Postbuckling of circular rings: An analytical solution

Abstract

Articles you may be interested inPostbuckling behaviors of nanorods including the effects of nonlocal elasticity theory and surface stressThe paper gives the analytical solution for postbuckling of an initially circular inextensible elastic ring under uniform external pressure.

Softly constrained films

“…After buckling, increasing the surface tension results in a reduction of the area enclosed by the rod, with the convex regions of the curve moving away from the center of mass and the convex regions shifting toward the interior, consistently with Eq. (22). This trend continues until one reaches a second n−dependent critical surface tensionσ cn at which two or more points on the opposite sides of the boundary come into contact.σ cn is called the contact surface-tension (or contact pressure) and was first introduced by Flaherty and coworkers [24].…”

“…Some fifty years after the original work by Lévy, Carrier resumed the problem, found an expression for the curvature of the buckled ring in terms of Jacobi elliptic functions and worked out an approximate solution near the onset of buckling. This analysis as been recently reconsidered by Adams [22] who further extended Carrier's work. Tadjbakhsh and Odeh [23], provided a rigor- ous study of the boundary-value problem describing the shape of the ring and the associated variational problem.…”

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

“…where σ = p/D and µ is an arbitrary constant of integration. On the other hand, combining equations (20), (11) and (12) we obtain the relation…”

“…He expressed the curvature of the deformed ring in terms of Jacobi cosine function [17] involving several unknown parameters to be determined by a system of algebraic equations. However, he succeeded to find approximate solutions to this system only for small deflections from the undeformed circular ring shape (see the exhaustive analysis provided recently by Adams [11] who has criticized and developed Carrier's work [1]).…”

Analytic Description of the Equilibrium Shapes of Elastic Rings Under Uniform Hydrostatic Pressure

“…He expressed the curvature of the deformed ring in terms of the Jacobi cosine function [14][15][16][17] involving several unknown parameters to be determined by a system of algebraic equations. However, he succeeded to find approximate solutions to this system only for small deflections from the undeformed circular ring shape (see the analysis provided recently by Adams [1] who has criticized and developed Carrier's work [6]). …”