The influence of interacting small defects on the fatigue limits of a pure iron and a bearing steel

This study investigates the effect of interacting small cracks on the fatigue limits of two types of pure iron of varying strengths and of a bearing steel. The experimental results revealed that although the fatigue limits were essentially controlled by the mechanics of interacting cracks, the non-propagating crack features and the severity of crack coalescence varied greatly among the different materials. In addition, it is demonstrated that the material-independent, analytical, interaction-criterion alone is not sufficient to estimate interaction effects in reality. This research shows that the material effect can be considered in terms of hardness and non-propagating crack size.


Introduction
Metallic engineering components possess numerous small natural defects which could potentially become sites for fatigue crack initiation. For example, such defects may be the result of manufacturing, machining and/or surface finishing. The effect of a single small defect on fatigue strength has been comprehensively investigated in past studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. According to a basic theory of elasticity, the sole factor to determine the fatigue limit is stress concentration factor. However, it is not crucial for controlling the fatigue limit, since in many metallic materials, the fatigue limit is defined by the non-propagation condition of cracks which have emanated from small defects. Once a crack emanates from the defect, grows till a certain length and finally becomes non-propagating, then the final state is a crack. In such a case, fatigue limit can be evaluated by fracture mechanics parameters, i.e. stress intensity factors, instead of stress concentrations. Therefore, a small defect can be considered to be mechanically equivalent to a small crack from the viewpoint of the fatigue limit. [1].
Although the afore-mentioned concept considers a single defect, real materials contain multiple defects that can interact if they are in close proximity. Ultimately, the occurrence of such an interaction results in a decreased fatigue limit. Defect interaction can happen under diverse circumstances, such as in the case of casting defects [1,2,4,14], weld imperfections [16][17][18], surface roughness [1,[19][20][21], non-metallic inclusions [1,[22][23][24] and so forth. In recent times, the presence of multiple defects in additively manufactured components has led to a significant reduction in their fatigue strength, consequently underscoring the importance of understanding the precise interaction effect in fatigue [25][26][27][28][29][30].
Some analytical studies of interaction problems of three-dimensional surface cracks have previously been conducted. For example, Murakami & Nemat-Nasser [31,32], Nisitani & Murakami [33], Åman [34] and Noda et al. [35] have evaluated stress intensity factors for interacting 3D cracks using the body force method, whereas Patel et al. [36] and Newman & Raju [37] have utilized finite element method. In addition, Zhu et al. simulated the propagation and coalescence of multiple cracks by a probabilistic method [38]. Also, experimental studies of interaction problems have been published. For instance, Li et al. [39] and Sonsino et al. [40] investigated the influence of porosity on the fatigue strength of aluminum alloys. Chen et al. [41], Silva et al. [42] and Al-Oiwaisi et al. [43] examined the interacting corrosion-induced cracks in pipes. Galatolo & Lazzeri [44] studied not only the interaction of two cracks but also the interaction of a crack and a circular hole, which sometimes arises in aircraft components. Deguchi T et al. [45] reported that the coalescence of non-propagating cracks emanating from graphite particles of a plain ductile cast iron specimen did not decrease the fatigue limit. Despite the existence of aforementioned studies, the authors are not aware of experimental works that have systematically studied the effect of interacting defects on the fatigue limit -except the previous research by authors themselves [46].
Among afore-mentioned previous researches, one of the most important analytical finding has been the concept of critical distance, i.e., the space between cracks at which the interaction effect becomes negligible [32]. The analytical critical distance can be explained as follows: if there is sufficient space between two cracks to allow for the insertion of an additional crack of a size similar to the smaller one, the maximum Mode I stress intensity factor, K I, max , is approximately equal to that of the larger crack in isolation, see Fig. 1 [1,32,34].
Since its proposal by Murakami et al. [47], the area parameter model has been widely applied for the prediction of fatigue limits influenced by small cracks or defects. The fatigue limit under a fullyreversed condition can be predicted using the area parameter model, as expressed by the following equation [47]: where, w is the fatigue limit, HV is the Vickers hardness and area is the square root of the defect area, projected to the plane perpendicular to the maximum principal stress. In order to include the effect of interaction into Eq. (1), the effective area is defined in consideration of the analytical critical distance ( Fig. 1) [1,32,34]. Specifically, if the spacing between defects was smaller than the analytical critical distance, i.e., if s < d 2 , where s is the spacing between defects and d 2 is the diameter of the smaller defect, the defects were assumed from the outset to behave like a coalesced single defect, formed by the fusion of two defects with a smooth contour. Otherwise, only the area of the larger defect was taken into account in the fatigue limit prediction process. The afore-mentioned interaction criterion was derived based on the increase in stress intensity factor(s) caused by the neighboring crack. However, this did not consider the crack growth at and below the fatigue limit, which subsequently decreases the distance between two interacting cracks, even though the cracks become non-propagating after a length of crack growth. Such a fact can lead to a discrepancy between the analytical critical distance and the real critical distance, which can only be determined experimentally. Therefore, in order to establish a comprehensive rule of interaction, empirical verification of existing criteria is necessary for various materials.
With respect to the validity of the area parameter model, this study provisionally explores the effect of defect interaction on the fatigue limits in diverse materials. The new experimental results relative to pure irons and a bearing steel all complement previous research involving medium carbon steel [46]. It had been concluded that the materials microstructure influenced the interaction effect primarily via the characteristics of non-propagating cracks. In order to elucidate the material outcomes, several data from the existing literature were also analyzed in support of suggestions advanced in this study. It is evident that the analytical and material-independent interaction criteria (see also BSI standard [48]) are on their own insufficient for the prediction of interaction phenomena and their real-life influence on fatigue strength.

Materials
The materials investigated were pure iron JIS-SUY1, hereafter referred to as SUY1, and bearing steel JIS-SUJ2, henceforth labeled as SUJ2. Since the HV of the as-received SUY1 (HV = 165) was close to the HV of a previously-examined, medium-carbon steel, JIS-S45C [46], henceforward identified as S45C (HV = 186), some SUY1 specimens were annealed to obtain a lower HV. Specimens were annealed at 600°C for one hour, followed by furnace-cooling at room temperature. The resulting HV was 110. In order to distinguish the differently heattreated SUY1 specimens, annealed SUY1 specimens are hereafter labeled as A-SUY1 and non-annealed ones as NA-SUY1. Initially, the SUJ2 was heat-treated at 840°C for one hour in a deoxidizing gas, then subsequently oil-quenched and tempered at 240°C for two hours to obtain the HV of 710. The microstructures of all investigated materials are displayed in Fig. 2. Fig. 2 (a) and (b) illustrate that the grain size of SUY1 remains the same both before and after annealing. Fig. 2 (c) shows that SUJ2 had typical martensitic microstructure including precipitated carbides around 10 µm in diameter. Tables 1 and 2 document the mechanical properties and chemical compositions of the materials, respectively.

Specimens and artificial defects
The specimen geometry is exhibited in Fig. 3. The gage-section surface was mirror-polished with a diamond paste in the case of SUY1 and electro-polished for SUJ2. In most cases, two small holes, diameter = depth = 200 µm in SUY1; 100 µm in SUJ2, were drilled onto the specimen surfaces at different spacings, s, varying as 0.5d 2 , d 2 or 1.5d 2 , as shown in Fig. 4(a). In order to investigate the effect of stress gradient near the notch tip, sharp notches, having the same area as the drilled hole, were also introduced onto the surface of SUJ2 Threshold stress intensity factor range for a small crack ΔK th, lc Threshold stress intensity factor range for a long crack σ w Fatigue limit σ w, exp Experimentally-determined fatigue limit σ w, pred Predicted fatigue limit specimens, as revealed in Fig. 4(b). Sharp notches were made by Electric Discharge Machining (EDM) method. The notch root radius, ρ, of EDM notches ranges from 3.6 to 4.0 µm, whereas the ρ of drilled holes is half of the diameter.

Fatigue testing
Fatigue tests were performed using servo-hydraulic testing machines under fully-reversed, tension-compression loading (stress ratio, R = −1), at test frequencies of 10-20 Hz. Fatigue limits were determined by testing at stress steps of 10-20 MPa. Each fatigue limit was defined as the maximum stress amplitude at which a specimen ceased to fail after 10 7 cycles. Each specimen was used only once at the constant stress amplitude. The fatigue limit was predicted using the area parameter model referenced in Eq.1 [47]. The effective area was established based on the analytical critical distance criteria, as explained in the previous section.  Table 3. Regarding the NA-SUY1 samples, the fatigue limit at the analytical critical distance (s = d 2 ) was the same as that of a single defect (160 MPa). When s = 0.5d 2 , the fatigue limit decreased to 140 MPa, due to crack coalescence. Moreover, when s = 1.5d 2 , the fatigue limit was marginally lower than that of a single defect (150 MPa). This is most probably due to a normal scatter, which typically results from differences in local microstructure. Similar scatter can be observed also in other materials, see e.g. [46]. The local microstructure causes scatter not only in the fatigue limit but also in the fatigue life. For example, in Fig. 5 (a), the life of a single hole specimen is shorter than that of a specimen with two holes under the same loading condition. On the other hand, the fatigue limit of A-SUY1 was 140 MPa regardless of the spacing between the defects, see Fig. 5 (b). The fatigue limit of SUJ2 was greatly dependent on the shape of the particular defect, see Fig. 5 (c). With regard to the drilled holes, the fatigue limit was 500 MPa, whereas in the presence of sharp notches, the fatigue limit was only 380 MPa (s = 0.5d 2 in both cases). It should be noted that the fatigue limit of SUJ2 was determined to be the crack

Non-propagating cracks at fatigue limits
The final state of defects interacting at the fatigue limits varied radically, depending on the materials in question. In the case of SUY1, despite heat treatment, the non-propagation of cracks was evident at the fatigue limit, as illustrated in Fig. 6 (a-f). In order to observe the shape of non-propagating cracks, the specimens were broken by impact loading at −196°C, at which the steel breaks in a brittle manner because of low-temperature embrittlement. Unfortunately, some of them were broken from other location than the non-propagating crack plane and therefore the shapes of those non-propagating cracks are not shown. The non-propagating cracks in A-SUY1 (HV = 110) were approximately 1-mm-long and artificial defects coalesced despite the spacing (s = d 2 or 1.5d 2 ). Furthermore, with respect to A-SUY1, the fatigue limit was the same (140 MPa) regardless of the spacing between defects. Experiments with smaller spacings were not considered relevant, since the interaction was already confirmed to occur at the critical distance s = d 2 and even when s = 1.5d 2 .
Regarding NA-SUY1 (HV = 165), the defects behaved as individual cracks without coalescence at the fatigue limit when the spacing between defects was greater than the critical distance, i.e., when s > d 2 , whereas at s ≤ d 2 , the defects behaved together as a single larger crack, with non-propagating cracks coalescing at the fatigue limit. It is noteworthy that cracks emanating from NA-SUY1 (HV = 165) grew transgranularly, while in A-SUY1 (HV = 110), the cracks essentially grew along grain boundaries, see Fig. 6.
In contrast, in the context of bearing steel SUJ2 (HV = 710), no non-propagating cracks were observed at the fatigue limit in all cases, see Fig. 6 (g-h). Therefore, the crack initiation limit was also considered to be the fatigue limit. Interaction did not occur even at s = 0.5d 2 , implying that the real critical distance for SUJ2 is considerably smaller. The authors therefore elected not to investigate larger spacings. It is clearly demonstrated that the area parameter model [47] can accurately predict fatigue limits, even in the framework of interaction problems. Throughout this research, the effective prediction area was determined according to an assumption of the analytical critical distance in all cases. Fig. 7 shows that the area parameter model can apparently also predict the SUJ2 drilled-hole fatigue limit, that is, the crack initiation limit. However, since the area parameter model is based on fracture mechanics, it may be just pure coincidence that it can also predict the crack-initiation limit of a drilled hole. In reality, since a drilled hole behaved precisely like a blunt notch in SUJ2, it would therefore have been more appropriate to use some of the well-established, fatigue-notch methods. The SUJ2 results are discussed in further detail later in this section.

Comparison of the predicted and experimental fatigue limits
To investigate the minimum size of the detrimental defect in SUY1, which behaves as a fracture origin instead of the persistent slip bands in large grains, grain sizes were inspected based on the statistics of extremes [49] in a manner similar to that recorded in the literature [1]. Grain size analysis was undertaken for NA-SUY1. It was not repeated for A-SUY1 since grain size was not influenced by annealing, see Fig. 2(a) and (b). The results of the grain size analysis are documented in Fig. 8. As shown in Fig. 2, the average grain size of SUY1 was approximately 60 µm, while the largest grain size was approximately 200 µm, according to Fig. 8. The result inferred that a defect smaller than 200 µm would not be expected to become a fracture origin. Under fatigue loading, the fracture origin in low-and moderate-strength materials is typically the largest grain on the surface. The persistent slip bands create cumulative dislocations within a grain, eventually penetrate the entire grain and become a crack [1]. Therefore, larger grains would naturally result in larger naturally occurring cracks and defects smaller than the largest grain would not become fracture origins [1]. Consequently, drilled holes of 200 µm were employed during the SUY1 experiments. It is noteworthy to mention that the absolute size of defects does not influence the analytical interaction criteria, since the interaction (or not) of cracks is determined by the ratio, s/d 2, (i.e., if s/ d 2 > 1, the interaction effect is negligible).

Table 3
Summarized results of this research and results from [46]. Notations are shown in Fig. 4.

The influence of non-propagating crack size on crack coalescence and fatigue limit
The material-independent analytical models are based on variations in elastic stress distributions and an increase in stress intensity factors due to the existence of neighboring defects. In view of the experimental findings presented in this study, it is clear that the analytical interaction criteria must be considered with caution in practical applications. This is because the behavior of interacting small defects at the fatigue limit varies between materials due to differences in the characteristics of non-propagating cracks. For example, it has been suggested that defect orientation [12], stress ratio [11,50] and dual-phase microstructure [46] can influence the size and location of non-propagating cracks. However, notwithstanding other factors, a non-propagating crack usually tends to appear as a result of plasticity-induced crack-closure [51]. Thus, HV, which reflects materials resistance to plastic deformation, can be one of the most relevant material properties in the description of non-propagating crack characteristics. To illustrate this phenomenon, several data from literature [12,46,[52][53][54][55][56][57][58][59][60] have been plotted in Fig. 9. All data were tested at R = −1. The data featured in Fig. 9 are reproduced in Table A1 in Appendix A. Fig. 9 reveals that the size of a non-propagating crack tends to diminish with an increase in hardness, although a large scatter exists. Moreover, it is evident that a non-propagating crack can hardly exist when HV > 400. During the course of this study, non-propagating cracks were also not observed at the fatigue limit in SUJ2 (HV = 710). Since experimental data are limited and not every material can be tested individually, it is very difficult to provide the exact values of HV and/or non-propagating crack size for which certain interaction criteria are always applicable. This greatly complicates the defect interaction problem. However, regarding the current experimental results presented herein and in the previous by Åman et al. [46], it is possible to consider the extreme cases of very low and very high HV, as well as intermediate HV cases separately.

The interaction effect in soft steels
For instance, in the context of the softest material tested, A-SUY1 (HV = 110), the non-propagating cracks are of particular interest, on display in Fig. 6 (e) and (f). The total length of coalesced non-propagating cracks, regardless of the spacing between defects, was approximately 1 mm, i.e., five times larger than the initial defect size. According to analytical interaction criteria, the defects should be acknowledged to be individual if the spacing between them is larger than the diameter of the smaller defect, i.e., when s > d 2 . Since the defects definitely behaved in the manner of a larger single defect, even when s = 1.5d 2 , the analytical interaction criteria apparently does not apply to materials with non-propagating cracks as long as those observed for A-SUY1. It is interesting to note that the fatigue limit was the same (140 MPa) regardless of the spacing between defects and despite crack coalescence. Nevertheless, if the spacing between defects were to increase beyond a certain value, crack coalescence might begin to decrease the fatigue limit, as the newly-formed coalesced crack may exceed the material's crack-propagation threshold simply due to its size. From a fatigue limit viewpoint, this implies that the real critical distance of such a material is in fact longer than the analytical one.

The interaction effect in hard steels
On the contrary, in the high-strength bearing steel SUJ2 (HV = 710), unlike in every other material tested, the interacting defects behaved clearly as individual defects, even when the spacing between them was smaller than the analytical critical distance. No defect interaction was observed for s = 0.5d 2 , which is significantly smaller than the proposed analytical critical distance, s = d 2 . If the spacing between the defects would have been further decreased, it could be expected that the stress in the region between the defects would reach the crack initiation limit and defects would have coalesced. As referenced in Fig. 9, since non-propagating cracks can hardly exist when HV > 400, the fatigue limit of such higher-strength steels is determined more or less by crack initiation. Consequently, the real critical distance for high-strength steels correlates with the crack-initiation threshold.

Small crack or large crack?
The evaluation of fatigue crack growth and its threshold via linear elastic fracture mechanics (LEFM) postulates that the small scale yielding condition holds. This requires that the plastic zone size is sufficiently small compared to the crack size and remains constant under a constant stress intensity. In such a regime, ΔK th becomes independent of crack size. On the other hand, when the defect is small and applied stress level is high, the plastic zone is no longer small compared to the crack size. In such a regime, the crack-tip yielding satisfies the large scale yielding condition, where ΔK th is decreased with    a decrease in crack size. The area parameter model targets the latter case. Namely, this model is applicable to the small crack, but not to the large crack. As exhibited in Fig. 7, the fatigue limit of the SUJ2 having drilled hole was in good agreement with the prediction by the area parameter model (i.e., σ w,exp /σ w,pred = 0.89). On the contrary, the fatigue limit of sharp-notched SUJ2 was well below the predicted value (i.e., σ w,exp / σ w,pred = 0.68). This phenomenon can be understood as follows. Fig. 10 depicts the ΔK th for various materials as a function of the defect/crack size [46,56,[61][62][63][64][65]. As pointed out by Chapetti [66], the small/large crack-transition size is more or less dependent on HV, i.e., the higher the HV, the smaller the transition size. According to Fig. 10, the transition point of area is approximately 50 μm for SUJ2, which is smaller than the size of the defects used in the present tests. Therefore, the area parameter model overestimates the crack growth threshold in the large crack regime (dashed lines in Fig. 10). In addition, in the present experiments, the fatigue limits of the drill-holed and sharp-notched specimens were both determined from crack initiation that is controlled by the stress concentration of the defect. As a result, the fatigue limit of a drill-holed specimen is higher than that of a sharp-notched specimen.

Crack or notch?
As expounded by Nisitani [67] and shown in Fig. 11, the critical notch-root radius, ρ 0 , is a material-dependent parameter that determines whether a defect behaves like a blunt notch or as a crack. If the notch root radius ρ is smaller than ρ 0 , two fatigue limits, one for crack initiation and the other for crack propagation, can be distinguished. In a variety of materials, the ρ 0 typically measures about 0.4-0.5 mm, but can decrease to less than 0.1 mm as tensile strength increases [67]. The experimental results infer that the ρ 0 of SUJ2 (σ 0.2 = 2131 MPa) is likely to be so small that the crack initiation and propagation limits were not possible to distinguish and that the fatigue limit was determined from crack initiation. The values of ρ 0 may also vary depending on the notch size for small defects, i.e., length or depth of the notch [67]. In fact, Schönbauer [64] performed ultrasonic fatigue tests of stainless steels (UTS = 878-1030 MPa) and established the ρ 0 to be less than 25 µm. The authors are not aware of any method to reliably predict ρ 0 other than experimentally, especially for the defects smaller than 1 mm. In cases where the notch root radius in a component is larger than ρ 0 , notch-based methods to assess fatigue strength are preferred rather than fracture mechanics-based approach. However, in reality, as was pointed out by Murakami [1], the defects generally include locally higher stress concentrations and therefore fracture mechanics-based evaluation can provide a reasonable prediction in many instances.

The interaction effect in moderate strength steels
In the moderate-strength steels NA-SUY1 (HV = 165) and S45C (HV = 186) [46], the analytical critical distance criteria seem to hold well when compared with the two afore-mentioned extreme cases. Yet, the dual-phase microstructure especially tends to exhibit larger scatter in non-propagating crack size and location(s), leading to larger scatter in fatigue limit [46]. For a material with such a complex microstructure, the selection of conservative criteria is recommended, i.e., the interaction should be assumed to occur even when s = d 2 .

Future works
In the authors' previous work [46], it was revealed that if d 1 = 2d 2 , only the larger defect would control the fatigue limit, regardless of the existence of the smaller defect and of the spacing between the defects. For this reason, two defects of identical size were employed in this study. Notwithstanding, it is yet unclear whether this applies to any material and what actually are the boundary conditions for the size effect in interaction. Another interesting future work would be to determine the threshold condition for crack initiation and to estimate the real critical distance in materials exhibiting non-propagating cracks larger than the initial defect size. In addition, the interaction effect under diverse loading conditions is of interest, as well as the interaction study considering problems including different defect shapes, orientations and location configurations.

Conclusions
The two defects were introduced onto the surface of pure iron SUY1 and bearing steel SUJ2 specimens in order to investigate the fatigue limit interaction effect and its material dependency. The experimental results complemented those previously obtained on the medium carbon steel JIS-S45C. The summarized results underlined the fact that the analytical interaction criteria are insufficient for identifying the interaction effect in all materials. The analytical models are based on the variations in elastic stress distributions due to the existence of neighboring defects, neglecting the features of non-propagating cracks. The following conclusions were drawn from the experimental findings: • Defects coalesced at the fatigue limit in annealed JIS-SUY1 (HV = 110), regardless of the spacing between them. Non-propagating cracks were approximately five times larger than the initial defect size. The fatigue limit and non-propagating crack sizes were independent of the spacing between the defects.
• Since no non-propagating cracks were observed in JIS-SUJ2, the fatigue limit was determined from crack initiation. Neither defect type, drilled holes nor sharp notches, interacted at the fatigue limit even when the spacing between the defects was less than the analytical critical distance.
• The analytical critical distance applies only for moderate HV materials. In reality, it appears that the critical distance is smaller than the analytical one in high-strength steels (HV > 400) and vice versa, when low-strength steels are concerned.
• Murakami's area parameter model also accurately predicted the fatigue limits of interacting small defects. The only exception involved the bearing steel SUJ2, the fatigue limit of which was identified as the crack initiation limit. Thus, the fracture mechanicsbased area parameter model was not applicable.
• The characteristics of non-propagating cracks significantly affect crack coalescence and consequently, the behavior of interacting cracks, thereby rendering the interaction phenomena complicated. Non-propagating crack size tends to decrease with the increased hardness of a material, although there is a large scatter.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  Table A1.