Universal Wrinkling of Supported Elastic Rings

Abstract

An exactly solvable family of models describing the wrinkling of substrate-supported inextensible elastic rings under compression is identified. The resulting wrinkle profiles are shown to be related to the buckled states of an unsupported ring and are therefore universal. Closed analytical expressions for the resulting universal shapes are provided, including the one-to-one relations between the pressure and tension at which these emerge. The analytical predictions agree with numerical continuation results to within numerical accuracy, for a large range of parameter values, up to the point of self-contact.

The θ-formulation of the two-dimensional elastica: buckling and boundary layer theory

“…where F t and F n are external axial and shear force per unit length, respectively. Recently, for instance, an analytically solvable model for elastic wrinkling on a nonlinear foundation was introduced, in which F t = 0 while F n scales as some power of the local displacement of the elastica [26][27][28] (see also [29][30][31][32][33] for more experimental and theoretical results on the subject). We may continue to define H as Bκ 2 /2 + t. It now satisfies…”