Gaps, challenges and possible solution for prediction of wheel–rail rolling contact fatigue crack initiation

It is essential to establish a relationship between the friction coefficient and the maximum contact pressure among the four types of material response regions shown in Fig. 3 [24]. For this purpose, diagrams called “shakedown maps” that provide the material response showing the location for the greatest fatigue damage were first proposed by Johnson based on 2D numerical calculation (Fig. 4) [35, 36]. This shakedown map is aimed at the line contact situation, and the shakedown limit by the Tresca criterion is reached when P₀/k = 1/μ. Then Ponter et al. [37] proposed the shakedown map for the more realistic case of point contact as shown in Fig. 5. This map is more representative of railway practice and is widely used in later research works. Surface or subsurface material damage is illustrated by the critical location of fatigue damage. The ordinate value, treated as a load factor, is the ratio of the maximum normal contact pressure p₀ to the ductility of material represented by the shear yield strength k. Meanwhile, the friction forces in the contact are characterized by traction coefficient μ on the abscissa of the map [15]. It should be mentioned that, although the shear yield strength k is commonly obtained using the Tresca criterion \(k={\sigma }_{\text{y}}/\sqrt{3}\) (\({\sigma }_{\text{y}}\) is the material tensile yield strength) [38, 39], some researchers pointed out that k can be estimated from the hardness measurement of the material. Jones et al. [40] considered the variation of the shear yield strength with depth after the material had been strain hardened, and put forward a relationship between k and the Vickers hardness HV: \(k={HV}/3\sqrt{3}\). The crack tends to initiate when the investigated point falls into the plastic shakedown and ratcheting region. Because the shakedown map can be used to obtain a first prediction of the material response and the expected location of the most serious fatigue damage under a specific contact load and geometry, it is reliable to predict the location of the initiation of the RCF crack based on the results of various wheel/rail models. Some researchers have used the input parameter (normal contact pressure) and the output parameter (friction coefficient) from experimental studies to estimate the material response using the shakedown map [41, 42]. However, as the field dynamic characteristics (traction force, track excitation, etc.) are difficult to determine in the experimental study, the finite element method (FEM) and multi-body system (MBS) analysis are the common and valuable tools used to analyze the wheel–rail contact problems in recent research studies.

The 2D FE model is the basic and simplified wheel/rail simulation model established in ABAQUS as shown in Fig. 6 according to Ref. [19]. The model only includes a rail disc sample, driven by a moving load which represents the function of the wheel disc. The full-slip contact condition was considered because it represents more damaging to the material, and no changes in the contact patch caused by plasticity were introduced in the model. Different combinations of contact pressure and friction coefficient are performed by simulations to obtain elastic shakedown, plastic shakedown and ratcheting regions in the shakedown map (Fig. 4). Then, the crack initiation criteria (e.g., crack initiation angle) are evaluated in the different regions of the shakedown map [19].

Comparing with 2D FEM, the 3D model is closer to the real wheel and rail which can better express the stress state and the contact patch properties of the rolling contact. Figure 7a displays a 3D FEM of the wheel/rail interface proposed by Vo [33] in the finite element code ANASYS/LS-DYNA. Due to the consideration of the wheel vibration, wheelset–vehicle and the track–vehicle interaction in wheel/rail contact stress were analyzed since they might increase the magnitude of the stress. The wheel was simulated as a rigid body with a rotational speed under a vertical load exerted on it. Also, a constant coefficient of friction was used. Then the properties of wheel/rail materials and track components were applied into the FEM. It was necessary for the mesh size of the model to be fully taken into consideration as it would affect the accuracy of the simulation results [1]. The simulation results for the maximum contact load p₀ and the friction coefficient (0.45) were input together into the shakedown map (Fig. 5) and plotted in Fig. 7b. This showed that all the investigated point fall into the ratcheting region (the area that contains the triangle in red). This meant that the damage tended to occur on the material, especially on the surface rather than in the sub-surface.

The main purpose of introducing the multi-body system (MBS) into the wheel–rail RCF analysis is that the vehicle dynamic characteristic can be taken into account in MBS, involving some parameters of vehicle components (e.g., mass, moment of inertia, suspension parameters, wheel profiles) and track parameters (curve radius, super elevation, rail profiles, track excitation). Meanwhile, the high efficiency and time-saving of MBS make it suitable to find a working condition which can lead to RCF initiation. The creep rate, creep force, normal force, contact patch size and maximum contact stress obtained by MBS provide the basic data for wheel/rail RCF prediction modeling. Lu [43] and Wang et al. [44] analyzed the location that is more likely to initiate cracks based on a vehicle system dynamic model using the dynamics software SIMPACK. Various vehicle and track parameters were included in this model. The parameter of the abscissa on the shakedown map is the traction coefficient \(\mu \), calculated by Eq. (1). Maximum contact pressure P₀, F_x and F_y are obtained from dynamic simulation while k is the constant value of the material.where F_x and F_y are the vertical and lateral creep force in analyzed step respectively, and F_n is the normal force between wheel and rail.

Anyway, the differences of the shakedown map input parameters (P₀, k, \(\mu \)) in different research methods (experiment, 2D FE model, 3D FE model and MBS model) are summarized in Table 1. It also indicates the benefits and limitations of analyzing the location where cracks initiate by the shakedown map in different research methods.

Various crack initiation criteria were used in the plastic shakedown region. The core of these criteria is to study how they perform when they are applied on RCF problems. The comparisons of criteria are illustrated in Table 3.

A simple form of SWT (Smith Watson Topper) damage parameter for predicting life under uni-axial loading was first proposed by Smith et al. [58] in 1976. However, the SWT parameter was used after a series of modifications in multi-axial proportional and non-proportional loading for materials with cracks. The most popular form of the SWT model is mentioned by Socie [34] with the consideration of mean stress as described in Eq. (8). The material plane with the largest value of the fatigue parameter, FP_max, is recognized as the critical plane.where \({\sigma }_{1}^{\mathrm{max}}=\left({\sigma }_{1}^{\text{m}}+\frac{\Delta {\upsigma }_{1}}{2}\right)\) is the maximum normal stress in the plane with normal n (critical plane which is mentioned in Sect. 4.3); \(\frac{\Delta {\varepsilon }_{1}}{2}\) is the maximum principal strain amplitude (Fig. 3); \({\sigma }_{\mathrm{f}}^{\mathrm{^{\prime}}}\) is the fatigue strength coefficient; \({\varepsilon }_{\mathrm{f}}^{\mathrm{^{\prime}}}\) is the fatigue ductility coefficient; E is Young’s modulus; b is the fatigue strength exponent; c is the fatigue ductility exponent; N_f is the number of cycles to failure [59, 60].

It should be noted that this criterion is only valid for \({\sigma }_{1}^{\mathrm{max}}\)>0, but in real rolling contact fatigue situations, \({\sigma }_{1}^{\mathrm{max}}\) often refers to a negative value due to the compressive loads. Therefore, a modification of \({\sigma }_{1}^{\mathrm{max}}\) was suggested by Beste in 1981. Only the stress amplitude \(\frac{\Delta {\sigma }_{1}}{2}\) is included and the compressive mean stress \({\sigma }_{1}^{\mathrm{m}}\) is neglected. This criterion was used as SWT/BESTE for predicting the crack initiation life [24].

Later in 1988, the additional hardening and mean stress effects were taken into account through the normal stress by Fatemi and Socie [61] based on axial fatigue data:where \(\Delta {\gamma }_{\mathrm{max}}\) is the maximum shear strain range; \({\sigma }_{\text{n}}^{\text{max}}\) is the maximum normal stress on the plane of \(\Delta {\gamma }_{\mathrm{max}}\); \({\sigma }_{\mathrm{y}}\) is the yield strength; k is a constant obtained from axial and torsional fatigue experimental data.

An adapted Coffin-Manson equation Eq. (10) was developed by Bannantine et al. [62] in 1990. In order to obtain the life predictions where material strain governs the life rather than stress, they calculated \(\frac{\Delta \gamma }{2}\) on the plane with the largest range of cyclic shear strain.where \(\Delta \gamma \) is the change in shear strain; \({\tau }_{\mathrm{f}}^{\mathrm{^{\prime}}}\) is the shear fatigue strength coefficient; \({\gamma }_{\mathrm{f}}^{\mathrm{^{\prime}}}\) is the shear fatigue ductility coefficient; G is the shear elastic modulus.

In order to predict the crack initiation thoroughly, in 1999, Chen et al. [63] postulated that both normal and shear components of stress and strain on the critical plane is significant in the study of material damage [64]. Two criteria (CXH equations) about material characterized by normal fracture (mode I) and shear type failure (mode II) are described in Eq. (11).where \(\Delta {\varepsilon }_{1}^{\mathrm{max}}\) is the maximum principal strain range; \(\Delta {\sigma }_{1}\), \(\Delta {\gamma }_{1}\) and \(\Delta {\tau }_{1}\) are the shear stress range that occur on the maximum principal strain range plane, respectively; \(\Delta {\gamma }_{\mathrm{max}}\) is the maximum shear strain range; \(\Delta \tau \), \(\Delta {\varepsilon }_{\mathrm{n}}\) and \(\Delta {\sigma }_{\mathrm{n}}\) are the shear stress range, normal strain range and normal stress range on the maximum shear strain plane, respectively [64].

Also in the year 1999, Jiang and Sehitoglu [34] proposed a crack initiation model which has a total strain energy density based formulation. It indicates that both normal and shear components of stress and strain contribute to the damage of the material on the critical plane. The model is expressed aswhere \(\langle {} \rangle\) is the MacCauley bracket, and \(\langle x\rangle =0.5\left(\left|x\right|+x\right)\); J is a constant from tests and commonly use J = 0.2 [34].

Note that a proper selection of J plays a vital role in modeling a material that displaying mixed normal and shear cracking. For instance, J = 0 reduces the criterion to the SWT model for the normal cracking behavior. And the model depicts shear cracking fatigue when J is a large number. Meanwhile, Jiang and Sehitoglu [34] came up with the relationship between life N_f and FP aswhere \({{FP}}_{0}\), m, and C are material fatigue properties determined from baseline experiments [23, 67].

Although some literature publications [23, 29] predict crack initiation life using this original relationship, researchers prefer to combine the J–S model with SWT criterion [17, 18, 23, 25, 28, 29, 35, 36, 42, 54, 55, 70‐74]. This updated criterion integrated the advantages in the above criterion. Also, it take into account of modeIand mode II fracture type according to CXH criterion. Therefore, the crack initiation life is more often calculated bywhere E and G are the elastic and shear modulus, respectively; \({\sigma }_{\text{f}}^{\mathrm{^{\prime}}}\) and \({\tau }_{\text{f}}^{\mathrm{^{\prime}}}\) are the tensile and shear fatigue strength coefficients; \({\varepsilon }_{\text{f}}^{^{\prime}}\) and \({\gamma }_{\text{f}}^{^{\prime}}\) are the tensile and shear fatigue ductility coefficients [53].

The model for predicting the crack initiation life in the ratcheting region was postulated by Kapoor [32]. The number of cycles to crack initiation is calculated aswhere \({\varepsilon }_{\mathrm{c}}\) is a material constant from experiments; \(\Delta {\varepsilon }_{i}\) and \(\Delta {\gamma }_{i}\) are the incremental normal and shear strain on the crack plane per load cycle.

One of the material responses mentioned above occurs when the material is subjected to cyclic constant amplitude loading. The prediction of the RCF life to crack initiation is determined by one of these criteria related with the material response clarified [28].

In many publications [17‐19, 23, 24, 28, 29, 53‐55, 66, 72, 74, 75], after the determination of FP_max, the crack initiation life is directly calculated using the prediction criteria mentioned above, and the value of stresses or strains needed in the criteria are consistent with the value in the critical plane. It is obvious that the maximum value of FP corresponds to the shortest life to crack initiation according to Ref. [29]. Otherwise, some researchers [25, 42, 71‐73, 76, 77] put the fatigue damage effect into the consideration of the period to crack initiation prediction. Radim et al. [42] and Ringsberg and Josefson [25] assumed that fatigue damage is accumulated linearly, and the damage within one cycle as a result of RCF can be expressed aswhere FP₀, K, and m are the material fatigue constants.

Furthermore, the evolution of the RCF cracks and the growth of wheel/rail wear reducing the material and changing the wheel/rail profiles. Thus, the coexistence of RCF and wear should be considered together. It should be mentioned that during simultaneous wear and RCF pre-existing cracks will lead to stress concentrations which will influence the crack initiation parameter. The Archard model is a wear model that is commonly being used in research activities [78, 79]:where V_m is the volume of material worn away; D is the sliding distance; N is the wheel–rail normal force; H is the hardness of the material; K is a wear coefficient [28, 80].

Bassevile and Gailletaud [65] investigated the evaluation of the competition between wear and crack initiation for alloy with Archard wear model based on SWT criterion. Zhou and his co-authors [71‐73, 76] proposed a calculation procedure of the period to crack initiation combined with fatigue damage effect as shown in Fig. 17. Firstly, the distribution of contact pressure and shear stresses were calculated using the software CONTACT based on the dynamic simulation results. And the damage analysis was according to the Archard damage theory. When wheels pass through, the equivalent wear of any point in the contact spot of the rail surface is calculated using Eq. (22):

where \(\Delta {z}_{y}\) is the abrasion amount of the longitudinal shadow cell of the contact spot when the abscissa is y; \(\Delta z\left(x,y\right)\) is the abrasion at cell (x, y); m and l are the number of vertical and horizontal cells of contact spot, respectively.

Then, each rail profile and the distribution of the normal pressure and tangential traction are implemented on the 3D rail FE model. The FE model includes a long global rail and four wheels in two vehicle bogie. And the stress/strain calculated from FE model is used to estimate the crack initiation life. Later, once the accumulated wear at any point of the rail head reaches a certain value, the accumulated wear within the sliding area of the contact spot is superposed on the rail profile. And the profile is smoothed with cubic interpolation spline curve to obtain the wear profile [81]. Also, the accumulated number of wheel passing cycles n_i in the phase of profile replacement is obtained. Then, the stage fatigue damage corresponding to wear stage n_i is

And according to the Miner theory [82], when the fatigue damage accumulation at any point of rail head in different wear stages reaches critical fatigue damage (Fig. 18),

At this point, it is considered that the fatigue crack is initiated and the initiation life iswhere N_f is the service life to crack initiation; and M is the replacement times of worn profile.

It should be noted that the Archard wear coefficient is being divided into four regions according to the sliding speed and contact pressure. And the specific value of each region needs to be determined by a large number of tests. Therefore, besides the Archard wear model, Bolton and Clayton [83] found that the wear rate of rail has a certain relationship with Tγ/A value (T tangential force at the wheel–rail interface; γ the creep coefficient; A the wheel–rail contact area) after rolling tests. Then, by analyzing a large number of wheel/rail material wear data from different test methods, Lewis et al. [84‐86] established the relationship between wheel/rail material wear rate and Tγ/A value under different working conditions. It can be seen that the value of Tγ/A—wear rate curve changes continuously with the Tγ/A value, which could be more accurate than Archard wear model in simulation calculation [87]. Later, Meli and his co-authors [88‐90] established an innovative model for the prediction of wheel/rail wear and RCF. Tγ/A was used and being improved to estimate the wheel/rail wear properly combined with the RCF model. The innovation model was used to update the wheel/rail profile, and provides a basis for the crack initiation life prediction when considering wear in future study.

Besides the plastic shakedown material response which caused by LCF as mentioned above, ratcheting effect is another main part that contribute to the fatigue damage. Generally, ratcheting deformation leads to the large shear deformation material microstructure, while the plastic shakedown produces less visible damage [15]. Therefore, Kapoor pointed out that LCF and ratcheting are the independent and competitive mechanisms for fatigue crack initiation. Equation (13) should be compared with a RCF crack initiation criterion described above [17].

It should be noted that Eq. (13) is aimed at the material that experiencing a constant ratcheting rate. However, for most pearlitic rail steels, the ratcheting rate is typically not constant: when the cumulative deformation increases, the ratcheting rate decay can be observed. Then, a more general expression for the consideration of ratcheting-induced damage per cycle is proposed by Jiang [34] aswhere \({D}_{\mathrm{r}}\) is the ratcheting damage; \({\gamma }_{\mathrm{cri}}\) is a material constant about the ductility of the material; \({\gamma }_{\mathrm{r}}\) is the orthogonal shear ratcheting strain; \(\mathrm{d}{\gamma }_{\mathrm{r}}/\mathrm{d}N\) represent the ratcheting rate.

Most of the literature publications [19, 23, 24, 29] just predict the crack initiation life under ratcheting effect directly using the Kapoor criterion as Eq. (13) and compare the results with the LCF service life to crack initiation. The crack initiation criterion which is satisfied in the smaller number of cycles will make the material tend to failure. Ringsberg et al. [19] proposed a ratcheting model based on elastic analysis which is supplemented by the elastic–plastic material behavior based on empirical, and the plastic deformation in a single cycle is predicted:where \(\mathrm\Delta Rd\) is the increment of plastic deformation in a single cycle; \({P}_{\mathrm{r}}\) is the ratcheting load; p is the applied contact pressure; \({\tau }_{\mathrm{max}}\) is the maximum shear stress determined for each layer in the model; \({k}_{\mathrm{eff}}\) is the “current” shear yield stress of the material; C is a material constant which is independent of stress [91]. For each pass of the load (cycle), the increment of one-way strain (\(\Delta Rd\)) is added to the total cumulative plastic deformation Rd. Failure is defined as Rd exceeding the critical value of strain \({\varepsilon }_{\mathrm{c}}\) determined by experiments. Furthermore, the results from this ratcheting model agree well with the results from the calculation using Kapoor criterion.

Other researchers [25, 42, 67] tended to take the ratcheting damage into account in the prediction of crack initiation life according to the assumption made by Jiang and Sehitoglo as shown in Eq. (26). If the material has low plasticity, LCF may occur in the form of crack initiation and propagation in a limited number of cycles. For ductile materials, extensive ratcheting strains can lead to the extrusion of thin surface slices. In both cases, the combination of ratcheting and fatigue can be used as a mean of failure prediction under these conditions. Jiang and Sehitoglo [34] proposed that the total damage is the sum of fatigue damage D_RCF and ratcheting damage D_r (Eq. (28)). When the total damage D reaches a unity, material fails.

However, Ringsberg and Josefson [25] mentioned that the damage of two mechanisms (LCF and ratcheting) should not be added in the same load cycle. It is accentuated if the damage caused by the two mechanisms is close in magnitude, which will lead to over conservative life prediction as Jiang and Sehitoglo did. Then a competitive approach is put forward in the summation of total damage. The largest damage component for the current load cycle is governed by the contribution of either RCF or ratcheting. The damage is assumed to accumulate linearly for every load cycle as

Material failure occurs when damage D reaches unity. This prediction approach seems more reliable and comprehensive, and is often used in the service life to crack initiation prediction. Radim et al. [42] mentioned that the comparison of ratcheting failure life and LCF failure life is needed in a life study. Once the former is bigger than the latter, material would experience the LCF failure life. And they came to the conclusion that surface and subsurface cracks were mainly caused by ratcheting and LCF, respectively.

Hence, if the shakedown map is used in the first step of crack initiation prediction as mentioned in Sect. 3; then it can be clear that the material is under the ratcheting effect or plastic deformation, and the accurate criterion will be used in the corresponding region.