Figure 8- Relationship between the size of
ODA and cycles to failure Nf
Figure 9- Secondary ion image of hydrogen trapped
hv the inclusion at fatigue fract1~re ori~rin ~[].12).
Figure 1 O- Secondary ion image of hydrogen trapped by
hydrogen is trapped in the vicinity of nonmetallic inclusions both in specimen QT and m specimen VQ of ASTM A295 52100.
As shown in Figure 8, in the case of ASTM A295 52100,the ratio of ODA to inclusion size at the identical Nf is smaller in specimens VQ than in specimens QT. However, the difference is not as dramatic as SCM435. The ODA size difference previously mentioned can be explained by considering the difference in hydrogen content for the materials. Table 3 shows the hydrogen content for all material/heat treatment combinations. During heat treatment, hydrogen is primarily trapped by inclusions that have a large trapping force. Hydrogen is saturated in the vicinity of inclusions, the remainder of the hydrogen is trapped by dislocations or at grain boundaries. Hydrogen content for SCM435 was 0.7-0.9 ppm in specimen QT and only 0.01 ppm in specimen VQ. In the later case the inclusions are not saturated with hydrogen, so significant difference must exist in the content of hydrogen trapped primarily by inclusions. For the ASTM A295 52100 steel, the specimen VQ (0.07 ppm) still has enough residual
Table Content of hydrogen in specimens (ppm).
Material QT VQ
SAE52100 0.80 0.07
SCM435 0.7~'~0.9 0.01
hydrogen to reach near saturation of the inclusions. The result is that there is only a small difference in behavior between the two heat treatments for ASTM A295 52100.
Because the hydrogen content contained in steels ASTM A295 52100 and in SCM435 causes a change in the size of the ODA, the ODA is a result of cyclic crack growth under the influence of hydrogen trapped by an inclusion [7-13]. In other words, ODA growth is not a pure fatigue mechanism but is a synergistic effect combining both cyclic stress and the hydrogen environment. Once the size of the ODA becomes large enough, the stress intensity exceeds the intrinsic threshold value and the fatigue crack propagates only due to cyclic loading without the assistance of hydrogen.
In order to prevent the development of ODA, and thus extend the fatigue life of actual machine parts, the trapped hydrogen at the inclusion must be reduced. The current study indicates that the 0.07 ppm hydrogen content for ASTM A295 52100 is still too great.
To indirectly confirm the above mentioned hypothesis, the effective size of an 1.5 O
0.5 _ - - - -
I
or Sires alttplltudr
~r~' Fatzgue hlmt calculated by,l-area pa~rnetet- model
I lakmg ODA into eonsldcralaon
O ~ 10 6 lO' lO ~ 5 < 10' N u m b e r o f cycles to failure Nr O :Specimen QT: Quenched and tempered Q :Specimen VQ: Heat treated in a vacuum followed
by quenching and tempering
MURAKAMI AND YOKOYAMA ON INFLUENCE OF HYDROGEN 121
inclusion was evaluated by adding the size of the ODA to the original size of the inclusion, another modified S-N data of type 2 can be drawn and is shown in Figure 11. The value oftr/aw' for fractured specimens exceeded approximately 1.0 in all cases. The results imply that after very slow fatigue crack growth inside the ODA near the inclusion, the crack size exceeds the critical dimension for the mechanical threshold value estimated by the ax/~-~rea parameter model. The fatigue crack then grows without the assistance o f hydrogen and produces a fatigue fracture surface typical o f a martensite lath structure. On the other hand, even if the size o f inclusion is small enough to satisfy a/aw < 1.0, specimens have a possibility of fail as the size of the ODA grows with number of cycles. For example, even though th e applied stress for the specimen with the mark No. 1 in Figure 5 satisfies ohrw < 1.0, the specimen is presumed to fail if the fatigue test is continued up to N ~10s-5 • 108 at the first stress level (see Figure 8). The prediction is based on the observation of the large ODA which was probably produced at the first low amplitude.
Effect of Dimensions of Specimens and Test Method on S-N curve
As previously mentioned, the controled volume of this specimen is V1_=770 mm 3. The control volume V(mm 3) o f the rotating bending specimen (Figure 12), which is used by a Japanese research group [18], is computed by calculating the volume in which the applied stress exceeds a definite values such as a>yao, as [16, 19]
V= 0.251t(1-y)(d+d02Zl where
d --the minimum diameter of the test part (mm) and R = the radius of curvature of the test specimen (mm).
(2)
Figure 12-Rotating bending fatigue specimen (ram) [18].
The test specimen of the ASTM A295 52100 (with the critical dimensions d=3 mm~ R=7.0 mm [18], and if 7=0/oo=0.90 was assumed) has a critical volume V2_=2.57 m m . A comparison of these specimens gives Vl/V2z-_300, so the control volume o f a single tension-compression specimen is equal to that of 300 rotating bending specimens with diameter 3 m m In other words, a single tension-compression specimen will have an inclusion as the fracture origin that corresponds to the largest inclusion leading to failure in 300 rotating bending specimens. The tension-compression specimen will also be broken in fewer cycles to failure, or at much lower stress amplitude, than for a rotating bending specimen. Conversely, a double S-N curve measured from rotating bending specimens with diameter 3 mm [18] would be significantly influenced by the small size and the large stress gradient. Therefore, it is possible that an S-N curve obtained from as many as 300 rotating bending specimens will not take on the same 2-step S-N curve. At least the double S-N curve will be not so pronounced. Such a possible variation is illustrated in Figure 13. Thus, the following inter-dependent factors that influence the
double S-N curve cannot be ignored:
(1) loading types, specimen size and shape (include the influence o f stress gradient), (2) distribution o f inclusion (scatter o f inclusion sizes),
(3) residual stress, and (4) number of specimens.
Based on the above, the results of rotating bending specimen tests with diameter 3 mm should be treated carefully when applied to engineering design, as they are not necessarily suitable when considering the mechanism of ultra-long fatigue failure.
t.* r ~
\
F a i l u r e f r o m [ R o t a t i n g b e n d i n g { s u r f a c e o r i g / n ~ I S m a l l s p e c i m e n N o t c h e d s p e c i m e n " . . " " ~ I n c r e a s e in n e m b e r E - - -; - - - ~ . . . . .- - , "-- ,r F a i l u r e f r o m , T c n S l o n - C o m p r e s s l o n , I . ~ ~ ~ .~ mlerrtal m r . - - 9 * L "o n t L a r g e s p e c t m e n ~ ,~ , s~ _oot_h_ ~r _~_~_ _ _, " .... 1 0 3 1 0 4 1 0 5 1 0 6 1 0 7 1 0 a 1 0 9 Number of cycles to failure NfFigure 13- Variation o f S-N curve by loading
types and specimen sizes.
Conclusions
To investigate the influence of hydrogen trapped by inclusion on ultra-long fatigue failure of ASTM A295 52100, tension-compression fatigue tests were carried out with two different hydrogen content specimens by changing the heattreatments. These two kinds o f specimens are specimens QT (hydrogen content 0.80 ppm) and specimens VQ (hydrogen content 0.07 ppm). The fiacture surfaces, especially in the vicinity of inclusions, were observed, and the relation with hydrogen content was considered. The results compared with those of SCM435 were as follows,
(1) As the number of cycles increased, the ODA (optically dark area) size as compared to inclusion size also increased. This relationship is the same as previously observed for specimens QT of SCM435.
(2) Specimens VQ of ASTM A295 52100 have smaller ODAs at the idntical fatigue life Nf than specimens QT. This tendency is the same as for SCM435. However, the difference is not as dramatic as that of SCM435 (specimens QT with 0.7-0.9 ppm, specimens VQ with 0.01 ppm content of hydrogen).
(3) Even in the lower hydrogen specimens VQ for ASTM A295 52100, hydrogen was trapped in the vicinity of nonmetallic inclusions. This is because during heat treatment hydrogen is primarily trapped by inclusions that have large trapping force, and after hydrogen is saturated in the vicinity of inclusions, the remainder of the hydrogen is trapped by dislocations or grain boundaries.
(4) In order to avoid fatigue failure in machine parts it is necessary to reduce the trapped hydrogen content and thus eliminate the formation of ODA. A hydrogen content of 0.07 ppm for ASTM A295 52100 is still too large.
(5) In order to investigate the mechanism o f ultra-long fatigue failure, small size rotating bending specimens are not suitable and it is best to test large size tension- compression specimens.
MURAKAMI AND YOKOYAMA ON INFLUENCE OF HYDROGEN 123
References
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[2] Naito, T., Ueda, I4_ and Kikuchi, M., "Fatigue Behavior of Carburized Steel with
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[3] Emura, H. and Asami, K., "Fatigue Strength Characteristics of High Strength Steel,"
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[4] Kuroshima, Y., Shimizu, M. and Kawasaki, K., "Fracture Mode Transition in High
Cycle Fatigue of High Strength Steel," Transactions of the Japan Society of
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[5] Abe, T. and Kanazawa, K., "Influence of Non-metallic Inclusions and Carbides on
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[6] Nakamura, T., Kaneko, M., Noguchi, T. and Jinbo, K., "Relation between High Cycle Fatigue Characteristics and Fracture Origins in Low-temperature-tempered
Cr-Mo Steel," Transactions of the Japan Society of Mechanical Engineers, A~ Vol.
64, No. 623, 1998, p. 1820.
[7] Murakami, Y., Nomoto, T., Ueda, T., Murakami, Y. and Ohori, M., "Analysis of the Mechanism of Supedong Fatigue Failure by Optical Microscope and SEM/AFM
Observations," Journal of Society of Material Science, Japan, Vol. 48, No. 10, 1999,
p. 1112.
[8] Murakami, Y., Nomoto, T. and Ueda, T., "Factors Influencing the Mechanism of
Superlong Fatigue Failure in Steels," FatigTte Fract, Engng. Mater. Struct.,
Blackwell Science Ltd., Vol. 22, 1999, p. 581.
[9] Murakami, Yo, Nomoto, T., Ueda, T. and Murakami Y., "Mechanism of Superlong Fatigue Failure in the Regime of N>I07 Cycles ancl Fractography of the Fracture
Surface," Transactions of the Japan Society of Mechanical Engineers, A, Vol. 66,
No. 642, 2000, p. 311.
[10]Murakami, Y., Nomoto, T., Ueda, T. and Murakami, Y., "On the Mechanism of Fatigue Failure in the Superlong Life Regime (N>107 Cycles). Part 1: Influence of
Hydrogen Trapped by Inclusions," Fatigue Fract. Engng. Mater. Struct., Blackwell
Science Ltd., Vol. 23, 2000, p. 893.
[11]Murakami, Y., Nomoto, T., Ueda, T. and Murakami, Y., "On the Mechanism of Fatigue Failure in the Superlong Life Regime (N>107 Cycles). Part2: A
Fractographic Investigation," Fatigue Fract. Engng. Mater. Struct., Blackwell
Science Ltd., Vol. 23, 2000, p. 903.
[12] Murakami, Y., Konishi, H. and Takai, K., "Effect of Hydrogen Trapped by
Inclusions on Supedong Fatigue Failure," Proceedings of the 49th Annual Meeting,
Society of Material Science, Japan, 2000, p.534.
[13]Murakami, Y., Konishi, H., Takai, K. and Murakami, Y., "Acceleration of Superlong Fatigue Failure by Hydrogen Trapped by Inclusions and Elimination on
Conventional Fatigue Limit," Tetsu-to-Hagand, The Iron and Steel Institute of Japan,
Vol. 86, No. 11,2000, p. 777.
[14] Takai, K., Homma, Y., hutsu, Ko and Nagumo, M., "Identification of Trapping Sites in High-Strength Steels by Secondary Ion Mass Spectrometry for Thermally
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[15]Takai, K., "Visualization of Hydrogen in Steels by Secondary Ion Mass
Spectrometry," Zairyo-to-Kankyo, Japan Society of Corrosion Engineering, Vol. 49,
No. 5, 2000, p. 271.
Yokendo Ltd., Tokyo, 1993.
[17]Murakami, Y. and Endo, M., "Effects of Hardness and Crack Geometry on AKth of
Small Cracks," Journal of Society of Material Science, Japan, VoL 35, No. 395,
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[18]Sakai, T., Takeda, M., Shiozawa, K., Ochi, Y., Nakajima, M, Nakamura, T. and
Oguma, N., "'Experimental Reconfirmation of Characteristic S-N Property for High
Carbon Chromium Bearing Steel in Wide Life Region in Rotating Bending,"
Journal of Society of Material Science, Japan, Vol. 49, No. 7, 2000, p. 779. [19]Murakami, Y., Toriyama, T. and Coudert, E. M., "'Instructions for a New Method of
Inclusion Rating and Correlations with the Fatigue Limit," Journal of Testing and
G. Shi, 1 H. V. Atkinson, ~ C. M. Sellars, l C. W. Anderson, 2 and J. R. Y a t e s ~
Statistical Prediction o f the M a x i m u m Inclusion Size in B e a r i n g Steels
Reference: Shi, G., Atkinson, H. V., Sellars, C. M., Anderson, C. W. and Yates, J. R.,
"Statistical Prediction of the Maximum Inclusion Size in Bearing Steels," Bearing
Steel Technology, ASTMSTP 1419, J. M. Beswick, Ed., American Society for Testing and Materials International, West Conshohocken, PA, 2002.
Abstract: Large and brittle oxide indusions may initiate fatigue failure in bearing
steels. The size of the maximum inclusion in a large volume must be predicted by statistical analysis because only small samples can directly be analysed and there are limitations on non-destructive testing methods. A new method based on the Generalized Pareto distribution (GPD) was recently proposed by the Sheffield group. This allows data on inclusion sizes in small samples to be used to predict the maximum inclusion size in a large volume of steel. The method has advantages over other statistics of extremes methods. The number of sources of failure and the failure rate of practical bearings can be estimated from the predictions of the GPD and the stress distribution in the bearing. Here the GPD method is compared with the Statistics of extreme values (SEV) method developed by Murakami and co-workers. The application of predictions from the GPD in the safe design of bearings will be illustrated.
Keywords: bearing steels, inclusion rating, statistics of extremes, generalized pareto
distribution
For engineering bearings where high fatigue strength and long lifetimes under dynamic loading are required, inclusions have been associated with the initiation of
the failure process [1-3]. Failure of engineering bearings usually originates from the
few large inclusions when the local stress amplitude is above a critical value. These few large oxide inclusions are randomly distributed in vast volumes of steel.
1Research Associate, 1Reader and 1Professor, respectively, in Engineering Materials Department, University of Sheffield, Mappin Street, Sheffield, S1 3JD, United Kingdom G. Shi is now Section Leader at TWI Limited, Abington, Cambridge, CB1 6AL, United Kingdom 2professor, School of Mathematics and Statistics, University of Sheffield, Hounsfield Road, Sheffield, $3 7RH, United Kingdom. 3Professor, Mechanical Engineering Department, University of Sheffield, Mappin Street, Sheffield, S1 3JD, United Kingdom.
125
The combination of low concentration and relatively small size of inclusions requires the examination of unrealistically large areas or volumes of steel to give a statistically significant result. This is often difficult because small volumes can only be detected by conventional methods and the size of large inclusions is below the resolution of
,t
non-destructiv'e testing methods such as ultrasonic tests. The size and number of large inclusions in large volumes of steel, say 1 tonne, must be predicted by statistical analysis based on information from small samples.
Recently, the steel industry has started using statistics of extreme value (SEV)
methods, which were initially developed by Murakami and co-workers [4-10], using
the size of the maximum inclusion in each sample. Thus data on inclusion sizes in small samples can be used to predict the maximum inclusion size in a large volume of steel. A new method based on the Generalized Pareto distribution (GPD), which is a branch of statistics of extremes, has recently been proposed by the authors [11-13]. The methodology for the GPD method has been developed along with computer simulation of the effects of the parameters on the estimation and the confidence intervals [14-15] and the effectiveness of the method in discriminating between steels of different cleanness demonstrated [16]. Predictions from the GPD have been compared with the SEV method and extrapolating the log-normal distribution [11-12]. In addition, the authors have shown that results from cold crucible remelted samples and polished optical cross-sections are consistent [17]. One of the key features for the GPD method is that the estimated size is below an upper limit in some circumstances. It deals with the size distribution of inclusions above a certain size in a sample, not only the maximum inclusion in a sample areaone as in the SEV method. It has been found by the authors that the single largest inclusion is unlikely to lie in a highly stressed volume, but rather that the more frequently occurring slightly smaller inclusions have the highest probability of leading to a fatigue failure [14]. Therefore, successful design of beating components must consider the size and distribution of large inclusions (and not just the largest) in a certain volume of steel. The GPD method developed by the Sheffield group is very powerful in this respect, in contrast with the SEV method, which does not have this capability given by the GPD method.
Here the application GPD method for estimating the size of the maximum inclusion in large volumes of bearing steels is summarised and the estimation is compared with the SEV method. These statistical methods focus on endogenous rather than exogenous inclusions. In addition, the use of the GPD to estimate the number of sources of failure and the safe design of steel components is illustrated for a bearing steel component.
Prediction of the Characteristic Size of the Maximum Inclusion in a Large Volume of Steel by the SEV and GPD Methods
The procedure for the SEV method has been standardized for the estimation of the maximum inclusion size by Murakami and co-workers [9]. The basic concept of the SEV method is that when a large number of data points following a basic distribution are collected, the maximum of each of these sets also follows a distribution. The distribution function was given by Gumbel [18] as follows
SHI ET AL. ON STATISTICAL PREDICTION 127
G(z)=exp(-exp(-(z-A )/ a) (1)
where G(z) is the probability that the largest inclusion is no larger than size z, and a and 3, are the scale and location parameters.
The characteristic size of the maximum inclusion in a large volume of steel (zv),
which is defined as the size of inclusion which it is expected will be exceeded exactly once in V, can be estimated by the following equation [7,9,12]
zv =2- a Ln(-Ln( ( T- 1)/I')) (2)
where T is the return period, defined as
T=V/V o (3)
V is the volume of steel for the extrapolation and Vo is the standard inspection volume that can be estimated according to the size of the standard testing area and the size of inclusions [7-9]. The values of a and 2 can be estimated by the maximum likelihood method [12] from the maximum sized inclusions (square root of the area) in N samples, z~ ... zN.
The procedure for the GPD method has been standardised for the estimation of the maximum inclusion size in clean steels by the authors [11-12]. Suppose u is the threshold and x is the size ( a~-r-~rea ) of inclusions larger than the threshold. Then, the probability of finding an inclusion no larger than x, F(x), given that it exceeds u, is
approximated by the GPD with the following equation [19]
F(x)= 1-( l + ~x-u)/ cr) -'~ (4)
where tr' >0 is a scale parameter and ~ (-oo<~,<oo) is a shape parameter. The range of
(x-u) is 0<x-u<oo if ~ >0, and O<x-u<-trT~ if ~ <0.
The characteristic size of the maximum inclusion xv in a large volume V (defined as the size expected to be exceeded by exactly one inclusion in l0 can be estimated using [11-12]
x v = u - i x '
{1-(Nffu)g)~}
(5)
where Nv(u) denotes the expected number of exceedances of u in unit volume, which
can be determined approximately according to the number of intercepted inclusions per unit area NA on the polished surface [11].
When ~<0, (Nv(u)V)r 1 when Vis very large, then
x V = u-tr'/~. (6)
(u-trT~) will be the upper limit of the inclusion size in the steel, whatever the volume considered.
The data needed for the SEV and GPD method are different as shown in Table 1. There are three parameters in the GPD function (Equation 4), u, cr" and ~. The authors have found in earlier work [11-12] that the estimated results from the GPD method are relatively insensitive to the choice of the critical' threshold u. However,'the
choice of as low a threshold as possible leads to the most precise estimation [14]. The
values of or'and ~ at the chosen threshold were estimated by the maximum likelihood method.
The confidence intervals of the estimated result are estimated by the profile likelihood method [19].
Table 1 compares the SEV and GPD methods.
Table 1- Comparison of the GPD and SEV methods
Method Data needed Parameters Size distribution Upper limit
SEV Size of the ec and ~, Size distribution of No upper
maximum ~ is set to zero the maximum limit
inclusion in inclusion
each sample
GPD Size and o', ~ and u Distribution of large Upper limit