# On the Influence of Loading Order in Nanostructural Fatigue Crack Propagation in BCC Iron—A Molecular Dynamics Study

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## Abstract

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## 1. Introduction

## 2. Simulation Method

## 3. Simulation Setup

## 4. Results

#### 4.1. Determination of Crack Length

#### 4.2. Constant Strain Amplitude

#### 4.3. High-Low and Low-High Loading

#### 4.3.1. High-Low

#### 4.3.2. Low-High

#### 4.4. Single Under- and Over-Load

#### 4.4.1. Single Under-Load

#### 4.4.2. Single Over-Load

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

MD | Molecular dynamics |

S-N | Stress life approach to fatigue |

$\epsilon $-N | Strain life approach to fatigue |

LEFM | Linear elastic fracture mechanics |

da | Increment increase of crack length |

dN | Increment increase of cycles |

C, m | Material parameters for the Paris equation |

$\Delta K$ | Range of stress intensity factor |

${K}_{max}$ | Maximum stress intensity factor |

${K}_{max}$ | Minimum stress intensity factor |

$\Delta {K}_{th}$ | Threshold value for fatigue crack propagation |

${D}_{d}$ | Damage sum according to Miner’s rule |

${n}_{Ei}$ | Number of cycles occurring at the stress range i |

${N}_{Ri}$ | Number of cycles to failure at the stress range i |

EAM | Embedded atom method |

${U}_{EAM}$ | EAM potential |

${r}_{ij}$ | Scalar distance between atoms |

${U}_{ij}\left({r}_{ij}\right)$ | Pairwise additive contributions to the potential |

${\rho}_{i}$ | Electron density |

${a}_{0}$ | Initial crack length |

H | Height of the specimen |

W | Width of the specimen |

a | Lattice constant |

$\epsilon $ | Strain |

bcc | Body centered cubic crystal structure |

fcc | Face centered cubic crystal structure |

hcp | Hexagonal closed packed crystal structure |

t | Time |

SOL | Single overload |

SUL | Single underload |

Mode I | Crack opening perpendicular to the crack plane |

Mode II | In plane sliding of the crack |

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**Figure 3.**Various crack tip geometries. The colouring is presented according to common neighbour analysis. The blue colour indicates bcc, whereas the green notates fcc, and the red indicates the hcp iron structure. Particles belonging to an unidentified structure are visualised in grey.

**Figure 4.**Evolution of the crack length. The blue signs indicate the peaks of the loading cycles, whereas the red triangles indicate the peaks of the unloading steps.

**Figure 5.**Evolution of the crack during cyclic loading. The images show the peak of the loading and unloading cycle for each of the eight cycles. The second row is a prolongation of the first. The colouring is presented according to common neighbour analysis. The blue colour indicates bcc, whereas the green notates fcc, and the red indicates the hcp iron structure. Particles belonging to an unidentified structure are visualised in grey.

**Figure 8.**Self-interaction due to the periodic boundary conditions. The particles leaving the left side of the model re-enter the system through the boundary on the opposite side.

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**MDPI and ACS Style**

Ladinek, M.; Hofer, T.
On the Influence of Loading Order in Nanostructural Fatigue Crack Propagation in BCC Iron—A Molecular Dynamics Study. *Metals* **2019**, *9*, 684.
https://doi.org/10.3390/met9060684

**AMA Style**

Ladinek M, Hofer T.
On the Influence of Loading Order in Nanostructural Fatigue Crack Propagation in BCC Iron—A Molecular Dynamics Study. *Metals*. 2019; 9(6):684.
https://doi.org/10.3390/met9060684

**Chicago/Turabian Style**

Ladinek, Markus, and Thomas Hofer.
2019. "On the Influence of Loading Order in Nanostructural Fatigue Crack Propagation in BCC Iron—A Molecular Dynamics Study" *Metals* 9, no. 6: 684.
https://doi.org/10.3390/met9060684