We first develop a continuum framework to predict the energies of topological defects in two-dimensional membranes embedded in a three-dimensional world. The topological defects discussed are the two-dimensional equivalent of dislocations -- the primary mediators of plasticity in bulk materials -- and our framework can be used to describe the deformation and plastic/elastic response of two-dimensional systems such as sp2 carbon systems (nanotubes and graphene). The theory is a simple, quantitative, transferable description that makes no a priori assumptions about the analytical form of the dislocation strain fields, while explicitly accounting for boundary conditions and long-ranged defect-defect interactions, and reproduces trends observed in atomistic simulations remarkably well.
Using this framework, we implement Monte Carlo simulations to study dislocation-glide-mediated plasticity in carbon nanotubes and graphene, which leads to the prediction of dislocation worms - a new type of a defect comprising of a dislocation screened by multiple dislocation dipoles - as a mediator of plastic deformation. The appearance of these dislocation worms is rationalized in terms of a competition between defect core energy and the buckling inherent to two-dimensional membranes deforming in a three-dimensional world. Monte Carlo simulations within the isoenthalpic-isotension-isobaric ensemble are also employed to compute the elastic properties and relevant thermodynamic functions of freestanding monolayer graphene in order to explore the influence of thermal ripples on the elastic properties. Thermal ripples soften the elastic constants, and lead to the negative thermal expansion coefficient observed experimentally at low temperatures.