  We investigate the effect of thermal fluctuations on the mechanical
properties of nanotubes by employing tools from statistical physics. For 2D
sheets it was previously shown that thermal fluctuations effectively
renormalize elastic moduli beyond a characteristic temperature-dependent
thermal length scale (a few nanometers for graphene at room temperature), where
the bending rigidity increases, while the in-plane elastic moduli reduce in a
scale-dependent fashion with universal power law exponents. However, the
curvature of nanotubes produces new phenomena. In nanotubes, competition
between stretching and bending costs associated with radial fluctuations
introduces a characteristic elastic length scale, which is proportional to the
geometric mean of the radius and effective thickness. Beyond elastic length
scale, we find that the in-plane elastic moduli stop renormalizing in the axial
direction, while they continue to renormalize in the circumferential direction
beyond the elastic length scale albeit with different universal exponents. The
bending rigidity, however, stops renormalizing in the circumferential direction
at the elastic length scale. These results were verified using molecular
dynamics simulations.
