Non-affine bending mode of thin cylindrical tubes

We discuss the response of thin cylindrical tubes to small indentations. The stretching-free embedding of these surfaces is completely determined by the embedding of one edge curve, thus reducing their bending response to an effectively one dimensional problem. We obtain that in general thin cylindrical tubes will bend non-uniformly in response to an indentation normal to their surface. As a consequence, their local bending stiffness, as measured by indenting at various locations along their axis, is predicted to peak at the center and decrease by a universal factor of 4 at either end of the tube. This characteristic spatial profile of the bending stiffness variation allows a direct determination of the bending modulus of very thin tubes that are shorter than their pinch persistence length, $l_{p}\propto R^{3/2}/t^{1/2}$, where $R$ is the tube diameter and $t$ its wall thickness. We compare our results with experiments performed on rolled-up stainless steel sheets.