## 3 Results
### Analysis of the Fracture Mechanisms
To find equilibrium structures at different tensile strains and analyze the evolution of bond lengths, all structures are considered at 0 K. Small random perturbations to coordinates of all atoms with the magnitude \(10^{-5}\) are introduced to check the structure for stability. For graphene, before point A, bond \(a\) is continuously stretching and achieves a constant value of 1.7 A  until point C. In contrast, bond \(b\) almost not changed before point A, but increase until point B. After point B, bond \(a\) cannot be stretched more, and deformation took  
Figure 6: Snapshots of graphene, graphene with SW defect and with DD\({}_{2}\) under tension along armchair direction at \(T\) = 0 K. Snapshots are presented in a perspective view and in a projection on \(yz\) plane. The defect is shown by a black circle. Corresponding dots A–D, B’, D’ are from Figure 5.  
place because of a further increase of the ripple amplitude and changes in the valent angles and bond \(b\). Moreover, bond length can be decreased because of the transformation of valent angles. After point C, again bond \(a\) starts to increase and then break. From Figure 6 it is seen that graphene became a plane without any ripples and the prevailing deformation mechanism is bond stretching. If structure with DD\({}_{n}\) is considered, better to analyze all the bonds on the defect and also around it. However, for clearance, only three bonds are presented in Figure 5c--bond 1 between 5- and 7-atoms ring, bond 2 on the 5-atomic ring, and bond 3 near the defect DD\({}_{2}\). As can be seen, bond 1 stretched before point A in the same way as bond \(a\) in graphene, while bonds 2 and 3 almost not changing. At B', the rapid increase of bond 3 took place which allow the further stretching of graphene. In fact, not only bond 3 is changing at this point. The same transformations appeared for some other orientations of the bonds in a hexagonal lattice. When the bonds are oriented along the tension direction (armchair in this case) they can provide tension along with the simultaneous ripple formation. However, when these bonds are critically strained, bonds with other orientations start to play the main role. These results are not trivial except for the case of defect-free graphene: it is expected that bond 1 in dislocation is the most strengthened, but the main changes in the lengths of the bonds took place near the defect, not right on it. Interestingly, that defect position affects the ripple distribution. SW defect is set on the side of one of the ripples, while DD\({}_{2}\) is placed right between two ripples. The increase of the dipole arm results in a slight redistribution in the structure: each dislocation is placed on a different ripple, on the corresponding side. After the transition from rippled graphene to the plane, another structural transformation took place for all the considered graphenes at points C, C': almost equilibrium hexagons transform to elongated ones and this is a pre-critical transformation. In defect-free graphene, fracture starts at any place in its basal plane (an example is shown in Figure 6). For graphene with defects, ripples start to disappear right from the defect, and then the second transformation takes place. The fracture starts close to the defect and the crack extends diagonally from the 7-ring defect. It should be mentioned that, for graphene with DD\({}_{6}\), a third ripple tends to appear, but the presence of a defect prevents this. Moreover, for DD\({}_{6}\)-DD\({}_{10}\) graphene does not transform to the totally planar before fracture. Also, the new defect appears under tension--defect 5-8-5, which is also well-known for graphene. In Figure 7a, the comparison between stress-strain curves (a) for tension at 0 K along zigzag direction for graphene, graphene with SW and DD\({}_{2}\) is presented in comparison with the average amplitude of ripples (b) and bond length (c) as the function of strain.