Interplay of gross and fine structures in strongly-curved sheets

TL;DR

This paper investigates the relationship between large-scale gross shapes and small-scale fine features in strongly curved thin sheets, using a balloon model to unify global and local behaviors and explain observed phenomena.

Contribution

It introduces a minimal elastic filament model constrained by a mean shape, bridging the gap between global geometry and local buckling features in curved sheets.

Findings

The geometric model captures the mean shape behavior.

The filament model reproduces wrinkle and fold patterns.

The approach explains morphology changes with pressure.

Abstract

Although thin films are typically manufactured in planar sheets or rolls, they are often forced into three-dimensional shapes, producing a plethora of structures across multiple length-scales. Existing theoretical approaches have made progress by separating the behaviors at different scales and limiting their scope to one. Under large confinement, a geometric model has been proposed to predict the gross shape of the sheet, which averages out the fine features. However, the actual meaning of the gross shape, and how it constrains the fine features, remains unclear. Here, we study a thin-membraned balloon as a prototypical system that involves a doubly curved gross shape with large amplitude undulations. By probing its profiles and cross sections, we discover that the geometric model captures the mean behavior of the film. We then propose a minimal model for the balloon cross sections, as independent elastic filaments subjected to an effective pinning potential around the mean shape. This approach allows us to combine the global and local features consistently. Despite the simplicity of our model, it reproduces a broad range of phenomena seen in the experiments, from how the morphology changes with pressure to the detailed shape of the wrinkles and folds. Our results establish a new route to understanding finite buckled structures over an enclosed surface, which could aid the design of inflatable structures, or provide insight into biological patterns.