Issue34

P. Hess, Frattura ed Integrità Strutturale, 34 (2015) 341-346; DOI: 10.3221/IGF-ESIS.34.37 345 average bond length of 0.142 nm observed for the single-crystal material [15]. This maximum bond extension is used to display the reduction of the fracture strength by large angle grain boundaries in Fig. 1. If defects are present, as is usually the case for large-area engineering graphene sheets, the concept of fracture toughness can be employed to characterize the real engineering strength of the material, using the Griffith relationship. For example, the Griffith relation has been employed for failure analysis in the case of artificial pre-cracks. The aim of this model is to explain the discrepancy between the theoretically predicted pristine strength and the actually observed real fracture strength. Griffith’s brittle fracture model depends on the same mechanical properties as Eq. (1) and deviates only by a numerical factor and the meaning of the length scale involved in the brittle cleavage process between the weakest crystallographic cleavage planes [3]  cr = [(2  2D E 3D ) /(  a 0 )] 1/2 (2) Here  cr is the actual critical fracture strength of real crystals governed by the stress amplification effect generated by defects or flaws, E 3D the bulk Young’ modulus, and  2D the surface energy. The length scale a 0 is connected with half the length of an internal failure-inducing microcrack or the depth of a surface-breaking crack responsible for inducing crack nucleation. Note that the intrinsic length scale occurs in Eqs. (1) and (2) for dimensionality reasons. For example, in Eq. (2) the Young’s modulus has the dimension ‘energy per volume’ and the surface energy the dimension ‘energy per surface area’. Experimental values reported recently for the effective fracture stresses and corresponding crack lengths, observed for various artificial central cracks introduced with a focused ion beam, show for the first time the applicability of the Griffith relation to defective 2D crystals [16]. The extrapolation of the measured fracture strength values to the intrinsic value yields a surprisingly good agreement, as can be seen in Fig. 1. Thus, it is important to stress that the 2D bond-breaking model applied to the intrinsic fracture behavior of perfect graphene and the Griffith analysis of defective graphene deliver consistent results covering four decades of flaw size from the subnanometer to the micrometer range, despite the relatively large scatter in the data. This conclusion is supported by recent MD simulation indicating that the difference between the Griffith criterion and MD simulation is small for cracks propagating along the zigzag direction with the lowest critical fracture stress. For an energetically less favorable crack path, e.g., a kinked path approaching during propagation the zigzag direction from a different chirality, deviations of more than 15% may be observed for small crack sizes below 10 nm [17]. This conclusion has been obtained by comparing the apparent fracture resistance used as a fitting parameter in the relationship of the Griffith stress to crack length and the evaluation of the fracture resistance by MD calculations of the line or edge energy for different chiralities. Such an overestimation of the fracture strength in the nanometer region by the Griffith criterion is in agreement with the results observed in the previously discussed QM/MM calculations [12]. C ONCLUSIONS ith the two 2D fracture models discussed above a complete and consistent set of mechanical properties of graphene has been derived that is in good agreement with the extensive published data. The pristine strength is consistent with most modeled and the measured critical fracture strengths of graphene containing crack-like defects, stretching from vacancies with subnanometer size to artificial pre-cracks extending to micrometer size. The observed correlation of defect size behavior in the nanometer range is in disagreement with a general flaw tolerance in the nanoscale, where essentially theoretical strengths should be observed in the presence of cracks of nanometer size [13,14]. The present findings clearly support the alternative point of view discussed in more detail in [18]. A further argument against general flaw tolerance is that the pathway and direction of crack propagation in graphene can be controlled under tensile load by the presence of atomistic defects in the form of vacancies placed deliberately in the monolayer. This has been demonstrated by applying a first-principles-based reactive force field potential (ReaxFF) that features specific patterns of vacancies to simulate the fracture paths in graphene [19]. In fact, it was possible to cut graphene monolayers along these selected patterns simply according to the spatial arrangement of vacancies. The mechanical reliability and ability of defective covalent monolayers to resist failure is an important practical issue, because graphene and other 2D solids may play a crucial role in the post-silicon era. The critical strength–defect size dependence presented in Fig. 1 provides essential information on the degree of degradation of the engineering strength with the square root of half defect length taking into consideration very different structural imperfections. Furthermore, it confirms that the combination of the 2D bond-breaking model, describing the perfect monolayer, and the Griffith model, W