Static bistability of spherical caps

TL;DR

This study investigates the conditions under which spherical shell caps exhibit bistability, identifying geometric thresholds and analyzing how indentation affects their stability and transition between states.

Contribution

It combines experiments, simulations, and theory to determine bistability thresholds and explores how indentation influences stability in spherical caps.

Findings

Bistability depends on solid angle geometry.

Modified shallow shell theory accurately predicts bistability.

Indentation can induce transitions or buckling depending on shell thickness.

Abstract

Depending on its geometry, a spherical shell may exist in one of two stable states without the application of any external force: there are two `self-equilibrated' states, one natural and the other inside out (or `everted'). Though this is familiar from everyday life -- an umbrella is remarkably stable, yet a contact lens can be easily turned inside out -- the precise shell geometries for which bistability is possible are not known. Here, we use experiments and finite element simulations to determine the threshold between bistability and monostability for shells of different solid angle. We compare these results with the prediction from shallow shell theory, showing that, when appropriately modified, this offers a very good account of bistability even for relatively deep shells. We then investigate the robustness of this bistability against pointwise indentation. We find that indentation provides a continuous route for transition between the two states for shells whose geometry makes them close to the threshold. However, for thinner shells, indentation leads to asymmetrical buckling before snap-through, while also making these shells more `robust' to snap-through. Our work sheds new light on the robustness of the `mirror buckling' symmetry of spherical shell caps.