Shape Transitions in Hyperbolic Non-Euclidean Plates

TL;DR

This paper investigates shape transitions in hyperbolic non-Euclidean plates, revealing the existence of flat, saddle, and wavy configurations, with findings relevant to both theoretical models and experimental observations.

Contribution

It provides a comprehensive analysis of shape transitions in hyperbolic plates using F"oppl - von Kármán and Kirchhoff models, highlighting new types of minimizers and isometric immersions.

Findings

Only flat and saddle shapes are global minimizers in F"oppl - von Kármán model.

Existence of local n-wave minimizers with inflection regions.

Periodic isometric immersions with exponentially growing waves in Kirchhoff model.

Abstract

We present and summarize the results of recent studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both the F\"oppl - von K\'arm\'an and Kirchhoff approximations. Motivated by experimental results we focus on annuli with a periodic profile. We show that in the F\"oppl - von K\'arm\'an approximation there are only two types of global minimizers -- flat and saddle shaped deformations with localized regions of stretching near the boundary of the annulus. We also show that there exists local minimizers with $n$-waves that have regions of stretching near their lines of inflection. In the Kirchhoff approximation we show that there exist exact isometric immersions with periodic profiles. The number of waves in these configurations is set by the condition that the bending energy remains finite and grows approximately exponentially with the radius of the annulus. For large radii, these shape are energetically favorable over saddle shapes and could explain why wavy shapes are selected by crocheted models of the hyperbolic plane.