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Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel
International Journal of Fatigue xxx (2016) xxx–xxx
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Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel Yoichi Yamashita a,⇑, Yukitaka Murakami b a b
Research Laboratory, IHI Corporation, 1, Shin-Nakahara-Cho, Isogo-Ku, Yokohama 235-8501, Japan Kyushu University, 774 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan
a r t i c l e
i n f o
Article history: Received 12 January 2016 Received in revised form 4 April 2016 Accepted 12 April 2016 Available online xxxx Keywords: Fatigue crack growth Hydrogen effect Nonmetallic inclusion pffiffiffiffiffiffiffiffiffiffi area parameter model ODA
a b s t r a c t Fatigue failure of high strength steels mostly originates at nonmetallic inclusions. An optically dark area (ODA) beside the inclusion can be observed in specimens fractured at very high cycle fatigue (VHCF) regime. The present paper proposes fatigue life prediction models from low to VHCF regime. The fatigue life prediction model inside ODA has been constructed in the VHCF regime based on the master curve of the growth of ODA where fatigue failure is caused by cyclic loading assisted by hydrogen trapped by inclusion. The fatigue crack growth law is proposed for a small crack outside ODA within the framework of the pffiffiffiffiffiffiffiffiffiffi area parameter model where the concept of ‘‘continuously variable fatigue limit” for small crack is introduced. The life and scatter of fatigue life originating at inclusions can be well evaluated by the proposed models. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue failure of high strength steels mostly originates at nonmetallic inclusions [1–5]. This paper reveals the crucial factors for solving this problem and proposes the model for predicting the fatigue life of components. First, fatigue failure mechanism of low or medium strength steels will be reviewed where fatigue cracks usually initiate at free surface of materials. And then, internal fatigue failure mechanism of high strength steels will be reviewed. Even in very high cycle fatigue (VHCF) regime beyond 107 cycles of low strength steel, crack would not originate at inclusion but at persistent slip band in the surface of specimen or component. It is known [6–10] that the persistent slip bands are irreversible during fatigue cycles and some of them can be the origin of crack initiation. But in this case, the effects of moisture and oxygen in the air on the fatigue mechanism cannot be ignored during long test period. This irreversibility should be different from the mechanism of very high cycle fatigue originating at subsurface inclusions. At and even below the fatigue limit of low strength steels many microcracks can be observed and so the fatigue limit should be defined as the threshold stress for the growth of cracks which are nucleated under the stress. These cracks initiate at very early stage of fatigue test and stop propagation much before 107 cycles ⇑ Corresponding author.
if environmental factors such as humidity, oxygen, temperature, etc do not affect by continuing test for long period [1]. The threshold condition (or fatigue limit) of a material is determined by the non-propagation of these microcracks that initiated in the original microstructure [1,11,12] on the surface of unnotched specimens where the most crucial and rational mechanism to explain the non-propagation phenomenon is the plasticity-induced crack closure found by Elber [13,14]. The oxide induced crack closure [15] and the surface roughness induced crack closure [16] have additional effects on crack closure behavior depending on the case. When cracks initiate along slip bands or at grain boundaries, good correlations have been obtained among ultimate tensile strength, rU , hardness, HV and fatigue limit [17–20]. The fatigue limit rFL is proportional to the Vickers hardness regardless of various microstructures with different grain sizes, e.g.: in steels rFL = 1.6HV for HV = <400 where rFL is in MPa and HV in kgf/mm2. On the other hand, in high strength steels of HV > 400, material defects such as nonmetallic inclusions are often observed at fracture origin. The fatigue limit is not simply proportional to the Vickers hardness for the materials with HV > 400. The fatigue limit is strongly influenced by defect size. The quantitative fatigue analysis of the effect of defect size using artificial small defects was systematically studied by Murakami and Endo [21]. Here, the fatigue threshold is also determined by the threshold condition of cracks emanating from such defects [22]. The early works of Naito et al. [23] and Asami and Sugiyama [24] have shown that fatigue failure does occur at lives longer than
E-mail address: [email protected] (Y. Yamashita). http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016 0142-1123/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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N = 107 at stress levels lower than the conventional fatigue limit. For high strength steels especially in rotary bending fatigue tests, fatigue failure can be initiated both at the surface and the interior presenting two knees in S–N diagram [25,26]. This two-stage material behavior in S–N diagram is called as the stepwise or duplex S–N property. Since the duplex S–N diagram is not observed in tension–compression fatigue tests [27], it can be understood that the duplex S–N diagram is resulted from stress gradient in bending. The loading frequency has been always a concern in the test of VHCF [28–35]. Stanzl-Tschegg [29] and Bathias [30,31] showed the VHCF data obtained by ultrasonic fatigue testing machines with very high frequency of 20 kHz. Some researchers found that the effect of loading frequency is very small in high strength steels by Furuya et al. [34], titanium alloy by Ritchie et al. [32] and aluminium alloy by Mayer et al. [33]. In contrast, some papers show the load frequency effects on fatigue strength for mild steels [28,35]. Qian et al. [36] and Sun et al. [37] reviewed the VHCF behavior and its life prediction models. Many of them are based on for example, Tanaka and Mura model [38] for fatigue crack initiation life [39] and are based on the Paris law [40] for fatigue crack growth life [25,41–46]. Recently, Sun et al. [37] studied the cumulative damage model for fatigue life estimation of high-strength steels in HCF and VHCF regime based on the ultrasonic fatigue test data. But this model has not taken into account the small crack effects [47] that the crack growth rate and the threshold stress intensity factor range definitely depends on its crack size. Fatigue strength prediction models for small cracks [3–5,48–50] pffiffiffiffiffiffiffiffiffiffi have been studied by many researchers using the area parameter model [21]. Murakami et al. proposed the fatigue life prediction pffiffiffiffiffiffiffiffiffiffi method using the area parameter model in combination with the statistics of extreme values of internal defects in VHCF regime [22]. Most of the researches have not taken into account the hydrogen effect coupled with cyclic stresses on fatigue failure originating at non-metallic inclusion. The fact that hydrogen is densely trapped by inclusions cannot be ignored in the discussion of fatigue mechanism. Also it must be noted that the fatigue failure from small defects such as nonmetallic inclusions is essentially the small crack problem and the models must be based on the mechanics of small crack. Especially in the VHCF regime of high strength steels, interior crack initiation emanating from internal inclusion often occurs with a distinct fracture surface morphologies beside the inclusions called the Optically Dark Area (ODA) (see Fig. 1, [1]). Murakami et al. [3–5] detected densely trapped hydrogen around inclusions. They pointed out that the fatigue failure accompanying a peculiar morphology, ODA, at the fracture origin of the center of fish-eye mark is caused by the assistance of the hydrogen trapped by inclusion with cyclic stresses. This assistance of hydrogen results in the elimination of the conventional fatigue limit. Considering the very different morphologies of the fracture surfaces inside ODA and outside ODA, it is presumed that the fatigue mechanisms inside ODA and outside ODA are substantially different. It has been found by Murakami et al. [4] that there is a good correlation between the ODA size normalized by the inclusion size and the number of fatigue cycles to failure originating at nonmetallic inclusions. This relationship was named the master curve of the growth of ODA. One of the design approaches was performed in [50]. However, these studies have not taken into account the difference of the fatigue crack growth mechanism inside ODA and outside ODA but based only on the master curve of the growth of ODA [4]. Fatigue crack growth models for small cracks inside ODA and outside ODA have not been well developed in the practical fatigue
design procedure. It must be noted that these two models should be different. The objective of this study is to elucidate the basic fatigue mechanism in these two domains, inside ODA and outside ODA, and to develop fatigue life prediction method for components of high strength steels from low to very high cycle fatigue. On the basis of the experimental findings, the present paper proposes a fatigue life prediction model based on the-mechanism-inside-ODA and the-mechanism-outside-ODA for high strength steels which fails from subsurface nonmetallic inclusions. 2. Mechanism of fatigue failure originating at nonmetallic inclusion 2.1. Optically dark area Murakami et al. [2–5] pointed out the presence of a particular morphology called Optically Dark Area (ODA) beside the inclusion at the center of the fish-eye mark as shown in Fig. 1(a) [4]. This finding was firstly reported in [2]. It is surprising that specimens having a longer life have a larger ODA relative to original inclusion size. They suggested [27] that the formation of ODA is presumed to be influenced by hydrogen trapped by inclusions. The chemical compositions of the inclusions have been mostly Al2O3, Al2O3(CaO)x, TiN, and MnS [4,5,27]. 2.2. Fracture surface morphology The surface at the optically dark area shows a very rough morphology [51] which is quite different from that outside ODA showing a typical structure of martensite lath. When an ODA is observed by SEM with the electron beam normal to fracture surface, ODA surface is observed as granular as shown in Fig. 1(b). This granular morphology is called by other namings of ODA as FGA (Fine Granular Area) [52] or GBF (Granular bright facet) [53]. However, the exact surface morphology exhibits not as granular but as sharp zigzag surface with observing under the inclined-direction by SEM as shown in Fig. 1(c). This indicates that the mechanism-inside-ODA is different from that outside ODA. When fatigue failure originates at artificial hole in the surface, an ODA cannot be observed. In this case, a non-propagating crack can be observed at definite fatigue limit where the most crucial mechanism is the plasticity-induced crack closure [13,14] and it must be recognized that a small crack problem is essential [1]. 2.3. Hydrogen trapped by nonmetallic inclusions Fig. 2(a) shows the hydrogen trapped by inclusions with the aid of autoradiographic observations of tritium absorption in JIS SCM435 [54,55]. It should be noted that the ODA cannot be found in the fracture surface beside the inclusions in low cycle fatigue regime (see Fig. 1(a)). Nonmetallic inclusions strongly trap hydrogen [54,56,57]. The ODA is formed only around nonmetallic inclusions. It is significant that a decrease in hydrogen content of a high strength steel leads to a reduction in the size of the ODA. When failure originates at other microstructures such as bainite [1] and defects such as small artificial defects, ODA cannot be observed, because hydrogen is not trapped by bainite and artificial defects. An important evidence of hydrogen effect on breaking non-propagation behavior of cracks emanating only from nonmetallic inclusion and not from an artificial defect is shown in the well planned experiment [58] in which a fatigue crack emanating from an artificial surface defect showed non-propagation behavior without ODA though a crack emanating from subsurface inclusion smaller in size than the artificial defect showed ODA and led the specimen to final fracture.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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(a) ODA
y x
z
Inclusion 10 µm
Inclusion : Al2O3·(CaO)x Inclusion z
y
x
ODA
10µm
The zigzag morphologies of ODA observed with the high resolution SEM by tilting the specimen
(b)
(c)
Fig. 1. (a) Optical micrographs of ODA [4]; (b) granular surface morphology near inclusion inside ODA that was the site of fatigue fracture; SCM435, Nf = 1.11 108, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi r = 560 MPa, area = areaInc: ¼ 29 lm, areaODA ¼ area þ ODA = 55 lm [51]; (c) the zigzag surface morphology of ODA observed with the high resolution SEM with tilting the specimen.
pffiffiffiffiffiffiffiffiffiffi parameter model can be modified by using areaODA to calculate the modified fatigue limit, r0W , at the border of ODA as the following:
This is a new phenomenon to be paid attention, because hydrogen causes hydrogen enhanced fatigue crack growth [59] and this problem has not been solved in the field of hydrogen embrittlement (HE) research. In terms of the fatigue failure originating from nonmetallic inclusions, we need to discuss this problem from the viewpoint of a coupled problem of small fatigue crack and hydrogen trapped by nonmetallic inclusions. The coupling of these two factors naturally influences the model to be constructed.
pffiffiffiffiffiffiffiffiffiffi where HV in kgf/mm2 and areaODA in microns. This equation is derived from the followings:
2.4. Master curve of the growth of ODA
DK th ¼ 2:77 103 ðHV þ 120Þ
pffiffiffiffiffiffiffiffiffiffi If the effective size of an inclusion, areaODA , by adding the size of the dark area to the original size of the inclusion is used, a modpffiffiffiffiffiffiffiffiffiffi ified S–N diagram can be drawn as shown in Fig. 3(a) [4]. The area
pffiffiffiffiffiffiffiffiffiffi 1=2 DK ODA ¼ 0:5Dr p areaODA ; pffiffiffiffiffiffiffiffiffiffi areaODA ¼ ðInclusion þ ODAÞsize;
pffiffiffiffiffiffiffiffiffiffi
r0W ¼ 1:56ðHV þ 120Þ=ð areaODA Þ
1=6
ð1Þ
pffiffiffiffiffiffiffiffiffiffi 1=3 areaODA
Dr ¼ 2 r
ð2Þ
ð3Þ
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Fig. 2. The hydrogen trapped by inclusions: autoradiographic observations of tritium absorption in SCM435; (a) Al2O3, (b) Al, Ca duplex oxide [54,55].
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where DK th and DK ODA in MPa m, Dr in MPa and areaODA of Eq. pffiffiffiffiffiffiffiffiffiffi (3) in meters. Thus, areaODA is the function of inclusion size, pffiffiffiffiffiffiffiffiffiffi area, stress amplitude r, HV and hydrogen content as expressed by Eqs. (2) and (3). In other words, we can say that ODA increases its size until the size reaches the critical value expressed by Eq. (1) under the applied stress r being equal to r0W . Fig. 3(b) shows the correlation between the ratio of the ODA size to the inclusion, pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi areaODA = area, and the life, Nf , called the master curve of the growth of ODA [4,8] for general heat treatment QT (Quenching & Tempering) steels of the hydrogen content 0.2–0.9 (p.p.m.). In pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Fig. 3(b), areaODA = area increases with the increase of failure life, Nf . In order to verify the influence of hydrogen, fatigue tests were carried out using specimen VA1, VA2 and VQ which contain less hydrogen (0.01 p.p.m.) than specimen QT (0.7–0.9 p.p.m.) [4]. Specimens VA1 and VA2 were prepared by annealing specimens QT at 300 °C for 1 h and 2 h in a vacuum, respectively. Heating the specimen at 300 °C is necessary for removing the hydrogen trapped by inclusions. Specimens VQ were prepared by heat treatment in a vacuum followed by quenching. The marks r and j in Fig. 3(c) show the ODAs in specimen VA1 and VA2, respectively. The mark N shows the ODAs in specimen VQ. The ODAs in specimens VA1 and VA2 are smaller than those for specimen QT. The ODAs in specimen VQ are much smaller than other specimens. Thus, we can conclude that the hydrogen trapped by inclusions crucially influences the formation of an ODA, the particular fracture morphology around the inclusions at the fracture origin. In Fig. 3(c), it is evident that lower hydrogen content leads to a slower growth of ODA, especially in the very high cycle fatigue regime (e.g. N > 107) and increases fatigue life. This result indicates that the hydrogen trapped by nonmetallic inclusion plays a key role on the ODA formation [64]. 2.5. Mechanism-inside-ODA Murakami et al. proposed a model for VHCF failure of high strength steels [50]. Based on the VHCF mechanism, recently, Murakami and Yamashita [60] proposed a basic concept for fatigue crack growth model to predict fatigue failure originating at nonmetallic inclusion. The schematic representation is shown in Fig. 4(a). The model proposes that the growth of the ODA does not occur cycle-by-cycle as conventional fatigue crack growth; rather it is caused by the combined effects of hydrogen trapped by the inclusion along with cyclic stress. When fatigue life Nf of a specimen of 10 mm in size is longer than 108, the average growth rate is less than lattice spacing (0.1 Å or 0.01 nm or 101112 m/cycle). Further, we can understand the reason why the crack does not stop inside ODA. During crack growth inside ODA, the value of DK should be lower than DK th for the crack size in normal environ-
ment. However, the crack does not stop even at DK lower than DK th in air, because hydrogen enhances fatigue crack growth due to hydrogen embrittlement mechanism where the crack growth is not in cycle by cycle. This is called as the-mechanism-insideODA in this paper. The hydrogen trapped by inclusion diffuses to crack tip zone with high stress concentration as crack grows. This is a kind of environmentally assisted fatigue crack growth [4].
2.6. Dependence of threshold stress intensity of ODA small crack on crack size and the mechanism-outside-ODA It is well known that the fatigue threshold stress intensity factor range DK th for small cracks is not a material constant as measured by large specimens with long crack. DK th decreases with decreasing crack size [1,15,21,47,61–63]. Murakami and Endo [21] and Murakami [1] investigated the crack size effect on DK th using specimens containing very small artificial defects and developed a quantitative model for DK th for small cracks. The model was termed ‘‘the pffiffiffiffiffiffiffiffiffiffi area parameter model”. Inside ODA, even under the loading condition that DK < DK th , the crack grows with the assistance of hydrogen trapped by inclusion. The basic model is based on that ODA border is the critical boundary where conventional cycle-by-cycle small fatigue crack growth begins after non-cycle-by-cycle fatigue crack growth with a synergistic effect between cyclic loading and hydrogen trapped by the inclusion through the ODA as shown in Fig. 4(a). This is called as the mechanism-outside-ODA in this paper. It should be noted that the ODA cannot be found in the fracture surface beside the inclusions at low cycle fatigue regime (see Fig. 1) because the stress level is high and DK > DK th . The results obtained with Eq. (2) are shown in Fig. 4(b) [37,64,65]. Fig. 4(b) shows the relationship between the ODA size pffiffiffiffiffiffiffiffiffiffi ( areaODA ) and the stress intensity factor range at the periphery of the ODA (DK ODA ). It is noted that, in Fig. 4(b), DK ODA is normalized by the (HV + 120) to take the difference in hardness into account. The values of DK ODA certainly exhibit the crack size dependence, just as the case with the other small crack problems reported in the literature [1,21,47]. Figs. 3(a) and 4(b) support the model of very high cycle fatigue life by illustrating the link between the ODA size and DK th for small cracks, along with the relationship between ODA size and fatigue limit.
3. Fatigue life prediction model 3.1. Fatigue crack growth models from low to very high cycle fatigue regime Fatigue failure originating at nonmetallic inclusion has two stages in VHCF regime. Especially the viewpoint which separates
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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(a)
5
(b)
Fig. 3. (a) Modified S–N data, ODA grows as a crack ? rw decreases and approaches to stress amplitude r ? failure where r0W is modified fatigue limit for R = 1 as pffiffiffiffiffiffiffiffiffiffi areaODA ¼ ðInclusion þ ODAÞsize [4]; (b) the master curve of the growth of ODA of general heat treatment QT (Quenching & Tempering) steels with hydrogen content 0.2 0.9 (p.p.m.); (c) the relationship between the size of ODA and the cycles to failure: The marks r and j show the ODAs in specimen VA1 and VA2 and the mark N shows the ODAs in specimen VQ (heat treated in a vacuum followed by quenching). The lower hydrogen content leads to a slower growth of ODA, especially in the very high cycle fatigue regime (e.g. N > 107) and increases fatigue life.
the fatigue crack growth processes inside ODA and that outside ODA is crucial for developing the fatigue crack growth model. In the early stage of very high cycle fatigue, fatigue crack grows according to the mechanism-inside-ODA represented by the master curve of the growth of ODA. In the second stage, cycle-by-cycle fatigue crack growth occurs based on the mechanism-outside-ODA. The fatigue life prediction models for the both stages are needed to develop. Total life Nf is the sum of the fatigue crack growth life based on the mechanism-inside-ODA, N insideODA , and the fatigue crack growth life based on the mechanism-outside-ODA, N outsideODA , as Nf = N insideODA + N outsideODA . In low cycle fatigue regime, N outsideODA is predominant in Nf. In very high cycle fatigue regime, N insideODA is predominant.
3.2. Fatigue life prediction model based on the mechanism-inside-ODA Based on the relationship of ODA growth normalized with pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi respect to inclusion size, areaODA = area, versus the number of cycles to failure, N f , the following relationship can be constructed,
Nf ffi N insideODA ¼ A
pffiffiffiffiffiffiffiffiffiffi B areaODA pffiffiffiffiffiffiffiffiffi ffi 1 area
ð4Þ
where A and B are material constants for high strength steels of pffiffiffiffiffiffiffiffiffiffi HV > 400. When areaODA equals to the initial size of inclusion, pffiffiffiffiffiffiffiffiffiffi area, that is, ODA cannot be observed, N insideODA is zero. A = 5 107 and B = 2 are determined as the material constants as
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where Dr ¼ 2r in MPa for R = 1, a ¼ areai = p; areai in meters pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi but in Eq. (6) areai in microns. In this model, areai is defined as the square root of growing crack area at the loading cycle N = i. The essential meaning of Eq. (5) is that the fatigue limit of a specimen continuously decreases with crack growth during fatigue test as shown in Fig. 5(c). The internal fatigue failure of high strength steel originating at nonmetallic inclusion is essentially ‘‘the small crack problem” [1,21]. The fatigue limit intensely pffiffiffiffiffiffiffiffiffiffi depends on the inclusion size as expressed by the area parameter model [1,21]. This fatigue limit varies momentarily in the fatigue crack growth process. Thus ‘‘the continuously variable fatigue limit” shown in Fig. 5(c) can be defined in this paper as the function of pffiffiffiffiffiffiffiffiffiffi the growing crack size. In other word, as a crack grows, areai increases and Drw decreases continuously resulting the increase in the crack growth driving force in terms of ðDr=Drw 1Þm . After analysing previously obtained data, i.e. the fatigue life and its scatter of JIS SCM435, C ¼ 1:0 104 , m = 4 and n = 1 are determined as the unified values for Eq. (5). pffiffiffiffiffiffiffiffiffiffi In Eq. (5), Drw is a function of the ‘‘variable” crack size, areai , that is, a function of the ‘‘variable” crack size, a, and means the ‘‘continuously variable fatigue limit” which decreases with increase in crack size. In Eq. (6), Drw is ‘‘variable” fatigue limit for a penny pffiffiffiffiffiffiffiffiffiffi shape crack of the crack radius size, a, converted from areai using pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi the relationship of a ¼ areai = p. Drw decreases with respect to the growing crack size. The final crack size ac does not necessarily correspond to the radius size of fish eye boundary experimentally found in fatigue fractured surfaces. This assumption is reasonable from the experimental facts that the most part of fatigue crack growth life outside ODA is spent in very small crack size region near ODA border. The final crack size, ac , is not crucial to the calculated crack growth life and is set to the value of ac = 1 mm for simplicity, because the crack growth rate around fish eye or ac = 1 mm is very high and the remaining life can be ignored. Fig. 5(a) illustrates the total process of fatigue crack growth from nonmetallic inclusion inside ODA and outside ODA. This process is composed with the master curve of the growth of ODA shown in Fig. 5(b) and ‘‘continuously variable fatigue limit” schematically shown in Fig. 5(c).
Fig. 4. (a) A cross-section view of an inclusion and the ODA surrounding the pffiffiffiffiffiffiffiffiffiffi inclusion; (b) relationship between areaODA and DK ODA for SAE52100 without pffiffiffiffiffiffiffiffiffiffi 1=2 pffiffiffiffiffiffiffiffiffiffi hydrogen charging: DK ODA ¼ 0:5Drðp areaODA Þ , areaODA ¼ ðInclusion þ ODAÞsize, Dr ¼ 2r [37,64,65].
shown within Fig. 3(b). In VHCF regime, fatigue crack growth life inside ODA can be estimated based on Eq. (4) under constant amplitude loading. 3.3. Fatigue crack growth law based on the mechanism-outside-ODA At the boundary between the mechanism-inside-ODA and the mechanism-outside-ODA, cycle-by-cycle fatigue crack growth without being assisted by hydrogen begins. With taking into account that DK th of small crack is not a constant which is obtained for long crack, the DK th depends on the crack size [1,21,47] which is well pffiffiffiffiffiffiffiffiffiffi expressed by the area parameter model. The fatigue crack growth law for small crack from ODA border to fish-eye border is proposed pffiffiffiffiffiffiffiffiffiffi within the framework of the area parameter model for R = 1 as follows:
m da Dr 1 an ¼ CX m Y n ¼ C dN Dr w
ð5Þ
pffiffiffiffiffiffiffiffiffiffi 1=6 Drw ¼ 2 1:56ðHV þ 120Þ=ð areai Þ
ð6Þ
4. Results of fatigue life prediction and discussions Fig. 6(a) and (b) shows the predicted N outsideODA and N f , respectively, with previously obtained tension–compression fatigue test data at R = 1. The calculated fatigue crack growth life, Np2, outside ODA performed using the Paris law [40], da=dN ¼ C DK m with material constants [5] of C = 2.92 1012 and m = 2 in MPa, m have also been exhibited in Fig. 6(a). Table 1 shows the summary of the fatigue test results and the predicted fatigue crack growth lives to understand the each life ratio of N insideODA =N f and N outsideODA =N f . The predicted results by the proposed model of Eq. (5) have the better agreements with the life and scatter of tests results than those by the Paris law. This is because the present model considers ‘‘the variable fatigue limit” during crack growth process. In the practical design, although the crack growth process from subsurface inclusions cannot be chased explicitly, the scatter of life can be evaluated in combination with the proposed model and the data of statistics of extremes for nonmetallic inclusion distribution [1,66,67]. When the cyclic loading sequence includes the stress amplitudes larger than the fatigue limit, the crack growth equation from a inclusion is needed to be expressed in relation to the ‘‘variable” fatigue limit, because the DK th for small cracks is the function of crack size [1,21,47] and accordingly continuously varies with crack length during the process of crack growth until final fracture. This nature of small crack growth and threshold is crucially
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Fig. 5. Framework of the proposed fatigue life prediction models emanating from a nonmetallic inclusion inside ODA and outside ODA: (a) crack growth behavior originating at nonmetallic inclusion, (b) the master curve of the growth of ODA and (c) continuously variable fatigue limit.
Fig. 6. Predicted fatigue failure lives compared with test results: (a) N outsideODA ; (b) N f , (JIS SCM435).
important for the case of variable amplitude loading, because the contribution of stress and number of cycles to fatigue damage always varies with crack growth, namely crack length. 5. Conclusion The following conclusions can be drawn from this study: (1) Based on the master curve of the growth of ODA where fatigue crack growth is caused by cyclic loading assisted by hydrogen trapped by nonmetallic inclusion, fatigue crack growth life inside ODA can be approximately estimated in the very high cycle fatigue regime although the hypothetical cycle-by-cycle fatigue crack growth rate is smaller than the lattice space.
(2) The fatigue crack growth model outside ODA is proposed with introducing the concept of ‘‘continuously variable fatigue limit” for small cracks where the ‘‘variable” fatigue limit can pffiffiffiffiffiffiffiffiffiffi be estimated within the framework of the area parameter model because the threshold stress intensity factor range DK th for small cracks is the function of crack size and accordingly continuously varies with crack length during the process of crack propagation until final fracture. The sequence effect of stress amplitude can be evaluated in this scheme when the cyclic loading sequence includes the stress amplitudes larger than the fatigue limit. This nature of small crack growth and threshold is crucially important, because the contribution of stress and number of cycles to fatigue damage always varies with crack growth, namely crack length.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Table 1 Comparison between fatigue test results and the predicted results by the proposed small crack growth models. Nf (fatigue test) cycles
Nf (predict) cycles
Nf (predict)/Nf (fatigue test)
NinsideODA/Nf (predict)
NoutsideODA/Nf (predict)
1.44E+05 3.82E+05 5.40E+05 1.11E+06 2.74E+06 4.88E+06 6.54E+06 1.31E+07 1.55E+07 3.44E+07 4.39E+07 5.17E+07 6.15E+07 7.73E+07 7.74E+07 1.00E+08 1.11E+08 2.17E+08
9.80E+04 1.04E+05 1.29E+06 5.50E+05 2.01E+06 2.18E+06 2.01E+06 1.49E+07 1.20E+07 1.40E+07 1.57E+07 2.56E+07 2.41E+07 3.32E+07 3.49E+07 5.56E+07 3.04E+07 2.06E+08
0.68 0.27 2.40 0.50 0.73 0.45 0.31 1.14 0.77 0.41 0.36 0.50 0.39 0.43 0.45 0.56 0.27 0.95
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.11 0.45 0.42 0.58 0.60 0.75 0.54 0.93
1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.94 1.00 1.00 0.89 0.55 0.58 0.42 0.40 0.25 0.46 0.07
(3) The proposed fatigue life prediction models predict well the total life and scatter of fatigue failures originating at nonmetallic inclusions from low to very high cycle fatigue regime for high strength steels.
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[55] Otsuka T, Tanabe T. Hydrogen diffusion and trapping process around MnS precipitates in aFe examined by tritium autoradiography. J Alloy Compd 2007;446–447:655–9. [56] Murakami Y, Nagata J. Effect of hydrogen on high cycle fatigue failure of high strength steel, SCM435. J Soc Mater Sci Jpn 2005;54:420–7 [in Japanese]. [57] Murakami Y, Matsunaga H. The effect of hydrogen on fatigue properties of steels used for fuel cell system. Int J Fatigue 2006;28:1509–20. [58] Murakami Y, Nagata J. Elimination of conventional fatigue limit due to fatigue crack originated at nonmetallic inclusion, and non-propagating of fatigue crack originated at artificial small hole. Trans Jpn Soc Mech Eng, Ser A 2006;72:1123–30. [59] Birnbaum HK, Sofronis P. Hydrogen-enhanced localized plasticity: a mechanism for hydrogen-related fracture. Mater Sci Eng A 1994;176:191–202. [60] Murakami Y, Yamashita Y. Prediction of life and scatter of fatigue failure originated at nonmetallic inclusions. Procedia Eng 2014;74:6–11. [61] Frost NE, Pook LP, Denton K. A fracture mechanics analysis of fatigue crack growth data for various materials. Eng Fract Mech 1971;3:109–26. [62] Kobayashi H, Nakazawa H. A stress criterion for fatigue crack propagation in metals. Proc 1st Int Conf Mech Behav Mater, vol. II. p. 199–208. [63] El Haddad MH, Smith KN, Topper TH. Fatigue crack propagation of short crack. J Eng Mater Technol, Trans ASME 1979;101:42–6. [64] Murakami Y, Nagata J, Matsunaga H. Factors affecting ultralong life fatigue and design method for components. In: Proceedings of the 9th International Congress on Fatigue. [65] Sakai T, Sato Y, Oguma N. Stepwise S–N curve and fish-eye failure in gigacycle fatigue. Trans Jpn Soc Mech Eng A 2001;67:1980–7. [66] Beretta S, Murakami Y. Statistical analysis of defects for fatigue strength prediction and quality control of materials. Fatigue Fract Eng Mater Struct 1998;21:1049–65. [67] Beretta S, Murakami Y. Largest-extreme-value distribution analysis of multiple inclusion types in determining steel cleanliness. Metall Mater Trans B 2001;32:517–23.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel Yoichi Yamashita a,⇑, Yukitaka Murakami b a b
Research Laboratory, IHI Corporation, 1, Shin-Nakahara-Cho, Isogo-Ku, Yokohama 235-8501, Japan Kyushu University, 774 Moto-oka, Nishi-ku, Fukuoka 819-0395, Japan
a r t i c l e
i n f o
Article history: Received 12 January 2016 Received in revised form 4 April 2016 Accepted 12 April 2016 Available online xxxx Keywords: Fatigue crack growth Hydrogen effect Nonmetallic inclusion pffiffiffiffiffiffiffiffiffiffi area parameter model ODA
a b s t r a c t Fatigue failure of high strength steels mostly originates at nonmetallic inclusions. An optically dark area (ODA) beside the inclusion can be observed in specimens fractured at very high cycle fatigue (VHCF) regime. The present paper proposes fatigue life prediction models from low to VHCF regime. The fatigue life prediction model inside ODA has been constructed in the VHCF regime based on the master curve of the growth of ODA where fatigue failure is caused by cyclic loading assisted by hydrogen trapped by inclusion. The fatigue crack growth law is proposed for a small crack outside ODA within the framework of the pffiffiffiffiffiffiffiffiffiffi area parameter model where the concept of ‘‘continuously variable fatigue limit” for small crack is introduced. The life and scatter of fatigue life originating at inclusions can be well evaluated by the proposed models. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fatigue failure of high strength steels mostly originates at nonmetallic inclusions [1–5]. This paper reveals the crucial factors for solving this problem and proposes the model for predicting the fatigue life of components. First, fatigue failure mechanism of low or medium strength steels will be reviewed where fatigue cracks usually initiate at free surface of materials. And then, internal fatigue failure mechanism of high strength steels will be reviewed. Even in very high cycle fatigue (VHCF) regime beyond 107 cycles of low strength steel, crack would not originate at inclusion but at persistent slip band in the surface of specimen or component. It is known [6–10] that the persistent slip bands are irreversible during fatigue cycles and some of them can be the origin of crack initiation. But in this case, the effects of moisture and oxygen in the air on the fatigue mechanism cannot be ignored during long test period. This irreversibility should be different from the mechanism of very high cycle fatigue originating at subsurface inclusions. At and even below the fatigue limit of low strength steels many microcracks can be observed and so the fatigue limit should be defined as the threshold stress for the growth of cracks which are nucleated under the stress. These cracks initiate at very early stage of fatigue test and stop propagation much before 107 cycles ⇑ Corresponding author.
if environmental factors such as humidity, oxygen, temperature, etc do not affect by continuing test for long period [1]. The threshold condition (or fatigue limit) of a material is determined by the non-propagation of these microcracks that initiated in the original microstructure [1,11,12] on the surface of unnotched specimens where the most crucial and rational mechanism to explain the non-propagation phenomenon is the plasticity-induced crack closure found by Elber [13,14]. The oxide induced crack closure [15] and the surface roughness induced crack closure [16] have additional effects on crack closure behavior depending on the case. When cracks initiate along slip bands or at grain boundaries, good correlations have been obtained among ultimate tensile strength, rU , hardness, HV and fatigue limit [17–20]. The fatigue limit rFL is proportional to the Vickers hardness regardless of various microstructures with different grain sizes, e.g.: in steels rFL = 1.6HV for HV = <400 where rFL is in MPa and HV in kgf/mm2. On the other hand, in high strength steels of HV > 400, material defects such as nonmetallic inclusions are often observed at fracture origin. The fatigue limit is not simply proportional to the Vickers hardness for the materials with HV > 400. The fatigue limit is strongly influenced by defect size. The quantitative fatigue analysis of the effect of defect size using artificial small defects was systematically studied by Murakami and Endo [21]. Here, the fatigue threshold is also determined by the threshold condition of cracks emanating from such defects [22]. The early works of Naito et al. [23] and Asami and Sugiyama [24] have shown that fatigue failure does occur at lives longer than
E-mail address: [email protected] (Y. Yamashita). http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016 0142-1123/Ó 2016 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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N = 107 at stress levels lower than the conventional fatigue limit. For high strength steels especially in rotary bending fatigue tests, fatigue failure can be initiated both at the surface and the interior presenting two knees in S–N diagram [25,26]. This two-stage material behavior in S–N diagram is called as the stepwise or duplex S–N property. Since the duplex S–N diagram is not observed in tension–compression fatigue tests [27], it can be understood that the duplex S–N diagram is resulted from stress gradient in bending. The loading frequency has been always a concern in the test of VHCF [28–35]. Stanzl-Tschegg [29] and Bathias [30,31] showed the VHCF data obtained by ultrasonic fatigue testing machines with very high frequency of 20 kHz. Some researchers found that the effect of loading frequency is very small in high strength steels by Furuya et al. [34], titanium alloy by Ritchie et al. [32] and aluminium alloy by Mayer et al. [33]. In contrast, some papers show the load frequency effects on fatigue strength for mild steels [28,35]. Qian et al. [36] and Sun et al. [37] reviewed the VHCF behavior and its life prediction models. Many of them are based on for example, Tanaka and Mura model [38] for fatigue crack initiation life [39] and are based on the Paris law [40] for fatigue crack growth life [25,41–46]. Recently, Sun et al. [37] studied the cumulative damage model for fatigue life estimation of high-strength steels in HCF and VHCF regime based on the ultrasonic fatigue test data. But this model has not taken into account the small crack effects [47] that the crack growth rate and the threshold stress intensity factor range definitely depends on its crack size. Fatigue strength prediction models for small cracks [3–5,48–50] pffiffiffiffiffiffiffiffiffiffi have been studied by many researchers using the area parameter model [21]. Murakami et al. proposed the fatigue life prediction pffiffiffiffiffiffiffiffiffiffi method using the area parameter model in combination with the statistics of extreme values of internal defects in VHCF regime [22]. Most of the researches have not taken into account the hydrogen effect coupled with cyclic stresses on fatigue failure originating at non-metallic inclusion. The fact that hydrogen is densely trapped by inclusions cannot be ignored in the discussion of fatigue mechanism. Also it must be noted that the fatigue failure from small defects such as nonmetallic inclusions is essentially the small crack problem and the models must be based on the mechanics of small crack. Especially in the VHCF regime of high strength steels, interior crack initiation emanating from internal inclusion often occurs with a distinct fracture surface morphologies beside the inclusions called the Optically Dark Area (ODA) (see Fig. 1, [1]). Murakami et al. [3–5] detected densely trapped hydrogen around inclusions. They pointed out that the fatigue failure accompanying a peculiar morphology, ODA, at the fracture origin of the center of fish-eye mark is caused by the assistance of the hydrogen trapped by inclusion with cyclic stresses. This assistance of hydrogen results in the elimination of the conventional fatigue limit. Considering the very different morphologies of the fracture surfaces inside ODA and outside ODA, it is presumed that the fatigue mechanisms inside ODA and outside ODA are substantially different. It has been found by Murakami et al. [4] that there is a good correlation between the ODA size normalized by the inclusion size and the number of fatigue cycles to failure originating at nonmetallic inclusions. This relationship was named the master curve of the growth of ODA. One of the design approaches was performed in [50]. However, these studies have not taken into account the difference of the fatigue crack growth mechanism inside ODA and outside ODA but based only on the master curve of the growth of ODA [4]. Fatigue crack growth models for small cracks inside ODA and outside ODA have not been well developed in the practical fatigue
design procedure. It must be noted that these two models should be different. The objective of this study is to elucidate the basic fatigue mechanism in these two domains, inside ODA and outside ODA, and to develop fatigue life prediction method for components of high strength steels from low to very high cycle fatigue. On the basis of the experimental findings, the present paper proposes a fatigue life prediction model based on the-mechanism-inside-ODA and the-mechanism-outside-ODA for high strength steels which fails from subsurface nonmetallic inclusions. 2. Mechanism of fatigue failure originating at nonmetallic inclusion 2.1. Optically dark area Murakami et al. [2–5] pointed out the presence of a particular morphology called Optically Dark Area (ODA) beside the inclusion at the center of the fish-eye mark as shown in Fig. 1(a) [4]. This finding was firstly reported in [2]. It is surprising that specimens having a longer life have a larger ODA relative to original inclusion size. They suggested [27] that the formation of ODA is presumed to be influenced by hydrogen trapped by inclusions. The chemical compositions of the inclusions have been mostly Al2O3, Al2O3(CaO)x, TiN, and MnS [4,5,27]. 2.2. Fracture surface morphology The surface at the optically dark area shows a very rough morphology [51] which is quite different from that outside ODA showing a typical structure of martensite lath. When an ODA is observed by SEM with the electron beam normal to fracture surface, ODA surface is observed as granular as shown in Fig. 1(b). This granular morphology is called by other namings of ODA as FGA (Fine Granular Area) [52] or GBF (Granular bright facet) [53]. However, the exact surface morphology exhibits not as granular but as sharp zigzag surface with observing under the inclined-direction by SEM as shown in Fig. 1(c). This indicates that the mechanism-inside-ODA is different from that outside ODA. When fatigue failure originates at artificial hole in the surface, an ODA cannot be observed. In this case, a non-propagating crack can be observed at definite fatigue limit where the most crucial mechanism is the plasticity-induced crack closure [13,14] and it must be recognized that a small crack problem is essential [1]. 2.3. Hydrogen trapped by nonmetallic inclusions Fig. 2(a) shows the hydrogen trapped by inclusions with the aid of autoradiographic observations of tritium absorption in JIS SCM435 [54,55]. It should be noted that the ODA cannot be found in the fracture surface beside the inclusions in low cycle fatigue regime (see Fig. 1(a)). Nonmetallic inclusions strongly trap hydrogen [54,56,57]. The ODA is formed only around nonmetallic inclusions. It is significant that a decrease in hydrogen content of a high strength steel leads to a reduction in the size of the ODA. When failure originates at other microstructures such as bainite [1] and defects such as small artificial defects, ODA cannot be observed, because hydrogen is not trapped by bainite and artificial defects. An important evidence of hydrogen effect on breaking non-propagation behavior of cracks emanating only from nonmetallic inclusion and not from an artificial defect is shown in the well planned experiment [58] in which a fatigue crack emanating from an artificial surface defect showed non-propagation behavior without ODA though a crack emanating from subsurface inclusion smaller in size than the artificial defect showed ODA and led the specimen to final fracture.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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(a) ODA
y x
z
Inclusion 10 µm
Inclusion : Al2O3·(CaO)x Inclusion z
y
x
ODA
10µm
The zigzag morphologies of ODA observed with the high resolution SEM by tilting the specimen
(b)
(c)
Fig. 1. (a) Optical micrographs of ODA [4]; (b) granular surface morphology near inclusion inside ODA that was the site of fatigue fracture; SCM435, Nf = 1.11 108, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi r = 560 MPa, area = areaInc: ¼ 29 lm, areaODA ¼ area þ ODA = 55 lm [51]; (c) the zigzag surface morphology of ODA observed with the high resolution SEM with tilting the specimen.
pffiffiffiffiffiffiffiffiffiffi parameter model can be modified by using areaODA to calculate the modified fatigue limit, r0W , at the border of ODA as the following:
This is a new phenomenon to be paid attention, because hydrogen causes hydrogen enhanced fatigue crack growth [59] and this problem has not been solved in the field of hydrogen embrittlement (HE) research. In terms of the fatigue failure originating from nonmetallic inclusions, we need to discuss this problem from the viewpoint of a coupled problem of small fatigue crack and hydrogen trapped by nonmetallic inclusions. The coupling of these two factors naturally influences the model to be constructed.
pffiffiffiffiffiffiffiffiffiffi where HV in kgf/mm2 and areaODA in microns. This equation is derived from the followings:
2.4. Master curve of the growth of ODA
DK th ¼ 2:77 103 ðHV þ 120Þ
pffiffiffiffiffiffiffiffiffiffi If the effective size of an inclusion, areaODA , by adding the size of the dark area to the original size of the inclusion is used, a modpffiffiffiffiffiffiffiffiffiffi ified S–N diagram can be drawn as shown in Fig. 3(a) [4]. The area
pffiffiffiffiffiffiffiffiffiffi 1=2 DK ODA ¼ 0:5Dr p areaODA ; pffiffiffiffiffiffiffiffiffiffi areaODA ¼ ðInclusion þ ODAÞsize;
pffiffiffiffiffiffiffiffiffiffi
r0W ¼ 1:56ðHV þ 120Þ=ð areaODA Þ
1=6
ð1Þ
pffiffiffiffiffiffiffiffiffiffi 1=3 areaODA
Dr ¼ 2 r
ð2Þ
ð3Þ
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Fig. 2. The hydrogen trapped by inclusions: autoradiographic observations of tritium absorption in SCM435; (a) Al2O3, (b) Al, Ca duplex oxide [54,55].
pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where DK th and DK ODA in MPa m, Dr in MPa and areaODA of Eq. pffiffiffiffiffiffiffiffiffiffi (3) in meters. Thus, areaODA is the function of inclusion size, pffiffiffiffiffiffiffiffiffiffi area, stress amplitude r, HV and hydrogen content as expressed by Eqs. (2) and (3). In other words, we can say that ODA increases its size until the size reaches the critical value expressed by Eq. (1) under the applied stress r being equal to r0W . Fig. 3(b) shows the correlation between the ratio of the ODA size to the inclusion, pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi areaODA = area, and the life, Nf , called the master curve of the growth of ODA [4,8] for general heat treatment QT (Quenching & Tempering) steels of the hydrogen content 0.2–0.9 (p.p.m.). In pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi Fig. 3(b), areaODA = area increases with the increase of failure life, Nf . In order to verify the influence of hydrogen, fatigue tests were carried out using specimen VA1, VA2 and VQ which contain less hydrogen (0.01 p.p.m.) than specimen QT (0.7–0.9 p.p.m.) [4]. Specimens VA1 and VA2 were prepared by annealing specimens QT at 300 °C for 1 h and 2 h in a vacuum, respectively. Heating the specimen at 300 °C is necessary for removing the hydrogen trapped by inclusions. Specimens VQ were prepared by heat treatment in a vacuum followed by quenching. The marks r and j in Fig. 3(c) show the ODAs in specimen VA1 and VA2, respectively. The mark N shows the ODAs in specimen VQ. The ODAs in specimens VA1 and VA2 are smaller than those for specimen QT. The ODAs in specimen VQ are much smaller than other specimens. Thus, we can conclude that the hydrogen trapped by inclusions crucially influences the formation of an ODA, the particular fracture morphology around the inclusions at the fracture origin. In Fig. 3(c), it is evident that lower hydrogen content leads to a slower growth of ODA, especially in the very high cycle fatigue regime (e.g. N > 107) and increases fatigue life. This result indicates that the hydrogen trapped by nonmetallic inclusion plays a key role on the ODA formation [64]. 2.5. Mechanism-inside-ODA Murakami et al. proposed a model for VHCF failure of high strength steels [50]. Based on the VHCF mechanism, recently, Murakami and Yamashita [60] proposed a basic concept for fatigue crack growth model to predict fatigue failure originating at nonmetallic inclusion. The schematic representation is shown in Fig. 4(a). The model proposes that the growth of the ODA does not occur cycle-by-cycle as conventional fatigue crack growth; rather it is caused by the combined effects of hydrogen trapped by the inclusion along with cyclic stress. When fatigue life Nf of a specimen of 10 mm in size is longer than 108, the average growth rate is less than lattice spacing (0.1 Å or 0.01 nm or 101112 m/cycle). Further, we can understand the reason why the crack does not stop inside ODA. During crack growth inside ODA, the value of DK should be lower than DK th for the crack size in normal environ-
ment. However, the crack does not stop even at DK lower than DK th in air, because hydrogen enhances fatigue crack growth due to hydrogen embrittlement mechanism where the crack growth is not in cycle by cycle. This is called as the-mechanism-insideODA in this paper. The hydrogen trapped by inclusion diffuses to crack tip zone with high stress concentration as crack grows. This is a kind of environmentally assisted fatigue crack growth [4].
2.6. Dependence of threshold stress intensity of ODA small crack on crack size and the mechanism-outside-ODA It is well known that the fatigue threshold stress intensity factor range DK th for small cracks is not a material constant as measured by large specimens with long crack. DK th decreases with decreasing crack size [1,15,21,47,61–63]. Murakami and Endo [21] and Murakami [1] investigated the crack size effect on DK th using specimens containing very small artificial defects and developed a quantitative model for DK th for small cracks. The model was termed ‘‘the pffiffiffiffiffiffiffiffiffiffi area parameter model”. Inside ODA, even under the loading condition that DK < DK th , the crack grows with the assistance of hydrogen trapped by inclusion. The basic model is based on that ODA border is the critical boundary where conventional cycle-by-cycle small fatigue crack growth begins after non-cycle-by-cycle fatigue crack growth with a synergistic effect between cyclic loading and hydrogen trapped by the inclusion through the ODA as shown in Fig. 4(a). This is called as the mechanism-outside-ODA in this paper. It should be noted that the ODA cannot be found in the fracture surface beside the inclusions at low cycle fatigue regime (see Fig. 1) because the stress level is high and DK > DK th . The results obtained with Eq. (2) are shown in Fig. 4(b) [37,64,65]. Fig. 4(b) shows the relationship between the ODA size pffiffiffiffiffiffiffiffiffiffi ( areaODA ) and the stress intensity factor range at the periphery of the ODA (DK ODA ). It is noted that, in Fig. 4(b), DK ODA is normalized by the (HV + 120) to take the difference in hardness into account. The values of DK ODA certainly exhibit the crack size dependence, just as the case with the other small crack problems reported in the literature [1,21,47]. Figs. 3(a) and 4(b) support the model of very high cycle fatigue life by illustrating the link between the ODA size and DK th for small cracks, along with the relationship between ODA size and fatigue limit.
3. Fatigue life prediction model 3.1. Fatigue crack growth models from low to very high cycle fatigue regime Fatigue failure originating at nonmetallic inclusion has two stages in VHCF regime. Especially the viewpoint which separates
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
Y. Yamashita, Y. Murakami / International Journal of Fatigue xxx (2016) xxx–xxx
(a)
5
(b)
Fig. 3. (a) Modified S–N data, ODA grows as a crack ? rw decreases and approaches to stress amplitude r ? failure where r0W is modified fatigue limit for R = 1 as pffiffiffiffiffiffiffiffiffiffi areaODA ¼ ðInclusion þ ODAÞsize [4]; (b) the master curve of the growth of ODA of general heat treatment QT (Quenching & Tempering) steels with hydrogen content 0.2 0.9 (p.p.m.); (c) the relationship between the size of ODA and the cycles to failure: The marks r and j show the ODAs in specimen VA1 and VA2 and the mark N shows the ODAs in specimen VQ (heat treated in a vacuum followed by quenching). The lower hydrogen content leads to a slower growth of ODA, especially in the very high cycle fatigue regime (e.g. N > 107) and increases fatigue life.
the fatigue crack growth processes inside ODA and that outside ODA is crucial for developing the fatigue crack growth model. In the early stage of very high cycle fatigue, fatigue crack grows according to the mechanism-inside-ODA represented by the master curve of the growth of ODA. In the second stage, cycle-by-cycle fatigue crack growth occurs based on the mechanism-outside-ODA. The fatigue life prediction models for the both stages are needed to develop. Total life Nf is the sum of the fatigue crack growth life based on the mechanism-inside-ODA, N insideODA , and the fatigue crack growth life based on the mechanism-outside-ODA, N outsideODA , as Nf = N insideODA + N outsideODA . In low cycle fatigue regime, N outsideODA is predominant in Nf. In very high cycle fatigue regime, N insideODA is predominant.
3.2. Fatigue life prediction model based on the mechanism-inside-ODA Based on the relationship of ODA growth normalized with pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi respect to inclusion size, areaODA = area, versus the number of cycles to failure, N f , the following relationship can be constructed,
Nf ffi N insideODA ¼ A
pffiffiffiffiffiffiffiffiffiffi B areaODA pffiffiffiffiffiffiffiffiffi ffi 1 area
ð4Þ
where A and B are material constants for high strength steels of pffiffiffiffiffiffiffiffiffiffi HV > 400. When areaODA equals to the initial size of inclusion, pffiffiffiffiffiffiffiffiffiffi area, that is, ODA cannot be observed, N insideODA is zero. A = 5 107 and B = 2 are determined as the material constants as
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi pffiffiffiffiffiffiffiffiffiffi where Dr ¼ 2r in MPa for R = 1, a ¼ areai = p; areai in meters pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi but in Eq. (6) areai in microns. In this model, areai is defined as the square root of growing crack area at the loading cycle N = i. The essential meaning of Eq. (5) is that the fatigue limit of a specimen continuously decreases with crack growth during fatigue test as shown in Fig. 5(c). The internal fatigue failure of high strength steel originating at nonmetallic inclusion is essentially ‘‘the small crack problem” [1,21]. The fatigue limit intensely pffiffiffiffiffiffiffiffiffiffi depends on the inclusion size as expressed by the area parameter model [1,21]. This fatigue limit varies momentarily in the fatigue crack growth process. Thus ‘‘the continuously variable fatigue limit” shown in Fig. 5(c) can be defined in this paper as the function of pffiffiffiffiffiffiffiffiffiffi the growing crack size. In other word, as a crack grows, areai increases and Drw decreases continuously resulting the increase in the crack growth driving force in terms of ðDr=Drw 1Þm . After analysing previously obtained data, i.e. the fatigue life and its scatter of JIS SCM435, C ¼ 1:0 104 , m = 4 and n = 1 are determined as the unified values for Eq. (5). pffiffiffiffiffiffiffiffiffiffi In Eq. (5), Drw is a function of the ‘‘variable” crack size, areai , that is, a function of the ‘‘variable” crack size, a, and means the ‘‘continuously variable fatigue limit” which decreases with increase in crack size. In Eq. (6), Drw is ‘‘variable” fatigue limit for a penny pffiffiffiffiffiffiffiffiffiffi shape crack of the crack radius size, a, converted from areai using pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi the relationship of a ¼ areai = p. Drw decreases with respect to the growing crack size. The final crack size ac does not necessarily correspond to the radius size of fish eye boundary experimentally found in fatigue fractured surfaces. This assumption is reasonable from the experimental facts that the most part of fatigue crack growth life outside ODA is spent in very small crack size region near ODA border. The final crack size, ac , is not crucial to the calculated crack growth life and is set to the value of ac = 1 mm for simplicity, because the crack growth rate around fish eye or ac = 1 mm is very high and the remaining life can be ignored. Fig. 5(a) illustrates the total process of fatigue crack growth from nonmetallic inclusion inside ODA and outside ODA. This process is composed with the master curve of the growth of ODA shown in Fig. 5(b) and ‘‘continuously variable fatigue limit” schematically shown in Fig. 5(c).
Fig. 4. (a) A cross-section view of an inclusion and the ODA surrounding the pffiffiffiffiffiffiffiffiffiffi inclusion; (b) relationship between areaODA and DK ODA for SAE52100 without pffiffiffiffiffiffiffiffiffiffi 1=2 pffiffiffiffiffiffiffiffiffiffi hydrogen charging: DK ODA ¼ 0:5Drðp areaODA Þ , areaODA ¼ ðInclusion þ ODAÞsize, Dr ¼ 2r [37,64,65].
shown within Fig. 3(b). In VHCF regime, fatigue crack growth life inside ODA can be estimated based on Eq. (4) under constant amplitude loading. 3.3. Fatigue crack growth law based on the mechanism-outside-ODA At the boundary between the mechanism-inside-ODA and the mechanism-outside-ODA, cycle-by-cycle fatigue crack growth without being assisted by hydrogen begins. With taking into account that DK th of small crack is not a constant which is obtained for long crack, the DK th depends on the crack size [1,21,47] which is well pffiffiffiffiffiffiffiffiffiffi expressed by the area parameter model. The fatigue crack growth law for small crack from ODA border to fish-eye border is proposed pffiffiffiffiffiffiffiffiffiffi within the framework of the area parameter model for R = 1 as follows:
m da Dr 1 an ¼ CX m Y n ¼ C dN Dr w
ð5Þ
pffiffiffiffiffiffiffiffiffiffi 1=6 Drw ¼ 2 1:56ðHV þ 120Þ=ð areai Þ
ð6Þ
4. Results of fatigue life prediction and discussions Fig. 6(a) and (b) shows the predicted N outsideODA and N f , respectively, with previously obtained tension–compression fatigue test data at R = 1. The calculated fatigue crack growth life, Np2, outside ODA performed using the Paris law [40], da=dN ¼ C DK m with material constants [5] of C = 2.92 1012 and m = 2 in MPa, m have also been exhibited in Fig. 6(a). Table 1 shows the summary of the fatigue test results and the predicted fatigue crack growth lives to understand the each life ratio of N insideODA =N f and N outsideODA =N f . The predicted results by the proposed model of Eq. (5) have the better agreements with the life and scatter of tests results than those by the Paris law. This is because the present model considers ‘‘the variable fatigue limit” during crack growth process. In the practical design, although the crack growth process from subsurface inclusions cannot be chased explicitly, the scatter of life can be evaluated in combination with the proposed model and the data of statistics of extremes for nonmetallic inclusion distribution [1,66,67]. When the cyclic loading sequence includes the stress amplitudes larger than the fatigue limit, the crack growth equation from a inclusion is needed to be expressed in relation to the ‘‘variable” fatigue limit, because the DK th for small cracks is the function of crack size [1,21,47] and accordingly continuously varies with crack length during the process of crack growth until final fracture. This nature of small crack growth and threshold is crucially
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Fig. 5. Framework of the proposed fatigue life prediction models emanating from a nonmetallic inclusion inside ODA and outside ODA: (a) crack growth behavior originating at nonmetallic inclusion, (b) the master curve of the growth of ODA and (c) continuously variable fatigue limit.
Fig. 6. Predicted fatigue failure lives compared with test results: (a) N outsideODA ; (b) N f , (JIS SCM435).
important for the case of variable amplitude loading, because the contribution of stress and number of cycles to fatigue damage always varies with crack growth, namely crack length. 5. Conclusion The following conclusions can be drawn from this study: (1) Based on the master curve of the growth of ODA where fatigue crack growth is caused by cyclic loading assisted by hydrogen trapped by nonmetallic inclusion, fatigue crack growth life inside ODA can be approximately estimated in the very high cycle fatigue regime although the hypothetical cycle-by-cycle fatigue crack growth rate is smaller than the lattice space.
(2) The fatigue crack growth model outside ODA is proposed with introducing the concept of ‘‘continuously variable fatigue limit” for small cracks where the ‘‘variable” fatigue limit can pffiffiffiffiffiffiffiffiffiffi be estimated within the framework of the area parameter model because the threshold stress intensity factor range DK th for small cracks is the function of crack size and accordingly continuously varies with crack length during the process of crack propagation until final fracture. The sequence effect of stress amplitude can be evaluated in this scheme when the cyclic loading sequence includes the stress amplitudes larger than the fatigue limit. This nature of small crack growth and threshold is crucially important, because the contribution of stress and number of cycles to fatigue damage always varies with crack growth, namely crack length.
Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016
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Table 1 Comparison between fatigue test results and the predicted results by the proposed small crack growth models. Nf (fatigue test) cycles
Nf (predict) cycles
Nf (predict)/Nf (fatigue test)
NinsideODA/Nf (predict)
NoutsideODA/Nf (predict)
1.44E+05 3.82E+05 5.40E+05 1.11E+06 2.74E+06 4.88E+06 6.54E+06 1.31E+07 1.55E+07 3.44E+07 4.39E+07 5.17E+07 6.15E+07 7.73E+07 7.74E+07 1.00E+08 1.11E+08 2.17E+08
9.80E+04 1.04E+05 1.29E+06 5.50E+05 2.01E+06 2.18E+06 2.01E+06 1.49E+07 1.20E+07 1.40E+07 1.57E+07 2.56E+07 2.41E+07 3.32E+07 3.49E+07 5.56E+07 3.04E+07 2.06E+08
0.68 0.27 2.40 0.50 0.73 0.45 0.31 1.14 0.77 0.41 0.36 0.50 0.39 0.43 0.45 0.56 0.27 0.95
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.11 0.45 0.42 0.58 0.60 0.75 0.54 0.93
1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.94 1.00 1.00 0.89 0.55 0.58 0.42 0.40 0.25 0.46 0.07
(3) The proposed fatigue life prediction models predict well the total life and scatter of fatigue failures originating at nonmetallic inclusions from low to very high cycle fatigue regime for high strength steels.
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Please cite this article in press as: Yamashita Y, Murakami Y. Small crack growth model from low to very high cycle fatigue regime for internal fatigue failure of high strength steel. Int J Fatigue (2016), http://dx.doi.org/10.1016/j.ijfatigue.2016.04.016