  Understanding thin sheets, ranging from the macro to the nanoscale, can allow
control of mechanical properties such as deformability. Out-of-plane buckling
due to in-plane compression can be a key feature in designing new materials.
While thin-plate theory can predict critical buckling thresholds for thin
frames and nanoribbons at very low temperatures, a unifying framework to
describe the effects of thermal fluctuations on buckling at more elevated
temperatures presents subtle difficulties. We develop and test a theoretical
approach that includes both an in-plane compression and an out-of-plane
perturbing field to describe the mechanics of thermalised ribbons above and
below the buckling transition. We show that, once the elastic constants are
renormalised to take into account the ribbon's width (in units of the thermal
length scale), we can map the physics onto a mean-field treatment of buckling,
provided the length is short compared to a ribbon persistence length. Our
theoretical predictions are checked by extensive molecular dynamics simulations
of thin thermalised ribbons under axial compression.
