Hidden Area and Mechanical Nonlinearities in Freestanding Graphene.

We investigated the effect of out-of-plane crumpling on the mechanical response of graphene membranes. In our experiments, stress was applied to graphene membranes using pressurized gas while the strain state was monitored through two complementary techniques: interferometric profilometry and Raman spectroscopy. By comparing the data obtained through these two techniques, we determined the geometric hidden area which quantifies the crumpling strength. While the devices with hidden area ∼0% obeyed linear mechanics with biaxial stiffness 428±10  N/m, specimens with hidden area in the range 0.5%-1.0% were found to obey an anomalous nonlinear Hooke's law with an exponent ∼0.1.