The Fatigue Behavior and Mechanism of large FV520B-I specimens in a Very High Cycle Regime

We examined the fatigue properties in very high cycle regime of large FV520B-I specimens in an ultrasonic fatigue test. The fatigue mechanism in very high cycle regime didn’t change and the fatigue properties obviously degraded as the specimen size enlarged. The fatigue life decreased and the S-N curve moved downward due to the increase of inclusion size in large specimens. The maximum inclusion sizes in specimens were predicted by the method of statistics of extreme value. The prediction of fatigue strength using the modified Murakami model was closer to the test result, and the fitting of fatigue life using the corrosion fatigue crack initiation life model was less effective compared with the fitting of small specimen test results


Introduction
Several failure cases show that fatigue is the main failure mode of centrifugal compressor impellers [1,2]. With the economy development and the constraint enhancement of the environment and resources, there is a higher demand for the impeller life. The impeller fatigue life should be exceed 10 7 cycles and into the very high cycle fatigue (VHCF) regime. Current research on impeller VHCF is rare. FV520B-I is an important material of centrifugal compressor impellers and is belong to martensitic precipitated hardening stainless steel. We had studied the VHCF of small FV520B-I specimens, which revealed that the cracks in the VHCF regime usually initiate from subsurface inclusions [3][4][5]. The maximum inclusion size (MIS) in the control volume of specimens is different to determine as the specimen size varies, and the VHCF properties of specimens with different sizes also seem different [6][7][8]. So it is important to study the effects of specimen size on the VHCF properties. The VHCF properties of a large FV520B-I specimen were tested and compared with previous experiment results of a small FV520B-I specimen.
The Fatigue Behavior and Mechanism of large FV520B-I specimens in a Very High Cycle Regime ·3· 2 Material and test method 2

.1. Material
The chemical components and mechanical properties of FV520-I are showed in Table 1 and  Table 2.

Test method
Relevant fatigue tests were conducted on a Shimadzu USF-2000. The resonance frequency is 20 kHz, and the stress ratio R is -1. We also adopted a specimen with funnel shape that we had previously used for the ultrasonic fatigue tests, and the minimum cross-section diameter of specimen enlarged from 3mm to 5mm. The specimen dimension determined by analytic calculation as shown in Figure1 (b) [5,9], while the dimension of a small specimen which we previously tested is shown in Fig. 1 (a). The work time and interval were set as 200ms and 600ms.
The tests were conducted in air ambient and at room temperature, and the specimen during testing was cooled by the compressed cold air. The specimens were sequential polished with 600, 1000, 2000 mesh emery papers.
(a) Small specimen (b) Large specimen Figure 1 Specimen dimensions (in mm).

Microstructure
The ferrites were distributed in the tempered sorbites, and the granular carbides were observed distributed within the matrix as shown in Figure  2

VHCF S-N Curve
The VHCF S-N curves of different size specimens are shown in Figure 3 The small specimens generally fractured in the subsurface, and one specimen fractured on the surface [4], while all large specimens fractured in the subsurface. Compared with the test results of small specimens, the S-N curve of large specimens moved downward and the fatigue life of large specimens decreased. The difference of fatigue life between the two types of specimens had an increasing trend as the stress declined, and the maximum difference was up to 10 times. The maximum stress level without fracturing for small specimens was 550MPa, and for large specimen it was 525MPa.

Figure 3
The VHCF S-N curves

The Observation of Fracture Surface
The fractures were observed by a Scanning Electron Microscope. All cracks initiated from subsurface inclusions, and the cracks mostly initiated from single inclusions while some cracks initiated from clusters as shown in Figure 4 All inclusions were identified as Al2O3 by energy spectrum analysis, as shown in Figure 5.  The cracks propagated and formed a granular bright facet (GBF) after the cracks initiated from inclusions, as shown in Figure 6 The GBF seemed relatively rough and different from the smooth and flat region of the usual crack origin. Also, there was a relatively flat region close to the specimen surface which looked like a fish eye, as shown in Figure 7.

Characteristic Area Size
The distance from the crack origin to the specimen surface, the fish-eye diameter, the GBF diameter, and the inclusion diameter were measured and listed in Table 3.

The Statistics of Inclusion Sizes in FV520B-I
The sight area S0=0.0195mm 2 observed by an optical microscope with a magnification of 200 times was defined as the standard inspection area. FV520B-I VHCF cracks usually initiated from spherical Al2O3, so the sizes of MnS inclusions with the shape of a thin strip were not involved. The square root of the maximum inclusion area in S0 is about the maximum inclusion diameter. There were six specimens, and the observation was repeated 10 times for each specimen. Every observation was selected at random and they did not overlap. The statistical results of the maximum inclusion diameters are shown in Table  4, and the photo of the 40th observation is shown in Figure 8.  Note：No. 1-10, 11-20, 21-30, 31-40, 41-50, and 51-60 are the statistical results for specimen 1, 2, 3, 4, 5, and 6, respectively 4 Discussion

The Statistics of Extreme Values of the Maximum Inclusion Sizes in VHCF Specimens
A subsurface inclusion is the usual VHCF crack origin [10][11][12][13][14], which has significant effects on the VHCF properties. Due to the low number of large inclusions and the limitation of traditional detection methods, the detection of large inclusions in steel is difficult. At present, the MIS in a large volume is estimated by the extrapolation of inclusion data in a small volume, and the fatigue properties and reliability of steel structure are further predicted by the maximum ·6· .
Han, Zhang et al.
inclusion size [15,16]. The statistics of extreme values (SEV) is an important method to estimate the MIS in a large volume, which is more suitable for steel with large inclusions [9]. SEV [9] predicts the MIS in a large range of steel based on the measured inclusion size in a small randomly selected area or volume [13,17]. The basic idea of SEV is that the maximum values of each group of data are subject to the same distribution as all data points subject to a distribution. This distribution is usually the Gumbel distribution. The distribution function is as follows: Where T is the return period: Where S0 is the standard inspection area and h is the average diameter of inclusions A is about the maximum inclusion diameter in the standard inspection area obtained by the ith observation, in units of μm. The control volume V has great effects on the maximum inclusion size zv predicted by the SEV method, so the determination of the control volume of VHCF specimen is important. VHCF specimens are funnel-shaped in this paper. A schematic diagram of its shape and size are shown in Figure 9.

Figure 9
The shape of a funnel specimen Strain ε(x) and stress σ(x) in the specimen crosssection which is along the x axis is shown in Eq. (6): The displacement of specimen in the x-axis direction U(x) is shown in Eq. (7): Where A0 is the displacement of the specimen end. Variables in Eq. (7) , 2 , Where E is the elasticity modulus, ρ is the material density, and f is the resonance frequency. For funnel shaped specimens, the maximum stress σmax appears in the minimum cross-section of the specimen where x=0, and the test specimen is usually broken at this position. The stress is decreased as the position shifts to each side. The control volume is supposed to be the volume where the stress is higher than 0.9σmax. When the stress is equal to 0.9σmax, x-axis coordinates are -lx and lx, respectively. Then the control volume 2 x1 2 V l R = [16]. For FV520B-I, when E=194GPa, ρ=7820kg· m -3 , and f=20kHz, the sizes of two types of specimens are shown in Fig. 1 The Gumbel distribution is usually used in the SEV analysis. The maximum inclusion diameters in Table 4 were fitted by Gumbel distribution using the gum.fit function in ISMEV package of R software which is a statistical analysis software, and we found that when μ=3.100 and θ=1.410, The Fatigue Behavior and Mechanism of large FV520B-I specimens in a Very High Cycle Regime ·7· and T=6.2×10 5 , T=2.4×10 6 , θ=1.410, and μ=3.100 are substituted into Eq. (2), the maximum inclusion diameters in VHCF specimens can be estimated as zv1=21.91μm and zv2=23.82μm when R1 is equal to 1.5mm and 2.5mm. The possible maximum inclusion diameter in VHCF specimens increases with an increase of the control volume. zv1 is larger than the maximum single inclusion diameter 15.6μm and cluster inclusion diameter 16.4μm on the fatigue fractures of small specimens [5]. zv2 is larger than the maximum single inclusion diameter 16.3μm and cluster inclusion diameter 16.4μm on the fatigue fractures of large specimens, as shown in Table 3.

Effects of Specimen Size on FV520B-I VHCF
The VHCF cracks of small specimens mostly initiated from the subsurface inclusions and only one crack initiated from the surface, while two cracks initiated from the subsurface matrix [5]. All VHCF cracks of large specimens initiated from the interface of the matrix-inclusion, and the inclusions were not cracked, as shown in Figure  4. This phenomenon is in accordance with the test results of small specimen [4,5] and the usual initiation mode of subsurface cracks in high strength steel [18]. The cracks propagated and formed the GBF after the cracks initiated from the interface of the matrix-inclusion, as shown in Figure 6. There are different views about the GBF formation [19,20], but most scholars generally think that the hydrogen is an important reason [21,22]. The effects of hydrogen seem to weaken as the crack grows [23]. The cracks may stably grow only under the mechanical load as the effects of hydrogen disappear. A subsurface crack quickly transforms into a surface crack and the stress intensity factor of the crack suddenly increases. The large difference of crack growth rates in the two parts results in the fish-eye formation, as shown in Figure 7. From section 4.1, the possible maximum inclusion diameter in specimens is increased because the control volume increased with an increase of VHCF specimen size. The actual inclusion sizes at the crack origins of the two groups of specimens were further examined. For large specimens, the maximum inclusion diameter was 16.4μm, the average inclusion diameter was 12.6μm, and the minimum inclusion diameter was 8.2μm as shown in Table  3. For small specimens, the maximum inclusion diameter was 16.4μm, the average inclusion diameter was 12.1μm, and the minimum inclusion diameter was only 4.4μm [5]. The inclusion diameter increased with an increase of specimen size, and which increased the probability of a crack initiating from the inclusion. Therefore, all VHCF cracks in large specimens initiated from the subsurface inclusions, and no crack initiated from surface or subsurface matrix. In addition, an increase of specimen size had no obvious effects on the FV520B-I VHCF mechanism. The S-N curve moved downward as the specimen size increased as shown in Figure 3. The fatigue life of the large specimens was reduced and the reduction seems more obvious in the lower stress, and the fatigue strength up to 10 9 cycles was also decreased. Above phenomenon is caused by two factors. First, the MIS in a specimen increases with an increase of control volume, and the increase of the MIS reduces the fatigue strength of the specimen. Second, the heat effects of large specimens may also affect the fatigue properties.

The Relationships of Characteristic Area Sizes and VHCF Life
The relationships of characteristic area sizes with VHCF life are shown in Figure 10.  It is similar to the previous small specimen test results that the relationships of the VHCF life of large specimens with regard to the distance from the crack origin to the specimen surface, and with the fish-eye diameter being not obvious as shown in Figure 10 (a), (b). Above results suggest that the crack grows fast outside the GBF and the VHCF life is mainly consumed in the GBF formation. The distribution of data points in both figures are similar, the fish-eye diameter is about two times the distance from the crack origin to the specimen surface, which further testifies that the fish-eye formation is caused by the transition from a subsurface crack to a surface crack. There is no significant relationship between the VHCF life and the inclusion diameter as illustrated in Figure 10 (c). The subsurface inclusion diameter has a decreasing trend with an increase of fatigue life if the load is constant. When the load is 600MPa or 550MPa, there is a data point at which that its inclusion diameter increases with an increase of fatigue life. Related researches showed that the initiation life of a crack which initiates from the subsurface inclusion appears nearly inversely proportional to the inclusion diameter [18]. So the initiation life may be not the main part of the VHCF life. This conclusion is consistent with the conclusion of small specimen test results [5]. The relationship of VHCF life with the GBF diameter was further investigated as illustrated in Figure 10 (d). The GBF diameter basically increases with an increase of fatigue life. But this trend is not obvious when the load is constant, and sometimes the GBF diameter decreases with an increase of fatigue life, which is very different from the small specimen test results [5]. Small specimen test results showed that the inclusion diameter has a certain effect on the GBF diameter, and the test results of large specimens also showed this. Therefore, we further investigated the relationship of VHCF life of large specimen and the ratio of the GBF diameter to the inclusion diameter, as illustrated in Figure  11. This ratio basically increases with an increase of fatigue life. This increasing trend is more obvious if the load is constant. But with loads of 575MPa and 550MPa, there are data points at which above ratio decrease with an increase of fatigue life. There is an increasing trend of VHCF life when above ratio increases. The crack initiation life increases with a decrease of the inclusion diameter, and the crack propagation life increases with an increase of the GBF diameter.
The Fatigue Behavior and Mechanism of large FV520B-I specimens in a Very High Cycle Regime ·9· Figure 11 Relationships of VHCF life of large FV520B-I specimens with the ratio of the GBF diameter to the inclusion diameter

The Predictions of VHCF Strength and VHCF Life
The Murakami model is widely used to predict the VHCF strength. In the case of containing subsurface inclusions, the fatigue strength can be expressed as [10,11]: ( ) Where HV is the Vickers hardness, in units of kgf· mm -2 and a is the inclusion radius, in units of μm. For large specimens, HV of 380 kgf· mm -2 and a of 8.2μm are substituted into Eqs. (11) and (12), and σw equals 500MPa and 510MPa, respectively. The fatigue strength at 10 9 cycles of large specimens was about 525MPa which was obtained by the S-N curve in Figure 3, and the prediction value of the modified model was closer to the test result. From section 4.1, the possible maximum inclusion diameters obtained by SEV were 21.91μm and 23.82μm for small specimens and large specimens. The Vickers hardness of 380kgf· mm -2 and the possible maximum inclusion radii of 10.96μm and 11.91μm were substituted into Eq. (11), and the low limits of fatigue strength of the two groups of specimens were 477MPa and 470MPa, respectively. The possible maximum inclusion diameter slightly increased and the fatigue strength slightly decreased because the control volume slightly increased. Considering the effects of the inclusion shape, the Vickers hardness of 380kgf· mm -2 , and the possible maximum inclusion radii of 10.96μm and 11.91μm were substituted into the Eq. (12), and the low limits of fatigue strength of two groups of specimens were 487MPa and 480MPa, respectively. The VHCF life is mainly consumed in the GBF formation, and the GBF formation is due to the synergistic actions of hydrogen embrittlement and fatigue. Also, the maximum GBF diameter of small specimens was 121μm [5] and the maximum GBF diameter of large specimens was 130μm as obtained from Table 3, which are smaller than the usual definition size (250-300μm) of a corrosion fatigue initiation crack [25]. So we assumed that the GBF formation is similar to the initiation of the corrosion fatigue crack. As most VHCF cracks initiate from subsurface inclusions, the subsurface inclusion was further regarded as a spherical cavity. Then we tried to predict the VHCF life by the initiation life of the corrosion fatigue crack around the spherical cavity.
Concerning the corrosion fatigue of high strength steel, crack initiation is usually caused by hydrogen embrittlement. Under the action of cyclic stress, the movement of dislocation transports hydrogen atoms to the triaxial stress region in the notch root. Then hydrogen embrittlement occurs and results in corrosion fatigue. Hirose thought that the initiation of the corrosion fatigue crack (CFC) is similar to the initiation of the stress corrosion crack, and derived a CFC initiation life model of a specimen with a notch by the dislocation dipole model [26]: (13) B1 and B2 are the constants related to the hydrogen diffusion coefficient and the loading frequency, is the stress intensity factor amplitude in the notch root, is the curve radius of the notch, Ni is the CFC initiation life, and the CFC initiation threshold is equal to the yield shear strength of high strength steel [26]. In the situation of a VHCF crack initiating from a subsurface inclusion, we assumed that the inclusion was a spherical cavity and approximately calculated by regarding the spherical cavity as a circular crack of infinite volume [27], then: Where a is the inclusion radius, ρ is equal to the spherical cavity radius and the inclusion radius a: Han, Zhang et al. (15) Assume Ni equals the GBF formation life, and is approximately equal to the VHCF life Nf. The yield shear strength is estimated by 0.7 times yield strength. Then we can obtain the equation: (16) There is a linear relationship between and according to Eq. (16). The relevant test data of small specimens were linearly fitted via the least squares principle and the result is shown in Figure 12. We obtained B1=333.2, B2=-16.80, and r2=0.871. Compared with the fitting result based on crack propagation in the GBF following the Paris law [5], the CFC initiation life model coincides well with the test data, and indirectly confirms the feasibility of research on the VHCF life by CFC initiation. Also, the CFC initiation life model only reflects the relationship of stress amplitude and fatigue life, so it was determined that the stress amplitude seems to be the main factor influencing the VHCF life.

Figure 12
The fitting results of test data of small specimens by the corrosion fatigue crack initiation life model.
The fitting results of the test data of large specimens by the corrosion fatigue crack initiation life model is shown in Figure 13. From Figure 13 and Eq. (16), we obtained B1=310.35, B2=-17.13, and r2=0.563. Compared with the fitting for the VHCF life of small specimens by the above model, the fitting deviation of the VHCF life of large specimens seems large. There is an obvious linear relationship of σa and lnNf for small specimens and the weaker linear relationship of σa and lnNf for large specimen result in the above phenomenon, and the weaker linear relationship of σa and lnNf for large specimens is caused even less by the fatigue life under 575MPa than the fatigue life under 600MPa. The heat effects of large specimens were significantly higher than that of small specimens, which may be the main reason for the above abnormal phenomenon.

Figure 13
The fitting results of test data of large specimens by the CFC initiation life model

Conclusions
It can be concluded that: (1) For FV520B-I, the VHCF mechanism did not change as specimen size increases. The S-N curve moves downward, and fatigue life is reduced and the fatigue strength up to 10 9 cycles decreases for large specimens. The main reason for the above phenomenon is that the control volume increases with an increase of specimen size, and the maximum inclusion size in large specimens is increased. (2) There are no obvious relationships between the VHCF life with the fish-eye diameter, with the distance from the crack origin to the specimen surface. When the load is constant, the VHCF life basically increases with an increase of the ratio of the GBF diameter to the inclusion diameter and with a decrease of inclusion diameter. The VHCF life is mainly consumed in the GBF formation.