Discrete dislocations in graphene

In this work, we present an application of the theory of discrete dislocations of Ariza and Ortiz (2005) to the analysis of dislocations in graphene. Specifically, we discuss the specialization of the theory to graphene and its further specialization to the force-constant model of Aizawa et al. (1990). The ability of the discrete-dislocation theory to predict dislocation core structures and energies is critically assessed for periodic arrangements of dislocation dipoles and quadrupoles. We show that, with the aid of the discrete Fourier transform, those problems are amenable to exact solution within the discrete-dislocation theory, which confers the theory a distinct advantage over conventional atomistic models. The discrete dislocations exhibit 5-7 ring core structures that are consistent with observation and result in dislocation energies that fall within the range of prediction of other models. The asymptotic behavior of dilute distributions of dislocations is characterized analytically in terms of a discrete prelogarithmic energy tensor. Explicit expressions for this discrete prelogarithmic energy tensor are provided up to quadratures.