Wrinkling reveals a new isometry of pressurized elastic shells

Abstract

We consider the point indentation of a pressurized, spherical elastic shell. Previously it was shown that such shells wrinkle once the indentation reaches a threshold value. Here, we study the behaviour of this system beyond the onset of instability. We show that rather than simply approaching the classical 'mirror-buckled' shape, the wrinkled shell approaches a new, universal shape that reflects a nontrivial type of isometry. For a given indentation depth, this "asymptotic isometry", which is only made possible by wrinkling, is reached in the doubly asymptotic limit of weak pressure and vanishing shell thickness.

Wrapping Liquids, Solids, and Gases in Thin Sheets

“…6c. Theoretical calculations reveal a wrinkled isometry that agrees with finite-element simulations (63). As with the indented floating film (Eq.…”

“…Vella et al (63) recently demonstrated a similar response in the indentation of a pressurized shell. In a well-known isometry called "mirror buckling", a portion of the shell is inverted into a concave spherical cap that is a mirror reflection of its original shape (74).…”

“…The research reviewed here has considered the response of a wrapper and its contents to their mutual interaction. We paid special attention to two recent developments: a new class of deformations called "asymptotic isometries" (60,61,63) and a geometric model for highly bendable sheets bound to liquid interfaces (86,94). These advances are already bearing fruit: the geometric model separates geometry from mechanics, making a class of highly nonlinear problems tractable.…”

“…4), only a negligible fraction of the total energy is stored in the sheet -nearly all the work is done on the gas, which is compressed. Here the doubly-asymptotic limit is of vanishing thickness and pressure, requiring the gas pressure to vanish as t α with 1 < α < 2 (63). If the pressure is any smaller, bending energies are expected to dominate and yet other morphologies may arise (75,76).…”

From Cylindrical to Stretching Ridges and Wrinkles in Twisted Ribbons

“…We will use this framework not only for the FIRs but also for the other shapes obtained upon increasing the tension. A similar approach has been used in other studies of sheets (or shells) on which a Gaussian curvature is imposed in the presence of a small tension [14][15][16]. The twist ηðθ; ϕ; R c Þ and contraction χðθ; ϕ; R c Þ of a FIRs can be obtained from geometrical arguments (see Ref.…”