Anomalous Hooke's law in disordered graphene
The discovery of graphene, a single monolayer of graphite, has closed the discussion on stability of 2D crystals. Although thermal fluctuations of such crystals tend to destroy the long range order in the system, the crystal can be stabilized by strong anharmonicity effects. This competition is the central issue of the crumpling transition, i.e. a transition between flat and crumpled phases. We show that anharmonicity controlled fluctuations of a graphene membrane around equilibrium flat phase lead to unusual elastic properties. In particular, we demonstrate that stretching \xi$ of a flake of graphene is a nonlinear function of the applied tension at small tension: \xi\propto\sigma^\eta/(2 \eta) and \xi\propto\sigma^\eta/(8 \eta) for clean and strongly disordered graphene, respectively. Conventional linear Hooke's law, \xi\propto\sigma} is realized at sufficiently large tensions: \sigma\gg\sigma_* where \sigma_* depends both on temperature and on the disorder strength.