Issue34

P. Hess, Frattura ed Integrità Strutturale, 34 (2015) 341-346; DOI: 10.3221/IGF-ESIS.34.37 342 studied in engineering devices and compared with the behavior of the corresponding perfect solids to judge the mechanical quality and reliability of real materials. Graphene is the paragon of the new class of 2D covalently bonded solids with unsurpassed intrinsic strength and strong in-plane but very weak out-of-plane stiffness. Together with h-BN monolayers (boronitrene) the one-atom-thick sp 2 - bonded hexagonal graphene structure is one of the thinnest known materials, however, without the partially ionic character of boronitrene, which makes the isolation of boronitrene monolayers more difficult. Compared to these single- atom layer crystals, the presently actively studied transition-metal dichalcogenide (TMD) monolayers consist of several atomic layers, similar to the most widely studied MoS 2 sheets, which contain a central hexagonal plane of Mo sandwiched between two hexagonal S planes. For these multilayer materials the exact layer thickness still is a controversial issue with differences in the assumed thickness of a factor of two and the failure behavior is considerably less understood than for the single-atom monolayers. Graphene is an ideal model system for studying the limitations of continuum mechanics, where the 2D crystal can no longer be characterized by the usual macroscopic mechanical properties. This is of interest for the application of the Griffith criterion, which is the most important basic relationship for investigating the role played by defects in engineering materials [3]. Therefore, an estimate of the critical size, where the model breaks down at the nanoscale, is of paramount importance for practical applications. Restrictions comparable to those for the Griffith relation may not be expected in the case of the 2D bond-breaking model introduced below, which contains the same mechanical quantities, however, a different meaning of the length scale. Presently, no contributions from experiments can be expected to solve this fundamental question concerning the range of validity of these basic models in applications to nanoobjects. Molecular dynamics (MD) simulations, which allow the treatment of fracture at the atomic scale, are strongly limited by the feature size. However, since a graphene monolayer is only one atom thick, larger sample areas may be investigated at reasonable computational cost, where size effects should be negligible. The MD simulations give access to microscopic details of the fracture process in graphene such as the difference between uniaxial and biaxial tension, the role played by chirality, e.g., the difference of strength in the zigzag and the armchair directions, and the temperature dependence of failure. Unfortunately, the uncertainties involved in these simulations are still large and in fact may be larger than the difference between the specific types of tension or the effect of chirality. For these reasons, it is useful to consider mean values of mechanical properties to extract the main characteristic behavior, such as the decrease of the critical fracture stress with the actual size of defects. To date contradictory results have been obtained by MD simulations at the nanoscale, yielding, for example, on one hand flaw tolerance in the nanometer range and on the other hand the applicability of continuum models at the nanometer length scale. T HEORY 2D fracture model of 2D mechanical properties of graphene he 2D bond-breaking model can be derived [4] from the corresponding 3D cleavage model [5] that combines the basic elastic and fracture properties with a subnanometer length. The analytical 2D bond-breaking model connects the relevant in-plane mechanical quantities of 2D solids belonging to continuum mechanics with a subnanometer length scale, describing the atomistic fracture behavior of chemical bonds by introducing a Morse-type interaction function  2D = [(  1D E 2D ) /(4 r 0 )] 1/2 (1) Here  2D is the 2D intrinsic strength or critical stress in (N/m), E 2D the 2D Young’s modulus in (N/m),  1D the breaking force in (N) or line (edge) energy or (J/m), and r 0 the nanoscopic length scale representing the equilibrium bond length of the covalent monolayer. This parameter-free 2D fracture model provides a versatile tool to determine any missing intrinsic mechanical property if the other two and the length scale are known. The so-called 2D solids are not really 2D in the strict mathematical sense but possess a finite thickness and volume determined in the single-atom layers by the size of the constituent atoms. For this reason bulk mechanical properties can be assigned to this particular atomic volume, despite the subnanometer thickness of the monolayer. The thickness of the building blocks in the layered graphite material agrees well with a definition of the active nanoscopic volume in such a way that the average electron density of the isolated monolayer matches that in the parent bulk material [6]. As expected, this T