Fatigue crack growth of a railway wheel

Engineering Failure Analysis xxx (xxxx) xxx–xxx

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Fatigue crack growth of a railway wheel Daniel F.C. Peixoto, Paulo M.S.T. de Castro⁎ Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

AB S T R A CT Typically, fatigue crack propagation in railway wheels is initiated at some subsurface defect and occurs under mixed mode (I–II) conditions. For a Spanish AVE train wheel, fatigue crack growth characterization of the steel in mode I, mixed mode I–II, and evaluation of crack path starting from an assumed flaw are presented and discussed. Mode I fatigue crack growth rate measurement were performed in compact tension C(T) specimens according to the ASTM E647 standard. Three different load ratios were used, and fatigue crack growth thresholds were determined according to two different procedures. Load shedding and constant maximum stress intensity factor with increasing load ratio R were used for evaluation of fatigue crack growth threshold. To model a crack growth scenario in a railway wheel, mixed mode I–II fatigue crack growth tests were performed using CTS specimens. Fatigue crack growth rates and propagation direction of a crack subjected to mixed mode loading were measured. A finite element analysis was performed in order to obtain the KI and KII values for the tested loading angles. The crack propagation direction for the tested mixed mode loading conditions was experimentally measured and numerically calculated, and the obtained results were then compared in order to validate the used numerical techniques. The modelled crack growth, up to final fracture in the wheel, is consistent with the expectation for the type of initial damage considered.

1. Introduction The paper presents a study of the fatigue crack growth in a high speed train wheel. Its aims are to evaluate the fatigue crack growth (FCG) properties of the material, and to illustrate their use in the prediction of FCG in the wheel, in the presence of an assumed initial defect. Two main issues are encountered when studying fatigue of Spanish AVE high speed train wheels: (i) near threshold fatigue crack growth (FCG), (ii) mixed mode FCG. Fatigue crack growth tests involve cyclic loading of notched specimens which have been fatigue pre-cracked. During these experimental tests the crack length (a) is recorded as a function of the number of cycles (N). The obtained results can be represented as a function of the crack-tip stress-intensity factor range (ΔK), in da/dN = f(ΔK) plots providing results independent from the geometry. This enables comparison of results obtained from a variety of specimen configurations and loading conditions assuming the similitude concept which implies that cracks of different lengths, subjected to the same nominal ΔK value, will grow by equal increments of crack extension per cycle [1]. Several studies of fatigue crack growth rates in wheel and rail steels are published in the technical literature. Among these, El-

⁎

Corresponding author. E-mail address: [email protected] (P.M.S.T. de Castro).

http://dx.doi.org/10.1016/j.engfailanal.2017.07.036 Received 31 March 2017; Received in revised form 28 July 2017; Accepted 31 July 2017 1350-6307/ © 2017 Published by Elsevier Ltd.

Please cite this article as: Peixoto, D.F., Engineering Failure Analysis (2017), http://dx.doi.org/10.1016/j.engfailanal.2017.07.036

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Shabasy and Lewandowski [2] present the effect of changes in load ratio R (min./max load ratio), and test temperature on the fatigue crack growth behavior of fully pearlitic eutectoid steel. This study revealed a significant effect of load ratio on the Paris law slope for a given test temperature and an increase in ΔKth as the test temperature decreases. The fatigue crack growth threshold is the asymptotic value of ΔK at which da/dN approaches zero, or according to [1] da/ dN < 10–10 m/cycle, and if the stress intensity factor for a given crack is below the threshold value, the crack is assumed to be nonpropagating. At near-threshold levels, several factors, such as microstructure, environment, loading condition, and crack size, significantly affect crack propagation rates, [3]. The fatigue crack growth threshold ΔKth is experimentally defined in [1], where a load reduction methodology is applied. Using this technique it is observed that ΔKth decreases as the (positive) load ratio is increased. Crack closure is generally considered to be the main reason for the load ratio effect on the fatigue threshold value in metallic materials [4]. This could be explained by the fact that the methodology described in the ASTM E647 standard uses a load reduction technique where the maximum and minimum loads are reduced, and, when the threshold is being reached, during the unloading process the crack will close first at some point along the wake or blunt at the crack tip, reducing the load effect at the crack tip [5,6]. According to this interpretation, the K-decreasing (ASTM E647) methodology leads to overestimates of FCG threshold since the load is shed in steps and the amount of crack-wake plastic deformation produced during a test is directly related to the magnitude of previously applied loads leading to remote plasticity-induced crack closure, which could generate artificially high threshold values, [5]. An experimental study on two structural steels (normalized C45 and 25CrMo4 grades) conducted by Carboni and Regazzi to determine the influence of the adopted technique onto the ΔKth value, [7], lead to conclude that in the threshold region, traditional approaches based on K-decreasing tests tend to systematically overestimate the ΔKth. Despite of this, the test method defined by ASTM is the only standardized test designed to produce the range of fatigue crack thresholds. Among others, the constant Kmax with increasing Kmin method, [8], was implemented to solve this problem, as this maintains high R–ratio levels that keep the entire crack open. Single-mode loading rarely occurs in practice. Under mixed mode loading conditions a crack will deviate from its original direction, e.g. Biner, [9]. Several researchers indicate that rolling contact fatigue cracks are subjected to mixed mode I and II loading cycles, see e.g. Wong et al., [10]. Wheel shelling and rail squats are examples of defects originated in cracks that cause loss of large pieces of metal from wheel treads and rail head as a result of wheel-rail rolling contact fatigue. Fatigue tests performed to obtain the fatigue crack growth rate under the mixed loading (mode I–II) can be helpful to increase safety and reduce railway industry costs related with maintenance of wheels and rails. Fatigue cracks tend to deviate so that the crack plane is perpendicular to the maximum principal stress direction. This was shown for mixed I + II mode for which the initial crack plane was perpendicular the minimum principal stress direction or under shear only loading, see e.g. Qian and Fatemi, [11]. Different specimen geometries and testing methodologies have been used to perform mixed mode tests. Some examples of specimens that can be used to perform mixed mode tests are the compact tension and shear specimen, [12], three- or four-point bending specimens with an offset edge crack [13], plate specimen with inclined central or edge crack loaded under tension, as e.g. [14]. These specimens were developed to be tested on uniaxial testing machines. However, there is the possibility to use in-plane biaxial testing machines specially designed to perform this type of tests, and in this case the most used specimen is the cruciform specimen with central crack, see e.g. [15]. Until now there is no standard methodology for mixed mode testing, making it difficult to compare experimental results from different specimen geometries or testing apparatus. Wheel and rail materials were tested by Akama [16] using an in-plain biaxial testing machine. Wong et al., [10] investigated the mechanics of crack growth under non-proportional mixed mode loading using cruciform specimens made by BS 11 normal grade rail steel tested in a biaxial testing machine. The fatigue crack growth behavior under mixed mode of a 60 kg rail steel, commonly used as a railroad track in Korea, was experimentally investigated by Kim and Kim, [17]. The authors reported that fatigue crack growth rate under mixed mode is slower than under mode I, and this difference decreases with the increase of the load R-ratio. In this study a special loading device proposed by Richard, [12] was used to obtain the mixed mode loading on a uniaxial testing machine. Tanaka, [18] presented a study on sheet specimens of aluminum in which the mixed mode is obtained by using an initial crack inclined to the tensile axis. Tests in compact mixed mode specimens (CTS) were carried out by Borrego et al. in AlMgSi1-T6 aluminum alloy, for several values of KI/KII, [19]. AISI-304 stainless steel samples were tested under mixed-mode I–II loading conditions using Compact Tension Shear Specimen (CTS) by Biner [9]. To evaluate the characteristics of mixed mode fatigue crack propagation, it is necessary to introduce a comparative, equivalent, stress intensity factor KV that considers the effect of mode I (KI) and II (KII) simultaneously. Several equivalent stress intensity factors have been proposed along times. Among them those presented by Tanaka, [18], Richard, [12,20], Richard/Henn [21] and Henn et al. [22], Tong et al. [23] and Yan et al. [24]. Tanaka [18] dealt with the FCG behavior under mixed mode loadings using the KV as presented in Eq. (1), which was derived from the dislocation model for fatigue crack propagation proposed by Weertman, [25].

KV =

4

KI4 + 8KII4

(1)

Tong et al. [23] and Yan et al. [24] combined mode I and mode II loadings based on maximum tangential stress criterion (MTS) proposed by Erdogan and Sih, [26], as:

KV = KI cos3

θ θ θ − 3KII cos2 sin 2 2 2

(2) 2

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where θ is the initial branch crack angle. Richard, [12,20], proposed another comparative stress intensity factor KV:

KV =

1 1 KI + KI2 + 4(ζKII )2 2 2

(3)

where ζ denotes the fracture toughness ratio (KIc/KIIc). According to Richard/Henn [21], and Henn et al. [22], the comparative stress intensity factor KV is calculated as:

KV =

KI 1 KI2 + 6KII2 + 2 2

(4)

In this work the subsurface crack propagation under rolling contact fatigue (RCF) of a train wheel is modelled with particular attention dedicated to the propagation direction of the crack at every increment of its length. Wheel shelling and rail squats are examples of defects originated in cracks that cause loss of large pieces of metal from wheel treads and rail head as a result of wheel-rail rolling contact fatigue. It was found that a crack would turn to the direction perpendicular to the higher tensile load if it was initially perpendicular to the lower tensile load. Under shear only loading, the crack turned to the direction perpendicular to the maximum principal stress, Qian and Fatemi, [11]. As rolling contact induces complex non-proportional mixed mode conditions at crack tips, the evolution of mode I and mode II stress intensity factors was followed along the loading cycle. Considering the results and numerical methodology validated by the experimental work on mixed mode fatigue crack propagation, [27], the Erdogan and Sih [26] maximum tangential stress criterion (MTS) was used to calculate the crack propagation direction along the crack tips loading cycle. The commercial finite element package ABAQUS was used to build and analyze the model. It can be assumed that the crack is far enough to be out of the near surface layer that is heavily plastically deformed by rolling contact. According to this, linear elastic fracture mechanics concepts can be considered and the crack propagation can be analyzed under these assumptions, Dubourg and Lamacq, [28]. The paper is organized as follows: after the present introduction, mode I FCG and threshold evaluation are carried out. A study of mixed mode I–II follows, and finally an approximate modelling of the crack path for an assumed initial flaw in a wheel is presented. 2. Mode I FCG and threshold Fatigue crack growth tests according to the ASTM E647 standard [1] were performed from the non-propagation, near-threshold regime, up to limits related to the extent of plasticity ahead of the crack tip, on the steel of an AVE used train wheel that reached the geometrical limits for continued usage. Firstly, chemical composition, mechanical properties, microstructure and hardness measurements, were performed on samples taken from the wheel. The chemical composition, presented in Table 1 was obtained by optical emission spectrometry. The microstructure of the wheel steel was observed using an optical microscope. A 2% nital solution was used as a reagent to obtain contrast between the different phases that can be identified in this type of materials. The pearlitic microstructure is shown in Fig. 1. Several hardness measurements were performed in a wheel cross-section sample to obtain the hardness distribution. The hardness varies along the wheel profile as shown in Fig. 2, where the location for the microstructure examination (Fig. 1) is shown. The location for the microstructure sample resulted from better usage of available material and cuts; it is foreseen that microstructure does not vary. At that location, the average hardness is indicated in Fig. 1. Analyzing the obtained hardness distribution and the location where the wheel material samples used to observe the microstructure were taken, shown in Fig. 2, it can be concluded that the higher hardness value is due to the fact that the sample was taken from an out-of-service wheel in a zone where the material was hardened by usage. Tensile tests according to NP EN 10002-1 standard were made with the studied materials, using round cross-section specimens with 10 mm diameter. The specimens were extracted from the wheel tire in the tangential direction and from the rail head in the longitudinal direction. The monotonic mechanical properties are listed in Table 2. The analysis revealed a ER7/ER8 grade wheel steel according to the EN 13262 standard. C(T) specimens according to the standard ASTM E647, Fig. 3, were used because this type of specimen has the advantage of requiring a smaller amount of material than the M(T) specimen. The 23.4 mm long and 0.3 mm thick notch was opened by electric discharge machining (EDM). Fig. 4 shows the extraction specimens from the wheel; a) shows the extraction of C(T) specimens, and b) concerns CTS specimens, to be discussed later. Two different orientations for the C(T) specimens taken from the wheel were chosen; in one of them the notch is oriented in the Table 1 Wheel and rail material chemical composition [% weight]. C

Mn

Si

P

S

Ni

Mo

Al

Cr

Cu

0.49

0.74

0.25

0.01

< 0.005

0.18

0.06

0.03

0.26

0.12

3

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Fig. 1. Wheel steel microstructure.

Fig. 2. Wheel hardness distribution.

Table 2 Mechanical properties of wheel steel. Young modulus, [GPa]

Yield strength, [MPa]

Tensile strength, [MPa]

Elongation [%]

Reduction in area [%]

197

503

859

18

51

radial direction (R) and in the other the notch is oriented in a direction parallel to the tangent of the rolling surface of the wheel (T). The ASTM E647 standard demands that before the test a fatigue pre-crack must be made in order to provide a sharpened fatigue crack which ensures that the effect of the machined notch is removed from the K-calibration, and the effects on subsequent crack growth rate data caused by changing crack front shape or pre-crack load history are eliminated. The pre-crack length is a function of specimen dimensions and, according to the ASTM E647 standard, for the considered geometry it must be at least 1.3 mm. 4

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Fig. 3. C(T) specimen dimensions [mm].

In the present study two different methodologies were used to characterize the fatigue crack propagation behavior in the Paris and near threshold regimes. K-increasing with constant load range (ΔP) method was used in the Paris law regime. Fatigue crack growth rates can be calculated from the experimental data a vs. N using two approaches: i) the secant or ii) the incremental polynomial method. Both methods are suitable for the K-increasing, constant load test. However, for the K-decreasing tests where force is shed in decremental steps the secant method is recommended. All FCG tests were performed using an MTS servo-hydraulic machine of 100kN capacity, at room temperature. A sinusoidal waveform was used and three different load ratios R = 0.1, R = 0.4 and R = 0.7 were tested in the Paris law regime. R = 0.1 and R = 0.4 load ratios were studied in the more time consuming tests for near-threshold behavior characterization. The used servo-hydraulic machine was equipped with two traveling microscopes in order to measure crack length variations down to 0.01 mm, using digital rulers and 20 × magnifiers. Because a visual measurement technique was used, the average value of the two surface crack lengths was considered to make all calculations, as recommended in the followed standard. To calculate the stress intensity factor range (ΔK) the stress intensity factor calibration presented in the ASTM E647 standard for C(T) specimens was used, as follows:

ΔK =

ΔP 2+α (0.886 + 4.64α − 13.32α 2 + 14.72α3 − 5.6α 4 ) B W (1 − α )3 2

(5)

where: α = a/W, a: crack length; W: specimen width; B: specimen thickness; ΔP: load range. This calibration is valid only when a/W ≥ 0.2, which was verified in the presented tests. Table 3 gives the Paris law parameters for the several R values and crack orientations considered. Concerning near threshold regime, the K-decreasing test procedure, described in the ASTM E647 standard, and the constant Kmax with increasing Kmin procedure were used to characterize fatigue crack propagation near the threshold. According to the ASTM E647 standard the K-decreasing test procedure is suited for rates below 10− 5 mm/cycle. The constant Kmax with increasing Kmin method was used in comparison to the K-decreasing test procedure, in order to evaluate which is able to obtain the more conservative value of threshold. The K-decreasing procedure is started by load cycling at a stress intensity factor range (ΔK) and maximum stress intensity factor (Kmax) levels equal or greater than the terminal pre-cracking values. Subsequently, forces are decreased (shed) as the crack grows until the lowest ΔK or crack growth rate of interest is achieved. This test method is described in detail by the ASTM E647 standard to obtain the fatigue crack growth rates in the near-threshold. During the force shedding procedure the normalized K-gradient value was kept algebraically equal or greater than - 0.08 mm− 1:

K gradient =

1 dΔK ⋅ > − 0.08mm−1 ΔK da

(6)

as recommended by the ASTM E647 standard. This requirement aims at ensuring that the step under a given load range does not correspond to such a large crack length variation (Δa) increment that the ΔK of the previous step is exceeded. This will prevent 5

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Fig. 4. Location for specimen extraction; a) C(T); b) CTS.

Table 3 Paris law parameters, and R2 correlation coefficient (ΔK in MPa √ m, and da/dN in mm/cycle). R ratio

0.1 0.4 0.7

Notch orientation

Radial Tangential Radial Tangential Radial Tangential

Paris law constants (fitting) C

m

R2

6.33E − 10 2.29E − 10 3.16E − 09 9.89E − 09 1.24E − 10 1.05E − 08

3.48 3.87 3.17 2.87 4.41 2.84

0.984 0.992 0.993 0.997 0.992 0.980

anomalous data resulting from reductions in the stress-intensity factor and associated transient growth rates. Force shedding of adjacent steps does not exceeded 10% and new fatigue crack growth rates were calculated only after a minimum crack extension of 0.50 mm, in order to avoid effects associated with discrete variations of load range: during the extension of up to 0.50 mm no crack growth data is recorded since it may be influenced by transient growth phenomena. For the K-decreasing tests where force is shed in decremental steps the secant method was used to calculate the fatigue crack 6

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Fig. 5. Mode I fatigue crack growth rates.

growth rates. Three different load ratios R = 0.1, 0.4 and 0.7 were tested, as the value of the threshold is function of the load ratio when the K-decreasing test is used. As an alternative to the K-decreasing method, the constant Kmax with increasing Kmin methodology – just briefly mentioned in section 8.7 ‘Alternative K-control test procedures’ of ASTM E647 - was used to determine the fatigue crack growth thresholds. For the constant Kmax with increasing Kmin test procedure, the applied loads are changed to maintain the Kmax value constant, while the Kmin value is increased until the crack growth rate reaches 1 × 10− 7 mm/cycle, the value specified by the ASTM E647 standard for the threshold. Similarly to the K-decreasing test, during the constant Kmax with increasing Kmin test the normalized K-gradient value was kept algebraically equal or greater than - 0.08 mm− 1. As it could be expected, since the Kmax is kept constant and the Kmin is increased along the test, the load ratio increases as the Kmin is increased. This behavior enables to obtain a threshold value that is independent from the load ratio. Peixoto and Ferreira give in [29] further details of the specimens' location in the wheel. The ΔKth values were calculated from the load and crack length at which the crack growth is < 0.1 mm per 106 cycles (da/ dN < 10− 7 mm/cycles). Fig. 5 shows the obtained near-threshold fatigue crack growth rates for the K-decreasing test and the Paris law regime results. Although the secant method was used for near threshold data, and polynomial technique for the Paris law regime, a good agreement in the transition between the two regimes is noticed. The obtained ΔKth values are shown in Table 4 for the K-decreasing test and in Table 5 for the Kmax with increasing Kmin test. For completeness of data presentation, the final value of Kmax, the Kmax th, is also included in both tables, [30]. The data presented is based on two techniques using positive R-ratio values both for the pre-cracking as well as for the actual FCG threshold measurement. The obtained results were compared with those presented in [31]. The comparison of the ΔKth was based on load ratio (R) as shown in Table 6. Given the direct comparability of the obtained results and data of Table 6, lower, more conservative, values were obtained in the

Table 4 ASTM E647 K-decreasing test results. R

ΔKth [MPa m1/2]

Kmax

0.1 0.4 0.7

9.25 6.19 3.42

10.28 10.32 11.41

7

th

[MPa m1/2]

K-gradient [1/mm] −0.080 −0.080 −0.070

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Table 5 Constant Kmax with increasing Kmin test method results. Rstart

Rth

ΔKth [MPam1/2]

Kmax

0.5

0.67

4.05

12

th

[MPa m1/2]

K-gradient [1/mm] − 0.075

Table 6 ΔKth results from [31]. Material

R

ΔKth [MPa·m1/2]

Cast wheel (rim-radial) Cast wheel (rim-circumferential) Forged wheel (rim-radial) Forged wheel (rim-circumferential)

0.1 0.1 0.1 0.1

12.97 12.65 12.07 12.18

present work. These differences may however be due to differences in the chemical composition or wheel fabrication process and the specific method of measuring the crack length, in [31] the compliance technique using COD gauges – an indirect method – was used to measure the crack length, and the load shedding procedure was used. 3. Mixed mode fatigue crack growth Mixed-mode I–II fatigue crack growth tests were conducted on Compact Tension Shear (CTS) specimens with thickness B = 9 mm, width W = 90 mm and an initial notch of length an = 42.5 mm and 2 mm thickness opened by EDM, see Fig. 6. The specimen dimensions are: width: 90 mm; height: 148 mm; hole diameter 13.5 mm; b = 54 mm; c = 54 mm. These specimens were taken from the wheel with the notch oriented in the radial direction, as shown in Fig. 4b. These experiments were performed using the mentioned MTS machine with the loading device shown in Fig. 7. This apparatus is based on the

Fig. 6. CTS specimens: applied load and loads on specimen holes; b = 54 mm; c = 54 mm.

8

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Fig. 7. Experimental set-up for fatigue crack growth measurements.

mixed mode testing technique proposed by Richard, [12]. Before testing, the specimen surfaces were polished manually in order to facilitate the measurement of the crack length. This loading device allows to apply pure mode I, pure mode II, as well as mixed mode loading to the CTS specimen using a uniaxial testing machine just by changing the loading angle between the longitudinal axis of the specimen and the load direction applied by the testing machine. The pre-cracking of CTS specimens was performed under mode I loading with a sinusoidal waveform until an a/W ratio of 0.55 was achieved. During pre-cracking the load range was decreased in steps of 20%. During the pre-cracking and until the loading device is rotated to obtain the mixed mode loading, the mode I and mode II crack tip stress intensity factors can be calculated as Richard, [12]:

KI =

KII =

F πa cos α WB 1 − W a

0.26 + 2.65(a (W − a)) 1 + 0.55(a (W − a)) − 0.08(a (W − a))2

(7)

F πa cos α WB 1 − W a

−0.23 + 1.40(a (W − a)) 1 − 0.67(a (W − a)) + 2.08(a (W − a))2

(8)

where F is the applied load, B is the thickness of specimen, and α is the applied loading angle in radians. These equations are valid only in the range of a/W between 0.55 and 0.7. Due to small amount of wheel material only 3 different angles were tested: 30°, 45° and 60°. All tests were conducted in air and at room temperature, in load control mode and the load ratio for all loading angles and pre-cracking was kept constant and equal to 0.1. The loads were applied with a sinusoidal waveform. Because a visual measurement technique was used, the average value of the two surface crack lengths was considered to make all calculations. Two traveling microscopes were used to measure the crack length variations down to 0.01 mm, using digital rulers and 20 × magnifiers. However, these traveling microscopes only allow to measure precisely the crack length in the horizontal direction, so along the mixed mode tests only the horizontal component of the crack propagation was measured. In these circumstances, the crack profile was drawn using a microscope equipped with two digital micrometers, and the fatigue crack growth data was calculated using trigonometry considering the angle of propagation determined from the drawn crack profile and from the loading device rotation angle, as shown in Fig. 6. The stress intensity factor solutions presented in Eqs. (7) and (8) are only adequate to be used for the pre-crack, before the crack suffers any deflection due to mixed mode loading. Therefore, a numerical analysis was performed in order to obtain KI and KII stress intensity factors and the initial crack deflection angle under mixed mode fatigue loading. Plane stress quadrilateral elements were used to build a 2D finite element model of the tested CTS specimens. Due to lack of loading and geometry symmetry a complete model of the specimens was built. The ABAQUS 6.12-3 commercial finite element package was used to build and analyze the model. To apply the load and the boundary conditions reference points (RP) were positioned at the center of the specimen holes and then each one was “coupled” to its respective hole. ABAQUS commands “*COUPLING” and “*DISTRIBUTING” were used. This coupling 9

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Fig. 8. Mixed mode fatigue crack growth rates and comparison with mode I.

constraint in ABAQUS provides coupling between a reference point and the nodes located on the hole surface by using weight factors that “distribute” the applied load or constraint to the hole circumference nodes. The uniaxial load F is related with loads applied on the holes as follows, Richard, [12]:

(

c b

sin α

)

(

c b

sin α

)

⎧ F1 = F6 = F 12 cos α + ⎪ F2 = F5 = F sin α ⎨ ⎪ F3 = F4 = F 1 cos α − 2 ⎩

(9)

The geometrical parameter, β, depicted in Fig. 6 is the crack growth direction under mixed mode load. For every experimentally measured point the corresponding KI and KII factors were calculated using finite element models and the Richard/Henn criterion, Richard et al. [21] and Henn et al. [22], was used to calculate the equivalent stress intensity factor (KV), see Eq. (4). The mixed mode fatigue crack growth rates obtained are graphically shown in Fig. 8 as da/dN = f(ΔKV), Peixoto [27]. The obtained Paris law constants, C and m, for mixed mode loading were obtained by fitting the results using a power function as:

da = C (ΔKV )m dN

(10) 2

These constants, including the correlation coefficient R , for the tested loading angles are listed in Table 7. Fig. 8 shows a comparison between the obtained mixed mode fatigue crack growth with the mode I fatigue crack growth rates, [27]. Table 8 shows a comparison between the experimentally measured and numerically calculated crack propagation angle beta for the tested mixed mode loading angle α. The numerical crack propagation angle was calculated using the Erdogan and Sih equation [26], as:

KI sin β + KII (3 cos β − 1) = 0

(11)

To apply this equation the KI and KII factors were calculated using Eqs. 7 and 8 respectively. The considered experimental propagation angle is the average of measurements on both surfaces of the specimen. 4. Modelling of fatigue crack propagation of a railway wheel The subsurface crack initiation and propagation under rolling contact fatigue (RCF) in wheels is studied e.g. in [32–35]. In the present work, subsurface crack path under RCF is modelled with particular attention to the propagation direction of the crack at Table 7 Paris law constants for the mixed mode loading. (ΔK in MPa √ m, and da/dN in mm/cycle). α

C

m

R2

30° 45° 60°

2.87E −11 7.49E −11 3.24E −11

4.81 4.45 4.82

0.748 0.765 0.739

10

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Table 8 Comparison between the experimentally measured and numerically calculated crack propagation angle β. (MTS: maximum tangential stress criterion). α β

30° 26° 25° 23°

Numerical MTS (ABAQUS) Experimental

45° 37° 36° 34°

60° 49° 48° 46°

every increment of its length. Wheel shelling and rail squats are examples of defects originated in cracks that cause loss of large pieces of metal from wheel treads and rail head as a result of wheel-rail rolling contact fatigue. It was found that a crack would turn to the direction perpendicular to the higher tensile load if it was initially perpendicular to the lower tensile load. Under shear only loading, the crack turned to the direction perpendicular to the maximum principal stress, Qian and Fatemi [11]. Considering the results and numerical methodology validated by the experimental work on the mixed mode fatigue crack propagation, [27], the MTS criterion, [26], was used to study the crack propagation direction along the crack tips loading cycle. The commercial finite element package ABAQUS was used to build and analyze the model. As rolling contact induces complex non-proportional mixed mode conditions at crack tips, the evolution of mode I and mode II stress intensity factors was followed along the loading cycle. It can be assumed that the crack is far enough to be out of the near surface layer that is heavily plastically deformed by rolling contact. According to this, linear elastic fracture mechanics concepts can be considered and the crack propagation can be analyzed under these assumptions, Dubourg and Lamacq, [28]. 4.1. Finite element model The commercial finite element package ABAQUS 6.12-3 was used to build and analyze the 2D model. To improve the performance of the simulation, it was decided to build a different part were the crack will growth and were the mesh is more refined apart from wheel model and then this part was “tied” to the wheel. This construction is shown in Fig. 9. Plane strain quadrilateral with 8-node elements (CPE8) were used to build a 2D finite element mesh of the wheel. Since the

Fig. 9. a) Finite element model construction; b) Un-cracked model mesh build with 841009 nodes and 278474 quadratic quadrilateral elements of type CPE8.

11

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Table 9 Un-cracked model results. μ

FN [kN]

11,5

0,10

P0 [GPa]

acontact [mm]

1,39

5,52

Tresca criterion ZS [mm]

τTresca

3,98

905,10

max

[MPa]

objective of this work is to simulate the propagation of a subsurface crack in the wheel, the rail was modelled as a rigid line. Singular elements with nodes at quarter-point positions were considered at the crack tip. As the crack will be loaded in compression it was necessary to use self-contact formulations to avoid interpenetration of the crack faces node. In this study the penalty method was considered as contact enforcement in the crack faces contact. No hydrodynamic or entrapment fluid effect or interfacial crack friction was considered between crack faces. As boundary conditions, the rail was fixed and the wheel was loaded against the rail by a vertical force of 11.5 kN and translated 40 mm in small increments. This translation associated with the friction force generated by the friction makes the wheel to turn around its geometric center that is free to rotate. The penalty method was also used has contact enforcement on the contact between the wheel and the rail and a friction coefficient of μ = 0.1 was considered. The material was assumed to be homogeneous, isotropic with linear elastic behavior. The elastic properties considered were Young modulus E = 210 GPa and Poisson ratio ν = 0.3. 4.2. Methodologies Since the mesh refinement of the wheel and cracked parts (terminology of Fig. 9) is substantially different, it was decided to perform an analysis of the contact stresses distribution on an un-cracked model to verify if that transition has any important influence on the stress field. Convergence was confirmed. Table 9 shows some variables as the applied normal force (FN), the considered friction coefficient (μ). Other variables obtained with the un-cracked model are also listed as the maximum Hertz pressure (P0), the contact width (acontact), the maximum value of the Tresca stress (τTresca max) and the depth at which it occurs (Zs). The depth at the maximum value of the Tresca stress occurs (Zs) is an important variable for this work, as the initial crack was positioned at that depth. From the un-cracked model, the same applied load and the friction coefficient were used in the cracked model to guarantee the same contact conditions. To exemplify the analysis methodology, the propagation from an initial macroscopic crack subsurface crack is presented. The crack is 10 mm long and located at the depth of the maximum value of the Tresca stress, in this case at 4 mm. In this crack path modelling initiation and the early propagation are not addressed. At every increment mode I and mode II stress intensity factors were calculated at each crack tip. The maximum tangential stress criterion, available in the software ABAQUS, was used to calculate the direction of the crack extension, 1 mm, at each crack tip. For homogeneous, isotropic elastic materials the direction of cracking propagation can be calculated using the maximum tangential stress criterion as

θ = cos−1

2 2 2 2 ⎛ 3KII + 4KI + 8KI KII ⎞ 2 2 ⎜ ⎟ KI + 9KII ⎝ ⎠

(12)

The Richard/Henn criterion, [21], and Henn et al., [22], was considered to calculate the equivalent stress intensity factor (KV), as in Eq. (4) above

KV =

KI 1 KI2 + 6KII2 + 2 2

(13)

It was defined that the process of increasing the crack will be repeated until the maximum mixed mode equivalent stress intensity factor reaches the threshold or reaches a value that can be considered that is within the unstable crack propagation zone. 5. Results The obtained crack path is shown in Fig. 10. The evolution of KI, KII and Kv during one rotation of the wheel is exemplified by Fig. 11, concerning the initial crack situation. In Fig. 11, ‘load distance’ means the distance, measured in the rolling direction, between the contact point and the initial crack center. In Table 10 the obtained crack propagation angles are listed; these angles are

Fig. 10. Crack propagation path.

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Fig. 11. Stress intensity factors evolution during load passage, initial crack; a) left tip; b) right tip.

Table 10 Crack propagation angles; (crack length is overall length, not projected length). Crack length [mm]

Left crack tip angle [°]

Right crack tip angle [°]

10 12 14 16 18 20

67 − 58 − 53 61 − 58 − 54

− 58 62 − 60 68 − 43 59

measured relatively to the horizontal plane. In this case the calculation process ended when the maximum mixed mode equivalent stress intensity factor reached a value considered in the unstable propagation zone, see Fig. 12, which implies that the crack will growth rapidly until it reaches the wheel surface; in this case, considering the propagation angle the crack will reach the wheel tread. In Fig. 12 the maximum value of KV recorded for each crack length is represented as a function of crack length. As can be observed in the presented results, the crack changes its direction from approximately + 60° to −60° at every increment of the crack length. This will promote a very irregular crack surface until the final fracture occurs. As rolling contact induces complex non-proportional mixed mode conditions at crack tips, the evolution of mode I and mode II stress intensity factors was followed along the loading cycle and no dominant mode at the crack tip was observed despite of the equivalent stress intensity factor being very sensitive to the variation of the mode II stress intensity factor. Values of toughness of the order of KIc = 70 to 85 MPa √ m are assumed here, typical for railway wheels, e.g. [36]. The calculation process ended when the maximum mixed mode equivalent stress intensity factor reached a value considered in the unstable 13

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Fig. 12. KV

max

vs. crack length. (a is the actual length of the crack, not the projected length, which is smaller).

propagation zone, which implies that the crack will grow up to complete failure. Starting from an assumed crack length, the prediction of crack path up to final failure is intended to illustrate a methodology. Important approximations were involved in the analyses, and should be highlighted. At some instances during the modelled mixed mode FCG, the left or the right tip may have experienced KV values close to threshold, invalidating the use of Paris law and implying some temporary slowdown or arrest at that tip. In these circumstances, instead of a rigorous analysis, where for a given number of cycles ΔN the crack growth at each tip would be evaluated, as a simplifying assumption identical crack extension steps were considered for both tips, i.e. an average identical KV for both tips was implied in this simplification. The direction of crack propagation was predicted according to Eq. (12) for the several steps of the analysis, leading to the saw-like effect shown in Fig. 11, which illustrates the propagation modelled on the basis of the simplifying assumptions above. Given the approximate nature of the model, the sensitivity of the analysis to the crack extension step used was not studied. If detailed analyses of crack growth are required, they may be implemented taking into account the full da/dN vs ΔK curve including the threshold regime, and using steps of crack increment at the left or right tip evaluated using that experimental data; the calculation of direction of each step of crack extension, however, is the same as that used here. Further to the approximation consisting in the large crack increment considered, the initial defect considered is itself an idealized situation, for the sake of a case study. The behavior found is based on KI and KII calculations at each tip, together with a criterion for crack growth direction prediction. More precise evaluations of crack path would require an analysis of the influence of crack growth increment on the results, and a detailed analysis of the stress state in each tip region could complement the interpretation of the results. Methodologies such as X-FEM would spare the re-meshing effort, whilst also revealing features as ‘zigzag’ crack paths in mixed mode situations, e.g. [37]. 6. Concluding remarks FCG studies were conducted on specimens extracted from a Spanish AVE train wheel. Concerning mode I threshold behavior. The ASTM method (load shedding) leads to higher ΔKth values than the Kmax constant procedure. The measured threshold stress-intensity range for fatigue-crack propagation, ΔKth, decreases as the load ratio is increased. In the Kmax constant procedure it is assumed that the crack stops growing in a closure free situation due to the high R–ratio. For the high R–ratio tested according to the ASTM procedure the obtained results are very similar to the results obtained with Kmax constant procedure. A comparison of the present data with literature values indicates that the ΔKth values are approximately similar, although the present ones are slightly lower. Concerning mixed mode I–II fatigue crack growth, FCG rates and the propagation angle were evaluated on Compact Tension Shear specimens. The used apparatus was based on the mixed mode testing technique that allows to perform mixed mode loading using a uniaxial testing machine just by changing the loading angle between the longitudinal axis of the specimen and the load direction applied by the testing machine. Three different loading angles were tested 30°, 45° and 60°. Since no numerical solution exists to calculate the KI and KII values a finite element analysis was also done in order to obtain them for the tested conditions. It was observed that the fatigue crack growth direction changed immediately from the initial fatigue mode I pre-crack orientation when load direction was changed. The experimental growth direction of the cracks for different load mixities were compared with the predictions based on numerical approaches which provide a similar estimation of the crack growth direction and a good agreement with the experimental 14

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results. It was observed that for the tested ΔK range the mixed mode fatigue crack growth rates are higher than the mode I fatigue crack growth rates. Concerning modelling of the crack path of an assumed initial flaw, the commercial finite element package ABAQUS 6.12-3 was used to build and analyze a 2D model of a subsurface crack propagation on the wheel/rail contact. The maximum tangential stress criterion was used to calculate the mode I and II stress intensity factors and the crack propagation direction along the crack tips loading cycle. Particular attention was dedicated to the propagation direction of the crack at every increment of its length. As can be observed in the presented results the crack changes its direction from approximately + 60° to −60° at every increment of the crack length. This will promote a very irregular crack surface until the final fracture occurs. As rolling contact induces complex non-proportional mixed mode conditions at crack tips, the evolution of mode I and mode II stress intensity factors was followed along the loading cycle and no dominant mode at the crack tip was observed despite of the equivalent stress intensity factor being very sensitive to the variation of the mode II stress intensity factor. In this study the calculation process ended when the maximum mixed mode equivalent stress intensity factor reached a value considered in the unstable propagation zone, which implies that the crack will grow rapidly until it reaches the wheel surface. Acknowledgements Daniel Peixoto acknowledges a Calouste Gulbenkian Foundation PhD grant, number 104047-B. The Portuguese Foundation for Science and Technology FCT project PTDC/EME-PME/100204/2008 “Railways” is acknowledged. ALSTOM kindly supplied the Spanish AVE high speed train wheel for this study. The advice of Prof. J.A.M. Ferreira (Univ. Coimbra) and of Prof. L. Borrego (ISEC, Coimbra) on the CTS testing, and the help of Dr. S.M.O. Tavares (FEUP, Porto) in aspects of the revised version of the manuscript, are gratefully acknowledged. References 1 ASTM, Standard test method for measurement of fatigue crack growth rates, E647, American Society for Testing and Materials (2008). [2] A. El-Shabasy, J. Lewandowski, Effects of load ratio, R, and test temperature on fatigue crack growth of fully pearlitic eutectoid steel (fatigue crack growth of pearlitic steel), Int. J. Fatigue 26 (2004) 305–309. [3] P. Liaw, ‘Overview of crack closure at near-threshold fatigue crack growth levels’, Mechanics of fatigue crack closure, ASTM Spec. Tech. Publ. 982 1988, pp. 62–92. [4] B. Boyce, R. Ritchie, Effect of load ratio and maximum stress intensity on the fatigue threshold in Ti–6Al–4V, Eng. Fract. Mech. 68 (2) (2001) 129–147. [5] S. Forth, J. Newman Jr., R. Forman, On generating fatigue crack growth thresholds, Int. J. Fatigue 25 (1) (2003) 9–15. [6] S. Smith, R. Piascik, An indirect technique for determining closure-free fatigue crack growth behaviour, ASTM Spec. Tech. Publ. 1372 (2000) 109–122. [7] M. Carboni, D. Regazzi, Effect of the experimental technique onto R dependence of Δ Kth, Procedia Engineering, 11th International Conference on the Mechanical Behaviour of Materials (ICM11), vol. 10, 2011, pp. 2937–2942. [8] R. Davenport, R. Brook, The stress intensity range in fatigue, Fatigue Fract. Eng. Mater. Struct. 1 (1979) 151–158. [9] S. Biner, Fatigue crack growth studies under mixed-mode loading, Int. J. Fatigue 23 (2001) 259–263. [10] S. Wong, P. Bold, M. Brown, R. Allen, Fatigue crack growth rates under sequential mixed-mode I and II loading cycles, Fatigue Fract. Eng. Mater. Struct. 23 (8) (2000) 667–674. [11] J. Qian, A. Fatemi, Mixed mode fatigue crack growth: a literature survey, Eng. Fract. Mech. 55 (6) (1996) 969–990. [12] H. Richard, Bruchvorhersagen bei überlagerter Normal – und Schubbeanspruchung von Rissen, VDI Forschungsheft 631 (1985) 1–60. [13] J. Qian, A. Fatemi, Fatigue crack growth under mixed mode I and II loading, Fatigue Fract. Eng. Mater. Struct. 19 (10) (1996) 1277–1284. [14] C.A.C.C. Rebelo, A.T. Marques, P.M.S.T. de Castro, Fracture characterization of composites in mixed mode loading, in: H.C. van Elst, A. Bakker (Eds.), Fracture Control of Engineering Structures’, Proceedings of the 6th Biennial European Conference on Fracture ECF6, EMAS, 1986, pp. 2195–2204. [15] V. Richter-Trummer, X. Zhang, P.E. Irving, M. Pacchione, M. Beltrão, J.F. dos Santos, Fatigue crack growth behaviour in friction stir welded aluminium–lithium alloy subjected to biaxial loads, Exp. Tech. 40 (3) (2016) 921–935. [16] M. Akama, Fatigue crack growth under mixed loading of tensile and in-plane shear modes, Quarterly Report of RTRI 44 (2) (2003) 65–71. [17] J. Kim, C. Kim, Fatigue crack growth behavior of rail steel under mode I and mixed mode loadings, Mater. Sci. Eng. A 338 (1) (2002) 191–201. [18] K. Tanaka, Fatigue crack propagation from a crack inclined to the cyclic tensile axis, Eng. Fract. Mech. 6 (3) (1974) 493–507. [19] L. Borrego, F. Antunes, J. Costa, J. Ferreira, Mixed-mode fatigue crack growth behavior in aluminium alloy, Int. J. Fatigue 28 (2006) 618–626. [20] H. Richard, Crack problems under complex loading, in: G.S. Sih, H. Nisitani, T. Ishihara (Eds.), Role of Fracture Mechanics in Modern Technology’, Proceedings of the Int. Conf., Fukuoka, Japan, June 2–6, 1986, North-Holland, Amsterdam, 1987, pp. 577–588. [21] H. Richard, W. Linnig, K. Henn, Fatigue crack propagation under combined loading, For. Eng. 3 (1991) 99–109. [22] K. Henn, H. Richard, W. Linnig, Fatigue crack growth under mixed mode and mode II cyclic loading, in: E. Czoboly (Ed.), Fracture Analysis - Theory and Practice, 2 EMAS Ltd, Warley, 1988, pp. 1104–1113. [23] J. Tong, J. Yates, M. Brown, The formation and propagation of mode I branch cracks in mixed mode fatigue failure, Eng. Fract. Mech. 56 (2) (1997) 213–231. [24] X. Yan, D. Shanyi, Z. Zehua, Mixed-mode fatigue crack growth prediction in biaxially stretched sheets, Eng. Fract. Mech. 43 (3) (1992) 471–475. [25] J. Weertman, Rate of growth of fatigue cracks calculated from the theory of infinitesimal dislocations distributed on a plane, Int. J. Fract. Mech. 2 (1966) 460–467. [26] F. Erdogan, G. Sih, On the crack extension in plates under plane loading and transverse shear, J. Basic Eng. 85 (1963) 519–525. [27] D.F.C. Peixoto, P.M.S.T. de Castro, Mixed mode fatigue crack propagation in a railway wheel steel, Procedia Struct. Integr. 1 (2016) 150–157. [28] M. Dubourg, V. Lamacq, A predictive rolling contact fatigue crack growth model: onset of branching, direction, and growth-role of dry and lubricated conditions on crack patterns, J. Tribol. 124 (4) (2002) 680–688. [29] D.F.C. Peixoto, L.A. Ferreira, Fatigue crack propagation behavior in railway steels, Int. J. Struct. Integr. 4 (4) (2013) 487–500. [30] D.F.C. Peixoto, P.M.S.T. de Castro, Near threshold fatigue crack propagation in railways' steels: comparison of two testing techniques, Theor. Appl. Fract. Mech. 80 (2015) 73–78. [31] S. Sivaprasad, S. Tarafder, V. Ranganath, N. Parida, Fatigue and fracture behaviour of forged and cast railway wheels, 11th International Conference on Fracture, Turin, Italy, March 2005, pp. 20–25. [32] M. Guagliano, L. Vergani, Experimental and numerical analysis of sub-surface cracks in railway wheels, Eng. Fract. Mech. 72 (2005) 255–269. [33] Y. Liu, L. Liu, S. Mahadevan, Analysis of subsurface crack propagation under rolling contact loading in railroad wheels using FEM, Eng. Fract. Mech. 74 (2007) 2659–2674. [34] J. Sandström, Subsurface rolling contact fatigue damage of railway wheels – a probabilistic analysis, Int. J. Fatigue 37 (2012) 146–152. [35] D. Zeng, L. Lu, J. Zhang, X. Jin, M. Zhu, Effect of micro-inclusions on subsurface-initiated rolling contact fatigue of a railway wheel, Proc IMechE Part F: J Rail and Rapid Transit, 2014. [36] S.-J. Kwon, D.H. Lee, J.-W. Seo, H.-K. Jun, Failure analysis for power car wheels based on contact positions and tread slope, Eng. Fail. Anal. vol. 230, (2) (2017) 544–553. [37] Z. Zhuang, Z. Liu, B. Cheng, J. Liao, Extended Finite Element Method, Academic Press, 2014.

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