# The Uncertainty Propagation for Carbon Atomic Interactions in Graphene under Resonant Vibration Based on Stochastic Finite Element Model

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## Abstract

**:**

## 1. Introduction

## 2. Method Description

#### 2.1. Geometrical Configuration

- (a)
- The finite plane element model projects the precise three-dimensional structure in Figure 1a into the two-dimensional x-y plane, which is more computationally economic than three-dimensional models, but is more sophisticated than the truss or beam finite element model;
- (b)
- The finite plane element model is an advanced method with a similar computational competence to the truss and beam finite element model of graphene, as shown in Figure 1b. However, the finite plane element model includes not only the carbon covalent bonds but also the carbon atoms;
- (c)
- The related geometrical parameters in the finite plane element model presented in Figure 2a are flexible to describe different special hexagons. Specifically, L, R
_{1}, and R_{2}are the length of the carbon covalent bonds, the radius of the carbon atoms, and twice the width of the carbon covalent bonds, respectively; - (d)
- Since the carbon atoms and carbon covalent bonds in graphene are described as different geometrical components, the corresponding material parameters can be assigned to them;
- (e)
- The carbon atoms and carbon covalent bonds, as presented in Figure 1c, share the common lines, ensuring the geometrical connection and mechanical compatibility. There will be common nodes on the shared lines after meshing the finite plane element model.

_{1}, and R

_{2}are equal to 0.27 nm, 0.05 nm, and 0.032 nm, respectively. The typical examples are provided according to the changes in related geometrical parameters. For example, when L is as short as 0.1 nm, the period characteristic hexagon in graphene is presented in Figure 2b. When L is extruded to 0.4 nm, the period characteristic hexagon in graphene is presented in Figure 2c. Furthermore, the change of geometrical parameter R

_{2}also evidently impacts the configuration of the period hexagon cell in graphene and then influences the entire intrinsic lattice description. In Figure 2d,e, R

_{2}equals 0.01 nm and 0.04 nm, respectively. Therefore, the proposed geometrical configuration of the finite plane element is flexible to represent different shapes and combinations.

#### 2.2. Material Parameters

_{1}and P

_{1}are presented in Figure 3a, and those for geometrical parameters of R

_{1}and L are presented in Figure 3b. In order to simplify the problem, the internal correlation between the material and geometrical parameters is supposed to be zero. The independent stochastic samples provided by the Monte Carlo simulation for the finite element computation are performed. The correlation between the corresponding parameters and the resonant frequencies of graphene is analyzed with the stochastic samples as input data and computational results of the finite element model as output data.

#### 2.3. Computational Method

_{ij}is the Cauchy stress component, ${e}_{ij}=\frac{1}{2}(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{i}}{\partial {x}_{j}})$ is the deformation tensor, and ${f}_{i}^{B}$ and ${f}_{i}^{S}$ represent the component of body force and surface traction, respectively. V and s are the volumes of the deformed body and the surface of the deformed body on which tractions are prescribed, respectively.

_{i}is the velocity and x

_{i}is the coordinates.

## 3. Results and Discussion

#### 3.1. Statistical Results

#### 3.2. Parameter Discussion

_{1}, R

_{2}, and L are the more critical factors to impact the resonant frequency of graphene. In other words, the length and the width of the carbon covalent bonds are essential in the finite plane element model of graphene. This consequence reaches good agreements with the assumption of the reported beam or truss finite element model of graphene [10,11,23].

_{1}, R

_{2}, and L in Figure 8a are steady, and those for E

_{1}and E

_{2}in Figure 8b are steady as well. However, the Monte Carlo relative errors in correlation computation for V

_{1}, V

_{2}, P

_{1}, and P

_{2}present evident variations in Figure 8c,d. Therefore, a sufficient number of stochastic samples is necessary to ensure the correlation coefficient accuracy.

#### 3.3. Vibration Modes

## 4. Conclusions

- (1)
- The commonly shared nodes in carbon atoms and carbon covalent bonds in the two-dimensional 8-node quadrilateral element keep the geometrical connection and mechanical compatibility well.
- (2)
- The interval ranges of resonant frequencies computed by the finite plane element model completely include the results in the reported literature.
- (3)
- The correlation coefficients computed by the Pearson and Spearman methods have substantial agreements with small discrepancies in the geometrical and material parameters.
- (4)
- The length and the width of the carbon covalent bonds in the finite plane element model of graphene are the essential factors that impact the resonant frequencies.
- (5)
- The proposed finite element model not only has merits in terms of computational expense and feasibility in the massive stochastic sampling process but also is flexible in presenting the precise vibration modes of graphene with consideration of both carbon atoms and covalent bonds.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**The geometrical configuration of the finite element model of graphene. (

**a**–

**c**) are the three-dimensional model, conventional two-dimensional model and proposed hybrid finite element model, respectively.

**Figure 2.**The geometrical parameters and typical examples in the finite plane element model of graphene.

**Figure 3.**The stochastic samples of corresponding parameters in the finite plane element model of graphene based on Monte Carlo simulation. (

**a**) are for the material parameters E1 and P1, (

**b**) are for the geometrical parameters R1 and L, respectively.

**Figure 5.**The flowchart of resonant frequency computation by the finite plane element model based on the Monte Carlo stochastic simulation.

**Figure 6.**The statistic results of resonant frequency in the finite plane element model of graphene.

**Figure 7.**The probability density distribution of resonant frequency in the finite plane element model of graphene. (

**a**–

**d**) are for the first-fourth resonant vibration, respectively.

**Figure 8.**The correlation coefficients of geometrical and material parameters in the finite plane element model of graphene. (

**a**) is for the geometrical parameters R1, R2 and L; (

**b**) is for the material parameters E1 and E2; (

**c**) is for the material parameters v1 and v2; (

**d**) is for the material parameters P1 and P2.

**Figure 9.**The vibration modes of graphene with the length of the carbon covalent bonds equal to 0.1 nm ((

**a**–

**d**) represent the first four resonant vibration modes).

**Figure 10.**The vibration modes of graphene with the length of the carbon covalent bonds equal to 0.4 nm ((

**a**–

**d**) represent the first four resonant vibration modes).

**Figure 11.**The vibration modes of graphene with the width of the carbon covalent bonds equal to 0.005 nm ((

**a**–

**d**) represent the first four resonant vibration modes).

**Figure 12.**The vibration modes of graphene with the width of the carbon covalent bonds equal to 0.02 nm ((

**a**–

**d**) represent the first four resonant vibration modes).

Symbols | Definitions | Value Intervals | Units |
---|---|---|---|

E_{1} | The Young’s modulus of carbon atoms | 10^{11}–10^{13} | Pa |

E_{2} | The Young’s modulus of carbon covalent bonds | 10^{6}–10^{8} | Pa |

v_{1} | The Poisson’s ratio of carbon atoms | 0.1–0.4 | - |

v_{2} | The Poisson’s ratio of carbon covalent bonds | 0.1–0.4 | - |

P_{1} | The physical density of carbon atoms | 500–5000 | Kg/m^{3} |

P_{2} | The physical density of carbon covalent bonds | 500–5000 | Kg/m^{3} |

R_{1} | The radius of carbon atoms | 0.04–0.09 | nm |

R_{2} | The two times width of carbon covalent bonds | (0.1–0.5) ∗ R_{1} | nm |

L | The length of carbon covalent bonds | 0.1–0.4 | nm |

F_{1} (THz) | F_{2} (THz) | F_{3} (THz) | F_{4} (THz) | |
---|---|---|---|---|

Mean | 3.4905 | 4.0902 | 4.1838 | 5.2333 |

Maximum | 16.894 | 19.554 | 21.362 | 26.402 |

Minimum | 0.2131 | 0.2808 | 0.2816 | 0.3319 |

Variance | 2.5899 | 2.7923 | 2.9301 | 3.8381 |

Liu [14] | 1.6081 | 3.7232 | 4.3172 | 6.4323 |

Kudin [15] | 1.5818 | 3.6623 | 4.2466 | 6.3271 |

Gupta [16] | 1.7581 | 4.0706 | 4.7201 | 7.0325 |

Lu [17] | 1.4311 | 3.3135 | 3.8422 | 5.7246 |

Wei [18] | 1.5946 | 3.6921 | 4.2811 | 6.3786 |

Cadelano [19] | 1.5649 | 3.6232 | 4.2012 | 6.2595 |

Reddy [20] | 1.3869 | 3.2111 | 3.7234 | 5.5475 |

Zhou [21] | 1.8716 | 4.3334 | 5.0248 | 7.4865 |

Khatibi [22] | 1.6030 | 2.4970 | 2.5980 | 3.5770 |

Chu [23] | 1.7282 | 3.2925 | 3.7442 | 5.1892 |

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**MDPI and ACS Style**

Shi, J.; Chu, L.; Ma, C.; Braun, R. The Uncertainty Propagation for Carbon Atomic Interactions in Graphene under Resonant Vibration Based on Stochastic Finite Element Model. *Materials* **2022**, *15*, 3679.
https://doi.org/10.3390/ma15103679

**AMA Style**

Shi J, Chu L, Ma C, Braun R. The Uncertainty Propagation for Carbon Atomic Interactions in Graphene under Resonant Vibration Based on Stochastic Finite Element Model. *Materials*. 2022; 15(10):3679.
https://doi.org/10.3390/ma15103679

**Chicago/Turabian Style**

Shi, Jiajia, Liu Chu, Chao Ma, and Robin Braun. 2022. "The Uncertainty Propagation for Carbon Atomic Interactions in Graphene under Resonant Vibration Based on Stochastic Finite Element Model" *Materials* 15, no. 10: 3679.
https://doi.org/10.3390/ma15103679