Analytic description and explicit parametrization of the equilibrium shapes of elastic rings and tubes under uniform hydrostatic pressure

TL;DR

This paper provides explicit analytic formulas for the equilibrium shapes of elastic rings and tubes under uniform pressure, simplifying the determination process and revising previous numerical findings.

Contribution

It introduces explicit parametric equations for equilibrium shapes and reduces shape determination to solving transcendental equations, improving upon prior numerical methods.

Findings

Explicit parametric equations for equilibrium shapes

Reduction to solving two transcendental equations

Revised results on shapes with lines of contact

Abstract

The parametric equations of the plane curves determining the equilibrium shapes that a uniform inextensible elastic ring or tube could take subject to a uniform hydrostatic pressure are presented in an explicit analytic form. The determination of the equilibrium shape of such a structure corresponding to a given pressure is reduced to the solution of two transcendental equations. The shapes with points of contact and the corresponding (contact) pressures are determined by the solutions of three transcendental equations. The analytic results presented here confirm many of the previous numerical results on this subject but the results concerning the shapes with lines of contact reported up to now are revised.