# Investigation into Multiaxial Character of Thermomechanical Fatigue Damage on High-Speed Railway Brake Disc

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Finite Element Model

#### 2.1. Theoretical Background

#### 2.1.1. Thermal Formulation

#### 2.1.2. Thermal–Mechanical Solution

#### 2.2. FE Model Description

_{L1}, r

_{L2}and r

_{M}indicate the inner, outer and average friction radii. The geometry dimension can be found in Table 1. A total of 29 elements (denoted as E

_{R1}-E

_{R29}) are evenly distributed along radial direction (between r

_{2}and r

_{3}) and 3 elements (represented by E

_{A1}-E

_{A3}) are uniformly arranged from the contact surface into the brake disc with a distance of 2 mm between each other.

^{2}is employed under a brake force of 17,000 N, the corresponding brake time is 50 s and the total brake distance is 1736.1 m. The environment temperature is 20 °C.

## 3. Multiaxial Fatigue Model

_{1}

^{T1′}and σ

_{2}

^{T1′}indicate the response status at moment T2 following the principal directions of T1. C and R are the center and radius of Mohr circle, and A is twice the angle between the loading plane and the principal plane. It can be seen that the changing of the material state can be described by the changing of C, R and A. Therefore, the material response status is described by three parameters (P1, P2 and P3), which are the equivalent shear parameter, equivalent normal parameter and out-of-phase parameter, respectively [21]. The changing of them (three fatigue basic units, i.e., f1(P1), f2(P2) and f3(P3)) can be used for fatigue damage evaluation, and the multiaxial fatigue space can be established by fatigue basic units (see Figure 4) [21]. The multiaxial fatigue parameter is described in Equation (6):

## 4. Results and Discussion

_{R15}), which is the same as the rotating loading calculation. The comparison of the results among uniform loading, rotating loading and equivalent methods at the maximum temperature position is shown in Table 5. It can be seen that the temperature, thermal strain and Huber–Mises stress are approximately equal with each other. The results with rotating loading is used as reference. The temperature and strain with the equivalent method are quite satisfactory and the relative error is less than 0.1%. The difference of Huber–Mises stress is around 5% because of the history effect and constraint from the extrusion by the surrounding elements. In general, the results with the equivalent method are acceptable and strain items are used for the fatigue evaluation with the MFS criterion.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Three-dimensional geometry model of the investigated disc brake system, (

**b**) simplified finite element model of the brake disc, (

**c**) description of brake disc dimension.

**Figure 5.**Temperature distribution with uniform loading at different moments (highest temperature 454 °C occurs at t4 = 35 s).

**Figure 7.**(

**a**) The changing of strain items following xx, yy and zz directions with rotating loading, (

**b**) the changing of strain items following xy, xz and yz directions with rotating loading.

**Figure 11.**The contribution of different parameters on multiaxial thermomechanical fatigue of the brake disc.

**Figure 12.**(

**a**) The relationship between the yield stress and temperature [15], (

**b**) extrapolation of fatigue curve.

r_{1} | r_{2} | r_{3} | r_{L1} | r_{L2} | r_{M} |
---|---|---|---|---|---|

93 mm | 175 mm | 320 mm | 187.5 mm | 312.5 mm | 250 mm |

Young’s Modulus (MPa) | Poisson’s Ratio | Density (kg/m ^{3}) | Thermal Expansion Coefficient | Specific Heat [J/(kg·K)] | Thermal Conductivity [W/(m·K)] |
---|---|---|---|---|---|

2.02 × 10^{5} | 0.29 | 7850 | 1.04 × 10^{−5} | 457.74 | 32 |

Wheel Radius | Brake Time | Transfer Efficiency | Acceleration |
---|---|---|---|

430 mm | 50 s | 0.9 | −1.389 m/s^{2} |

Brake force | Brake distance | Friction coefficient | Speed |

17,000 N | 1736.1 m | 0.4 | 250 km/h |

${\mathit{k}}_{\mathit{i}\mathit{n}}$ | ${\mathit{h}}_{\mathit{i}\mathit{n}}$ | ${\mathit{k}}_{\mathit{o}\mathit{u}\mathit{t}}$ | ${\mathit{h}}_{\mathit{o}\mathit{u}\mathit{t}}$ | ${\mathit{\chi}}_{\mathit{P}\mathbf{1}}$ | ${\mathit{\chi}}_{\mathit{P}\mathbf{2}}$ | ${\mathit{\chi}}_{\mathit{P}\mathbf{3}}$ | $\mathbf{A}$ | $\mathit{B}$ |

1.7 | 1 | 1.12 | 1 | 0.1 | 0 | 0 | 50.13 | −0.4411 |

Item | Rotating | Uniform | Equivalent | |||
---|---|---|---|---|---|---|

Value | Error % | Value | Error % | |||

Temperature (°C) | 406.73 | 417.82 | −2.73 | 406.38 | 0.086 | |

Thermal strain (%) | 0.40220 | 0.41373 | −2.87 | 0.40184 | 0.090 | |

Huber–Mises (MPa) | 1063.5 | 1082.8 | −1.81 | 1117.8 | −5.1 |

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**MDPI and ACS Style**

Lu, C.; Mo, J.; Sun, R.; Wu, Y.; Fan, Z.
Investigation into Multiaxial Character of Thermomechanical Fatigue Damage on High-Speed Railway Brake Disc. *Vehicles* **2021**, *3*, 287-299.
https://doi.org/10.3390/vehicles3020018

**AMA Style**

Lu C, Mo J, Sun R, Wu Y, Fan Z.
Investigation into Multiaxial Character of Thermomechanical Fatigue Damage on High-Speed Railway Brake Disc. *Vehicles*. 2021; 3(2):287-299.
https://doi.org/10.3390/vehicles3020018

**Chicago/Turabian Style**

Lu, Chun, Jiliang Mo, Ruixue Sun, Yuanke Wu, and Zhiyong Fan.
2021. "Investigation into Multiaxial Character of Thermomechanical Fatigue Damage on High-Speed Railway Brake Disc" *Vehicles* 3, no. 2: 287-299.
https://doi.org/10.3390/vehicles3020018