# Prediction of Rolling Contact Fatigue Behavior in Rails Using Crack Initiation and Growth Models along with Multibody Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Methodology

^{®}, where calls to several processes of the MBS software and the mathematical computing of the physical models are carried out. The flowchart of Figure 1 shows the methodology process, divided into five main sections explained in the subsections below.

#### 2.1. Input Data

- Vehicle load condition.
- Profile and degradation level of rails.
- The number of vehicles per day and the number of bogies of a unit.
- Track quality, i.e., irregularities.
- Vehicle speed and traction/braking effort.
- Wheel/rail friction coefficient.

#### 2.2. Multibody Simulations

#### 2.3. Crack Initiation Model

- For low values of $T\gamma $, the rail surface is not affected because there is not enough work to cause incremental plastic straining of the material.
- Afterwards, the risk of RCF cracks starts to increase linearly with energy dissipation.
- Then this risk decreases when wear becomes the dominant form of surface damage.
- Finally, high values of wear number prevent crack initiation. This occurs because the effect of wear damage is higher than the propagation of a theoretical crack, so this crack will never appear.

#### 2.4. Crack Growth Model

#### 2.5. Output Results

## 3. Experimental Validation

#### 3.1. Case Studies

^{®}, where calls to several processes of the MBS software and the mathematical computing of the physical models are carried out. The key conclusions are the following:

- No cracks appear at any straight lines for both sections.
- Section 1 shows a spread distribution of different depths at the circular curve with less effect at clothoids.
- Measurements for Section 2 are rather punctual without a clear pattern.

#### 3.2. Results

- No cracks have been predicted for any straight track.
- Different crack depths at the whole circular curve and some cracks at clothoids are predicted for Section 1
- Punctual cracks are predicted for Section 2

## 4. Discussion and Conclusions

- In the first section under analysis, the predictions show clearly the same RCF pattern that is experimentally observed in the real track. The methodology assumes the existence of RCF problems at specific locations. The predicted results at straight lines and circular curves show a good relationship with measurements. On the other hand, although similar conclusions might be obtained for clothoids, relationship is not so accurate. Consequently, these transitions must be treated carefully, because the vehicle dynamic behavior is subjected to changes between different track sections.
- In the second track section, only isolated cracks without a clear pattern are measured. The prediction of isolated cracks may be due to numerical effects owing to the MBS irregularities file. In this section, no RCF problems are predicted, since calculated cracks are few and very isolated. Nevertheless, the appearance of isolated cracks could be a sign of a potential breakage of the rail if preventive maintenance actions are not carried out on time.
- Predictions are good for small depth values because the initiation and minimum crack size are well controlled. Therefore, the scheduling of grinding maintenance task can be improved to avoid the propagation of cracks that could lead to rail failure.
- The difference in some crack growths could be due to different factors. First, a lack of information regarding the degradation state of the rails in the first measurement date. A second reason behind the difference for deeper cracks while the trend remains similar could be a lack of data to characterize the coefficients of the fourth-degree polynomial. Finally, the employed damage function was developed for normal grade rails but the material grade of installed rails could have small differences.
- $\Delta t$ and ${T}_{\mathrm{total}}$ have been considered equal. Smaller values of $\Delta t$ with an updating of track irregularities or rail profiles may vary results in ${T}_{\mathrm{total}}$.
- The methodology was implemented using models available in the literature. The database for crack propagation, i.e., ${a}_{\mathrm{min}}$ values and coefficients of fourth-degree can be updated by XFEM calculations [47]. This kind of numerical solution is suitable as has been proven by Bergara et al. [48].

- The location of predicted cracks and their growth allow predicting RCF behavior. The track sections can then be sorted against the chance of failure, reducing the LCC with the adequate scheduling of the consequent maintenance programs. The high reliability in the prediction of the initiation of cracks makes possible a reduction of maintenance costs.
- The track quality and the evolution of rail wear can be considered along with the assessment of cracks. In the same way, this methodology can be used to support the decision of changes in running conditions and the design of track sections.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${V}_{\mathrm{wear}}$ | volume of worn material |

${k}^{\prime}$ | wear coefficient |

$P$ | normal force |

$s$ | sliding distance |

$H$ | material hardness |

${T}_{\mathrm{total}}$ | total prediction time |

$\Delta t$ | prediction time step |

$DI$ | damage index |

$x$ | longitudinal coordinate of any point of the rail |

$y$ | lateral coordinate of any point of the rail |

$n$ | types of vehicles |

$m$ | vehicles of the same type |

$wh$ | wheelsets on each vehicle |

$h$ | elliptic function over the contact patch |

$d$ | maximum damage on $x$ as a function of $\mu $ and $T\gamma $ |

$\mu $ | shear force coefficient |

$T\gamma $ | wear number |

$\gamma $ | creepages |

$d{a}_{\mathrm{net}}$ | net crack growth rate |

$T{\gamma}_{\mathrm{TP}}$ | wear number value at turning points |

${\sigma}_{\mathrm{y}}$ | yield strength |

${\sigma}_{\mathrm{UTS}}$ | ultimate tensile strength |

$A$ | contact area |

${\mathsf{\gamma}}_{\mathrm{TP}}$ | creepage at turning points |

$i$ | cross-sections with wheel-rail contact along the analyzed track |

$j$ | discretization of the cross-sections to store different parameters |

$W{f}_{\mathrm{cp}}$ | weighting function for contact point $cp$ |

$cp$ | contact point |

$ycr$ | lateral position of the contact patch center |

$wp$ | width of the contact patch |

$D{I}_{\mathrm{i},\mathrm{j}}$ | damage index of the analyzed cell |

$D{I}_{\mathrm{cp}}$ | damage index for contact point $cp$ |

$W{f}_{\mathrm{p},\mathrm{k}}$ | weighting function for contact point $p$ at location of $k$ |

$k$ | discretization of the analyzed cells |

${N}_{\mathrm{d}}$ | number of discrete points of the analyzed cells |

${N}_{\mathrm{c}.\mathrm{i}}$ | number of contact points on each cross-section i |

$TotalDI$ | total damage index matrix |

${T}_{\mathrm{calc}}$ | calculated time |

$da$ | crack growth rate |

$\theta $ | inclined angle from normal to rail surface to the crack face |

$dw$ | wear rate |

$tc$ | traction coefficients |

$a$ | crack depth |

${p}_{q}$ | polynomial coefficients, $q=0,\dots ,4$ |

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**Figure 2.**Rail rolling contact fatigue damage function for normal grade rails. Values are taken from [14].

**Figure 3.**DI distribution for each division of the rail cross-section. (ycr: lateral position of the contact patch center; wp: width of the contact patch).

**Figure 5.**Net Crack considering crack growth and rail wear. Adapted from [21].

**Figure 6.**Database values from [21] results: (

**a**) Crack growth fourth-degree polynomial considering contact stress and wear (tc = 0.2 and γ = 0.1%); (

**b**) Minimum crack length for different creepages and traction coefficients.

**Figure 11.**Number of cracks measured as a function of their depth, grouped by curvature type: (

**a**) Section 1; (

**b**) Section 2.

**Figure 12.**Location comparison between predicted and measured cracks for 4.752 MTn: (

**a**) Section 1; (

**b**) Section 2.

**Figure 13.**Comparison of the trend-lines between measurements and predictions for three different sections. Assessment with one measured cross-section per yard: (

**a**) Measurements; (

**b**) Predictions.

$\mathit{t}\mathit{c}$ | γ | ${\mathit{a}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\left[\mathbf{mm}\right]$ | ${\mathit{p}}_{4}$ | ${\mathit{p}}_{3}$ | ${\mathit{p}}_{2}$ | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{0}$ |
---|---|---|---|---|---|---|---|

0.2 | 0.1% | 0.26 | 7.53 × 10^{−6} | −3.05 × 10^{−4} | 2.28 × 10^{−3} | 2.49 × 10^{−2} | 3.10 × 10^{−2} |

Components | Parameters | Values |
---|---|---|

Vehicle | Load condition | 11.25 T/axle |

Number of cars per unit | 4 | |

Primary suspension | Rubber-metallic | |

Secondary suspension | Air-spring | |

Maximum speed | 160 km/h | |

Wheels | Type | P8 |

Diameter | 860 mm | |

Back-to-back distance | 1360 mm | |

Rails | Type | BS113a |

Inclination | 1/20 | |

Track gauge | 1435 mm | |

Track Section 1 | Speed | 95 km/h |

Mean curve radio | 720 m | |

Mean curvature | 138·× 10^{−5} × 1/m | |

Mean cross-level | 95 mm | |

Cant deficiency | 53 mm | |

Track Section 2 | Speed | 120 km/h |

Mean curve radio | 1100 m | |

Mean curvature | 91·× 10^{−5} × 1/m | |

Mean cross-level | 65 mm | |

Cant deficiency | 90 mm |

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**MDPI and ACS Style**

Rodríguez-Arana, B.; San Emeterio, A.; Alvarado, U.; Martínez-Esnaola, J.M.; Nieto, J.
Prediction of Rolling Contact Fatigue Behavior in Rails Using Crack Initiation and Growth Models along with Multibody Simulations. *Appl. Sci.* **2021**, *11*, 1026.
https://doi.org/10.3390/app11031026

**AMA Style**

Rodríguez-Arana B, San Emeterio A, Alvarado U, Martínez-Esnaola JM, Nieto J.
Prediction of Rolling Contact Fatigue Behavior in Rails Using Crack Initiation and Growth Models along with Multibody Simulations. *Applied Sciences*. 2021; 11(3):1026.
https://doi.org/10.3390/app11031026

**Chicago/Turabian Style**

Rodríguez-Arana, Borja, Albi San Emeterio, Unai Alvarado, José M. Martínez-Esnaola, and Javier Nieto.
2021. "Prediction of Rolling Contact Fatigue Behavior in Rails Using Crack Initiation and Growth Models along with Multibody Simulations" *Applied Sciences* 11, no. 3: 1026.
https://doi.org/10.3390/app11031026