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Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a Bearing Steel in a Very High Cycle Regime

Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a Bearing Steel in a... applied sciences Article Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a Bearing Steel in a Very High Cycle Regime 1 , 2 3 4 Tatsuo Sakai *, Akiyoshi Nakagawa , Yuki Nakamura and Noriyasu Oguma Research Organization of Science and Technology, Ritsumeikan University, Kusatsu 525-8577, Japan Hitachi Industrial Products, Ltd., Osaka 530-0005, Japan; akiyoshi.nakagawa.ue@hitachi.com National Institute of Technology, Toyota College, Toyota 471-8525, Japan; nakamura@toyota-ct.ac.jp Faculty of Engineering, University of Toyama, Toyama 930-8555, Japan; oguma@eng.u-toyama.ac.jp * Correspondence: sakai@se.ritsumei.ac.jp Abstract: In S-N diagrams for high strength steels, the duplex S-N curves for surface-initiated failure and interior inclusion-initiated failure were usually confirmed in the very high cycle regime. This trend is more distinct in the loading type of rotating bending, due to the stress distribution across the section. In the case of interior failure mode, the fish-eye is usually observed on the fracture surface and an inclusion is also observed at the center of the fish-eye. In the present work, the authors attempted to construct a probabilistic model on the statistical fatigue property in the interior failure mode, based on the distribution characteristics of the location and the size of the interior inclusion at the crack initiation site. Thus, the P-S-N characteristics of the bearing steel (SUJ2) in the very high cycle regime were successfully explained. Keywords: probabilistic model; very high cycle fatigue; duplex S-N curves; rotating bending; inclusion; fish-eye; inclusion size; inclusion depth Citation: Sakai, T.; Nakagawa, A.; Nakamura, Y.; Oguma, N. Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a 1. Introduction Bearing Steel in a Very High Cycle From both viewpoints of economical design of mechanical structures and reduction Regime. Appl. Sci. 2021, 11, 2889. of the environmental load, elongation of the design life for every product in the wide https://doi.org/10.3390/app11072889 range of industries, is an important subject. In such a circumstance, for train axles and wheels, load bearing parts of automobiles, turbine rotors of the power plants and so on, Academic Editor: Massimo Rossetto the number of loading cycles reaches over 10 cycles, during the service period of the respective mechanical components [1–4]. One of the typical aspects of fatigue property in Received: 24 February 2021 the very high cycle regime for structural steels is the fact that the duplex S-N characteristics Accepted: 19 March 2021 consisting of the respective S-N curves for the surface-initiated and the interior-initiated Published: 24 March 2021 failures are observed, as reported by many researchers [3,5–7]. The term “S-N curve” refers to the relationship between the stress level [S] and the number of stress cycles to failure Publisher’s Note: MDPI stays neutral [N]. Expressions of S-N property and S-N characteristics are also used in this paper, based with regard to jurisdictional claims in on a similar sense. In the case of interior-initiated failure, an inclusion is usually found at published maps and institutional affil- the core portion of the fish-eye formed on the fracture surface [3,5,8]. Thus, the interior iations. inclusion at the crack initiation site plays a dominant role to govern the fatigue strength and fatigue life of the specimen. It is well-known that the fatigue limit of the metallic material can be evaluated by combining the material hardness and the concept of area for the defect size proposed Copyright: © 2021 by the authors. by Murakami et al. [9]. Another important factor to control the fatigue property is the Licensee MDPI, Basel, Switzerland. location of the inclusion, since the stress distribution across the section has a steep slope in This article is an open access article rotating bending [10]. Accordingly, the fatigue property of those metallic materials should distributed under the terms and be analyzed as a function of the size and the location of the interior inclusions, together conditions of the Creative Commons with the hardness. However, both the size and the location of the inclusions inside the Attribution (CC BY) license (https:// material have particular distributions, depending on the fabrication process of the actual creativecommons.org/licenses/by/ metallic materials [11–13]. 4.0/). Appl. Sci. 2021, 11, 2889. https://doi.org/10.3390/app11072889 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 2889 2 of 15 From this point of view, the authors attempted to construct a probabilistic model on the very high cycle fatigue property for the bearing steel of SUJ2 in rotating bending, by combining the distribution characteristics of the size and the location of the interior inclusion. In this model, a two-parameter Weibull distribution was accepted for the distribution pattern of the inclusion size, whereas the random distribution was accepted for the location of the inclusion. Since the scatter of the hardness is sufficiently small, the hardness of this steel was supposed to be constant for the sake of simplicity. Although some number of papers analyzing the statistical aspects of S-N property were published [14–19], the main target of those research is the statistical property within the usual life region, which is less than 10 cycles. Considering this point, the main aim of the present study was set to the physical interpretation of the statistical aspects of the S-N characteristics of a bearing steel, in the very high cycle regime. 2. Experimental Fatigue Test Data Referred in This Study Sufficient number of fatigue test data are required to analyze the statistical fatigue property of metallic materials. However, the fatigue test in the very high cycle regime occupies a long time. For example, at the usual loading frequency of 50–60 Hz, it takes around 200 days to reach the loading cycles of N = 10 . In such a circumstance, the authors extracted a series of fatigue test results obtained as round robin tests by the Research Group on Statistical Aspects of Materials Strength [RGSAMS] [3]. These fatigue tests were performed in rotating bending in very high cycle regime over N = 10 cycles, for a bearing steel of JIS:SUJ2, by using the same type of testing machine and common specimens. Since such fatigue test results give useful data for discussion on the very high cycle fatigue property of metallic materials, they are often referred to by many researchers in this area [3,7,20,21]. All experimental results are plotted as an S-N diagram in Figure 1, in which fatigue test data obtained at different laboratories in seven universities in Japan were indicated altogether. Open symbols represent the S-N data for the surface-induced fracture, whereas solid symbols indicate the results for the interior-induced fracture. Several data points attached arrows around N = 10 , which indicate runout specimens without fracture. The solid and dotted curves indicate the S-N curves fitted to the respective fracture modes, using the JSMS Standard of “Standard Regression Method of S-N Curves”, accepting the semi-logarithmic bilinear model [22]. The fatigue limit for the surface-induced fracture was s = 1278 MPa, as given by the horizontal portion of the solid S-N curve. This conventional fatigue limit cannot provide the true fatigue limit, since fatigue fractures take place in the very high cycle region, at stress levels lower than the horizontal portion of the solid S-N curve. For the experimental data in the interior-induced fracture mode, the linear regression line was indicated by a dotted line. Although the horizontal line was also indicated from the point of N = 10 along the dotted S-N curve, further experimental Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16 reconfirmation would be required in order to make clear the appearance of the horizontal portion in the fracture mode by continuing fatigue tests, until further long life region. Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Typical examples of fracture surfaces in the respective fracture modes are shown in Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-in- duced fracture mode and (b) is the fracture surface in the interior-induced fracture mode [23]. The fatigue crack occurs at the specimen surface in the short life region at a relatively high stress level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture surface in the very high cycle region, at low stress levels, as shown in Figure 2b. In such a case, an inclusion is also found at the core site of the fish-eye. This fact means that the fatigue crack takes place around the inclusion at the crack initiation site, in a very high cycle regime. The main aim of this study was to analyze the statistical distribution char- acteristics of the S-N property in the interior inclusion-induced fracture mode, by combin- ing distribution characteristics of the inclusion size and the inclusion depth. 100μ 100μ (a) Surface-initiated fracture (b) Interior inclusion-initiated fracture 𝜎 = 1800 MPa, N = 2.16 × 10 𝜎 = 1300 MPa, N = 2.66 × 10 Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. For the sake of important reference, static mechanical properties such as the tensile strength and the Vickers’ hardness were examined, prior to performing fatigue tests of this bearing steel. Experimental results of these properties are indicated in Table 1. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16 Appl. Sci. 2021, 11, 2889 3 of 15 Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Typical examples of fracture surfaces in the respective fracture modes are shown in Typical examples of fracture surfaces in the respective fracture modes are shown in Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-in- Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-induced duced fracture mode and (b) is the fracture surface in the interior-induced fracture mode fracture mode and (b) is the fracture surface in the interior-induced fracture mode [23]. The [23]. The fatigue crack occurs at the specimen surface in the short life region at a relatively fatigue crack occurs at the specimen surface in the short life region at a relatively high stress high stress level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture surface in surface in the very high cycle region, at low stress levels, as shown in Figure 2b. In such a the very high cycle region, at low stress levels, as shown in Figure 2b. In such a case, an case, an inclusion is also found at the core site of the fish-eye. This fact means that the inclusion is also found at the core site of the fish-eye. This fact means that the fatigue crack fatigue crack takes place around the inclusion at the crack initiation site, in a very high takes place around the inclusion at the crack initiation site, in a very high cycle regime. cycle regime. The main aim of this study was to analyze the statistical distribution char- The main aim of this study was to analyze the statistical distribution characteristics of the acteristics of the S-N property in the interior inclusion-induced fracture mode, by combin- S-N property in the interior inclusion-induced fracture mode, by combining distribution ing distribution characteristics of the inclusion size and the inclusion depth. characteristics of the inclusion size and the inclusion depth. 100μ 100μ (a) Surface-initiated fracture (b) Interior inclusion-initiated fracture 𝜎 = 1800 MPa, N = 2.16 × 10 𝜎 = 1300 MPa, N = 2.66 × 10 Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. For the sake of important reference, static mechanical properties such as the tensile For the sake of important reference, static mechanical properties such as the tensile strength and the Vickers’ hardness were examined, prior to performing fatigue tests of this strength and the Vickers’ hardness were examined, prior to performing fatigue tests of bearing steel. Experimental results of these properties are indicated in Table 1. this bearing steel. Experimental results of these properties are indicated in Table 1. Table 1. Static mechanical properties of the material. Tensile strength 2316 (MPa) Vickers’ hardness 778 (HV) 3. Probabilistic Model to Explain the Statistical Fatigue Property in Interior-Induced Fracture 3.1. Effects of Size and Depth of Inclusions on the Fatigue Strength According to the area model proposed by Y. Murakami [9], the fatigue limit of a metallic material, s , was well provided by the defect size r and the Vickers’ hardness HV (kgf/mm ), as follows; 1/6 s = a( HV + 120)/ area , (1) where area indicates the area of the defect projected perpendicular to the longitudinal direction of the specimen, and the factor of a was given by a = 1.43 for surface defect and a = 1.56 for interior defect, respectively. When this concept of area was applied to the fatigue strength at N = 10 cycles as an attempt, assuming a spherical interior inclusion with a radius of r, the fatigue strength at N = 10 , s , was given by the following expression, w9 q 1/6 s = 1.56( HV + 120)/ pr . (2) w9 Appl. Sci. 2021, 11, 2889 4 of 15 Since the Vickers’ hardness of this bearing steel (JIS:SUJ2) was reported as HV = 778 [3], Equation (2) was reduced as follows; 1/6 s = 1273r (3) w9 Equation (3) indicates the relationship between the fatigue strength at N = 10 and the inclusion size (radius) for this steel. Such a relationship is depicted in Figure 3. When the inclusion size is small, those inclu- sions have no harmful effect on the fatigue strength, as reported by many researchers [24,25]. From this point of view, if the critical small size is assumed as r = 3 m in this study, the upper bound of the fatigue strength given by Equation (3) becomes s = 1060 MPa, as w9 indicated in Figure 3. On the other hand, an inclusion larger than 40 m (r > 40 m) is seldom found in the present steel. In such an assumption, the lower bound of the fatigue Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16 strength given by Equation (3) becomes s = 689 MPa, as indicated by the horizontal w9 dotted line in Figure 3. Thus, the distribution range of the fatigue strength at N = 10 due to the distribution of the inclusion size would be 689 MPa~1060 MPa. Figure 3. Relationship between fatigue strength s and inclusion radius r. w9 Figure 3. Relationship between fatigue strength  and inclusion radius  . w9 In the case of the rotating bending, the stress distribution across the specimen section has a distinct gradient, such that the stress on the surface gives the maximum and the 3.2. Distribution Characteristics of Inclusion Size and Inclusion Depth stress becomes zero at the center of the section. Accordingly, the inclusion depth is another The inclusion size is not constant, and therefore, the size has its own distribution important factor to govern the fatigue strength of s , even if the inclusion size is fixed w9 to char a certain acteris value. tics pec When uliar the fatigue to the str fabri ength cati at the on depth process x is denoted of the as met s allic , the fatigue material [11–13]. Of w9 strength evaluated by nominal bending stress on the specimen surface, s , is given as course, since the inclusion size 𝜌 is always positive, we havew 9𝜌 > 0. In other words, the follows [26]; lower bound of the inclusion size is assumed “0”. In addition, a size larger than 40 μm is r 1.5 s = s = s (4) w9 w9 w9 seldom found in the bearing steel. Distribution pattern of the inclusion size with the above r x 1.5 x feature is well-represented by a two-parameter Weibull distribution [27], without a loca- where r indicates the radius of the critical portion of the specimen. Here, substituting tion parameter. The probability density function and the cumulative distribution function Equation (3) into Equation (4), we have the following expression; are given as follows: 1.5 1910 1/6 1/6 s =  1273r = r (5) w9 𝑎 −1 𝑎 1.5 x 1.5 x 𝑎 𝜌 𝜌 (6) ( ) 𝑓 𝜌 = ( ) 𝑒𝑥𝑝 {− ( ) }, 𝑏 𝑏 𝑏 Consequently, the fatigue strength of s for the specimen with an inclusion size of r w9 and a depth of x could be calculated by Equation (5). and 3.2. Distribution Characteristics of Inclusion Size and Inclusion Depth ( ) (7) 𝐹 𝜌 = 1 − 𝑒𝑥𝑝 {− ( ) }, The inclusion size is not constant, and therefore, the size has its own distribution characteristics peculiar to the fabrication process of the metallic material [11–13]. Of where a is the shape parameter and b is the scale parameter, respectively. course, since the inclusion size r is always positive, we have r > 0. In other words, the When the number of inclusions included within the critical volume of a specimen is lower bound of the inclusion size is assumed “0”. In addition, a size larger than 40 m is seldom found in the bearing steel. Distribution pattern of the inclusion size with the denoted by n, the inclusion with the maximum size would be the crack starter under the cyclic loadings. Accordingly, the distribution characteristics of the maximum size among n of inclusions in the critical volume become the most important factor to control the fa- tigue strength and fatigue life of the metallic material. Based on the concept of extremes’ distribution [28], the probability density function of the maximum size among n inclu- ( ) sions, 𝑓 𝜌 , is provided by 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 (8) ( ) { ( )} 𝑓 𝜌 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 𝑏 𝑏 𝑏 Although we have to know the original distribution of the inclusion size 𝜌 itself, it is difficult to directly observe such a distribution pattern. In such a circumstance, the au- thors determined the respective parameters of the original size distribution, so as to satisfy the following conditions: [ ] 𝑃 1 μm < 𝜌 < 15 μm = 0.8, 𝐹 = 0.1 at 𝜌 = 1 μm and 𝐹 = 0.9 at 𝜌 = 15 μm. 0 0 This condition for the inclusion size was introduced here as a possible assumption supposed from the observation results by Toriyama et al. [12]. Thus, we obtain the results for the Weibull parameters as 𝑎 = 1.139 and 𝑏 = 7.214, respectively. In addition, since the number of inclusions inside the critical volume, n, is also unknown, the authors ana- lyzed the distribution of the extremes by supposing various values as the number of in- clusions, like n = 2, 3, 5, 6, 10, 20, and 40, respectively. Probability density functions of the maximum inclusion size thus calculated by Equation (8) are depicted in Figure 4, where Appl. Sci. 2021, 11, 2889 5 of 15 above feature is well-represented by a two-parameter Weibull distribution [27], without a location parameter. The probability density function and the cumulative distribution function are given as follows: n   o a1 a a r r f (r) = ex p , (6) b b b and n   o F (r) = 1 ex p , (7) where a is the shape parameter and b is the scale parameter, respectively. When the number of inclusions included within the critical volume of a specimen is denoted by n, the inclusion with the maximum size would be the crack starter under the cyclic loadings. Accordingly, the distribution characteristics of the maximum size among n of inclusions in the critical volume become the most important factor to control the fatigue strength and fatigue life of the metallic material. Based on the concept of extremes’ distribution [28], the probability density function of the maximum size among n inclusions, f (r ), is provided by n   o a1 a a r r n1 n n f (r ) = nfF (r )g ex p (8) n n b b b Although we have to know the original distribution of the inclusion size r itself, it is difficult to directly observe such a distribution pattern. In such a circumstance, the authors determined the respective parameters of the original size distribution, so as to satisfy the following conditions: P[1 m < r < 15 m] = 0.8, F = 0.1 at r = 1 m and F = 0.9 at r = 15 m. 0 0 This condition for the inclusion size was introduced here as a possible assumption supposed from the observation results by Toriyama et al. [12]. Thus, we obtain the results for the Weibull parameters as a = 1.139 and b = 7.214, respectively. In addition, since the number of inclusions inside the critical volume, n, is also unknown, the authors analyzed the distribution of the extremes by supposing various values as the number of inclusions, like n = 2, 3, 5, 6, 10, 20, and 40, respectively. Probability density functions of the maximum inclusion size thus calculated by Equation (8) are depicted in Figure 4, where f (r ) plotted by a fine dotted curve in the most left hand side indicates the original probability density function of the inclusion size. The mode of the largest inclusion size, i.e., the inclusion size at each peak of the probability density function, tends to shift into the right hand side, with an increase in the number of inclusions in the critical volume of the specimen. As described in Section 3.1, the inclusion depth at the crack initiation site, x , plays an important role to control both the fatigue strength and fatigue life of the metallic material, in the case of the rotating bending, due to the steep slope of the stress distribution across the section. Figure 5 indicates the distribution feature of the inclusions projected to the specimen section, assuming random distribution (uniform distribution) [23]. Here, if the inclusion depth from the surface is denoted as x , x , . . . , x , from the most shallow 1 2 n inclusion, one can obtain the one-dimensional distribution of the inclusion depth. As reported in another paper [26], the probability density function of the inclusion depth x can be represented by the following expression: 2 x f x = 1 , 0 < x < r . (9) ( ) [ ] r r Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 16 ( ) 𝑓 𝜌 plotted by a fine dotted curve in the most left hand side indicates the original prob- ability density function of the inclusion size. The mode of the largest inclusion size, i.e., the inclusion size at each peak of the probability density function, tends to shift into the Appl. Sci. 2021, 11, 2889 6 of 15 right hand side, with an increase in the number of inclusions in the critical volume of the specimen. Size of the largest inclusion, 𝜌 (𝜇 m) Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 16 Figure 4. Probability density functions of the largest inclusion size among various numbers of inclusions. Figure 4. Probability density functions of the largest inclusion size among various numbers of inclusions. As described in Section 3.1, the inclusion depth at the crack initiation site, ξ, plays an important role to control both the fatigue strength and fatigue life of the metallic material, in the case of the rotating bending, due to the steep slope of the stress distribution across the section. Figure 5 indicates the distribution feature of the inclusions projected to the specimen section, assuming random distribution (uniform distribution) [23]. Here, if the inclusion depth from the surface is denoted as 𝜉 , 𝜉 , · · , 𝜉 , from the most shallow inclu- 1 2 𝑛 sion, one can obtain the one-dimensional distribution of the inclusion depth. As reported in another paper [26], the probability density function of the inclusion depth ξ can be rep- resented by the following expression: 2 𝜉 𝑓 (𝜉 ) = (1 − ), [0 < 𝜉 < 𝑟 ]. (9) 𝑟 𝑟 In Equation (9), r indicates the radius of the specimen. The probability density of Figure 5. Definition of inclusion depth from surface. Equation Figure 5. Definition (9) becomes of inclusion a maxim depth um at from the surface. specimen surface (ξ = 0), whereas the density becomes zero at the center of the specimen section. As reported by several researchers In Equation (9), r indicates the radius of the specimen. The probability density of [5,29], the depth of the crack initiation site is restricted within the thin surface layer of ξ < Equation (9) becomes a maximum at the specimen surface (x = 0), whereas the density 250 μm in rotating bending. Therefore, the probability density function of Equation (9) becomes zero at the center of the specimen section. As reported by several researchers [5,29], should be normalized as the conditional distribution, under the condition of ξ < 250 μm. the depth of the crack initiation site is restricted within the thin surface layer of x < 250 m Thus, we have the actual probability density function of ξ, as follows: in rotating bending. Therefore, the probability density function of Equation (9) should be 2 𝜉 normalized as the conditional distribution, under the condition of x < 250 m. Thus, we ( ) 𝑓 𝜉 = (1 − ), [0 < 𝜉 < 250 μm], (10) 𝑟 𝐹 𝑟 have the actual probability density function of x, as follows: where 𝐹 indicates the probability giving the condition of 0 < 𝜉 < 250 μm and its prob- 2 x ability becomes 𝐹 = 0.3055 in the condition of the present work. The probability density f (x) = 1 , [0 < x < 250 m], (10) rF r function thus normalized by Equation (10) is depicted in Figure 6. where F indicates the probability giving the condition of 0 < x < 250 m and its proba- Depth of inclusion (mm) bility becomes F = 0.3055 in the condition of the present work. The probability density function thus normalized by Equation (10) is depicted in Figure 6. Figure 6. Probability function of inclusion depth f  . ( ) 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength Distribution In the previous section, distribution characteristics on the size and the depth of the inclusion are discussed, and the probability density functions of 𝑓 (𝜌 ) and 𝑓 (𝜉 ) are de- 𝑛 0 rived as Equations (8) and (10). Since both 𝜌 and ξ are statistically independent random ( ) variables, the joint probability density function, ℎ 𝜌 , 𝜉 , is given by 𝑎 −1 𝑎 𝑎 𝜌 𝜌 2 𝜉 𝑛 𝑛 𝑛 −1 ℎ(𝜌 , 𝜉 ) = 𝑓 (𝜌 ) · 𝑓 (𝜉 ) = 𝑛 {𝐹 (𝜌 )} ( ) 𝑒𝑥𝑝 {− ( ) } · (1 − ) (11) 𝑛 𝑛 0 0 𝑛 𝑏 𝑏 𝑏 𝑟 𝐹 𝑟 By putting n = 5, 𝑎 = 1.139, 𝑏 = 7.214, r = 1.5, and 𝐹 = 0.3055 along the previous ( ) example, the joint probability density function of ℎ 𝜌 , 𝜉 was numerically calculated. ( ) The result of ℎ 𝜌 , 𝜉 thus obtained is depicted in Figure 7, where the joint probability function of ℎ(𝜌 , 𝜉 ) gives the curved surface like a mountain range. In the figure, ∆𝐻 (𝜌 , 𝜉 ) corresponding to the volume of the vertical column indicates the probability that 𝜌 and 𝜉 are within the square region of ∆𝜌 × ∆𝜉 . The dashed line appearing on 𝑛 𝑛 ( ) the 𝜉 − ℎ 𝜌 , 𝜉 plane corresponds to the marginal distribution for the inclusion depth 𝜉 , as given by Equation (10). On the other hand, another dashed curve appearing on the 𝜌 − ℎ(𝜌 , 𝜉 ) plane corresponds to the marginal distribution for the inclusion size of 𝜌 , 𝑛 𝑛 𝑛 given by Equation (12). 𝑐 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 ( ) { ( )} (12) 𝑓 𝜌 = ∫ ℎ(𝜌 , 𝜉 )𝑑𝜉 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 𝑏 𝑏 𝑏 Probability density, ( ) Probability density Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 16 Appl. Sci. 2021, 11, 2889 7 of 15 Figure 5. Definition of inclusion depth from surface. Depth of inclusion (mm) Figure 6. Probability function of inclusion depth f (x). Figure 6. Probability function of inclusion depth f  . ( ) 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength Distribution In the previous section, distribution characteristics on the size and the depth of 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength the inclusion are discussed, and the probability density functions of f r and f x are ( ) ( ) n 0 Distribution derived as Equations (8) and (10). Since both r and x are statistically independent random variables, the joint probability density function, h(r , x), is given by In the previous section, distribution characteristics on the size and the depth of the n   o inclusion are discussed, and the probability density functions of 𝑓 (𝜌 ) and 𝑓 (𝜉 ) are de- a1 a a r r 2 x 𝑛 0 n n n1 h(r , x) = f (r ) f (x) = nfF (r )g ex p  1 (11) n n 0 0 n b b b rF r rived as Equations (8) and (10). Since both 𝜌 and ξ are statistically independent random ( ) variables, the joint probability density function, ℎ 𝜌 , 𝜉 , is given by By putting n = 5, a = 1.139, b = 7.214, r = 1.5, and F = 0.3055 along the previous example, the joint probability density function of h(r , x) was numerically calculated. The 𝑎 −1 𝑎 𝑎 𝜌 𝜌 2 𝜉 𝑛 𝑛 result of h(r , x) thus obtained is depic 𝑛 te − d 1in Figure 7, where the joint probability function { } ℎ(𝜌 , 𝜉 ) = 𝑓 (𝜌 ) · 𝑓 (𝜉 ) = 𝑛 𝐹 (𝜌 ) ( ) 𝑒𝑥𝑝 {− ( ) } · (1 − ) (11) 𝑛 𝑛 0 0 𝑛 of h(r , x) gives the curved surface like a mountain range. In the figure, D H(r , x) corre- n n 𝑏 𝑏 𝑏 𝑟 𝐹 𝑟 sponding to the volume of the vertical column indicates the probability that r and x are within the square region of Dr  Dx. The dashed line appearing on the x h(r , x) plane By putting n = 5, 𝑎 = 1.139, 𝑏 = 7.214, r = 1.5, and 𝐹 = 0.3055 along the previous n n corresponds to the marginal distribution for the inclusion depth x, as given by Equation (10). example, the joint probability density function of ℎ(𝜌 , 𝜉 ) was numerically calculated. On the other hand, another dashed curve appearing on the r h(r , x) plane corresponds n n ( ) The result of ℎ 𝜌 , 𝜉 thus obtained is depicted in Figure 7, where the joint probability to the marginal distribu 𝑛 tion for the inclusion size of r , given by Equation (12). function of ℎ(𝜌 , 𝜉 ) gives the curved surface like a mountain range. In the figure, n   o a1 a a r r ∆𝐻 (𝜌 , 𝜉 ) corresponding to the volume of the vertical column indicates the probability n1 n n f (r ) = h(r , x)dx = nfF (r )g ex p (12) n n n b b b that 𝜌 and 𝜉 are within the square region of ∆𝜌 × ∆𝜉 . The dashed line appearing on 𝑛 0 𝑛 the 𝜉 − ℎ(𝜌 , 𝜉 ) plane corresponds to the marginal distribution for the inclusion depth 𝜉 , In this place, let us divide the r x plane into fine meshes of Dr  Dx and consider n n as given by Equation (10). On the other hand, another dashed curve appearing on the the mesh at the arbitrary point of r , x , as shown in Figure 7. Then, the probability that ( ) the inclusion size r and the inclusion depth x yield within the area of Dr  Dx is provided ( ) n n 𝜌 − ℎ 𝜌 , 𝜉 plane corresponds to the marginal distribution for the inclusion size of 𝜌 , 𝑛 𝑛 𝑛 by the volume of the vertical column raising at this mesh of Dr  Dx . In other words, the given by Equation (12). volume of this column corresponds to the probability that the fatigue strength at N = 10 is given by Equation (5). Thus, we have the following equation, 𝑐 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 ( ) { ( )} (12) 𝑓 𝜌 = ∫ ℎ(𝜌 , 𝜉 )𝑑𝜉 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 P[r 2 Dr \ x 2 Dx] 𝑛 = h(r , x)Dr Dx = P[s 2 Ds ] (13) n n n n w9 w9 𝑏 𝑏 𝑏 Based on the repetition of numerical calculations by Equation (13), the probability density function f (s ) and the cumulative distribution function F(s ) can be analyzed numerically. w9 w9 Probability density Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 16 In this place, let us divide the 𝜌 − 𝜉 plane into fine meshes of ∆𝜌 × ∆𝜉 and con- 𝑛 𝑛 sider the mesh at the arbitrary point of (𝜌 , 𝜉 ), as shown in Figure 7. Then, the probability that the inclusion size 𝜌 and the inclusion depth 𝜉 yield within the area of ∆𝜌 × ∆𝜉 is 𝑛 𝑛 provided by the volume of the vertical column raising at this mesh of ∆𝜌 × ∆𝜉 . In other words, the volume of this column corresponds to the probability that the fatigue strength at N = 10 is given by Equation (5). Thus, we have the following equation, ∗ ∗ 𝑃 [𝜌 ∈ ∆𝜌 ∩ 𝜉 ∈ ∆𝜉 ] = ℎ(𝜌 , 𝜉 )∆𝜌 ∆𝜉 = 𝑃 [𝜎 ∈ ∆𝜎 ] (13) 𝑛 𝑛 𝑛 𝑛 𝑤 9 𝑤 9 Based on the repetition of numerical calculations by Equation (13), the probability ∗ ∗ ( ) ( ) density function 𝑓 𝜎 and the cumulative distribution function 𝐹 𝜎 can be ana- 𝑤 9 𝑤 9 lyzed numerically. Appl. Sci. 2021, 11, 2889 8 of 15 Equation (11) Figure Figure7. 7. Sc Schematics hematics oof f joint joint probabil probability ity d density ensity fun function ction of of 𝜌r and and x𝜉. . 4. Results and Discussions 4. Results and Discussions 4.1. Distribution Characteristics of Fatigue Strength s w ∗9 4.1. Distribution Characteristics of Fatigue Strength 𝜎 𝑤 9 Since the number of inclusions included within the critical volume of a specimen Since the number of inclusions included within the critical volume of a specimen is is unknown, the authors analyzed the distribution characteristics of the fatigue strength, unknown, the authors analyzed the distribution characteristics of the fatigue strength, s , assuming several numbers of inclusions like n = 2, 3, 5, 6, 10, 20, and 40, respectively. w9 𝜎 , assuming several numbers of inclusions like n = 2, 3, 5, 6, 10, 20, and 40, respectively. 𝑤 9 Among them, the analytical results of the fatigue strength distribution obtained for n = 2, 5, Among them, the analytical results of the fatigue strength distribution obtained for n = 2, and 20, by the method in the previous section are indicated in Figure 8 as typical examples. 5, and 20, by the method in the previous section are indicated in Figure 8 as typical exam- Figure 8a shows the probability density functions under these three cases of n = 2, 5, and ples. Figure 8a shows the probability density functions under these three cases of n = 2, 5, 20, whereas Figure 8b indicates the cumulative distribution functions corresponding to and 20, whereas Figure 8b indicates the cumulative distribution functions corresponding the respective cases. Weibull parameters a and b in Equation (8) are provided to satisfy to the respective cases. Weibull parameters a and b in Equation (8) are provided to satisfy the condition of P[1 m < r < 15 m] = 0.8. It is found that the peak (mode) and the Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16 [ ] the condition of 𝑃 1 μm < 𝜌 < 15 μm = 0.8. It is found that the peak (mode) and the standard deviation of the fatigue strength distribution tends to decrease with an increase standard deviation of the fatigue strength distribution tends to decrease with an increase in the inclusion number. in the inclusion number. 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 800 1000 1200 1400 1600 Fatigue strength (MPa) (a) Probability density functions 1.0 Figure 8. Cont. 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Cumulative distribution functions Figure 8. Distribution patterns of fatigue strength. In order to investigate the effect of the condition for the dispersion of the inclusion size of 𝜌 , similar analyses were performed, giving some other conditions of 𝑃 [1 μm < 𝜌 < 10 μm] = 0.8 and 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8. Among these analyses, only the results of the cumulative distribution functions are indicated in Figure 9, due to article page limit. As seen in Figures 8b and 9, median of the fatigue strength distribution tends to increase with a decrease of the dispersion of the inclusion size, while standard deviation of the fatigue strength distribution tends to decrease a little, depending on the decrease of the dispersion of the inclusion size. Comparing these analytical distribution characteristics of the fatigue strength 𝜎 in 𝑤 9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the [ ] fatigue strength 𝜎 under the conditions of 𝑃 1 μm < 𝜌 < 15 μm = 0.8 and n = 5 was 𝑤 9 roughly in agreement with the experimental distribution aspect of the fatigue strength at N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. Probability density Cumulative probability Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 Appl. Sci. 2021, 11, 2889 9 of 15 1000 1200 1400 1600 Fatigue strength (MPa) (a) Probability density functions 1.0 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Cumulative distribution functions Figure 8. Distribution patterns of fatigue strength. Figure 8. Distribution patterns of fatigue strength. In order to investigate the effect of the condition for the dispersion of the inclusion size of r, In order to investigate the effect of the condition for the dispersion of the inclusion similar analyses were performed, giving some other conditions of P[1 m < r < 10 m] = 0.8 size of 𝜌 , similar analyses were performed, giving some other conditions of 𝑃 [1 μm < 𝜌 < and P 1 m < r < 5 m = 0.8. Among these analyses, only the results of the cumulative [ ] 10 μm] = 0.8 and 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8. Among these analyses, only the results of distribution functions are indicated in Figure 9, due to article page limit. As seen in the cumulative distribution functions are indicated in Figure 9, due to article page limit. Figures 8b and 9, median of the fatigue strength distribution tends to increase with a As seen in Figures 8b and 9, median of the fatigue strength distribution tends to increase decrease of the dispersion of the inclusion size, while standard deviation of the fatigue Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 16 with a decrease of the dispersion of the inclusion size, while standard deviation of the strength distribution tends to decrease a little, depending on the decrease of the dispersion fatigue strength distribution tends to decrease a little, depending on the decrease of the of the inclusion size. dispersion of the inclusion size. Comparing these analytical distribution characteristics of the fatigue strength 𝜎 in 1.0 𝑤 9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the 0.8 fatigue strength 𝜎 under the conditions of 𝑃 [1 μm < 𝜌 < 15 μm] = 0.8 and n = 5 was 𝑤 9 roughly in agreement with the experimental distribution aspect of the fatigue strength at 0.6 N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. 0.4 0.2 0.0 600 800 1000 1200 1400 Fatigue strength (MPa) [ ] (a) Results in 𝑃 1 μm < 𝜌 < 10 μm = 0.8 1.0 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Results in 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8 Figure 9. Comparison of cumulative distribution functions in respective conditions. Figure 9. Comparison of cumulative distribution functions in respective conditions. 4.2. Expansion of the Probabilistic Model to Analyze P-S-N Characteristics in Interior-Induced Fracture Target of the probabilistic model described in Section 3 is the distribution character- istics of the fatigue strength only at stress cycles of N = 10 . Accordingly, the analytical model should be conceptionally expanded to interpret the whole statistical aspect of the S-N property, in the interior inclusion-induced fracture mode. From this point of view, denoting the fatigue strength at any stress cycle, 𝑁 , by 𝜎 , Equation (3) is rewritten as follows: −1/6 𝜎 = 1273 · 𝜌 + 𝛽 , (14) 𝛽 = 𝜆 {𝑙𝑜𝑔 (𝑁 )} + 𝛾 . (15) Based on the dotted S-N curve in Figure 1, parameters λ and γ in Equation (15) were 6 ∗ 9 ∗ ( ) ( determined from two points 𝑁 = 10 , 𝜎 = 1424 MPa and 𝑁 = 10 , 𝜎 = 𝑤 9 𝑤 9 920 MPa ) as λ = −168 and γ = 1512, respectively. Substituting these values into Equations (14) and (15), Equation (14) is rewritten as −1/6 𝜎 = 1273 · 𝜌 − 168 · {𝑙𝑜𝑔 (𝑁 )} + 1512. (16) 6 7 Here, 𝜎 − 𝜌 relationships under the fixed numbers of the fatigue life of N = 10 , N = 10 , 8 9 N = 10 , and N = 10 are depicted in Figure 10. As suggested from Equations (14)–(16), analytical curves in Figure 10 tend to shift in parallel along the ordinate. Of course, the curve tends to shift downwards with an increase in the fixed number of stress cycles. Ap- plying the concept of Equation (4), one could calculate the fatigue strength 𝜎 by 1.5 ∗ −1/6 𝜎 = [1273 · 𝜌 − 168 · {𝑙𝑜𝑔 (𝑁 )} + 1512]. (17) 1.5−𝜉 Probability density Cumulative probability Cumulative probability Cumulative probability 𝑓𝑖𝑥 𝑓𝑖𝑥 𝑓𝑖𝑥 𝑓𝑖𝑥 Appl. Sci. 2021, 11, 2889 10 of 15 Comparing these analytical distribution characteristics of the fatigue strength s in w9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the fatigue strength s under the conditions of P[1 m < r < 15 m] = 0.8 and n = 5 was w9 roughly in agreement with the experimental distribution aspect of the fatigue strength at N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. 4.2. Expansion of the Probabilistic Model to Analyze P-S-N Characteristics in Interior-Induced Fracture Target of the probabilistic model described in Section 3 is the distribution character- istics of the fatigue strength only at stress cycles of N = 10 . Accordingly, the analytical model should be conceptionally expanded to interpret the whole statistical aspect of the S-N property, in the interior inclusion-induced fracture mode. From this point of view, denoting the fatigue strength at any stress cycle, N , by s , Equation (3) is rewritten f ix as follows: 1/6 s = 1273r + b, (14) n  o b = l log N + g. (15) f ix Based on the dotted S-N curve in Figure 1, parameters  and in Equation (15) were de- 6  9 termined from two points N = 10 , s = 1424 MPa and N = 10 , s = 920 MPa w9 w9 as l =168 and g = 1512, respectively. Substituting these values into Equations (14) and (15), Equation (14) is rewritten as n  o 1/6 s = 1273r 168 log N + 1512. (16) f ix Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 16 6 7 Here, s r relationships under the fixed numbers of the fatigue life of N = 10 , N = 10 , 8 9 N = 10 , and N = 10 are depicted in Figure 10. As suggested from Equations (14)–(16), analytical curves in Figure 10 tend to shift in parallel along the ordinate. Of course, the Equation (17) includes arbitrarily given number of stress cycles 𝑁 , together with curve tends to shift downwards with an increase in the fixed number of stress cycles. the inclusion size 𝜌 and the inclusion depth ξ. Therefore, combining the joint probability Applying the concept of Equation (4), one could calculate the fatigue strength s by function in Figure 7 with Equation (17), one could analyze the distribution characteristics h n  o i of the fatigue strength at any number of stress cycles 𝑁 by repeating the numerical 1.5 1/6 s = 1273r 168 log N + 1512 . (17) w f ix calculations. 1.5 x N=10 N=10 N=10 N=10 10 20 30 40 Inclusion radius 𝜌 (μm) Figure 10. s - r relationships at several given numbers of stress cycles. Figure 10. σw- 𝜌 relationships at several given numbers of stress cycles. Equation (17) includes arbitrarily given number of stress cycles N , together with the f ix 6 7 8 Probability density functions of the fatigue strength thus obtained at N = 10 , 10 , 10 , inclusion size r and the inclusion depth . Therefore, combining the joint probability func- and 10 are indicated at the respective numbers of the stress cycles in Figure 11. These analytical results are roughly in agreement with the overall feature of the experimental aspect on the statistical fatigue characteristics. The scale of the axis for the probability density is adjusted to correspond to each value listed in the table attached in the right- hand side in Figure 11. The percentile points of 𝐹 = 1%, 10%, 50%, 90%, and 99% for the fatigue strength are indicated by marks of ◇’s along the vertical axis of the probability density functions. The respective thin dotted lines passing through the percentile points corresponding to the same probability give the P-S-N curves representing the statistical fatigue characteristics of this steel. All data points yield within the range of 1–99% in Fig- ure 11. This fact suggests that the probabilistic model developed in this study has an avail- ability to interpret the physical meaning of the statistical fatigue property in the very high cycle regime. Fatigue strength σw (MPa) 𝑓𝑖𝑥 𝑓𝑖𝑥 Appl. Sci. 2021, 11, 2889 11 of 15 tion in Figure 7 with Equation (17), one could analyze the distribution characteristics of the fatigue strength at any number of stress cycles N by repeating the numerical calculations. f ix 6 7 8 Probability density functions of the fatigue strength thus obtained at N = 10 , 10 , 10 , and 10 are indicated at the respective numbers of the stress cycles in Figure 11. These analytical results are roughly in agreement with the overall feature of the experimental aspect on the statistical fatigue characteristics. The scale of the axis for the probability density is adjusted to correspond to each value listed in the table attached in the right-hand side in Figure 11. The percentile points of F = 1%, 10%, 50%, 90%, and 99% for the fatigue strength are indicated by marks of 3’s along the vertical axis of the probability density functions. The respective thin dotted lines passing through the percentile points corresponding to the same probability give the P-S-N curves representing the statistical fatigue characteristics of this steel. All data points yield within the range of 1–99% in Figure 11. This fact suggests that the probabilistic model developed in this study has an Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 16 availability to interpret the physical meaning of the statistical fatigue property in the very high cycle regime. Probability density at peak of distribution ◇: Analyzed percentile points Nfix f(𝜎 ) 1600 𝑤 6 −3 10 3.76×10 7 −3 10 4.10×10 8 −3 10 4.52×10 9 −3 10 4.96×10 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode Although distribution characteristics of the fatigue strength s at the given number Although distribution characteristics of the fatigue strength 𝜎 at the given number of stress cycles were analyzed in the previous section, the fatigue life distributions could of stress cycles were analyzed in the previous section, the fatigue life distributions could also be analyzed based on Equation (17), by giving any value of the stress amplitude also be analyzed based on Equation (17), by giving any value of the stress amplitude 𝜎 . s . Applying this method, fatigue life distributions were calculated at s = 1400 MPa, w ∗ w Applying this method, fatigue life distributions were calculated at 𝜎 = 1400 MPa, 1200 1200 MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus ob- obtained were indicated at the respective stress levels in Figure 12. Since the fatigue tained were indicated at the respective stress levels in Figure 12. Since the fatigue life in- life increased logarithmically with decrease of the stress level, the probability density at creased logarithmically with decrease of the stress level, the probability density at the the peak of the density function tended to decrease such that the integrated value of the peak of the density function tended to decrease such that the integrated value of the den- density function maintained unity. Values of the peak density at the respective stress sity function maintained unity. Values of the peak density at the respective stress levels levels are indicated on the right-hand side, in Figure 12. As shown in this table, the peak are indicated on the right-hand side, in Figure 12. As shown in this table, the peak density density varied drastically, depending on the stress level, due to the above described reason, varied drastically, depending on the stress level, due to the above described reason, but but the distribution curves were drowned when they had the same height for the sake the distribution curves were drowned when they had the same height for the sake of con- of convenience. venience. In order to distinguish the fatigue strength distribution and the fatigue life distribution, the notations of f (s ) and F(s ) are used for the fatigue strength distribution, whereas w w other notations of g s and G s are accepted for the fatigue life distribution. Similar ( ) ( ) w w to Figure 11, the percentile points of G = 1%, 10%, 50%, 90%, and 99% for the fatigue Probability density at life are indicated by marks of 3’s along the horizontal lines, at the respective stress levels. peak of distribution ◇: Analyzed percentile points The respective thin dotted lines passing through the percentile points corresponding to 𝜎 (MPa) −7 1400 6.14×10 −8 1200 4.61×10 −9 1000 3.57×10 −10 900 9.91×10 2 3 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. Stress amplitude, (MPa) Stress amplitude, (MPa) Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 17 Probability density at peak of distribution ◇: Analyzed percentile points Nfix f( ) 6 −3 10 3.76×10 7 −3 10 4.10×10 8 −3 10 4.52×10 9 −3 10 4.96×10 9 10 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode Although distribution characteristics of the fatigue strength at the given number of stress cycles were analyzed in the previous section, the fatigue life distributions could also be analyzed based on Equation (17), by giving any value of the stress amplitude . Applying this method, fatigue life distributions were calculated at = 1400 MPa, 1200 MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus obtained were indicated at the respective stress levels in Figure 12. Since the fatigue life Appl. Sci. 2021, 11, 2889 12 of 15 increased logarithmically with decrease of the stress level, the probability density at the peak of the density function tended to decrease such that the integrated value of the density function maintained unity. Values of the peak density at the respective stress the same probability give the P-S-N curves of this steel. All experimental data in the levels are indicated on the right-hand side, in Figure 12. As shown in this table, the peak interior-induced fracture appear within the range of 1–99% in Figure 12. Thus, it was finally density varied drastically, depending on the stress level, due to the above described noted that the statistical fatigue property in the interior-induced fracture of this steel could reason, but the distribution curves were drowned when they had the same height for the be well explained from either one of the fatigue strength distribution and the fatigue life sake of convenience. distribution, through the probabilistic model proposed in this study. Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 16 Probability density at In order to distinguish the fatigue strength distribution and the fatigue life distribu- peak of distribution ◇: Analyzed percentile points tion, the notations of 𝑓 (𝜎 ) and 𝐹 (𝜎 ) are used for the fatigue strength distribution, 𝑤 𝑤 MPa whereas other notations of 𝑔 (𝜎 ) and 𝐺 (𝜎 ) are accepted for the fatigue life distribu- 𝑤 𝑤 1600 −7 1400 6.14×10 tion. Similar to Figure 11, the percentile points of 𝐺 = 1%, 10%, 50%, 90%, and 99% for −8 1200 4.61×10 the fatigue life are indicated by marks of ◇’s along the horizontal lines, at the respective −9 1000 3.57×10 stress levels. The respective thin dotted lines passing through the percentile points corre- −10 900 9.91×10 sponding to the same probability give the P-S-N curves of this steel. All experimental data in the interior-induced fracture appear within the range of 1–99% in Figure 12. Thus, it was finally noted that the statistical fatigue property in the interior-induced fracture of this steel could be well explained from either one of the fatigue strength distribution and the fatigue life distribution, through the probabilistic model proposed in this study. 4.4. Reconfirmation of the Number of Inclusions in the Critical Volume 2 3 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 10 10 As described in Section 4.1, since the number of inclusions included in the critical Number of stress cycles to failure, N volume is unknown, the analysis was carried out by setting three conditions of n = 2, 5, Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. and 20, as the inclusion number. Then, the condition of n = 5 was tentatively selected as a Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. preferable number of inclusions. However, in order to confirm the most reasonable num- 4.4. Reconfirmation of the Number of Inclusions in the Critical Volume ber of inclusions, the fatigue life distributions were additionally analyzed, under condi- As described in Section 4.1, since the number of inclusions included in the critical tions of n = 6, 8, and 10, respectively. Based on the series of analytical results, percentile volume is unknown, the analysis was carried out by setting three conditions of n = 2, 5, points of the fatigue life distribution corresponding to 𝐺 = 1%, 10%, 50%, 90%, and 99% and 20, as the inclusion number. Then, the condition of n = 5 was tentatively selected as a were determined; the relationship between those percentile points and the number of in- preferable number of inclusions. However, in order to confirm the most reasonable number clusions are depicted in Figure 13. Comparing five solid curves carefully with the disper- of inclusions, the fatigue life distributions were additionally analyzed, under conditions sion aspect of the experimental results in an interior-induced fracture in Figure 12, it was of n = 6, 8, and 10, respectively. Based on the series of analytical results, percentile points finally noted that the analytical result at n = 6 was well fitted compared to the result at n of the fatigue life distribution corresponding to G = 1%, 10%, 50%, 90%, and 99% were = 5. determined; the relationship between those percentile points and the number of inclusions Based on this evidence, the statistical fatigue properties (P-S-N properties) in Figures are depicted in Figure 13. Comparing five solid curves carefully with the dispersion aspect 11 and 12 were numerically analyzed by putting n = 6 for the number of inclusions in- of the experimental results in an interior-induced fracture in Figure 12, it was finally noted cluded in the critical volume of the specimen. that the analytical result at n = 6 was well fitted compared to the result at n = 5. 𝜎 = 900MPa 7 8 9 10 10 10 10 10 Number of stress cycles to failure, N Figure 13. Number of inclusions versus number of stress cycles to failure. Figure 13. Number of inclusions versus number of stress cycles to failure. 4.5. Mutual Relationship between Both Distributions of Fatigue Strength and Fatigue Life In this study, the authors showed that both fatigue strength distribution and fatigue ( ) life distribution could be derived from the joint probability density function of ℎ 𝜌 , 𝜉 , given by Equation (11). It was also confirmed that the analytical results were in good agreement with the statistical aspect of the experimental fatigue data in the very high cycle regime. In this place, mutual relationship between the distribution aspects of the fatigue Stress amplitude, (MPa) Stress amplitude, (MPa) Number of inclusions, n Appl. Sci. 2021, 11, 2889 13 of 15 Based on this evidence, the statistical fatigue properties (P-S-N properties) in Appl. Sci. 2021, 11, x FOR PEER REVIEW Figur es 11 and 12 were numerically analyzed by putting n = 6 for the number of inclusions 14 of 16 included in the critical volume of the specimen. 4.5. Mutual Relationship between Both Distributions of Fatigue Strength and Fatigue Life strength and the fatigue life is further discussed to reconfirm the reasonability of the pre- In this study, the authors showed that both fatigue strength distribution and fatigue sent probabilistic model for the P-S-N characteristics. The mutual relationship between life distribution could be derived from the joint probability density function of h(r , x), both distributions is illustrated in Figure 14, where the ordinate is the fatigue strength given by Equation (11). It was also confirmed that the analytical results were in good and t agreement he absciwith ssa isthe the statistical fatigue life aspect N. When of the any p experimental oint P(Nfix, fatigue ) is ta dat ken a in in th the is d very iagram, high cycle regime. In this place, mutual relationship between * the distribution aspects of the the probability that the fatigue life at the stress level is less than Nfix always corre- fatigue strength and the fatigue life is further discussed to reconfirm the reasonability of the sponds to the probability that the fatigue strength at the definite stress cycles Nfix is lower present probabilistic model for the P-S-N characteristics. The mutual relationship between * * than  . In other words, the cumulative probability of Nfix, G(Nfix, ), always corre- w w both distributions is illustrated in Figure 14, where the ordinate is the fatigue strength s * *  * spon andd the s toabscissa the cumis ulative the fatigue probability life N .of When  , F any (Nfipoint x, ) P [3 (0 N ]. G ,(s Nfi) x,is take ) gives n in th this e di diagram, stribu- w w fix w w the probability that the fatigue life at the stress level s is less than N always corresponds * * fix tion function of the fatigue life at  and F(Nfix, ) gives the distribution function of the w w to the probability that the fatigue strength at the definite stress cycles N is lower than s . fix w fatigue strength at Nfix. Thus, we have In other words, the cumulative probability of N , G(N ,s ), always corresponds to the fix fix * * cumulative probability of s , F(N ,s ) [30]. G(N ,s ) gives the distribution function of w fix w fix w   (18) G(Nfix, ) = F(Nfix, ) w w the fatigue life at s and F(N ,s ) gives the distribution function of the fatigue strength at fix w w N . Thus, we have fixEquation (18) implies that the dashed area G(Nfix, ) is always equal to the other G N , s ) = F N , s (18) ( ( ) * fix fix w w dashed area F(Nfix, ) in Figure 14. Figure 14. Relationship between fatigue strength distribution and fatigue life distribution. Figure 14. Relationship between fatigue strength distribution and fatigue life distribution. Equation (18) implies that the dashed area G(N ,s ) is always equal to the other fix w Based on this equality of Equation (18), one could obtain the S-N curve corresponding dashed area F(N ,s ) in Figure 14. fix to any level of the fracture probability. Thus, the probabilistic S-N curves at the respective Based on this equality of Equation (18), one could obtain the S-N curve correspond- fracture probabilities could be analyzed as P-S-N curves. As described in Sections 4.2 and ing to any level of the fracture probability. Thus, the probabilistic S-N curves at the 4.3, the distribution characteristics of the experimental S-N data were well explained from respective fracture probabilities could be analyzed as P-S-N curves. As described in either viewpoint of the fatigue strength distribution or the fatigue life distribution. The Sections 4.2 and 4.3, the distribution characteristics of the experimental S-N data were well equality of Equation (18) is the reason why the reasonable P-S-N curves could be equally explained from either viewpoint of the fatigue strength distribution or the fatigue life obtained from different viewpoints of the fatigue strength distribution and the fatigue life distribution. The equality of Equation (18) is the reason why the reasonable P-S-N curves distribution. could be equally obtained from different viewpoints of the fatigue strength distribution and the fatigue life distribution. 5. Conclusions 5. Conclusions Main conclusions obtained in this study are summarized as follows. Main conclusions obtained in this study are summarized as follows. 1. The probability density functions of the inclusion size at the crack initiation site, 𝑓 (𝜌 ), was successfully derived by combining the Weibull distribution and the con- 1. The probability density functions of the inclusion size at the crack initiation site, f (r ), cept of extreme distribution. In addition, the probability density function of the in- was successfully derived by combining the Weibull distribution and the concept of ( ) clusion depth, 𝑓 𝜉 , was also derived from the uniform distribution of the location of the inclusion in the material space. 2. Since the inclusion size 𝜌 and the crack depth ξ are statistically independent, the joint probability density function of these random variables, ℎ(𝜌 , 𝜉 ), is given by the 𝑛 Appl. Sci. 2021, 11, 2889 14 of 15 extreme distribution. In addition, the probability density function of the inclusion depth, f (x), was also derived from the uniform distribution of the location of the inclusion in the material space. 2. Since the inclusion size r and the crack depth  are statistically independent, the joint probability density function of these random variables, h(r , x), is given by the direct multiplication of the above two probability density functions, such as h(r , x) = f (r ) f (x). n n 0 3. For the hourglass type specimen of a bearing steel with the definite hardness, the rotating bending fatigue strength of the specimen with any size r and depth  of the inclusion at the arbitrarily given number of stress cycles N is analytically provided fix in the very high cycle regime. 4. Based on the above joint probability density function of h(r , x), repeating the numer- ical calculations following P[r 2 Dr \ x 2 Dx] = h(r , x)Dr Dx = P[s 2 Ds ], n n n n w9 w9 one could obtain the fatigue strength distribution at any number of stress cycles and the fatigue life distribution at any stress level. The analytical results thus obtained were in good agreement with the statistical feature of the experimental fatigue test data. In the construction of the probabilistic model in this study, several assumptions were introduced for the sake of simplicity. In order to reconfirm the reasonability of such assumptions, sufficient number of experimental results should be further filed up in the future. In addition, it is still unknown whether the fatigue limit exists for the interior inclusion-induced fracture mode. This is one of the most important issues to be solved in the near future. Author Contributions: T.S. has performed fatigue tests and organized whole of this work. A.N. and Y.N. have analyzed experimental data and prepared all of diagrams. N.O. has prepared fatigue specimens and performed fatigue tests. All 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 1. Bathias, C. There is no infinite fatigue life in metallic materials. Fatigue Fract. Eng. Mater. Struct. 1999, 22, 559–565. [CrossRef] 2. Mughrabi, H. Specific features and mechanisms of fatigue in the ultrahigh-cycle regime. Int. J. Fatigue 2006, 28, 1501–1508. [CrossRef] 3. Sakai, T.; Takeda, M.; Shiozawa, K.; Ochi, Y.; Nakajima, M.; Nakamura, T.; Oguma, N. Experimental reconfirmation of character- istic S-N property for high carbon chromium bearing steel in wide life region in rotating bending. J. Soc. Mat. Sci. Jpn. 2000, 49, 779–785. [CrossRef] 4. Bathias, C.; Paris, P.C. Gigacycle Fatigue in Mechanical Practice; Marcel Deckker: New York, NY, USA, 2005. 5. Sakai, T.; Sato, Y.; Oguma, N. Characteristic S-N properties of high-carbon-chromium-bearing steel under axial loading in long-life fatigue. Fatigue Fract. Eng. Mater. Struct. 2002, 25, 765–773. [CrossRef] 6. Sakai, T.; Sato, Y.; Nagano, Y.; Takeda, M.; Oguma, N. Effect of stress ratio on long life fatigue behavior of high carbon chromium bearing steel under axial loading. Int. J. Fatigue 2006, 28, 1547–1554. [CrossRef] 7. Sakai, T.; Lian, B.; Takeda, M.; Shiozawa, K.; Oguma, N.; Ochi, Y.; Nakajima, M.; Nakamura, T. Statistical duplex S-N characteristics of high carbon chromium bearing steel in rotating bending in very high cycle regime. Int. J. Fatigue 2010, 32, 497–504. [CrossRef] 8. Oguma, N.; Lian, B.; Sakai, T.; Watanabe, K.; Odake, Y. Long life fatigue fracture induced by interior inclusions for high carbon chromium bearing steels under rotating bending. J. ASTM Int. 2010, 7, 1–9. [CrossRef] 9. Murakami, Y. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions; Elsevier: Oxford, UK, 2002; pp. 88–94. 10. Takeda, M.; Sakai, T.; Oguma, N. Rotating bending fatigue property and fractography for high strength steels and carbon steel over ultra wide life region. Trans. JSME Ser. A 2002, 68, 977–984. [CrossRef] 11. Murakami, Y.; Kodama, S.; Konuma, S. Quantitative evaluation of effects of nonmetallic inclusions on fatigue strength of high strength steel. Trans. JSME Ser. A 1988, 54, 688–696. [CrossRef] 12. Toriyama, T.; Murakami, Y.; Makino, T. Database of nonmetallic inclusions and its application to the fatigue strength prediction method of high strength steels. J. Soc. Mat. Sci. Jpn. 1991, 40, 1497–1503. [CrossRef] 13. Adachi, A.; Shoji, H.; Kuwabara, A.; Inoue, Y. Rotating bending fatigue phenomenon of JIS SUJ2 bearing steel. DENKI SEIKO 1975, 46, 176–182. [CrossRef] 14. Barbosa, J.F.; Correia, J.A.F.O.; Junior, R.C.S.F.; De Jesus, A.M.P. Fatigue life prediction of metallic materials considering mean stress effects by means of an artificial neural network. Int. J. Fatigue 2020, 135, 105527. [CrossRef] Appl. Sci. 2021, 11, 2889 15 of 15 15. Gope, P.C.; Mahar, C.S. Evaluation of fatigue damage parameters for Ni-based super alloys Inconel 825 steel notched specimen using stochastic approach. Fatigue Fract. Eng. Mater. Struct. 2021, 44, 427–443. [CrossRef] 16. Lehner, P.; Krejsa, M.; Parenica, P.; Krivy, V.; Brozovsky, J. Fatigue damage analysis of a riveted steel overhead crane support truss. Int. J. Fatigue 2019, 128, 105190. [CrossRef] 17. Tomaszewski, T.; Strzelecki, P.; Wachowski, M.; Stopel, M. Fatigue life prediction for acid-resistant steel plate under operating loads. Bull. Pol. Acad. Tech. 2020, 68, 913–921. [CrossRef] 18. Feng, L.; Zhang, L.; Liao, X.; Zhang, W. Probabilistic fatigue life of welded plate joints under uncertainty in arctic areas. J. Constr. Steel Res. 2021, 176, 106412. [CrossRef] 19. Caiza, P.D.T.; Ummenhofer, T. A probabilistic Stussi function for modelling the S-N curves and its application on specimens made of steel S355J2+N. Int. J. Fatigue 2018, 117, 121–134. [CrossRef] 20. Mughrabi, H. Zur Dauerschwingfestigkeit im Bereich extrem hoher bruchlastspielzahlen: Mehrstufige lebensdauerkurven. HTM/Harterei-Tech. Mitt. 2001, 56, 300–303. 21. Harlow, D.G.; Wei, R.P.; Sakai, T.; Oguma, N. Crack growth based probability modeling of S-N response for high strength steel. Int. J. of Fatigue 2006, 28, 1479–1485. [CrossRef] 22. Sakai, T. Chair of Editorial Committee. In Standard Evaluation Method of Fatigue Reliability for Metallic Materials—Standard Regression Method of S-N Curves; JSMS-SD-11-07; The Society of Materials Science: Kyoto, Japan, 2007. 23. Nakagawa, A.; Sakai, T.; Harlow, D.G.; Oguma, N.; Nakamura, Y.; Ueno, A.; Kikuchi, S.; Sakaida, A. A probabilistic model on crack initiation modes of metallic materials in very high cycle fatigue. Procedia Struct. Integr. 2016, 2, 1199–1206. [CrossRef] 24. Kitagawa, H.; Takahashi, S. Fracture mechanics study on small fatigue crack growth and the condition of its threshold. Trans. JSME Ser. A 1979, 45, 1289–1303. 25. Tanaka, K.; Nakai, Y.; Yamashita, M. Fatigue growth threshold of small cracks. Int. J. Fract. 1981, 17, 519–533. 26. Nakamura, Y.; Sakai, T.; Harlow, D.G.; Oguma, N.; Nakajima, M.; Nakagawa, A. Probabilistic model on statistical fatigue property in very high cycle regime based on distributions of size and location of interior inclusions. In Proceedings of the VHCF-7, Dresden, Germany, 3–5 July 2017; pp. 81–86. 27. Weibull, W. A statistical distribution function of wide applicability. J. Appl. Mech. 1951, 18, 293–297. 28. Gumbel, E.J. Statistics of Extremes; Columbia University Press: New York, NY, USA, 1958; pp. 75–78. 29. Shiozawa, K.; Lu, L.-T.; Ishihara, S. Subsurface fatigue crack initiation behavior and S-N curve characteristics in high carbon- chromium bearing steel. J. Soc. Mat. Sci. Jpn. 1999, 48, 1095–1100. [CrossRef] 30. Sakai, T.; Sakaida, A.; Fujitani, K.; Tanaka, T. A study on statistical fatigue properties of metallic materials with particular attention to relation between fatigue life and fatigue strength distributions. Mem. Res. Inst. Sci. Eng. 1982, 42, 79–91. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Sciences Multidisciplinary Digital Publishing Institute

Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a Bearing Steel in a Very High Cycle Regime

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applied sciences Article Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a Bearing Steel in a Very High Cycle Regime 1 , 2 3 4 Tatsuo Sakai *, Akiyoshi Nakagawa , Yuki Nakamura and Noriyasu Oguma Research Organization of Science and Technology, Ritsumeikan University, Kusatsu 525-8577, Japan Hitachi Industrial Products, Ltd., Osaka 530-0005, Japan; akiyoshi.nakagawa.ue@hitachi.com National Institute of Technology, Toyota College, Toyota 471-8525, Japan; nakamura@toyota-ct.ac.jp Faculty of Engineering, University of Toyama, Toyama 930-8555, Japan; oguma@eng.u-toyama.ac.jp * Correspondence: sakai@se.ritsumei.ac.jp Abstract: In S-N diagrams for high strength steels, the duplex S-N curves for surface-initiated failure and interior inclusion-initiated failure were usually confirmed in the very high cycle regime. This trend is more distinct in the loading type of rotating bending, due to the stress distribution across the section. In the case of interior failure mode, the fish-eye is usually observed on the fracture surface and an inclusion is also observed at the center of the fish-eye. In the present work, the authors attempted to construct a probabilistic model on the statistical fatigue property in the interior failure mode, based on the distribution characteristics of the location and the size of the interior inclusion at the crack initiation site. Thus, the P-S-N characteristics of the bearing steel (SUJ2) in the very high cycle regime were successfully explained. Keywords: probabilistic model; very high cycle fatigue; duplex S-N curves; rotating bending; inclusion; fish-eye; inclusion size; inclusion depth Citation: Sakai, T.; Nakagawa, A.; Nakamura, Y.; Oguma, N. Proposal of a Probabilistic Model on Rotating Bending Fatigue Property of a 1. Introduction Bearing Steel in a Very High Cycle From both viewpoints of economical design of mechanical structures and reduction Regime. Appl. Sci. 2021, 11, 2889. of the environmental load, elongation of the design life for every product in the wide https://doi.org/10.3390/app11072889 range of industries, is an important subject. In such a circumstance, for train axles and wheels, load bearing parts of automobiles, turbine rotors of the power plants and so on, Academic Editor: Massimo Rossetto the number of loading cycles reaches over 10 cycles, during the service period of the respective mechanical components [1–4]. One of the typical aspects of fatigue property in Received: 24 February 2021 the very high cycle regime for structural steels is the fact that the duplex S-N characteristics Accepted: 19 March 2021 consisting of the respective S-N curves for the surface-initiated and the interior-initiated Published: 24 March 2021 failures are observed, as reported by many researchers [3,5–7]. The term “S-N curve” refers to the relationship between the stress level [S] and the number of stress cycles to failure Publisher’s Note: MDPI stays neutral [N]. Expressions of S-N property and S-N characteristics are also used in this paper, based with regard to jurisdictional claims in on a similar sense. In the case of interior-initiated failure, an inclusion is usually found at published maps and institutional affil- the core portion of the fish-eye formed on the fracture surface [3,5,8]. Thus, the interior iations. inclusion at the crack initiation site plays a dominant role to govern the fatigue strength and fatigue life of the specimen. It is well-known that the fatigue limit of the metallic material can be evaluated by combining the material hardness and the concept of area for the defect size proposed Copyright: © 2021 by the authors. by Murakami et al. [9]. Another important factor to control the fatigue property is the Licensee MDPI, Basel, Switzerland. location of the inclusion, since the stress distribution across the section has a steep slope in This article is an open access article rotating bending [10]. Accordingly, the fatigue property of those metallic materials should distributed under the terms and be analyzed as a function of the size and the location of the interior inclusions, together conditions of the Creative Commons with the hardness. However, both the size and the location of the inclusions inside the Attribution (CC BY) license (https:// material have particular distributions, depending on the fabrication process of the actual creativecommons.org/licenses/by/ metallic materials [11–13]. 4.0/). Appl. Sci. 2021, 11, 2889. https://doi.org/10.3390/app11072889 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 2889 2 of 15 From this point of view, the authors attempted to construct a probabilistic model on the very high cycle fatigue property for the bearing steel of SUJ2 in rotating bending, by combining the distribution characteristics of the size and the location of the interior inclusion. In this model, a two-parameter Weibull distribution was accepted for the distribution pattern of the inclusion size, whereas the random distribution was accepted for the location of the inclusion. Since the scatter of the hardness is sufficiently small, the hardness of this steel was supposed to be constant for the sake of simplicity. Although some number of papers analyzing the statistical aspects of S-N property were published [14–19], the main target of those research is the statistical property within the usual life region, which is less than 10 cycles. Considering this point, the main aim of the present study was set to the physical interpretation of the statistical aspects of the S-N characteristics of a bearing steel, in the very high cycle regime. 2. Experimental Fatigue Test Data Referred in This Study Sufficient number of fatigue test data are required to analyze the statistical fatigue property of metallic materials. However, the fatigue test in the very high cycle regime occupies a long time. For example, at the usual loading frequency of 50–60 Hz, it takes around 200 days to reach the loading cycles of N = 10 . In such a circumstance, the authors extracted a series of fatigue test results obtained as round robin tests by the Research Group on Statistical Aspects of Materials Strength [RGSAMS] [3]. These fatigue tests were performed in rotating bending in very high cycle regime over N = 10 cycles, for a bearing steel of JIS:SUJ2, by using the same type of testing machine and common specimens. Since such fatigue test results give useful data for discussion on the very high cycle fatigue property of metallic materials, they are often referred to by many researchers in this area [3,7,20,21]. All experimental results are plotted as an S-N diagram in Figure 1, in which fatigue test data obtained at different laboratories in seven universities in Japan were indicated altogether. Open symbols represent the S-N data for the surface-induced fracture, whereas solid symbols indicate the results for the interior-induced fracture. Several data points attached arrows around N = 10 , which indicate runout specimens without fracture. The solid and dotted curves indicate the S-N curves fitted to the respective fracture modes, using the JSMS Standard of “Standard Regression Method of S-N Curves”, accepting the semi-logarithmic bilinear model [22]. The fatigue limit for the surface-induced fracture was s = 1278 MPa, as given by the horizontal portion of the solid S-N curve. This conventional fatigue limit cannot provide the true fatigue limit, since fatigue fractures take place in the very high cycle region, at stress levels lower than the horizontal portion of the solid S-N curve. For the experimental data in the interior-induced fracture mode, the linear regression line was indicated by a dotted line. Although the horizontal line was also indicated from the point of N = 10 along the dotted S-N curve, further experimental Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16 reconfirmation would be required in order to make clear the appearance of the horizontal portion in the fracture mode by continuing fatigue tests, until further long life region. Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Typical examples of fracture surfaces in the respective fracture modes are shown in Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-in- duced fracture mode and (b) is the fracture surface in the interior-induced fracture mode [23]. The fatigue crack occurs at the specimen surface in the short life region at a relatively high stress level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture surface in the very high cycle region, at low stress levels, as shown in Figure 2b. In such a case, an inclusion is also found at the core site of the fish-eye. This fact means that the fatigue crack takes place around the inclusion at the crack initiation site, in a very high cycle regime. The main aim of this study was to analyze the statistical distribution char- acteristics of the S-N property in the interior inclusion-induced fracture mode, by combin- ing distribution characteristics of the inclusion size and the inclusion depth. 100μ 100μ (a) Surface-initiated fracture (b) Interior inclusion-initiated fracture 𝜎 = 1800 MPa, N = 2.16 × 10 𝜎 = 1300 MPa, N = 2.66 × 10 Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. For the sake of important reference, static mechanical properties such as the tensile strength and the Vickers’ hardness were examined, prior to performing fatigue tests of this bearing steel. Experimental results of these properties are indicated in Table 1. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 16 Appl. Sci. 2021, 11, 2889 3 of 15 Figure 1. Rotating bending S-N characteristics of bearing steel in very high cycle regime. Typical examples of fracture surfaces in the respective fracture modes are shown in Typical examples of fracture surfaces in the respective fracture modes are shown in Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-in- Figure 2, where (a) gives the SEM observation of the fracture surface in the surface-induced duced fracture mode and (b) is the fracture surface in the interior-induced fracture mode fracture mode and (b) is the fracture surface in the interior-induced fracture mode [23]. The [23]. The fatigue crack occurs at the specimen surface in the short life region at a relatively fatigue crack occurs at the specimen surface in the short life region at a relatively high stress high stress level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture level, as shown in Figure 2a. However, a clear fish-eye is found on the fracture surface in surface in the very high cycle region, at low stress levels, as shown in Figure 2b. In such a the very high cycle region, at low stress levels, as shown in Figure 2b. In such a case, an case, an inclusion is also found at the core site of the fish-eye. This fact means that the inclusion is also found at the core site of the fish-eye. This fact means that the fatigue crack fatigue crack takes place around the inclusion at the crack initiation site, in a very high takes place around the inclusion at the crack initiation site, in a very high cycle regime. cycle regime. The main aim of this study was to analyze the statistical distribution char- The main aim of this study was to analyze the statistical distribution characteristics of the acteristics of the S-N property in the interior inclusion-induced fracture mode, by combin- S-N property in the interior inclusion-induced fracture mode, by combining distribution ing distribution characteristics of the inclusion size and the inclusion depth. characteristics of the inclusion size and the inclusion depth. 100μ 100μ (a) Surface-initiated fracture (b) Interior inclusion-initiated fracture 𝜎 = 1800 MPa, N = 2.16 × 10 𝜎 = 1300 MPa, N = 2.66 × 10 Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. Figure 2. SEM observations of fracture surfaces in surface-induced fracture and interior inclusion-induced fracture. For the sake of important reference, static mechanical properties such as the tensile For the sake of important reference, static mechanical properties such as the tensile strength and the Vickers’ hardness were examined, prior to performing fatigue tests of this strength and the Vickers’ hardness were examined, prior to performing fatigue tests of bearing steel. Experimental results of these properties are indicated in Table 1. this bearing steel. Experimental results of these properties are indicated in Table 1. Table 1. Static mechanical properties of the material. Tensile strength 2316 (MPa) Vickers’ hardness 778 (HV) 3. Probabilistic Model to Explain the Statistical Fatigue Property in Interior-Induced Fracture 3.1. Effects of Size and Depth of Inclusions on the Fatigue Strength According to the area model proposed by Y. Murakami [9], the fatigue limit of a metallic material, s , was well provided by the defect size r and the Vickers’ hardness HV (kgf/mm ), as follows; 1/6 s = a( HV + 120)/ area , (1) where area indicates the area of the defect projected perpendicular to the longitudinal direction of the specimen, and the factor of a was given by a = 1.43 for surface defect and a = 1.56 for interior defect, respectively. When this concept of area was applied to the fatigue strength at N = 10 cycles as an attempt, assuming a spherical interior inclusion with a radius of r, the fatigue strength at N = 10 , s , was given by the following expression, w9 q 1/6 s = 1.56( HV + 120)/ pr . (2) w9 Appl. Sci. 2021, 11, 2889 4 of 15 Since the Vickers’ hardness of this bearing steel (JIS:SUJ2) was reported as HV = 778 [3], Equation (2) was reduced as follows; 1/6 s = 1273r (3) w9 Equation (3) indicates the relationship between the fatigue strength at N = 10 and the inclusion size (radius) for this steel. Such a relationship is depicted in Figure 3. When the inclusion size is small, those inclu- sions have no harmful effect on the fatigue strength, as reported by many researchers [24,25]. From this point of view, if the critical small size is assumed as r = 3 m in this study, the upper bound of the fatigue strength given by Equation (3) becomes s = 1060 MPa, as w9 indicated in Figure 3. On the other hand, an inclusion larger than 40 m (r > 40 m) is seldom found in the present steel. In such an assumption, the lower bound of the fatigue Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 16 strength given by Equation (3) becomes s = 689 MPa, as indicated by the horizontal w9 dotted line in Figure 3. Thus, the distribution range of the fatigue strength at N = 10 due to the distribution of the inclusion size would be 689 MPa~1060 MPa. Figure 3. Relationship between fatigue strength s and inclusion radius r. w9 Figure 3. Relationship between fatigue strength  and inclusion radius  . w9 In the case of the rotating bending, the stress distribution across the specimen section has a distinct gradient, such that the stress on the surface gives the maximum and the 3.2. Distribution Characteristics of Inclusion Size and Inclusion Depth stress becomes zero at the center of the section. Accordingly, the inclusion depth is another The inclusion size is not constant, and therefore, the size has its own distribution important factor to govern the fatigue strength of s , even if the inclusion size is fixed w9 to char a certain acteris value. tics pec When uliar the fatigue to the str fabri ength cati at the on depth process x is denoted of the as met s allic , the fatigue material [11–13]. Of w9 strength evaluated by nominal bending stress on the specimen surface, s , is given as course, since the inclusion size 𝜌 is always positive, we havew 9𝜌 > 0. In other words, the follows [26]; lower bound of the inclusion size is assumed “0”. In addition, a size larger than 40 μm is r 1.5 s = s = s (4) w9 w9 w9 seldom found in the bearing steel. Distribution pattern of the inclusion size with the above r x 1.5 x feature is well-represented by a two-parameter Weibull distribution [27], without a loca- where r indicates the radius of the critical portion of the specimen. Here, substituting tion parameter. The probability density function and the cumulative distribution function Equation (3) into Equation (4), we have the following expression; are given as follows: 1.5 1910 1/6 1/6 s =  1273r = r (5) w9 𝑎 −1 𝑎 1.5 x 1.5 x 𝑎 𝜌 𝜌 (6) ( ) 𝑓 𝜌 = ( ) 𝑒𝑥𝑝 {− ( ) }, 𝑏 𝑏 𝑏 Consequently, the fatigue strength of s for the specimen with an inclusion size of r w9 and a depth of x could be calculated by Equation (5). and 3.2. Distribution Characteristics of Inclusion Size and Inclusion Depth ( ) (7) 𝐹 𝜌 = 1 − 𝑒𝑥𝑝 {− ( ) }, The inclusion size is not constant, and therefore, the size has its own distribution characteristics peculiar to the fabrication process of the metallic material [11–13]. Of where a is the shape parameter and b is the scale parameter, respectively. course, since the inclusion size r is always positive, we have r > 0. In other words, the When the number of inclusions included within the critical volume of a specimen is lower bound of the inclusion size is assumed “0”. In addition, a size larger than 40 m is seldom found in the bearing steel. Distribution pattern of the inclusion size with the denoted by n, the inclusion with the maximum size would be the crack starter under the cyclic loadings. Accordingly, the distribution characteristics of the maximum size among n of inclusions in the critical volume become the most important factor to control the fa- tigue strength and fatigue life of the metallic material. Based on the concept of extremes’ distribution [28], the probability density function of the maximum size among n inclu- ( ) sions, 𝑓 𝜌 , is provided by 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 (8) ( ) { ( )} 𝑓 𝜌 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 𝑏 𝑏 𝑏 Although we have to know the original distribution of the inclusion size 𝜌 itself, it is difficult to directly observe such a distribution pattern. In such a circumstance, the au- thors determined the respective parameters of the original size distribution, so as to satisfy the following conditions: [ ] 𝑃 1 μm < 𝜌 < 15 μm = 0.8, 𝐹 = 0.1 at 𝜌 = 1 μm and 𝐹 = 0.9 at 𝜌 = 15 μm. 0 0 This condition for the inclusion size was introduced here as a possible assumption supposed from the observation results by Toriyama et al. [12]. Thus, we obtain the results for the Weibull parameters as 𝑎 = 1.139 and 𝑏 = 7.214, respectively. In addition, since the number of inclusions inside the critical volume, n, is also unknown, the authors ana- lyzed the distribution of the extremes by supposing various values as the number of in- clusions, like n = 2, 3, 5, 6, 10, 20, and 40, respectively. Probability density functions of the maximum inclusion size thus calculated by Equation (8) are depicted in Figure 4, where Appl. Sci. 2021, 11, 2889 5 of 15 above feature is well-represented by a two-parameter Weibull distribution [27], without a location parameter. The probability density function and the cumulative distribution function are given as follows: n   o a1 a a r r f (r) = ex p , (6) b b b and n   o F (r) = 1 ex p , (7) where a is the shape parameter and b is the scale parameter, respectively. When the number of inclusions included within the critical volume of a specimen is denoted by n, the inclusion with the maximum size would be the crack starter under the cyclic loadings. Accordingly, the distribution characteristics of the maximum size among n of inclusions in the critical volume become the most important factor to control the fatigue strength and fatigue life of the metallic material. Based on the concept of extremes’ distribution [28], the probability density function of the maximum size among n inclusions, f (r ), is provided by n   o a1 a a r r n1 n n f (r ) = nfF (r )g ex p (8) n n b b b Although we have to know the original distribution of the inclusion size r itself, it is difficult to directly observe such a distribution pattern. In such a circumstance, the authors determined the respective parameters of the original size distribution, so as to satisfy the following conditions: P[1 m < r < 15 m] = 0.8, F = 0.1 at r = 1 m and F = 0.9 at r = 15 m. 0 0 This condition for the inclusion size was introduced here as a possible assumption supposed from the observation results by Toriyama et al. [12]. Thus, we obtain the results for the Weibull parameters as a = 1.139 and b = 7.214, respectively. In addition, since the number of inclusions inside the critical volume, n, is also unknown, the authors analyzed the distribution of the extremes by supposing various values as the number of inclusions, like n = 2, 3, 5, 6, 10, 20, and 40, respectively. Probability density functions of the maximum inclusion size thus calculated by Equation (8) are depicted in Figure 4, where f (r ) plotted by a fine dotted curve in the most left hand side indicates the original probability density function of the inclusion size. The mode of the largest inclusion size, i.e., the inclusion size at each peak of the probability density function, tends to shift into the right hand side, with an increase in the number of inclusions in the critical volume of the specimen. As described in Section 3.1, the inclusion depth at the crack initiation site, x , plays an important role to control both the fatigue strength and fatigue life of the metallic material, in the case of the rotating bending, due to the steep slope of the stress distribution across the section. Figure 5 indicates the distribution feature of the inclusions projected to the specimen section, assuming random distribution (uniform distribution) [23]. Here, if the inclusion depth from the surface is denoted as x , x , . . . , x , from the most shallow 1 2 n inclusion, one can obtain the one-dimensional distribution of the inclusion depth. As reported in another paper [26], the probability density function of the inclusion depth x can be represented by the following expression: 2 x f x = 1 , 0 < x < r . (9) ( ) [ ] r r Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 16 ( ) 𝑓 𝜌 plotted by a fine dotted curve in the most left hand side indicates the original prob- ability density function of the inclusion size. The mode of the largest inclusion size, i.e., the inclusion size at each peak of the probability density function, tends to shift into the Appl. Sci. 2021, 11, 2889 6 of 15 right hand side, with an increase in the number of inclusions in the critical volume of the specimen. Size of the largest inclusion, 𝜌 (𝜇 m) Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 16 Figure 4. Probability density functions of the largest inclusion size among various numbers of inclusions. Figure 4. Probability density functions of the largest inclusion size among various numbers of inclusions. As described in Section 3.1, the inclusion depth at the crack initiation site, ξ, plays an important role to control both the fatigue strength and fatigue life of the metallic material, in the case of the rotating bending, due to the steep slope of the stress distribution across the section. Figure 5 indicates the distribution feature of the inclusions projected to the specimen section, assuming random distribution (uniform distribution) [23]. Here, if the inclusion depth from the surface is denoted as 𝜉 , 𝜉 , · · , 𝜉 , from the most shallow inclu- 1 2 𝑛 sion, one can obtain the one-dimensional distribution of the inclusion depth. As reported in another paper [26], the probability density function of the inclusion depth ξ can be rep- resented by the following expression: 2 𝜉 𝑓 (𝜉 ) = (1 − ), [0 < 𝜉 < 𝑟 ]. (9) 𝑟 𝑟 In Equation (9), r indicates the radius of the specimen. The probability density of Figure 5. Definition of inclusion depth from surface. Equation Figure 5. Definition (9) becomes of inclusion a maxim depth um at from the surface. specimen surface (ξ = 0), whereas the density becomes zero at the center of the specimen section. As reported by several researchers In Equation (9), r indicates the radius of the specimen. The probability density of [5,29], the depth of the crack initiation site is restricted within the thin surface layer of ξ < Equation (9) becomes a maximum at the specimen surface (x = 0), whereas the density 250 μm in rotating bending. Therefore, the probability density function of Equation (9) becomes zero at the center of the specimen section. As reported by several researchers [5,29], should be normalized as the conditional distribution, under the condition of ξ < 250 μm. the depth of the crack initiation site is restricted within the thin surface layer of x < 250 m Thus, we have the actual probability density function of ξ, as follows: in rotating bending. Therefore, the probability density function of Equation (9) should be 2 𝜉 normalized as the conditional distribution, under the condition of x < 250 m. Thus, we ( ) 𝑓 𝜉 = (1 − ), [0 < 𝜉 < 250 μm], (10) 𝑟 𝐹 𝑟 have the actual probability density function of x, as follows: where 𝐹 indicates the probability giving the condition of 0 < 𝜉 < 250 μm and its prob- 2 x ability becomes 𝐹 = 0.3055 in the condition of the present work. The probability density f (x) = 1 , [0 < x < 250 m], (10) rF r function thus normalized by Equation (10) is depicted in Figure 6. where F indicates the probability giving the condition of 0 < x < 250 m and its proba- Depth of inclusion (mm) bility becomes F = 0.3055 in the condition of the present work. The probability density function thus normalized by Equation (10) is depicted in Figure 6. Figure 6. Probability function of inclusion depth f  . ( ) 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength Distribution In the previous section, distribution characteristics on the size and the depth of the inclusion are discussed, and the probability density functions of 𝑓 (𝜌 ) and 𝑓 (𝜉 ) are de- 𝑛 0 rived as Equations (8) and (10). Since both 𝜌 and ξ are statistically independent random ( ) variables, the joint probability density function, ℎ 𝜌 , 𝜉 , is given by 𝑎 −1 𝑎 𝑎 𝜌 𝜌 2 𝜉 𝑛 𝑛 𝑛 −1 ℎ(𝜌 , 𝜉 ) = 𝑓 (𝜌 ) · 𝑓 (𝜉 ) = 𝑛 {𝐹 (𝜌 )} ( ) 𝑒𝑥𝑝 {− ( ) } · (1 − ) (11) 𝑛 𝑛 0 0 𝑛 𝑏 𝑏 𝑏 𝑟 𝐹 𝑟 By putting n = 5, 𝑎 = 1.139, 𝑏 = 7.214, r = 1.5, and 𝐹 = 0.3055 along the previous ( ) example, the joint probability density function of ℎ 𝜌 , 𝜉 was numerically calculated. ( ) The result of ℎ 𝜌 , 𝜉 thus obtained is depicted in Figure 7, where the joint probability function of ℎ(𝜌 , 𝜉 ) gives the curved surface like a mountain range. In the figure, ∆𝐻 (𝜌 , 𝜉 ) corresponding to the volume of the vertical column indicates the probability that 𝜌 and 𝜉 are within the square region of ∆𝜌 × ∆𝜉 . The dashed line appearing on 𝑛 𝑛 ( ) the 𝜉 − ℎ 𝜌 , 𝜉 plane corresponds to the marginal distribution for the inclusion depth 𝜉 , as given by Equation (10). On the other hand, another dashed curve appearing on the 𝜌 − ℎ(𝜌 , 𝜉 ) plane corresponds to the marginal distribution for the inclusion size of 𝜌 , 𝑛 𝑛 𝑛 given by Equation (12). 𝑐 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 ( ) { ( )} (12) 𝑓 𝜌 = ∫ ℎ(𝜌 , 𝜉 )𝑑𝜉 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 𝑏 𝑏 𝑏 Probability density, ( ) Probability density Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 16 Appl. Sci. 2021, 11, 2889 7 of 15 Figure 5. Definition of inclusion depth from surface. Depth of inclusion (mm) Figure 6. Probability function of inclusion depth f (x). Figure 6. Probability function of inclusion depth f  . ( ) 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength Distribution In the previous section, distribution characteristics on the size and the depth of 3.3. Joint Distribution of Inclusion Size and Inclusion Depth and Analysis of Fatigue Strength the inclusion are discussed, and the probability density functions of f r and f x are ( ) ( ) n 0 Distribution derived as Equations (8) and (10). Since both r and x are statistically independent random variables, the joint probability density function, h(r , x), is given by In the previous section, distribution characteristics on the size and the depth of the n   o inclusion are discussed, and the probability density functions of 𝑓 (𝜌 ) and 𝑓 (𝜉 ) are de- a1 a a r r 2 x 𝑛 0 n n n1 h(r , x) = f (r ) f (x) = nfF (r )g ex p  1 (11) n n 0 0 n b b b rF r rived as Equations (8) and (10). Since both 𝜌 and ξ are statistically independent random ( ) variables, the joint probability density function, ℎ 𝜌 , 𝜉 , is given by By putting n = 5, a = 1.139, b = 7.214, r = 1.5, and F = 0.3055 along the previous example, the joint probability density function of h(r , x) was numerically calculated. The 𝑎 −1 𝑎 𝑎 𝜌 𝜌 2 𝜉 𝑛 𝑛 result of h(r , x) thus obtained is depic 𝑛 te − d 1in Figure 7, where the joint probability function { } ℎ(𝜌 , 𝜉 ) = 𝑓 (𝜌 ) · 𝑓 (𝜉 ) = 𝑛 𝐹 (𝜌 ) ( ) 𝑒𝑥𝑝 {− ( ) } · (1 − ) (11) 𝑛 𝑛 0 0 𝑛 of h(r , x) gives the curved surface like a mountain range. In the figure, D H(r , x) corre- n n 𝑏 𝑏 𝑏 𝑟 𝐹 𝑟 sponding to the volume of the vertical column indicates the probability that r and x are within the square region of Dr  Dx. The dashed line appearing on the x h(r , x) plane By putting n = 5, 𝑎 = 1.139, 𝑏 = 7.214, r = 1.5, and 𝐹 = 0.3055 along the previous n n corresponds to the marginal distribution for the inclusion depth x, as given by Equation (10). example, the joint probability density function of ℎ(𝜌 , 𝜉 ) was numerically calculated. On the other hand, another dashed curve appearing on the r h(r , x) plane corresponds n n ( ) The result of ℎ 𝜌 , 𝜉 thus obtained is depicted in Figure 7, where the joint probability to the marginal distribu 𝑛 tion for the inclusion size of r , given by Equation (12). function of ℎ(𝜌 , 𝜉 ) gives the curved surface like a mountain range. In the figure, n   o a1 a a r r ∆𝐻 (𝜌 , 𝜉 ) corresponding to the volume of the vertical column indicates the probability n1 n n f (r ) = h(r , x)dx = nfF (r )g ex p (12) n n n b b b that 𝜌 and 𝜉 are within the square region of ∆𝜌 × ∆𝜉 . The dashed line appearing on 𝑛 0 𝑛 the 𝜉 − ℎ(𝜌 , 𝜉 ) plane corresponds to the marginal distribution for the inclusion depth 𝜉 , In this place, let us divide the r x plane into fine meshes of Dr  Dx and consider n n as given by Equation (10). On the other hand, another dashed curve appearing on the the mesh at the arbitrary point of r , x , as shown in Figure 7. Then, the probability that ( ) the inclusion size r and the inclusion depth x yield within the area of Dr  Dx is provided ( ) n n 𝜌 − ℎ 𝜌 , 𝜉 plane corresponds to the marginal distribution for the inclusion size of 𝜌 , 𝑛 𝑛 𝑛 by the volume of the vertical column raising at this mesh of Dr  Dx . In other words, the given by Equation (12). volume of this column corresponds to the probability that the fatigue strength at N = 10 is given by Equation (5). Thus, we have the following equation, 𝑐 𝑎 −1 𝑎 𝑎 𝜌 𝜌 𝑛 𝑛 𝑛 −1 ( ) { ( )} (12) 𝑓 𝜌 = ∫ ℎ(𝜌 , 𝜉 )𝑑𝜉 = 𝑛 𝐹 𝜌 ( ) 𝑒𝑥𝑝 {− ( ) } 𝑛 0 𝑛 P[r 2 Dr \ x 2 Dx] 𝑛 = h(r , x)Dr Dx = P[s 2 Ds ] (13) n n n n w9 w9 𝑏 𝑏 𝑏 Based on the repetition of numerical calculations by Equation (13), the probability density function f (s ) and the cumulative distribution function F(s ) can be analyzed numerically. w9 w9 Probability density Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 16 In this place, let us divide the 𝜌 − 𝜉 plane into fine meshes of ∆𝜌 × ∆𝜉 and con- 𝑛 𝑛 sider the mesh at the arbitrary point of (𝜌 , 𝜉 ), as shown in Figure 7. Then, the probability that the inclusion size 𝜌 and the inclusion depth 𝜉 yield within the area of ∆𝜌 × ∆𝜉 is 𝑛 𝑛 provided by the volume of the vertical column raising at this mesh of ∆𝜌 × ∆𝜉 . In other words, the volume of this column corresponds to the probability that the fatigue strength at N = 10 is given by Equation (5). Thus, we have the following equation, ∗ ∗ 𝑃 [𝜌 ∈ ∆𝜌 ∩ 𝜉 ∈ ∆𝜉 ] = ℎ(𝜌 , 𝜉 )∆𝜌 ∆𝜉 = 𝑃 [𝜎 ∈ ∆𝜎 ] (13) 𝑛 𝑛 𝑛 𝑛 𝑤 9 𝑤 9 Based on the repetition of numerical calculations by Equation (13), the probability ∗ ∗ ( ) ( ) density function 𝑓 𝜎 and the cumulative distribution function 𝐹 𝜎 can be ana- 𝑤 9 𝑤 9 lyzed numerically. Appl. Sci. 2021, 11, 2889 8 of 15 Equation (11) Figure Figure7. 7. Sc Schematics hematics oof f joint joint probabil probability ity d density ensity fun function ction of of 𝜌r and and x𝜉. . 4. Results and Discussions 4. Results and Discussions 4.1. Distribution Characteristics of Fatigue Strength s w ∗9 4.1. Distribution Characteristics of Fatigue Strength 𝜎 𝑤 9 Since the number of inclusions included within the critical volume of a specimen Since the number of inclusions included within the critical volume of a specimen is is unknown, the authors analyzed the distribution characteristics of the fatigue strength, unknown, the authors analyzed the distribution characteristics of the fatigue strength, s , assuming several numbers of inclusions like n = 2, 3, 5, 6, 10, 20, and 40, respectively. w9 𝜎 , assuming several numbers of inclusions like n = 2, 3, 5, 6, 10, 20, and 40, respectively. 𝑤 9 Among them, the analytical results of the fatigue strength distribution obtained for n = 2, 5, Among them, the analytical results of the fatigue strength distribution obtained for n = 2, and 20, by the method in the previous section are indicated in Figure 8 as typical examples. 5, and 20, by the method in the previous section are indicated in Figure 8 as typical exam- Figure 8a shows the probability density functions under these three cases of n = 2, 5, and ples. Figure 8a shows the probability density functions under these three cases of n = 2, 5, 20, whereas Figure 8b indicates the cumulative distribution functions corresponding to and 20, whereas Figure 8b indicates the cumulative distribution functions corresponding the respective cases. Weibull parameters a and b in Equation (8) are provided to satisfy to the respective cases. Weibull parameters a and b in Equation (8) are provided to satisfy the condition of P[1 m < r < 15 m] = 0.8. It is found that the peak (mode) and the Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16 [ ] the condition of 𝑃 1 μm < 𝜌 < 15 μm = 0.8. It is found that the peak (mode) and the standard deviation of the fatigue strength distribution tends to decrease with an increase standard deviation of the fatigue strength distribution tends to decrease with an increase in the inclusion number. in the inclusion number. 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 800 1000 1200 1400 1600 Fatigue strength (MPa) (a) Probability density functions 1.0 Figure 8. Cont. 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Cumulative distribution functions Figure 8. Distribution patterns of fatigue strength. In order to investigate the effect of the condition for the dispersion of the inclusion size of 𝜌 , similar analyses were performed, giving some other conditions of 𝑃 [1 μm < 𝜌 < 10 μm] = 0.8 and 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8. Among these analyses, only the results of the cumulative distribution functions are indicated in Figure 9, due to article page limit. As seen in Figures 8b and 9, median of the fatigue strength distribution tends to increase with a decrease of the dispersion of the inclusion size, while standard deviation of the fatigue strength distribution tends to decrease a little, depending on the decrease of the dispersion of the inclusion size. Comparing these analytical distribution characteristics of the fatigue strength 𝜎 in 𝑤 9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the [ ] fatigue strength 𝜎 under the conditions of 𝑃 1 μm < 𝜌 < 15 μm = 0.8 and n = 5 was 𝑤 9 roughly in agreement with the experimental distribution aspect of the fatigue strength at N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. Probability density Cumulative probability Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 16 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0.000 Appl. Sci. 2021, 11, 2889 9 of 15 1000 1200 1400 1600 Fatigue strength (MPa) (a) Probability density functions 1.0 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Cumulative distribution functions Figure 8. Distribution patterns of fatigue strength. Figure 8. Distribution patterns of fatigue strength. In order to investigate the effect of the condition for the dispersion of the inclusion size of r, In order to investigate the effect of the condition for the dispersion of the inclusion similar analyses were performed, giving some other conditions of P[1 m < r < 10 m] = 0.8 size of 𝜌 , similar analyses were performed, giving some other conditions of 𝑃 [1 μm < 𝜌 < and P 1 m < r < 5 m = 0.8. Among these analyses, only the results of the cumulative [ ] 10 μm] = 0.8 and 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8. Among these analyses, only the results of distribution functions are indicated in Figure 9, due to article page limit. As seen in the cumulative distribution functions are indicated in Figure 9, due to article page limit. Figures 8b and 9, median of the fatigue strength distribution tends to increase with a As seen in Figures 8b and 9, median of the fatigue strength distribution tends to increase decrease of the dispersion of the inclusion size, while standard deviation of the fatigue Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 16 with a decrease of the dispersion of the inclusion size, while standard deviation of the strength distribution tends to decrease a little, depending on the decrease of the dispersion fatigue strength distribution tends to decrease a little, depending on the decrease of the of the inclusion size. dispersion of the inclusion size. Comparing these analytical distribution characteristics of the fatigue strength 𝜎 in 1.0 𝑤 9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the 0.8 fatigue strength 𝜎 under the conditions of 𝑃 [1 μm < 𝜌 < 15 μm] = 0.8 and n = 5 was 𝑤 9 roughly in agreement with the experimental distribution aspect of the fatigue strength at 0.6 N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. 0.4 0.2 0.0 600 800 1000 1200 1400 Fatigue strength (MPa) [ ] (a) Results in 𝑃 1 μm < 𝜌 < 10 μm = 0.8 1.0 0.8 0.6 0.4 0.2 0.0 600 800 1000 1200 1400 1600 Fatigue strength (MPa) (b) Results in 𝑃 [1 μm < 𝜌 < 5 μm] = 0.8 Figure 9. Comparison of cumulative distribution functions in respective conditions. Figure 9. Comparison of cumulative distribution functions in respective conditions. 4.2. Expansion of the Probabilistic Model to Analyze P-S-N Characteristics in Interior-Induced Fracture Target of the probabilistic model described in Section 3 is the distribution character- istics of the fatigue strength only at stress cycles of N = 10 . Accordingly, the analytical model should be conceptionally expanded to interpret the whole statistical aspect of the S-N property, in the interior inclusion-induced fracture mode. From this point of view, denoting the fatigue strength at any stress cycle, 𝑁 , by 𝜎 , Equation (3) is rewritten as follows: −1/6 𝜎 = 1273 · 𝜌 + 𝛽 , (14) 𝛽 = 𝜆 {𝑙𝑜𝑔 (𝑁 )} + 𝛾 . (15) Based on the dotted S-N curve in Figure 1, parameters λ and γ in Equation (15) were 6 ∗ 9 ∗ ( ) ( determined from two points 𝑁 = 10 , 𝜎 = 1424 MPa and 𝑁 = 10 , 𝜎 = 𝑤 9 𝑤 9 920 MPa ) as λ = −168 and γ = 1512, respectively. Substituting these values into Equations (14) and (15), Equation (14) is rewritten as −1/6 𝜎 = 1273 · 𝜌 − 168 · {𝑙𝑜𝑔 (𝑁 )} + 1512. (16) 6 7 Here, 𝜎 − 𝜌 relationships under the fixed numbers of the fatigue life of N = 10 , N = 10 , 8 9 N = 10 , and N = 10 are depicted in Figure 10. As suggested from Equations (14)–(16), analytical curves in Figure 10 tend to shift in parallel along the ordinate. Of course, the curve tends to shift downwards with an increase in the fixed number of stress cycles. Ap- plying the concept of Equation (4), one could calculate the fatigue strength 𝜎 by 1.5 ∗ −1/6 𝜎 = [1273 · 𝜌 − 168 · {𝑙𝑜𝑔 (𝑁 )} + 1512]. (17) 1.5−𝜉 Probability density Cumulative probability Cumulative probability Cumulative probability 𝑓𝑖𝑥 𝑓𝑖𝑥 𝑓𝑖𝑥 𝑓𝑖𝑥 Appl. Sci. 2021, 11, 2889 10 of 15 Comparing these analytical distribution characteristics of the fatigue strength s in w9 Figures 8 and 9 with the experimental result in Figure 1, the distribution feature of the fatigue strength s under the conditions of P[1 m < r < 15 m] = 0.8 and n = 5 was w9 roughly in agreement with the experimental distribution aspect of the fatigue strength at N = 10 . The most preferable number of inclusions, n, is further discussed again in Section 4.4. 4.2. Expansion of the Probabilistic Model to Analyze P-S-N Characteristics in Interior-Induced Fracture Target of the probabilistic model described in Section 3 is the distribution character- istics of the fatigue strength only at stress cycles of N = 10 . Accordingly, the analytical model should be conceptionally expanded to interpret the whole statistical aspect of the S-N property, in the interior inclusion-induced fracture mode. From this point of view, denoting the fatigue strength at any stress cycle, N , by s , Equation (3) is rewritten f ix as follows: 1/6 s = 1273r + b, (14) n  o b = l log N + g. (15) f ix Based on the dotted S-N curve in Figure 1, parameters  and in Equation (15) were de- 6  9 termined from two points N = 10 , s = 1424 MPa and N = 10 , s = 920 MPa w9 w9 as l =168 and g = 1512, respectively. Substituting these values into Equations (14) and (15), Equation (14) is rewritten as n  o 1/6 s = 1273r 168 log N + 1512. (16) f ix Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 16 6 7 Here, s r relationships under the fixed numbers of the fatigue life of N = 10 , N = 10 , 8 9 N = 10 , and N = 10 are depicted in Figure 10. As suggested from Equations (14)–(16), analytical curves in Figure 10 tend to shift in parallel along the ordinate. Of course, the Equation (17) includes arbitrarily given number of stress cycles 𝑁 , together with curve tends to shift downwards with an increase in the fixed number of stress cycles. the inclusion size 𝜌 and the inclusion depth ξ. Therefore, combining the joint probability Applying the concept of Equation (4), one could calculate the fatigue strength s by function in Figure 7 with Equation (17), one could analyze the distribution characteristics h n  o i of the fatigue strength at any number of stress cycles 𝑁 by repeating the numerical 1.5 1/6 s = 1273r 168 log N + 1512 . (17) w f ix calculations. 1.5 x N=10 N=10 N=10 N=10 10 20 30 40 Inclusion radius 𝜌 (μm) Figure 10. s - r relationships at several given numbers of stress cycles. Figure 10. σw- 𝜌 relationships at several given numbers of stress cycles. Equation (17) includes arbitrarily given number of stress cycles N , together with the f ix 6 7 8 Probability density functions of the fatigue strength thus obtained at N = 10 , 10 , 10 , inclusion size r and the inclusion depth . Therefore, combining the joint probability func- and 10 are indicated at the respective numbers of the stress cycles in Figure 11. These analytical results are roughly in agreement with the overall feature of the experimental aspect on the statistical fatigue characteristics. The scale of the axis for the probability density is adjusted to correspond to each value listed in the table attached in the right- hand side in Figure 11. The percentile points of 𝐹 = 1%, 10%, 50%, 90%, and 99% for the fatigue strength are indicated by marks of ◇’s along the vertical axis of the probability density functions. The respective thin dotted lines passing through the percentile points corresponding to the same probability give the P-S-N curves representing the statistical fatigue characteristics of this steel. All data points yield within the range of 1–99% in Fig- ure 11. This fact suggests that the probabilistic model developed in this study has an avail- ability to interpret the physical meaning of the statistical fatigue property in the very high cycle regime. Fatigue strength σw (MPa) 𝑓𝑖𝑥 𝑓𝑖𝑥 Appl. Sci. 2021, 11, 2889 11 of 15 tion in Figure 7 with Equation (17), one could analyze the distribution characteristics of the fatigue strength at any number of stress cycles N by repeating the numerical calculations. f ix 6 7 8 Probability density functions of the fatigue strength thus obtained at N = 10 , 10 , 10 , and 10 are indicated at the respective numbers of the stress cycles in Figure 11. These analytical results are roughly in agreement with the overall feature of the experimental aspect on the statistical fatigue characteristics. The scale of the axis for the probability density is adjusted to correspond to each value listed in the table attached in the right-hand side in Figure 11. The percentile points of F = 1%, 10%, 50%, 90%, and 99% for the fatigue strength are indicated by marks of 3’s along the vertical axis of the probability density functions. The respective thin dotted lines passing through the percentile points corresponding to the same probability give the P-S-N curves representing the statistical fatigue characteristics of this steel. All data points yield within the range of 1–99% in Figure 11. This fact suggests that the probabilistic model developed in this study has an Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 16 availability to interpret the physical meaning of the statistical fatigue property in the very high cycle regime. Probability density at peak of distribution ◇: Analyzed percentile points Nfix f(𝜎 ) 1600 𝑤 6 −3 10 3.76×10 7 −3 10 4.10×10 8 −3 10 4.52×10 9 −3 10 4.96×10 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode Although distribution characteristics of the fatigue strength s at the given number Although distribution characteristics of the fatigue strength 𝜎 at the given number of stress cycles were analyzed in the previous section, the fatigue life distributions could of stress cycles were analyzed in the previous section, the fatigue life distributions could also be analyzed based on Equation (17), by giving any value of the stress amplitude also be analyzed based on Equation (17), by giving any value of the stress amplitude 𝜎 . s . Applying this method, fatigue life distributions were calculated at s = 1400 MPa, w ∗ w Applying this method, fatigue life distributions were calculated at 𝜎 = 1400 MPa, 1200 1200 MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus ob- obtained were indicated at the respective stress levels in Figure 12. Since the fatigue tained were indicated at the respective stress levels in Figure 12. Since the fatigue life in- life increased logarithmically with decrease of the stress level, the probability density at creased logarithmically with decrease of the stress level, the probability density at the the peak of the density function tended to decrease such that the integrated value of the peak of the density function tended to decrease such that the integrated value of the den- density function maintained unity. Values of the peak density at the respective stress sity function maintained unity. Values of the peak density at the respective stress levels levels are indicated on the right-hand side, in Figure 12. As shown in this table, the peak are indicated on the right-hand side, in Figure 12. As shown in this table, the peak density density varied drastically, depending on the stress level, due to the above described reason, varied drastically, depending on the stress level, due to the above described reason, but but the distribution curves were drowned when they had the same height for the sake the distribution curves were drowned when they had the same height for the sake of con- of convenience. venience. In order to distinguish the fatigue strength distribution and the fatigue life distribution, the notations of f (s ) and F(s ) are used for the fatigue strength distribution, whereas w w other notations of g s and G s are accepted for the fatigue life distribution. Similar ( ) ( ) w w to Figure 11, the percentile points of G = 1%, 10%, 50%, 90%, and 99% for the fatigue Probability density at life are indicated by marks of 3’s along the horizontal lines, at the respective stress levels. peak of distribution ◇: Analyzed percentile points The respective thin dotted lines passing through the percentile points corresponding to 𝜎 (MPa) −7 1400 6.14×10 −8 1200 4.61×10 −9 1000 3.57×10 −10 900 9.91×10 2 3 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. Stress amplitude, (MPa) Stress amplitude, (MPa) Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 17 Probability density at peak of distribution ◇: Analyzed percentile points Nfix f( ) 6 −3 10 3.76×10 7 −3 10 4.10×10 8 −3 10 4.52×10 9 −3 10 4.96×10 9 10 2 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 Number of stress cycles to failure, N Figure 11. Fatigue strength distributions and P-S-N curves in very high cycle regime. 4.3. Analysis of the Fatigue Life Distributions in Interior-Induced Fracture Mode Although distribution characteristics of the fatigue strength at the given number of stress cycles were analyzed in the previous section, the fatigue life distributions could also be analyzed based on Equation (17), by giving any value of the stress amplitude . Applying this method, fatigue life distributions were calculated at = 1400 MPa, 1200 MPa, 1000 MPa, and 900 MPa, respectively. The probability density functions thus obtained were indicated at the respective stress levels in Figure 12. Since the fatigue life Appl. Sci. 2021, 11, 2889 12 of 15 increased logarithmically with decrease of the stress level, the probability density at the peak of the density function tended to decrease such that the integrated value of the density function maintained unity. Values of the peak density at the respective stress the same probability give the P-S-N curves of this steel. All experimental data in the levels are indicated on the right-hand side, in Figure 12. As shown in this table, the peak interior-induced fracture appear within the range of 1–99% in Figure 12. Thus, it was finally density varied drastically, depending on the stress level, due to the above described noted that the statistical fatigue property in the interior-induced fracture of this steel could reason, but the distribution curves were drowned when they had the same height for the be well explained from either one of the fatigue strength distribution and the fatigue life sake of convenience. distribution, through the probabilistic model proposed in this study. Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 16 Probability density at In order to distinguish the fatigue strength distribution and the fatigue life distribu- peak of distribution ◇: Analyzed percentile points tion, the notations of 𝑓 (𝜎 ) and 𝐹 (𝜎 ) are used for the fatigue strength distribution, 𝑤 𝑤 MPa whereas other notations of 𝑔 (𝜎 ) and 𝐺 (𝜎 ) are accepted for the fatigue life distribu- 𝑤 𝑤 1600 −7 1400 6.14×10 tion. Similar to Figure 11, the percentile points of 𝐺 = 1%, 10%, 50%, 90%, and 99% for −8 1200 4.61×10 the fatigue life are indicated by marks of ◇’s along the horizontal lines, at the respective −9 1000 3.57×10 stress levels. The respective thin dotted lines passing through the percentile points corre- −10 900 9.91×10 sponding to the same probability give the P-S-N curves of this steel. All experimental data in the interior-induced fracture appear within the range of 1–99% in Figure 12. Thus, it was finally noted that the statistical fatigue property in the interior-induced fracture of this steel could be well explained from either one of the fatigue strength distribution and the fatigue life distribution, through the probabilistic model proposed in this study. 4.4. Reconfirmation of the Number of Inclusions in the Critical Volume 2 3 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 10 10 As described in Section 4.1, since the number of inclusions included in the critical Number of stress cycles to failure, N volume is unknown, the analysis was carried out by setting three conditions of n = 2, 5, Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. and 20, as the inclusion number. Then, the condition of n = 5 was tentatively selected as a Figure 12. Fatigue life distributions and P-S-N curves in a very high cycle regime. preferable number of inclusions. However, in order to confirm the most reasonable num- 4.4. Reconfirmation of the Number of Inclusions in the Critical Volume ber of inclusions, the fatigue life distributions were additionally analyzed, under condi- As described in Section 4.1, since the number of inclusions included in the critical tions of n = 6, 8, and 10, respectively. Based on the series of analytical results, percentile volume is unknown, the analysis was carried out by setting three conditions of n = 2, 5, points of the fatigue life distribution corresponding to 𝐺 = 1%, 10%, 50%, 90%, and 99% and 20, as the inclusion number. Then, the condition of n = 5 was tentatively selected as a were determined; the relationship between those percentile points and the number of in- preferable number of inclusions. However, in order to confirm the most reasonable number clusions are depicted in Figure 13. Comparing five solid curves carefully with the disper- of inclusions, the fatigue life distributions were additionally analyzed, under conditions sion aspect of the experimental results in an interior-induced fracture in Figure 12, it was of n = 6, 8, and 10, respectively. Based on the series of analytical results, percentile points finally noted that the analytical result at n = 6 was well fitted compared to the result at n of the fatigue life distribution corresponding to G = 1%, 10%, 50%, 90%, and 99% were = 5. determined; the relationship between those percentile points and the number of inclusions Based on this evidence, the statistical fatigue properties (P-S-N properties) in Figures are depicted in Figure 13. Comparing five solid curves carefully with the dispersion aspect 11 and 12 were numerically analyzed by putting n = 6 for the number of inclusions in- of the experimental results in an interior-induced fracture in Figure 12, it was finally noted cluded in the critical volume of the specimen. that the analytical result at n = 6 was well fitted compared to the result at n = 5. 𝜎 = 900MPa 7 8 9 10 10 10 10 10 Number of stress cycles to failure, N Figure 13. Number of inclusions versus number of stress cycles to failure. Figure 13. Number of inclusions versus number of stress cycles to failure. 4.5. Mutual Relationship between Both Distributions of Fatigue Strength and Fatigue Life In this study, the authors showed that both fatigue strength distribution and fatigue ( ) life distribution could be derived from the joint probability density function of ℎ 𝜌 , 𝜉 , given by Equation (11). It was also confirmed that the analytical results were in good agreement with the statistical aspect of the experimental fatigue data in the very high cycle regime. In this place, mutual relationship between the distribution aspects of the fatigue Stress amplitude, (MPa) Stress amplitude, (MPa) Number of inclusions, n Appl. Sci. 2021, 11, 2889 13 of 15 Based on this evidence, the statistical fatigue properties (P-S-N properties) in Appl. Sci. 2021, 11, x FOR PEER REVIEW Figur es 11 and 12 were numerically analyzed by putting n = 6 for the number of inclusions 14 of 16 included in the critical volume of the specimen. 4.5. Mutual Relationship between Both Distributions of Fatigue Strength and Fatigue Life strength and the fatigue life is further discussed to reconfirm the reasonability of the pre- In this study, the authors showed that both fatigue strength distribution and fatigue sent probabilistic model for the P-S-N characteristics. The mutual relationship between life distribution could be derived from the joint probability density function of h(r , x), both distributions is illustrated in Figure 14, where the ordinate is the fatigue strength given by Equation (11). It was also confirmed that the analytical results were in good and t agreement he absciwith ssa isthe the statistical fatigue life aspect N. When of the any p experimental oint P(Nfix, fatigue ) is ta dat ken a in in th the is d very iagram, high cycle regime. In this place, mutual relationship between * the distribution aspects of the the probability that the fatigue life at the stress level is less than Nfix always corre- fatigue strength and the fatigue life is further discussed to reconfirm the reasonability of the sponds to the probability that the fatigue strength at the definite stress cycles Nfix is lower present probabilistic model for the P-S-N characteristics. The mutual relationship between * * than  . In other words, the cumulative probability of Nfix, G(Nfix, ), always corre- w w both distributions is illustrated in Figure 14, where the ordinate is the fatigue strength s * *  * spon andd the s toabscissa the cumis ulative the fatigue probability life N .of When  , F any (Nfipoint x, ) P [3 (0 N ]. G ,(s Nfi) x,is take ) gives n in th this e di diagram, stribu- w w fix w w the probability that the fatigue life at the stress level s is less than N always corresponds * * fix tion function of the fatigue life at  and F(Nfix, ) gives the distribution function of the w w to the probability that the fatigue strength at the definite stress cycles N is lower than s . fix w fatigue strength at Nfix. Thus, we have In other words, the cumulative probability of N , G(N ,s ), always corresponds to the fix fix * * cumulative probability of s , F(N ,s ) [30]. G(N ,s ) gives the distribution function of w fix w fix w   (18) G(Nfix, ) = F(Nfix, ) w w the fatigue life at s and F(N ,s ) gives the distribution function of the fatigue strength at fix w w N . Thus, we have fixEquation (18) implies that the dashed area G(Nfix, ) is always equal to the other G N , s ) = F N , s (18) ( ( ) * fix fix w w dashed area F(Nfix, ) in Figure 14. Figure 14. Relationship between fatigue strength distribution and fatigue life distribution. Figure 14. Relationship between fatigue strength distribution and fatigue life distribution. Equation (18) implies that the dashed area G(N ,s ) is always equal to the other fix w Based on this equality of Equation (18), one could obtain the S-N curve corresponding dashed area F(N ,s ) in Figure 14. fix to any level of the fracture probability. Thus, the probabilistic S-N curves at the respective Based on this equality of Equation (18), one could obtain the S-N curve correspond- fracture probabilities could be analyzed as P-S-N curves. As described in Sections 4.2 and ing to any level of the fracture probability. Thus, the probabilistic S-N curves at the 4.3, the distribution characteristics of the experimental S-N data were well explained from respective fracture probabilities could be analyzed as P-S-N curves. As described in either viewpoint of the fatigue strength distribution or the fatigue life distribution. The Sections 4.2 and 4.3, the distribution characteristics of the experimental S-N data were well equality of Equation (18) is the reason why the reasonable P-S-N curves could be equally explained from either viewpoint of the fatigue strength distribution or the fatigue life obtained from different viewpoints of the fatigue strength distribution and the fatigue life distribution. The equality of Equation (18) is the reason why the reasonable P-S-N curves distribution. could be equally obtained from different viewpoints of the fatigue strength distribution and the fatigue life distribution. 5. Conclusions 5. Conclusions Main conclusions obtained in this study are summarized as follows. Main conclusions obtained in this study are summarized as follows. 1. The probability density functions of the inclusion size at the crack initiation site, 𝑓 (𝜌 ), was successfully derived by combining the Weibull distribution and the con- 1. The probability density functions of the inclusion size at the crack initiation site, f (r ), cept of extreme distribution. In addition, the probability density function of the in- was successfully derived by combining the Weibull distribution and the concept of ( ) clusion depth, 𝑓 𝜉 , was also derived from the uniform distribution of the location of the inclusion in the material space. 2. Since the inclusion size 𝜌 and the crack depth ξ are statistically independent, the joint probability density function of these random variables, ℎ(𝜌 , 𝜉 ), is given by the 𝑛 Appl. Sci. 2021, 11, 2889 14 of 15 extreme distribution. In addition, the probability density function of the inclusion depth, f (x), was also derived from the uniform distribution of the location of the inclusion in the material space. 2. Since the inclusion size r and the crack depth  are statistically independent, the joint probability density function of these random variables, h(r , x), is given by the direct multiplication of the above two probability density functions, such as h(r , x) = f (r ) f (x). n n 0 3. For the hourglass type specimen of a bearing steel with the definite hardness, the rotating bending fatigue strength of the specimen with any size r and depth  of the inclusion at the arbitrarily given number of stress cycles N is analytically provided fix in the very high cycle regime. 4. Based on the above joint probability density function of h(r , x), repeating the numer- ical calculations following P[r 2 Dr \ x 2 Dx] = h(r , x)Dr Dx = P[s 2 Ds ], n n n n w9 w9 one could obtain the fatigue strength distribution at any number of stress cycles and the fatigue life distribution at any stress level. The analytical results thus obtained were in good agreement with the statistical feature of the experimental fatigue test data. In the construction of the probabilistic model in this study, several assumptions were introduced for the sake of simplicity. In order to reconfirm the reasonability of such assumptions, sufficient number of experimental results should be further filed up in the future. In addition, it is still unknown whether the fatigue limit exists for the interior inclusion-induced fracture mode. This is one of the most important issues to be solved in the near future. Author Contributions: T.S. has performed fatigue tests and organized whole of this work. A.N. and Y.N. have analyzed experimental data and prepared all of diagrams. N.O. has prepared fatigue specimens and performed fatigue tests. All 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 1. Bathias, C. There is no infinite fatigue life in metallic materials. Fatigue Fract. Eng. Mater. Struct. 1999, 22, 559–565. [CrossRef] 2. Mughrabi, H. Specific features and mechanisms of fatigue in the ultrahigh-cycle regime. Int. J. Fatigue 2006, 28, 1501–1508. [CrossRef] 3. Sakai, T.; Takeda, M.; Shiozawa, K.; Ochi, Y.; Nakajima, M.; Nakamura, T.; Oguma, N. Experimental reconfirmation of character- istic S-N property for high carbon chromium bearing steel in wide life region in rotating bending. J. Soc. Mat. Sci. Jpn. 2000, 49, 779–785. [CrossRef] 4. Bathias, C.; Paris, P.C. Gigacycle Fatigue in Mechanical Practice; Marcel Deckker: New York, NY, USA, 2005. 5. Sakai, T.; Sato, Y.; Oguma, N. Characteristic S-N properties of high-carbon-chromium-bearing steel under axial loading in long-life fatigue. Fatigue Fract. Eng. Mater. Struct. 2002, 25, 765–773. [CrossRef] 6. Sakai, T.; Sato, Y.; Nagano, Y.; Takeda, M.; Oguma, N. Effect of stress ratio on long life fatigue behavior of high carbon chromium bearing steel under axial loading. Int. J. Fatigue 2006, 28, 1547–1554. [CrossRef] 7. Sakai, T.; Lian, B.; Takeda, M.; Shiozawa, K.; Oguma, N.; Ochi, Y.; Nakajima, M.; Nakamura, T. Statistical duplex S-N characteristics of high carbon chromium bearing steel in rotating bending in very high cycle regime. Int. J. Fatigue 2010, 32, 497–504. [CrossRef] 8. Oguma, N.; Lian, B.; Sakai, T.; Watanabe, K.; Odake, Y. Long life fatigue fracture induced by interior inclusions for high carbon chromium bearing steels under rotating bending. J. ASTM Int. 2010, 7, 1–9. [CrossRef] 9. Murakami, Y. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions; Elsevier: Oxford, UK, 2002; pp. 88–94. 10. Takeda, M.; Sakai, T.; Oguma, N. Rotating bending fatigue property and fractography for high strength steels and carbon steel over ultra wide life region. Trans. JSME Ser. A 2002, 68, 977–984. [CrossRef] 11. Murakami, Y.; Kodama, S.; Konuma, S. Quantitative evaluation of effects of nonmetallic inclusions on fatigue strength of high strength steel. Trans. JSME Ser. A 1988, 54, 688–696. [CrossRef] 12. Toriyama, T.; Murakami, Y.; Makino, T. Database of nonmetallic inclusions and its application to the fatigue strength prediction method of high strength steels. J. Soc. Mat. Sci. Jpn. 1991, 40, 1497–1503. [CrossRef] 13. Adachi, A.; Shoji, H.; Kuwabara, A.; Inoue, Y. Rotating bending fatigue phenomenon of JIS SUJ2 bearing steel. DENKI SEIKO 1975, 46, 176–182. [CrossRef] 14. Barbosa, J.F.; Correia, J.A.F.O.; Junior, R.C.S.F.; De Jesus, A.M.P. Fatigue life prediction of metallic materials considering mean stress effects by means of an artificial neural network. Int. J. Fatigue 2020, 135, 105527. [CrossRef] Appl. Sci. 2021, 11, 2889 15 of 15 15. Gope, P.C.; Mahar, C.S. Evaluation of fatigue damage parameters for Ni-based super alloys Inconel 825 steel notched specimen using stochastic approach. Fatigue Fract. Eng. Mater. Struct. 2021, 44, 427–443. [CrossRef] 16. Lehner, P.; Krejsa, M.; Parenica, P.; Krivy, V.; Brozovsky, J. Fatigue damage analysis of a riveted steel overhead crane support truss. Int. J. Fatigue 2019, 128, 105190. [CrossRef] 17. Tomaszewski, T.; Strzelecki, P.; Wachowski, M.; Stopel, M. Fatigue life prediction for acid-resistant steel plate under operating loads. Bull. Pol. Acad. Tech. 2020, 68, 913–921. [CrossRef] 18. Feng, L.; Zhang, L.; Liao, X.; Zhang, W. Probabilistic fatigue life of welded plate joints under uncertainty in arctic areas. J. Constr. Steel Res. 2021, 176, 106412. [CrossRef] 19. Caiza, P.D.T.; Ummenhofer, T. A probabilistic Stussi function for modelling the S-N curves and its application on specimens made of steel S355J2+N. Int. J. Fatigue 2018, 117, 121–134. [CrossRef] 20. Mughrabi, H. Zur Dauerschwingfestigkeit im Bereich extrem hoher bruchlastspielzahlen: Mehrstufige lebensdauerkurven. HTM/Harterei-Tech. Mitt. 2001, 56, 300–303. 21. Harlow, D.G.; Wei, R.P.; Sakai, T.; Oguma, N. Crack growth based probability modeling of S-N response for high strength steel. Int. J. of Fatigue 2006, 28, 1479–1485. [CrossRef] 22. Sakai, T. Chair of Editorial Committee. In Standard Evaluation Method of Fatigue Reliability for Metallic Materials—Standard Regression Method of S-N Curves; JSMS-SD-11-07; The Society of Materials Science: Kyoto, Japan, 2007. 23. Nakagawa, A.; Sakai, T.; Harlow, D.G.; Oguma, N.; Nakamura, Y.; Ueno, A.; Kikuchi, S.; Sakaida, A. A probabilistic model on crack initiation modes of metallic materials in very high cycle fatigue. Procedia Struct. Integr. 2016, 2, 1199–1206. [CrossRef] 24. Kitagawa, H.; Takahashi, S. Fracture mechanics study on small fatigue crack growth and the condition of its threshold. Trans. JSME Ser. A 1979, 45, 1289–1303. 25. Tanaka, K.; Nakai, Y.; Yamashita, M. Fatigue growth threshold of small cracks. Int. J. Fract. 1981, 17, 519–533. 26. Nakamura, Y.; Sakai, T.; Harlow, D.G.; Oguma, N.; Nakajima, M.; Nakagawa, A. Probabilistic model on statistical fatigue property in very high cycle regime based on distributions of size and location of interior inclusions. In Proceedings of the VHCF-7, Dresden, Germany, 3–5 July 2017; pp. 81–86. 27. Weibull, W. A statistical distribution function of wide applicability. J. Appl. Mech. 1951, 18, 293–297. 28. Gumbel, E.J. Statistics of Extremes; Columbia University Press: New York, NY, USA, 1958; pp. 75–78. 29. Shiozawa, K.; Lu, L.-T.; Ishihara, S. Subsurface fatigue crack initiation behavior and S-N curve characteristics in high carbon- chromium bearing steel. J. Soc. Mat. Sci. Jpn. 1999, 48, 1095–1100. [CrossRef] 30. Sakai, T.; Sakaida, A.; Fujitani, K.; Tanaka, T. A study on statistical fatigue properties of metallic materials with particular attention to relation between fatigue life and fatigue strength distributions. Mem. Res. Inst. Sci. Eng. 1982, 42, 79–91.

Journal

Applied Sciences – Multidisciplinary Digital Publishing Institute

Published: Mar 24, 2021

Keywords: probabilistic model; very high cycle fatigue; duplex S-N curves; rotating bending; inclusion; fish-eye; inclusion size; inclusion depth

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