# Study of Quenched Crankshaft High-Cycle Bending Fatigue Based on a Local Sub Model and the Theory of Multi-Axial Fatigue

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Prediction Process

_{e}and M

_{A}are the prediction of the fatigue limit load and the certain load applied to the crankshaft, respectively; ${\tau}_{r}$ and ${\tau}_{A}$ are the effective stress of the residual stress field; and the effective stress under load M

_{A}, respectively, ${\tau}_{b}$ is the shear fatigue limit of the material.

#### 2.2. The Coordinate Transform Method

**O**is a random point in the three-dimensional space, as well as the original point of the coordinate system

**O-XYZ**. The plane $\Delta $ is a random plane in the space that crosses point

**O**, and

**n**is the normal vector of this plane. The angles between the projection of the normal vector

**n**in the coordinate system

**O-XYZ**and the

**X**,

**Z**axis are $\mathit{\theta}$ and $\mathit{\phi}$, respectively. For point

**O**, the 3D stress tensor in the coordinate

**O-XYZ**is expressed as:

**O-abn**is another coordinate that also crosses point

**O**. Therefore, the expressions of the normal vector

**n**and the

**a**,

**b**axis in the coordinate system can be expressed as:

**m**is a random line that crosses point O. The angle between this line and the axial

**a**is $\alpha $. Therefore, the expression of the unit vector along the line

**m**is:

**nom**can be expressed as:

**O**in Equation (2) can be easily determined by a finite element analysis. Therefore, the shear stress in any plane that crosses the same point can be determined by changing the values of the corresponding three angles from Equations (3)–(7). In this way, the maximum value of the shear stress and corresponding critical plane direction can be determined.

## 3. Results

#### 3.1. Heating Stage Analysis

#### 3.1.1. Mesh Model and Material Parameter

#### 3.1.2. Definition of the Boundary Condition

^{2}·°C) and 0.8. In addition, the symmetric cross section of the model was set to be adiabatic due to the architectural feature of the crankpin.

#### 3.1.3. Simulation Results and Analysis

#### 3.2. Cooling Process Analysis

#### 3.2.1. Temperature Field Analysis

#### 3.2.2. The Residual Stress Field Analysis

#### 3.3. Prediction and Experimental Verification

#### 3.3.1. Multi-Axial Fatigue Model Selection

- Determine the location of the maximum stress point of the crankshaft by a finite element model and record the stress tensor of the point.
- Calculate the values of the shear stress in every plane. In this way, the coordinate of the maximum shear stress critical plane can be determined, as well as the normal stress.

#### 3.3.2. Prediction Based on the Multi-Axial Fatigue Model

#### 3.3.3. Other Prediction Methods and Application

#### 3.3.4. Experimental Verification

#### 3.3.5. Prediction Errors and Discussion

- (1)
- As shown in Table 5, the types of the normal and shear stress in the critical plane caused by the residual stress field are both compressive; whereas, for the alternating stress caused by the load, the type of shear stress in this critical plane is tensile and the type of normal stress is compressive. In addition, the ratios of the shear stress and the tensile stress from different sources are not the same. As a result, the effective stresses from these two sources are not in the same direction, which is not suitable for the application of the mean stress models.
- (2)
- According to our previous study, the fatigue damage type of the crankshaft in this condition is shear fatigue damage, while the modified McDiarmid multi-axial fatigue model is just considered to be a typical effective model in researching this type of problem. This makes the application of this model more effective.

## 4. Conclusions and Further Work Plan

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Temperature (°C) | Relative Permeability (Mur) | Volumetric Heat Capacity (J·m^{3}·°C) | Heat Conductivity (m·°C) | Electrical Resistivity (Ω·m) |
---|---|---|---|---|

25 | 200 | 3,685,270 | 38.5 | 1.8 × 10^{−7} |

100 | 194 | 3,795,044 | 35.5 | 2.0 × 10^{−7} |

200 | 188 | 4,085,161 | 35.0 | 3.2 × 10^{−7} |

300 | 181 | 4,390,960 | 33.5 | 4.2 × 10^{−7} |

400 | 170 | 4,759,487 | 32.5 | 5.0 × 10^{−7} |

500 | 158 | 5,237,788 | 31.0 | 6.2 × 10^{−7} |

600 | 141 | 5,842,545 | 28.0 | 7.7 × 10^{−7} |

700 | 100 | 6,845,193 | 24.5 | 9.7 × 10^{−7} |

760 | 1 | 8,429,075 | 20.0 | 1.0 × 10^{−6} |

800 | 1 | 6,241,436 | 21.0 | 1.2 × 10^{−6} |

900 | 1 | 5,363,244 | 23.0 | 1.2 × 10^{−6} |

1000 | 1 | 5,308,357 | 22.5 | 1.2 × 10^{−6} |

Detailed Parameter | Value |
---|---|

Current frequency | 8000 Hz |

Fillet working time | 12 s |

Crankpin working time | 4 s |

Current intensity | 350 A |

Current density (fillet coil) | 9.9 × 10^{7} A/m^{2} |

Current density (crankpin coil) | 1.15 × 10^{8} A/m^{2} |

Parameter | Value |
---|---|

Convective heat transfer coefficient | 150.00 W/(m^{2}·°C) |

Radiative heat transfer coefficient | 0.8 |

Young’s modulus | 210,000 MPa |

Poisson’s ratio | 1.35 × 10^{−6}/K |

Coefficient of thermal expansion | 1.35 |

Stefan–Boltzmann constant | 5.67 × 10^{−8} W/(m^{−2}·K^{−4}) |

Parameter | From the Given Load | From the Residual Stress Field |
---|---|---|

S11 | 70.5 MPa | −72.5 MPa |

S22 | 123 MPa | −75.1 MPa |

S33 | 117 MPa | −323.8 MPa |

S12 | 1.5 MPa | 75.4 MPa |

S13 | −2.9 MPa | −1.6 MPa |

S23 | −118 MPa | 2.03 MPa |

Parameter | Residual Stress Field | Given Load |
---|---|---|

Shear stress | −24.4 MPa | 119.2 MPa |

Normal stress | −157.3 MPa | −118.6 MPa |

Model Type | Expression | Model Type | Expression |
---|---|---|---|

Goodman | ${\sigma}_{eq}=\frac{{\sigma}_{a}}{1-\frac{{\sigma}_{m}}{{\sigma}_{b}}}$ | Haigh | ${\sigma}_{eq}=\frac{{\sigma}_{a}}{1-{\left(\frac{{\sigma}_{m}}{{\sigma}_{b}}\right)}^{0.5}}$ |

Gerbera | ${\sigma}_{eq}=\frac{{\sigma}_{a}}{1-{\left(\frac{{\sigma}_{m}}{{\sigma}_{b}}\right)}^{2}}$ | Soderberg | ${\sigma}_{eq}=\frac{{\sigma}_{a}}{1-\frac{{\sigma}_{m}}{{\sigma}_{y}}}$ |

Model Name | Results (N·m) |
---|---|

Goodman | 2494 |

Haigh | 2956 |

Soderberg | 2082 |

Gerbera | 2683 |

Load Value/N·m | Media Rank |
---|---|

2893 | 0.067308 |

3025 | 0.163462 |

3255 | 0.259615 |

3280 | 0.355769 |

3321 | 0.451923 |

3343 | 0.548077 |

3352 | 0.644231 |

3386 | 0.740385 |

3738 | 0.836538 |

3759 | 0.932692 |

Model Name | Error |
---|---|

Goodman | 32% |

Haigh | 22.8% |

Soderberg | 24.8% |

Gerbera | 43.2% |

McDiarmid | 1.1% |

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**MDPI and ACS Style**

Sun, S.; Zhang, X.; Wan, M.; Gong, X.; Xu, X. Study of Quenched Crankshaft High-Cycle Bending Fatigue Based on a Local Sub Model and the Theory of Multi-Axial Fatigue. *Metals* **2022**, *12*, 913.
https://doi.org/10.3390/met12060913

**AMA Style**

Sun S, Zhang X, Wan M, Gong X, Xu X. Study of Quenched Crankshaft High-Cycle Bending Fatigue Based on a Local Sub Model and the Theory of Multi-Axial Fatigue. *Metals*. 2022; 12(6):913.
https://doi.org/10.3390/met12060913

**Chicago/Turabian Style**

Sun, Songsong, Xingzhe Zhang, Maosong Wan, Xiaolin Gong, and Xiaomei Xu. 2022. "Study of Quenched Crankshaft High-Cycle Bending Fatigue Based on a Local Sub Model and the Theory of Multi-Axial Fatigue" *Metals* 12, no. 6: 913.
https://doi.org/10.3390/met12060913