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Experimental validation of railway axle fatigue crack growth using operational loading
Engineering Fracture Mechanics 213 (2019) 142–152
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Experimental validation of railway axle fatigue crack growth using operational loading
T
M. Yamamoto , K. Makino, H. Ishiduka ⁎
Railway Technical Research Institute, 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, Japan
ARTICLE INFO
ABSTRACT
Keywords: Railway axle Crack growth behaviour Variable amplitude loading Full-scale test
Prediction of fatigue crack growth is an important tool for determining inspection intervals for railway axles. Previously, we described the correspondence between the theoretical crack growth behaviour exhibited by compact tension specimens and the observed results for full-scale axles under constant amplitude loading. In this study, the crack growth behaviour of full-scale axles was experimentally investigated under variable amplitude loading, which was based on the operational load–time history. Crack growth models were compared with full-scale testing results. Moreover, the effect of variable amplitude loading on crack growth behaviour is discussed.
1. Introduction Railway axle failure may cause critical accidents such as vehicle derailment, therefore axle integrity is a critical aspect of railway safety. Axle design guides such as EN 13103 [1], intended for the European railway network, and the Japanese JIS E 4501 [2] prescribe an infinite life design based on fatigue limits. Several studies have reported fatigue testing results using full-scale axles [3–7]. Fatigue tests are usually conducted on axles that do not exhibit any damage or defects [3–5]. However, in-service axles may exhibit damage or defects such as corrosion, ballast impacts, or non-metallic inclusion, and fatigue cracks may initiate from these defects. Few studies have reported fatigue testing results on full-scale axles including such defects [6,7]. Infinite life design cannot assess the safety for the axles with such defects, leading to an increasing demand for damage tolerance analyses that are able to estimate residual lifetime or appropriate inspection intervals for railway axles [8–15]. Beretta et al. [8] and Carboni et al. [9] conducted crack propagation tests using compact specimens and full-scale axles under constant stress amplitude, and they both compared the results obtained in the two different cases. Our previous report [10] indicated that the crack growth behaviour of full-scale axles under constant stress amplitude may be evaluated via crack growth test results obtained using compact tension specimens through the inclusion of the threshold stress intensity factor range, ΔKth, in a practical application. Several studies have reported damage tolerance analyses of railway axles using the relationship between crack growth rate, da/dN, and the stress intensity factor range, ΔK, obtained under constant stress amplitude [11–18]. Yasniy et al. [11] conducted a probabilistic analysis of railway axle lifetimes using crack growth behaviour observed under constant stress amplitude. Pokorný et al. [12,13] evaluated the influence of load spectra on residual fatigue lifetime of railway axles. Carboni and Beretta [15] analysed the effect of the probability of detection upon inspection intervals of railway axles. Railway axles are subjected to variable amplitude loading. It is well known that load sequence effects fatigue crack growth rate under variable amplitude loading [19]. For the accurate prediction of fatigue life or inspection intervals, some studies have focused on the application of a crack growth model for railway axles under variable amplitude loading [20–24]. Beretta et al. [20,21]
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Corresponding author. E-mail address: [email protected] (M. Yamamoto).
https://doi.org/10.1016/j.engfracmech.2019.04.001 Received 22 November 2018; Received in revised form 8 March 2019; Accepted 1 April 2019 Available online 02 April 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A C, C′, m, da/dN Kdy Kmax Kop Z ΔK
ΔKeff ΔKeff,dy
elongation m′ material constant crack growth rate stress intensity factor for individual σdy maximum stress intensity factor crack opening stress intensity factor reduction of area stress intensity factor range
ΔKeff,th ΔKth σB σdy σmax σst σy
effective stress intensity factor range effective stress intensity factor range for individual σdy threshold value of ΔKeff threshold stress intensity factor range tensile strength dynamic stress maximum applied stress static stress yield strength
conducted crack propagation tests under both constant and variable amplitude loading and analysed the application of crack growth prediction models. Their results show that a load interaction effect causes retardation of fatigue crack growth in full-scale axles, and that the estimation of the Strip–Yield model offers better correspondence than that of the no-interaction model. Wu et al. [23,24] developed a fatigue crack growth model that was derived from low cycle fatigue property. They predicted the fatigue life of axles with inside axle boxes through the crack growth simulation. A number of studies have addressed experimental validation of the effect of load sequence on the fatigue crack growth behaviour of railway axles [25–28]. Traupe et al. [25] conducted crack propagation tests on full-scale axles for the experimental validation of inspection intervals. The block load sequence derived from in-service stress measurements and a resonance-type fatigue testing rig were employed for the test. Mädler et al. [26] reported test results obtained using a full-scale wheel-rail roller test rig, which allows realistic bending moment conditions. Their study indicated that crack growth may be significantly slower when crack-retarding load changes occur relatively frequently. Luke et al. [27] obtained crack growth curves for middle tension specimens, one-third-scale axles, and full-scale axles under variable amplitude loading. The results showed that both the shape of the load sequence and the individual block length of the load sequence influence crack growth behaviour. Sander and Richard [28] investigated the effects of the reconstruction of load–time history using compact tension specimens and showed that these effects play an important role in fatigue crack growth. Despite the complication that crack propagation behaviour depends on load sequence effects, no studies have addressed the crack growth testing of full-scale axles using operational load–time history, owing to the technical restrictions of testing facilities [21,25,26,28]. This study presents an experimental investigation of the crack growth behaviour of full-scale axles under variable amplitude loading that is based on operational load–time history. The testing apparatus employed for the tests has been designed in order to facilitate fatigue testing of full-scale axles under four-point rotary bending conditions and allows immediate changes in stress levels. The crack growth models employed are evaluated based on the obtained results and the effect of variable amplitude loading on crack growth behaviour is also discussed. 2. Operational load–time history of railway axles Several test runs on conventional rail lines in Japan were conducted in order to measure variations in the axle stress accompanying the passage of time, which are referred to as the axle operational load–time history. The axle stress was measured using strain gauges attached on the axle body. Fig. 1 shows an example of part of the axle operational load–time history. In Fig. 1, the upper and lower vertical axes indicate normalised stress defined as the ratio of dynamic stress (σdy) to the static stress (σst) and the curvature
Fig. 1. Example of an operational load–time history. Normalised stress is defined as the ratio of dynamic stress (σdy) to static stress (σst). Curvature indicates the inverse of curve radius. 143
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given by the inverse of curve radius, respectively, and the horizontal axis indicates measurement time. When proceeding along a straight track, the σdy/σst amplitudes are approximately equal to 1. However, the ratio temporarily increases while passing on the crossing and continuously increases while running along a curved track. As shown in Fig. 1, axle operational load–time history is affected by the track conditions. Fig. 2 shows an example of block and random loading sequences derived from the operational load–time history shown in Fig. 1. The stress amplitude of the uniform magnitude continues in the block loading sequence and does not continue in the random one. A comparison of Figs. 1 and 2 indicates that the block and random loading sequences differ from the axle operational load–time history. Therefore, the crack growth behaviour of railway axles, which may vary depending on a selected load sequence as is illustrated in the previous section, should be evaluated using variable amplitude loading based on operational load–time history. Fig. 3 shows the histogram of the stress–frequency distribution derived from the axle operational load–time history. The histogram was obtained using zero crossing peak count method as shown in Fig. 3(a) [29]. This is because peak values of axle stress for each cycle are required in order to generate the load sequences using in this study. In order to clearly assess crack growth behaviour, the measured stress of the load–time history was discretised into 12 stress levels. First, as is indicated on the left vertical axis, the relationship between the normalised stress and relative frequency was analysed for the case that the axle stress was measured in a test run which carried no passengers. Then the relationship between the applied stress for the crack propagation tests and relative frequency was obtained by multiplying the normalised stress by a σst in an actual running condition. In this study, σst was set to be 50 MPa which corresponds to the maximum value considered in Japanese design guides [30]. The maximum applied stress, σmax, in the crack propagation test was to be 150 MPa, which was derived from the maximum value of σdy/σst = 3. The cumulative frequencies of one period of the operational load–time history consist of 80,000 loading cycles, which corresponds to approximately 210 km in running distance. 3. Experimental procedure of crack propagation tests for full-scale axles 3.1. Test rig The test rig employed in this study is shown in Fig. 4. A key advantage of the test rig used is that the applied stress, which is controlled by the electro-hydraulic actuator, can be changed immediately. Therefore, the rig is suitable for crack propagation testing under variable loading amplitude based on an operational load–time history. Each of tested axles had a starter artificial notch, which was 1 mm in both length and depth and located at axle central part. This is because the K solution is simple for a crack propagating on the axle body, which has no stress concentration and exhibits a linear stress gradient in the radial direction, although the T-transition is subjected to stress concentration and a nonlinear stress gradient. The fatigue pre-crack was generated from the notch in accordance with ASTM E 647 [31]. Our previous article [10] shows that the fatigue crack propagates semi-elliptically regardless of the starternotch shape, so that the starter-notch shape does not influence on the crack propagation test results for the examined condition. The final lengths of the pre-cracks are 30 mm in the experiment shown in Section 4.1 and 12 mm in the experiments shown in Section 4.2 and Section 5, respectively. The details of the rig and pre-cracking procedures are also detailed in the previous article [10]. The testing frequency was set to 600 rpm at the rotation speed. Full-scale axles were made of SFA 640 steel, which is widely used in Japanese railway axles [32]. The chemical composition and mechanical properties of the SFA640 used here are shown in Tables 1 and 2, respectively. 3.2. Crack propagation tests Crack propagation tests performed using operational load–time history require a very long observation time, i.e., verification of a residual life time of 500,000 km in running distance corresponds to more than 7 months at a rotation speed of 600 rpm. It is necessary
Fig. 2. Examples of block and random loading sequences translated from the operational load–time history shown in Fig. 1. 144
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Fig. 3. (a) Explanation of zero crossing peak count method for an axle operational load–time history. Stress–frequency distribution shown in figure (b) are calculated by counting σn+ and absolute value of σn−. (b) Stress–frequency distribution derived fro m the operational load–time history. σst and σmax are set to 50 MPa and 150 MPa, respectively. The load–time history is discretised into 12 stress levels.
Fig. 4. (a) Test rig employed for crack propagation tests under variable amplitude loading based on operational load–time history. (b) Shape of the starter artificial notch. Table 1 Chemical composition of full-scale axle specimens (wt%). C
Si
Mn
P
S
0.45
0.20
0.80
0.021
0.008
Table 2 Mechanical properties of full-scale axle specimens (σy: yield strength (MPa), σB: tensile strength (MPa), A: Elongation (%), Z: reduction of area (%)). σy
σB
A
Z
439
754
27
62
to employ omission methods such as those presented previously [25,26] in order to reduce the fatigue testing time. Hence, the load–time history with the stress–frequency distribution shown in Fig. 5 was generated. Hereinafter, the load–time histories related to the distributions shown in Figs. 3 and 5 are referred to as ‘normal’ and ‘shortened’ load histories, respectively. The omission method employed here differs from those employed in [25,26]; in these cases, the omission stress level is determined by ΔKth. The details of the omission method employed here are described below. The stress–frequency distributions of both the shortened and normal load histories are the same for stresses greater than or equal to 90 MPa; they have the same σdy values, and the sequence followed by the data points follows the same order. However, for σdy values ranging from 60 to 80 MPa in the case of the shortened load history, the frequencies are reduced compared to those of the normal load history and those for 40 and 50 MPa are omitted. An example of comparison between normal and shortened load histories is shown in Table 3. The number of cumulative frequencies in one period of the shortened load history was reduced to 16,000 cyclic loadings, which corresponds to one-fifth of that of the normal load history. Two full-scale axles were employed for the crack propagation tests. During the crack propagation tests, magnetic particle testing 145
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Fig. 5. Stress–frequency distribution for the shortened load–time history. The stress ranging from 60 to 80 MPa is reduced compared to that of the normal load history. Values below 60 MPa are omitted. Table 3 An example of comparison between normal and shortened load–time histories.
Fig. 6. Crack propagation test results obtained using the normal and shortened load histories. The normal and shortened load histories are alternatively applied every time the crack length increment of 2 mm is reached. 146
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was conducted periodically so as to measure the crack length. First, tests were conducted using the normal and shortened load histories in order to assess the effect of using both load histories on crack growth behaviour. This test was conducted for crack lengths ranging from 30 to 90 mm in the following manner. A crack of 30 mm was propagated using the shortened load history by an increment of approximately 2 mm. Following the crack propagation using the shortened load history, the crack was then further propagated using the normal load history by the same increment. In this manner, the normal and shortened load histories were alternatively applied every time the incremental crack length value of 2 mm was reached. The incremental value of 2 mm was determined considering the plastic zone size at the crack tip. Afterwards, similar tests were conducted using only the shortened load history in order to evaluate the crack growth behaviour for crack lengths ranging from 12 to 90 mm. The obtained results were compared with crack growth models. 4. Experimental results 4.1. The effect of two types of load histories on crack growth behaviour Fig. 6 shows the crack propagation test results obtained using the normal and shortened load histories. The horizontal axis denotes the number of the applied sequences for both the normal and shortened load histories. Here, 80,000 cyclic loadings are counted as one period in the case of the normal load history, and 16,000 cyclic loadings are counted as one period in the case of the shortened load history. As can be seen in Fig. 6, the experimental data obtained using both load histories are fitted, maintaining continuity. In order to clearly evaluate the propagation behaviour, the increment of crack length per period is calculated as shown in Fig. 7. The crack length increment curve obtained using the test data for the normal load history corresponds to that of the test data for the shortened load history. The ΔK value induced by a σdy equal to 80 MPa, the loading cycles of which are reduced in the case of the shortened load history, exceeds the ΔKth value after the crack length reaches 20 mm. Moreover, the ΔK value induced by a σdy equal to 50 MPa, the loading cycles of which are omitted in the case of the shortened load history, exceeds the ΔKth value after the crack length reaches 50 mm. However, the obtained results indicate that the impact of the normal load history on crack growth behaviour is equivalent to that of the shortened load history. These results demonstrate that crack propagation ceased below a certain stress amplitude regardless of the ΔKth value of the material. Our previous report indicated that the crack growth behaviour under constant amplitude loading could be estimated using the crack growth model considering ΔK or ΔKth [10]. However, the obtained results indicate that the crack growth model obtained under constant amplitude loading, which is expressed as the relationship between da/dN and ΔK or ΔKth, offers a conservative estimation of the residual lifetime of railway axles, which are subjected to variable amplitude loading, as shown in the previous literature [22,25–27]. 4.2. Crack propagation test using shortened load history As can be seen in the previous section, utilizing the shortened load history instead of the normal load history provides an equivalent impact on crack growth behaviour. Therefore, crack propagation tests were conducted using only the shortened load history for a crack ranging from 12 to 90 mm in length. Fig. 8 shows the crack length of the tested axle as a function of equivalent running distance. In Fig. 8, one period of the shortened load history was converted into a running distance of 210 km. A crack with a length of 12 mm was propagated to 90 mm after 500,000 km in equivalent running distance. Fig. 9 shows the relationship between crack length and its increment value per 1000 km of equivalent running distance. The results of our previous report indicated that the crack growth behaviour of full-scale axles under constant amplitude loading may be fitted in accordance with the following equation, employed by Klesnil and Lukáš [33]:
Fig. 7. Comparison of crack length increment value per period. The values obtained using the normal load history show a good correspondence with those obtained using the shortened load history. 147
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Fig. 8. Crack propagation test result using shortened load history.
Fig. 9. Relationship between crack length and its increment value per 1000 km in equivalent running distance.
da = C ( Km dN
Kthm),
(1)
where material parameters C = 1.7 × 10−13, m = 3.9 and ΔKth = 8.7 MPa·m1/2 obtained using full-scale axle in the previous report [10] are employed, requiring da/dN to be expressed in units of m/cycle. Fig. 10 shows the comparison between the obtained result shown in Fig. 9 and a crack growth model using Eq. (1). Note that the vertical and horizontal axes are displayed on a logarithmic
Fig. 10. Comparison of crack length increment values for test data and the crack growth model using Eq. (1). 148
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scale. The crack length increment values estimated using Eq. (1) are three to five times higher than those of the test data, regardless of crack size. This is because crack propagation ceased below a certain stress amplitude regardless of the ΔKth value, as mentioned in Section 4.1, whereas the crack growth model that employs Eq. (1) assumes that any stress amplitude that exceeds ΔKth contributes to crack propagation. In order to assess crack growth behaviour under variable amplitude loading within this range, a crack growth model considering an effective stress intensity factor range, ΔKeff, is adopted. ΔKeff is given by:
K eff = Kmax
(2)
K op,
where Kmax and Kop are the maximum applied stress intensity factor and crack opening stress intensity factor, respectively. Considering crack closure phenomena under variable amplitude loading, which is based not on a long-term block loading condition but instead on operational loading, the crack opening stress would be affected by the maximum applied stress of the load. However, there are no articles which investigate Kop for SFA640, although some articles have been published for steel grade used in European railway network [21,34]. Therefore, the relationship between Kmax and Kop for SFA640 steel was obtained using middle-tension specimens under tension–compression loading, as shown in Fig. 11. Kop was obtained using unloading elastic compliance method [35]. Middletension specimens were sampled from the tested full-scale axle. In other words, the dispersion due to material batches is not considered here. Kop increases with increasing Kmax, although there is some variation in the fit of the data. The crack growth behaviour using the shortened load history was determined in accordance with the following equation:
da m = C ( K eff dN
m K eff , th ),
(3) −12
where material parameters C′ and m′ are re-evaluated and set to 1.5 × 10 and 3.4, respectively. ΔKeff,th is the threshold of ΔKeff, and was obtained using ΔKth in Eq. (1) and the relationship shown in Fig. 11. Note that a physically short crack is not considered in this study, so that ΔKeff,th corresponds to that of long crack and was set to 6.0 MPa·m1/2. A crack growth model using Eq. (3) was compared with the crack propagation test results. As can be seen in Figs. 10 and 12, the crack length increment values estimated using Eq. (3) offer an improved correspondence with the experimental results compared to those estimated via Eq. (1) under the operational time–load history. As can be seen in Eq. (3), the crack growth model assumed here does not consider crack growth retardation or acceleration effects under variable amplitude loading. However, considering the fact that the crack growth rate is estimated with a wide margin of error, incorporating the dispersion of material constants such as C or m values in the damage tolerance analysis, Eq. (3) may be used to estimate the crack growth behaviour of axles under variable amplitude loading for practical applications. 5. Discussion As described in Section 4.2, Kop is affected by Kmax of the whole load–tome history, i.e., the crack opening stress is affected by the maximum applied stress of the whole load–time history. The K value for individual σdy in the load–time history and corresponding ΔKeff are expressed by the subscript “dy” to clarify the following explanation. The relationship between Kdy and ΔKeff,dy is given by:
K eff , dy = K dy
(4)
K op.
ΔKeff,dy depends on the individual σdy in the load–time history, although Kop depends on the σmax of the whole load–time history. As an example, we may examine the condition that the Kmax value at a σmax of 150 MPa is equal to 15 MPa·m1/2 and Kop is equal to 3 MPa·m1/2. If the stress suddenly decreases to 100 MPa, the Kdy value at σdy = 100 MPa becomes 10 MPa·m1/2, however, Kop remains at 3 MPa·m1/2 and ΔKeff,dy is equal to 7 MPa·m1/2. However, if σmax is equal to 120 MPa, the Kop value decreases to 2.5 MPa·m1/2. As a result, the ΔKeff,dy at σdy = 100 MPa becomes 7.5 MPa·m1/2. In this manner, even if Kmax (i.e., σmax) decreases, ΔKeff,dy might increase
Fig. 11. Relationship between maximum stress intensity factor, Kmax, and crack opening stress intensity factor, Kop, for the tested material. 149
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Fig. 12. Comparison of crack length increment values for the test data and crack growth model using Eq. (3).
as a consequence of the decreasing Kop. In order to validate the crack growth model, considering the interplay of ΔKeff and ΔKeff,th, a crack growth test was conducted using a modified stress–frequency distribution with a maximum stress of 120 MPa, as shown in Fig. 13. In the modified load history, the stress amplitudes ranging from 130 to 150 MPa of the shortened load history are converted to 120 MPa. In other words, the number of cycles for stress amplitudes ranging from 120 to 150 MPa of the shortened load history is equal to that of 120 MPa in the modified load history. Fig. 14 compares the resulting crack growth curves. Although the sum of the fatigue damage obtained using the modified load history with σmax = 120 MPa is less than that of the shortened load history with a σmax of 150 MPa, the crack tested using the modified load history grew more rapidly than that using the shortened load history. This outcome demonstrates the effectiveness of the crack growth model that considers ΔKeff and ΔKeff,dy in predicting residual lifetime or appropriate inspection intervals for axles subjected to variable amplitude loading. 6. Conclusion In the present study, crack growth behaviour under variable amplitude loading was assessed through crack propagation tests of full-scale axles. A four-point rotary bending facility, which controls the applied bending moment using an electro-hydraulic actuator and permits tests based on an operational load–time history, was employed. The following conclusions were reached. (1) The omission method was proposed in order to reduce fatigue testing time, and the shortened load history, in which the cumulative number of cycles corresponds to one-fifth of the operational load history, was generated. A crack propagation test using operational and shortened load histories was conducted. The result indicates that utilising either of these load histories has an equivalent impact on crack growth behaviour. The results also indicate that the crack growth model obtained under constant amplitude loading, which is expressed as the relationship of crack growth rate (da/dN) and stress intensity factor range (ΔK) or its
Fig. 13. Stress–frequency distribution of the shortened and modified load histories. The stress amplitude ranging from 130 to 150 MPa of the shortened load history is converted to 120 MPa in the case of the modified load history. 150
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Fig. 14. Comparison of crack growth curve using the shortened (σmax = 150 MPa) and modified (σmax = 120 MPa) load history. The crack tested using the modified load history grows more rapidly than that using the shortened load history, although σmax of the former is less than that of the latter.
threshold value (ΔKth), offers a conservative estimation as to the residual lifetime of railway axles, which are subject to variable amplitude loading. (2) The crack propagation tests using the shortened load history was conducted for a crack with a length ranging from 12 to 90 mm in order to assess crack growth behaviour under operational loading condition. The crack growth model that considered the effective stress intensity factor range (ΔKeff) and its threshold value (ΔKeff,th) offers a good correspondence with the experimental test results in practical terms. (3) In order to validate the crack growth model considering ΔKeff and ΔKeff,th, the crack growth test was conducted using a modified shortened load history with a maximum stress of 120 MPa and the result was compared to that of the test data obtained using the shortened load history with a maximum stress of 150 MPa. The outcomes indicate the effectiveness of the crack growth model that considers ΔKeff and ΔKeff,dy in predicting residual lifetime or appropriate inspection intervals for axles subjected to variable amplitude loading. The influence of operational time–load histories that have a different stress–frequency distribution than that employed in the current study will be the subject of future work. We will also examine the effectiveness of the proposed crack growth model when used with various different operational load–time histories. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Experimental validation of railway axle fatigue crack growth using operational loading
T
M. Yamamoto , K. Makino, H. Ishiduka ⁎
Railway Technical Research Institute, 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, Japan
ARTICLE INFO
ABSTRACT
Keywords: Railway axle Crack growth behaviour Variable amplitude loading Full-scale test
Prediction of fatigue crack growth is an important tool for determining inspection intervals for railway axles. Previously, we described the correspondence between the theoretical crack growth behaviour exhibited by compact tension specimens and the observed results for full-scale axles under constant amplitude loading. In this study, the crack growth behaviour of full-scale axles was experimentally investigated under variable amplitude loading, which was based on the operational load–time history. Crack growth models were compared with full-scale testing results. Moreover, the effect of variable amplitude loading on crack growth behaviour is discussed.
1. Introduction Railway axle failure may cause critical accidents such as vehicle derailment, therefore axle integrity is a critical aspect of railway safety. Axle design guides such as EN 13103 [1], intended for the European railway network, and the Japanese JIS E 4501 [2] prescribe an infinite life design based on fatigue limits. Several studies have reported fatigue testing results using full-scale axles [3–7]. Fatigue tests are usually conducted on axles that do not exhibit any damage or defects [3–5]. However, in-service axles may exhibit damage or defects such as corrosion, ballast impacts, or non-metallic inclusion, and fatigue cracks may initiate from these defects. Few studies have reported fatigue testing results on full-scale axles including such defects [6,7]. Infinite life design cannot assess the safety for the axles with such defects, leading to an increasing demand for damage tolerance analyses that are able to estimate residual lifetime or appropriate inspection intervals for railway axles [8–15]. Beretta et al. [8] and Carboni et al. [9] conducted crack propagation tests using compact specimens and full-scale axles under constant stress amplitude, and they both compared the results obtained in the two different cases. Our previous report [10] indicated that the crack growth behaviour of full-scale axles under constant stress amplitude may be evaluated via crack growth test results obtained using compact tension specimens through the inclusion of the threshold stress intensity factor range, ΔKth, in a practical application. Several studies have reported damage tolerance analyses of railway axles using the relationship between crack growth rate, da/dN, and the stress intensity factor range, ΔK, obtained under constant stress amplitude [11–18]. Yasniy et al. [11] conducted a probabilistic analysis of railway axle lifetimes using crack growth behaviour observed under constant stress amplitude. Pokorný et al. [12,13] evaluated the influence of load spectra on residual fatigue lifetime of railway axles. Carboni and Beretta [15] analysed the effect of the probability of detection upon inspection intervals of railway axles. Railway axles are subjected to variable amplitude loading. It is well known that load sequence effects fatigue crack growth rate under variable amplitude loading [19]. For the accurate prediction of fatigue life or inspection intervals, some studies have focused on the application of a crack growth model for railway axles under variable amplitude loading [20–24]. Beretta et al. [20,21]
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Corresponding author. E-mail address: [email protected] (M. Yamamoto).
https://doi.org/10.1016/j.engfracmech.2019.04.001 Received 22 November 2018; Received in revised form 8 March 2019; Accepted 1 April 2019 Available online 02 April 2019 0013-7944/ © 2019 Elsevier Ltd. All rights reserved.
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Nomenclature A C, C′, m, da/dN Kdy Kmax Kop Z ΔK
ΔKeff ΔKeff,dy
elongation m′ material constant crack growth rate stress intensity factor for individual σdy maximum stress intensity factor crack opening stress intensity factor reduction of area stress intensity factor range
ΔKeff,th ΔKth σB σdy σmax σst σy
effective stress intensity factor range effective stress intensity factor range for individual σdy threshold value of ΔKeff threshold stress intensity factor range tensile strength dynamic stress maximum applied stress static stress yield strength
conducted crack propagation tests under both constant and variable amplitude loading and analysed the application of crack growth prediction models. Their results show that a load interaction effect causes retardation of fatigue crack growth in full-scale axles, and that the estimation of the Strip–Yield model offers better correspondence than that of the no-interaction model. Wu et al. [23,24] developed a fatigue crack growth model that was derived from low cycle fatigue property. They predicted the fatigue life of axles with inside axle boxes through the crack growth simulation. A number of studies have addressed experimental validation of the effect of load sequence on the fatigue crack growth behaviour of railway axles [25–28]. Traupe et al. [25] conducted crack propagation tests on full-scale axles for the experimental validation of inspection intervals. The block load sequence derived from in-service stress measurements and a resonance-type fatigue testing rig were employed for the test. Mädler et al. [26] reported test results obtained using a full-scale wheel-rail roller test rig, which allows realistic bending moment conditions. Their study indicated that crack growth may be significantly slower when crack-retarding load changes occur relatively frequently. Luke et al. [27] obtained crack growth curves for middle tension specimens, one-third-scale axles, and full-scale axles under variable amplitude loading. The results showed that both the shape of the load sequence and the individual block length of the load sequence influence crack growth behaviour. Sander and Richard [28] investigated the effects of the reconstruction of load–time history using compact tension specimens and showed that these effects play an important role in fatigue crack growth. Despite the complication that crack propagation behaviour depends on load sequence effects, no studies have addressed the crack growth testing of full-scale axles using operational load–time history, owing to the technical restrictions of testing facilities [21,25,26,28]. This study presents an experimental investigation of the crack growth behaviour of full-scale axles under variable amplitude loading that is based on operational load–time history. The testing apparatus employed for the tests has been designed in order to facilitate fatigue testing of full-scale axles under four-point rotary bending conditions and allows immediate changes in stress levels. The crack growth models employed are evaluated based on the obtained results and the effect of variable amplitude loading on crack growth behaviour is also discussed. 2. Operational load–time history of railway axles Several test runs on conventional rail lines in Japan were conducted in order to measure variations in the axle stress accompanying the passage of time, which are referred to as the axle operational load–time history. The axle stress was measured using strain gauges attached on the axle body. Fig. 1 shows an example of part of the axle operational load–time history. In Fig. 1, the upper and lower vertical axes indicate normalised stress defined as the ratio of dynamic stress (σdy) to the static stress (σst) and the curvature
Fig. 1. Example of an operational load–time history. Normalised stress is defined as the ratio of dynamic stress (σdy) to static stress (σst). Curvature indicates the inverse of curve radius. 143
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given by the inverse of curve radius, respectively, and the horizontal axis indicates measurement time. When proceeding along a straight track, the σdy/σst amplitudes are approximately equal to 1. However, the ratio temporarily increases while passing on the crossing and continuously increases while running along a curved track. As shown in Fig. 1, axle operational load–time history is affected by the track conditions. Fig. 2 shows an example of block and random loading sequences derived from the operational load–time history shown in Fig. 1. The stress amplitude of the uniform magnitude continues in the block loading sequence and does not continue in the random one. A comparison of Figs. 1 and 2 indicates that the block and random loading sequences differ from the axle operational load–time history. Therefore, the crack growth behaviour of railway axles, which may vary depending on a selected load sequence as is illustrated in the previous section, should be evaluated using variable amplitude loading based on operational load–time history. Fig. 3 shows the histogram of the stress–frequency distribution derived from the axle operational load–time history. The histogram was obtained using zero crossing peak count method as shown in Fig. 3(a) [29]. This is because peak values of axle stress for each cycle are required in order to generate the load sequences using in this study. In order to clearly assess crack growth behaviour, the measured stress of the load–time history was discretised into 12 stress levels. First, as is indicated on the left vertical axis, the relationship between the normalised stress and relative frequency was analysed for the case that the axle stress was measured in a test run which carried no passengers. Then the relationship between the applied stress for the crack propagation tests and relative frequency was obtained by multiplying the normalised stress by a σst in an actual running condition. In this study, σst was set to be 50 MPa which corresponds to the maximum value considered in Japanese design guides [30]. The maximum applied stress, σmax, in the crack propagation test was to be 150 MPa, which was derived from the maximum value of σdy/σst = 3. The cumulative frequencies of one period of the operational load–time history consist of 80,000 loading cycles, which corresponds to approximately 210 km in running distance. 3. Experimental procedure of crack propagation tests for full-scale axles 3.1. Test rig The test rig employed in this study is shown in Fig. 4. A key advantage of the test rig used is that the applied stress, which is controlled by the electro-hydraulic actuator, can be changed immediately. Therefore, the rig is suitable for crack propagation testing under variable loading amplitude based on an operational load–time history. Each of tested axles had a starter artificial notch, which was 1 mm in both length and depth and located at axle central part. This is because the K solution is simple for a crack propagating on the axle body, which has no stress concentration and exhibits a linear stress gradient in the radial direction, although the T-transition is subjected to stress concentration and a nonlinear stress gradient. The fatigue pre-crack was generated from the notch in accordance with ASTM E 647 [31]. Our previous article [10] shows that the fatigue crack propagates semi-elliptically regardless of the starternotch shape, so that the starter-notch shape does not influence on the crack propagation test results for the examined condition. The final lengths of the pre-cracks are 30 mm in the experiment shown in Section 4.1 and 12 mm in the experiments shown in Section 4.2 and Section 5, respectively. The details of the rig and pre-cracking procedures are also detailed in the previous article [10]. The testing frequency was set to 600 rpm at the rotation speed. Full-scale axles were made of SFA 640 steel, which is widely used in Japanese railway axles [32]. The chemical composition and mechanical properties of the SFA640 used here are shown in Tables 1 and 2, respectively. 3.2. Crack propagation tests Crack propagation tests performed using operational load–time history require a very long observation time, i.e., verification of a residual life time of 500,000 km in running distance corresponds to more than 7 months at a rotation speed of 600 rpm. It is necessary
Fig. 2. Examples of block and random loading sequences translated from the operational load–time history shown in Fig. 1. 144
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Fig. 3. (a) Explanation of zero crossing peak count method for an axle operational load–time history. Stress–frequency distribution shown in figure (b) are calculated by counting σn+ and absolute value of σn−. (b) Stress–frequency distribution derived fro m the operational load–time history. σst and σmax are set to 50 MPa and 150 MPa, respectively. The load–time history is discretised into 12 stress levels.
Fig. 4. (a) Test rig employed for crack propagation tests under variable amplitude loading based on operational load–time history. (b) Shape of the starter artificial notch. Table 1 Chemical composition of full-scale axle specimens (wt%). C
Si
Mn
P
S
0.45
0.20
0.80
0.021
0.008
Table 2 Mechanical properties of full-scale axle specimens (σy: yield strength (MPa), σB: tensile strength (MPa), A: Elongation (%), Z: reduction of area (%)). σy
σB
A
Z
439
754
27
62
to employ omission methods such as those presented previously [25,26] in order to reduce the fatigue testing time. Hence, the load–time history with the stress–frequency distribution shown in Fig. 5 was generated. Hereinafter, the load–time histories related to the distributions shown in Figs. 3 and 5 are referred to as ‘normal’ and ‘shortened’ load histories, respectively. The omission method employed here differs from those employed in [25,26]; in these cases, the omission stress level is determined by ΔKth. The details of the omission method employed here are described below. The stress–frequency distributions of both the shortened and normal load histories are the same for stresses greater than or equal to 90 MPa; they have the same σdy values, and the sequence followed by the data points follows the same order. However, for σdy values ranging from 60 to 80 MPa in the case of the shortened load history, the frequencies are reduced compared to those of the normal load history and those for 40 and 50 MPa are omitted. An example of comparison between normal and shortened load histories is shown in Table 3. The number of cumulative frequencies in one period of the shortened load history was reduced to 16,000 cyclic loadings, which corresponds to one-fifth of that of the normal load history. Two full-scale axles were employed for the crack propagation tests. During the crack propagation tests, magnetic particle testing 145
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Fig. 5. Stress–frequency distribution for the shortened load–time history. The stress ranging from 60 to 80 MPa is reduced compared to that of the normal load history. Values below 60 MPa are omitted. Table 3 An example of comparison between normal and shortened load–time histories.
Fig. 6. Crack propagation test results obtained using the normal and shortened load histories. The normal and shortened load histories are alternatively applied every time the crack length increment of 2 mm is reached. 146
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was conducted periodically so as to measure the crack length. First, tests were conducted using the normal and shortened load histories in order to assess the effect of using both load histories on crack growth behaviour. This test was conducted for crack lengths ranging from 30 to 90 mm in the following manner. A crack of 30 mm was propagated using the shortened load history by an increment of approximately 2 mm. Following the crack propagation using the shortened load history, the crack was then further propagated using the normal load history by the same increment. In this manner, the normal and shortened load histories were alternatively applied every time the incremental crack length value of 2 mm was reached. The incremental value of 2 mm was determined considering the plastic zone size at the crack tip. Afterwards, similar tests were conducted using only the shortened load history in order to evaluate the crack growth behaviour for crack lengths ranging from 12 to 90 mm. The obtained results were compared with crack growth models. 4. Experimental results 4.1. The effect of two types of load histories on crack growth behaviour Fig. 6 shows the crack propagation test results obtained using the normal and shortened load histories. The horizontal axis denotes the number of the applied sequences for both the normal and shortened load histories. Here, 80,000 cyclic loadings are counted as one period in the case of the normal load history, and 16,000 cyclic loadings are counted as one period in the case of the shortened load history. As can be seen in Fig. 6, the experimental data obtained using both load histories are fitted, maintaining continuity. In order to clearly evaluate the propagation behaviour, the increment of crack length per period is calculated as shown in Fig. 7. The crack length increment curve obtained using the test data for the normal load history corresponds to that of the test data for the shortened load history. The ΔK value induced by a σdy equal to 80 MPa, the loading cycles of which are reduced in the case of the shortened load history, exceeds the ΔKth value after the crack length reaches 20 mm. Moreover, the ΔK value induced by a σdy equal to 50 MPa, the loading cycles of which are omitted in the case of the shortened load history, exceeds the ΔKth value after the crack length reaches 50 mm. However, the obtained results indicate that the impact of the normal load history on crack growth behaviour is equivalent to that of the shortened load history. These results demonstrate that crack propagation ceased below a certain stress amplitude regardless of the ΔKth value of the material. Our previous report indicated that the crack growth behaviour under constant amplitude loading could be estimated using the crack growth model considering ΔK or ΔKth [10]. However, the obtained results indicate that the crack growth model obtained under constant amplitude loading, which is expressed as the relationship between da/dN and ΔK or ΔKth, offers a conservative estimation of the residual lifetime of railway axles, which are subjected to variable amplitude loading, as shown in the previous literature [22,25–27]. 4.2. Crack propagation test using shortened load history As can be seen in the previous section, utilizing the shortened load history instead of the normal load history provides an equivalent impact on crack growth behaviour. Therefore, crack propagation tests were conducted using only the shortened load history for a crack ranging from 12 to 90 mm in length. Fig. 8 shows the crack length of the tested axle as a function of equivalent running distance. In Fig. 8, one period of the shortened load history was converted into a running distance of 210 km. A crack with a length of 12 mm was propagated to 90 mm after 500,000 km in equivalent running distance. Fig. 9 shows the relationship between crack length and its increment value per 1000 km of equivalent running distance. The results of our previous report indicated that the crack growth behaviour of full-scale axles under constant amplitude loading may be fitted in accordance with the following equation, employed by Klesnil and Lukáš [33]:
Fig. 7. Comparison of crack length increment value per period. The values obtained using the normal load history show a good correspondence with those obtained using the shortened load history. 147
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Fig. 8. Crack propagation test result using shortened load history.
Fig. 9. Relationship between crack length and its increment value per 1000 km in equivalent running distance.
da = C ( Km dN
Kthm),
(1)
where material parameters C = 1.7 × 10−13, m = 3.9 and ΔKth = 8.7 MPa·m1/2 obtained using full-scale axle in the previous report [10] are employed, requiring da/dN to be expressed in units of m/cycle. Fig. 10 shows the comparison between the obtained result shown in Fig. 9 and a crack growth model using Eq. (1). Note that the vertical and horizontal axes are displayed on a logarithmic
Fig. 10. Comparison of crack length increment values for test data and the crack growth model using Eq. (1). 148
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scale. The crack length increment values estimated using Eq. (1) are three to five times higher than those of the test data, regardless of crack size. This is because crack propagation ceased below a certain stress amplitude regardless of the ΔKth value, as mentioned in Section 4.1, whereas the crack growth model that employs Eq. (1) assumes that any stress amplitude that exceeds ΔKth contributes to crack propagation. In order to assess crack growth behaviour under variable amplitude loading within this range, a crack growth model considering an effective stress intensity factor range, ΔKeff, is adopted. ΔKeff is given by:
K eff = Kmax
(2)
K op,
where Kmax and Kop are the maximum applied stress intensity factor and crack opening stress intensity factor, respectively. Considering crack closure phenomena under variable amplitude loading, which is based not on a long-term block loading condition but instead on operational loading, the crack opening stress would be affected by the maximum applied stress of the load. However, there are no articles which investigate Kop for SFA640, although some articles have been published for steel grade used in European railway network [21,34]. Therefore, the relationship between Kmax and Kop for SFA640 steel was obtained using middle-tension specimens under tension–compression loading, as shown in Fig. 11. Kop was obtained using unloading elastic compliance method [35]. Middletension specimens were sampled from the tested full-scale axle. In other words, the dispersion due to material batches is not considered here. Kop increases with increasing Kmax, although there is some variation in the fit of the data. The crack growth behaviour using the shortened load history was determined in accordance with the following equation:
da m = C ( K eff dN
m K eff , th ),
(3) −12
where material parameters C′ and m′ are re-evaluated and set to 1.5 × 10 and 3.4, respectively. ΔKeff,th is the threshold of ΔKeff, and was obtained using ΔKth in Eq. (1) and the relationship shown in Fig. 11. Note that a physically short crack is not considered in this study, so that ΔKeff,th corresponds to that of long crack and was set to 6.0 MPa·m1/2. A crack growth model using Eq. (3) was compared with the crack propagation test results. As can be seen in Figs. 10 and 12, the crack length increment values estimated using Eq. (3) offer an improved correspondence with the experimental results compared to those estimated via Eq. (1) under the operational time–load history. As can be seen in Eq. (3), the crack growth model assumed here does not consider crack growth retardation or acceleration effects under variable amplitude loading. However, considering the fact that the crack growth rate is estimated with a wide margin of error, incorporating the dispersion of material constants such as C or m values in the damage tolerance analysis, Eq. (3) may be used to estimate the crack growth behaviour of axles under variable amplitude loading for practical applications. 5. Discussion As described in Section 4.2, Kop is affected by Kmax of the whole load–tome history, i.e., the crack opening stress is affected by the maximum applied stress of the whole load–time history. The K value for individual σdy in the load–time history and corresponding ΔKeff are expressed by the subscript “dy” to clarify the following explanation. The relationship between Kdy and ΔKeff,dy is given by:
K eff , dy = K dy
(4)
K op.
ΔKeff,dy depends on the individual σdy in the load–time history, although Kop depends on the σmax of the whole load–time history. As an example, we may examine the condition that the Kmax value at a σmax of 150 MPa is equal to 15 MPa·m1/2 and Kop is equal to 3 MPa·m1/2. If the stress suddenly decreases to 100 MPa, the Kdy value at σdy = 100 MPa becomes 10 MPa·m1/2, however, Kop remains at 3 MPa·m1/2 and ΔKeff,dy is equal to 7 MPa·m1/2. However, if σmax is equal to 120 MPa, the Kop value decreases to 2.5 MPa·m1/2. As a result, the ΔKeff,dy at σdy = 100 MPa becomes 7.5 MPa·m1/2. In this manner, even if Kmax (i.e., σmax) decreases, ΔKeff,dy might increase
Fig. 11. Relationship between maximum stress intensity factor, Kmax, and crack opening stress intensity factor, Kop, for the tested material. 149
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Fig. 12. Comparison of crack length increment values for the test data and crack growth model using Eq. (3).
as a consequence of the decreasing Kop. In order to validate the crack growth model, considering the interplay of ΔKeff and ΔKeff,th, a crack growth test was conducted using a modified stress–frequency distribution with a maximum stress of 120 MPa, as shown in Fig. 13. In the modified load history, the stress amplitudes ranging from 130 to 150 MPa of the shortened load history are converted to 120 MPa. In other words, the number of cycles for stress amplitudes ranging from 120 to 150 MPa of the shortened load history is equal to that of 120 MPa in the modified load history. Fig. 14 compares the resulting crack growth curves. Although the sum of the fatigue damage obtained using the modified load history with σmax = 120 MPa is less than that of the shortened load history with a σmax of 150 MPa, the crack tested using the modified load history grew more rapidly than that using the shortened load history. This outcome demonstrates the effectiveness of the crack growth model that considers ΔKeff and ΔKeff,dy in predicting residual lifetime or appropriate inspection intervals for axles subjected to variable amplitude loading. 6. Conclusion In the present study, crack growth behaviour under variable amplitude loading was assessed through crack propagation tests of full-scale axles. A four-point rotary bending facility, which controls the applied bending moment using an electro-hydraulic actuator and permits tests based on an operational load–time history, was employed. The following conclusions were reached. (1) The omission method was proposed in order to reduce fatigue testing time, and the shortened load history, in which the cumulative number of cycles corresponds to one-fifth of the operational load history, was generated. A crack propagation test using operational and shortened load histories was conducted. The result indicates that utilising either of these load histories has an equivalent impact on crack growth behaviour. The results also indicate that the crack growth model obtained under constant amplitude loading, which is expressed as the relationship of crack growth rate (da/dN) and stress intensity factor range (ΔK) or its
Fig. 13. Stress–frequency distribution of the shortened and modified load histories. The stress amplitude ranging from 130 to 150 MPa of the shortened load history is converted to 120 MPa in the case of the modified load history. 150
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Fig. 14. Comparison of crack growth curve using the shortened (σmax = 150 MPa) and modified (σmax = 120 MPa) load history. The crack tested using the modified load history grows more rapidly than that using the shortened load history, although σmax of the former is less than that of the latter.
threshold value (ΔKth), offers a conservative estimation as to the residual lifetime of railway axles, which are subject to variable amplitude loading. (2) The crack propagation tests using the shortened load history was conducted for a crack with a length ranging from 12 to 90 mm in order to assess crack growth behaviour under operational loading condition. The crack growth model that considered the effective stress intensity factor range (ΔKeff) and its threshold value (ΔKeff,th) offers a good correspondence with the experimental test results in practical terms. (3) In order to validate the crack growth model considering ΔKeff and ΔKeff,th, the crack growth test was conducted using a modified shortened load history with a maximum stress of 120 MPa and the result was compared to that of the test data obtained using the shortened load history with a maximum stress of 150 MPa. The outcomes indicate the effectiveness of the crack growth model that considers ΔKeff and ΔKeff,dy in predicting residual lifetime or appropriate inspection intervals for axles subjected to variable amplitude loading. The influence of operational time–load histories that have a different stress–frequency distribution than that employed in the current study will be the subject of future work. We will also examine the effectiveness of the proposed crack growth model when used with various different operational load–time histories. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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