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.