From wrinkling to global buckling of a ring on a curved substrate

TL;DR

This paper investigates the stability of a ring on a curved substrate under internal pressure, identifying conditions for wrinkling or buckling, and compares analytical predictions with numerical simulations to understand the influence of substrate curvature.

Contribution

It provides a combined analytical and numerical framework to analyze the stability and transition between wrinkling and buckling modes of a ring on a curved substrate, including a phase diagram and critical pressure expression.

Findings

Critical stress is insensitive to substrate curvature.

Wavelength is affected by the substrate's curvature due to geometric constraints.

Analytical predictions align well with finite element simulations.

Abstract

We present a combined analytical approach and numerical study on the stability of a ring bound to an annular elastic substrate, which contains a circular cavity. The system is loaded by depressurizing the inner cavity. The ring is modeled as an Euler-Bernoulli beam and its equilibrium equations are derived from the mechanical energy which takes into account both stretching and bending contributions. The curvature of the substrate is considered explicitly to model the work done by its reaction force on the ring. We distinguish two different instabilities: periodic wrinkling of the ring or global buckling of the structure. Our model provides an expression for the critical pressure, as well as a phase diagram that rationalizes the transition between instability modes. Towards assessing the role of curvature, we compare our results for the critical stress and the wrinkling wavelength to their planar counterparts. We show that the critical stress is insensitive to the curvature of the substrate, while the wavelength is only affected due to the permissible discrete values of the azimuthal wavenumber imposed by the geometry of the problem. Throughout, we contrast our analytical predictions against finite element simulations.