Wrinkling in Sheets with Nonuniform Growth and Bending Rigidity

TL;DR

This paper investigates how non-uniform growth and differing bending rigidities in thin elastic sheets cause wrinkling, revealing how these factors influence wrinkle patterns and their spatial extent.

Contribution

It introduces a theoretical framework for understanding wrinkling in bi-strips with non-uniform bending rigidity, highlighting the role of local undulations and critical distances.

Findings

Wrinkle wavelength scales as w^{2/3} when non-swollen sheet rigidity is infinite.

Wrinkle extent w_C is proportional to the radius of curvature R_0.

Local undulations reduce bending energy by increasing the radius of curvature.

Abstract

Thin elastic sheets bend easily, leading to mechanical instabilities such as wrinkling. Here, we investigate wrinkles at edges of bi-strips, which consist of two thin sheets, one that swells and one that does not, joined side-by-side. It is well known that when bending rigidity is uniform across an isolated bi-strip, swelling results in axisymmetric shapes like a wine bottle: two cylinders of different radii are joined by a smooth transition zone. However, when the bending rigidity of the swollen sheet differs from that of the non-swollen sheet, purely axisymmetric shapes are no longer energetically favorable, and wrinkles arise. When the bending rigidity of the non-swollen sheet is essentially infinite, the wrinkles coarsen with distance from the transition zone such that dimensionless wavelengths and widths are related by $\tilde{\lambda} \propto \tilde{w}^{2/3}$. If the bending rigidity of the non-swollen sheet is non-infinite (but~still significantly larger than that of the swollen sheet), then the non-swollen sheet assumes a non-infinite radius of curvature, $R_0$. We find that the wrinkles in this system extend a critical distance, $w_C$, beyond the junction of the two strips and that $w_C \propto R_0$. Local undulations of wrinkles are favorable in this system because they decrease the overall bending energy by allowing the non-swollen sheet to have a larger radius of curvature than would otherwise be dictated by its reference geometry. Our results are relevant to a wide range of sheets that experience non-uniform growth, whether in natural systems such as plants or in synthetic systems such as designed, responsive materials.