13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- fracture mechanics motivates further investigation of fracture of graphene using continuum concepts. Figure 5. Variation of ultimate strength with crack length at 1 K; σult = 120.9*(1/√a) + 5.7 and σult = 139.7*(1/√a) + 4.7 for armchair and zigzag sheets, respectively. Table 1: Variation of KI g and c with temperature. Temp (K) KI g (MPa m1/2) c (GPa) Armchair 1 4.8 5.7 300 4.2 11 Zigzag 1 5.6 4.7 300 4.6 4 4. Continuum Fracture Mechanics Inglis [15] and Griffith [16] proposed two fundamental theories in fracture mechanics. Inglis derived the stress concentration due to an elliptical hole in a linearly elastic material [15]. Considering the nonlinear σ-ε relation of graphene, the remote stress at failure (σf) can be written as σf = Ef γs 4a , (3) where Ef is the tangent modulus at failure; γs is the surface energy and a is the crack length. Inglis’ theory was followed by Griffith's work on the fracture of brittle solids [16]. Griffith’s energy balance says that failure occurs when the energy stored in the structure is sufficient to overcome the surface energy of the material. Failure stress of Griffith’s model is expressed as σf = 2Ef γs πa . (4) The value of γs is calculated by dividing the difference in energy of a graphene sheet before and after fracture by the area of newly created surface. The value of γs is 5.02 J/m2 for both armchair and zigzag graphene since the distance between two broken carbon-carbon bonds is similar in both sheets. The tangent modulus at failure is obtained from the σ-ε curves of pristine graphene sheets, and it can be expressed of as E(ε) = -5.89ε + 1.08 TPa and E(ε) = -3.50ε + 0.89 TPa for armchair
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