PREDICTING ROLLING CONTACT FATIGUE OF RAILWAY WHEELS

Transcription

1 Presented at the 13th International Wheelset Congress in Rome, September 17 21, 21 Revised version Ekberg, Kabo & Andersson 1 PREDICTING ROLLING CONTACT FATIGUE OF RAILWAY WHEELS Anders Ekberg*, Elena Kabo* & Hans Andersson** * Dept of Solid Mechanics/ ** Swedish National Testing and CHARMEC Research Institute Chalmers University of Technology Box 857 SE Göteborg SE Borås SWEDEN SWEDEN anek@solid.chalmers.se kabo@solid.chalmers.se hans.andersson@sp.se Summary An engineering model for rolling contact fatigue of railway wheels is developed. In this model, three well-known types of fatigue of wheels are accounted for: surface induced fatigue, subsurface fatigue and fatigue initiated at deep material defects. The fatigue response in a railway wheel is defined in terms of three fatigue indices and pertinent thresholds: FI surf = µ 2πabk > 3 (1) FI sub = ( 1 + µ 2 ) + a 4πab DV σ h, res > σ e FI def = > F th (2) (3) Components of the criteria above are described in the paper. If any of the inequalities above are fulfilled, fatigue is predicted to occur. Although primarily intended to compare two train/track configurations in terms of fatigue performance, the model also gives an indication of whether fatigue will appear and in which form. The model can easily be integrated in a multi-body dynamics code without increasing computational demands. This forms a very powerful tool for optimizing train/track configurations with respect to fatigue performance. Key words: rolling contact fatigue, railway wheels, engineering model, analytical model, CAE analysis

2 Ekberg, Kabo & Andersson 2 1 Introduction Wheel fractures caused by propagating fatigue cracks may manifest themselves in a violent manner where a part of the wheel (or the entire wheel) breaks off. Such failures may be very costly in economical terms as well as in terms of delays and human injuries. Consequently, there is an increasing need for predictive models and countermeasures to prevent such failures. In the following, an engineering model for rolling contact fatigue analysis is outlined. It is intended to be used in conjunction with a multi-body dynamics simulation of train/ track interaction. The model sets out by noting that, at least, three fairly different types of rolling contact fatigue damage in railway wheels exist: surface initiated fatigue subsurface initiated fatigue fatigue initiated at deep defects The model will address all of these mechanisms. 2 Surface initiated fatigue 2.1 Fatigue mechanisms Initiation of surface cracks is the result of ratchetting and/or low-cycle fatigue of the surface material. Once cracks initiate in the surface layer, they will propagate in a shallow angle to the surface, deviating first into an almost radial, and later into a circumferential direction of growth, see figure 1. Fracture will finally occur as a branching of the crack towards the surface breaking off a piece of the wheel tread. 5 m figure 1 Initiation of a surface crack and the final fractured surface, from [1] 2.2 Predictive model A fast and reasonably accurate way of identifying load levels corresponding to surface fatigue in rolling contact is the use of shakedown maps, see [2]. Such a diagram uses the following input parameters: the yield stress in pure shear (torsion), k vertical load magnitude, lateral load magnitude, F lat

3 Ekberg, Kabo & Andersson 3 semi-axes of the Hertzian contact, a and b The vertical load magnitude, contact geometry and the yield stress in shear, k normalized vertical load: v = 3 p πabk k which is combined with the utilized friction coefficient, defined as forms a (4) F lat µ = = 2 2 F x + F y (5) Here F x and F y are lateral loads in the wheel axle and rail direction. These measures define a work point ( µ, v) in the shakedown map. The equation of the boundary curve for surface flow in the shakedown map, see [3], is v = 1 µ (6) Normalized vertical load, ν=p /k = (3 )/(2πabk) Alternating plasticity (plastic shakedown) Elastic shakedown Subsurface BC FI surf BC-WP Incremental growth (ratchetting) Elastic Surface Utilized friction coefficient, µ WP figure 2 Shakedown map with work point ( WP ) indicated by X. Boundary curve for plastic surface deformations is denoted BC. The shortest distance between X and BC is indicated with a solid line. Surface fatigue index is indicated by a dashed line. In a proportional plot, BC is almost vertical. Hence, the surface fatigue index is taken as the horizontal projection of the shortest distance between BC and the current WP (positive if WP is to the right of BC ). The fatigue index is thus given by FI surf µ 1 -- = µ 2πabk (7) v 3 Surface fatigue is predicted to occur if FI surf > (8)

4 Ekberg, Kabo & Andersson 4 3 Subsurface initiated fatigue 3.1 Fatigue mechanisms Subsurface cracks initiate at depths of below some 3 mm from the wheel tread. Below some 1 mm, the fatigue resistance will be totally governed by the presence of macroscopic inclusions. This is discussed in sec 4. Typical features of subsurface fatigue are, see [4]: no signs of macroscopic inclusions or voids at the point of fatigue initiation crack propagation in an angle downwards to a depth of some 2 mm final fracture towards the surface circumferential crack length of some 15 to 3 mm at fracture fatigue crack propagation final fracture point of initiation initial wheel surface fatigue crack figure 3 Subsurface initiated fatigue crack, from [1] and fatigue initiated at a deep defect, from [4] 3.2 Predictive model In previous studies, see [5], this type of fatigue has been analyzed by means of numerical simulations employing the Dang Van multiaxial fatigue initiation criterion recalled as σ EQ = max[ σ EQ () t ] = max[ τ a () t + a DV σ h () t ] > σ e t t Here, τ a () t is the time-dependent shear stress amplitude. Details on the derivation of τ a () t are, for instance, given in [5]. Further, a DV is a material parameter, σ h () t is the hydrostatic stress positive in tension) and is the equivalent fatigue limit. σ e The fatigue index proposed in the current model is an approximation (based on the assumption of Hertzian contact) of the magnitude of the largest subsurface Dang Van equivalent stress FI sub ( 1 + µ 2 ) + a 4πab DV σ h, res This approximation of the Dang Van equivalent stress is remarkably accurate with an error, as compared to a thorough derivation, of less than some 8%, see [6] According to the Dang Van criterion, fatigue damage will be predicted by the inequality (9) (1)

5 Ekberg, Kabo & Andersson 5 FI sub > σ e (11) Here, σ e is the equivalent fatigue limit (normally taken equal to the fatigue limit in pure shear). Note however that in a railway wheel, there is a large material volume subjected to high stress magnitudes. Consequently, the design fatigue limit has to take into account the risk of an occurring material defect in this highly stressed volume. An approximative reduction of the fatigue limit due to an occurring defect is, see [4] σ w σ e d = d Here, σ w is the fatigue limit of the material containing a defect of diameter d. Further, σ e is the fatigue limit in the absence of large defects and d is the diameter of a defect corresponding to fatigue initiation at the stress level σ e. A further approximation would be to set σ e to the smooth specimen fatigue limit in torsion, d to some 1 2 µm and d to the largest defect size allowed in the material. Reasons for, and drawbacks of equation (12) are discussed in [1] and [4]. (12) 4 Fatigue initiated at deep defects 4.1 Fatigue mechanisms The partition between subsurface initiated fatigue and fatigue initiated at deep defects is somewhat dim. Here, fatigue initiated at deep defects denotes cracks initiated at relatively large material inclusions (in the order of 1 mm). Typical features are, see [4] fatigue crack initiation at a depth of 1 25 mm below the wheel tread crack propagation at an almost constant depth below the wheel tread (corresponding to depth of initiation) until fracture occurs final failure due to branching of the circumferentially growing crack at a length of 25 to 135 mm 4.2 Predictive model and fatigue index The fatigue impact at the defect is fairly unaffected by the contact geometry at moderate vertical load magnitudes, see [7]. This motivates adopting the vertical load magnitude as fatigue index FI def This index can be employed to compare the relative impact of different train/track configurations presumed the wheel material is unaltered. For a quantitative prediction, fatigue is predicted by the following inequality FI def > F th ( zdh,,, ) Currently, intensive research aiming at quantifying F th is underway, see [7] and [8]. It is, today, known that F th is a function of the position of the defect below the wheel tread, z, the size of the defect, d and of load history, H. It is also likely that F th depends on shape and metallurgical composition of the defect. It should also be noted that defects often appear in clusters. In such cases, the defects will interact and cause a larger decrease in than what an individual defect would give, see [8]. F th (13) (14)

6 Ekberg, Kabo & Andersson 6 5 Practical use 5.1 Implementation The simplicity of the current model facilitates its embedding in a multi-body dynamics code with nearly no increases in computational demands. For current evaluation purposes, the model is implemented in MATLAB and used as post-processor. The example below concerns a bogie passing a lateral misalignment on a rigid track and is further described in [9]. Evaluation of twelve thousand fatigue indices took sixteen seconds on a Macintosh Numerical results In figure 4a, work points for the back right wheel are introduced in a shakedown map. The yield stress in shear is here taken as k = 3 MPa. In figure 4b, histograms of FI surf for all wheels of the bogie are presented. It is seen from these figures that the risk of surface fatigue is low. Normalized vertical load, ν=p /k = (3 )/(2πabk) Elastic shakedown Alternating plasticity (Plastic shakedown) Surface fatigue Subsurface Incremental growth (Ratchetting) Elastic Surface Utilized friction coefficient, µ Front left Back left Front right Back right figure 4 a) Work points (indicated by the band of X:es) for the back right wheel plotted in a shakedown map. b) FI surf for all four wheels 4 Front left 4 Front right 4 Front left 4 Front right b x 1 8 Back left x 1 8 Back right x 1 4 Back left x 1 4 Back right x x x x 1 4 figure 5 a) FI sub, and b) FI def for all four wheels. Vertical axes denotes number of occurrences and horizontal line magnitude of the fatigue index.

7 Ekberg, Kabo & Andersson 7 Pertinent subsurface fatigue indices are presented in figure 5a. In order to prevent fatigue damage, we should have a material with a threshold magnitude for subsurface fatigue of some 2 MPa. Considering a material with an equivalent fatigue limit (equal to the fatigue limit in shear) of σ e = 35 MPa, equation (12) and following approximations give an allowable defect size of d.3 to.6 mm at a depth corresponding to the maximum shear stress.it should be noted that residual stresses are not accounted for in this example. As an example, a uniaxial compressive residual stress of σ x = 3 MPa would lead to a decrease in the fatigue index of FP sub = 1a DV 4 MPa, cf [1]. The resulting allowable defect size will be in the order of d = 1 to 2 mm. Finally, fatigue indices corresponding to deep defects are presented in figure 5b. It is seen that the response is fairly centered around FI def = 98 kn. However, the extreme values will reach up to FI def = 17 kn. The indications from ongoing research, see [8], is that the fatigue response will be dominated by these extreme load magnitudes. Acknowledgments The current work is part of the activity within the Swedish National Centre of Excellence CHARMEC Chalmers Railway Mechanics. The majority of the work has been carried out within the European project HIPERWHEEL funded by the European Commission under contract G3RD CT2 24. The provision of computer routines and aid mainly regarding shakedown map analysis from Dr Jonas Ringsberg, vehicle dynamic analyses from Mr Clas Andersson and insightful comments from Profs Stefano Bruni and Roger Lundén are gratefully acknowledged along with the help from friends and colleagues. Bibliography 1. EKBERG A & SOTKOVSZKI P, Anisotropy and fatigue of railway wheels, International Journal of Fatigue, vol 23, no 1, pp 29 43, JOHNSON K L, The strength of surfaces in rolling contact, Proceedings of the Institution of Mechanical Engineers (IMechE), vol 23, pp , PONTER R S, HEARLE A D & JOHNSON K L, Application of the kinematical shakedown theorem to rolling and sliding point contacts, Journal of the Mechanics of Physics and Solids, vol 41, pp , EKBERG A & MARAIS J, Effects of imperfections on fatigue initiation in railway wheels, IMechE, Journal of Rail and Rapid Transit, vol 214, pp 45 54, EKBERG A, Rolling contact fatigue of railway wheels towards tread life prediction through numerical modelling considering material imperfections, probabilistic loading and operational data, PhD-thesis, Chalmers University of Technology, Department of Solid Mechanics, Göteborg, Sweden, 2 6. EKBERG A, KABO E & ANDERSSON H, An engineering model for prediction of rolling contact fatigue of railway wheels, in preparation 7. KABO E & EKBERG A, Fatigue initiation in railway wheels - on the influence of defects, Proceedings of 5th International Conference on Contact Mechanics and Wear of Wheel/Rail Systems in Tokyo, Japan, July 25 28, 2 8. KABO E, Material defects in rolling contact fatigue influence of overloads and defect clusters, in preparation 9. ANDERSSON C & ABRAHAMSSON T, Simulation of interaction between a train in general motion and a track, submitted for international publication 1. DANG VAN K, CAILLETAUD G, FLAVENOT J F, LE DOUARON A & LIEURADE H P, Criterion for high cycle fatigue failure under multiaxial loading, in Biaxial and Multiaxial Fatigue (Brown M W & Miller K J, eds), Mechanical Engineering Publications, London, pp , 1989