Anomalous Hooke's law in disordered graphene

The discovery of graphene, a single monolayer of graphite, has provided an experimental demonstration of 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. Weshow that anharmonicity-controlled fluctuations of a graphene membrane around equilibrium flat phase lead to unusual elastic properties. In particular, we demonstrate that stretching. of a flake of graphene is a nonlinear function of the applied tension at small tension: xi proportional to sigma(eta/(2-eta)) and xi proportional to sigma(eta/(8-eta)) for clean and strongly disordered graphene, respectively. Conventional linear Hooke's law, xi proportional to sigma , is realized at sufficiently large tensions: sigma >> sigma(*), where sigma(*) depends both on temperature and on the disorder strength.