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Thermal cracking of railway wheels: Towards experimental validation
Tribology International 94 (2016) 409–420
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Tribology International journal homepage: www.elsevier.com/locate/triboint
Thermal cracking of railway wheels: Towards experimental validation Sara Caprioli a, Tore Vernersson a,b,n, Kazuyuki Handa c, Katsuyoshi Ikeuchi c a b c
CHARMEC / Department of Applied Mechanics, Chalmers University of Technology 412 96 Gothenburg, Sweden ÅF Industry AB, Gothenburg, Sweden Railway Technical Research Institute (RTRI), Tokyo, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 27 May 2015 Received in revised form 16 September 2015 Accepted 25 September 2015
Thermal cracking of railway wheel treads is investigated using a combined experimental and numerical approach. Results from controlled brake rig tests of repeated stop braking cycles for a railway wheel in rolling contact with a so-called railwheel are presented. Test conditions are then numerically analysed using finite element (FE) simulations that account for the thermomechanical loading of the wheel tread. For the studied stop braking case, thermal cracks are found in the wheel tread after few brake cycles. Results from thermal imaging reveal a frictionally excited thermoelastic instability phenomenon called “banding” where the contact between brake block and wheel occurs only over a fraction of the block width. This condition results in locally high temperatures. The numerical analysis assesses ratcheting response of the wheel tread material under operational conditions corresponding to two types of banding patterns and also a case of uniform heating. Fatigue life predictions are estimated from the evaluated ratcheting response using a simplified accumulation rule. Predicted lives are found to be in reasonable agreement with experimental results. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Tread braking Rolling contact fatigue Brake rig experiments Elastoplastic finite element simulations
1. Background and aim of the analysis
2. Experimental set-up
During tread braking, railway wheel treads are subjected to simultaneous impact from wheel–rail contact loads and thermally induced stresses. In previous experimental full-scale studies, it was found that thermal cracks equivalent to those observed on operating railway wheels could only be generated under combined loading from braking and wheel–rail rolling contact, in contrast to thermal cracks on brake discs [1]. Moreover, thermal cracking has been found to be influenced mainly by tangential (rather than normal) forces in the wheel–rail contact and by the build-up of residual tensile stresses caused by localized heating and subsequent cooling, see [2]. In previous numerical studies [3,4] the effect of combined thermal and mechanical loading is investigated. For cases of drag braking cycles (relatively low braking power over a long time) with tread temperatures below 300 °C, the impact from braking was studied with respect to the evolution of effective strain. Moreover, differences in response between pure rolling and tractive rolling were investigated. In the present work, numerical modelling is combined with full-scale testing in order to validate numerical models and identify conditions under which thermal cracks are induced in railway wheel treads.
Full-scale tests featuring repeated stop braking have been performed in a full-scale brake dynamometer (see Fig. 1), at the Railway Technical Research Institute (RTRI) in Japan. The test apparatus consists of an electrically powered tread braked railway wheel in rolling contact with a so-called railwheel. Flywheels are installed on the axle of the railwheel to provide required equivalent inertia. This arrangement allows the proper wheel–rail tractive forces to be transmitted via the rolling contact both at acceleration and deceleration. Each stop braking test starts with enforcing a prescribed wheel–rail contact force. The rig is thereafter accelerated to the test speed. The engine is then disconnected and the braking action is imposed by pushing the brake block towards the wheel tread with a prescribed brake force. Speed, braking torque, braking force and normal contact force between the two wheels are measured during braking. Temperatures in wheel and brake block are measured by three thermocouples in the wheel and two thermocouples in the block. These are positioned at a depth of 10 mm below the contact surfaces. Tread temperatures are also monitored using a thermocamera that observes the full tread width and about 1/10 of the circumference. At every tenth braking cycle the wheel is inspected for cracks at three cross sections using liquid penetrant and magnetic particle inspection methods, see also [5]. Repeated stop braking was performed using a single sinter brake block from an initial speed of 160 km/h with a deceleration of about 0.75 m/s2. Inertia and mechanical wheel–rail loading
n Corresponding author at: Department of Applied Mechanics / Dynamics, Chalmers University of Technology 412 96 Gothenburg, Sweden. Tel.: þ 46 31 772 8501; fax: þ 46 31 772 3827 E-mail address: [email protected] (T. Vernersson).
http://dx.doi.org/10.1016/j.triboint.2015.09.042 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
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Fig. 2. Schematic representation of thermal resistances in the wheel–block interface.
described below. The evaluated thermal loading on the wheel tread is introduced in a three-dimensional (3D) FE model of a wheel sector. In addition, the wheel sector is loaded by frictional rolling contact stress distributions corresponding to Hertz [7] and Carter [8] theories. The stress analysis is carried out using a Chaboche material model calibrated towards cyclic stress–strain data from isothermal experiments on ER7 wheel material for temperatures up to 650 °C, see [9]. All simulations are carried out in Abaqus, version 6.11–1 [10]. 3.1. Thermal analysis
Fig. 1. Set-up of brake rig employed in the test.
corresponding to 20 tonnes axle load were employed. The wheel considered in the study is a standard S-shaped wheel. The tread braking here considered corresponds to an initial brake power of 333 kW ramped down to zero in 59.3 s, load case pertinent to an Intercity passenger train. The next stop braking cycle is performed when the wheel temperature is below 60 °C. The experimental set-up and procedure are described in detail in [5]. The test conditions are as follows:
General:
○ Standard S-shaped wheel with a diameter of 855 mm and material ER7 [6]. ○ Sinter brake block with a nominal width of 80 mm. ○ Radii of curvature of the contacting bodies: n Wheel: r x;1 ¼ 855=2 mm, r y;1 ¼ 1: n Railwheel: r x;2 ¼ 1000=2 mm, r y;2 ¼ 600 mm: ○ Braking energy per stop: 10.6 MJ. ○ Moment of inertia of flywheels: 2156 kg m2. ○ Total moment of inertia: 2694 kg m2. Initial speed: 160 km/h. Nominal braking time: 59.3 s. Nominal maximum braking power: 333 kW. Initial temperature (10 mm below wheel tread): 60 °C. Heat convection transfer coefficient (in numerical simulations): h¼ 20 W °C 1 m 2 Rolling contact conditions: ○ Contact load (between wheels): 98 kN (20 tonnes axle load). ○ Maximum theoretical Hertzian pressure: 1245 MPa. ○ Theoretical contact patch size (semi-axes): a¼ 8.4 mm (rolling direction), b¼4.5 mm (lateral direction). ○ Average tangential force: 7.3 kN. ○ Ratio of tangential force to normal force: 0.075.
3. Numerical modelling Heat partitioning between brake block, wheel and rail during stop braking is evaluated in an axisymmetric thermal analysis as
The braking conditions featured in the test rig are evaluated using a numerical model previously calibrated for freight [11] and metro applications [9]. The modelling aims at reproducing the temperature distribution observed in full-scale rig experiments under given braking conditions. In short, wheel and block temperatures are evaluated in a FE simulation under the assumption of rotational symmetry (axisymmetry) accounting for cooling by convection and radiation. In addition, rail chill effect (heat transfer from the hot wheel to the cold rail) is included. The average heat input at the wheel–block interface is calculated from brake rig data as qbrake ¼
Q brake Btot Lb
ð3:1Þ
where Q brake is the brake power (in W) given by brake deceleration and brake rig inertia, Lb is the length of the brake block and Btot is the total assumed width of the wheel–block contact. The heat flux generated during the braking process is partitioned between the wheel, qw , and the block qb at the brake block– wheel interface as (see also Fig. 2) wheel T block Rb cont T cont þ qbrake Rw þ Rb Rw þ Rb
ð3:2Þ
block T wheel Rw cont T cont þ qbrake Rw þ Rb Rw þ Rb
ð3:3Þ
qw ¼
qb ¼
wheel In Eqs. (3.2) and (3.3), T block cont , Rb and T cont , Rw are the contact temperatures and thermal contact resistances of the block and the wheel, respectively. The thermal model accounts for the so-called banding during tread braking in a simplified way: contact between block and tread is assumed to take place only for prescribed areas, with total assumed width Btot , and hence heat is only generated and transferred between block and wheel tread in those areas. The heat transferred from wheel to railwheel is assessed under the assumption that the railwheel has a constant temperature during the braking cycle and that perfect wheel–rail thermal contact prevails. The heat transferred from the wheel to the railwheel is then given by rffiffiffiffiffiffi bλ av p ffiffiffi Q perfect ð3:4Þ 6:99 T Þ ðT w0 r0 wr 8π κ
where a and b are the semi-axes of the contact patch in the rolling and in the lateral direction respectively, λ the thermal conductivity,
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Fig. 3. Contact load coordinate system indicated on the mesh refined part of the wheel tread. The contact load traversal is discretized into a number of quasi-static simulations.
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In Eq. (3.5), p0 is the peak pressure, a and b are the semi-axes of the contact ellipse in the rolling and lateral direction respectively and x and y are local coordinates with x ¼0 and y¼ 0 at the centre of the contact (Fig. 3). Interfacial shear stresses are introduced in the contact assuming partial slip or a full slip [4,13]. Rolling is simulated by traversing the pre-evaluated contact stress distribution along the wheel tread in discrete steps. The FE mesh features hexahedral 20-node elements (DC3D20 for heat transfer analysis and C3D20 for structural analysis). A coarse mesh is used for the hub and web of the wheel whereas a finer mesh is introduced in the vicinity of the wheel–rail contact region, see Fig. 4. Parts with incompatible meshes are connected with surfacebased tie constraints. The toroidal shape of the densely meshed volume (following [4]) is designed to capture the stress field under Hertzian contact loading with a mesh density bias towards the centre of wheel–railwheel contact band. This improves computational efficiency and result accuracy at the centre of the wheel rim. In the transient thermal analysis, the wheel hub and the sector boundaries are modelled as thermally insulated. In the structural analysis, displacements at the sector boundaries are constrained in the circumferential direction (in a cylindrical coordinate system) and the nodes of the wheel hub are fully constrained in all directions. Note that due to computational limits, the entire Table 1 Material parameters for ER7 including initial kinematic hardening modulus C and the non-linear kinematic hardening parameter γ.
Fig. 4. Geometry of the S-shaped railway wheel sector. The mesh refined part of the wheel rim is extracted. Typical element size in this part is 1 mm.
κ thermal diffusivity, v train speed, T w0 wheel tread temperature and T r0 railwheel temperature. Dissipation of heat to the surroundings is accounted for by including convection and radiation cooling during and after the braking process. Details on the thermal model are given in [12]. The axisymmetric model of the wheel is not suitable for assessing the effect of the thermal loading when combined with the effect of mechanical loading as imposed by the wheel–rail contact [3]. Instead a 3D FE model of the wheel sector is used as detailed in the following section. The temperature distribution in the 3D model is assumed to follow rotational symmetry, i.e. temperature variations in the circumferential direction are neglected and only the average (around the circumference) temperature is considered. To this end, the heat flux evaluated from axisymmetric thermal simulations is approximated by polynomial functions over time in the 3D model. This is implemented using a Fortran user subroutine in Abaqus. 3.2. Thermomechanical finite element model The computational model for the thermomechanical analysis consists of a 30° sector of the wheel. The analyses are performed in two steps. First, temperatures during braking and cooling are calculated using prescribed heat in-flux to the wheel tread as determined by the axisymmetric thermal model. Nodal temperature histories are then applied in a structural analysis together with the mechanical rolling contact stresses. Rolling contact pressure distribution is derived using Hertzian theory as [7] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 y2 ð3:5Þ pðx; yÞ ¼ p0 1 a b
Temperature (°C) Modulus of elasticity (GPa)
Poisson's ratio (–)
Yield stress, σ0 (MPa)
C (GPa) γ (–)
0 250 300 325 350 400 500 650
0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29
275 275 275 273 255 238 188 93
155 155 155 152 146 127 80 43
195.0 195.0 193.0 189.0 185.0 183.0 177.0 144.0
615 615 615 603 621 602 548 662
Fig. 5. Experimental results and numerical predictions for the last (sixth) stress– strain cycle used in the calibration at T ¼ 350 1C. Table 2 Temperature dependent thermal properties of ER7T. Temperature (°C) Thermal conductivity, λ (W K 1 m 1) 0 200 400 600 800
43.7 44.1 39.3 32.9 25.0
Specific heat capacity, cp (J kg 1 K 1)
Thermal expansion, α ( 10 6 K 1)
440 510 570 630 700
10.8 11.8 13.5 15.8 18.5
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Fig. 6. Typical variations of (a) brake power, (b) block–wheel coefficient of friction, (c) wheel speed, (d) wheel contact load, (e) block temperatures, and (f) wheel temperatures during a brake cycle. Thermocouples 1, 2 and 3 are located 45, 75, and 15 mm from field side, respectively. Note the different time scale in the graphs.
thermomechanical braking cycle (featuring thousands of wheel revolutions) cannot be included in the numerical simulations. Instead the mechanical loading is introduced at chosen instances during the braking cycle. During such instances the temperature is kept constant.
isotropic hardening is insignificant and thus excluded. Calibrated material parameters are given in Table 1 and an example of experimental results and numerical predictions is given in Fig. 5. Temperature dependent thermal properties are given in Table 2 [9].
3.3. Material model and calibration An elastoplastic constitutive model featuring non-linear kinematic hardening following Lemaitre and Chaboche [14] as implemented in Abaqus is employed. The material model has been calibrated using results from isothermal cyclic stress–strain tests [9]. The tests are performed on specimens taken from the rim part of a ER7 railway wheel. The highest test temperature has been 650 °C. The calibration is based on the first six strain test cycles. The first two cycles were run at 0.02 Hz per cycle and the subsequent at 0.2 Hz per cycle. For the cyclic data at 400 °C, the first two cycles were run at 0.0125 Hz and the subsequent at 0.125 Hz. The calibration is made with respect to the yield strength σ0, elasticity modulus E, the initial kinematic hardening modulus C and the saturation value B1 . The non-linear kinematic hardening parameter γ is defined as γ ¼ C=B1 . For the studied material, the
4. Results 4.1. Full-scale brake rig experiment Typical variations of wheel speed, friction coefficient in the block–wheel contact, brake power, wheel contact load, wheel and block temperatures are shown in Fig. 6. The graphs show that the coefficient of friction in the block–wheel interface is increasing from about 0.2 to 0.25 at the start of the brake cycle up to about 0.35 at end. In addition, as seen in Fig. 6(a), the maximum brake power in the test is about 311 kW. This is lower than the maximum nominal brake power 333 kW due to the lower block–wheel friction coefficient at the beginning and due to the time it takes to obtain the correct brake (normal) force. After stop braking the
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the results for braking with sinter blocks clearly reveal a characteristic banding pattern on the tread. This pattern generally changes from one brake cycle to the next. One or two bands are generated and for most of the brake cycles the axial position of these bands remain fixed. During banding, the tread will locally be subjected to higher (and lower) temperatures than in a situation with evenly distributed heat over the entire block width. The evolution of crack width found at inspections is shown in Fig. 8. Corresponding pictures of the wheel tread are shown in Fig. 9. Typical thermal cracks can be seen in the part of the tread subjected to rolling contact between wheel and railwheel. These cracks were observed already after 11 brake cycles. This corresponds to an accumulated total brake energy of approximately 116 MJ. This is considerably lower than the accumulated energy (approximately 174 MJ) corresponding to crack initiation at similar experiments under lower wheel load and lower inertia, see [2]. The depth of the thermal cracks after 40 brake cycles was found to be approximately 2.0 mm. This was evaluated by stepwise turning of the wheel tread; after 2.0 mm turning only very fine discrete cracks could be observed by magnetic inspection. 4.2. Finite element analysis results
Fig. 7. Wheel tread temperatures during the final 15 brake cycles (times between brake cycles have been removed) as detected using thermocamera. Temperatures have been assessed presuming an emissivity equal to 1.0.
4.2.1. Thermal analysis Results are presented for two assumed types of contact banding. The aim of these analyses is to simulate the experimental conditions. In addition, the case of wheel–block contact over the whole block width is studied. This is an ideal case that could result if no frictionally induced instabilities (banding) were present. This case is added since it is known from previous brake rig experience using organic composite brake blocks (which commonly cause no banding) that thermal cracking in such cases takes substantially longer time to initiate. Fig. 10 shows the considered braking cases with band widths indicated on the wheel tread. The two cases of banding – one central band, or two bands – have both a total band width of 50 mm (block– wheel contact over 62.5% of tread width). The case without banding presumes thermal contact over the entire block width of 80 mm. In the following, the thermal loading cases will be denoted as a Two 25 mm wide bands. b One 80 mm wide band. c One 50 mm wide band.
Fig. 8. Crack length along the wheel tread measured at inspections after 11, 20, 30 and 40 cycles. A, B and C indicate the different measurement positions.
wheel was cooled to 60 °C during pure rolling with a wheel–rail contact load of 30 kN. The maximum temperature measured by thermocouples embedded 10 mm below the tread surface was approximately 300 °C. In addition, a thermocamera was used to evaluate variation of temperature during braking. The resulting videos are postprocessed by partitioning the tread width into 20 bands. Each band thus corresponds to a given axial position on the tread. The average temperature in each band at each time frame of the video is determined by image processing using Matlab image processing toolbox [15]. Some examples of tread temperatures evaluated from thermocamera data for an assumed surface emissivity of 1.0 are given in Fig. 7. Since the actual surface emissivity is lower than the 1.0, the temperatures of the tread are higher than indicated in Fig. 7. Moreover these temperatures are subjected to additional uncertainties due to variations in emissivity during braking. Nevertheless,
Calculated temperatures in the wheel rim at the instants in time corresponding to the maximum tread temperatures are presented in Fig. 11. It can be noted that the maximum temperature in the region of the tread traversed by the rolling contact loading is highest (467 °C) for case c. Load cases a and b result in peak temperatures of 441 °C and 336 °C, respectively. The calculated temperature histories for a point on the wheel tread and a point 10 mm below the tread during the first 100 s (stop braking 58.5 s followed by cooling) are shown in Fig. 12. The solid circles on the graphs indicate time instances when rolling contact loads are traversed along the tread in the simulations. At each such instance two wheel revolutions (rolling contact load passages) are simulated. Instance (1) gives rolling contact conditions prior to braking, instance (2) gives response after a short braking time at intermediate temperature conditions, instance (3) is time when the maximum temperature is reached and instance (4) is at end of braking. It is to be reminded that in the simulation one braking cycle is defined by brakingþ cooling time. As shown in Fig. 6(f), the maximum temperature measured by the thermocouples is about 300 1C. This is in good agreement with simulated case b (Fig. 12(b)). The other simulated cases predict slightly higher temperatures than as measured by the thermocouples.
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Fig. 9. Wheel appearance and penetrant tested tread after (a) 11 cycles, (b) 20 cycles, (c) 30 cycles and (d) 40 cycles.
4.2.2. Thermomechanical analysis In the analysis, a partial slip distribution for the tangential stresses at the wheel–rail contact is considered. The tangential force during braking is Q¼7.3 kN. In the baseline case, f ¼ 0:075 ¼ Q =P and μ ¼ 0:5. The simulations feature surface tangential stresses for a braking wheel, i.e. the shear stresses are opposite to the rolling direction, see Fig. 13. A sensitivity analysis is conducted to assess the effect of the tangential tractions by comparing this baseline case with partial slip f ¼0.075 and μ ¼ 0:3, full slip qðx; yÞ ¼ fpðx; yÞ, and pure rolling, i.e. qðx; yÞ ¼ 0. Tangential traction distributions along the centre of
the rolling contact patch are shown in Fig. 14 together with traction distributions for assumed full slip conditions. An empirical model to predict fatigue crack initiation owing to ratcheting is proposed by Kapoor [16]. The number cycles to initiation of ratcheting induced fatigue, Nr, is predicted from the ratcheting strain increment Δεr as Nr ¼
εc Δεr
ð4:1Þ
where εc is a material parameter that could e.g. be determined by twin disc tests [17]. It should be noted that since there is no
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approximated as
Δεr ¼
4 X
Δεi
ð4:4Þ
i¼1
Fig. 10. Block–wheel contact zones on wheel tread for assumed brake cases.
information on the value of the material constant εc for the material and load conditions considered in this study, a qualitative evaluation is made in which εc is kept as an unknown parameter. In the current analysis, the ratcheting is evaluated from the effective strain as (see also [18]) pffiffiffi 2 εeff ¼ ½ðεxx εyy Þ2 þ ðεxx εzz Þ2 þ ðεyy εzz Þ2 þ 6ðε2xy þ ε2yz þ ε2xz Þ1=2 3 ð4:2Þ In Fig. 15, the effective strain defined in Eq. (4.2) is evaluated along a radial path at the centre of the wheel–railwheel contact at the end of each point marked in Fig. 12. As shown, for the cases studied the maximum effective strain occurs at approximately 1 mm below the surface. As observed in Fig. 15(c), the position corresponding to the highest residual effective strain magnitude has shifted to the surface for the most severe thermal load case (case c). Fig. 16 shows the time evolution of the effective strain at depths 0.5 mm and 1.0 mm below the centre of wheel–railwheel contact band (y¼0 in Fig. 3). The results show large accumulation of strain for the rolling contact passages at instances (3) and (4) in Fig. 12. Recall that the maximum temperatures at the centre of the wheel– railwheel contact are (at instance (3) in Fig. 12) 441 °C, 363 °C and 467 °C for cases a, b and c, respectively. In general higher temperatures at positions corresponding to the wheel–railwheel contact give significantly larger ratcheting increments. Note from Table 1 that the material parameters show significant variations between 400 °C and 500 °C. This, together with the direct influence of thermal strain, explains the variation in responses between the different thermal load cases. The results from Fig. 16 are used to assess the total ratcheting strain for one complete brake cycle for the three thermal load cases analysed. To this end, the number of wheel revolutions that actually took place in the time intervals pertaining to the calculated ratcheting strains need to be accounted for. For each load cycle marked by a solid circle in Fig. 12 the ratcheting increment of the effective strain for the first wheel passage is denoted Δεi;1 . For the second wheel, it is denoted Δεi;2 and is also taken as representative for all subsequent wheel revolutions in the interval. Thus, the total ratcheting strain in the interval can be evaluated as
Δεi ¼ 1 Δεi;1 þ ðN i 1Þ Δεi;2
ð4:3Þ
where i represent time intervals corresponding to each of the mechanical load passages, i.e. time instants (1)–(4) marked in Fig. 12 and Ni is the number of wheel revolutions in the interval i, see the example in Fig. 17. During cooling, the contact load between wheel and railwheel in the experimental study is low (about 30 kN). For this reason no substantial ratcheting is assumed to be induced during this stage (cf discussion on initial shakedown below). Consequently the total effective strain increment for an entire braking cycle may be
It is informative to introduce the working point for the mechanical load cycles prior to braking in a shakedown map [19] as done in Fig. 18. It is found that the working point is not in the (surface) ratcheting area of the map, but rather well into the elastic or plastic shakedown area.1 For this reason, cycles prior to braking are not expected to give any additional plastic strain once shakedown has occurred. The numerical simulations presume an initially stress-free wheel material. In reality the wheel rim will have circumferential compressive residual stresses of about 100 MPa [6] stemming from the manufacturing process. This residual stress contributes to the shakedown. In the simulations, two rolling cycles prior to braking have been simulated to obtain an initial state of shakedown. These are thus not considered in the life analysis. Results for predicted cycles to crack initiation are given in Table 3. To compare the different thermal cases, the number of cycles for thermal case c and μ ¼ 0:5 evaluated from the accumulated strain 0.5 mm below the surface is taken as a reference life denoted Nnr . For comparison, the ratcheting fatigue life evaluated 1.0 mm below the surface is also presented. It is however not considered representative for the current damage pattern since observed cracks initiate at (or near) the surface. In addition subsurface initiation would require the presence of a material defect and likely corresponds to higher effective strain magnitudes (e.g. a higher εc ) due to the material confinement. From Table 3, it is seen that the load case corresponding to uniform heating has a predicted ratcheting fatigue life that is 7.49 times higher than the reference case (thermal case c). Taking εc 10 (cf [17]) crack initiation corresponding to ratcheting fatigue is predicted after 2 brake cycles for case a and 6 cycles for case b. Thermal load case c is predicted to cause crack initiation already during the first brake cycle. 4.2.3. Effect of tangential traction in wheel–railwheel contact For thermal load cases a and c, the previously analysed baseline case is compared to the following mechanical load cases: A Partial slip, f ¼0.075, μ ¼ 0:30: B Full slip, f ¼0.075. C Pure rolling, f ¼0. Results are presented in Fig. 19 and Tables 4–6. It is seen that for the thermal load case a the ratcheting fatigue life in pure rolling is more than 8 times the fatigue life for the mechanical baseline case under thermal load case c. Adopting εc 10, the number of cycles to initiation varies between 1 (case a–A) and 8 (case a–C). As also noted in previous studies, see e.g. [4], the results in Tables 4–6 show that the material response is rather sensitive to the presumptions regarding tangential traction distributions (partial slip vs full slip) and on the friction coefficient. In addition, experimental observations in [5] indicated that crack formation under less severe braking loads was related to high traction rolling rather than pure rolling. In the current study, differences – in particular between pure rolling and tractive rolling – are appreciable despite the low magnitude of the applied traction. 1 Inpffiffiffi the shakedown map, p0 is the peak Hertzian pressure according and k ( ¼ σ y = 3 according to the von Mises yield criterion) is the yield strength in pure shear for the work-hardened material. For the given material, σ y ( ¼ B1 þ σ 0 , where B1 ¼ C=γ ) ¼ 527 MPa at T ¼ 0 °C following material data in Table 1.
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Fig. 11. Temperature distributions at instants in time corresponding to the highest surface temperatures for (a) two 25 mm wide bands, (b) one 80 mm wide band and (c) one 50 mm wide band. The region traversed by the rolling contact load is indicated on the tread.
Fig. 12. Temperature distribution for the uppermost node at the centre of the contact (solid line) and at a node about 10 mm below the surface (dashed line) for (a) 2 bands 2×25 mm, (b) 1 band 80 mm and (c) 1 band 50 mm. The first 100 s are shown. Braking power varies according to Fig. 6(a) during the initial 58.5 s, after which the power is zero. Solid circles indicate time instances when rolling contact loads are traversed along the tread.
Fig. 13. Tangential stresses for a braking wheel.
Fig. 14. Distribution of tangential tractions.
4.2.4. Influence of FE mesh A mesh sensitivity study is carried out based on thermal load case a. A simplified analysis is performed where mechanical load passages are introduced only at the points marked (1) and (3) in Fig. 12. Two different mesh densities of the refined part are used, see Fig. 4. Data for “mesh 1” used in the previous analyses (with a
characteristic element size of 1 mm) are compared to the more refined “mesh 2” that has a characteristic element size of 0.5 mm, see Table 7. The remaining part of the wheel sector consists in both cases of 13 600 elements and 63 333 nodes. As shown in Fig. 20, good convergence of results is found for effective residual strain magnitudes at depths below some 0.5 mm. The difference in the peak value exactly at the surface is due to the resolution (discrete step size) of the translation of the tangential contact load: The traversal of the contact loads follow the bias of the mesh. The number of steps is the same in “mesh 1” and “mesh 2” but the element size in the longitudinal direction varies between the two meshes. Thus stress magnitudes at the surface are not fully resolved and results between two meshes at the surface may differ. This issue cannot be mitigated unless the mesh is extremely refined which would result in extremely high computational costs. 4.2.5. Influence of material model It is known that the material model used in the analyses may overpredict ratcheting [20]. In order to control the amount of ratcheting for higher strain levels, additional backstresses can be added in the model, see e.g. [21]. In this paper, a sensitivity analysis is conducted: one additional backstress is added to the material model and the material response is evaluated for thermal cases a and c. The evolution of the second backstress is assumed to be linear with hardening modulus C 2 ¼ C 1 =10. This allows for a better representation of the stress–strain evolution at larger strains. In fact, as seen in Fig. 21, the material response with two backstresses matches in a better way the experimental data at 7 0.6% strain.
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Fig. 15. Calculated residual effective strains after rolling contact load passages (time instances (1)–(4) in Fig. 12) for (a) 2 bands 2×25 mm, (b) 1 band 80 mm and (c) 1 band 50 mm.
Fig. 16. Effective strain evolution for a node located (a) 0.5 mm and (b) 1.0 mm below the centre of the wheel–railwheel contact.
Fig. 17. Example of assessment of ratcheting strain based on effective strain evolution for the equivalent braking cycle in the FE simulations.
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Table 3 Ratcheting fatigue life Nr expressed as a function of critical strain εc and in normalized form. Location
2 bands 2 25 mm
1 band width 80 mm
1 band width 50 mm
0.5 mm below surface Nr
εc =3:57 3:25N nr
εc =1:55 7:49N nr
εc =11:61
1.0 mm below surface
εc =3:67
εc =1:94
εc =10:40
N nr
Fig. 18. Shakedown map for actual loading conditions prior to braking.
Table 4 Mechanical load case A (f¼ 0.075, μ ¼ 0:3) predicted ratcheting fatigue life. Location
a (2 25 mm)
c (50 mm)
0.5 mm below surface Nr
εc =2:81 4:13N nr
εc =9:15 1:27N nr
1.0 mm below surface
εc =3:84
εc =10:71
Table 5 Mechanical load case B ðμ ¼ f ¼ 0:075Þ predicted ratcheting fatigue life. Location
a ð2 25 mmÞ
c (50 mm)
0.5 mm below surface Nr
εc =1:55 7:49N nr
εc =6:86 1:69N nr
1.0 mm below surface
εc =2:73
εc =8:42
Table 6 Mechanical load case C (f¼ 0) predicted ratcheting fatigue life. Location
a (2 25 mm)
c (50 mm)
0.5 mm below surface Nr
εc =1:34 8:66N nr
εc =4:29 2:71N nr
1.0 mm below surface
εc =2:48
εc =6:15
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Table 7 Mesh data and computational times a simplified brake cycle analysis. Model
Mesh 1 Mesh 2
Model data for refined part of the wheel tread Characteristic elem. size (mm)
No. of nodes
No. of elements
CPU time (h)
1 0.5
33 242 93 691
7000 21 400
109 265
Fig. 19. Effective strain evolution for node below the centre of the wheel–railwheel contact for case 2 bands 2×25 mm at (a) depth 0.5 mm below surface and (b) 1.0 mm, and for case 1 band 50 mm at (c) depth 0.5 mm and (d) 1.0 mm.
Fig. 21. Predicted response of material model with one and two backstresses together with experimental test data for ER7T (test temperature 350 1C). Last cycle only.
With εc 10, the predicted number of cycles to initiation are 5 (thermal load case c) and 10 (thermal load case a). Fig. 20. Effective residual strain magnitudes for banded wheel–block contact as calculated using two different mesh densities.
5. Conclusions and outlook As shown in Fig. 22, the material model has a significant effect on predicted ratcheting rates. For the material model with 2 backstresses it is seen that the initial rolling cycle results in (more or less) shakedown for the second rolling cycles (at room temperature) and also for the rolling cycles at the beginning of the brake cycle. This indicates that it is a reasonable choice to neglect the contributions from the first rolling cycles in the estimation of number of cycles to initiation as made in the previous section. The ratcheting strain for the brake cycle can now be seen to be dominated by the rolling cycles introduced at the time of maximum tread temperature. With εc 10, the results in Table 8 yield 5 brake cycles to initiation for case c and 10 for case a.
This paper is a first step towards full-scale test rig experimental validation of FE predictions of thermomechanical cracking. Ratcheting fatigue life based on thermomechanical numerical simulations for brake cycles is evaluated. Different thermal contact conditions between block–wheel found by post-processing of experimental data have been employed. It has been shown that one 50 mm wide thermal band is a more severe case than two 25 mm wide bands. Further, both of these thermal load cases are more severe than uniform heating over the entire brake block width (a thermal load case that was not found in the experiments). Predicted number of brake cycles to crack initiation presuming a critical ratcheting strain
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Fig. 22. Effective strain evolution for a node 0.5 mm below the tread surface for (a) thermal load case a (2 25 mm) and (b) thermal load case c (50 mm). Table 8 Predicted ratcheting fatigue lives with a modified material model. Location
a ð2 25 mmÞ
c (50 mm)
0.5 mm below surface Nr
εc =1:07 10:85N nr
εc =2:07 5:61N nr
1.0 mm below surface
εc =0:83
εc =1:65
http://www.charmec.chalmers.se in Gothenburg, Sweden and the Railway Technical Research Institute (RTRI) in Tokyo, Japan (http:// www.rtri.or.jp/eng/). Prof. Roger Lundén is acknowledged for coordinating the cooperation with RTRI. Many thanks go to Prof. Anders Ekberg at Chalmers for useful discussions and inputs during this work. Special thanks to Rebecka Brommesson at Chalmers for fruitful discussions in particular on the material modelling and her help in compiling the data for the material response with two backstresses in Fig. 21.
εc of approximately 10 was found to be in the order of 1–8 depending on thermal and mechanical (wheel–railwheel) contact conditions. Adoption of a more refined material model led to a prediction in the range 5–10 brake cycles. These predictions are in line with the experimental observations which detected 2.0 mm wide cracks in the wheel tread after 11 braking cycles. Some approximations in the current analyses tend to make the estimation conservative. Examples are the presumptions of a constant banding pattern between brake cycles (implicitly presumed in the damage accumulation) and that progressing shakedown is not captured due to the truncation of mechanical load cycles. In addition the material model has been shown to have a significant effect on the predicted ratcheting fatigue life. The current analyses can be improved by
Addition of at least one more experimental condition, i.e. braking from another initial speed.
Addition of experiments and simulations for braking with
organic composite blocks where no (or less severe) banding is expected. Further enhancements in material models and pertinent material parameters, in particular related to life predictions. In a more detailed study calibration with two or more backstresses should be considered. Including load rate effects. In [9] it has been found that viscous effects are significant at temperatures above some 500 °C. The use of more advanced material models that account for this effect can improve the accuracy of the predictions. Including compressive residual stresses from manufacturing. Evaluating low cycle fatigue (LCF) damage. Accumulation procedure for LCF should be included and LCF life should be compared with ratcheting life. The model which predicts the shortest life to fatigue is considered to determine the fatigue life [16,17]. Account for wear effects.
Acknowledgements This work is part of the activities within the Swedish National Centre of Excellence CHARMEC (Chalmers Railway Mechanics,
References [1] Dufrénoy P, Bodovillé G, Dagallaix G. Damage mechanisms and thermomechanical loading. Temp. Fatigue Interact. 2002;29:167–76. [2] Handa K, Kimura Y, Mishima Y. Surface cracks initiation on carbon steel railway wheels under concurrent load of continuous rolling contact and cyclic frictional heat. Wear 2010;268:50–8. [3] Vernersson T, Caprioli S, Kabo E, Hansson H, Ekberg A. Wheel tread damage: a numerical study of railway wheel tread plasticity under thermomechanical loading. Proc Inst Mech Eng Part F: J Rail Rapid Transit 2010;224:435–43. [4] Caprioli S, Ekberg A. Numerical evaluation of the material response of a railway wheel under thermomechanical braking conditions and Corrigendum. Wear 2014;314(319) 181–8, 245–7. [5] Handa K, Morimoto F. Influence of wheel/rail tangential traction force on thermal cracking of railway wheels. Wear 2012;289:112–8. [6] European Committee for Standardization, European Standard–EN 13262 – Railway applications – Wheelsets and bogies – Wheels – Product requirement, Technical report; 2004. [7] Hertz H. Ueber die Beruehrung fester elastischer Körper (On the contact of elastic solids). J Reine Angew Math 1882;92:156–71. [8] Carter FW. On the action of a locomotive driving wheel. Proc R Soc A: Math Phys Eng Sci 1926;112:151–7. [9] Teimourimanesh S. Thermal capacity of railway wheels [Thesis for the degree of doctor of philosophy in solid and structural mechanics]. Gothenburg: Chalmers University of Technology; 2014. [10] Abaqus. Abaqus analysis user's manual, 6.11 ed. Providence, Rhode Island, USA: Simulia; 2011. [11] Vernersson T. Temperatures at railway tread braking. Part 2: calibration and numerical examples. Proc Inst Mech Eng Part F: J Rail Rapid Transit 2007;221:429–41. [12] Vernersson T. Temperatures at railway tread braking. Part 1: modelling. Proc Inst Mech Eng Part F: J Rail Rapid Transit 2007;221:167–82. [13] Johnson KL. Contact mechanics, 9th ed. Cambridge, UK: Cambridge University Press; 1985. [14] Lemaitre J, Chaboche JL. Mechanics of solid materials. Cambridge, UK: Cambridge University Press; 1990. [15] MathWorks. Matlab image processing toolbox; 2014. [16] Kapoor A. A re-evaluation of the life to rupture of ductile metals by cyclic plastic strain. Fatigue Fract Eng Mater Struct 1994;17:201–19. [17] Ringsberg J. Life prediction of rolling contact fatigue crack initiation. Int J Fatigue 2001;23:575–86. [18] Sandström J, Ekberg A. Numerical study of the mechanical deterioration of insulated rail joints. Proc Inst Mech Eng Part F: J Rail Rapid Transit 2009;223:265–73. [19] Johnson KL. The strength of surfaces in rolling contact. Proc Inst Mech Eng Part C: J Mech Eng Sci 1989;203:151–63. [20] Chaboche JL. A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 2008;24:1642–93. [21] Bandula-Heva T, Dhanasekar M. Failure of discontinuous railhead edges due to plastic strain accumulation. Eng Fail Anal 2014;44:110–24.
Contents lists available at ScienceDirect
Tribology International journal homepage: www.elsevier.com/locate/triboint
Thermal cracking of railway wheels: Towards experimental validation Sara Caprioli a, Tore Vernersson a,b,n, Kazuyuki Handa c, Katsuyoshi Ikeuchi c a b c
CHARMEC / Department of Applied Mechanics, Chalmers University of Technology 412 96 Gothenburg, Sweden ÅF Industry AB, Gothenburg, Sweden Railway Technical Research Institute (RTRI), Tokyo, Japan
art ic l e i nf o
a b s t r a c t
Article history: Received 27 May 2015 Received in revised form 16 September 2015 Accepted 25 September 2015
Thermal cracking of railway wheel treads is investigated using a combined experimental and numerical approach. Results from controlled brake rig tests of repeated stop braking cycles for a railway wheel in rolling contact with a so-called railwheel are presented. Test conditions are then numerically analysed using finite element (FE) simulations that account for the thermomechanical loading of the wheel tread. For the studied stop braking case, thermal cracks are found in the wheel tread after few brake cycles. Results from thermal imaging reveal a frictionally excited thermoelastic instability phenomenon called “banding” where the contact between brake block and wheel occurs only over a fraction of the block width. This condition results in locally high temperatures. The numerical analysis assesses ratcheting response of the wheel tread material under operational conditions corresponding to two types of banding patterns and also a case of uniform heating. Fatigue life predictions are estimated from the evaluated ratcheting response using a simplified accumulation rule. Predicted lives are found to be in reasonable agreement with experimental results. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Tread braking Rolling contact fatigue Brake rig experiments Elastoplastic finite element simulations
1. Background and aim of the analysis
2. Experimental set-up
During tread braking, railway wheel treads are subjected to simultaneous impact from wheel–rail contact loads and thermally induced stresses. In previous experimental full-scale studies, it was found that thermal cracks equivalent to those observed on operating railway wheels could only be generated under combined loading from braking and wheel–rail rolling contact, in contrast to thermal cracks on brake discs [1]. Moreover, thermal cracking has been found to be influenced mainly by tangential (rather than normal) forces in the wheel–rail contact and by the build-up of residual tensile stresses caused by localized heating and subsequent cooling, see [2]. In previous numerical studies [3,4] the effect of combined thermal and mechanical loading is investigated. For cases of drag braking cycles (relatively low braking power over a long time) with tread temperatures below 300 °C, the impact from braking was studied with respect to the evolution of effective strain. Moreover, differences in response between pure rolling and tractive rolling were investigated. In the present work, numerical modelling is combined with full-scale testing in order to validate numerical models and identify conditions under which thermal cracks are induced in railway wheel treads.
Full-scale tests featuring repeated stop braking have been performed in a full-scale brake dynamometer (see Fig. 1), at the Railway Technical Research Institute (RTRI) in Japan. The test apparatus consists of an electrically powered tread braked railway wheel in rolling contact with a so-called railwheel. Flywheels are installed on the axle of the railwheel to provide required equivalent inertia. This arrangement allows the proper wheel–rail tractive forces to be transmitted via the rolling contact both at acceleration and deceleration. Each stop braking test starts with enforcing a prescribed wheel–rail contact force. The rig is thereafter accelerated to the test speed. The engine is then disconnected and the braking action is imposed by pushing the brake block towards the wheel tread with a prescribed brake force. Speed, braking torque, braking force and normal contact force between the two wheels are measured during braking. Temperatures in wheel and brake block are measured by three thermocouples in the wheel and two thermocouples in the block. These are positioned at a depth of 10 mm below the contact surfaces. Tread temperatures are also monitored using a thermocamera that observes the full tread width and about 1/10 of the circumference. At every tenth braking cycle the wheel is inspected for cracks at three cross sections using liquid penetrant and magnetic particle inspection methods, see also [5]. Repeated stop braking was performed using a single sinter brake block from an initial speed of 160 km/h with a deceleration of about 0.75 m/s2. Inertia and mechanical wheel–rail loading
n Corresponding author at: Department of Applied Mechanics / Dynamics, Chalmers University of Technology 412 96 Gothenburg, Sweden. Tel.: þ 46 31 772 8501; fax: þ 46 31 772 3827 E-mail address: [email protected] (T. Vernersson).
http://dx.doi.org/10.1016/j.triboint.2015.09.042 0301-679X/& 2015 Elsevier Ltd. All rights reserved.
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Fig. 2. Schematic representation of thermal resistances in the wheel–block interface.
described below. The evaluated thermal loading on the wheel tread is introduced in a three-dimensional (3D) FE model of a wheel sector. In addition, the wheel sector is loaded by frictional rolling contact stress distributions corresponding to Hertz [7] and Carter [8] theories. The stress analysis is carried out using a Chaboche material model calibrated towards cyclic stress–strain data from isothermal experiments on ER7 wheel material for temperatures up to 650 °C, see [9]. All simulations are carried out in Abaqus, version 6.11–1 [10]. 3.1. Thermal analysis
Fig. 1. Set-up of brake rig employed in the test.
corresponding to 20 tonnes axle load were employed. The wheel considered in the study is a standard S-shaped wheel. The tread braking here considered corresponds to an initial brake power of 333 kW ramped down to zero in 59.3 s, load case pertinent to an Intercity passenger train. The next stop braking cycle is performed when the wheel temperature is below 60 °C. The experimental set-up and procedure are described in detail in [5]. The test conditions are as follows:
General:
○ Standard S-shaped wheel with a diameter of 855 mm and material ER7 [6]. ○ Sinter brake block with a nominal width of 80 mm. ○ Radii of curvature of the contacting bodies: n Wheel: r x;1 ¼ 855=2 mm, r y;1 ¼ 1: n Railwheel: r x;2 ¼ 1000=2 mm, r y;2 ¼ 600 mm: ○ Braking energy per stop: 10.6 MJ. ○ Moment of inertia of flywheels: 2156 kg m2. ○ Total moment of inertia: 2694 kg m2. Initial speed: 160 km/h. Nominal braking time: 59.3 s. Nominal maximum braking power: 333 kW. Initial temperature (10 mm below wheel tread): 60 °C. Heat convection transfer coefficient (in numerical simulations): h¼ 20 W °C 1 m 2 Rolling contact conditions: ○ Contact load (between wheels): 98 kN (20 tonnes axle load). ○ Maximum theoretical Hertzian pressure: 1245 MPa. ○ Theoretical contact patch size (semi-axes): a¼ 8.4 mm (rolling direction), b¼4.5 mm (lateral direction). ○ Average tangential force: 7.3 kN. ○ Ratio of tangential force to normal force: 0.075.
3. Numerical modelling Heat partitioning between brake block, wheel and rail during stop braking is evaluated in an axisymmetric thermal analysis as
The braking conditions featured in the test rig are evaluated using a numerical model previously calibrated for freight [11] and metro applications [9]. The modelling aims at reproducing the temperature distribution observed in full-scale rig experiments under given braking conditions. In short, wheel and block temperatures are evaluated in a FE simulation under the assumption of rotational symmetry (axisymmetry) accounting for cooling by convection and radiation. In addition, rail chill effect (heat transfer from the hot wheel to the cold rail) is included. The average heat input at the wheel–block interface is calculated from brake rig data as qbrake ¼
Q brake Btot Lb
ð3:1Þ
where Q brake is the brake power (in W) given by brake deceleration and brake rig inertia, Lb is the length of the brake block and Btot is the total assumed width of the wheel–block contact. The heat flux generated during the braking process is partitioned between the wheel, qw , and the block qb at the brake block– wheel interface as (see also Fig. 2) wheel T block Rb cont T cont þ qbrake Rw þ Rb Rw þ Rb
ð3:2Þ
block T wheel Rw cont T cont þ qbrake Rw þ Rb Rw þ Rb
ð3:3Þ
qw ¼
qb ¼
wheel In Eqs. (3.2) and (3.3), T block cont , Rb and T cont , Rw are the contact temperatures and thermal contact resistances of the block and the wheel, respectively. The thermal model accounts for the so-called banding during tread braking in a simplified way: contact between block and tread is assumed to take place only for prescribed areas, with total assumed width Btot , and hence heat is only generated and transferred between block and wheel tread in those areas. The heat transferred from wheel to railwheel is assessed under the assumption that the railwheel has a constant temperature during the braking cycle and that perfect wheel–rail thermal contact prevails. The heat transferred from the wheel to the railwheel is then given by rffiffiffiffiffiffi bλ av p ffiffiffi Q perfect ð3:4Þ 6:99 T Þ ðT w0 r0 wr 8π κ
where a and b are the semi-axes of the contact patch in the rolling and in the lateral direction respectively, λ the thermal conductivity,
S. Caprioli et al. / Tribology International 94 (2016) 409–420
Fig. 3. Contact load coordinate system indicated on the mesh refined part of the wheel tread. The contact load traversal is discretized into a number of quasi-static simulations.
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In Eq. (3.5), p0 is the peak pressure, a and b are the semi-axes of the contact ellipse in the rolling and lateral direction respectively and x and y are local coordinates with x ¼0 and y¼ 0 at the centre of the contact (Fig. 3). Interfacial shear stresses are introduced in the contact assuming partial slip or a full slip [4,13]. Rolling is simulated by traversing the pre-evaluated contact stress distribution along the wheel tread in discrete steps. The FE mesh features hexahedral 20-node elements (DC3D20 for heat transfer analysis and C3D20 for structural analysis). A coarse mesh is used for the hub and web of the wheel whereas a finer mesh is introduced in the vicinity of the wheel–rail contact region, see Fig. 4. Parts with incompatible meshes are connected with surfacebased tie constraints. The toroidal shape of the densely meshed volume (following [4]) is designed to capture the stress field under Hertzian contact loading with a mesh density bias towards the centre of wheel–railwheel contact band. This improves computational efficiency and result accuracy at the centre of the wheel rim. In the transient thermal analysis, the wheel hub and the sector boundaries are modelled as thermally insulated. In the structural analysis, displacements at the sector boundaries are constrained in the circumferential direction (in a cylindrical coordinate system) and the nodes of the wheel hub are fully constrained in all directions. Note that due to computational limits, the entire Table 1 Material parameters for ER7 including initial kinematic hardening modulus C and the non-linear kinematic hardening parameter γ.
Fig. 4. Geometry of the S-shaped railway wheel sector. The mesh refined part of the wheel rim is extracted. Typical element size in this part is 1 mm.
κ thermal diffusivity, v train speed, T w0 wheel tread temperature and T r0 railwheel temperature. Dissipation of heat to the surroundings is accounted for by including convection and radiation cooling during and after the braking process. Details on the thermal model are given in [12]. The axisymmetric model of the wheel is not suitable for assessing the effect of the thermal loading when combined with the effect of mechanical loading as imposed by the wheel–rail contact [3]. Instead a 3D FE model of the wheel sector is used as detailed in the following section. The temperature distribution in the 3D model is assumed to follow rotational symmetry, i.e. temperature variations in the circumferential direction are neglected and only the average (around the circumference) temperature is considered. To this end, the heat flux evaluated from axisymmetric thermal simulations is approximated by polynomial functions over time in the 3D model. This is implemented using a Fortran user subroutine in Abaqus. 3.2. Thermomechanical finite element model The computational model for the thermomechanical analysis consists of a 30° sector of the wheel. The analyses are performed in two steps. First, temperatures during braking and cooling are calculated using prescribed heat in-flux to the wheel tread as determined by the axisymmetric thermal model. Nodal temperature histories are then applied in a structural analysis together with the mechanical rolling contact stresses. Rolling contact pressure distribution is derived using Hertzian theory as [7] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 y2 ð3:5Þ pðx; yÞ ¼ p0 1 a b
Temperature (°C) Modulus of elasticity (GPa)
Poisson's ratio (–)
Yield stress, σ0 (MPa)
C (GPa) γ (–)
0 250 300 325 350 400 500 650
0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29
275 275 275 273 255 238 188 93
155 155 155 152 146 127 80 43
195.0 195.0 193.0 189.0 185.0 183.0 177.0 144.0
615 615 615 603 621 602 548 662
Fig. 5. Experimental results and numerical predictions for the last (sixth) stress– strain cycle used in the calibration at T ¼ 350 1C. Table 2 Temperature dependent thermal properties of ER7T. Temperature (°C) Thermal conductivity, λ (W K 1 m 1) 0 200 400 600 800
43.7 44.1 39.3 32.9 25.0
Specific heat capacity, cp (J kg 1 K 1)
Thermal expansion, α ( 10 6 K 1)
440 510 570 630 700
10.8 11.8 13.5 15.8 18.5
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Fig. 6. Typical variations of (a) brake power, (b) block–wheel coefficient of friction, (c) wheel speed, (d) wheel contact load, (e) block temperatures, and (f) wheel temperatures during a brake cycle. Thermocouples 1, 2 and 3 are located 45, 75, and 15 mm from field side, respectively. Note the different time scale in the graphs.
thermomechanical braking cycle (featuring thousands of wheel revolutions) cannot be included in the numerical simulations. Instead the mechanical loading is introduced at chosen instances during the braking cycle. During such instances the temperature is kept constant.
isotropic hardening is insignificant and thus excluded. Calibrated material parameters are given in Table 1 and an example of experimental results and numerical predictions is given in Fig. 5. Temperature dependent thermal properties are given in Table 2 [9].
3.3. Material model and calibration An elastoplastic constitutive model featuring non-linear kinematic hardening following Lemaitre and Chaboche [14] as implemented in Abaqus is employed. The material model has been calibrated using results from isothermal cyclic stress–strain tests [9]. The tests are performed on specimens taken from the rim part of a ER7 railway wheel. The highest test temperature has been 650 °C. The calibration is based on the first six strain test cycles. The first two cycles were run at 0.02 Hz per cycle and the subsequent at 0.2 Hz per cycle. For the cyclic data at 400 °C, the first two cycles were run at 0.0125 Hz and the subsequent at 0.125 Hz. The calibration is made with respect to the yield strength σ0, elasticity modulus E, the initial kinematic hardening modulus C and the saturation value B1 . The non-linear kinematic hardening parameter γ is defined as γ ¼ C=B1 . For the studied material, the
4. Results 4.1. Full-scale brake rig experiment Typical variations of wheel speed, friction coefficient in the block–wheel contact, brake power, wheel contact load, wheel and block temperatures are shown in Fig. 6. The graphs show that the coefficient of friction in the block–wheel interface is increasing from about 0.2 to 0.25 at the start of the brake cycle up to about 0.35 at end. In addition, as seen in Fig. 6(a), the maximum brake power in the test is about 311 kW. This is lower than the maximum nominal brake power 333 kW due to the lower block–wheel friction coefficient at the beginning and due to the time it takes to obtain the correct brake (normal) force. After stop braking the
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the results for braking with sinter blocks clearly reveal a characteristic banding pattern on the tread. This pattern generally changes from one brake cycle to the next. One or two bands are generated and for most of the brake cycles the axial position of these bands remain fixed. During banding, the tread will locally be subjected to higher (and lower) temperatures than in a situation with evenly distributed heat over the entire block width. The evolution of crack width found at inspections is shown in Fig. 8. Corresponding pictures of the wheel tread are shown in Fig. 9. Typical thermal cracks can be seen in the part of the tread subjected to rolling contact between wheel and railwheel. These cracks were observed already after 11 brake cycles. This corresponds to an accumulated total brake energy of approximately 116 MJ. This is considerably lower than the accumulated energy (approximately 174 MJ) corresponding to crack initiation at similar experiments under lower wheel load and lower inertia, see [2]. The depth of the thermal cracks after 40 brake cycles was found to be approximately 2.0 mm. This was evaluated by stepwise turning of the wheel tread; after 2.0 mm turning only very fine discrete cracks could be observed by magnetic inspection. 4.2. Finite element analysis results
Fig. 7. Wheel tread temperatures during the final 15 brake cycles (times between brake cycles have been removed) as detected using thermocamera. Temperatures have been assessed presuming an emissivity equal to 1.0.
4.2.1. Thermal analysis Results are presented for two assumed types of contact banding. The aim of these analyses is to simulate the experimental conditions. In addition, the case of wheel–block contact over the whole block width is studied. This is an ideal case that could result if no frictionally induced instabilities (banding) were present. This case is added since it is known from previous brake rig experience using organic composite brake blocks (which commonly cause no banding) that thermal cracking in such cases takes substantially longer time to initiate. Fig. 10 shows the considered braking cases with band widths indicated on the wheel tread. The two cases of banding – one central band, or two bands – have both a total band width of 50 mm (block– wheel contact over 62.5% of tread width). The case without banding presumes thermal contact over the entire block width of 80 mm. In the following, the thermal loading cases will be denoted as a Two 25 mm wide bands. b One 80 mm wide band. c One 50 mm wide band.
Fig. 8. Crack length along the wheel tread measured at inspections after 11, 20, 30 and 40 cycles. A, B and C indicate the different measurement positions.
wheel was cooled to 60 °C during pure rolling with a wheel–rail contact load of 30 kN. The maximum temperature measured by thermocouples embedded 10 mm below the tread surface was approximately 300 °C. In addition, a thermocamera was used to evaluate variation of temperature during braking. The resulting videos are postprocessed by partitioning the tread width into 20 bands. Each band thus corresponds to a given axial position on the tread. The average temperature in each band at each time frame of the video is determined by image processing using Matlab image processing toolbox [15]. Some examples of tread temperatures evaluated from thermocamera data for an assumed surface emissivity of 1.0 are given in Fig. 7. Since the actual surface emissivity is lower than the 1.0, the temperatures of the tread are higher than indicated in Fig. 7. Moreover these temperatures are subjected to additional uncertainties due to variations in emissivity during braking. Nevertheless,
Calculated temperatures in the wheel rim at the instants in time corresponding to the maximum tread temperatures are presented in Fig. 11. It can be noted that the maximum temperature in the region of the tread traversed by the rolling contact loading is highest (467 °C) for case c. Load cases a and b result in peak temperatures of 441 °C and 336 °C, respectively. The calculated temperature histories for a point on the wheel tread and a point 10 mm below the tread during the first 100 s (stop braking 58.5 s followed by cooling) are shown in Fig. 12. The solid circles on the graphs indicate time instances when rolling contact loads are traversed along the tread in the simulations. At each such instance two wheel revolutions (rolling contact load passages) are simulated. Instance (1) gives rolling contact conditions prior to braking, instance (2) gives response after a short braking time at intermediate temperature conditions, instance (3) is time when the maximum temperature is reached and instance (4) is at end of braking. It is to be reminded that in the simulation one braking cycle is defined by brakingþ cooling time. As shown in Fig. 6(f), the maximum temperature measured by the thermocouples is about 300 1C. This is in good agreement with simulated case b (Fig. 12(b)). The other simulated cases predict slightly higher temperatures than as measured by the thermocouples.
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Fig. 9. Wheel appearance and penetrant tested tread after (a) 11 cycles, (b) 20 cycles, (c) 30 cycles and (d) 40 cycles.
4.2.2. Thermomechanical analysis In the analysis, a partial slip distribution for the tangential stresses at the wheel–rail contact is considered. The tangential force during braking is Q¼7.3 kN. In the baseline case, f ¼ 0:075 ¼ Q =P and μ ¼ 0:5. The simulations feature surface tangential stresses for a braking wheel, i.e. the shear stresses are opposite to the rolling direction, see Fig. 13. A sensitivity analysis is conducted to assess the effect of the tangential tractions by comparing this baseline case with partial slip f ¼0.075 and μ ¼ 0:3, full slip qðx; yÞ ¼ fpðx; yÞ, and pure rolling, i.e. qðx; yÞ ¼ 0. Tangential traction distributions along the centre of
the rolling contact patch are shown in Fig. 14 together with traction distributions for assumed full slip conditions. An empirical model to predict fatigue crack initiation owing to ratcheting is proposed by Kapoor [16]. The number cycles to initiation of ratcheting induced fatigue, Nr, is predicted from the ratcheting strain increment Δεr as Nr ¼
εc Δεr
ð4:1Þ
where εc is a material parameter that could e.g. be determined by twin disc tests [17]. It should be noted that since there is no
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approximated as
Δεr ¼
4 X
Δεi
ð4:4Þ
i¼1
Fig. 10. Block–wheel contact zones on wheel tread for assumed brake cases.
information on the value of the material constant εc for the material and load conditions considered in this study, a qualitative evaluation is made in which εc is kept as an unknown parameter. In the current analysis, the ratcheting is evaluated from the effective strain as (see also [18]) pffiffiffi 2 εeff ¼ ½ðεxx εyy Þ2 þ ðεxx εzz Þ2 þ ðεyy εzz Þ2 þ 6ðε2xy þ ε2yz þ ε2xz Þ1=2 3 ð4:2Þ In Fig. 15, the effective strain defined in Eq. (4.2) is evaluated along a radial path at the centre of the wheel–railwheel contact at the end of each point marked in Fig. 12. As shown, for the cases studied the maximum effective strain occurs at approximately 1 mm below the surface. As observed in Fig. 15(c), the position corresponding to the highest residual effective strain magnitude has shifted to the surface for the most severe thermal load case (case c). Fig. 16 shows the time evolution of the effective strain at depths 0.5 mm and 1.0 mm below the centre of wheel–railwheel contact band (y¼0 in Fig. 3). The results show large accumulation of strain for the rolling contact passages at instances (3) and (4) in Fig. 12. Recall that the maximum temperatures at the centre of the wheel– railwheel contact are (at instance (3) in Fig. 12) 441 °C, 363 °C and 467 °C for cases a, b and c, respectively. In general higher temperatures at positions corresponding to the wheel–railwheel contact give significantly larger ratcheting increments. Note from Table 1 that the material parameters show significant variations between 400 °C and 500 °C. This, together with the direct influence of thermal strain, explains the variation in responses between the different thermal load cases. The results from Fig. 16 are used to assess the total ratcheting strain for one complete brake cycle for the three thermal load cases analysed. To this end, the number of wheel revolutions that actually took place in the time intervals pertaining to the calculated ratcheting strains need to be accounted for. For each load cycle marked by a solid circle in Fig. 12 the ratcheting increment of the effective strain for the first wheel passage is denoted Δεi;1 . For the second wheel, it is denoted Δεi;2 and is also taken as representative for all subsequent wheel revolutions in the interval. Thus, the total ratcheting strain in the interval can be evaluated as
Δεi ¼ 1 Δεi;1 þ ðN i 1Þ Δεi;2
ð4:3Þ
where i represent time intervals corresponding to each of the mechanical load passages, i.e. time instants (1)–(4) marked in Fig. 12 and Ni is the number of wheel revolutions in the interval i, see the example in Fig. 17. During cooling, the contact load between wheel and railwheel in the experimental study is low (about 30 kN). For this reason no substantial ratcheting is assumed to be induced during this stage (cf discussion on initial shakedown below). Consequently the total effective strain increment for an entire braking cycle may be
It is informative to introduce the working point for the mechanical load cycles prior to braking in a shakedown map [19] as done in Fig. 18. It is found that the working point is not in the (surface) ratcheting area of the map, but rather well into the elastic or plastic shakedown area.1 For this reason, cycles prior to braking are not expected to give any additional plastic strain once shakedown has occurred. The numerical simulations presume an initially stress-free wheel material. In reality the wheel rim will have circumferential compressive residual stresses of about 100 MPa [6] stemming from the manufacturing process. This residual stress contributes to the shakedown. In the simulations, two rolling cycles prior to braking have been simulated to obtain an initial state of shakedown. These are thus not considered in the life analysis. Results for predicted cycles to crack initiation are given in Table 3. To compare the different thermal cases, the number of cycles for thermal case c and μ ¼ 0:5 evaluated from the accumulated strain 0.5 mm below the surface is taken as a reference life denoted Nnr . For comparison, the ratcheting fatigue life evaluated 1.0 mm below the surface is also presented. It is however not considered representative for the current damage pattern since observed cracks initiate at (or near) the surface. In addition subsurface initiation would require the presence of a material defect and likely corresponds to higher effective strain magnitudes (e.g. a higher εc ) due to the material confinement. From Table 3, it is seen that the load case corresponding to uniform heating has a predicted ratcheting fatigue life that is 7.49 times higher than the reference case (thermal case c). Taking εc 10 (cf [17]) crack initiation corresponding to ratcheting fatigue is predicted after 2 brake cycles for case a and 6 cycles for case b. Thermal load case c is predicted to cause crack initiation already during the first brake cycle. 4.2.3. Effect of tangential traction in wheel–railwheel contact For thermal load cases a and c, the previously analysed baseline case is compared to the following mechanical load cases: A Partial slip, f ¼0.075, μ ¼ 0:30: B Full slip, f ¼0.075. C Pure rolling, f ¼0. Results are presented in Fig. 19 and Tables 4–6. It is seen that for the thermal load case a the ratcheting fatigue life in pure rolling is more than 8 times the fatigue life for the mechanical baseline case under thermal load case c. Adopting εc 10, the number of cycles to initiation varies between 1 (case a–A) and 8 (case a–C). As also noted in previous studies, see e.g. [4], the results in Tables 4–6 show that the material response is rather sensitive to the presumptions regarding tangential traction distributions (partial slip vs full slip) and on the friction coefficient. In addition, experimental observations in [5] indicated that crack formation under less severe braking loads was related to high traction rolling rather than pure rolling. In the current study, differences – in particular between pure rolling and tractive rolling – are appreciable despite the low magnitude of the applied traction. 1 Inpffiffiffi the shakedown map, p0 is the peak Hertzian pressure according and k ( ¼ σ y = 3 according to the von Mises yield criterion) is the yield strength in pure shear for the work-hardened material. For the given material, σ y ( ¼ B1 þ σ 0 , where B1 ¼ C=γ ) ¼ 527 MPa at T ¼ 0 °C following material data in Table 1.
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Fig. 11. Temperature distributions at instants in time corresponding to the highest surface temperatures for (a) two 25 mm wide bands, (b) one 80 mm wide band and (c) one 50 mm wide band. The region traversed by the rolling contact load is indicated on the tread.
Fig. 12. Temperature distribution for the uppermost node at the centre of the contact (solid line) and at a node about 10 mm below the surface (dashed line) for (a) 2 bands 2×25 mm, (b) 1 band 80 mm and (c) 1 band 50 mm. The first 100 s are shown. Braking power varies according to Fig. 6(a) during the initial 58.5 s, after which the power is zero. Solid circles indicate time instances when rolling contact loads are traversed along the tread.
Fig. 13. Tangential stresses for a braking wheel.
Fig. 14. Distribution of tangential tractions.
4.2.4. Influence of FE mesh A mesh sensitivity study is carried out based on thermal load case a. A simplified analysis is performed where mechanical load passages are introduced only at the points marked (1) and (3) in Fig. 12. Two different mesh densities of the refined part are used, see Fig. 4. Data for “mesh 1” used in the previous analyses (with a
characteristic element size of 1 mm) are compared to the more refined “mesh 2” that has a characteristic element size of 0.5 mm, see Table 7. The remaining part of the wheel sector consists in both cases of 13 600 elements and 63 333 nodes. As shown in Fig. 20, good convergence of results is found for effective residual strain magnitudes at depths below some 0.5 mm. The difference in the peak value exactly at the surface is due to the resolution (discrete step size) of the translation of the tangential contact load: The traversal of the contact loads follow the bias of the mesh. The number of steps is the same in “mesh 1” and “mesh 2” but the element size in the longitudinal direction varies between the two meshes. Thus stress magnitudes at the surface are not fully resolved and results between two meshes at the surface may differ. This issue cannot be mitigated unless the mesh is extremely refined which would result in extremely high computational costs. 4.2.5. Influence of material model It is known that the material model used in the analyses may overpredict ratcheting [20]. In order to control the amount of ratcheting for higher strain levels, additional backstresses can be added in the model, see e.g. [21]. In this paper, a sensitivity analysis is conducted: one additional backstress is added to the material model and the material response is evaluated for thermal cases a and c. The evolution of the second backstress is assumed to be linear with hardening modulus C 2 ¼ C 1 =10. This allows for a better representation of the stress–strain evolution at larger strains. In fact, as seen in Fig. 21, the material response with two backstresses matches in a better way the experimental data at 7 0.6% strain.
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Fig. 15. Calculated residual effective strains after rolling contact load passages (time instances (1)–(4) in Fig. 12) for (a) 2 bands 2×25 mm, (b) 1 band 80 mm and (c) 1 band 50 mm.
Fig. 16. Effective strain evolution for a node located (a) 0.5 mm and (b) 1.0 mm below the centre of the wheel–railwheel contact.
Fig. 17. Example of assessment of ratcheting strain based on effective strain evolution for the equivalent braking cycle in the FE simulations.
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Table 3 Ratcheting fatigue life Nr expressed as a function of critical strain εc and in normalized form. Location
2 bands 2 25 mm
1 band width 80 mm
1 band width 50 mm
0.5 mm below surface Nr
εc =3:57 3:25N nr
εc =1:55 7:49N nr
εc =11:61
1.0 mm below surface
εc =3:67
εc =1:94
εc =10:40
N nr
Fig. 18. Shakedown map for actual loading conditions prior to braking.
Table 4 Mechanical load case A (f¼ 0.075, μ ¼ 0:3) predicted ratcheting fatigue life. Location
a (2 25 mm)
c (50 mm)
0.5 mm below surface Nr
εc =2:81 4:13N nr
εc =9:15 1:27N nr
1.0 mm below surface
εc =3:84
εc =10:71
Table 5 Mechanical load case B ðμ ¼ f ¼ 0:075Þ predicted ratcheting fatigue life. Location
a ð2 25 mmÞ
c (50 mm)
0.5 mm below surface Nr
εc =1:55 7:49N nr
εc =6:86 1:69N nr
1.0 mm below surface
εc =2:73
εc =8:42
Table 6 Mechanical load case C (f¼ 0) predicted ratcheting fatigue life. Location
a (2 25 mm)
c (50 mm)
0.5 mm below surface Nr
εc =1:34 8:66N nr
εc =4:29 2:71N nr
1.0 mm below surface
εc =2:48
εc =6:15
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Table 7 Mesh data and computational times a simplified brake cycle analysis. Model
Mesh 1 Mesh 2
Model data for refined part of the wheel tread Characteristic elem. size (mm)
No. of nodes
No. of elements
CPU time (h)
1 0.5
33 242 93 691
7000 21 400
109 265
Fig. 19. Effective strain evolution for node below the centre of the wheel–railwheel contact for case 2 bands 2×25 mm at (a) depth 0.5 mm below surface and (b) 1.0 mm, and for case 1 band 50 mm at (c) depth 0.5 mm and (d) 1.0 mm.
Fig. 21. Predicted response of material model with one and two backstresses together with experimental test data for ER7T (test temperature 350 1C). Last cycle only.
With εc 10, the predicted number of cycles to initiation are 5 (thermal load case c) and 10 (thermal load case a). Fig. 20. Effective residual strain magnitudes for banded wheel–block contact as calculated using two different mesh densities.
5. Conclusions and outlook As shown in Fig. 22, the material model has a significant effect on predicted ratcheting rates. For the material model with 2 backstresses it is seen that the initial rolling cycle results in (more or less) shakedown for the second rolling cycles (at room temperature) and also for the rolling cycles at the beginning of the brake cycle. This indicates that it is a reasonable choice to neglect the contributions from the first rolling cycles in the estimation of number of cycles to initiation as made in the previous section. The ratcheting strain for the brake cycle can now be seen to be dominated by the rolling cycles introduced at the time of maximum tread temperature. With εc 10, the results in Table 8 yield 5 brake cycles to initiation for case c and 10 for case a.
This paper is a first step towards full-scale test rig experimental validation of FE predictions of thermomechanical cracking. Ratcheting fatigue life based on thermomechanical numerical simulations for brake cycles is evaluated. Different thermal contact conditions between block–wheel found by post-processing of experimental data have been employed. It has been shown that one 50 mm wide thermal band is a more severe case than two 25 mm wide bands. Further, both of these thermal load cases are more severe than uniform heating over the entire brake block width (a thermal load case that was not found in the experiments). Predicted number of brake cycles to crack initiation presuming a critical ratcheting strain
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Fig. 22. Effective strain evolution for a node 0.5 mm below the tread surface for (a) thermal load case a (2 25 mm) and (b) thermal load case c (50 mm). Table 8 Predicted ratcheting fatigue lives with a modified material model. Location
a ð2 25 mmÞ
c (50 mm)
0.5 mm below surface Nr
εc =1:07 10:85N nr
εc =2:07 5:61N nr
1.0 mm below surface
εc =0:83
εc =1:65
http://www.charmec.chalmers.se in Gothenburg, Sweden and the Railway Technical Research Institute (RTRI) in Tokyo, Japan (http:// www.rtri.or.jp/eng/). Prof. Roger Lundén is acknowledged for coordinating the cooperation with RTRI. Many thanks go to Prof. Anders Ekberg at Chalmers for useful discussions and inputs during this work. Special thanks to Rebecka Brommesson at Chalmers for fruitful discussions in particular on the material modelling and her help in compiling the data for the material response with two backstresses in Fig. 21.
εc of approximately 10 was found to be in the order of 1–8 depending on thermal and mechanical (wheel–railwheel) contact conditions. Adoption of a more refined material model led to a prediction in the range 5–10 brake cycles. These predictions are in line with the experimental observations which detected 2.0 mm wide cracks in the wheel tread after 11 braking cycles. Some approximations in the current analyses tend to make the estimation conservative. Examples are the presumptions of a constant banding pattern between brake cycles (implicitly presumed in the damage accumulation) and that progressing shakedown is not captured due to the truncation of mechanical load cycles. In addition the material model has been shown to have a significant effect on the predicted ratcheting fatigue life. The current analyses can be improved by
Addition of at least one more experimental condition, i.e. braking from another initial speed.
Addition of experiments and simulations for braking with
organic composite blocks where no (or less severe) banding is expected. Further enhancements in material models and pertinent material parameters, in particular related to life predictions. In a more detailed study calibration with two or more backstresses should be considered. Including load rate effects. In [9] it has been found that viscous effects are significant at temperatures above some 500 °C. The use of more advanced material models that account for this effect can improve the accuracy of the predictions. Including compressive residual stresses from manufacturing. Evaluating low cycle fatigue (LCF) damage. Accumulation procedure for LCF should be included and LCF life should be compared with ratcheting life. The model which predicts the shortest life to fatigue is considered to determine the fatigue life [16,17]. Account for wear effects.
Acknowledgements This work is part of the activities within the Swedish National Centre of Excellence CHARMEC (Chalmers Railway Mechanics,
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