Tensional wrinkling of thin elastic sheets with two circular holes

TL;DR

This study combines analytical modeling and experiments to understand how two circular holes in thin elastic sheets influence tension-induced wrinkling, revealing the impact of geometric symmetry breaking on wrinkle behavior.

Contribution

It extends the classical wrinkling analysis to sheets with multiple holes using bipolar coordinates and validates predictions through experiments.

Findings

The presence of a second hole significantly affects wrinkle nucleation and orientation.

Analytical solutions accurately predict the wrinkling threshold as a function of hole spacing.

Experimental observations confirm the influence of geometry on wrinkle patterns.

Abstract

A paradigm for the study of wrinkling in elastic sheet is the Lam\'{e} configuration, in which azimuthal wrinkles form in an annular sheet subjected to tensile loads at both edges. Since wrinkles are spatially extended, this instability provides a mechanism for stress transmission over long distances. A natural extension of this problem is wrinkling in sheets with multiple holes or broken symmetry. Here, we investigate tension-induced wrinkling in thin elastic sheets containing two circular holes by combining analytical modeling and experiments. The pre-buckled state is solved analytically using bipolar coordinates, enabling identification of the wrinkling threshold as a function of the distance between the two holes. Near-threshold wrinkling and interactions between wrinkles are analyzed, and we validate our theoretical predictions against experimental observations obtained through video imaging of spin-coated polystyrene sheets floating on liquid surfaces with controlled surface tension. Our results demonstrate that geometric symmetry breaking, such as the presence of a second hole, strongly influences wrinkle nucleation, orientation, and spatial extent. Beyond mechanics, these findings might provide a simple mechanism for cellular mechanosensing, where force transmission is amplified by mechanical instabilities.