1 Introduction

Wheel/rail contact fatigue is always a serious problem for railways, especially for high-speed railways, but it is difficult to solve so far [1]. Wheel/rail contact fatigue increases the operating costs and endangers the safety of trains. The failure mechanism in wheel/rail contacts is very complicated, and many vague aspects remain to be studied. The main damage form of high-speed rails is governed by fatigue crack growth [2]. Plastic deformation layers will form and accumulate in rails after repeated rolling compaction. When the plastic deformation reaches a threshold value, micro-cracks are generated, which may further grow into macro-cracks [3]. The crack propagation rate of rail surface would become smaller when the crack propagated to a certain level [4].

The crack propagation of rails has been an important research direction in the field of wheel/rail contact fatigue. Criterions to predict crack propagation direction were proposed by many previous works, such as the maximum circumferential tensile stress criterion [5], the minimum strain energy density factor criterion [6], the maximum energy release rate criterion [7] and an empirical formula [8]. These criterions can be used to predict crack propagation direction under proportional monotonic loads, but they cannot be applied to random loads. For the crack propagation problem under complex loads, an infinitesimal branch crack needs to be established at the tip of the main crack, and the crack propagation direction can be determined by the stress intensity factor or the propagation rate of the branch crack. This method was applied to predict crack propagation direction under complex loads by some researchers [9,10,11,12,13,14]. However, the applicable conditions of this method remain disputable and unclear. Hence, it is not a mature method for predicting crack propagations.

The load paths of the rail crack in wheel/rail contact are different from conventional experimental load paths in that the crack propagation direction under wheel/rail contact is uncertain. In this paper, the probabilistic method is applied to predict the crack propagation direction. The results preliminarily demonstrate that it is reasonable to use the average value of possible crack propagation directions as the crack propagation direction.

2 Methodology

2.1 Research model

A research model of wheel/rail contact as shown in Fig. 1 is built. In this model, the wheel rolls forward with a speed of v and without acceleration. Although there is no whole sliding between wheel and rail, the local sliding and adhesion still exist in the contact zone. G is the weight of the wheel. M e is the driving moment. F w is wind resistance. The rail surface contains a micro-crack before the wheel rolls over the rail. The contact pressure is p, and the contact friction is f.

Fig. 1
figure 1

Model for wheel/rail in rolling contact

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2.2 Strain-rate effect of wheel/rail contact

The strain rate in the contact zone is relatively large because the wheel rolls on the rail at a high speed. The U71Mn steel, as the rail material, shows an obvious strain-rate effect when the strain rate is relatively large. The strain-rate characteristics of U71Mn steel can be given as follows [15, 16]:

$$\sigma _{{\text{s}}} = \left\{ {\begin{array}{ll} {\sigma _{{{\text{s}}0}} } & {{\text{when}}\quad \dot{\varepsilon }_{{{\text{eq}}}} \le 1\,{\text{s}}^{{ - 1}} ,} \\ {\sigma _{{{\text{s}}0}} + \frac{{(\sigma _{{{\text{s}}1}} - \sigma _{{{\text{s}}0}} ){\mkern 1mu} {\text{lg}}\,\dot{\varepsilon }_{{{\text{eq}}}} }}{{{\text{lg}}\,\dot{\varepsilon }_{{{\text{eq1}}}} }}} & {{\text{when}}\quad 1\,{\text{s}}^{{ - 1}} \le \dot{\varepsilon }_{{{\text{eq}}}} \le 300\ {\text{s}}^{{ - 1}} ,} \\ {\sigma _{{{\text{s1}}}} + \frac{{(\sigma _{{{\text{s2}}}} - \sigma _{{{\text{s1}}}} ){\text{lg}}\,\dot{\varepsilon }_{{{\text{eq}}}} }}{{({\text{lg}}\,\dot{\varepsilon }_{{{\text{eq2}}}} - {\text{lg}}\,\dot{\varepsilon }_{{{\text{eq1}}}} )}}} & {{\text{when}}\quad 300\ {\text{s}}^{{ - 1}} \le \dot{\varepsilon }_{{{\text{eq}}}} \le 450\ {\text{s}}^{{ - 1}} ,} \\ \end{array} } \right.$$
(1)

where σ s is the yield stress at the strain rate \(\dot{\varepsilon }_{\text{eq}}\); σ s0 = 550 MPa is the yield stress in quasi-static state; σ s1 = 637 MPa is the yield stress at the strain rate \(\dot{\varepsilon }_{{{\text{eq}}1}} = 300\,{\text{s}}^{ - 1}\); σ s2 = 738 MPa is the yield stress at the strain rate \(\dot{\varepsilon }_{\text{eq1}} = 450\,{\text{s}}^{ - 1}\); \(\dot{\varepsilon }_{\text{eq}}\) is the total strain rate of the material deformation, and it can be expressed as

$$\begin{aligned} \dot{\varepsilon }_{\text{eq}} & = \frac{1}{\sqrt 2 (1 + \nu )}[(\dot{\varepsilon }_{x} - \dot{\varepsilon }_{y} )^{2} + (\dot{\varepsilon }_{y} - \dot{\varepsilon }_{z} )^{2} \\ \, & + (\dot{\varepsilon }_{z} - \dot{\varepsilon }_{x} )^{2} + \frac{3}{2}(\dot{\gamma }_{xy}^{2} + \dot{\gamma }_{yz}^{2} + \dot{\gamma }_{zx}^{2} )]^{{\frac{1}{2}}} , \\ \end{aligned}$$
(2)

where \(\dot{\varepsilon }_{x}\), \(\dot{\varepsilon }_{y}\) and \(\dot{\varepsilon }_{z}\) are components of the linear strain rate; \(\dot{\gamma }_{xy}\), \(\dot{\gamma }_{yz}\) and \(\dot{\gamma }_{zx}\) are components of the shear strain rate; and v is Poisson’s ratio of the material.

2.3 Maximum circumferential tensile stress criterion

Erdogan and Sih [5] proposed the maximum circumferential tensile stress criterion in 1963. Based on the criterion, the crack propagation direction can be given by

$$\left. \begin{aligned} \theta = 2\tan^{ - 1} \left[ {\frac{1}{4}\frac{{K_{\text{I}} }}{{K_{\text{II}} }} - \frac{1}{4}\sqrt {\left( {\frac{{K_{\text{I}} }}{{K_{\text{II}} }}} \right)^{2} + 8} } \right],\quad K_{\text{II}} > 0 \hfill \\ \theta = 2\tan^{ - 1} \left[ {\frac{1}{4}\frac{{K_{\text{I}} }}{{K_{\text{II}} }} + \frac{1}{4}\sqrt {\left( {\frac{{K_{\text{I}} }}{{K_{\text{II}} }}} \right)^{2} + 8} } \right],\quad K_{\text{II}} < 0 \hfill \\ \end{aligned} \right\},$$
(3)

where θ is the crack propagation direction defined with a positive value in the counterclockwise direction and a negative value in the clockwise direction; K I and K II are stress intensity factors of types I and II, respectively. The singular element is employed in the crack tip, as shown in Fig. 2. The stress intensity factors at the crack tip can be obtained by the displacement extrapolation method [17]:

$$\left. \begin{aligned} K_{\text{I}} = \frac{\mu }{\kappa + 1}\sqrt {\frac{2\uppi }{L}} \,\left[ {4(v_{b} - v_{d} ) + v_{e} - v_{c} } \right] \hfill \\ K_{\text{II}} = \frac{\mu }{\kappa + 1}\sqrt {\frac{2\uppi }{L}}\, \left[ {4(u_{b} - u_{d} ) + u_{e} - u_{c} } \right] \hfill \\ \end{aligned} \right\},$$
(4)
$$\mu = \frac{E}{2(1 + \nu )},$$
(5)
$$\kappa = \left\{ \begin{aligned} 3 - 4\nu \quad {\text{plane strain}} \hfill \\ {{(3 - \nu )} \mathord{\left/ {\vphantom {{(3 - \nu )} {(1 + \nu )}}} \right. \kern-0pt} {(1 + \nu )}}\quad {\text{plane stress}} \hfill \\ \end{aligned} \right.,$$
(6)

where E is elasticity modulus; L is the length of the element; u i is the nodal displacement in direction x and v i is the nodal displacement in direction y in the local coordinate system, in which i = b, c, d, e represents the number of nodes, as shown in Fig. 2.

Fig. 2
figure 2

Quarter-points elements at the crack tip

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2.4 Weibull distribution

The probability density function f(x) and cumulative probability function F(x) of Weibull distribution are given as follows:

$$f(x) = \left\{ {\begin{array}{ll} {\frac{\beta}{\alpha }(x - \gamma)^{\beta - 1} \exp \left[ {-\frac{{\left( {x - \gamma } \right)^{\beta }}}{\alpha}} \right]} \hfill & {x \ge \gamma } \hfill \\ 0 \hfill & {x< \gamma } \hfill \\ \end{array} ,} \right.$$
(7)
$$F(x) = \left\{ {\begin{array}{l} {1 - \exp \left[ { - \frac{{\left( {x - \gamma } \right)^{\beta } }}{\alpha }} \right]} \hfill \\ 0 \hfill \\ \end{array} } \right.\quad \gamma\,<\, x\, <\, \infty ,$$
(8)

where γ is the location parameter, β is the shape parameter and α is the scale parameter. The probability coordinate paper method [18] is employed to determine whether the data are satisfied to Weibull distribution. Equation (8) can be transformed into

$$\ln \left[ {\ln \frac{1}{1 - F(x)}} \right] = \beta \ln \left( {x - \gamma } \right) - \ln \alpha ,$$
(9)

by setting

$$Y_{i} = \ln \left[ {\ln \frac{1}{{1 - F(x_{i} )}}} \right],\quad X_{i} = \ln \left( {x_{i} - \gamma } \right),$$
(10)

where x i is a random variable; the data satisfy the Weibull distribution when (X i , Y i ) complies with a linear distribution.

2.5 Nonparametric bootstrap method

For the problem that the sample capacity n from a certain distribution is known but the overall distribution is unknown, its overall statistical distribution can be inferred by the bootstrap method [19]. Let

$$x = (x_{1} ,x_{2} , \ldots ,x_{n} ) ,$$
(11)

where x is a known sample from the overall F. The bootstrap sample can be obtained by sampling with replacement successively and independently from the sample x. The mean values of bootstrap samples are calculated and sequenced as follows:

$$\omega_{(1)} \le \omega_{(2)} \le \cdots \le \omega_{(B)} .$$
(12)

Setting

$$k_{1} = \left[ {B \times \frac{\alpha }{2}} \right],\quad k_{2} = \left[ {B \times \left( {1 - \frac{\alpha }{2}} \right)} \right],$$
(13)

the bootstrap confidence interval can be obtained as

$$\left( {\bar{X} - \omega_{{(k_{2} )}} \frac{S}{\sqrt n },\quad \bar{X} - \omega_{{(k_{1} )}} \frac{S}{\sqrt n }} \right) ,$$
(14)

where \(\bar{X}\) and S are the mean value and standard deviation of the sample x, respectively. For the confidence interval, the confidence coefficient is 1 − α.

3 Finite element simulation

U71Mn steel is applied as the rail material in this work, and its mechanical properties [16, 20] are listed in Table 1.

Table 1 Material parameters of U71Mn steel
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The contact forces and crack propagations in rails are investigated comparatively at two different train speeds: a high speed of 350 km/h and a low speed of 50 km/h. The weight of each wheel is 5 t. The wheels roll on the rail steadily without any relative slip. The equivalent wind resistance in every wheel is 38 N for trains at the low speed (50 km/h) and 1018 N for trains at the high speed (350 km/h), which are obtained in CRH3 high-speed trains with eight carriages [21, 22].

A two-dimensional finite element model is established. In this model, the height is 176 mm and the length is 1000 mm. The rail bottom is fully restrained. As shown in Fig. 3, there is an inclined edge crack in the surface, and a singular element is employed in the crack tip.

Fig. 3
figure 3

Finite element model of rail crack

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The crack angle is defined as the angle between the crack line and the rolling direction. The initial length of the crack is 100 μm, and the initial angle is 30°.

In the process that the wheel rolls over the crack, the crack tip is subjected to a random fatigue load spectrum with multiple peaks, which cannot be expressed into a sine wave or a saw tooth wave as usual.

On the one hand, the low-stress amplitude has a little effect on the crack propagation. On the other hand, the crack propagation rate decreases sharply after a wave peak due to the overload retardation effect. Therefore, the load segments of low-stress amplitude are removed from the fatigue analysis [23].

During the process that the wheel rolls over the crack, the direction of the maximum circumferential tensile stress around the crack tip changes continuously, making the crack propagate in an indeterminate direction. For this reason, the probability and statistics method is applied to analyze the crack propagation direction, by using the load segment after being cut as the target of sampling. For every cycle, the same number of samples are taken in the processes of loading and unloading.

The analysis of crack propagation process is conducted as follows: First, the initial crack is established with a length of 100 μm and an angle of 30°. Then, the stress intensity factors are obtained in the process of the wheel rolling over the crack. The crack propagation direction, in which the crack grows 100 μm, is determined by the probability and statistics method. Repetitive computations are made in the same way till the crack path is obtained.

4 Results and discussion

4.1 Distribution of contact forces

Considering the strain-rate effect and the wind resistance, the contact forces between wheel and rail are obtained as shown in Fig. 4, where the abscissa axis denotes the size of the contact zone, and the vertical coordinate denotes the contact forces p and f. Figure 4 reveals that the contact pressure with low-speed trains is different from that with high-speed trains. The distribution of contact pressure for low-speed trains is close to a Hertz pressure distribution, while the distribution of contact pressure for high-speed trains has two peaks and hence cannot be replaced directly by Hertz pressure distribution. In addition, there also exists obvious difference in friction force between high-speed trains and low-speed trains, which is thought due to the wind resistance. This difference will further affect the crack propagation.

Fig. 4
figure 4

Distribution of contact and friction forces between wheel and rail

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4.2 Micro-crack propagation

The equivalent stress intensity factor K eff is calculated by considering the combined action of K I and K II as follows [18]:

$$K_{\text{eff}} = 0.5K_{\text{I}} + 0.5\sqrt {K_{\text{I}}^{2} + 4(1.155K_{\text{II}} )^{2}} .$$
(15)

The variation of K eff at the crack tip versus the crack length for different speeds is depicted in Fig. 5. According to Paris’ formula, the rate of the crack propagation increases with ΔK eff increasing. K eff is zero after the wheel rolls over the crack. Therefore, the maximum value of ΔK eff is equal to the maximum value of K eff. The results in Fig. 5 show that the K eff for high-speed trains (350 km/h) is larger than that for low-speed trains (50 km/h) in most of the time. This means that the crack propagation rate for high-speed trains is faster than that for low-speed trains.

Fig. 5
figure 5

Variation of K eff with the crack length

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The variation of K I and K II at the crack tip versus the crack length is depicted in Fig. 6. It shows that at the 50 km/h train speed (Fig. 6a), K I is larger than K II when the crack length is less than 0.8 mm, while K I is less than K II when the crack length is larger than 0.8 mm. At the 350 km/h train speed (Fig. 6b), however, K I is always larger than K II.

Fig. 6
figure 6

Variation of stress intensity factor at crack tip with the crack length for different train speeds: a 50 km/h; b 350 km/h

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The results in Figs. 5 and 6a show that there is an inflection point when the crack length is 1.3 mm, which is due to the crack swerving in propagation. The stress intensity factor K II increases with the crack length increasing, and hence, the crack face slides intensively. After the crack swerves, the crack propagates in the opposite direction of the train. The sliding effect of crack surface decreases immediately. Accordingly, K II and K eff decrease sharply.

For the process that the wheel rolls over the crack, the possible crack propagation directions are analyzed by the probability and statistic method. The analysis results are listed in Tables 2 and 3, where the correlation coefficient l is obtained by data fitting based on the probability paper to show the possibility that the data satisfy the Weibull distribution. For instance, the correlation coefficient value of 0.955 means that the possibility of the data satisfying a Weibull distribution is 95.5%.

Table 2 Statistical results of rail crack propagation directions for 50 km/h train
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Table 3 Statistical results of rail crack propagation directions for 350 km/h train
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Table 2 shows the results of low-speed trains (50 km/h). The correlation coefficient is larger than 95% when the crack length is less than 1 mm, implying that the possibility of the data satisfying Weibull distribution is relatively high at this stage. The correlation coefficient is less than 86% when the crack length is larger than 1 mm and less than 1.4 mm, which means that the possibility of the data satisfying Weibull distribution is relatively small at this stage. The correlation coefficient is larger than 97.5% when the crack length is larger than 1.4 mm, so there is a great possibility for the data to satisfy the Weibull distribution at this stage.

Table 3 shows the results of high-speed trains (350 km/h). The correlation coefficient is always larger than 90%, and even larger than 95% at the most time, which indicates a high possibility of the data satisfying the Weibull distribution in the process of crack propagation for high-speed trains (350 km/h).

The simulated crack propagation paths in rail and the experimental crack paths in rail of Datong–Qinhuangdao line [24] are shown in Fig. 7. From Fig. 7a, b, one can see that the crack paths for both 50 and 350 km/h trains present approximately the same trend: In the first stage, the crack propagates in a direction with an acute angle between the crack propagation direction and the train running direction. In the second stage, the crack propagates in a direction nearly perpendicular to the train running direction. Finally, the crack propagates in the opposite direction of the train running. The crack path simulated is consistent with the experimental crack path in the rail from Datong–Qinhuangdao line, which proves the rationality to use the average value of crack propagation as crack propagation direction.

Fig. 7
figure 7

Comparison of crack paths between simulation and experiment: a Simulated crack path for 50 km/h train. b Simulated crack path for 350 km/h train. c Experimental crack path in Datong–Qinhuangdao line rail [24]

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In addition, the deflection of crack propagation for high-speed trains occurs earlier than that for the low-speed trains, implying a faster strip failure in rails for high-speed trains.

5 Conclusions

In the paper, the crack propagation of wheel/rail contact fatigue is investigated using the strain-rate effect and statistics and probabilistic method. Some conclusions can be summarized as follows:

  1. (1)

    Distributions of wheel/rail contact forces are different between low-speed trains and high-speed trains. The contact pressure produced by high-speed trains cannot be replaced by Hertz pressure.

  2. (2)

    The mode of rail crack propagation is different between low-speed trains and high-speed trains. Under 50 km/h trains, the rail crack propagates mainly in an opening mode in the beginning and then develops mainly in a sliding mode. However, under 350 km/h trains, the opening mode always plays a leading role in the process of crack propagation. Furthermore, the crack propagation rate in rails for high-speed trains is faster than that for low-speed trains.

  3. (3)

    For 50 km/h trains, there is a high possibility that the rail crack propagation direction satisfies the Weibull distribution only when the crack length is less than 1 mm and larger than 1.4 mm. For 350 km/h trains, however, the possibility of the crack propagation direction satisfying the Weibull distribution is always high.

  4. (4)

    The simulated rail crack propagation paths are consistent with the experimental ones, which proves that it is reasonable to use the average value of possible crack propagation directions as the actual crack propagation direction.