Exact non-Hookean scaling of cylindrically bent elastic sheets and the large-amplitude pendulum

TL;DR

This paper demonstrates that the elastic force of a cylindrically bent elastic sheet exhibits an exact non-Hookean power-law behavior, analogous to a large-amplitude pendulum, supported by analytical solutions and experimental validation.

Contribution

It establishes an exact analytical description of the non-Hookean elastic force in bent sheets using a pendulum analogy, extending understanding of large deformation elasticity.

Findings

Elastic force follows an inverse square law for large deformations.

Analytical solution matches experimental measurements across deformation ranges.

The problem is equivalent to a gravitational pendulum with large amplitude.

Abstract

A sheet of elastic foil rolled into a cylinder and deformed between two parallel plates acts as a non-Hookean spring if deformed normally to the axis. For large deformations the elastic force shows an interesting inverse squares dependence on the interplate distance [Siber and Buljan, arXiv:1007.4699 (2010)]. The phenomenon has been used as a basis for an experimental problem at the 41st International Physics Olympiad. We show that the corresponding variational problem for the equilibrium energy of the deformed cylinder is equivalent to a minimum action description of a simple gravitational pendulum with an amplitude of 90 degrees. We use this analogy to show that the power-law of the force is exact for distances less than a critical value. An analytical solution for the elastic force is found and confirmed by measurements over a range of deformations covering both linear and non-Hookean behavior.