Issue34

P. Hess, Frattura ed Integrità Strutturale, 34 (2015) 341-346; DOI: 10.3221/IGF-ESIS.34.37 343 general definition provides layer thicknesses in very close agreement with those known for the building blocks in the layered structure of the corresponding bulk crystals, however, takes into consideration the real shape of the nanoobject. The 2D Young’s modulus and 2D intrinsic strength of perfect graphene have been measured by nanoindentation under biaxial tension and have also been studied by density-functional theory (DFT) calculations and MD simulations, considering fracture by uniaxial tension. From this information the breaking force and line or edge energy can be obtained by using Eq. (1). Furthermore, the fracture toughness and critical strain energy release rate can be determined. Owing to the large scatter of published experimental and theoretical data on graphene, which is larger than the differences due to chirality and the type of tension, the following mean 2D values were used in the present analysis: E 2D = 333 N/m,  2D = 34.6 N/m, and  1D = 2.0 nJ/m, using r 0 = 0.142 nm as bond length in graphene [4]. In addition, the values of the fracture toughness and critical strain energy release rate were estimated from these quantities: 2 D IC K =  2D (8 r 0 ) 1/2 = 1.2  10 -3 N/m 1/2 and 2 D IC G = 2  1D = 4.0 nJ/m. For comparison with literature data, bulk properties were calculated by dividing these properties by the thickness of the graphene monolayer of h = 0.334 nm, extracted from the interlayer thickness of the covalent layers in bulk graphite [7]. The layer thickness is generally employed to connect the 2D monolayer properties with the corresponding bulk properties of the 3D solid. These formal bulk properties are: E 3D = 997 N/m 2 ,  3D = 104 N/m 2 ,  2D = 6.0 J/m 2 , K IC = 3.5 MPa m 1/2 , and G IC = 12 J/m 2 . Atomic-scale mechanisms of fracture pathways in graphene Elucidation of the elementary steps of graphene fracture by first-principles calculations and experimental investigations employing high-resolution imaging are largely lacking. For this reason, atomistic simulations have been performed to investigate the fracture pathways in graphene. For example, an analytical bond-order potential has been used to describe the interaction of the covalent C  C bonds in graphene [8]. With this model two competing atomic failure processes have been identified, namely bond breaking at the crack-tip and C  C bond rotation by 90°, leading to the formation of two pentagon/heptagon Stone–Wales (SW) defects. Usually, brittle fracture via bond breaking is expected to prevail at low temperatures, whereas plastic deformation based on dislocations and the nucleation and motion of SW defects should dominate at high temperatures. Simulations of the minimum energy paths indicate that bond rotation may be energetically and kinetically more favorable with a 1  2 eV lower energy barrier. However, the generation of the SW defect changes the local environment at the crack tip, leading to a crossover point, where symmetric and asymmetric bond breaking may dominate kinetically. Crack extension is expected to remain straight according to the condition of quasi-static growth on the plane of maximum normal stress. Consequently, the model suggests a mixed mechanism involving an alternating sequence of bond breaking and bond rotation under quasi-static loading conditions near the energetic limit of fracture [8]. The non-uniform bond deformation and rupture processes with highly localized stresses at the tip may create a variety of edge morphologies including reconstructed edges, which influence the electronic properties of graphene [8]. Thus, alternatively to a simple bond-breaking process as the atomic-scale fracture pathway in the presence of pre-existing cracks in graphene, an alternating bond-breaking and bond-rotation mechanism may occur. Besides the temperature, the fracture mechanism depends also on the loading rate. In the case of fast strong loading, crack kinking and branching will take place in graphene. Atomistic MD simulations on the dynamics of rapid fracture show that under pure opening load (mode I) the crack moves straight along the zigzag direction with atomically smooth edges at a speed of up to 8.2 km/s, which corresponds to 65% of the Rayleigh wave phase velocity that is 12.56 km/s in graphene [9]. Above this critical speed, kinking and instability with the formation of rough and irregular edges involving pentagons and heptagons is observed, owing to the strong energy input. R ESULTS AND DISCUSSION Theoretical analysis of the influence of defects in graphene s mentioned before, structural defects such as vacancies may play an important role in deteriorating the structural integrity and mechanical performance, especially of large graphene layers. In the subnanometer length scale range the effects of (multiple) vacancies on the fracture strength have been studied by MD simulations [10,11]. Fig. 1 shows the critical fracture strength of n =1, 2, and 3 vacancy units with a crack length of l = na , where a is the lattice spacing a = 3 r 0 = 0.246 nm, normalized to the mean intrinsic strength of  2D = 34.6 N/m [10] as a function of the A