Theoretical and experimental analysis of a thin elastic cylindrical tube acting as a non-Hookean spring

We analyze the (large) deformation and energy of a thin elastic cylindrical tube compressed between two plates parallel to the tube axis. The deformation is studied theoretically using a numerical calculation and the variational approach. The results are used to interpret the experimental data obtained by pressing tubes made from plastic-foil transparencies. We obtain a universal scaling relation that characterizes the response of the tube. Our results may serve as a benchmark for the application of variational methods to thin-walled nanoscale systems in order to obtain functional relations between the energy and the deformation.

Figure 1

Illustration of the experimental setup.

Figure 2

Theoretical predictions for the spring energies. The circles show the numerical results, the dotted line is the prediction of the variational method based on the stadium profile, and the dashed line corresponds to the elliptic profile. The solid line shows the stadium profile energy multiplied by 0.912.

Figure 3

Half of the separation between the plates $b$ vs scale reading $m_e$ for four springs made from identical foils. The dashed line shows the slope expected for the $b\propto m_e^{-1/2}$ dependence. The thin dotted line shows $b=0.7\langle b_0\rangle$. See text for details.

Figure 4

Scaled forces as specified by Eqs. (9) (line) and (8) (symbols). Circles and crosses correspond to the two sheets from set 1 rolled along the longer and shorter sides, respectively. Triangles and squares correspond to the two sheets from set 2 rolled along the longer and shorter sides, respectively. Pluses correspond to a measurement on a sheet from set 3 rolled along its longer side. The solid line shows the numerical results. The inset shows the calculated profiles of the spring for different pressing forces.