Evaluation of the Stress-Strain of Single-Layer Graphene Sheet Including a Crack

2.1 Molecular dynamics

Most nanomaterials, including graphene, have a large specific surface area. The phenomena occurring on the surface greatly affect physical properties, of materials. If the structure changes minutely on the surface, it is difficult to control the change of the physical properties. This limitation can be solved by using MD technique which solves the motion of molecules based on Newton's equations. It is the only way to analyze the dynamic properties of a non-equilibrium system. MD are calculated in conjunction with the following four aspects.

For interatomic potentials, the evolution of the static and dynamic properties of each atom over time is defined. In the present study, graphene was calculated using the AIREBO (Adaptive Intermolecular Reactive Empirical Bond Order) potential.¹³⁾ AIREBO enables simulations of the formation and rupture of covalent bonds for various atoms. This potential also determines the onset of cracking and allows for the precise calculation of bond formation and rupture based on the equation (1).

where $$ {E_{ij}^{rebo}}^{}$$ is the energy stored in the bonds between atoms, $$ E_{ij}^{lj}$$ is the nonbonding interaction between atoms and $$ E_{kijl}^{torsion}$$ is the torsional interaction between atoms. Using the AIREBO potential value, the interatomic, nonbonding and torsional interactions in graphene can be determined.

Next, the using algorithm with MD simulation in this study was CG (Conjugate Gradient) method. It is used to calculate the new direction of modified atoms perpendicular to the previous direction by combining the force gradient with data obtained from the prior iteration. The highly accurate results are obtained by minimizing numerical errors during prolonged iterative calculations of the trajectories of all atoms in the system.

In a periodic system, atoms can interact across boundaries and exit one end of the box which make atoms move and reenter the other end. Therefore, PBC (Periodic Boundary Conditions) were used.

Also, the ensemble concept is used to simulate the motion of atoms over time. Initially, random molecular positions or lattice structures can be created and random positions of atoms and molecules are often arranged in a lattice structure because of the high risk of overlapping molecules. Representative ensembles include NVE(Number of atoms(N), Volume(V), and Energy(E) are isolated from changes in the system), NVT(Number of atoms(N), Volume(V), and Temperature(T) are conserved in the system) and NPT(Number of atoms(N), Pressure(P), and Temperature(T) are conserved in the system). The present work employed the NVT ensemble, which is the only one that can be used for controlling temperature condition of atoms similar to general experiments or systems. And the NVT ensemble is also called a canonical ensemble.

2.2 Analytical model

The graphene analytical model is illustrated n in Fig. 1. The horizontal and vertical dimensions of this model are 341.6 and 100.6 Å, respectively. The sample sheet was stretched at both ends in the x-axis direction, and the tensile velocity was set at 0.17 Å/ps.

Fig. 1

An analytical model of the graphene sheet

The total number of atoms used to model the sample sheet was 13,296 and the conditions of molecular dynamic analysis applied to the graphene sheet are provided in Table 1.

Table 1

The parameter of molecular dynamics analysis

The temperature was fixed at 300 K and was controlled using a Nose-Hoover thermostat which is an analytical method using in MD simulation to realize NVT ensemble as an actual experiment conditions. As for the potential, CH.AIREBO was used. CH.AIREBO is the potential to calculate the interactions between carbon atoms, specifying carbon atoms. The cut-off distance, which affects the calculations by specifying the distance between atoms, was set at 1.92 based on the typical range of 1.7–2.0 Å designated by the AIREBO, and because 1.92 was the most suitable value for use in the model. The CG algorithm was employed to minimize this distance. All calculations employed Newton's equations, with the units set to the metal options so that the stress values could be converted to units of GPa.

The initial shape of the crack inserted for the crack analysis was a combination of zig-zag and arm-chair morphologies with 18 atoms removed, as illustrated in Fig. 2. A total of three analytical models were employed, varying the positions, distances and angles at which these cracks were inserted, as presented in Fig. 3. Information regarding the position of the initial cracks is illustrated in Fig. 3. The analysis was performed for a graphene sheet divided into four parts symmetrically, so as to increase the efficiency of the analysis.

Fig. 2

A schematic diagram of the inserted crack shape

Fig. 3

The model types Ⅰ, Ⅱ, Ⅲ for the crack analysis

In Model Ⅰ, the graphene sample sheet was divided into four parts symmetrically and the crack was inserted at each of the four vertex positions as illustrated in Fig. 3 (a). Model Ⅱ analyzed nine cases with variations according to the x-axis relative to the center of the graphene sheet, as in Fig. 3(b). Finally, Model Ⅲ, as presented in Fig. 3(c), was divided into three cases of 0°, 30° and 45° in which the angle of the initial crack varied with respect to the load direction.

3.1 The stress-strain curve according to the crack position

Stress-strain curves were obtained using the LAMMPS algorithm, and Fig. 4 shows the results of the analysis for the cracks at the four points of the graphene sheet in Model Ⅰ. In this model, the stress gradually increases to a maximum value in all of case after which it rapidly decreases. These maximum stresses indicate the failure strength. In addition, the graphene, which breaks immediately after reaching its failure strength without stress stagnation, evidently undergoes brittle fracture. Also the maximum stress value can indicate the initial cracking instability point in the graphene sheet. Consequently, a load should be applied below the maximum stress that the material can bear to prevent the failure of graphene due to cracking. In Model Ⅰ, the maximum stresses for the C1-Ⅰ, C2-Ⅰ and C4-Ⅰ cases were 114, 101 GPa, and 122 GPa respectively, while the C3-Ⅰ was 319 GPa. In the C3-Ⅰcrack of high stress 319 GPa, it is supposed the effect of saint-venant’s principle because of the localized stress concentration.

Fig. 4

The stress-strain curves for Model Ⅰ

Fig. 5 shows the crack propagation leading to fracture. These images demonstrate two distinct patterns. One involves a single path which is divided two parts, as in the case of the C1-Ⅰ and C3-Ⅰ results, while the other is a dual path that divides into three parts, as in the C2-Ⅰ and C4-Ⅰ.

Fig. 5

The initial crack propagations in Model Ⅰ

Next, a total of nine analyses were performed with Model Ⅱ, in which more initial cracks were inserted between the C3-Ⅰ and C4-Ⅰ distances used in Model Ⅰ, to determine the effect of the initial crack position in the center of the graphene sheet. The interval between these new distances was approximately 18 Å.

There maximum stress expresses the fracture strength. The both of stress and strain are decreasing according to the position of the initial crack tip. In particular, the decrease rate of the stress is faster than that of the strain, demonstrating that crack position affects highly the stress variation.

When the maximum stresses are compared with tensile strength of the smooth graphene sheet, the internal cracks have a great influence on the fracture strength. In Fig. 6, the variation of the maximum stress and strain have two phases that consist of two inclined lines. The values decrease swiftly until the position of about 90 Å from graphene sheet foundary. And after, it keeps stable stress value. However strains decrease slightly.

Fig. 6

The maximum stress and strain as functions of the crack position along the center line (the x direction)

3.2 Stress variation according to the crack angle

The cracks generated in a structure do not always occur vertically or horizontally. Therefore, the effects of changes in the angle of the crack also need to be considered. The angular orientation of the initial crack was changed based on the length of the x-axis.

Fig. 7 shows the stress-strain curves obtained from the analysis of the C1-Ⅲ, C2-Ⅲ and C3-Ⅲ cases. The breaking strength for the C1-Ⅲ, with an initial crack angle of 0°, was 122 GPa, while those for the C2-Ⅲ, C3-Ⅲ with the initial crack inserted at 30° and 45°, were 126 GPa and 142 GPa. Thus, the breaking strength was increased due to angular orientation, indicating that the initial crack at 0° resulted in easier breaking of the material.

Fig. 7

The stress-strain curves for Model Ⅲ

These results show that the initial crack at 45° is harder to fracture than that of at 0° and 30°. The results from Model Ⅲ evaluation express that the loading orientation for the graphene sheet with a crack need to be considered to avoid the vertical direction for the crack.

Author contributions

O. H. Kwon; Conceptualization, Supervision and Writing-review & editing. W. R. Park; Project administration and Visualiztion. S. M. Kim; Investigation, Writing-original draft, Data curation and Visualiztion.