A study into crack growth in a railway wheel under thermal stop brake loading spectrum

Engineering Failure Analysis 25 (2012) 280–290

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A study into crack growth in a railway wheel under thermal stop brake loading spectrum D. Peng a,⇑, R. Jones a, T. Constable b a b

CRC for Rail Innovation, Department of Mechanical and Aerospace Engineering, Monash University, P.O. Box 31, Monash University, Victoria 3800, Australia Asset Engineering, Operational Excellence, QR National, RC 1-11, 305 Edward Street, Brisbane 4000, Queensland, Australia

a r t i c l e

i n f o

Article history: Received 14 November 2011 Received in revised form 21 May 2012 Accepted 28 May 2012 Available online 9 June 2012 Keywords: Stop braking Nonlinear stress analysis Stress intensity factor Fatigue crack growth

a b s t r a c t This paper provides a method for solving thermal fatigue crack growth in the rail wheel under stop braking spectrum. The analysis was performed in three stages. In the first stage, a finite element model of the rail wheel is used for all braking applications. For each application, a non-linear thermal stress analysis is performed. The second stage of the sequential analysis is carried out to calculate the stress intensity factor of thermal cracks, by using a semi-analytical solution technique that involves the use of an analytical solution combined with a numerical algorithm to assess fracture strength. In the third stage, a generalised Frost–Dugdale approach has been used to modelling thermal fatigue crack growth. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Railway wheels, used in freight service, perform three functions: support the cars, steer the cars and serve as brake drums. Thermal loading has various types, but it is generally a product of braking. Railway cars for both passengers and freight are quite commonly braked by using blocks in Australia, which contribute to the thermal load on the rail wheel. During wheel tread braking, heat generated by frictional forces is distributed as severe thermal gradients within the near surface or more gradually over the wheel rim and plate, depending on the intensity of the heating source [1,2]. The stresses experienced by the railway wheel during service are due to mechanical and thermal loads. Rail wheel failures in service have been reported by Michael and Sehitoglu [3], Stone and Carpenter [4], Sakamoto and Hirakawa [5], Kwon et al. [6] and Ramanan et al. [7]. The fatigue life of wheel treads has a strong bearing on the economy and safety of rail transportation. An understanding of fatigue mechanisms and a prediction of lifetimes are of interest to both manufacturers and operators. The thermal damage caused by braking can be classified into two major categories – damage from the thermal input distributed around the circumference of the tread from friction during on-tread braking, and that associated with the extreme, thermal input that occurs locally between wheels and rails during skids caused by wheels locking. Moyar and Stone [8,9] used a multiaxial fatigue criterion developed by Fatemi and Socie [10] to quantify fatigue damage induced at the running surface. According to Moyar and Stone, no fatigue damage is induced at the surface during free running of a cold wheel. When the brakes are applied and the temperature rises, the fatigue strength of the material drops. Also, the induced shear stress range and maximum normal stress are increased. This will increase the fatigue damage. According to their paper, the thermal cycles play a fundamental role in crack nucleation and in the growth of the crack until the threshold value of the equivalent stress intensity factor is reached. Also, thermal cycles play an important role in the generation of ⇑ Corresponding author. E-mail address: [email protected] (D. Peng). 1350-6307/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfailanal.2012.05.018

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residual stress fields. The same point is also remarked in [11] when thermal loading is the dominant cause of fatigue, the resulting surface cracks tend to be radial. Some tests and experimental studies of hot spotting phenomenon have been reported in [12]. In their paper, the damage analysis results based on linear damage rule within braking block have also been provided. A simplified elastic analysis of an idealised braked rail–car wheel subjected to periodic brake-shoe thermal shock, rail chill and realistic tractive rail contact stresses has been used to demonstrate the important thermal contributions to surface fatigue cracking using a critical plane fatigue initiation theory [13]. Contact region fatigue of railway wheels under combined mechanical rolling pressure and thermal loads has been extensively studied by Lunden [14]. This problem has been considered as an axisymmetric model. The mechanical load acting on the wheel was approximated as a time-variant axisymmetric pressure. This pressure was calculated based on the Hertzian contact between the wheel and the rail. The combined mechanical and thermal loads are realised in two ways. Specified numbers of mechanical loading cycles are followed by one thermal cycle. Thermal cycles have been applied as disturbances during the execution of a specified mechanical loading programme. However, Lunden [14] has not addressed certain issues in his work. Mechanical load calculations do not account for the lateral load that is realised by the wheels due to track irregularities and perturbations. The effect of friction between the wheel and rail during contact and the contact region stress, which promote plastic deformation in the tread surface, have not been considered. Further, the effect of combined thermal and mechanical loads on a three-dimensional model has not been addressed. A fatigue crack propagation prediction models based on fracture mechanics method has been referenced in [15,16]. In these papers, Hertz theory has been used to calculate wheel/rail contact stress field and an axisymmetric finite element model has been employed to obtain thermal distribution. A superposition principle [17] has been applied to estimate the approximate local maximum stress intensity factor. Then, a cycle by cycle procedure based on Paris law is used to calculate cracks propagation. This paper presents the results of a study into methods for estimating the thermal fatigue life of the rail S-shape plate rail wheel under braking loading. The thermal crack growth model considered in this approach is focused on the thermal input distributed around the circumference of the tread from friction during on-tread braking. This study consists of the following areas of analysis. For each application in the thermal loading spectrum, a 3D finite element nonlinear transient analysis based on heat transfer principle has been used for estimating the temperature variation of the rail wheel. The sequential nonlinear static analysis calculates thermal stress distribution in the rail wheel. A simple and accurate formula [18–25] for the stress-intensity factors associated with surface has been employed in the present paper [18–25]. Solutions to stress-intensity factors can then be obtained for a variety of cracks using the original finite element analysis quickly and easily. An equivalent block method, based on the Generalised Frost–Dugdale approach [26–39], was used to modelling crack growth.

2. Thermal fatigue crack growth model 2.1. Thermal load and stress analysis During the braking process, the friction generated by the brake-shoe on the moving tread produces a heating rate. The heat generation at the braking shoe-wheel interface and heat transfer to the rail is shown schematically in Fig. 1. The coefficient Qwheel(=Qshoe) is the average instantaneous braking (friction) power (kW) and Qrail is indicated the heat into rail from wheel/rail contact surface. A spectrum of braking loads that would typically be applied to the wheel will be analysed in this paper over a trip, see Fig. 2. There are 29 applications in this thermal loading spectrum (we have named it spectrum no.1). The further details may be found in [40].

Wheel Braking shoe

Qshoe

Qwheel

Moving

Qrail

Rail

Fig. 1. Graphical representation of the rail chill effect.

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Fig. 2. A spectrum of braking loads.

As the wheel rotates, there are heating and cooling periods for the wheel tread. Conduction in the rail wheel itself occurs during wheel tread braking process. The unsteady heat conduction is governed by the equation [41,42]

r2 T ¼

qcp @T _  qv j @t

ð1Þ

In which q = material density, cp = specific heat, j = thermal conductivity q_ v = heat generated during the phase transformation from austenite. As the wheel rotates, there are heating and cooling periods for the wheel tread. The condition of two thermal contact interfaces of wheel–rail and brake-wheel are very complex. Many factors, such as roughness of the surfaces, oxides, lubricant, organic material and sand, may cause a nonzero thermal contact (imperfect contact). In this analysis, interface thermal contact resistance has been ignored and a widely used assumption [8,43–48] of perfect thermal contact between the two surfaces has been accepted. At high Peclet number, the detailed distribution of the fast moving heat sources in the wheel tread is not important. For each application in the thermal spectrum, average braking power load was applied to the tread region of the wheel around its circumference. The heat source description was based on braking at a constant deceleration rate from initial speed to a full stop. A linear ramp loading pattern is used in non linear transient analysis, similar in [49]. In this analysis, a justifiable assumption that only 90% of heat generated goes into the railway wheel has been used. In order to account for the dissipation of thermal heat to the surroundings, a convectional boundary condition is applied to the surface of the wheel. As the wheel rotates, tread surface is subjected to periodic heating and cooling. A method to account for the cooling influence from the rail is transformed to convection cooling to the area of wheel–rail contact. As a first approximation, the Hertz’s theory is used to evaluate contact area [50] in this paper. The rail convection heat transfer coefficient is taken from [41,46]. In this approach, the transient cool down with a convection (in the air) boundary condition is applied to the surface, where the area of wheel/rail and wheel/braking shoe is excluded. These temperatures were used to calculate the resulting thermal stress field in the wheel following thermal loading. 2.2. Stress intensity factor and crack growth calculation method The present paper use a semi-analytical solution technique that involves the use of an analytical solution combined with a numerical algorithm to assess fracture strength.

h i ad qa KI ¼ K1 I F e þ ðF s  F e Þe

ð2Þ

Here K 1 I is the infinite body solution developed by Vijayakumar and Atluri [52], ad is constant, q is the local curvature, a is a length of the surface elliptical crack; Fe and Fs are the boundary correction factors are taken from Newman and Raju [53]. A generalised Frost and Dugdale [54] crack growth law has been used to estimate the surface crack growth. This law presented in Jones et al. [26–39], namely:

da=dN ¼ C  að1m 



 =2Þ



ðDKÞm

ð3Þ

Where C and m are constants. Here it should be noted that da/dN is a function of both a and DK (which when combined give the crack growth a DK dependence with a correction that makes the crack growth history log-linearly (i.e. exponentially) dependent on crack depth). Jones et al. [31] subsequently expanded the generalised Frost and Dugdale law (Eq. (2)) to account for R-ratio as follows:

D. Peng et al. / Engineering Failure Analysis 25 (2012) 280–290

 n da=dN ¼ C  að1n=2Þ DK g ðK max Þð1gÞ

283

ð4Þ

Where C, n and g are constants. Kmax is the maximum value of the stress intensity factor in a block. This formulation has been subsequently supported by measuring the fractal dimension of a wheel specimen supplied by QR National. Here it has been found that the fatigue surface has a fractal dimension D of approximately 1.2. This finding has been further substantiated via fractal dimension measurements on cracking, see Fig. 3 and Table 1. These results are particularly important as they imply that laws based on the Paris based crack growth law are inconsistent with the physical processes driving crack growth. The advantage of approaching the problem in this way negates the need to model the crack explicitly in the finite element model. In this way a crack of any size can be chosen arbitrarily in the original finite element model. As the crack is not modelled explicitly a coarser mesh can be used minimising the number of degrees of freedom, and thereby improving analysis time. Solutions to stress-intensity factors can then be obtained for a variety of cracks using the original finite element analysis quickly and easily.

3. FEM model and results analysis 3.1. Mesh, boundary conditions and thermal loads In this section, two numerical results of a study into methods for estimating the fatigue crack growth in a 920 mm diameter rail S-shape plate rail wheel associated with stop braking and drag braking are presented. The influence on the crack growth in the rail wheel with rail chill effect has been investigated. The half wheel cut face has been constrained to construct the finite element model. The resultant mesh has 57,980 nine-noded elements and 63,096 nodes (with 189,288 degree of freedom) by using software FEMAP [56], see Fig. 4. A sufficient fine mesh is taken in the vicinity of the wheel–rail contact for putting cool flux to model the rail chill effect. This model uses symmetry boundary conditions in the XY plane. All other surfaces of the wheel exposed to the atmosphere are considered with convective heat transfer, except for the symmetry boundary condition surface and tread surface. The thermal load band was 66 mm wide, to reflect the use of a brake shoe. In service conditions, a small band of the wheel tread was loaded in short fast repeated cycles. The thermal load corresponds to approximately a gross bogie load of 128 tonne.

3.2. Material properties In this analysis, the material is assumed to be a high carbon steel wheel AAR grade B. For the initial verification study, the material properties are assumed to be equivalent to Microalloyed AAR class B wheel steel and are extracted from [51]. The room temperature yield strength of the material is set at 800 MPa, material stiffness at 206 GPa, possions ratio at 0.286 and density at 7870 kg/m3. The material thermal properties used are specific heat – 490 J/kg °C, coefficient of thermal expansion – 14  105, thermal conductivity – 47.5 W/m °C and free convection heat transfer coefficient – 25  106 W/°C m2. For the non-linear analysis, the effect of temperature on the stress strain characteristics is considered. The variations of the yield strength and the elastic–plastic property with temperature have been shown in Fig. 5. The coefficient of thermal expansion, thermal conductivity and specific heat also varied with temperature.

Fig. 3. Location of fractal dimension measurements.

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Table 1 Fractal dimension D associated with in-service cracking. 1

2

3

4

5

6

7

8

9

10

11

1.194 1.193

1.204 1.185

1.212 1.286

1.167 1.180

1.190 1.242

1.312 1.323

1.342 1.372

1.306 1.339

1.212 1.179

1.202 1.254

1.267 1.234

Fig. 4. 3D mesh of the 920 mm rail wheel.

900 800

Yield Strength (MPa) Elastic Modulus (GPa) Plastic Modulus (MPa x 10)

Stress (MPa)

700 600 500 400 300 200 100 0 200

300

400

500

600

700

Temperature (0C) Fig. 5. Variation in stress characteristics over predicted temperature range.

Thermal Transient Analysis 200 180 160

Temperature (0C)

X Y

140 120 100 80 60 40 20 0 0

20

40

60

80

100

120

140

Time (Seconds) Fig. 6. Maximum tread region temperature of the wheel versus braking time for applications.

LT3 LT1 LT2 LT4 LT5 LT6 LT7 LT8 LT9 LT10 LT11 LT12 LT13 LT14 LT15 LT16 LT17 LT18 LT19 LT20 LT21

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Fig. 7. rz Stress distribution local detail of the temperature cooling to 25 °C.

Thermal Hoop Stress

Depth Below Tread Surface (mm)

-100 0.00

-50

0

50

100

150

200

250

300

350

400

450

-10.00

-20.00

-30.00

-40.00 Average Braking Power = 50.4 KW -50.00

-60.00

(MPa) Fig. 8. Hoop stress distribution along T to T0 of the temperature cooling to 25 °C.

3.3. Heat transfer transient analysis The thermal load was applied for a duration time, after which, the temperature variations throughout the wheel were calculated by using nonlinear transient analysis [57]. The ambient temperature was set at 25 °C. The temperature distribution on the tread throughout the wheel of the current model in different braking time was shown in Fig. 6. Note that there are eight applications were not included in Fig. 6 as those applications may have a litter effect on the life calculation. Two meshes, with 189,288 (mesh 1) and 729,534 (mesh 2) degree of freedom respectively, were employed to check the convergence of the rail wheel temperature under thermal load results. The maximum difference between those meshes for the temperature was less than 1%. 3.4. Non-linear thermal stress analysis The sequential analysis uses the temperatures in the wheel from nonlinear transient analysis to calculate the resulting thermal stress field by using a Newton–Raphson method to do nonlinear static stress analysis [57]. In this analysis, 50 increments have been used to carry out non-linear analysis. Fig. 7 shows the rz stress profile through the wheel during the thermal loading cycle at periods of cooling to 25 °C for application LT1. Add the number of increment to 60, there were almost no

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Stress viz. Temperature 250

Temperature (0C)

200

150

100

50

0 100

150

200

250

300

350

400

450

500

Stress (MPa) Fig. 9. Maximum rz stress viz. temperature for applications in the thermal loading spectrum.

Braking Loading Spectrum No.1 100.0 90.0

Percentage (%)

80.0

70.0 60.0 50.0

40.0 30.0 20.0

10.0 0.0 Loading Distribution (%) Repeat Times

1 29.4 1

2 55.4 2

3 67.0 3

4 93.0 1

5 100.0 1

6 71.8 1

7 62.7 1

8 46.6 1

Application Number Fig. 10. Duty cycles for braking loading spectrum no.1.

different from current model. In convergence check, for the Von-Mises stress and maximum principle stress results, difference between mesh 1 and mesh 2 were less than 2%. It was verified that present mesh was sufficiently accurate for estimating the thermal stress of the wheel during tread braking. The thermal loading cycle that the wheel is subject to through its life results in what is termed a stress reversal within the material. Over time, the cyclic loading causes the residual stress state of the wheel to change from one of compressive stress to tensile stress. This is due to the small amounts of plastic deformation, which occur during each thermal loading cycle. The material is unable to recover from the plastic deformation, caused through thermal expansion when cooling occurs. As cooling occurs from peak temperature down to room temperature, tensile stress occurs in area 1. Whilst in area 2 and 3, tensile stress occurs as temperature progresses from room temperature up to peak level. In area 1, the maximum stress level is around 400 MPa and in area 2 and 3, the maximum stress level is around 70 MPa. The stress distribution along T to T0 is represented by Fig. 8. For each application, the maximum temperature and corresponding stress at tread is outlined by Fig. 9. Maximum rz stress viz. temperature for applications in the thermal loading spectrum is shown in Fig. 9. There are eight applications did not occur in this figure as their maximum thermal stress were lower than 50 MPa. Only duty cycles in braking spectrum were thus considered in this paper given there are virtually no difference in the computed fatigue life and that the running time is markedly reduced. A stress threshold can be applied to additionally remove applications in the thermal loading spectrum where this estimated stress is below a threshold. This sets an absolute limit such as a 50 MPa fatigue limit. The results of the duty cycle analysis for the braking spectrum no.1 is provided in Fig. 10. From this analysis it is observed that some applications can be omitted from the duty cycle analysis as the damage due to these applications is minimal, i.e. less than 0.03% of the total damage.

D. Peng et al. / Engineering Failure Analysis 25 (2012) 280–290

Fig. 11. The positions of the postulated cracks.

Z Y

γ

P

a

Q’ c

o

Crack Plane

α β

X Q

Fig. 12. Semi elliptical surface crack, QQ0 is a free edge.

Crack Length viz. Number of Block 45.0

Crack Length (mm)

40.0 35.0 30.0 25.0

Crack Length

20.0

Crack Depth

15.0 10.0 5.0 0.0 0

500

1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 6500

Number of Block Fig. 13. Crack propagation results for the wheel under braking loading spectrum no.1 at location 1.

3.5. Fatigue life estimating The material (micro-alloyed class B) properties [55] have been used to calculate crack growth in this paper is:    

n = 3.0.

g = 1.0. C = 3.38  1012. Initial crack length: ai = 0.05 mm; ci = 0.2 mm.

287

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Crack Length viz. Number of Blocks 20.0 18.0

Crack Length (mm)

16.0 14.0 12.0 10.0 8.0 6.0 Crack Length Crack Depth

4.0 2.0 0.0 0

5000

10000

15000

20000

25000

Number of Blocks Fig. 14. Crack propagation results for the wheel under braking loading spectrum no.1 at location 2.

Crack Length viz. Number of blocks 20.0 18.0

Crack Length (mm)

16.0 14.0 12.0 10.0 8.0 6.0 Crack Length

4.0

Crack Depth 2.0 0.0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Number of Blocks Fig. 15. Crack propagation results for the wheel under braking loading spectrum no.1 at location 3.

Table 2 Crack growth results for the wheel under braking loading spectrum no.1. Case

ai (mm)

ci (mm)

af (mm)

cf (mm)

NLife (years)

1

0.05 1 0.05 0.05

0.2 1 0.2 0.2

12.4 12.4 15.0 15.0

20.3 20.3 18.3 18.2

19.4 11.5 >50 >50

2 3

 Utilisation = 200,000 km/year (average annual usage).  The definitions associated with these RCF induced initial surface defects are shown in Fig. 11.  Orientation in this paper: a = b = c = 0 (degree), see Fig. 12. Three cases of thermal fatigue induced crack growth were considered. The thermal fatigue locations associated with these three cases were set at points 1 (case 1), 2 (case 2) and 3 (case 3), see Figs. 13–15. Here, one block includes all the duty cycles in braking loading spectrum no.1. The resultant crack growth results are given in Table 2. These results appear to be in qualitative agreement with fleet experience.

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4. Conclusions This paper has shown how heat transfer methods and fracture mechanics tools can be used to predict the growth of thermal fatigue induced cracks. In the paper we have considered three possible locations of crack growth. It can be concluded that under service conditions, the alterative thermal stress resulting in tensile circumferential stresses can be enhanced fatigue crack initiation and growth. In area 2 and 3, during pure thermal loading, no failure is likely to occur. It should be noted that no mechanical loading and residual stresses (due to the manufacturing process) are included in this analysis. Acknowledgement This work was funded by the Commonwealth Research Centre for Railway Innovation through CRC project BR11. References [1] Hackenberger DE, Lonsdale CP. An initial feasibility study to develop a wayside cracked railroad wheel detector. In: 1998 ASME/IEEE joint railway conference, America; 1998. p. 65–76. [2] Cole I. 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