Non-affine bending mode of thin cylindrical tubes

TL;DR

This paper analyzes how thin cylindrical tubes bend non-uniformly under small indentations, revealing a universal stiffness profile that enables direct measurement of their bending modulus.

Contribution

It introduces a simplified one-dimensional model for the bending response of thin cylindrical tubes and validates it with experimental data.

Findings

Bending stiffness peaks at the center and drops by a factor of 4 at the ends.

The model predicts a universal stiffness profile for thin tubes.

Experimental results on stainless steel sheets confirm the theoretical predictions.

Abstract

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.