Elastic strips

TL;DR

This paper develops a theoretical framework for elastic strips, deriving conservation laws and identifying new classes of integrable solutions, including connections to spherical curves and Hopf tori, advancing understanding of developable elastic structures.

Contribution

The paper introduces two new conservation laws for elastic strips and discovers two classes of integrable solutions linked to spherical elastic curves.

Findings

Derived two conservation laws for elastic strips.

Identified two new classes of integrable elastic strips.

Established a connection between Hopf tori and force-free strips.

Abstract

Motivated by the problem of finding an explicit description of a developable narrow Moebius strip of minimal bending energy, which was first formulated by M. Sadowsky in 1930, we will develop the theory of elastic strips. Recently E.L. Starostin and G.H.M. van der Heijden found a numerical description for an elastic Moebius strip, but did not give an integrable solution. We derive two conservation laws, which describe the equilibrium equations of elastic strips. In applying these laws we find two new classes of integrable elastic strips which correspond to spherical elastic curves. We establish a connection between Hopf tori and force--free strips, which are defined by one of the integrable strips, we have found. We introduce the P--functional and relate it to elastic strips.