## 3 Results  
### Fracture Strength of Graphene with Defects  
Figure 2 displays the course of the stress-strain curves for defect-free graphene and that featuring a number of different dislocation dipoles, under tension along armchair/zigzag orientations. For simplicity, all the values for tension along the zigzag direction will be defined by superscript \(z\) (zigzag) and, for tension along the armchair direction, by superscript \(a\) (armchair). All the critical values such as ultimate tensile strength \(\sigma_{UITS}\), fracture strength \(\varepsilon_{F}\) and elastic modulus \(E\) for all considered structures under tension along armchair and zigzag direction are presented in Figure 3. Critical values for graphene under tension are shown by open circles. All the results are presented for three temperatures--0 K (theoretical strength), 300 K (room temperature), and 1000 K (finite temperatures). For a better understanding, both Figures 2 and 3 should be described in connection with each other. \begin{table}
\begin{tabular}{c c c} \hline
**Defect** & \(\mathbf{l}\), \(\mathbf{\hat{A}}\) & \(\mathbf{h}\), \(\mathbf{\hat{A}}\) \\ \hline SW & 0 & 0 \\ \hline DD\({}_{2}\) & 6 & 1.89 \\ \hline DD\({}_{4}\) & 11 & 2.0 \\ \hline DD\({}_{6}\) & 15 & 1.25 \\ \hline DD\({}_{8}\) & 20 & 1.22 \\ \hline DD\({}_{10}\) & 25 & 1.78 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of dipole arm \(l\) and buckling height \(h\) for graphene with SW and DD\({}_{n}\). Figure 1: Typical structures of graphene in two projections: (**a**) graphene with SW defect, (**b**–**f**) graphene with dislocation dipole DD\({}_{n}\) with the different arm. Geometrical parameters for dipoles are presented in Table 1. Pentagons and heptagons are colored in dark and light blue, respectively. First, let us discuss graphene strength. Commonly, in the literature, the graphene strength is very high, close to 100 GPa [19; 41; 42; 43]. However, results obtained by different numerical methods are considerably dependent on the simulation technique, which is reviewed in [19]. For example, as it is mentioned in [44], under dynamic load covalent networks exhibit brittle fracture instead of ductile because of insufficient structure relaxation. How the tensile strain is applied results in a different scenario for the composites based on graphene network [45]: values of the fracture strength of covalent systems for dynamic loading can be overestimated. In the present work, crumpling of graphene is allowed with the aim to consider the effect of natural graphene rippling on fracture strength. It is known that the deformation energy in free-standing graphene can be easily released through the formation of out-of-plane wrinkles [31]. This is in line with [32; 33], where it was shown that at positive strain values along one direction and negative strain values along the other direction, graphene loses its stability, and ripples with different orientations appear. In the present work, when graphene is stretched along one direction, it simultaneously can shrinkage along the normal direction, which results in positive stress along one axis (tension) and negative stress normal (compression). Thus, this scenario is known from the literature and in good agreement. It should be also mentioned, that type of ripples considerably affects the elastic modulus and fracture strain and strength [34; 35]: if there  
Figure 2: Stress-strain curves for graphene, graphene with SW defect and with DD\({}_{\mu}\) under tension along armchair/zigzag direction at three temperatures. are small-amplitude corrugations of graphene, the fracture strength and elastic modulus are lower, but not so notably in comparison with the high-amplitude ripples. Wrinkles and ripples can significantly reduce the magnitude of in-plane stresses, generated by the defects [38]. Thus, such a low \(\sigma_{UITS}^{a}\) = 8.1 GPa (\(\sigma_{UITS}^{z}\) = 7.1 GPa) and such a high \(\varepsilon_{F}^{a}\) = 0.51 (\(\varepsilon_{F}^{z}\) = 0.43) is quite understandable.