# A Comparison of Two Methods Modeling High-Temperature Fatigue Crack Initiation in Ferrite–Pearlite Steel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Methods

## 3. Results

#### 3.1. The First Approach for Crack Initiation Prediction

#### 3.1.1. Generation of Geometric Model

#### 3.1.2. Crack Initiation Model

#### 3.1.3. Computational Model Parameters’ Determination

#### 3.1.4. Results of the First Approach

#### 3.2. The Second Approach for Crack Initiation Prediction

#### 3.2.1. Generation of Microstructures

#### 3.2.2. Constitutive Equations

_{th}slip system. The evolution of back stress $\chi $ and deformation resistance Q is characterized by

#### 3.2.3. Parameter Calibration

#### 3.2.4. Simulation and Results

#### 3.2.5. 3D Geometric Models

## 4. Discussion

- Both approaches are capable of predicting the fatigue crack initiation positions and initiation life in ferrite–pearlite steel; the crack density evolution is in good agreement with experiment results.
- The first approach determines whether a crack initiates by basal energy, which makes the simulation process much faster than the finite element method. However, stress and strain fields cannot be obtained from this approach.
- The second approach is based on CPFEM and more variables can be calculated, such as the stress and strain field, slip rate, resolved shear stress, etc., but a higher computational cost is required.
- Although the parameters are calibrated in 2D models, they can be applied to 3D models by recalibrating only one parameter.
- Both approaches are promising. For example, multiple slip systems can be introduced into the first approach, and more slip controlling parameters can be included in the second approach, such as temperature.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 15.**The first cycle, 6th cycle, 11th cycle, and 16th cycle of simulation, representing the first cycle, 1000th cycle, 2000th cycle, and 3000th cycle of experiment, with D = 20,000.

**Figure 16.**The first cycle, 3th cycle, 7th cycle, and 11th cycle of simulation, representing the first cycle, 600th cycle, 1800th cycle, and 3000th cycle of experiment, with D = 30,000.

**Figure 17.**Distribution of $FI{P}_{\u03f5}$; the first cycle in this figure represents the first cycle in the experiment, and the 6th cycle represents the 1000th cycle in the experiment, etc.

**Figure 18.**Distribution of $FI{P}_{E}$; the first cycle in this figure represents the first cycle in the experiment; the 6th cycle represents the 1000th cycle in the experiment, etc.

**Figure 19.**Distribution of $FI{P}_{eff}$; the first cycle in this figure represents the first cycle in the experiment; the 6th cycle represents the 1000th cycle in the experiment, etc.

**Figure 20.**The crack density evolution calculated by $FI{P}_{eff}$ of the 2nd cycle of simulation, and the stress and strain evolution in this cycle.

**Figure 21.**The 3D geometric models. Model (

**A**) and Model (

**B**) consist of 200 grains but with different grain structures, while Model (

**C**) consists of 300 grains.

**Figure 22.**The evolution of peak and valley stress of Model (A) with different orientations, compared with experiment.

C | Si | Mn | Cr | S | P |
---|---|---|---|---|---|

$0.2$ | $0.23$ | $0.51$ | $0.013$ | $0.011$ | $0.024$ |

Name of Parameters | Units | Room Temperature | High Temperature |
---|---|---|---|

Temperature | °C | 20 | 500 |

Yield strength | MPa | $335.5$ | $316.1$ |

Tensile strength | MPa | $471.98$ | $412.37$ |

Elongation | % | 32 | 45 |

Elastic modulus | MPa | $2.28\times {10}^{5}$ | $2.05\times {10}^{5}$ |

Poisson’s ratio | $0.303$ | $0.301$ | |

Fracture energy density | KJ/m${}^{2}$ | 2 | 2 |

Shear modulus | GPa | 94 | 81 |

Parameters | Units | Values |
---|---|---|

Temperature | °C | 500 |

Strain rate | s${}^{-1}$ | $0.0024$ |

Strain ratio | $-1$ | |

Strain amplitude | $0.12\%$ |

Parameters | Units | Values |
---|---|---|

G | GPa | 81 |

$\mu $ | 0.3 | |

${\tau}_{c}$ | MPa | 81 |

${W}_{c}$ | KJ/m${}^{2}$ | 2 |

${C}_{1}$ | $1.3\times {10}^{-10}$ | |

Initial basal energy of PSB | 0.3∼0.5 | |

Initial basal energy of F-FGB | 0.4∼0.6 | |

Initial basal energy of F-PGB | 0.45∼0.65 |

**Table 5.**Calibrated parameters in the second approach; ${C}_{11}^{f}$, ${C}_{12}^{f}$, ${C}_{44}^{f}$ are elastic constants of ferrite and ${C}_{11}^{p}$, ${C}_{12}^{p}$, ${C}_{44}^{p}$ are elastic constants of pearlite.

Parameters | Values | Units |
---|---|---|

${C}_{11}^{f}$, ${C}_{12}^{f}$, ${C}_{44}^{f}$ | $210.36,122.07,106.31$ | GPa |

${C}_{11}^{p}$, ${C}_{12}^{p}$, ${C}_{44}^{p}$ | $300.51,174.39,151.87$ | GPa |

A | $102.8$ | MPa |

${A}_{d}$ | $7.8$ | |

q | $1.4$ | |

Q | $108.55$ | MPa |

S | $0.35$ | MPa |

K | $0.032$ | MPa |

B | $-0.001$ | |

D | 20,000 | |

${h}_{0}$ | $4.0$ | MPa |

${h}_{s}$ | $4.0$ | MPa |

N | $22.8$(2D) 110(3D) | MPa |

${N}_{d}$ | $18.8$ |

Experiment | 1 | 200 | 400 | 600 | 800 | 1000 | 1200 | 1400 | 1600 | 1800 | 2000 | 2200 | 2400 | 2600 | 2800 | 3000 |

Simulation | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |

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**MDPI and ACS Style**

Fang, Z.; Wang, L.; Wang, Z.; He, Y.
A Comparison of Two Methods Modeling High-Temperature Fatigue Crack Initiation in Ferrite–Pearlite Steel. *Crystals* **2022**, *12*, 718.
https://doi.org/10.3390/cryst12050718

**AMA Style**

Fang Z, Wang L, Wang Z, He Y.
A Comparison of Two Methods Modeling High-Temperature Fatigue Crack Initiation in Ferrite–Pearlite Steel. *Crystals*. 2022; 12(5):718.
https://doi.org/10.3390/cryst12050718

**Chicago/Turabian Style**

Fang, Zheng, Lu Wang, Zheng Wang, and Ying He.
2022. "A Comparison of Two Methods Modeling High-Temperature Fatigue Crack Initiation in Ferrite–Pearlite Steel" *Crystals* 12, no. 5: 718.
https://doi.org/10.3390/cryst12050718