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P. Hess, Frattura ed Integrità Strutturale, 34 (2015) 341-346; DOI: 10.3221/IGF-ESIS.34.37 344 square root of half defect length. These values are for tensile load at 300 K, where graphene is expected to fail by brittle fracture either by bond breaking or combined bond-rupture and bond-rotation processes. Larger slits in the range of 0.7  4 nm have been studied by a quantum mechanical/molecular mechanical/continuum mechanical (QM/MM/CM) model and compared with the Griffith formula [12]. Quantum mechanics was used to describe a small region around the slit and the coupled continuum-atomistic model was applied elsewhere. The electronic structure calculations agree with continuum fracture mechanics within 10% for the longest crack investigated. The Griffith relation, assuming linear behavior of the stress–strain relationship and neglecting lattice trapping, allows a reasonable description of graphene fracture initiated by crack-like defects as small as 1 nm in size. In fact, DFT calculations of the energy release rate exceeded the lower bound value of G IC = 2  2D by only 10%, indicating modest lattice trapping [12]. Fig. 1 presents the normalized critical fracture strengths of the nanometer slits obtained by the QM/MM/CM calculations versus the square root of half defect lengths. The decrease of the critical strength values is nearly paralel to the main correlation, however, with higher fracture strengths. As stated by the authors, the shape of such slits bears little relationship to sharp cracks assumed in the Griffith model. This could be a simple explanation for the higher strength values. Molecular dynamics simulations indicate that tensile fracture in nanocrystalline graphene may become insensitive to a pre- existing hole or notch below a characteristic critical length of 30 nm [13,14]. Circular holes with 1  6 nm radius created in the center of a nanostrip of nanocrystalline graphene with 2 nm average grain size did not initiate crack nucleation by stress concentration at the flaw but nucleation of cracks started some distance away from the hole [14]. This increase of the characteristic length scale of flaw tolerance in nanocrystalline graphene was connected with a decrease in the elastic modulus, fracture strength, and the strain energy release rate to 8 J/m 2 compared to about 12 J/m 2 for single-crystalline graphene. As stated above, higher critical fracture strengths are expected for circular holes in comparison with atomically sharp tips due to reduced stress amplification. 0.1 1 10 50 ( Half defect length) 1/2 1/2 (nm) 10 50 Strength (N/m) Hess (2015) this work Daly et al. (201 5) Wang et al. (2012) Khare et al. (2007) Yin et al. (2015) Rasool et al. (2013) 1 Zhang et al. (2014) Figure 1 : Intrinsic stress of perfect [4] and critical engineering stress of defective graphene containing vacancies [10,20], slits [12], grain boundaries [15], and pre-cracks studied theoretically by DFT and MD calculations [17] and experimentally by nanoindentation [16]. The straight line is a guide to the eye. Experimental analysis of graphene with defects Measurements of the intrinsic strength of crystalline and polycrystalline graphene membranes have been performed with a custom-fabricated diamond tip of 115 nm radius, to ensure a uniform stress distribution [15]. The strongest single-crystal membrane failed at 31.5 N/m, which is slightly lower than our mean value of 34.6 N/m presented in Fig. 1. For most large angle grain boundaries a strength of 80 GPa has been measured that corresponds to 26.8 N/m. Normalization of this value to the mean intrinsic strength considered here yields 29.4 N/m. By high-resolution transmission electron microscopy the atomic-scale strain field has been mapped showing bond lengths of up to 0.185 nm in comparison to the