ROLLING CONTACT FATIGUE OF RAILWAY WHEELS COMPUTER MODELLING AND IN-FIELD DATA

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1 ROLLING CONTACT FATIGUE OF RAILWAY WHEELS COMPUTER MODELLING AND IN-FIELD DATA A. Ekberg Division of Solid Mechanics and CHARMEC Chalmers University of Technology, S-4 96 Göteborg, Sweden ABSTRACT The fatigue life of wheel treads has a strong bearing on the economy and safety of rail transportation. An understanding of fatigue mechanisms and a prediction of lifetimes are of interest to both manufacturers and operators. The issue addressed in this paper is how to design a suitable mathematical model for fatigue life prediction for railway wheels. In the past, several models have been developed for the analysis of multiaxial fatigue in general and for the study of rolling contact fatigue in particular []. However, due to the complexity of the problem, there is still no standard model available for prediction of the lifetime of a railway wheel subject to high cycle rolling contact fatigue. The aim of the model to be presented and discussed in the following is that it should serve as a fast and sufficiently accurate tool in the practical study of rolling contact fatigue. To this end, a semi-analytical approach and an equivalent stress criterion are employed. The problem of how to select an appropriate equivalent stress criterion is treated in some detail. Some calculated results are compared to statistical data obtained for passenger trains from workshops doing wheel reprofiling. Finally, the influence of some parameters on the predicted damage and lifetime is studied. Keywords: rolling contact fatigue, high cycle multiaxial fatigue, fatigue life prediction, equivalent stress criterion, railway wheels, reprofiling statistics. THE FATIGUE LIFE MODEL WLIFE An initiation model for high cycle rolling contact fatigue with application to wheel/rail damage has been reported previously []. The model was implemented in the computer code WLIFE, and it is based on a semi-analytical procedure for analysing the Hertzian contact stress field. The Dang Van fatigue criterion, the Wöhler diagram and the Palmgren Miner damage accumulation rule are combined to evaluate the fatigue damage in discrete points and directions over a selected material volume of the wheel rim. Factors such as rotating principal stresses, high hydrostatic pressures, randomness of applied loads, and orientation effects are included. A linear elastic material is assumed. No thermal or inertia effects are considered. The different components of the fatigue life model were penetrated in the previous studies [] and []. Below, only the fatigue initiation criterion is discussed. This is a key issue in the analysis of multiaxial high cycle fatigue. CHAlmers Railway MEChanics The Swedish National Centre of Excellence in Railway Mechanics Research located at Chalmers University of Technology, see Acknowledgements

2 . EQUIVALENT STRESS CRITERIA Can a multiaxial stress field be satisfactorily represented through an equivalent scalar measure (an equivalent stress) such that the fatigue damage can be predicted by use of this measure for a unit material volume in a general state of cyclically timevarying stress?. Critical Plane Approaches Equivalent Stresses In answer to the question above, a shear stress criterion based on the Dang Van criterion [3] was attempted in WLIFE. According to this criterion, damage occurs if the combination of the value τ a () t of a shear stress (in the most dangerous shear plane) and the value σ h () t of the hydrostatic stress at the material point considered fulfils one or both of the following two inequalities during some time portion(s) t < t < t of the full stress cycle, Here, τ a () t σ h () t τ e a DV τ τ () t + a σ () t > τ EQ a DV h e τ τ () t a σ () t < τ EQ a DV h e (a) (b) is the time-dependent value of the shear stress at the specified material point and in the specified shear plane through this point. The value τ a () t (here called shear stress amplitude ) is calculated as the difference between the current magnitude at time t of the (rotating) shear stress in the specified plane and its mid magnitude during one cycle is the time-dependent value of the hydrostatic stress (positive when tensile) at the same material point is a material parameter (positive) usually taken equal to the fatigue limit of the material in pure shear is a dimensionless material parameter (positive) representing the influence of the hydrostatic stress The criterion (a,b) is graphically displayed in Fig.. Since the critical shear plane is unknown at the outset, the fulfilment of the criterion must, in principle, be examined for all possible shear planes through each material point considered. In order to reduce the computational effort, the largest shear stress in a material point (i.e. the Tresca shear stress), instead of the shear stress in a specified plane, can be used [4]. The Tresca shear stress is positive and defined as τtresca()= t [ σ () t σ () t 3 ] where σ and σ 3 are the algebraically largest and smallest principal stress, respectively. Using the amplitude of the Tresca shear stress, τ a,tresca () t (defined in a similar manner as τ a ()) t, a simplified fatigue criterion can be formulated as τ τ () t + a σ () t > τ EQ3 a,tresca DV h e It should be noted that damage calculated by criterion () is a scalar quantity. Only the largest damage in a material point is thus identified and different shear planes are not considered. In the case of a pulsating compressive loading, the criteria (a,b) and () have some drawbacks as will be described in some detail below. In an attempt to ()

3 overcome these problems (some portion of) the full magnitude of the Tresca shear stress τ Tresca () t can be introduced. A modified fatigue criterion could thus be expressed as τeq4 ( γ) τa,tresca() t + aσh() t + γτtresca() t > τe (3) Here, γ = means that only the full magnitude of the shear stress is employed whereas γ = yields the criterion (). A desirable extension of (3) could be to use a parameter γ which is dependent on the nature of loading. This could yield a criterion which is equivalent to () for alternating and repeated tension and which uses the full shear stress magnitude for repeated compression. However, this is not a trivial task. NO FATIGUE τa τ e FATIGUE τ a = τ e a DV σ h τ a = a DV σ h τ e τ e a DV σ h τ e FATIGUE Fig.. Dang Van criterion, see [3]. Stress paths for uniaxial loading with repeated compression, with alternating tension/compression and with repeated tension are marked with solid, dashed and dashed-dotted lines, respectively Shear Stress Amplitude In the criteria above a shear stress amplitude is used in order to eliminate the influence of a mean shear stress. How such an amplitude should be defined is not obvious. According to [5] this amplitude is usually calculated as τa() t = ( τa() t ) = ( τ() t ) τ i i ( mid). Here τ() t σ n σ n n n i ( ) = i ij j jk k j i is the shear stress vector with n i being the normal to the shear plane studied. Further ( τ mid ) i is the mid shear stress in this shear plane as defined by the centre of the smallest circle containing the path of ( τ( t) ) i in the plane, see Fig. a. According to [6] the origin of the shear stress path can be defined as the tensor ρ ij that fulfils the max-min formulation Min Max( J [ dev( σij( t)) dev( ρij )]), where J ρ t is the second stress invariant. Also, for stresses varying in a sinusoidal way (in-phase or out-of-phase), dev( ρ ij ) can be defined as the mean value of dev( σ ij ( t )). In WLIFE, a simplified approach is adopted as described in (). It should be noted that this simplification is only justifiable in the case of pulsating loading, as in wheel/rail contact problems. The simplification leads to a reduction in computational efforts.

4 (a) (b) t(t) t (t) a σ^ σ a σ a t t mid O Fig.. (a) Usual procedure to define shear stress amplitude, see [5] (b) Stresses in uniaxial repeated compressive loading. Stress amplitude σ a and maximum magnitude of applied stress ˆσ Comparison of Criteria When comparing the different criteria, it is convenient to specialise to uniaxial loading. Let us consider the stress cycle in Fig. b. In accordance with the discussion above, we can define τ a,tresca ( σ () t σ ) σ () t σ = ( ),mid 3 3,mid where σ and σ 3 are the algebraic values of the largest and smallest principal stress, respectively, and index mid denotes the mid value during a stress cycle. The hydrostatic stresses are considered positive in tension. Using () at instants Œ and in Fig. b, the equivalent stresses can be calculated as σˆ σˆ σ τ EQ3 = adv and τeq3 4 3 = ˆ 4 (5a,b) Since a DV is positive, the largest equivalent stress is found at instant. Uniaxial fatigue threshold values, ˆσ th, for repeated compression using Eqs. () and (3) (with γ = ) are calculated as ˆσth = τe and ˆσ th τe 4 3 a = (6a,b) Note that the largest equivalent stress is found in and Œ, respectively, see Fig. b. In this case, Eq. () will not take into account any direct effects of the compressive hydrostatic stress. This could be in contradiction with experimental data, see [7] and [8]. In addition, by using (3), we have two parameters (τ e and a DV ) which can be selected in an optimal way so as to enhance the performance of the model for the considered loading. It is seen from Table that (3) may be a suitable criterion for repeated compression (provided that accurate material parameters are used), but not for a general loading situation. For example, the criterion does include an influence of a superposed static shear stress. This is not in good agreement with experimental data [5] and calls for attention when dealing with residual stresses. Since none of the criteria discussed above can be deemed to be best in all situations, both () and (3) are used in the following, and differences in the calculated results are discussed. (4)

5 Table. Uniaxial fatigue thresholds values ˆσ th and ˆτ th. The threshold values are normalised relative to the measured threshold value in alternating tension/compression (A ) Predicted threshold values according to Eq.() and Eq.(4) (B) Predicted threshold values according to Eq.(3) (C) Measured threshold values for a rail steel, see [8] Employed material parameters are τ e = 5 MPa, a a DV = = 56. and γ = A B C Alternating tension/compression Repeated tension.6.6 Repeated compression.7.. Alternating shear Repeated shear Other Approaches to Modelling of Fatigue Initiation All approaches discussed above are so-called critical plane (CP) approaches, i.e. they include the shear stress acting on a (hypothetical) shear plane through a material point. Another option is to use a stress invariant (SI) criterion, such as the Sines and Crossland criteria. Since these criteria involve the second invariant of the deviatoric stress tensor and the hydrostatic stress (which is also an invariant), only one equivalent stress value per material point is calculated. A comparison between different CP and SI criteria is given in [9]. A completely different approach is to use fracture mechanics []. Since all materials have some kind of defects, the aim is to identify a small critical initial crack and analyse its growth. Some problems with this approach are how to theoretically treatof small fatigue cracks, how to deal with the multiaxial states of stress, and how to take high hydrostatic stresses and corresponding crack friction into account. Regardless of these problems, the reason why a fracture mechanics approach is not applied in WLIFE is that neither shape nor position of a critical initial crack for railway wheels under complex loading is known. A parametric search would require a large computational effort and the model would not be a fast predictive tool. However, fracture mechanics analysis is valuable in a more thorough analysis of rolling contact fatigue of railway wheels, see [] and []..3 Some Complicating Factors Anisotropy The rim of a railway wheel normally has some degree of anisotropy. When using a critical plane criterion, this anisotropy could be dealt with by assigning different material parameters in different directions. Note that the criterion () has to be used here since it is no longer evident that maximum damage is induced in the plane of maximum shear (amplitude). When using a stress invariant criterion, no direction of the equivalent stress can be defined. A possible approach for this case is to use the material parameters corresponding to the weakest direction, which will lead to a conservative approximation. Magnitude of Applied Loads The accuracy of predicted damage is strongly dependent on reliable information on wheel/rail contact forces. It is difficult to establish, both theoretically and experimentally, a characteristic load spectrum and still more difficult to determine and analyse its high frequency content. Much current research is therefore dealing with the dynamics of wheel/rail interaction.

6 Verification of Models Once a theoretical model has been developed, its usefulness should be analysed by means of experiments. High cycle fatigue experiments, especially multiaxial, are always costly and difficult to perform. When dealing with fatigue of railway wheels, it is extraordinarily difficult to verify the model. Full scale tests would be very expensive. Also, they would not include environmental factors that affect operating wheels. Here, an attempt has been made to verify the present model by use of infield reprofiling statistics. However, also this approach has its drawbacks as described below. 3. NUMERICAL RESULTS AND COMPARISON TO IN-FIELD DATA In order to verify the fatigue life model, calculated numerical results are compared to data from workshops having done wheel reprofiling of Swedish passenger trains between March 99 and June 995. The trains have three different sets of wheel diameters and nominal axle loads (in metric tonnes), see table below. Notation Wheel diameter Axle load W mm 8 tonnes W 88 mm 5.5 tonnes W3 88 mm tonnes Comparing predicted results with wheel reprofiling statistics is a difficult task. For example, there is a great deal of uncertainty in the classification of damage (made by the personnel in the workshop) and in the influence of interacting factors (such as wear). Another aspect is that nondamaged wheels are not accounted for in the statistics. However, a rough judgement of the validity of the model can be made. 3. Predicted Damage As input data to the present mathematical models, actual wheel/rail contact forces, measured in full scale tests performed by the Swedish State Railways, were used. Here, loads corresponding to probability fractions s =.4% and s =.% are used for comparison to wheel reprofiling statistics. This means that.4% (.%) of the rolling cycles induce loads with a magnitude equal to or larger than the magnitudes used in the damage calculations. The frequency content of the measurements is, in these cases, up to about Hz. No horizontal forces or residual stresses are included in the calculations. R F z F z R F x F y F y F x R R R = Young s modulus = MPa a DV =. 3 R = Poisson s ratio =.3 a = 3. R = 3. m γ = τ e = 6 MPa Fig. 3. Contact geometry of wheel/rail interface. Employed material and geometric parameters

7 In the calculations below (if nothing else is explicitly stated), the material and geometrical parameters are as given in Fig. 3. The wheel diameters, the vertical forces (from full scale tests) and the corresponding predicted damages per cycle are Type Wheel diameter F z Damage Eq.(3) Damage Eq.().4%.%.4%.%.4%.% W 7 kn 9 kn 4E-6 5E-6 E-6 W 88 5 kn 74 kn 6E-6 E-6 E-6 W kn 4 kn E-6 9E-6 3. Reprofiling Statistics Some statistical data from reprofiling are presented in Fig. 4. A problem with the statistics is that many reprofilings are classified as based on other causes. Further, no cause is specified when a wheel is scrapped and replaced W 5 reprofilings W 9 reprofilings W3 53 reprofilings Fraction of reprofilings due to fatigue: 9% Mean distance to reprofiling due to fatigue: 3. km Fraction of reprofilings due to fatigue: 9% Mean distance to reprofiling due to fatigue: 33. km Fraction of reprofilings due to fatigue: % Mean distance to reprofiling due to fatigue: 3. km Fig. 4. Reported distances to reprofiling due to fatigue for the three different types of wheels. Note that 4 3 km is the circumference of the earth 3.3 Comparison of Results Normalised values of predicted and observed distances to reprofiling due to fatigue are given below. The predicted travelled distance is calculated as W d = π D sd (7) where W D is the diameter of the wheel, s is the probability fraction of the load (s =.4% and s =.%, see section 3.) and D is the magnitude of the damage per revolution of the wheel. Note that (7) is a gross approximation and that a more elaborate estimation would result in lower values of the distance d. Table. Comparison between reported and predicted distances to reprofiling due to fatigue. Distances given in multiples of km Mean distance to reprofiling due to fatigue Predicted distance Eq.(3) s =.4% Predicted distance Eq.(3) s =.% Predicted distance Eq.() s =.4% Predicted distance Eq.() s =.% W W W From Fig. 4 and Table, it is seen that the criterion in Eq. (3) gives the most reasonable magnitudes of damage. Too large a damage is predicted for the wheels of

8 type W. An explanation may be that beneficial residual stresses and plastic effects are not accounted for. Further, in 49% of the reprofilings of the wheels of types W, no cause was specified. For wheels of types W and W3, the corresponding values are 7% and 3%, respectively. For wheels of type W3, the predicted life time is longer than the observed one from reprofiling data. Two possible explanations could be the simplifications made in Eq. (7) and the large scatter in reprofiling data for wheel type W3. A reasonable supposition would be that reprofilings for wheels of type W3 with rolling distances shorter than, say km, are not based on damages due to high cycle fatigue. 3.4 Influence of Certain Parameters In this section, the influence of some parameters on the magnitude of predicted damage per cycle using Eqs. () and (3) is presented. The examples are analysed using a HP J workstation with a MHz processor. Each calculation required less than two minutes CPU-time. Vertical Loads and Diameter of Wheels An increased magnitude of the applied loads will, as expected, give an increase in predicted damage, see Fig. 5. This increase will be more significant for heavy loading. However, at these load levels, plastic effects are important (but not considered in the present model). Predicted Damage per Cycle D Predicted Damage per Cycle D x (a) Wheel diameter:.88 m Wheel diameter:. m Vertical load [kn] Predicted Damage per Cycle D 4 x (b) Wheel diameter:.88 m Vertical load [kn] Wheel diameter:. m Fig. 5. Calculated damage vs vertical load using equivalent stresses according to (a) Eq.(3) and (b) Eq.().5 x (a) Vertical load: kn Vertical load: 75 kn Wheel Diameter [m] Predicted Damage per Cycle D x 5 (b) Vertical load: 75 kn Vertical load: kn Wheel Diameter [m] Fig. 6. Calculated damage vs diameter of wheel using of equivalent stresses according to (a) Eq.(3) and b) Eq.()

9 The well-known fact that a decrease in wheel diameter will increase the damage is shown in Fig. 6. This increase will be more significant for small diameters (where, not considered, plastic effects will be more significant). An interesting feature is that the predicted damage has a local increase for wheel diameters of approximately. meters. Material Parameters In Fig. 7 it is demonstrated that a has a strong influence both on the magnitude and the depth of the maximum predicted damage. When using the criterion (), a DV (and the hydrostatic stress) has no influence and the predicted depth of maximum damage is identical to the depth of maximum shear stress amplitude. Predicted Damage per Cycle D (a) Vertical load: kn Vertical load: 75 kn a Depth to maximum damage [m] 5.5 x (b) Vertical load: kn wheel diameter.88 and. m Vertical load: 75 kn a Fig. 7. Calculated influence of material parameter a in Eq.(3). Dashed lines correspond to the wheel diameter.88 m. and solid lines to. m. (a) Predicted magnitude of damage and (b) Predicted depth below wheel tread of maximum damage In Fig. 8, the calculated influence of τ e on the magnitude of predicted damage is displayed. Predicted Damage per Cycle.5 x (a) Vertical load: 75 kn Vertical load: kn τ x e [Pa] 8 Predicted Damage per Cycle.5 x (b) Vertical load: kn Vertical load: 75 kn τ e [Pa] x 8 Fig. 8. Calculated influence of material parameter τ e. Magnitude of damage predicted by (a) Eq.(3) and (b) Eq.()

10 4. CONCLUDING REMARKS AND FUTURE WORK A critical-plane approach has been employed to model high cycle fatigue of railway wheels. A previously presented fatigue life model [] has been modified and the calculated numerical results have been compared to workshop statistics from wheel reprofiling. The modified model is very fast and the results are somewhat encouraging even though several simplifications have been employed. In future studies, the effect of plastic deformations should be included. A possible approach would be to study the elastoplastic contact problem by use of a FEprogram. The calculated residual stresses would then also be included in an elastic fatigue analysis using WLIFE. Acknowledgements This work has been performed within the National Centre of Excellence CHARMEC, which has been established by the Swedish National Board for Industrial and Technical Development, NUTEK. The study is part of a joint research project involving the wheelset manufacturer Adtranz Wheelset, the train manufacturer Adtranz (Mechanical Systems Division), and the Division of Solid Mechanics at Chalmers University of Technology. The work has been supervised by Dr Roger Lundén and Professor Bengt Åkesson. Valuable discussions have been had with Mr Nils Nilstam of Adtranz and with Mr Magnus Johansson and Professor Hans Andersson at the Division of Solid Mechanics. The author would also like to thank several other friends and colleagues for help and comments. 5. REFERENCES [] Ekberg, A. Bjarnehed, H.: Rolling Contact Fatigue of Wheel/Rail Systems A Literature Survey. Report F8, Chalmers University of Technology, Division of Solid Mechanics, Göteborg, 995. (5 pp.) [] Ekberg, A., Bjarnehed, H. and Lundén, R.: A Fatigue Life Model for General Rolling Contact with Application to Wheel/Rail Damage, Fatigue & Fracture of Engineering Materials & Structures, 8(), 995. (pp.89-99) [3] Dang Van, K., et al., Criterion for High Cycle Fatigue Failure under Multiaxial Loading, in Biaxial and Multiaxial Fatigue. Mechanical Engineering Publications, London, 989. (pp ) [4] Dang Van, K.: Macro-micro Approach in High Cycle Multiaxial Fatigue, in Advances in Multiaxial Fatigue ASTM STP 9, 993. (pp.-3) [5] Lemaitre, J. Chabouche, J.L.: Mechanics of Solid Materials, Cambridge University Press, Avon, 99. (556 pp.) [6] Dang Van, K., Griveau, B. and Message, O.: On a New Multiaxial Fatigue Limit Criterion: Theory and Application, in Biaxial and Multiaxial Fatigue. Mechanical Engineering Publications, London, 989. (pp ) [7] Frost, N.E., Marsh, K.J. and Pook, L.P.: Metal Fatigue, Clarendon Press, Oxford, 974. (499 pp.) [8] Checking of Fatigue Criterion Proposed by Mr. Dang Van. Question C 53, Report 8, Pt. 3, European Rail Research Institute, Utrecht, 973. (pp.-3) [9] Stenström, R.: A Continuum Mechanics Approach to High Cycle Fatigue Modelling. Licentiate Thesis, Lund Institute of Technology, Division of Solid Mechanics, Lund, 996. (63 pp.) [] Suresh, S.: Fatigue of Materials, Cambridge University Press, Cambridge, 99. (67 pp.) [] Lundén, R.: Cracks in Railway Wheels under Rolling Contact Load, in th International Wheelset Congress, Sydney, 99. (pp.63-67) [] Giménez, J.G. Sobejano, H.: Theoretical Approach to the Crack Growth and Fracture of Wheels, in th International Wheelset Congress, Paris, 995. (pp.5-)