.; Maria De Arrabida Farelo, J.; Correia, J.A.F.O. Analysis of the Deceleration Methods of Fatigue Crack Growth Rates under Mode I Loading Type in Pearlitic Rail Steel. *Metals* **2021**, *11*, 584. https:// doi.org/10.3390/met11040584

Academic Editor: Thomas Niendorf

Received: 10 February 2021 Accepted: 1 April 2021 Published: 2 April 2021

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

**Copyright:** © 2021 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/).

application, many different tests are required. There are many studies on the wear rate for different types of steel. These tests are conducted in laboratory conditions [2,3] and, importantly, in real conditions [4,5]. Comparing the results achieved in laboratories with real conditions greatly helps in achieving the complete characteristics of the tested materials. Other studies mainly focus on fracture toughness [6–8] or focus on fatigue crack growth (FCG) [9]. For example, in [7] it was shown that the fracture toughness of very good quality pearlitic steels is at the level of 30.4 MPa√m at the temperature of <sup>−</sup><sup>20</sup> ◦C.

Additionally, another study, [6], investigated how the high shear deformation affects the fracture toughness Kq. The tests were carried out for R260 steel, and the obtained Kq value of the undeformed material varied from 53 to 42 MPa√m. These values were obtained after one pass in flush-channel angular pressing (ECAP). It should be noted that for these tests, the samples used were relatively thin (B = 2 mm). Therefore, unfortunately, it is not possible to determine the actual KIc. Moreover, it is worth noting that the degree of deformation and orientation are of great importance for fracture toughness, which was shown in [8].

The paper presents a comparison of the results of the fatigue crack growth rate for R260 steel, steel reinforced with composite material, and also in the case of counteracting crack growth using the stop-hole technique. Generally speaking, all methods are widely used as deceleration methods of fatigue crack growth. Carbon fiber-reinforced polymer (CFRP) patches are still of particular interest in various applications in civil engineering [9–12]. However, there are not many papers devoted to degradation problems in a real operational environment [13–15]. Thus, in some cases, the historic stop-hole technique is still attractive based on simple crack "blunting" and re-initiation period [16–18].

On the other hand, the fatigue crack growth process is strongly associated with the crack driving force and its local condition. In [19–21], authors explained crack growth rate decreasing with an effective stress intensity factor (crack closure concept [22]) based on ΔKeff concept. This assumption allows us to explain, simply using a closure parameter, the role of crack closure in analytical formulas. Recently, the concept of crack closure triggering due to the application of fluid [23] into the crack was successfully validated for low-carbon steel [19]. Therefore, this paper's main goal is to compare several different (physically) approaches in crack growth rate deceleration in pearlitic steel, which might be used for rail manufacturing [24].

Additionally, for selected materials, visual and microscopic inspection of the fracture surfaces were carried out to characterize them and understand how the fracture development and degradation occurred.

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

#### *2.1. Materials*

The material from which all the samples presented in the study were prepared was taken from the rail that had previously been withdrawn from use. The rail was delivered directly from the manufacturer in a fully operational condition, and its profile conformed to UIC60 [24]. Samples for examination were cut out under cooling with liquid in order to avoid microstructural changes. The chemical composition of the material was spectrally analyzed. The results are summarized in Table 1. Additionally, Table 2 lists the basic static mechanical properties of the tested R260 steel.

**Table 1.** Chemical composition of the investigated R260 steel.



**Table 2.** Mechanical properties of the investigated steel.

The microstructure of the tested steel is shown in Figure 1. In addition to the samples made of the base material, samples were also made that were reinforced with a carbon fiber material. CFRP polymer is an extremely strong and light fiber-reinforced plastic that contains fibers. In general, CFRP composites use thermosetting resins such as epoxy, polyester, or vinyl ester [10]. The Sika® CarboDur® S1014/180 bands were applied to reinforce steel specimens, pultruded carbon fiber plates on an epoxy matrix for structural strengthening of structures. CFRP is highly used in aeronautics because of its weight (much lighter than steel and aluminum) and strength. Carbon fiber is up to five times stronger than typical steel material, however only in specific conditions (along the direction of fibers). The specimens (Figure 2) were reinforced with CFRP considering how the samples were subjected to the cyclic loading and the direction of fibers in composite material. Figure 3 shows steel specimen (base) and Figure 4 reinforced with CFRP. Two types of reinforced specimens, with wider and narrower CFRP strips, were prepared.

**Figure 1.** Microstructure of tested steel; non-etched state (on the **left**) with noticeable small nonmetallic inclusions and pearlite structure (on the **right**) after etching 3% HNO3.

**Figure 2.** Design of compact tension specimen for fatigue crack growth rate experiment (all dimensions in mm).

**Figure 3.** Specimens ready to be glued with carbon fiber-reinforced polymer (CFRP) patches (strips).

**Figure 4.** Real specimens devoted to the experimental campaign; (**a**) base metal compact tension (CT), (**b**) specimen after crack growth test—with drilled hole—stope hole technique, (**c**) full-face one-side CFRP patch.

The Sikadur®-31 CF Normal glue was used to connect the CFRP to the steel specimen. This adhesive is a 2-component thixotropic epoxy adhesive. This product is moisture tolerant and based on a combination of epoxy resins and extra filler. The bonding can be utilized in temperatures between +10 ◦C and +30 ◦C. It can be applied to join concrete elements, natural stone, ceramics, bricks, mortar, masonry, steel, iron, aluminum, wood, epoxy, and glass. The adhesive has the following advantages:


#### *2.2. Methodology*

The tests were carried out on five types of samples, which are listed below:


For the experimental campaign, compact tension (CT) specimens were prepared in accordance with ASTM E647 [25] standard. The scheme of the specimen is shown in Figure 2. Figures 3 and 4 show prepared specimens ready for tests.

Fatigue tests were carried out on the MTS testing machine with 100 kN capacity. During all tests, data were acquired using MTS TestSuite™ Multipurpose Software (Series 793, MTS Systems, Corporation, Eden Prairie, MN, USA). The straight-through notches of 2.5 mm width and 12.5 mm length were machined and pre-cracked. The pre-crack frequency and maximum stress intensity range were 10 Hz and 15 MPa√m, respectively. For specimens without CFRP stress intensity factor was calculated based on linear elastic fracture mechanics [25]:

$$
\Delta K = \frac{\Delta P}{B\sqrt{W}} \frac{(2+a)}{(1-a)^{1.5}} \left[ 0.886 + 4.64a - 13.32a^2 + 14.72a^3 - 5.6a^4 \right],
\tag{1}
$$

where *α* represents the normalized crack length *a*/*W* with *a* as the corresponding crack length observed during the test; Δ*P* is the applied range of force; *B* is the thickness of the specimen, and *W* is the width of specimen defined as in ASTM E647 standard [25]. Crack length was calculated using elastic compliance [25] method automated with MTS system special software for data analysis—Figure 5. Additionally, for specimens with CFRP, crack length was observed in one side (without patch) of the specimen (this strategy was also successfully applied in previous studies based on the beach marking technique in [26]) in order to confirm registered values of crack length.

**Figure 5.** Sinusoidal waveform of loading and measured hysteresis loops and elastic compliance for crack length calculations.

The crack length was monitored using MTS FCGR modular software for hysteresis loop analysis and calculating the current crack length using the compliance method. The exemplary graphical user interface is shown in Figure 5.

This experimental campaign began with standard steel specimens preparation and started with fatigue crack growth rate testing with a maximum stress intensity factor Kmax = 15 MPa\*m0.5 (sinusoidal waveform, where R = 0.1) in pre-cracking phase. All types of specimens were pre-cracked up to 14 mm crack length. Then specimens were selected and divided into two groups; one for composite strips (gluing) and another one, "pure metal" and "stop hole" were devoted for direct testing up to a specific number of 75,000 cycles. In this FCGR (fatigue crack growth rate) experiment phase, a constant amplitude loading method was used. Fmax = 8 kN and R = 0.1 were kept during the experiment for all types of specimens in order to maintain the same testing conditions for all types of specimens.

After reaching mentioned 75,000 cycles in specimen marked as "stop hole" a hole of 5.6 mm diameter was drilled. After drilling the hole, the specimen was placed again on the testing machine, and cyclic loading was continued in order to receive information about fatigue crack growth within the use of the stop-hole technique.

Specimen marked as "with\_fluid" (Figures 6 and 7) was tested in special environmental liquid solutions. Unique, patented fluid [23] was tested in order to trigger the artificial crack closure phenomenon. The liquid matter, as the particular technological environment (STE), containing an active component patented in [23] was used in the experiment. The solvent serves as a transport medium of the active component into fatigue crack. After applying the fluid into a crack cavity, it starts to interact with the crack's metal surfaces chemically.

**Figure 6.** Specimen with fluid injected into the machined notch tip of CT specimen mounted in the hydraulic pulsator.

**Figure 7.** Fatigue crack growth curves for all tested specimens.

In the case of these investigations, the sample did not require any additional operations prior to testing, besides well finishing of sides surfaces of the specimen. The sample was subjected to the same cycling loading values as in the previously mentioned methods. After a specified number of cycles (70,000 cycles), the injections were stopped, and the sample was cycled until the break.

#### **3. Fatigue Crack Growth Results and Fractography**

Figure 7 presents fatigue lifetime curves for all specimens tested in the same loading conditions. As it is noticeable comparable effect is obtained for the designed stop hole technique and composite CFRP patches. Noticeable is no extra difference in CFRPs (large and small patch). A similar effect was also observed and described in the Authors' paper [25] with a numerical analysis of the CFRP effect on reducing SIF.

For direct comparison of the fluid activation effect on the crack deceleration for metallic specimens (without CFRP) kinetic fatigue fracture diagram (KFFD) was constructed— Figure 8.

**Figure 8.** Comparison of fatigue crack growth rates for pure metallic specimens (stop hole, CT specimen with injected fluid).

Noticeable is the high deceleration effect caused by injected fluid—much stronger than in stop hole technique. The crack growth rate decreases at least ten times in the initial injection. As the crack length increases (and thus injections were interrupted, the crack was dried, for approx. ΔK = 37–38 MPa\*m0.5), the crack growth rate (characterized by slope of the linear part of KFFD) was similar to the initial slope and initial FCGR—like that obtained for pure metal (large dots in Figure 8). Based on the above, it was decided to analyze fatigue fracture surfaces using light and scanning electron (SEM) microscopy in order to analyze the crack growth mechanism in both cases (with and without fluid) due to the observed significant retardation effect of fatigue crack growth rate -before and after of the injection. Additionally, the same type of observations was done for the "stop hole" specimen. In Figures 9 and 10 are presented fatigue crack surfaces in the vicinity of drilled stop-hole in a macroscopic view. Detailed SEM analysis before drilling hole (crack tip position 3 mm after pre-crack) does not differ from the original crack path in this steel [24]. As the crack grows (under constant load amplitude) an increasing number of secondary cracks is observed as a natural result of increased KI values under tensile crack growth mode—Figure 11. After drilling the hole (Figure 12) 75,000 cycles to 110,000 cycles were required for re-initiation of fatigue crack growth. After that, the crack starts to grow similarly as in pure CT specimen under the tensile mode, which is reflected in Figure 13 on fractograms with a large number of secondary cracks and noticeable fatigue striations.

**Figure 9.** Specimen after stop hole technique testing procedure with re-initiated crack after the hole (crack growth direction from left to right).

**Figure 10.** Initial fatigue crack path (15 mm) of the tested specimen with stop hole (ΔK = 24 MPa <sup>×</sup> m0.5), crack growth direction from bottom to top.

**Figure 11.** Initial fatigue crack path (22 mm) of the tested specimen with stop hole (ΔK = 34.8 MPa <sup>×</sup> m0.5), crack growth direction from bottom to top, fracture surface located close to drilled hole.

**Figure 12.** Initial fatigue crack path (24.3 mm) of the tested specimen with stop hole (ΔK = 40 MPa × m0.5), crack growth direction from bottom to top, fracture surface located close to drilled hole with reinitiated fatigue crack.

**Figure 13.** Initial fatigue crack path (24.5 mm) of the tested specimen with stop hole (ΔK = 40.4 MPa <sup>×</sup> <sup>m</sup>0.5), crack growth direction from bottom to top, fracture surface located close to the drilled hole with reinitiated fatigue crack—noticeable fatigue striations.

On the contrary to the typical crack growth mechanism in specimen tested in fluid injections—Figure 14—an etched fracture surface was observed due to a possible mechanism of interaction fluid with crack surfaces and chemical reactions [19]. As a result of interaction between the fluid and crack surfaces, a solid product of substantial volume appears, which fills the crack cavity. The chemical mechanism on which this method is based is similar to intrinsic crack closure caused by products of interaction between metal and humid air or the corrosive environment due to fretting corrosion. The active

fluid components interact with ferrous ions and form insoluble complexes. When steel is exposed to an electrolyte (in this case to fluid), metal ions leave the lattice and enter the electrolyte as ferrous ions. Ferrous ions and active fluid ions react to form insoluble compounds, chelate Fe(II).

**Figure 14.** SEM macroscopic view on the "wet" and "dry" zone of crack propagation mechanism in specimens tested with fluid injections.

However, natural crack closure is peculiar to fatigue crack growth at low ΔK. Therefore, the task consisted of searching for such substance that would rapidly provide much more intensive interaction with the metallic specimen's crack surfaces.

The fatigue fracture surface in Figure 10 is mainly shaped by noticeable longitudinal ridges (normal to the direction of maximum tensile stress) and facets associated with various pearlite colony orientations.

With increasing crack length, fatigue fracture surface (Figure 11) is mainly shaped by numerous secondary cracks resulting from increasing tensile stress in front of the growing crack for higher values of stress intensity factor.

Large ridges mostly shaped macroscopic view on close area to drilled hole—called "re-initiation region" (Figure 12). The fracture surface for increased crack length (Figure 13) was characterized by microcracks in the interphase zone (plates of cementite and ferrite) on the background of fatigue striations.

After testing in environmental—fluid—conditions, the region on the border between dry and wet crack was particularly interesting for SEM investigations. In Figure 14, a macroscopic view of the crack front between "wet" and "dry" zone (approx. after 150,000 cycles) is shown with marked by frames microscopic SEM images.

As it was expected in the region without fluid influence "dry zone"—Figure 14—crack growth mechanism was typical for this steel as it was evidenced in previous specimens (i.e., stop hole technique) and results in a similar final slope of the da/dN–ΔK diagram. This region is also shaped mainly by the number of secondary cracks and fatigue striations. The "wet zone" is characterized by many metal–fluid reaction products and wear parts of the fracture surface as a consequence of the oxidization and finally artificial crack closure

effect. Initially, based on PICC (Plasticity Induced Crack Closure). Elber [22] defined the effective stress intensity factor range as:

$$
\Delta \mathcal{K}\_{eff} = \mathcal{K}\_{\text{max}} - \mathcal{K}\_{op\_{\prime}} \tag{2}
$$

where *Kmax* is the maximal value of stress intensity factor; *Kop* is the stress intensity factor at crack opening during fatigue cycle. Finally, the closure parameter *U* can be defined as:

$$
\mathcal{U} = \frac{\Delta \mathcal{K}\_{eff}}{\Delta \mathcal{K}\_{app}}.\tag{3}
$$

For the tested specimen, crack closure parameter *U* was calculated based on LQSM (Linear Quadratic Spline Method) successfully applied in previous Authors' papers [26]. As observed from fracture surfaces, many fluid chemical reactions products caused a significant drop of *U*-value, which constitutes effective crack driving force and finally deceleration of fatigue crack growth. A significant decrease of *U*-parameter was observed after fluid injection—Figure 15. This fact may explain the combined physico-chemical artificial crack closure effect (PCMACC).

**Figure 15.** Elber closure parameter variation for "fluid" specimen before and after injection (including 2 mm crack length increment).

#### **4. Conclusions**

Comparison of fatigue lifetime curves for all specimens tested in the same loading conditions shown that a noticeable comparable effect was obtained for the designed stop hole technique and composite CFRP patches.

Moreover, a comparison of FCGR methods shown a noticeable higher deceleration effect caused by injected fluid than in the stop hole technique. The crack growth rate decreases at least ten times in the initial injection.

The detailed SEM analysis of the stop hole technique's fractogram showed no differences between paths of this sample and the raw steel before drilling the hole. After drilling the hole, crack starts to grow in a similar manner as in pure CT specimen under the tensile mode, which is reflected on fractograms with a large number of secondary cracks and noticeable fatigue striations.

On the contrary to the typical crack growth mechanism in (raw specimen), in the specimen tested with fluid injections, an etched fracture surface was observed. This chemical treatment triggers the mechanism on which this method is based and which is similar to intrinsic crack closure caused by products of interaction between metal and humid air or the corrosive environment due to fretting corrosion.

Based on experimental results presented in this paper, as well as in previous paper devoted to low-carbon steel [19], it can be concluded that the fluid should be thin enough to fall into the crack easily and fill it. What is more, the chemical substance should be dissoluble to concentrations that effectively delay crack growth. It is worth noting that this effect should be tested on a wider group of materials to validate its usefulness in engineering practice combined with analytical modelling of crack growth retardation effect.

**Author Contributions:** Conceptualization, H.N. and J.A.F.O.C.; methodology, G.L.; software, S.D.; validation, G.L., J.M.D.A.F. and R.M.; formal analysis, G.L.; investigation, G.L., B.B., J.M.D.A.F. and O.Z.; data curation, R.M.; writing—original draft preparation, G.L., J.M.D.A.F. and R.M.; writing review and editing, G.L.; visualization, S.D.; supervision, J.A.F.O.C.; project administration, O.Z.; funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** The project was supported in part by the Polish National Agency for Academic Exchange (Polish–Ukrainian bilateral agreement) grant number PPN/BUA/2019/1/00086.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

#### **References**


## *Article* **Fatigue Crack Growth Behaviour and Role of Roughness-Induced Crack Closure in CP Ti: Stress Amplitude Dependence**

**Mansur Ahmed 1,\*, Md. Saiful Islam 2, Shuo Yin 1, Richard Coull <sup>1</sup> and Dariusz Rozumek <sup>3</sup>**


**Abstract:** This paper investigated the fatigue crack propagation mechanism of CP Ti at various stress amplitudes (175, 200, 227 MPa). One single crack at 175 MPa and three main cracks via sub-crack coalescence at 227 MPa were found to be responsible for fatigue failure. Crack deflection and crack branching that cause roughness-induced crack closure (RICC) appeared at all studied stress amplitudes; hence, RICC at various stages of crack propagation (100, 300 and 500 μm) could be quantitatively calculated. Noticeably, a lower RICC at higher stress amplitudes (227 MPa) for fatigue cracks longer than 100 μm was found than for those at 175 MPa. This caused the variation in crack growth rates in the studied conditions.

**Keywords:** CP Ti; stress amplitude; fatigue crack propagation; crack growth rate; roughness-induced crack closure

#### **1. Introduction**

Commercially pure titanium (CP Ti) possesses high ductility as well as excellent corrosion resistance and biocompatibility properties; hence, it has been used in the chemical and biomedical industries, especially in reactor container in chemical plants, and in power station heat exchangers [1]. In these environments, cyclic loading is applied to the components. Therefore, investigation of the fatigue behaviour of CP Ti becomes an important research subject. Fatigue crack initiation, growth, closure and fractography are critical features describing fatigue behaviour. A fatigue crack path may be a powerful resource determining all the aforementioned features. For instance, roughness-induced crack closure (RICC) is attributed to crack path deflection, especially near the threshold range, at which a serrated or zigzag crack path is induced by microstructure-sensitive crack growth [2–4]. The tilt angle of the crack path in 2124 Al alloy is reported to be a key controlling factor for RICC [5]. Wang and Müller [2,6] reported that RICC occurs due to a serrated crack path, which significantly affects crack growth rates in Ti-2.5 Cu (wt%) and TIMETAL 1100 alloys. Fatigue crack growth rates are reported to be decreased by crack path deflection [3,4,7], as the crack path causes a direct reduction of the local driving force for crack propagation and an increase in the total crack path length, which results in lower crack growth rates and induces RICC. Antunes et al. [8] mentioned that cracks with larger path deflection may result in higher mode II displacement between the two fracture surfaces, causing higher crack closure stress intensity [9]. Ding et al. [10] correlated the fatigue crack profile with fatigue crack growth rate in Ti–6Al–4V and Ti–4.5Al–3V–2Mo–2Fe alloys. Okayasu et al. [11] investigated the influence of fatigue loading conditions such as, stress amplitude and stress ratio on the contact features of fracture surfaces in annealed Carbon Steel via two

**Citation:** Ahmed, M.; Islam, M.S.; Yin, S.; Coull, R.; Rozumek, D. Fatigue Crack Growth Behaviour and Role of Roughness-Induced Crack Closure in CP Ti: Stress Amplitude Dependence. *Metals* **2021**, *11*, 1656. https://doi.org/10.3390/ met11101656

Academic Editor: Denis Benasciutti

Received: 16 September 2021 Accepted: 18 October 2021 Published: 19 October 2021

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

**Copyright:** © 2021 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/).

methods: (1) Collecting fracture debris fallen from the crack surfaces; and (2) Observing the fracture surface directly through the replica technique. They showed that the fatigue stress amplitude and stress ratio played major factors in determining the contact status between the mating fracture surfaces, e.g., a larger stress amplitude and smaller stress ratio lead to stronger fracture surface contact or interaction. Student et al. [12] studied the role of fracture surface roughness on crack closure in long-term exploited heat-resistant steel. They reported that shear processes at the tip of a fatigue crack significantly affect crack closure and contribute to the roughness of the fracture surface.

The above discussions evidently show that fatigue crack path can be effectively used to investigate numerous phenomena. Unfortunately, study on the fatigue behaviour of CP Ti related to crack path and its role in detecting fatigue phenomena such as RICC is totally lacking in the literature. To fill this gap, the fatigue crack growth mechanism and corresponding RICC from fatigue crack path in CP Ti has been revealed here. This will further help in understanding fatigue crack growth at the small crack regime and the role of RICC in such cracking.

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

The composition of the studied CP Ti is given in Table 1. A vacuum furnace heattreatment of the as received hot rolled sample was performed at 700 ◦C for 30 min for obtaining equiaxed α grains. Following heat treatment, samples were machined in order to prepare fatigue specimen with the dimensions shown in Figure 1. Because the surface condition of the specimen in fatigue testing is very sensitive, the samples were then carefully mechanically polished using various grades of SiC emery papers followed by colloidal silica with hydrogen peroxide solution. The polished surfaces were then etched using Kroll's reagent (3HF:6HNO3:91H2O) to reveal the microstructure.

**Table 1.** Chemical composition (in wt%) of the CP Ti.

**Figure 1.** (**a**) Schematic of sample with the dimensions used for the rotating bending test (dimensions in mm); (**b**) Ycoordinates along the crack path profile with equidistant spacing.

Tensile tests of CP Ti samples were performed employing an Instron Universal Tensile Machine. The dimensions of the samples were in accordance with the ASTM E8. For each condition, three samples are tested, at room temperature and at a strain rate of 1 mm/min.

An Ono-type rotating bending fatigue machine was used for fatigue tests, at a frequency of 30 Hz. A stress interval of 25 MPa was chosen, from the fatigue limit to a stress at which the distinction of the stress amplitude effect could be numerated. Consequently, fatigue tests at the stresses of 175, 200 and 227 MPa were conducted. The stress ratio (R) was −1 with sinusoidal wave form. All tests were conducted at temperature from 15–20 ◦C. During testing, a cooling fan was used to cool the specimens, as heat production affects the deformation mechanism of Ti. During fatigue testing, the replica technique was intermittently used at different cycles to trace out crack initiation and propagation; this technique involved immersing replica films into methyl acetate solution and subsequently pasting them onto the specimen surface. To then acquire the image from the replica sheet, an optical microscope (VHX-2000 series, Keyence, Osaka, Japan) was used. One sample of each condition was investigated. Following the fatigue tests, fractographic analysis was conducted using a JEOL IT-300 (JEOL, Tokyo, Janpa) scanning electron microscope (SEM) at an acceleration voltage of 30 kV.

Roughness parameters were quantitatively measured using the surface crack paths along the fractured specimens. Two roughness parameters, (i) linear roughness parameter (RL) and (ii) arithmetic mean deflection angle (θ), were evaluated using the equidistant spacing method [3,4,13]. Figure 1b displays the equidistant spacing method used to calculate these roughness parameters. The details of this method can be found elsewhere [3].

The linear roughness parameter, RL of a crack profile is defined as follows:

$$\mathbf{R}\_{\rm L} = \mathbf{L} / \mathbf{L}' \tag{1}$$

where L and L correspond to the true length and projected length of the crack profile, respectively. The arithmetic mean deflection angle of a crack profile is given below:

$$\overline{\Theta} = \frac{1}{\mathbf{n}} \sum\_{i=1}^{n} |\Theta\_{i}| \tag{2}$$

where θ<sup>i</sup> represents the angle between the profile element and X-coordinate axis and can have either a positive or a negative value (90◦ ≤ θ ≤ −90◦), depending on the crack profile.

#### **3. Results**

#### *3.1. Initial Microstructure, and Tensile and Fatigue Properties*

Figure 2a shows the initial microstructure of the studied material. Uniformly distributed (hexagonal closed packed) α grains with an average diameter of ~35 μm can be observed. The size of the α grains was calculated using Image J. The engineering stress– strain curve plotted from the tensile test is shown in Figure 2b. Beyond the yielding point, work-hardening is followed by work-softening along the stress–strain curve. The yield strength (YS), ultimate tensile strength (UTS) and total elongation (El.) of the studied material were measured to be 293 ± 12 MPa, 383 ± 7 MPa and 67 ± 1%, respectively. Table 2 shows the fatigue test results at various stress amplitudes. As expected, the total life cycle was found to be reduced with increasing stress amplitude. For instance, the sample tested at 175 MPa survived for 6.94 × 105 cycles, while the sample tested at 227 MPa only lasted for 1.05 × <sup>10</sup><sup>5</sup> cycles. At the intermediate stress amplitude of 200 MPa, the sample failed after reaching 3.4 × <sup>10</sup><sup>5</sup> cycles.

**Figure 2.** (**a**) Initial microstructure; (**b**) Representative stress–strain curve of the studied CP Ti.


**Table 2.** Summary of fatigue tests results obtained in this study.

#### *3.2. Fatigue Crack Nucleation and Propagation, Propagation Rate and Fractography*

Figure 3 shows optical micrographs of the fatigue surface crack and its surroundings at various cycles at 175 MPa. The microstructure in Figure 3a shows an image taken before the test corresponding to a region where the main fatigue crack initiated. After <sup>2</sup> × <sup>10</sup><sup>5</sup> cycles, a micro-sized crack located inside an <sup>α</sup> grain can be seen (Figure 3b). This indicates that the crack initiated between <sup>1</sup> × <sup>10</sup>5–2 × <sup>10</sup><sup>5</sup> cycles, as no crack was found after <sup>1</sup> × <sup>10</sup><sup>5</sup> cycle (not shown here). Therefore, it is evident that most of the fatigue lifecycle was consumed by fatigue crack propagation considering total fatigue life (6.94 × <sup>10</sup><sup>5</sup> cycles). Crack propagation and its features after 4 × 105 cycles can be observed in Figure 3c, and was predominantly transgranular in nature and thus deflected by almost every grain. Following 6 × 105 cycles (Figure 3d), crack propagation was incremental, continuing in transgranular mode. Some important fatigue crack propagation features, such as crack branching (Figure 3f,h) and fine scale zig-zag (Figure 3e–h) can also be seen. Some lines, presumably slip bands and/or deformation twinning, according to [14], appear in some grains (Figure 3f,h). Interactions between the crack and those lines confirm that cracking propagated along or across these slip bands/deformation twinning. Ismarrubie and Sugano [15] have also reported such lines belonging to slip bands. While crack branching in Figure 3f is transgranular, intergranular cracking also appears in Figure 3h.

**Figure 3.** At 175 MPa: Images showing surface crack paths at different cycles: (**a**) Initial condition, i.e.,N=0 cycle; (**b**) After N=2 <sup>×</sup> 105 cycles; (**c**) After N = 4 <sup>×</sup> 105 cycles; (**d**) After N = 6 <sup>×</sup> 105 cycles. The free-shape object in (**a**) and (**b**) indicates a particular area where the crack started. The red arrows illustrate crack initiation sites, and black arrows delineate the crack tips. Examples of fine zig-zag (**e**,**g**), crack branching (**f**,**g**), and crack interactions with slip bands (**e**,**h**).

Figure 4 delineates the initiation and subsequent propagation of the main crack at various cycles at 227 MPa. Figure 4a illustrates the main fatigue crack, with a zig-zag pattern, after 1 × 105 cycles. This crack has been sectioned to understand its propagation mechanism. There are three sub-cracks labeled 1–3 connected to the main crack. Sub-cracks were labeled according to their connecting sequence with the main crack. Therefore, it can be seen that the fatigue crack of CP Ti tested at higher stress amplitude (227 MPa) grew by coalescing sub-cracks. Figure 4b shows the main crack after 0.4 × <sup>10</sup><sup>5</sup> cycles, with the crack initiation site marked by red arrows. It is worth mentioning that the crack initiated between 0.2 × <sup>10</sup>5–0.3 × 105 cycles, as there was no crack at 0.2 × <sup>10</sup><sup>5</sup> cycles (not shown here). Similar to the sample tested at 175 MPa, this condition also consumed majority of the total cycle of crack growth. Figure 4c–e show micrographs after 0.5 × <sup>10</sup>5, 0.7 × <sup>10</sup><sup>5</sup> and 0.9 × 105 cycles, respectively, where pre-coalescence of the sub-cracks with the main crack are shown. Unlike the crack at 175 MPa, the crack at 227 MPa preferably propagated via coalescing sub-cracks. This is the first time we saw such a difference in crack propagation mechanism with respect to the stress amplitude in CP Ti. Of the fatigue characteristics, a zig-zag nature and fine-scale crack branching are also visible under this condition. As at 175 MPa, a transgranular fracture mode was also predominant in this case. It is worth mentioning that crack propagation behavior at 200 MPa was identical to that at 227 MPa.

**Figure 4.** At 227 MPa: Images displaying crack paths at different cycles. After (**a**)N=1 <sup>×</sup> <sup>10</sup><sup>5</sup> cycles; (**b**) N = 0.4 <sup>×</sup> <sup>10</sup><sup>5</sup> cycles; (**c**) N = 0.5 <sup>×</sup> <sup>10</sup><sup>5</sup> cycles; (**d**) N = 0.7 <sup>×</sup> 105 cycles; (**e**) N = 0.9 <sup>×</sup> 105 cycles. Red arrows show the crack initiation line, while black arrows indicate crack tips; (**c**) shows the main crack and sub-crack 1 before coalescing, while the main crack and sub-crack 2 can be seen in (**d**); (**e**) shows the main crack and sub-crack 3 prior to coalescence. The yellow arrows in (**a**) indicate the junctions of crack coalescence.

> Figure 5 shows the relationship between crack length and number of cycles, as well as crack propagation rate with respect to the crack length. These data have been calculated based on the crack paths shown in Figures 3 and 4. While the crack length in the 175 MPa sample grows steadily, crack length at 227 MPa increases abruptly after around 300 μm (Figure 5a). On the other hand, the crack growth rate in the 227 MPa sample is significantly higher than that of the 175 MPa sample for cracks longer than 100 μm. Figure 6 shows the fracture surfaces of the samples tested at various stress amplitudes. Three main features, including (i) crack initiation site (marked with green boxes), (ii) crack propagation, and (iii) the final fracture as dimples can be observed in each sample. Higher magnification images of the crack initiation sites are shown in Figure 6d–f. Interestingly, the position of the dimples progressively moves toward centre of the sample with increased stress amplitude. This corresponds to the number of cracks responsible for fracture with stress amplitude; for instance, one single crack, two cracks and three cracks were responsible for fracture of samples tested at 175, 200 and 227 MPa, respectively.

**Figure 5.** (**a**) Relationship between fatigue crack length and number of cycles; (**b**) Crack growth rate with respect to fatigue crack length.

**Figure 6.** Fractography of the samples tested at (**a**) 175, (**b**) 200, and (**c**) 227 MPa. Green boxes indicate the nucleation sites of the main cracks. Crack nucleation sites for (**d**) 175, (**e**) 200, and (**f**) 227 MPa stress conditions.

#### **4. Discussion**

#### *4.1. Fatigue Crack Initiation and Propagation*

Fatigue cracks at various stress amplitudes evidently initiated from the surfaces of the specimens (Figure 6d–f). This is consistent with earlier claims that fatigue cracks in Ti and its alloys tend to initiate from surface in the case of continuous cyclic loading if the surface is free of residual stress [16]. In this study, mechanical polishing followed by chemical etching was performed in order to ensure a residual stress-free surface so that crack would start from specimen surface. The measured angle of ~50◦ (Figures 3b and 4b) between crack initiation and the loading axis is close to that of the maximum critical resolved shear stress (45◦). The cracks thereafter propagated along a direction of approximately 70◦ with respect to the loading axis, which is corresponded to mixed mode I and mode II crack growth. Close examination of crack path shows that a significant portion of the cracks propagated as zig-zag where the crack paths moved along or across slip bands/deformation twinning in short distances (Figure 4e–h). Such zig-zag phenomenon has been attributed to the alternative branching mode I and mode II in forged VT3-1 alloy [17]. Some portion of the crack propagated along the direction perpendicular to the loading axis, corresponding to mode II. It has been mentioned in [18,19] that a shift in direction perpendicular to the specimen may sometimes appear in crack branching. This is consistent with the results shown in Figure 4f. The nature of the crack path is likely not dependent on the stress amplitude, while the number of cracks causing failure is highly dependent upon the stress amplitude. At 175 MPa, a single crack was responsible for failure, whereas at higher stress amplitudes the number of cracks that are responsible for failure increases. For instance, two cracks and three cracks nucleated at different stages in samples tested at 200 and 227 MPa (Figure 3b,c), respectively. Each of the main crack propagated via the sub-crack coalescence mechanism (Figure 5). Therefore, it can be claimed that fatigue crack nucleation and their propagation in CP Ti are largely dependent on the stress amplitude.

#### *4.2. Role of Roughness Induced Crack Closure (RICC)*

It is established that crack tortuosity, crack branching or their combination induce crack closure, as they promote higher roughness [19]. In this study, we have seen crack deflections with a zig-zag nature, crack branching, and a directional shift perpendicular to the specimen surrounding crack branching. Therefore, it is assumed that RICC has played a role in crack propagation. As such, the RICC for 175 and 227 MPa samples has been calculated; the stress amplitude dependence of RICC is discussed below.

A model proposed by Pokluda and Pippan is used to quantitatively measure RICC [20]. The total maximum level of RICC, - δcl <sup>δ</sup>max RICC, can be expressed as follows:

$$(\frac{\delta\_{\rm cl}}{\delta\_{\rm max}})\_{\rm RICC} = \mathcal{G}\eta \sqrt{(\mathcal{R}\_{\Theta}^2 - 1)} + \frac{3\eta(\mathcal{R}\_{\Theta} - 1)}{[\sqrt{\delta} + 3(\mathcal{R}\_{\Theta} - 1)]} \tag{3}$$

where C ≈ <sup>10</sup>−<sup>1</sup> is a dimensionless constant independent of material, R<sup>θ</sup> = cos−1(θ), is the arithmetic mean of the angle that dictates the crack deflection with crack propagation (Figure 7b). η is a parameter that strongly depends on the size ratio, SR = dm/rp where dm is the mean grain size and rp is the static plastic zone size. Static plastic zone size varies depending on the maximal applied intensity factor as a function of the applied stress amplitude and the crack length. Therefore, the value of η differs for different stress amplitudes and, moreover, significantly changes during crack propagation. To assess the values of η at various crack lengths, the following equation [21] is applied:

$$\eta = \exp\left[-\left(0.886 \text{ rp/dm}\right)^{2.2}\right] \tag{4}$$

**Figure 7.** Roughness parameters of samples tested at the stress of 175 and 227 MPa based on the surface crack path profile: (**a**) Linear roughness parameter, RL; (**b**) Arithmetic mean of the deflection angle, (θ). Specific fatigue life denotes the ratio of instantaneous fatigue life (N) to total fatigue life (Nf); the ellipses in (**a**,**b**) show a comparison at a specific point.

For a small crack length, 2a = 100 μm, we obtain η<sup>175</sup> = 0.96, η<sup>227</sup> = 0.88 and η175/η<sup>227</sup> = 1.09. This means that there is still a substantial level of RICC for both stress amplitudes. For a longer crack of 2a = 300 μm, however, the result is η<sup>175</sup> = 0.64, η<sup>227</sup> = 0.25 and η175/η<sup>227</sup> = 2.56. Interestingly, the level of RICC for σa2 (=227 MPa) becomes significantly lower than that for σa1 (=175 MPa). In the case of 2a = 500 μm, it holds η<sup>175</sup> = 0.26, η<sup>227</sup> = 0.014, η175/η<sup>227</sup> = 18.6 and the level of RICC for σa2 (=227 MPa) already becomes negligible. The above comparative analysis clearly shows that there is a lower level of RICC related to the higher applied stress amplitude σa2 (=227 MPa) for all fatigue cracks longer than 100 μm, which corresponds to a higher crack growth rate under the applied stress amplitude σa2 (=227 MPa) than that for σa1 (=175 MPa). This clearly explains the slow crack growth up to around 100 μm of sample under the applied stress amplitude. Crack coalescence under the applied σa2 (=227 MPa) happened for cracks much longer than 100 μm with the level of RICC lower (or even negligible) compared to that under the stress σa1 (=175 MPa). Therefore, crack coalescence is considered to compensate for lower RICC level for σa2 (=227 MPa). On the other hand, the crack coalescence could compensate for the retardation of crack growth rate caused by a longer crack path due to a higher tortuosity (i.e., zig-zag growth) of the crack when σa2 (=227 MPa). This retardation is considered to be directly proportional to the linear roughness ratio, RL175/RL227 [21]. Indeed, the kinking geometry does not change the level of the maximum shear stress ahead of the crack front; therefore, the related decrease of KIa (geometrical shielding) has no considerable effect on the crack growth rate [22].

#### **5. Conclusions**

Fatigue crack growth behaviour such as crack propagation and its features, roughnessinduced crack closure and fatigue striation with respect to various stress amplitudes (175, 200 and 227 MPa), were studied here for CP Ti. The followings are the main outcomes of the investigation:


fatigue failure. Each of the cracks at 227 MPa propagated via sub-crack coalescence. In all conditions, crack deflection, crack branching and slip bands, which are characteristics of crack closure, were noticed. Beyond the initial 100 μm, crack growth rate for the 227 MPa sample was higher than that of the 175 MPa sample.

• RICC calculation for crack lengths of 100, 300 and 500 μm under the 175 and 227 MPa conditions showed a remarkable outcome. Up to 500 μm crack length a substantial RICC was calculated at 175 MPa, while the same level of RICC for 227 MPa was found up to 300 μm. Beyond 100 μm, the RICC level for 175 MPa displayed a higher value than at 227 MPa. This gives a reasonable explanation for the abrupt increase of crack growth rate under the 227 MPa condition.

**Author Contributions:** Conceptualization, M.A. and D.R.; validation, M.A.; investigation, M.A., M.S.I.; resources, R.C., S.Y.; data curation, M.A.; writing—original draft preparation, M.A., M.S.I.; writing—review and editing, D.R.; visualization, M.A., M.S.I.; supervision, D.R.; funding acquisition, M.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This project has received funding from Enterprise Ireland and the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No 847402.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

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

#### **References**


### *Article* **Fatigue Variability of Alloy 625 Thin-Tube Brazed Specimens**

**Seulbi Lee 1,†, Hanjong Kim 2,†, Seonghun Park 2,\* and Yoon Suk Choi 1,\***


† All authors contributed equally to this study.

**Abstract:** As an advanced heat exchanger for aero-turbine applications, a tubular-type heat exchanger was developed. To ensure the optimum performance of the heat exchanger, it is necessary to assess the structural integrity of the tubes, considering the assembly processes such as brazing. In this study, fatigue tests at room temperature and 1000 K were performed for 0.135 mm-thick alloy 625 tubes (outer diameter of 1.5 mm), which were brazed to the grip of the fatigue specimen. The variability in fatigue life was investigated by analyzing the locations of the fatigue failure, fracture surfaces, and microstructures of the brazed joint and tube. At room temperature, the specimens failed near the brazed joint for high *σ*max values, while both brazed joint failure and tube side failure were observed for low *σ*max values. The largest variability in fatigue life under the same test conditions was found when one specimen failed in the brazed joint, while the other specimen failed in the middle of the tube. The specimen with brazed joint failure showed multiple crack initiations circumferentially near the surface of the filler metal layer and growth of cracks in the tube, resulting in a short fatigue life. At 1000 K, all the specimens exhibited failure in the middle of the tube. In this case, the short-life specimen showed crack initiation and growth along the grains with large through thickness in addition to multiple crack initiations at the carbides inside the tube. The results suggest that the variability in the fatigue life of the alloy 625 thin-tube brazed specimen is affected by the presence of the brazed joint, as well as the spatial distribution of the grain size and carbides.

**Keywords:** fatigue variability; alloy 625; thin tube; fractography; microstructure

#### **1. Introduction**

Heat exchangers are one of the key components for environmentally friendly gasturbine engines with lower emissions and higher specific fuel consumption ratings to meet environmental requirements and airline operation conditions [1–5]. Advanced heat exchangers for aero-turbine applications require compact and complicated shapes to achieve high efficiency and size limitations. Such a design limitation sometimes necessitates the use of submillimeter-scale thin tubes to maximize the heat exchange rate in a limited space. However, the use of such thin tubes requires an additional assembly process called "brazing" to connect the thin tubes to the inlet and outlet of the heat exchanger. Here, the mechanical integrity of thin tubes (including the brazed joint) needs to be thoroughly evaluated to ensure the optimum performance of the heat exchanger. However, it is difficult to assess the thermo-mechanical strengths of thin tubes and brazed joints under fluctuating loads, which simulate actual service conditions [6–14].

In the present study, a thin-tube brazed fatigue specimen was designed to evaluate the fatigue properties of thin tubes including brazed joints at room and elevated (1000 K) temperatures, considering the actual operating conditions of heat exchangers for the aeroturbine engine. Here, solid-solution-strengthened Ni-based alloy 625 was chosen as the thin-tube material. Alloy 625 has been used for a variety of components in the aerospace,

**Citation:** Lee, S.; Kim, H.; Park, S.; Choi, Y.S. Fatigue Variability of Alloy 625 Thin-Tube Brazed Specimens. *Metals* **2021**, *11*, 1162. https:// doi.org/10.3390/met11081162

Academic Editor: Dariusz Rozumek

Received: 25 June 2021 Accepted: 19 July 2021 Published: 22 July 2021

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

**Copyright:** © 2021 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/).

aeronautics, marine, chemical, and nuclear industries because of its high-temperature strength, corrosion resistance in a variety of environments, excellent fabricability, and weldability (for tubing) [15–18]. Since alloy 625 components in industry fields were subjected to high temperature operations for a long duration, it is important to secure the mechanical properties at service temperature. V. Shankar et al. [19] extensively investigated the tensile properties at intermediate temperatures with various strain rates. L.M. Suave et al. [20] investigated the high temperature fatigue properties and clarified the thermal aging effect on the mechanical properties. Additionally, evaluation of the mechanical properties, considering the structural integrity, was important to estimate the reliable properties of the component. For the brazed superalloys, low-cycle fatigue [21,22], creep [23,24], and thermal cycling [25] have been studied, but high-cycle fatigue has been conducted in few studies [26,27]. In terms of the structural applications for brazed joints, J. Chen et al. [26,27] performed a high-cycle fatigue test of alloy 625 joints brazed with the Palnicro-36MTM filler metal to clarify the effect of single-lap joints under various lap distance-to-thickness ratios.

The alloy 625 thin-tube brazed fatigue specimens used in this study were designed similarly to the final configuration installed in an actual heat exchanger to reliably assess the fatigue properties of the alloy 625 thin tubes and brazed joints. A systematic study was conducted to understand the source of fatigue variability observed in alloy 625 thin-tube specimens. In particular, an effort was made to interpret the observed fatigue variability in terms of the microstructural variability of thin tubes, including the brazed joint, using microstructure analysis and fractography. The thermo-mechanical responses of the thin tube and brazed joint were also investigated to clarify the fracture mechanism.

#### **2. Experiments**

An alloy 625 tube with a thickness of 0.135 mm and an outer diameter of 1.5 mm was prepared by repeated drawing and heat treatment processes after initial tubing via tungsten inert gas welding of a 0.2 mm thick strip roll. Figure 1 shows the geometry of the newly designed thin-tube fatigue specimen. As can be seen, the thin tube was connected to the grip (also made of alloy 625) using brazing, in the same way that a tube is brazed to a tube sheet in an actual heat exchanger. This type of fatigue specimen allows us to investigate the fatigue behavior of the thin tube, including the influence of the brazed joint. The filler metal used for brazing was BNi-2, containing boron and silicon as melting point depressants. The chemical compositions of the alloy 625 tube and filler metal are listed in Table 1.

**Figure 1.** Geometry of the thin-tube brazed fatigue specimen.



Before the fatigue tests, the microstructure of the brazed specimen was analyzed. The sample was polished to a 0.04 μM finish using colloidal silica and etched in 15 mL of HCl, 10 mL of acetic acid, and 10 mL of HNO3. The microstructures were then characterized using optical microscopy and scanning electron microscopy (SEM) equipped with energy dispersive spectroscopy (EDS). To measure the local mechanical properties, the hardness across the brazed joint was measured using a micro-Vickers hardness tester. The applied load was 4.903 N for the dwell time of 10 s.

Fatigue tests were performed using the MTS 810 system (MTS Systems Corp., Eden Prairie, MN, USA). The fatigue specimen shown in Figure 1 was connected to a ball-joint grip with a pin for the tilt- and twist-free alignment of the specimen along the loading direction. The applied load was measured using a 2 kN load cell (BCA-200K, TESTA Corp., Gyeonggi-do, Korea). Fatigue tests were performed under cyclic loading with a stress ratio (R) of 0.1 and a frequency of 10 Hz. Fatigue tests were performed at six different maximum stresses (*σ*max) for the two temperatures as follows: 495, 526, 557, 587, 618, and 649 MPa for room temperature (RT), and 355, 371, 386, 402, 418, and 433 MPa for 1000 K. Here, three to seven tests were conducted for each test condition. Fatigue test conditions are summarized in Table 2. After the tests, fractography was performed on the failed specimens to clarify the failure mechanisms and fatigue variability of the thin-tube brazed specimens.

**Table 2.** Fatigue test conditions used in this study.


#### **3. Results and Discussion**

Figure 2 shows the SEM images of the longitudinal and radial cross-sections of an alloy 625 thin tube prior to the fatigue test. The EDS analysis indicated that carbides of (Nb, Ti)C were present in the tube. The size of the carbides distributed outside the tube was larger than those distributed inside the tube. Carbide streaks along the drawing direction were also observed, which originated from the drawing process. Figure 3 shows the metallographs of the radial cross-section of the tube, including the weld zone in Figure 3b. As can be seen, the grain size appears to be finer in the weld zone than in the other regions (base metal). Despite the repeated annealing process after every drawing step, microstructural differences between the weld zone and the base metal are still observed. Therefore, the grain size distribution through the thickness is quite heterogeneous in the range of approximately 1–8 grains.

**Figure 2.** SEM images of an alloy 625 thin tube prior to the fatigue test: (**a**) longitudinal and (**b**) radial cross-sections.

**Figure 3.** Metallographs at the radial cross-section of the tube: (**a**) typical base metal region and (**b**) region including the weld zone.

Figure 4 shows the SEM images of the cross-section for the as-brazed fatigue specimen. Here, Figure 4b,c presents enlarged images of the areas labeled in Figure 4a. A typical brazing microstructure of Ni-base alloys is apparent, comprising various intermetallic phases [28,29]. An EDS analysis in an athermal solidification zone (Figure 4b) demonstrated that the phases marked by Z1, Z2, and Z3 are nickel boride, Ni–Si–B ternary intermetallic, and eutectic of γ-nickel and fine nickel silicide, respectively. The microstructure at the interface between the filler metal and the base metal shown in Figure 4c consisted of phases marked by Z4 and Z5, which are chromium boride, and the γ-nickel solid solution, respectively. In addition, the microhardness profile across the brazed joint was measured, as displayed in Figure 5. The microhardness in the tube was approximately 182 HV. In the filler metal region, however, the hardness increased to as high as 710 HV. This high hardness value is attributed to intermetallic constituents, including nickel boride, chromium boride, and nickel silicide, which are known for hard and brittle phases.

Fatigue tests were conducted at RT and 1000 K on the thin-tube brazed specimens, and the S–N curves are plotted in Figure 6. Each fatigue test condition shows different variabilities in the fatigue life (*N*f). For the RT fatigue tests, the maximum variability in *N*<sup>f</sup> was found at *σ*max = 495 MPa, which exhibited maximum and minimum *N*<sup>f</sup> values of 2,891,024 and 444,361 cycles, respectively. For the 1000 K fatigue tests, however, an even higher maximum variability in *N*<sup>f</sup> was found at *σ*max = 402 MPa, which showed maximum and minimum *N*<sup>f</sup> values of 781,877 and 31,909 cycles, respectively. In addition, such a variability in *N*<sup>f</sup> does not seem to show any particular relationship with the level of the maximum stress applied, particularly for the 1000 K fatigue tests. The fatigue strengths determined at *<sup>N</sup>*<sup>f</sup> = 1 × <sup>10</sup><sup>6</sup> cycles were 511 and 371 MPa at RT and 1000 K, respectively. The modified fatigue strengths with various standard deviations are listed in Table 3.

**Figure 4.** (**a**) SEM image of the brazed region; (**b**,**c**) enlarged SEM images of the areas marked in (**a**).

**Figure 5.** Microhardness profile across the brazed region.

**Figure 6.** S–N plots for all fatigue data tested at RT and 1000 K.


**Table 3.** Modified fatigue strength at 106 cycles with various standard deviations.

Because the thin-tube fatigue specimen has an unusual geometry, i.e., a thin tube brazed to the grip (Figure 1), further analysis was performed to clarify whether the brazed joint affected the variability of the fatigue life. In Figure 7, the fatigue fracture locations of all tested specimens are plotted as a function of *N*<sup>f</sup> for different applied stresses (*σ*max). Here, the fracture location (%) is the percentile fracture location relative to the distance from the grip section: 0% for fracturing at the brazed joint and 50% for fracturing in the middle of the thin tube. As can be seen, the fatigue fracture at RT was mainly near the brazed joint for high *σ*max (approximately *σ*max ≥ 557 MPa and *N*<sup>f</sup> < 400,000 cycles), while the fatigue fracture was either near the brazed joint or in the thin tube for low *σ*max (*N*<sup>f</sup> > 400,000 cycles). However, the fatigue fracture at 1000 K was mainly in the thin tube away from the brazed joint, regardless of the magnitude of *σ*max. This result implies that the presence of the brazed joint affects the fatigue life mainly for RT fatigue at a high *σ*max.

**Figure 7.** Variations of the fracture location as a function of fatigue life (*N*f) for different σmax values at (**a**) RT and (**b**) 1000 K.

To further investigate the relationship between the fracture location and the fatigue life at RT, fractography was performed for the two specimens, which showed a difference of approximately seven times in fatigue life even under the same test conditions (*σ*max = 495 MPa). Here, the short-life tube specimen fractured near the brazed joint at 444,361 cycles, whereas the long-life tube specimen fractured in the middle of the tube at 2,891,024 cycles, as shown in Figure 8a,b, respectively. Figure 9 shows the fractography of the two specimens tested at RT and *σ*max = 495 MPa. In Figure 9, the crack initiation sites are indicated by arrows. The fracture surfaces of the short-life specimen presented in Figure 9a,b show typical brittle fractures with multiple crack initiations (indicated by the arrow in Figure 9b) almost everywhere near the surrounding filler metal surfaces. These initial cracks circumferentially formed on the surfaces of the filler metal seem to propagate toward the inside of the thin tube, leading to premature fatigue failure. The gradual change in the fracture type from the transgranular quasi-cleavage fracture in the filler metal layer of the outer tube to the relatively dimpled fracture of the inner tube indirectly supports crack propagation. The microhardness profile displayed in Figure 5 indicates that the filler metal layer in the vicinity of the brazed joint is brittle, as expected, compared with the thin tube, owing to the presence of various intermetallic compounds (see Figure 4). Moreover, a certain degree of stress concentration is expected at the brazed joint owing to the geometric discontinuity (a notch effect) between the thin tube and the grip. The stress concentration factor of the fillet in the brazed joint can be calculated quantitatively using the factor of *k <sup>f</sup>* shown in Equation (1) [30]:

$$k\_f = 0.268 \left(\frac{D\_0}{r}\right) + 0.998\tag{1}$$

where *k <sup>f</sup>* is the stress concentration factor and *D*<sup>0</sup> and *r* the outer diameter and radius of fillet, respectively. By Equation (1), the *k <sup>f</sup>* is calculated at 1.40, which is well-coincident with S.H. Kang et al.'s study [31]. They verified the local mechanical response of alloy 625 brazed tubes with BNi-2 filler metal by considering the geometry and the local material properties of the brazed part, using a finite element method. Under these circumstances, the stress caused by cycling loading tends to be unevenly distributed in the brazed joint, and the incompatible deformation response of the filler metal layer (due to the presence of different intermetallic compounds) facilitate crack initiation on the surfaces of the filler metal near the brazed joint, causing a relatively short fatigue life.

**Figure 8.** (**a**) Short-life (444,361 cycles) and (**b**) long-life (2,891,024 cycles) tube specimens at RT and *σ*max = 495 MPa.

**Figure 9.** Fracture surfaces for (**a**,**b**) short-life (444,361 cycles) and (**c**,**d**) long-life (2,891,024 cycles) specimens tested at RT and σmax = 495 MPa.

In contrast to the short-life specimen, the long-life specimen, which fractured in the middle of the tube (Figure 8b), exhibited single crack initiation near the outer surface, as shown in Figure 9c,d. It can be seen that the initial crack formed a facet perpendicular to the loading direction (Figure 9d), which is typical for high-cycle fatigue. Unlike the short-life specimen, which displays simultaneous crack propagation from the filler metal layer of the outer tube to the inner tube, the long-life specimen shows a single crack propagating from the outer tube surface through the thickness, then spreading out to the neighboring area.

The fatigue lives at 1000 K also exhibited large variability, as shown in Figure 6. However, almost all fatigue fractures were observed in the region of the thin tube and not in the brazed joint, as shown in Figure 7b. Accordingly, fractography was performed for the two specimens, which showed a difference of approximately 10 times in fatigue life (*N*<sup>f</sup> = 781,887 and 78,621 cycles for the long-life and short-life specimens, respectively) at 1000 K and *σ*max = 402 MPa. As shown in Figure 10, both specimens displayed failure in the tube region away from the brazed joint. Figure 11 presents the fracture surfaces

for the short- and long-life specimens tested at 1000 K and *σ*max = 402 MPa. Regarding the short-life specimen, multiple crack initiations both at the outer surface and carbides inside the tube were observed, as indicated by the arrows in Figure 11b,c. In particular, cracks initiated at the outer surface (arrow in Figure 11b) showed progress through the tube thickness. Metallographs (Figure 12a,c) taken directly underneath the fracture surface presented in Figure 11a indicate that the grain size at the crack initiation site (marked with an arrow) is very large (only approximately two grains through the thickness). In addition, based on the non-uniform microstructure distribution compared to the surrounding area, the crack initiation site corresponds to the weld zone. Hence, during cyclic loading at 1000 K, the heterogeneous grain distribution can cause premature fracture, particularly in the weld zone.

**Figure 10.** (**a**) Short-life (78,621 cycles) and (**b**) long-life (781,887 cycles) tube specimens at 1000 K and σmax = 402 MPa.

**Figure 11.** Fracture surfaces for (**a**–**c**) short-life (78,621 cycles) and (**d**,**e**) long-life (781,887 cycles) specimens tested at 1000 K and σmax = 402 MPa. The arrows in (**a**,**b**,**d**,**e**) indicate the crack initiation sites. (**c**) presents the enlarged image of the area highlighted in (**b**).

**Figure 12.** Microstructures directly underneath the fracture surfaces fatigue-tested at 1000 K and σmax = 402 MPa for (**a**,**c**) short-life (78,621 cycles) and (**b**,**d**) long-life (781,887 cycles) specimens. The arrows indicate the locations of the fatigue crack initiation.

For the long-life specimen, a single crack initiation was observed near the outer surface, as shown in Figure 11d,e. Here, the initial surface crack is indicated by an arrow in Figure 11e. Metallographs (Figure 12b,d) obtained directly underneath the fracture surface presented in Figure 11d show homogeneous grain distribution over the tube. This indicates that, for the long-life specimen, if the inhomogeneity of the weld zone does not directly lead to fracture at the beginning of the fatigue test, the effect of grain size on the fatigue life is reduced owing to homogenization by long-term exposure at 1000 K.

Combining the results of the fatigue life variability for the alloy 625 thin-tube brazed specimens tested at RT and 1000 K, the following factors were found to affect the fatigue variability: the brazed joint (particularly, the filler metal layer at the joint) and the spatial distribution of the grain size and carbides. The presence of the brazed joint shown in Figure 1 (and the filler metal layer provided in Figures 4 and 5) can cause a notch stress concentration effect. Hence, the filler metal layer in the brazed part can act as a crack initiation site, particularly for the RT fatigue and under high *σ*max, because the various intermetallic phases in the filler metal layer, as well as the geometrical effect of the brazed part, cause local deformation incompatibility under cycling loading. The fatigue crack initiation in the filler metal layer (near the brazed joint) occurred at high *σ*max values (approximately ≥ 557 MPa) and resulted in short fatigue lives (*N*<sup>f</sup> < 400,000 cycles), as shown in Figure 7a. In this case, multiple cracks initiated circumferentially in the filler metal layer and propagated inward into the thin tube, as shown in Figure 9a,b. The largest fatigue life variability at RT was found when one specimen failed near the brazed joint, whereas the other specimen failed in the tube region, as shown in Figure 8, at *σ*max = 495 MPa, which seems to be in a transient stress range between the brazed joint failure and the tube failure (see Figure 7a). This result indicates that the presence of the brazed joint causes variability in fatigue life, particularly for low *σ*max values.

For 1000 K fatigue, however, no apparent brazed joint failure was observed, as shown in Figure 7b. This is because the deformation incompatibility among different intermetallic phases in the filler metal layer was fully accommodated (even under cycling loading) at such a high temperature [32]. In this case, the spatial distribution of the grain size and carbides seems to affect the fatigue life variability. The presence of a large near-surface grain (corresponding to the weld zone), which has approximately 1–2 through-thickness grains and can facilitate the initiation of a fatigue crack near the outer surface, as shown in Figure 11b,c, leads to a short fatigue life. In particular, the fatigue life will be even shorter if multiple crack initiations at carbides inside the tube occur simultaneously, as shown in Figure 11b,c, in addition to the crack initiation at a large near-surface grain.

#### **4. Conclusions**

Variability in fatigue life at room temperature and 1000 K was investigated for brazed alloy 625 thin-tube specimens. The fatigue life variability was found to be influenced by the presence of the brazed joint (and its properties), as well as the spatial distribution of the grain size and carbides.

At room temperature, a correlation between the fracture location and fatigue life was observed. Specimens tested at σmax ≥ 557 MPa exhibited failure near the brazed joint and relatively short fatigue lives (typically, *N*<sup>f</sup> < 400,000 cycles). For σmax < 557 MPa, however, a short-life specimen failed at the brazed joint, whereas a long-life specimen failed in the middle of the tube. The brazed-joint failed specimens showed multiple crack initiations circumferentially in the filler metal layer and growth of cracks through the thickness of the tube, leading to a short fatigue life.

At 1000 K, all test specimens failed in the middle of the tube. Specifically, the short-life specimen showed fatigue crack initiation and growth at a location with only 1–2 throughthickness grains. Crack growth seemed to be further facilitated by multiple crack initiations at the carbides inside the tube. In conclusion, homogeneous grain distribution within the tube and small grains through the tube thickness can prevent premature fracture, leading to a long fatigue life.

**Author Contributions:** Research conceptualization and design, Y.S.C. and S.P.; evaluation of the mechanical properties, H.K.; microstructure analysis, S.L.; data analysis and writing S.L. and H.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20193310100050) and by a two-year research grant from Pusan National University.

**Data Availability Statement:** Not applicable.

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

#### **References**


## *Article* **Fatigue Behavior of Nonreinforced Hand-Holes in Aluminum Light Poles**

**Cameron R. Rusnak 1,\* and Craig C. Menzemer <sup>2</sup>**


**Abstract:** Hand-holes are present within the body of welded aluminum light poles. They are used to provide access to the electrical wiring for both installation and maintenance purposes. Wind is the main loading on these slender aluminum light poles and acts in a very cyclic way. In the field, localized fatigue cracking has been observed. This includes areas around hand-holes, most of which are reinforced with a cast insert welded to the pole. This study is focused on an alternative design, specifically hand-holes without reinforcement. Nine poles with 18 openings were fatigue tested in four-point bending at various stress ranges. Among the 18 hand-holes tested, 17 failed in one way or another as a result of fatigue cracking. Typically, fatigue cracking would occur at either the 3:00 or 9:00 positions around the hand-hole and then proceed to propagate transversely into the pole before failure. Finite element analysis was used to complement the experimental study. Models were created with varying aspect ratios to see if the hand-hole geometry had an effect on fatigue life.

**Keywords:** aluminum hand-hole; nonreinforced hand-hole; fatigue test; design S-N curve; high cycle fatigue

#### **1. Introduction**

Aluminum light poles support overhead light fixtures and are used to are illuminate sidewalks, roadways, parking lots, recreational areas, and others. This is due to its light weight, resistance to corrosion, high strength to weight ratio, and ease of handling and joining. Wind is the main contribution to loading to these light poles, which can be classified as slender structures. Fatigue cracking can occur in either steel or aluminum when exposed to any kind of repeated loads. In these aluminum light poles, electrical wires will run through conduit, into the hollow section of the pole, and then proceed up into the light [1,2].

Stress concentrations occur when there are changes within the cross-sectional area of a structural member, examples of which include connections, copes, keyways, cutouts, and others. In modern fatigue design, specifications will use the lower bound S-N curve established from full-scale test data as it will be the first to fail [3,4]. A series of S-N curves represents a ranking of the stress concentration condition that is associated with different mechanical and structural details. One way to increase fatigue life would be to minimize or eliminate abrupt changes in the cross-section or provide smooth, gradual transitions. In aluminum light poles, there are multiple structural details of interest. These include the base to pole connection, mono-tube or truss arm joints truss, and the hand-holes used for electrical access. The behavior of the fatigue in these electrical access hand-holes within these aluminum light poles is largely unknown. The majority of the existing data that have been collected were from large, welded steel poles [5].

Report number 176 from NCHRP (National cooperative Highway Research Program) web only found results of unreinforced and reinforced hand-hole fatigue tests for welded

**Citation:** Rusnak, C.R.; Menzemer, C.C. Fatigue Behavior of Nonreinforced Hand-Holes in Aluminum Light Poles. *Metals* **2021**, *11*, 1222. https://doi.org/10.3390/ met11081222

Academic Editor: Dariusz Rozumek

Received: 6 July 2021 Accepted: 28 July 2021 Published: 30 July 2021

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

**Copyright:** © 2021 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/).

steel structures. Lehigh University studied detail associated with steel light poles under fatigue. During these experiments, 13 of the specimens had handholes with different geometries (as compared with aluminum poles). During testing, zero of the hand-holes failed or cracked. A finite element study was conducted and was used to provide an estimate of how the stress concentration around the pole and hand-hole. On the basis of the analysis, the research found the fatigue resistance of both the unreinforced and reinforced hand-holes to align with AASHTO (American Association of State High and Transportation Officals) Category E of the design S-N curves [6].

Field observations have shown that some hand-hole details are susceptible to failure due to fatigue cracking. NCHRP report number 469 [5] described these fatigue cracks of welded steel structures near the hand-hole in multiple states. These included California, New Mexico, New York, and Minnesota. Subsequent inspections of different poles occurred after the failure of a high mast light pole in Iowa. It was found that another tower had crack associated with the hand-hole. Cracks were found in some of the aluminum light poles along the Mullica River Bridge after a violent storm in 2011 [7]. A picture of one of these failures was taken and can be seen in Figure 1.

**Figure 1.** Aluminum light pole containing a fatigue crack within the field.

At the University of Akron, a study was conducted on 20 light-pole specimens, with fatigue tests conducted under bending loads. In addition to fatigue test, several static ones were conducted in order to see how the strain distributed around the hand-hole. This study found that the data from the welded hand-hole fatigue tests fell above the category D and E design S-N curves of the Aluminum Design Manual [8]. Another study found that the change in diameter of the pole has a modest effect on the fatigue life. In this companion study, seven eight-inch poles with 14 details were tested [9].

Nine aluminum light poles, each containing two separate hand-holes, were tested in fatigue under four-point bending. All poles were supplied to the University of Akron and were manufactured to standards typical for the industry. Finite element models were created to help improve the understanding of the stress concentrated around the hand-holes. The models created had different aspect ratios.

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

#### *2.1. Pole Geometry and Material Properties*

Nine aluminum light poles were tested under cyclic loading to examine the behavior of the unreinforced hand-hole. These specimens consisted of a 10 in (25.4 cm) diameter aluminum alloy extruded tube with a 1/4 in (0.635 cm) thick wall. Each of the tubes was fabricated from aluminum alloy 6063. Typically, there is a reinforcement welded into the hand-hole opening, but these tests used only open holes (Figure 2). Hand-holes measured 6 in (150 mm) along the length of the pole in the longitudinal direction of the pole and 4 in (100 mm) in the transverse direction. Each specimen was 144 in (3.66 m), or 12 feet in length, with the hand-holes placed 54 in (1.37 m) in from either end. Support rollers for the specimens were inserted 6 in (15.2 cm) from each end [10].

**Figure 2.** Typical geometry of a welded aluminum hand-hole detail in four-point fatigue testing. (In this study there was no welded detail). (L = 3.66 m (144 in); D = 25.4 cm (10 in); a = 1.37 m (54 in); b = 15.2 cm (6 in); e = 150 mm (6 in); f = 100 mm (4 in)).

Table 1 is a summary of the minimum mechanical properties of the aluminum tube.

**Table 1.** Mechanical properties of the aluminum tube.


#### *2.2. Fatigue Tests*

Figure 3 depicts a photo of the four-point bending fatigue test setup in the lab. During testing, a 55 kips (245 KN) MTS servo-hydraulic actuator (MTS Systems Headquarters, Eden Prairie, MN 55344, USA) along with a control system was used to apply the loads to the specimens. The actuator itself was mounted to a load frame capable of safely supporting 300 kips (1335 KN). Loads were applied to each of the specimens through a spreader beam that was attached to the hydraulic system. The supports the specimens rested on consisted of rollers that were machined to fit the cylindrical profile of the specimens.

**Figure 3.** Four point bending fatigue test set-up.

Testing was conducted with a load control, while the strains were monitored using gages that were applied around the hand-holes, along with a single gage placed in the middle of the specimen. The typical location of the strain gages can be seen in Figure 4. Strain gages had a resistance of 350 ohms and were 1/8 in (3.175 mm) in length. Strain readings were taken every two hours intermittently for 10 seconds using a Micro-Measurements System 8000 data acquisition device (Micro-Measurements A VPG Brand Raleigh, NC 27611, USA) that was wired to the strain gages.

**Figure 4.** Typical strain gage location and position around the hand-holes.

All of the specimens were oriented with the hand-holes facing downwards direction so that they were in tension during cyclic testing. Failure was achieved when the hand-hole region was cracked to the point that the loading on the specimen could no longer be supported. A maximum displacement was placed on the specimens for each of the fatigue tests to ensure that both hand-holes could be tested. The upper limit was set 10% larger than the maximum static displacement. When this maximum displacement was exceeded, the test would automatically shut down. Once this limit was reached and the test was stopped, the detail that failed was reinforced with a moment clamp. An image half of the moment clamp can be seen in Figure 5. Two halves were placed around the handhole and mechanically secured. The test was restarted and continued until the other hand-hole failed. With the specimen being loaded in four-point bending, the moment and hence applied stress does not change at the undamaged handhole after the moment clamp was applied. Of the nine specimens tested, only one resulted in a catastrophic failure where repair of the detail was not possible.

**Figure 5.** Half of the moment clamp used to reinforce the hand-hole after initial failure.

Nine poles, each with two hand-holes, were tested at stress ranges between 17.24 MPa (2.5 ksi) and 58.61 MPa (8.5 ksi). Figure 6 depicts a sketch of where the strain gages were installed adjacent to the hand-hole. Strain gages were placed at the 3 and 9 o'clock positions, with the addition of a strain gage in the middle of the specimen. This gage was within 2 to 3 times the tube thickness away from the edge of each hand-hole. All of the strain gages were wired to the data acquisition system to measure the applied strains. Five specimens were cycled at 1 Hz and four were cycled at 2 Hz. Testing continued around the clock. Visual inspections of the hand-holes were conducted daily.

**Figure 6.** The position of strain gages installed around a hand-hole.

#### *2.3. Finite Element Models*

The finite element (FE) model was created for the four-point bending specimens in an attempt to gain a better understanding of the stress distribution around and adjacent to the hand-holes. All modeling was completed within Abaqus CAE (2018 version, Troy, MI 48084, USA). The material model was a general, linear elastic material with only the modulus of elasticity and Poisson's ratio specified. The mechanical response and the influence of geometry on local stresses was the primary concern and focus of the FE analysis. As such, an advanced material model was not needed. Models were classified as "shell" models. Loading was applied by selecting the outermost nodes on the transverse plane where the rollers made contact with the pole. In the model, 365 individual nodes were selected, with a concentrated load of 0.01926 N of force applied to each. A total of 7.03 N was applied

to the tube where the rollers were located. Multiple models were created with different aspect ratios. These included 2-1, 1.75-1, 1.5-1, 1-1, 1-1.25, and 1-1.5. The aspect ratio was calculated by dividing length of the hand-hole by the transverse dimension of the handhole. The purpose of the analyses was to determine whether a change in the aspect ratio had any effect on the local stress distribution around the hand-hole. Local stresses were mesh-dependent for this study. A finer mesh size typically increases the stresses local to important geometric details, whereas a more course mesh often results in a reduction in local stresses. The elements in the model consisted of a mix of both hexahedral and tetrahedral element types. Figure 7 depicts the mesh of the 1.5-1 model. In the field, the hand-holes with reinforcement have an aspect ratio of 1.5-1.

**Figure 7.** Overall FE model of 1.5-1.

#### **3. Results**

*3.1. Fatigue Tests Results*

Of the 18 hand-holes tested, 17 failed, with the results shown in Figure 8.

**Figure 8.** Fatigue test results.

All of the fatigue test data appears to follow the lower bound "E detail" S-N curve, even though the handholes themselves were not reinforced. Additional data would be needed to establish a lower bound for the unreinforced handholes but would be expected to be lower than Category E. Figure 9 shows a comparison between no insert specimens and a previous study conducted on reinforced hand-holes in 10 in diameter poles. This figure clearly demonstrates the benefit of having the welded reinforcement around the hand-holes [11].

**Figure 9.** No insert detail vs. old 10 in data.

During fatigue testing, cracks were observed at either the 3:00 or 9:00 position along the minor axis of the hand-hole. Typically, these cracks would initiate and then propagate transversely into the pole from the point of origin. Cracks would become visible and quickly progress into the pole, followed by failure. Figure 10 depicts a fatigue crack at the 3:00 position.

**Figure 10.** Fatigue crack through pole.

#### *3.2. Finite Element Results*

"Hot spots" are generally as local areas with elevated stresses and provide an indication where fatigue cracks may develop. Figures 11–16 depict the stress contour maps along the Z-axis (longitudinal stress) for handholes with different aspect ratios. In all cases, hot spots were most prevalent at either the 3:00 or 9:00 position. The stress concentrations make sense due to how the loading is applied and how the hand-holes themselves are simply an opening within the specimen. The maximum stress increased as the aspect ratio changed from 2–1 to 1–1.5.

**Figure 11.** Aspect ratio 2-1 in longitudinal (Z) direction.

**Figure 12.** Aspect ratio 1.5-1 in longitudinal (Z) direction.

**Figure 13.** Aspect ratio 1.25-1 in longitudinal (Z) direction.

**Figure 14.** Aspect ratio 1-1 in longitudinal (Z) direction.

**Figure 15.** Aspect ratio 1-1.25 in longitudinal (Z) direction.

**Figure 16.** Aspect ratio 1-1.5 in longitudinal (Z) direction.

Figures 17–22 show the maximum transverse stresses along the X-axis (transverse stress) around the hand-holes with different aspect ratios. The transverse stresses follow the same pattern as the longitudinal, with the 1-1.5 aspect ratio having the largest local stresses and the 2-1 having the mildest. These figures also show how stress accumulates inside of the pole apart from the hand-hole opening itself.

**Figure 17.** Aspect ratio 2-1 in transverse (X) direction.

**Figure 18.** Aspect ratio 1.5-1 in transverse (X) direction.

**Figure 19.** Aspect ratio 1.25-1 in transverse (X) direction.

**Figure 20.** Aspect ratio 1-1 in transverse (X) direction.

**Figure 21.** Aspect ratio 1-1.25 in transverse (X) direction.

**Figure 22.** Aspect ratio 1-1.5 in transverse (X) direction.

The maximum longitudinal stress was plotted against the aspect ratio in order to gain a better understanding of how the aspect ratio affects the local stress. The area was taken along the inside of the hand-hole at the 3:00 position. This location was chosen as this spot contained some of the largest stresses (Figure 23). Table 2 accompanies Figure 23 and provides not only longitudinal stresses, but transvers as well. Table 2 shows that the transverse stress was negligible at the location of interest, near the 3:00 position.

**Figure 23.** Max longitudinal stress vs. aspect ratio.


**Table 2.** Longitudinal and transverse stress at the 3 o'clock position for the different aspect ratios.

#### **4. Conclusions**

Four-point bending fatigue tests were conducted on aluminum light pole containing no reinforcement. Tests revealed that there was a negative effect on the fatigue life when the cast reinforcement was removed. This was most evident when the stress range was higher. This can be seen from Figure 9 from Section 3. A total of nine tests were conducted, resulting in 17 data points, stressed at various different stress ranges. Finite element models showed how different hand-hole aspect ratios affect the stress concentration. While a handhole aspect ratio of 2-1 may provide the lowest local longitudinal stress, it may not be the most practical out in the field. It is unknown how a hand-hole with an aspect ratio of 2-1 would behave if reinforced.

**Author Contributions:** For the duration of, C.R.R. did the conceptualization, methodology, software utilization, validation, formal analysis, investigation, resources, data curation, writing of the original draft, visualization with C.C.M. writing in the form of reviewing and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by HAPCO Light Pole Products.

**Data Availability Statement:** Not applicable.

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

#### **References**


## *Article* **Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results**

**Kenichi Ishihara 1,\*, Hayato Kitagawa 1, Yoichi Takagishi <sup>1</sup> and Toshiyuki Meshii <sup>2</sup>**


**Abstract:** Analyzing the structural integrity of ferritic steel structures subjected to large temperature variations requires the collection of the fracture toughness (*KJ*c) of ferritic steels in the ductile-tobrittle transition region. Consequently, predicting *KJ*<sup>c</sup> from minimal testing has been of interest for a long time. In this study, a Windows-ready *KJ*<sup>c</sup> predictor based on tensile properties (specifically, yield stress σYSRT and tensile strength *σ*BRT at room temperature (RT) and *σ*YS at *KJ*<sup>c</sup> prediction temperature) was developed by applying an artificial neural network (ANN) to 531 *KJ*<sup>c</sup> data points. If the *σ*YS temperature dependence can be adequately described using the Zerilli–Armstrong *σ*YS master curve (MC), the necessary data for *KJ*<sup>c</sup> prediction are reduced to *σ*YSRT and *σ*BRT. The developed *KJ*<sup>c</sup> predictor successfully predicted *KJ*<sup>c</sup> under arbitrary conditions. Compared with the existing ASTM E1921 *KJ*<sup>c</sup> MC, the developed *KJ*<sup>c</sup> predictor was especially effective in cases where *σ*B/*σ*YS of the material was larger than that of RPV steel.

**Keywords:** fracture toughness; machine learning; artificial neural network; predictor; yield stress; tensile strength; specimen size

#### **1. Introduction**

Both researchers and practitioners have characterized the fracture toughness (*KJ*c) of ferritic steels in the ductile-to-brittle transition (DBT) region, which is key for analyzing the structural integrity of cracked structures subjected to large temperature changes. *KJ*<sup>c</sup> is associated with (I) a large temperature dependence (a change of approximately 400% corresponding to a temperature change of 100 ◦C) [1–10]; (II) specimen-thickness dependence (roughly, *KJ*<sup>c</sup> ∝ 1/(specimen thickness)1/4) [8,11–21]; and (III) large scatter (approximately ±100% variation around the median value) [8,22,23]. Thus, understanding these three effects is necessary for efficient *KJ*<sup>c</sup> data collection.

Since Ritchie and Knott introduced the idea of using critical stress and distance to predict fracture toughness temperature dependence [4], researchers who explicitly or implicitly applied this idea have obtained results that demonstrate a strong correlation between the temperature dependence of fracture toughness and that of yield stress (*σ*YS) [5,6]. Wallin observed that the increase in fracture toughness with increasing temperature is not sensitive to steel alloying, heat treatment, or irradiation [7]. This observation led to the concept of a universal curve shape that applies to all ferritic steels, i.e., the difference in materials is reflected by the temperature shift. This concept is now known as the master curve (MC) method, as described by the American Society for Testing and Materials (ASTM) E1921 [8]. The existence of a *KJ*<sup>c</sup> MC was physically supported by Kirk et al. based on dislocation mechanics considerations [9,10]. They argued that the temperature dependence of *KJ*<sup>c</sup> is related to the temperature dependence of the strain energy density (SED). Furthermore,

**Citation:** Ishihara, K.; Kitagawa, H.; Takagishi, Y.; Meshii, T. Application of an Artificial Neural Network to Develop Fracture Toughness Predictor of Ferritic Steels Based on Tensile Test Results. *Metals* **2021**, *11*, 1740. https://doi.org/10.3390/ met11111740

Academic Editor: Dariusz Rozumek

Received: 14 October 2021 Accepted: 26 October 2021 Published: 30 October 2021

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

**Copyright:** © 2021 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/).

because all steels with body-centered cubic (BCC) lattice structures exhibit a unified *σ*YS temperature dependence, as described by the Zerilli–Armstrong (Z–A) constitutive model (i.e., Z–A *σ*YS MC) [24], the existence of a BCC iron lattice structure is the sole factor needed to ensure that *KJ*<sup>c</sup> in the DBT region has an MC. Note that Kirk et al. implicitly assumed that the tensile-to-yield stress ratio does not vary with materials, which is not true, and will be a source of deviation from the MC. For example, the failure of this MC to evaluate increases in *KJ*<sup>c</sup> at high temperatures has been reported for non-reactor pressure vessel (RPV) steels [25,26]. Despite the successful application of *KJ*<sup>c</sup> MC to RPV steels, a reexamination of the basis of *KJ*<sup>c</sup> MC existence and additional application limits must be reexamined for the application of ASTM E1921 MC to ferritic steels in general and not be limited to RPV steels.

The size dependence of *KJ*<sup>c</sup> has been understood based on the weakest link theory deduced as *KJ*<sup>c</sup> ∝ 1/(specimen thickness)1/4 [17], but because this relationship cannot describe the existence of a lower-bound *KJ*<sup>c</sup> for large specimens, researchers have begun to investigate the size dependence of *KJ*<sup>c</sup> as the critical stress distribution ahead of a cracktip requires a second parameter in addition to *J* (*J*-*A*, *J*-*T* approach, etc.) [18,19], which is categorized as a crack-tip constraint issue. Consequently, it appears that the development of a deterministic and data-driven size effect formula is possible. ASTM E1921 provides a semiempirical size effect formula based on the *KJ*<sup>c</sup> of a 1-inch-thick specimen, which considers a lower-bound *KJ*<sup>c</sup> of 20 MPa·m1/2 and proportionality to 1/(specimen thickness)1/4. There are various opinions regarding this lower-bound value [27–30]; thus, the establishment of a data-driven size effect formula that does not depend on the ∝ 1/(specimen thickness)1/4 relationship seems possible and necessary.

The statistical nature of fracture toughness has been modeled using the Weibull distribution; some researchers used stress [22] and some used *KJ*<sup>c</sup> [8] as the model mean parameter. The idea of using Weibull distributions stems from the understanding that the cleavage fracture can be modeled using the weakest link theory. ASTM E1921 [8] applies a three-parameter Weibull distribution, which assumes a shape parameter of four and a position parameter of 20 MPa·m1/2. The failure of this model to predict the scatter in *KJ*<sup>c</sup> has also been reported; Weibull parameters (shape and position) vary as functions of the specimen size and temperature, and the parameters differ from those specified in ASTM E1921 [31,32]. If the observed model parameters differ from the assumed parameters, the predicted *KJ*<sup>c</sup> and scatter deviate from the measured values. Hence, a more practical method that can potentially prevent the mismatch of the assumed statistical model, i.e., a data-driven approach, is necessary.

Considering the three aforementioned issues, it was considered that a data-driven *KJ*<sup>c</sup> predictor that captured features of a variety of BCC metals could improve *KJ*<sup>c</sup> prediction accuracy. Another idea was to replace time- and material-consuming fracture toughness tests with tensile tests, assuming that *KJ*<sup>c</sup> has a direct relationship with SED obtained via tensile tests. Thus, the artificial neural network (ANN) approach was applied to 531 *KJ*<sup>c</sup> data collected in our previous works [30,33] to construct a *KJ*<sup>c</sup> predictor based on tensile test properties, thereby eliminating the need to conduct fracture toughness tests. The data were obtained for five heats of RPV and seven heats of non-RPV steels. The widths *W* of the specimens ranged from 20 to 203.2 mm, and the thickness-to-width ratio *B*/*W* was limited to 0.5 (i.e., data obtained with PCCV specimens of *B*/*W* = 1 were excluded). As a result, a Windows-ready *KJ*<sup>c</sup> predictor, which enables *KJ*<sup>c</sup> prediction by giving specimen size, tensile and yield stress, was developed. Time- and material-consuming fracture toughness tests are no more necessary.

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

#### *2.1. Selection of Machine Learning Model*

Machine learning models are used in many fields, such as search engines, image classification, and voice recognition, and various methods have been proposed according to the application. In this study, a tool to predict the fracture toughness *KJ*<sup>c</sup> of a material under arbitrary conditions such as the specimen size and temperature, without performing the fracture toughness test, was conducted; this is treated as a regression issue. There are various algorithms for machine learning models for regression. In this study, a multilayer perceptron (MLP) was classified into an ANN that can express complex nonlinear relationships. The regression model was constructed using the MLP regressor, which is a scikit-learn library of the general-purpose programming language Python [34].

#### *2.2. Overview of Multilayer Perceptron in an Artificial Neural Network*

Figure 1 shows a schematic diagram of the MLP network. The MLP is a hierarchical network comprising an input layer, a hidden layer, and an output layer; the unit of the hidden layer is completely connected to the input and output layers [34,35].

**Figure 1.** Schematic diagram of multilayer perceptron in an ANN [35].

In Figure 1, only one hidden layer is schematically shown; however, in general, multiple hidden layers are used to enhance the expressiveness of the model. The unit in the hidden layer (hereinafter, referred to as the activation unit *aj* (*j* = 1~*k*)) is calculated using Equation (1), where *n* input values are *Xi* and the output values are *f*(*X*).

$$a\_j = \phi\left(\sum\_{i=0}^n w\_{j,i}^h X\_i\right) \tag{1}$$

Here, *w*<sup>h</sup> *<sup>j</sup>*,*<sup>i</sup>* is the connection weight, *X*<sup>0</sup> is a constant called bias, and *φ* of Equation (1) is a function called the activation function. For the activation function, a function with differentiable nonlinearity was selected to enhance the expressiveness of the model. In this study, the rectified linear unit (ReLU) function *φ*(*z*) = max(0, *z*) was used and *aj* was assigned to the hidden layer. The total number *k* of *aj* (the number of nodes in the hidden layer) and the number of hidden layers are parameters that were adjusted according to the learning accuracy. The output value *f*(*X*) can be obtained via Equation (2).

$$f(X) = \phi\left(\sum\_{j=0}^{k} w\_j^{\diamond} a\_j\right),\tag{2}$$

where *w*<sup>o</sup> *<sup>j</sup>* denotes the connection weight. In Equations (1) and (2), the connection weights *w*h *j*,*i* , *w*<sup>o</sup> *<sup>j</sup>* are unknown constants and can be obtained from the combination of known input and output values. By assuming that the known teaching data (true value) are *Y* (to distinguish it from *f*(*X*), predicted from the input value *Xi* from Equation (2)), the connection weights can be updated in Equation (3), using the loss function *E*.

$$E = \frac{1}{2} \sum\_{l} \left( Y\_l - f\_l(X) \right)^2 + \frac{\alpha}{2} \sum\_{l} |w\_l^o|^2 \tag{3}$$

Here, the first term in Equation (3) is the sum of the squared residuals of the teaching data *Y* and the output value *f*(*X*), and the second term is a regularization term using the *L*<sup>2</sup> norm to suppress overfitting. *α* is a parameter that is adjusted according to learning accuracy. Overfitting is a problem in which training data are overfitted and unknown data cannot be effectively generalized. Several effective optimization algorithms have been developed to avoid falling into a locally optimal solution for updating the connection weights. In this study, adaptive moment estimation (Adam) [36] was used. The connection weight *w* is updated using Equations (4)–(9).

$$\mathcal{W}^{(t)} = w^{(t-1)} - \eta \frac{m^{\hat{\jmath}\_{(t)}}}{\sqrt{\upsilon^{\hat{\jmath}\_{(t)}}} + \varepsilon} \tag{4}$$

$$m^{\hat{\ell}(t)} = \frac{m^{(t)}}{1 - \beta\_1^t} \tag{5}$$

$$v^{\hat{\ell}^{(t)}} = \frac{v^{(t)}}{1 - \beta\_2^t} \tag{6}$$

$$m^{(t)} = \beta\_1 m^{(t-1)} + (1 - \beta\_1) \frac{\partial E}{\partial w} \tag{7}$$

$$
\sigma^{(t)} = \beta\_2 \upsilon^{(t-1)} + (1 - \beta\_2) \left(\frac{\partial E}{\partial w}\right)^2 \tag{8}
$$

$$m^{(0)} = v^{(0)} = 0\tag{9}$$

The recommended values were used for the adjustment parameters *η*, *β*1, *β*2, and [36]. The error backpropagation method to update the connection weight was used, which calculates the gradient of the loss function by moving backward from the output layer. This method is known to be less computationally expensive than updating weights in the forward direction [37].

#### *2.3. Goodness Valuation of Constructed Learning Model*

The goodness of valuation of the constructed machine learning model is based on the coefficient of determination *R*<sup>2</sup> in Equation (10), where *n* is the amount of teaching data, *Yi* is the true objective value, *f*(*X*) is the predicted objective value, and the average value of the true objective values is *σ*Y.

$$R^2 = 1 - \frac{\sum\_{i} \left(\boldsymbol{\chi}\_{i} - f\_{i}(\boldsymbol{X})\right)^2}{\sum\_{i} \left(\boldsymbol{\chi}\_{i} - \boldsymbol{\mu}\_{\boldsymbol{Y}}\right)^2} \tag{10}$$

The coefficient of determination indicates the goodness of fit of the regression model and is an evaluation index for assessing how well the predicted and true values match. *R*<sup>2</sup> = 1 when the true and predicted values are the same. There is no clear standard for the coefficient of determination, but it can be considered compatible if it is approximately 0.5 or more.

#### *2.4. Dataset*

For machine learning, the fracture toughness test data of 531 ferritic steels in the DBTT region obtained by the authors or previous studies were used. Table 1 presents the chemical compositions of the test specimens of the materials considered in the teaching data.


**Table 1.** Chemical compositions of the test specimens (wt %) of the considered materials.

Tables 2–4 summarize the material heats (heat No. 1–12) used in this study, *nT* indicates the specimen thickness, and *n* is expressed in multiples of 25 mm. They are fundamentally extracted from previous work [30,33], but differ slightly in terms of the following: (1) *KJ*<sup>c</sup> > *KJ*c(ulimit) invalid data were excluded, (2) *KJ*<sup>c</sup> data were limited to cases obtained with standard specimens of thickness-to-width ratio *B*/*W* = 0.5, (3) When there were no *σ*YS data for the fracture toughness test temperature, it was obtained by using the following modified Z–A *σ*YS temperature-dependent MC [9]

$$\sigma\_{\rm OZA}(T) = \sigma\_{\rm OkT} + \mathcal{C}\_1 \exp\left[ \left( T + 273.15 \right) \left( -\mathcal{C}\_3 + \mathcal{C}\_4 \log \left( \dot{\varepsilon} \right) \right) \right] - 49.6 \text{ (MPa)}, \tag{11}$$

where *T* is the temperature (◦C), *C*<sup>1</sup> = 1033 (MPa), *C*<sup>3</sup> = 0.00698 (1/K), *C*<sup>4</sup> = 0.000415 (1/K), and . *ε* = 0.0004 (1/s). The three Miura heats (heat No. 1, 4, 5) were another exception for which linear interpolation of raw data was used because the fracture toughness and tensile test temperatures were different.



**Table 3.** *KJ*<sup>c</sup> data used to construct the proposed tensile property-based *KJ*<sup>c</sup> MC: RPV steel ASTM A533B and equivalent.

\*: Side-grooved specimens.

**Table 4.** *KJ*<sup>c</sup> data used to construct the proposed tensile property-based MC: non-RPV steels.


The objective variable was *KJ*c. Assuming a direct relationship between the SED temperature dependence and that of *KJ*c, *σ*<sup>B</sup> temperature dependence was the first candidate explanatory parameter. However, considering that (i) *σ*B/*σ*YS temperature dependence is small, (ii) ferritic steel has a *σ*YS temperature-dependent MC such as Z−A MC, and (iii) *σ*B/*σ*YS at RT is usually easily available, *σ*<sup>B</sup> and *σ*YS at RT, and *σ*YS at *KJ*<sup>c</sup> test temperatures and specimen width *W* were selected as the explanatory variables. To optimize the connection weight, 371 points, i.e., 70% of the 531 points in the known dataset, were used as the training data. The data were divided by "train\_test\_split" of Python's scikit-learn library. If the digits of the input value and output value to be learned are significantly different, the influence of variables with small digits may not be fully considered in learning. Therefore, in this study, the input values *W*, *σ*YS, *σ*YSRT, *σ*BRT, and output value *KJ*<sup>c</sup> were standardized, as shown in Equation (12).

$$
\begin{pmatrix}
\text{W (mm)} \\
\sigma\_{\text{YS (MPa)}} \\
\sigma\_{\text{Y8RT (MPa)}} \\
\sigma\_{\text{BRT (MPa)}} \\
K\_{\text{fc}} \text{ (MPa} \cdot \text{m}^{1/2})
\end{pmatrix}\_{\text{Normalized}} \begin{pmatrix}
\text{W}/50 \\
\sigma\_{\text{YS}}/550 \\
\sigma\_{\text{Y8RT}}/550 \\
\sigma\_{\text{BkT}}/550 \\
K\_{\text{fc}}/100
\end{pmatrix} \tag{12}
$$

Here, with reference to ASTM E1921, *W* was normalized using the width 50 mm of a 1T specimen, and the yield stress and tensile strength were normalized using the average value of 550 MPa of the yield stress of 275 to 825 MPa in the allowable temperature range targeted by the standard. *KJ*<sup>c</sup> was normalized to a fracture toughness of value 100 MPa·m1/2 at the reference temperature.

#### *2.5. Fracture Toughness Prediction by the Constructed Learning Model*

Table 5 presents a list of hyperparameters used for the machine learning model in this study. Using the data in Tables 2–4 and the parameters in Table 5, which is currently an invariant model, the coefficient of determination *R*<sup>2</sup> of the developed *KJ*<sup>c</sup> predictor was 0.61 for the training data and 0.53 for the test data. Table 6 presents the explanation variables for predicting fracture toughness *KJ*c.


**Table 5.** Hyperparameters used for the learning model.

**Table 6.** Explanatory variables for case studies applied to the developed tool.


The input data (*W*, *σ*YS, *σ*YSRT, *σ*BRT) for the developed *KJ*<sup>c</sup> predictor and output window after its execution (the coefficient of determination *R*<sup>2</sup> and the predicted *KJ*c) are shown in Figure 2. In Figure 3, the comparison of *KJ*<sup>c</sup> of ASTM E1921 MC and predicted *KJ*<sup>c</sup> by the predictor is shown. In Figure 3, the horizontal axis is *T*, the vertical axis *KJ*c(1T) is the test data, and the predicted *KJc* is converted to 1T thickness. The *KJ*<sup>c</sup> of the ASTM E1921 MC is plotted as a black solid line, the *KJ*<sup>c</sup> of the test data are plotted as open black symbols, and the predicted conditions listed in Table 6 are plotted as open red symbols. In Figure 3a, for RPV steel, both the *KJ*<sup>c</sup> by the ASTM E1921 MC and the predicted *KJ*<sup>c</sup> by this model are in agreement with the test results. However, in Figure 3b for SCM440, although the *KJ*<sup>c</sup> by the ASTM E1921 MC significantly differs from the test results at high temperatures, the predicted *KJ*<sup>c</sup> values by this model are in agreement with the test results.

(**a**) (**b**)

**Figure 3.** Comparison of *KJ*<sup>c</sup> of ASTM E1921 MC (solid line) and predicted *KJ*<sup>c</sup> by the predictor (open red symbols): Dataset used for training model and result of predicted *KJ*c. (**a**) RPV steel (Miura SFVQ1A); (**b**) Meshii FY2017SCM440. *KJ*<sup>c</sup> pre-dicted by the developed predictor accurately predicted *KJ*<sup>c</sup> regardless of materials.

#### **3. Discussion**

By applying the ANN, a *KJ*<sup>c</sup> predictor for ferritic steels that only requires tensile properties (i.e., *σ*YS at the desired temperature for predicting *KJ*c, and the RT values *σ*YSRT and *σ*BRT) were derived. This method eliminates the need for time- and material-consuming fracture toughness tests. The tool for predicting *KJ*<sup>c</sup> by considering the specimen size and material properties is based on 531 fracture toughness test data values obtained from five RPV steel heats and seven non-RPV steel heats. The specimen sizes ranged from 0.4T to 4T to learn the size effect, the yield stress ranged from 328 to 775, and the tensile strength ranged from 519 to 832 to learn the material properties. The data range used in the training was equal to the application limit of the predictor. The developed *KJ*<sup>c</sup> predictor successfully predicted training data with *R*<sup>2</sup> = 0.61 and test data with *R*<sup>2</sup> = 0.53.

To predict *KJ*<sup>c</sup> at a specific temperature of interest, the user needs *σ*YS at this temperature as well as *σ*YSRT and *σ*BRT at RT. If the material of interest is known to be well fitted by the Z–A *σ*YS MC, the quantities for which test data are necessary for *KJ*<sup>c</sup> prediction are only *σ*YSRT and *σ*BRT.

A considerable advantage of the proposed *KJ*<sup>c</sup> predictor is that fracture toughness tests are not necessary to predict *KJ*c. The key novel idea here is to use tensile properties (such as *σ*YS and *σ*B) and specimen size *W*.

Although the developed *KJ*<sup>c</sup> predictor predicts one *KJ*<sup>c</sup> for a combination of explanatory variables, the predicted *KJ*<sup>c</sup> fracture probability is predicted by assuming the probability distribution of the data to be learned (e.g., Weibull distribution). It is also possible to evaluate it together, which is a future issue.

According to Tables 2–4, the (*σ*B/*σ*YS)RT of non-RPV and RPV are different. Accepting Kirk's opinion that *KJ*<sup>c</sup> and SED correspond, ASTM E1921 MC may deviate from non-RPV. However, this *KJ*<sup>c</sup> predictor has an advantage in that it considers this. On this point, the developed *KJ*<sup>c</sup> predictor, compared with the existing ASTM E1921 *KJ*<sup>c</sup> MC, is expected to be especially effective in cases where *σ*B/*σ*YS of the material is larger than that of RPV steel.

The predictors that were generated and analyzed during the current study are available from the corresponding author upon reasonable request.

#### **4. Conclusions**

In this study, a tool was developed that can predict *KJ*<sup>c</sup> for an arbitrary specimen size *W* and material properties (*σ*YSRT, *σ*YS, *σ*BRT) via an ANN applied to 531 fracture toughness test data values. Currently, the conditions applicable to the tool are material properties ranging from *σ*YSRT = 328 to 775 MPa, *σ*BRT = 519 to 832 MPa, specimen size ranging from 0.4T to 4T and its types are CT and SEB. By using the tool developed through the application of data-driven ideas, it is possible to predict the fracture toughness at this temperature from the tensile test results and the specimen size at the target temperature of the fracture toughness without performing a fracture toughness test. In the future, it is planned to predict the predicted probability of fracture toughness.

**Author Contributions:** Conceptualization, T.M.; methodology, H.K. and Y.T.; software, H.K. and Y.T.; resource, T.M.; data curation, T.M.; writing-original draft preparation, K.I. and H.K. and Y.T. and T.M.; writing-review and editing, K.I. and T.M.; supervision, T.M.; project administration, T.M.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** The data that support the findings of this study are openly available in Appendix E at https://doi.org/10.1016/j.engfailanal.2020.104713 (accessed on 29 October 2021).

**Acknowledgments:** This work is part of the cooperative research between KOBELCO RESEARCH IN-STITUTE, INC., and the University of Fukui. Support from both organizations is greatly appreciated.

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

#### **Nomenclature**


	- hyper parameter for numerical stability in Adam
	- *β*<sup>1</sup> hyper parameter for *m*(*t*) in Adam

#### **Abbreviations**


#### **References**


## *Article* **Research on the Corrosion Fatigue Property of 2524-T3 Aluminum Alloy**

**Chi Liu 1, Liyong Ma 2,3,\*, Ziyong Zhang 3, Zhuo Fu <sup>1</sup> and Lijuan Liu <sup>2</sup>**


**Abstract:** The 2524-T3 aluminum alloy was subjected to fatigue tests under the conditions of *R* = 0, 3.5% NaCl corrosion solution, and the loading cycles of 106, and the S-N curve was obtained. The horizontal fatigue limit was 169 MPa, which is slightly higher than the longitudinal fatigue limit of 163 MPa. In addition, detailed microstructural analysis of the micro-morphological fatigue failure features was carried out. The influence mechanism of corrosion on the fatigue crack propagation of 2524-T3 aluminum alloy was discussed. The fatigue source characterized by cleavage and fracture mainly comes from corrosion pits, whose expansion direction is perpendicular to the principal stress direction. The stable propagation zone is characterized by strip fractures. The main feature of the fracture in the fracture zone is equiaxed dimples. The larger dimples are mixed with second-phase particles ranging in size from 1 to 5 μm. There is almost a one-to-one correspondence between the dimples and the second-phase particles. The fracture mechanism of 2524 alloy at this stage is transformed into a micro-holes connection mechanism, and the nucleation of micropores is mainly derived from the second-phase particles.

**Keywords:** 2524-T3 aluminum alloy; fatigue; corrosion; crack propagation; fracture

#### **1. Introduction**

2524-T3 aluminum alloy has the advantages of low density, high specific strength, excellent corrosion resistance, good formability, and low cost, so it is the main material in aircraft, vehicles, bridges, engineering equipment, and large pressure vessels [1–3]. In the service process, 2524-T3 aluminum alloy undergoes the alternating load for a long time, which has high requirements for the fatigue resistance of structural materials [4–6]. Especially in coastal areas and industrial areas with serious air pollution, structural parts are exposed to varying degrees of corrosive environments, such as salt spray and acid rain [4,7,8]. Under the interaction and synergy of alternating stress and corrosive environment, the fatigue resistance of components is significantly lower than that of ordinary mechanical fatigue, and the fatigue life is severely shortened. Corrosion fatigue is not a simple superposition of corrosion and fatigue damage but rather a process of synergy and promotion. Therefore, it has great destructive effects on the aluminum alloy structure [7,9].

Studies have been concentrated on the fatigue properties of aluminum alloys [10–12]. Zhang [13] conducted a fatigue test with A6005 aluminum alloy welded joints, demonstrating that the crack nucleation was a result of metallic oxides or discrete bar-like materials and that the crack propagation rates were inversely proportional to fractal dimension. C.S. Hattori et al. [14] studied the microstructure and fatigue properties of extruded aluminum alloys 7046 and 7108. The AA7046 displayed better tensile and fatigue properties than the AA7108. In addition, deep secondary cracks perpendicular to the growth direction of

**Citation:** Liu, C.; Ma, L.; Zhang, Z.; Fu, Z.; Liu, L. Research on the Corrosion Fatigue Property of 2524-T3 Aluminum Alloy. *Metals* **2021**, *11*, 1754. https://doi.org/ 10.3390/met11111754

Academic Editor: Dariusz Rozumek

Received: 28 September 2021 Accepted: 29 October 2021 Published: 1 November 2021

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

**Copyright:** © 2021 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/).

the main crack were visible on all fracture surfaces. In the medium and high cycle fatigue tests of the AA7108 and AA7046, the cracks advanced in a perpendicular direction to the elongated grains resulting from the extrusion process.

In recent years, research studies have been conducted on the corrosion fatigue performance of aluminum alloy materials, mostly in the aerospace field. Ye et al. [15] performed the plasma electrolytic oxidation (PEO) on 7A85 aluminum alloy, and the influence on fatigue behavior in air and 3.5% NaCl solution was studied, demonstrating that PEO treatments significantly reduced the corrosion fatigue life of 7A85 aluminum alloy. R.K. et al. [16] studied the effects of corrosion on mechanical properties and fatigue life of 8011 aluminum alloy. Through tensile test and fatigue test, the research found that the corrosion had the severest destroy on the fatigue life of 8011 aluminum alloy structures. Zhang et al. [17] used ultrasonic nanocrystal surface modification (UNSM) to rejuvenate the fatigue performance of pre-corroded 7075-T651 aluminum alloy, finding that the fatigue life of the pre-corroded and UNSM treated specimens was twenty times higher than that of the only corroded specimens. Meanwhile, the reduction of the corroded surface layer and surface work hardening is beneficial for the fatigue performance rejuvenation of the pre-corroded alloy.

The relationship between the texture and grains with the fatigue properties of 2524 aluminum alloy were studied [1,3], demonstrating that the increasing of the intensity ratio of Cube to Brass texture is beneficial to the fatigue properties of 2524 aluminum alloy. Grain sizes among 50 and 100 mm exhibited high fatigue crack propagation resistances. When it comes to the effect of localized corrosion environment on fatigue properties of aluminum alloys, the studies reported recently [18–21] put their stress on the action of corrosion environment, especially the electrochemical effect on the fatigue life and properties.

In this paper, the corrosion fatigue properties of 2524-T3 aluminum alloy are investigated. The effect of 3.5% NaCl solution on the transverse and longitudinal corrosion fatigue properties of the alloy is studied and compared. The crack initiation, crack propagation, and the fracture processes of corrosion fatigue of 25,24-T3 aluminum alloy are analyzed. The effects of microstructure, such as the lengths of cracks and the widths of fatigue striations are discussed. By measuring the length of the crack and observing the width of the fatigue striations at different stages, the change in the crack growth rate is explained.

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

The experimental material is a 2524-T3 aluminum alloy sheet with a thickness of 4 mm, and its chemical composition is shown in Table 1. Rectangular specimens were selected. The longitudinal specimens were taken along the rolling direction, and the transverse specimens are taken perpendicular to the rolling direction. The width of the working section is 10 mm, the radius of the uniform transition chamfer is 120 mm, and the width of the clamping end is 30 mm. The engineering drawing of the sample for the corrosion fatigue experiment is shown in Figure 1.

**Table 1.** Chemical compositions of 2524-T3 aluminum alloy (%, mass fraction).


**Figure 1.** Engineering drawing of sample for corrosion fatigue experiment.

First, the samples were subjected to a universal tensile test on the MTS-LPS-204 universal testing machine (10 kN, MTS Industrial Co., Ltd., Eden Prairie, MN, USA) at a temperature of 25 ◦C. Second, the corrosion fatigue tests were conducted on the MTS-858 testing machine. The standard for the corrosion fatigue test is GB/T 20120.1-2006 (Corrosion of metals and alloys Corrosion fatigue test Part 1: Cyclic failure test) of National Standardization Administration of P.R. China. Before the test, the sample was fixed in the box mounted on the fatigue testing machine, as shown in Figure 2. Then, the box was filled with 3.5% NaCl solution, and the corrosion fatigue test was conducted after the box was sealed. The load was an axial sine wave, and the stress ratio *R* = 0 was selected. The number of cycles is set to 106, the test frequency is 3 Hz, and the stress levels are 5. The stress levels are selected according to the tensile test results, and the minimum number of samples is confirmed by the coefficient of variation.

**Figure 2.** Setup for corrosion fatigue test: (**a**) MTS-LPS-204 universal testing machine; (**b**) box to provide corrosion experiment.

#### **3. Results and Discussion**

#### *3.1. S-N Curves*

The universal tensile test shows that the tensile limit of 2524-T3 aluminum alloy is 475 MPa, and the first stress level of the fatigue test is selected as 190 MPa, which is 40% of the tensile limit. The stress levels of 2524-T3 aluminum alloy corrosion fatigue tests are 190 MPa, 180 MPa, 170 MPa, 160 MPa, and 150 MPa, respectively. The effective test results of horizontal and longitudinal corrosion fatigue test of materials under different stress levels are shown in Table 2. When the stress level was 150 MPa, the fracture did not appear after loading for 10<sup>6</sup> cycles, and the experiments finished.



According to the test data in Table 2 and referring to Equations (1)–(3), the average value *x*, standard deviation *s*, and coefficient of variation *C* of the sub-samples to judge the validity of the data are calculated. The least-squares method [22–24] is used to obtain a safety fatigue life curve with reliability and a confidence of 50%, as shown in Equations (1)–(3).

$$\overline{\mathfrak{X}} = \frac{1}{n} \sum\_{i=1}^{n} \lg N\_i = \hat{\mu} = \log \hat{N} \tag{1}$$

$$\sigma = \sqrt{\frac{\sum\_{i=1}^{n} x\_i^2 - \frac{1}{n} \left(\sum\_{i=1}^{n} x\_i\right)^2}{n-1}} \tag{2}$$

$$C = \frac{\delta\_{\text{max}}\sqrt{n}}{p} \ge \frac{\sigma}{\overline{\chi}}\tag{3}$$

where *μ*ˆ is the population mean estimator; *σ* is the population standard deviation estimator; *δ*max is the error limit, usually 5%; *p* is the probability density, which can be determined by looking up the table for *n*. The fitted curves are shown in Figure 3.

**Figure 3.** Horizontal and longitudinal corrosion fatigue tests for 2524-T3 aluminum alloy.

The S-N curve of the effective corrosion test under the confidence of 50% and the reliability of 50%, with different horizontal and longitudinal stress levels are shown in Equations (4) and (5), respectively. The combined correlation coefficient for the curves are 0.9266 and 0.9711, respectively.

$$S = 502.7457 - 55.2365 \lg N \tag{4}$$

$$S = 490.8512 - 49.5742 \text{lgN} \tag{5}$$

In Equations (4) and (5), the horizontal fatigue limit of 2524-T3 aluminum alloy corresponding to 10<sup>6</sup> fatigue cycles is calculated to be 169 MPa, which is slightly higher than the longitudinal fatigue limit, which is 163 MPa. So, the fatigue lifetime of the longitudinal samples is slightly higher than that of the horizontal ones.

#### *3.2. Fracture Analysis*

Since the fracture process of the transverse and longitudinal specimens is similar, the longitudinal fracture of the specimen under 180 MPa is taken as an example to reveal the corrosion fatigue fracture process of the aluminum alloy.

Figure 4 is the macroscopic fatigue fracture morphology of the 2524-T3 aluminum alloy. It can be roughly divided into three areas from right to left:


**Figure 4.** Morphologies of fracture specimen for 2524-T3 aluminum alloy at 180 MPa stress level.

As observed from the macroscopic fracture, the initial crack is directly emitted from the corrosion pitting of the sample. The crack propagation rate in Zone A is very slow, and the fatigue cracks are all nucleated from the surface. The corrosion pitting morphology can be observed in part of the shell lines in the fatigue source area, revealing that the fatigue cracks are initiated by the pitting pits caused by corrosion. From Zone A to depth, the crack enters the stable propagation stage, forming a flat Zone B. The fatigue crack propagation zone is bright, indicating a brittle intergranular fracture morphology, and numerous secondary cracks can be found in this area. As the crack continues to grow, the stress intensity factor at the crack tip increases, the fatigue crack propagates sharply, and the alloy enters the rapid fracture Zone C, which is rough and fibrous. The boundary between Zone B and Zone C is an obvious arc-shaped crack front. Near the crack front, the fracture has the mixture characteristics of crack propagation and plastic tearing.

In the crack propagation stage, the crack front is first generated near the center of the fracture. When the crack propagates to the vicinity of the surface of the specimen, the unbroken area cannot withstand the action of external force, and it breaks along the shear direction 45◦ to the cyclic stress. When the crack extends to Zone C, the plane strain fracture amount gradually decreases, and fracture occurs when it is stressed at an angle of 45◦.

Figure 5 shows the 3D morphology of the fatigue fracture surface of the specimen in different crack regions. The dark red and dark blue in the figure indicate the highest and lowest positions, respectively.

Figure 5a shows the surface roughness of the fatigue source region. The fatigue section is relatively rough. At this time, the crack closure is mainly affected by the roughness.

As the crack expands, it can be clearly seen that the peaks produced by the crack deflection are smoothed and flat, as shown in Figure 5b. It can be inferred that the mechanism that affects the crack closure in the medium and high stress zone has changed from roughness induction to plastic zone induction. The existence of the plastic zone at the crack tip triggers cracks in advance or mismatched contact. The original convex wave peaks are constantly squeezed during the cyclic loading process. Pressed and rubbed, it becomes flat and shiny. The plastic-induced crack closure leads to the contact between the crack surfaces in advance, and the crack-opening displacement is reduced, which indirectly reduces the driving force of crack growth, resulting in a decrease in the crack growth rate.

**Figure 5.** Three-dimensional (3D) morphology of the fatigue fracture surface of 2524-T3 aluminum alloy: (**a**) fatigue crack initiation zone; (**b**) crack propagation zone; (**c**) instantaneous failure zone.

In the later stage of crack propagation, the crack is in a rapid expansion state where the expansion and tearing are mixed. The crack may directly tear through several grains under each load, so the section in a small area appears very flat, as shown in Figure 5c. Meanwhile, at the upper right corner of the figure, the fracture is rising, and the feature of tearing appears.

#### 3.2.1. Fatigue Source and the Initial Stage of Fatigue Crack Propagation

The SEM microscopic morphology of the fatigue source zone of the specimen after fracture at a stress level of 170 MPa is shown in Figure 6. The crack originates from the pits generated after the surface of the material is corroded, where stress concentration occurs under the action of fatigue loads. Excessive concentrated stress causes the dislocation movement of the material surface to intensify, forming a small slip zone where fatigue cracks are generated.

**Figure 6.** SEM micro-morphologies of fatigue source area.

In Figure 6, the fatigue source zone radiates toward the direction the cracks propagate and the crack front deviates in the propagation direction due to different resistances, so the crack starts to continue to expand along a series of planes with height differences, and the different fracture surfaces intersect to form steps, which constitute radial rays on the fatigue fracture. Near the corrosion pits, at Region I and Region II, the cracks originate at the corrosion pits, and there are obvious fan-shaped ray patterns along the crack propagation direction, showing ductile fracture characteristics. Region III may be affected by inclusions or twinning in the grains. The fracture characteristics feature a micro-zone cleavage mechanism under stress, and the crack source spreads around. After the main crack passes over both sides of the twinning at Region III, it continues to expand, forming a unique "tongue"-like pattern. The microscopic fracture morphology at Region IV presents the unique cleavage steps. Since the alloy has a high dislocation resistance at the grain boundary, when the crack propagates to the grain boundary, to minimize the energy consumed, the cleavage steps will expand along different crystal planes nearby. The face-centered cubic crystal structure (FCC) 2524-T3 aluminum alloy mainly slides along the direction <110> on the sliding surface {111} [12]. As a result of the excellent toughness of 2524-T3 alloy, the proportion of the region with a brittle fracture is small. In addition, a stepped crack developed into the material at Region V, indicating that the crack tip has lateral slippage during the propagation of the main crack.

Figure 7 shows the fracture morphology of the 2524-T3 alloy at the initial stage of fatigue crack propagation. The fatigue crack propagation zone in the initial stage is flat, and the obvious quasi-cleavage fracture characteristics and wave-like pattern morphology can be seen, which is the main feature of damage in the corrosive environment. The crack propagates along a favorable direction relative to the maximum shear stress and extends to the depth of one or several grains. There are many relatively flat facets on the

fractures with different heights. The small planes are connected by tear ridges (Figure 7a), demonstrating the crack propagations on different crystal planes. The tear ridges are deflected relative to the main crack direction, resulting in a certain angle difference in the main crack propagation direction. At the region shown by the cross-line in Figure 7b, the deflection angle is 32◦. In addition, there are many micro-holes scattered on the crosssection, and the size and depth of the are different. It can be inferred that these micro-holes are caused by the coarse particles in the matrix being squeezed and stretched continuously during the crack propagation and then peeled off from the matrix.

**Figure 7.** Morphology of fracture of 2524-T3 aluminum alloy in the initial stage of fatigue crack propagation: (**a**) tear ridges and crack propagation direction; (**b**) micro-holes and angle difference in the main crack propagation direction.

#### 3.2.2. Stable Stage of Fatigue Crack Propagation

The fatigue life is mainly determined by the time of crack initiation and stable growth [25,26], and the crack propagation zone is the key feature of the fatigue fracture. Figure 6 shows the stable crack propagation area at a distance of 3 mm and 10 mm from the crack initiation end. Under ×1000 magnification, the main features in Figure 8a,c are similar. Along the crack propagation direction, the crack propagates in the form of transgranular fracture, and there are obvious crack edges and small planes on the fracture. This is because there are differences in the local orientation of cracks when they propagate inside the alloy. The resistance and growth rates encountered at the crack front are also different. The cracks continue to deviate as they propagate along their respective favorable slip surfaces, leaving different fracture surfaces intersecting. In Figure 8, the number of steps at the crack length of 3 mm is larger, indicating that the crack deflection is more frequent in the initial stage of crack propagation. The fracture contains a large number of unevenly distributed micro-holes. These micro-holes are formed by the gradual separation, breaking, and peeling of the unmelted coarse second-phase particles from the matrix under the action of cyclic stress.

**Figure 8.** SEM images of stable crack propagation zone: (**a**,**b**) 3 mm from the crack initiation end; (**c**,**d**) 10 mm from the crack initiation end.

In Figure 8b,d, the relatively smooth areas of the fracture are further enlarged and observed, and obvious fatigue striations can be observed. Fatigue striations are the most typical microscopic feature of fatigue fracture. In Figure 8b,d, the thin and parallel fatigue striations are uniformly distributed, forming a group of parallel lines. The fatigue stripes on each small fracture plane are discontinuous and non-parallel, but their normal directions are along the crack propagation direction in the local fracture plane. In the process of crack propagation, the front of the crack is in an open plane strain state, and the crack propagation starts to proceed along the two slip systems at the same time or across. The double slip will cause the crack tip to be plastically passivated, i.e., fatigue striations. In Figure 8b, the width of the five fatigue striations is about 1.6 μm, and the distance of each ridge is about 0.32 μm, i.e., the microscopic crack propagation rate d*a*/d*N* = 0.32 μm/cycle. The width of the five fatigue striations is significantly enlarged, about 6 μm, and the crack propagation rate is about three times higher than that at 3 mm, indicating that the spacing of the fatigue striations increases with the increase in the magnitude of the stress intensity factor.

Although the formation of fatigue striations is a localized process, the general propagation trend is along the propagation direction of macroscopic cracks. As shown in Figure 9, the fatigue striations on the fracture are not all distributed in the direction of crack propagation, and some will deviate from the direction of fatigue crack propagation. As shown in Figure 9a,b, when the fatigue crack crosses from one plane to another, it will

also leave fatigue striations on planes with different directions and uneven heights [7]. 2524-T3 aluminum alloy is solid-solution strengthened and then subjected to natural aging treatment. The intragranular is mainly a shearable coherent phase GPB zone, and its cracks will propagate along the favorable slip surface. Due to the orientation differences between the two favorable slip planes in adjacent grains, the cracks continue to grow along the favorable slip plane after growing across the grain boundary, and they gradually deflect. The fatigue striations on both sides of the grain boundary present an angle, but there is no clear change in the size of the fatigue striations on both sides. The deflection process of fatigue cracks consumes the deformation and storage energy under the action of cyclic stress, effectively reduces stress concentration, and improves the fatigue resistance of the aluminum alloy. As shown in Figure 9c, when the fatigue crack encounters the twinning boundary, it is swallowed by the twinning boundary and continues to expand [27,28]. The fatigue striations show a symmetrical relationship along the twin boundary, but the widths are unchanged, indicating that the co-lattice twinning interface energy in 2524 aluminum alloy is relatively low, and the effect on the crack propagation rate is not obvious. When the orientation difference of adjacent grains is not large, the crack can expand through the grain boundary and enter another grain without changing the expansion angle too much, which is conducive to the transgranular expansion of the crack and the formation of obvious straight cracks. When the crack propagates in the grain, it will preferentially propagate along a certain cleavage plane, and the propagation path may form a "Z"-shaped crack, as shown in Figure 9d.

**Figure 9.** SEM images of different orientations in the stable crack propagation zone: (**a**,**b**) fatigue fringes on adjacent grains with different orientations; (**c**) twinning and fatigue fringes on both sides; (**d**) "Z"-shaped crack.

The 2524-T3 alloy sheet contains a large number of micron-sized second-phase particles which are generally Al2CuMg phase and Fe-rich phase. The impurity phase will leave obvious features on the fatigue fracture. As shown in Region A in Figure 10a, there are a large number of broken coarse second-phase particles and holes on the crack propagation path. This is because under the action of cyclic stress, some of the unmelted, coarse second-phase particles are torn during the crack propagation process to form broken particles, and some are separated from the matrix and leave holes. These broken particles and holes provide a preferential path for crack propagation. In Region B, because the coarse second-phase particles break under the action of cyclic stress, the fatigue striations choose to bypass the particles for expansion, indicating that the cracks tend to expand in the direction of more inclusions [29,30], bridging larger debonded inclusions, and thus, the fatigue resistance of the material is weakened.

**Figure 10.** SEM images of coarse particles and secondary cracks in the stable crack propagation zone: (**a**) coarse particles; (**b**) secondary cracks.

In addition, another important feature of 2524 aluminum alloy during the crack propagation process—secondary cracking—was observed on the fracture. As shown in Region C in Figure 10a, there are a large number of secondary cracks distributed along the direction of the glare on the fatigue fracture. They are cracks that expand from the surface of the fracture to the inside, which is distributed intermittently on the fracture, and the directions are often parallel to the fatigue striations, but the depths are much greater than the depths of the striations. Some secondary cracks are initiated and propagated along the second-phase particles (in Figure 10b). Golden et al. [31] explained this phenomenon, demonstrating that stress concentration leads to the accumulation of dislocations that generate along the weak position of the phase interface, and the stress is relieved after the second cracking. From the point of view of the energy method, the fatigue damage process is the accumulation of strain energy in the plastic zone under cyclic stress [32,33]. The formation of secondary cracks is beneficial to release the energy at the crack tip and slow down the crack propagation rate to a certain extent.

#### 3.2.3. Stage of Rapid Fatigue Crack Propagation and Fatigue Fracture

When the stress intensity factor amplitude of the crack tip is <sup>Δ</sup>*<sup>K</sup>* ≈ 25 MPa·m1/2, the crack enters the rapid growth zone, and the crack tip expands to the position shown by the dotted line of the crack front in Figure 3. Near the boundary, there is a mixed fracture morphology of the transition between the crack propagation zone and the transient zone. This area was observed under a high-power SEM electron microscope, and the results are shown in Figure 11.

**Figure 11.** Microscopic morphology of the fatigue rapid growth zone: (**a**) crack propagation and fracture transition zone; (**b**) instantaneous failure zone; (**c**,**d**) enlarged views of dimples, (**e**) EDX map of the coarse particles.

On the right side of the dividing line (shown in Figure 11a), the crack has just transferred from the stable propagation zone to the rapid propagation zone. There are a few fatigue striations on the fracture with a large distance of 2 μm. The fracture morphology shows the characteristics of a mixture of fatigue bands and dimples. On the left side of the dividing line, the material begins to lose stability, and fracture occurs. The microscopic morphology begins to change from a mixed dimple-fatigue striation zone to a dimple tear zone (shown in Figure 11b). Microscopically, the fracture of the dimple is honeycomb-like, composed of many holes and small pits. The fracture morphology is characterized by quasi-static ductile tensile fracture [34], and the fracture mechanism is a microporous connection.

Dimples are the main feature of fatigue fracture of 2524 alloy at this stage. Due to the normal tensile stress of the specimen in the test, the microscopic voids in the alloy will grow at the same rate along the three sides of the space, thus forming the equiaxed dimples in the figure. The size of the dimples is not the same. The large dimples are full of small dimples. A large number of sliding steps and tearing edges can be seen at the boundaries of the dimples. This indicates that during the rapid expansion stage, the 2524 alloy fractured after a relatively large plastic deformation. These characteristics were observed in a previous study by Magnusen et al. [10]. The process of dimple fracture is divided into three stages: microporous nucleation–growth–polymerization. In Figure 11b, it can be observed that the larger-sized dimples are all mixed with second-phase particles, with sizes ranging from 1 to 5 μm. The second-phase particles have an almost one-to-one relationship, which verifies the view that the second-phase particles are the source of micropore nucleation.

In the 2524 alloy, there are mainly impurity phases such as Mg2Si, Al7Cu2Fe, Al12FeSi, (MgFe)3SiAl12, (FeMn)Al3, (FeMn)Al6, and S (Al2CuMg) equilibrium phase. These coarse impurity phases are very brittle, and they are easily separated from the matrix under stress or crack to form micro-holes (Figure 11c). Some strengthening phases (Al2CuMg) that are relatively firmly combined with the matrix will eventually produce micro-holes due to inconsistent plastic deformation with the matrix under the severe stress concentration in the later stage of crack propagation, as shown in Figure 11d. Excessive coarse second phases inside the alloy will severely affect the fracture toughness of the alloy. Therefore, to improve the fracture toughness of the alloy, the size and quantity of the coarse second phase should be significantly reduced. Figure 11e is an EDX map of Spectrum 1 of the coarse particle in Figure 11c. The chemical composition of Spectrum 1 is 46.9% Al, 52.5% Cu, 0.6% Mg, which can be judged as Al2Cu phase, which is brittle. Under the action of alternating loads, the coarse second phase cannot simultaneously deform in coordination with the matrix, and dislocations will continue to accumulate at the particle–phase interface and cause stress concentration. When the peak stress in the local area exceeds the fracture strength of the alloy, the coarse phase will be separated from the matrix at the interface and peeled from the aluminum matrix to form cavities, which is also claimed in Ref. [35].

#### **4. Conclusions**


crack propagation/fracture mechanism is transformed into a micro-holes connection mechanism. The main characteristic morphology at this stage is equiaxed dimples. Larger dimples are mixed with second-phase particles ranging in size from 1 to 5 μm. The relationship between dimples and second-phase particles is almost one-to-one, indicating that the nucleation of micropores mainly comes from the second-phase particles in the alloy.

**Author Contributions:** Conceptualization, C.L. and L.M.; methodology, C.L. and L.M.; validation, C.L., L.M. and Z.Z.; formal analysis, L.L. and Z.F.; investigation, C.L., L.M. and L.L.; resources, C.L.; data curation, C.L.; writing—original draft preparation, C.L.; writing—review and editing, L.M.; project administration, C.L., L.M., Z.F. and L.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Scientific Research Fund of Hunan Provincial Education Department (grant number 20C0168 and 20B068), Changsha Municipal Natural Science Foundation (kq2007085), the Basic Scientific Research Business Project of Hebei University of Architecture (2021QNJS08), the Science and Technology Research and Development Command Plan of Zhangjiakou (1911031A), and "14th Five-Year Plan" Project of Hebei Higher Education Association (GJXH2021-109).

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** The authors acknowledge Xiaohong Sun for providing support in experiments.

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

#### **References**


## *Editorial* **Fracture Mechanics and Fatigue Design in Metallic Materials**

**Dariusz Rozumek**

Department of Mechanics and Machine Design, Opole University of Technology, Mikolajczyka 5, 45-271 Opole, Poland; d.rozumek@po.edu.pl; Tel.: +48-77-449-8410

#### **1. Introduction and Scope**

Devices, working structures and their elements are subjected to the influence of various loads. These can be static, cyclic or dynamic loads. The accumulation of damage and the development of fatigue cracks under the influence of loads is a common phenomenon that occurs in metals. To slow down crack growth and ensure an adequate level of safety and the optimal durability of structural elements, experimental tests and simulations are required to determine the influence of various factors. Such factors include, among others, the impact of microstructures, voids, notches, the environment, etc. Research carried out in this field and the results obtained are necessary to guide development toward the receipt of new and advanced materials that meet the requirements of the designers. This Special Issue aims to provide the data, models and tools necessary to provide structural integrity and perform lifetime prediction based on the stress (strain) state and, finally, the increase in fatigue cracks in the material, which would result in the application of advanced mathematical, numerical and experimental techniques.

#### **2. Contributions**

Fracture mechanics are present in most structures that work cyclically, e.g., in the automotive or aviation industry. To extend the life of structures, they must be properly fatigue-proofed and made of appropriate materials. This Special Issue shows the fatigue behavior of various alloys and the conditions under which these alloys work. A paper by Sharma et al. [1] reviews the research and development in the field of fatigue damage, focusing on the very high cycle fatigue (VHCF) of metals, alloys and steels. In addition, they showed the influence of various defects, crack initiation sites, fatigue models and simulation studies to understand the crack development in VHCF regimes. A paper by Wang et al. [2] investigates the influence of the crack behavior propagation process in welded joints and sheds light on the mechanism of their branching, and a paper by Wei Xu et al. [3] proposes an ultra-high-frequency (UHF) fatigue test of a titanium alloy TA11 based on an electrodynamic shaker to develop a feasible testing method in the VHCF regime. The results from UHF tests data show good consistency with those from the axial-loading fatigue and rotating bending fatigue tests. Moreover, the fatigue life obtained from an ultrasonic fatigue test with the loading frequency of 20 kHz is significantly higher than all the other fatigue test results.

Artola et al. [4] investigated the impact of quench and tempering and hot-dip galvanizing on the hydrogen embrittlement behavior of a high-strength steel. Slow-strain-rate tensile testing was employed to assess this influence. Two sets of specimens were tested, both in-air and immersed in synthetic seawater. It was found that the risk of rupture only arises due to hydrogen re-embrittlement in wet service.

The closure of the crack was discussed in three articles [5–7]. Zakavi et al. [5] presents new tools to evaluate the crack front shape of through-the-thickness cracks propagating in plates under quasi-steady-state conditions. A numerical approach incorporating simplified phenomenological models of plasticity-induced crack closure was developed and validated against experimental results.

**Citation:** Rozumek, D. Fracture Mechanics and Fatigue Design in Metallic Materials. *Metals* **2021**, *11*, 1957. https://doi.org/10.3390/ met11121957

Received: 1 December 2021 Accepted: 2 December 2021 Published: 6 December 2021

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**Copyright:** © 2021 by the author. 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/).

Lesiuk et al. [6] showed a comparison of the results of the fatigue crack growth rate for raw rail steel, steel reinforced with composite material—CFRP—and the case of counteracting crack growth using the stop-hole technique, as well as with an "anti-crack growth fluid". It has been shown that the fatigue crack grows fastest in the case of the raw material and slowest in the case of the "anti-crack growth fluid" application. As a result of fluid activity, the fatigue crack closure occurred, which reduced the growth of this crack.

Ahmed et al. [7] investigated the fatigue crack propagation mechanism of CP Ti at various stress amplitudes. One crack at 175 MPa and three main cracks via sub-crack coalescence at 227 MPa were found to be responsible for the fatigue failure. The crack deflection and crack branching that cause roughness-induced crack closure (RICC) appeared at all studied stress amplitudes; hence, RICC at various stages of crack propagation (100, 300 and 500 μm) could be quantitatively calculated. Noticeably, a lower RICC was found at higher stress amplitudes (227 MPa) for fatigue cracks longer than 100 μm than for those at 175 MPa. This caused the variation in crack growth rates under the studied conditions.

Lee et al. [8] conducted fatigue tests at room temperature and 1000 K for 0.135-mmthick alloy 625 tubes (outer diameter of 1.5 mm), which were brazed to the grip of the fatigue specimen. The variability in fatigue life was investigated by analyzing the locations of the fatigue failure, fracture surfaces and microstructures of the brazed joint and tube. At room temperature, the specimens failed near the brazed joint. Rusnak et al. [9] fatigue tested nine poles with 18 openings using four-point bending at various stress ranges. Among the 18 hand-holes tested, 17 failed in one way or another as a result of fatigue cracking. Typically, fatigue cracking would occur at either the three or nine o'clock positions around the hand-hole and then proceed to transversely propagate into the pole before failure. Finite element analysis was used to complement the experimental study.

Ishihara et al. [10] analyzed the structural integrity of ferritic steel structures subjected to large temperature variations, which required the collection of the fracture toughness (*K*Jc) of ferritic steels in the ductile-to-brittle transition region. In this study, a Windows-ready *K*Jc predictor based on tensile properties (specifically, yield stress and tensile strength at room temperature and yield stress at *K*Jc prediction temperature) was developed by applying an artificial neural network to 531 *K*Jc datapoints.

Liu et al. [11] subjected the 2524-T3 aluminum alloy to fatigue tests under the conditions of *R* = 0, a 3.5% NaCl corrosion solution and loading cycles of 106, and the S-N curve was obtained. The horizontal fatigue limit was 169 MPa, which is slightly higher than the longitudinal fatigue limit of 163 MPa. The influence mechanism of corrosion on the fatigue crack propagation of the 2524-T3 aluminum alloy was discussed. The fatigue source characterized by cleavage and fracture mainly comes from corrosion pits, whose expansion direction is perpendicular to the principal stress direction.

#### **3. Conclusions and Outlook**

In this Special Issue, there are various topics relating to the latest approach to fatigue crack growth. They relate to the influence of load, microstructure, friction, corrosion or to welded joints. However, many issues in this area of research have not yet been explored and the dissemination of these results should be continued. As a Guest Editor, I hope that the research results presented in this Special Issue will contribute to the further progression of research on the growth of fatigue cracks.

Finally, I would like to thank all the reviewers for their input and efforts in producing this Special Issue, and the authors for the papers they have prepared. I would also like to thank all the staff at the *Metals* Editorial Office, especially Toliver Guo, the Assistant Editor, who managed and facilitated the publication process.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


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