Perforations, disclination quadrapoles and crumpling of free-standing graphene

Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, characterized by a dimensionless coupling constant (the ``Foeppl-von Karman number'') that can easily reach vK = 10$^{7}$ in an ordinary sheet of writing paper. These equations lead to highly nonlinear force-extension curves associated with the buckling of partial disclinations, even for the simple case of a square sheet punctured by a large square hole. However, thermal fluctuations in thin elastic membranes fundamentally alter the long wavelength physics. We discuss the remarkable properties of free-standing graphene sheets (with vK = 10$^{13}$!) at room temperature, where enhancements of the bending rigidity by factors of $\sim$4000 compared to T = 0 values have now been observed. Thermalized elastic membranes can undergo a crumpling transition when the microscopic bending stiffness is comparable to kT. We argue that the crumpling temperature can be dramatically reduced by inserting a regular lattice of laser-cut perforations. These expectations are confirmed by extensive molecular dynamics simulations, which also reveal a remarkable "frame crumpling transition" triggered by a single large hole inserted into a graphene sheet.