# The Crack Angle of 60° Is the Most Vulnerable Crack Front in Graphene According to MD Simulations

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## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

**:**The angular crack front was modeled by the vertex angle of an isosceles triangle-shaped crack embedded in a single layer of graphene. The pre-cracks were obtained by removing carbon atoms in the confined zone of an isosceles triangle, as shown in Figure 1. The example system displayed is the pre-cracked single-layer graphene with the crack shape of a 90° isosceles triangle, which is denoted as Angle90 hereafter for convenience. We examined eight vertex angles: 30°, 36°, 60°, 72°, 90°, 108°, 120°, and 150°. The system size was 20.7 nm × 20.5 nm with 16,128 carbon atoms before the creation of the cracks. The carbon-carbon bond length was 0.142 nm at room temperature.

## 3. Results and Discussion

#### 3.1. Stress–Strain Relationship

#### 3.2. Angle Effect

#### 3.3. Size Effect

#### 3.3.1. System Size Effect (Same Crack Size but Different System Size)

#### 3.3.2. Crack Size Effect (Same System Size but Different Crack Size)

#### 3.4. Strain Rate Effect

^{9}, indicating the lesser need for effective stress at these strain rates. Moreover, the mechanical properties seem to converge as the strain rate decreases to a more realistic value and the system is given sufficient time to react to the applied tensile load. At room temperature and high strain rates $(\approx {10}^{9}{\mathrm{s}}^{-1})$, the fracture strength can be fairly compared to the values reported in [66], whereas it is substantially enhanced when compared to the strength of polycrystalline graphene at various temperatures (including room temperature) and analogous range strain rates, as reported in [67].

#### 3.5. Temperature Effect

^{−3}to 0.33 J·m

^{−3}and 0.253 to 0.086, respectively. This signifies a 94.8% reduction in toughness and a 66.0% reduction in fracture strain for this temperature range. The Young’s modulus obeyed a general negative correlation with temperature at ranges above 500 K, with a drop from 977 GPa to 914 GPa seen with a 6.45% reduction in fracture strain.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A snapshot of the configuration of Angle90, which is a single layer of graphene with a pre-crack in an isosceles triangle shape with a vertex angle of 90°. Left: the system size and the isosceles triangle sides. Right: a zoomed-in plot of the crack tip. The system of the pre-cracked single-layer graphene with the crack shape of an isosceles triangle with a vertex angle of 90° is denoted as Angle90 for convenience.

**Figure 2.**Stress–strain relationships of the eight SLGSs pre-cracked in the shape of an isosceles triangle with different vertex angles ranging from 30° to 150° compared with that of pristine graphene. The tensile simulations were performed under the following conditions: The temperature was 300 K. The pressure was 0.0001 GPa (1 atm). The strain rate was 10

^{9}s

^{−1}. The model I loading was applied along the x axis (armchair direction). The insets were snapshots for the angles of (

**a**) 30°, (

**b**) 72°, (

**c**) 120°.

**Figure 3.**The crack angle shape dependence of cracked SLGSs’ mechanical properties: (

**a**) toughness, (

**b**) Young’s modulus, (

**c**) fracture strength, and (

**d**) fracture strain. The tensile simulations were carried out under the following conditions: The temperature was 300 K. The pressure was 0.0001 GPa (1 atm). The strain rate was 10

^{9}s

^{−1}. The model I loading was applied along the x axis (armchair direction). The eight SLGSs were pre-cracked in the shape of an isosceles triangle with different vertex angles ranging from 30° to 150° compared with that of pristine graphene.

**Figure 4.**(

**a**) Stress–strain relationships of SLGSs with different sized cracks. Three systems all with the same vertex angle of 90° are compared. The tensile simulations were carried out under the following conditions: The temperature was 300 K. The pressure was 0.0001 GPa (1 atm). The strain rate was 10

^{9}s

^{−1}. The model I loading was applied along the x axis (armchair direction). Snapshots of the x-component atomic stress distributions in the system with the size of 4 × 4 (or 20.7 nm × 20.5 nm with 16,128 lattice sites) can be seen in (

**b**) for the system just before crack and (

**c**) after the crack propagation. The colormap is in the middle.

**Figure 5.**Stress–strain relationships of cracked SLGSs for different strain rates (${\mathrm{s}}^{-1})$. The strain rate ranges from 10

^{7}s

^{−1}to 3 × 10

^{10}s

^{−1}. The tensile simulations were carried out under the following conditions: The temperature was 300 K. The pressure was 0.0001 GPa (1 atm). The model I loading was applied along the x axis (armchair direction). The system size was 4 × 4 (or 20.7 nm × 20.5 nm with 16,128 lattice sites). All the systems had pre-cracks in the shape of an isosceles right triangle.

**Figure 6.**Strain rate dependence of cracked SLGSs’ mechanical properties: (

**a**) toughness, (

**b**) Young’s modulus, (

**c**) fracture strength, and (

**d**) fracture strain compared with that of pristine graphene. The strain rate ranges from 10

^{7}s

^{−1}to 3 × 10

^{10}s

^{−1}. The tensile simulations are under the following conditions: The temperature is 300 K. The pressure is 0.0001 GPa (1 atm). The model I loading is applied along the x axis (armchair direction). The system size is 4 × 4 (or 20.7 nm × 20.5 nm with 16,128 lattice sites). All the systems have the pre-crack in the shape of an isosceles right triangle.

**Figure 7.**The stress–strain relationships of defective SLGSs at eight examined temperatures. The tensile simulations were carried out under the following conditions: The strain rate was 10

^{9}s

^{−1}. The pressure was 0.0001 GPa (1 atm). The model I loading was applied along the x axis (armchair direction). The system size was 4 × 4 (or 20.7 nm × 20.5 nm with 16,128 lattice sites). All the systems had the pre-crack in the shape of an isosceles right triangle.

**Figure 8.**Temperature dependence of defective SLGSs’ mechanical properties, including: (

**a**) toughness, (

**b**) Young’s modulus, (

**c**) fracture strength, and (

**d**) fracture strain. The tensile simulations were carried out under the following conditions: The strain rate was 10

^{9}s

^{−1}. The pressure was 0.0001 GPa (1 atm). The model I loading was applied along the x axis (armchair direction). The system size was 4 × 4 (or 20.7 nm × 20.5 nm with 16,128 lattice sites). All the systems had a pre-crack in the shape of an isosceles right triangle.

**Table 1.**The setup of the eight models. The number of atoms removed and their percentage out of the total number of atoms in the system, as summarized according to the top angle. We aim to use the same area of the triangle for comparison.

Top Angle | Atoms Removed | Percentage of Defects | |
---|---|---|---|

1 | 30° | 150 | 0.930% |

2 | 36° | 153 | 0.949% |

3 | 60° | 154 | 0.959% |

4 | 72° | 153 | 0.949% |

5 | 90° | 158 | 0.980% |

6 | 108° | 145 | 0.900% |

7 | 120° | 156 | 0.967% |

8 | 150° | 156 | 0.967% |

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**MDPI and ACS Style**

Alahmed, I.I.; Altanany, S.M.; Abdulazeez, I.; Shoaib, H.; Alsayoud, A.Q.; Abbout, A.; Peng, Q.
The Crack Angle of 60° Is the Most Vulnerable Crack Front in Graphene According to MD Simulations. *Crystals* **2021**, *11*, 1355.
https://doi.org/10.3390/cryst11111355

**AMA Style**

Alahmed II, Altanany SM, Abdulazeez I, Shoaib H, Alsayoud AQ, Abbout A, Peng Q.
The Crack Angle of 60° Is the Most Vulnerable Crack Front in Graphene According to MD Simulations. *Crystals*. 2021; 11(11):1355.
https://doi.org/10.3390/cryst11111355

**Chicago/Turabian Style**

Alahmed, Ishaq I., Sameh M. Altanany, Ismail Abdulazeez, Hassan Shoaib, Abduljabar Q. Alsayoud, Adel Abbout, and Qing Peng.
2021. "The Crack Angle of 60° Is the Most Vulnerable Crack Front in Graphene According to MD Simulations" *Crystals* 11, no. 11: 1355.
https://doi.org/10.3390/cryst11111355