13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Mixed-mode loading in classical fracture mechanics is then imposed by initially assigning all atoms in the displacement field given by the crack-tip asymptotic solution of a specified initial Keff app. In Figure 1, atoms (pink) on the outer boundary layer are held fixed, while all the other atoms (green) are set free, and the atomic configuration is then relaxed. Then we implement the deformation-control method by applying displacement increments gradually to the fixed boundary layer separately every 500 MD steps. At each applied loading, the statically equilibrium lattice structure is calculated to minimize the total energy by the limited memory BFGS geometry optimization algorithm [26], thereby local energy minimum configurations are obtained. The velocity-Verlet time stepping scheme is used with a time step 1.0 fs at predominantly 300K with a Berendsen thermostat, and this yields a strain rate 0.0002 ps−1 primarily. We note that MD simulations are often sensitive to the temperature control and the loading rate, thus our results mainly provide a qualitative understanding of the fracture mechanisms. 3. Results and discussion The energy-balance criterion by Griffith is the fundamental fracture criterion for brittle continua, which states that a crack meets the critical growth condition when the net change in the total energy of the system vanishes upon crack extension by an infinitesimal distance [27]. Using the relationship between the critical SIF of Griffith Kth c and the energy release rate (twice of the surface energy density γs) for linear elastic materials, one has Kth c = (2Eγs) 1/2 [24]. Since E is assumed isotropic for graphene, Kth c will be mainly determined by γs. By use of γs = 1.041 eV/Å and 1.091 eV/Å [28, 29] for ZZ and AC cracks, we get Kth c = 3.162 nN Å-3/2 and 3.238 nN Å-3/2, respectively. Table 1. Effective critical stress intensity factors Keff c (nN Å-3/2) of zigzag and armchair cracks in graphene under far-field loading at various phase angles φ. φ 0° 15° 30° 45° 60° 75° 90° 90° 90° 90° 90° 90° Kc eff ZZ 3.06 2.75 2.63 2.90 3.15 3.02 3.05 3.16a 4.21b 2.64c 6.0 d 10.32e AC 2.87 3.30 3.28 2.87 2.78 2.85 3.38 3.24a 3.71b a Critical stress intensity factor of Griffith Kth c; b Ref. 6; c Ref. 7; d Ref. 5; e Ref. 30. In Table 1 and Figure 2, our results show that the effective critical SIF Keff c of I/II mixed-mode loading falls in the range between 2.63 nN Å-3/2 and 3.38 nN Å-3/2 varying with φ, relatively low compared to steel, which reveals that graphene is brittle at 300K opposing its ultrahigh strength. As the difference of geometric chiral angle between AC and ZZ edges is 30°, similar trends for Keff c are observed if φ is shifted by 30°, see Figure 2. Keff c along ZZ edges are slightly lower indicating smaller toughness, thus graphene is easier to break along ZZ direction. For ZZ cracks under pure opening tension (φ = 90°), our Keff c are reasonable with theoretical Kth c, and compared with available reported datum, Table 1, the discrepancy may be due to different crack models [5, 7] and potentials [6, 30].
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