Shape transitions in hyperbolic non-Euclidean plates

Abstract

A non-Euclidean plate is a thin elastic object whose intrinsic geometry is not flat and hence has residual stresses arising from being embedded in three dimensional space. Recently, there has been interest in using localized swelling to induce residual stresses that shape flat objects into desired three dimensional structures. A fundamental question is whether we can use the mathematical theory of non-Euclidean plates to deduce the three dimensional configuration of the swelling sheet given the exact knowledge of the imposed geometry. We present and summarize the results of recent mathematical studies on non-Euclidean plates with imposed constant negative Gaussian curvature in both annular and disc geometries. We show in the Föppl–von Kármán approximation to the elastic energy there are only two types of global minimizers – flat and saddle shaped deformations – with localized regions of stretching near the boundary of the domain. We also show that there exist n-wave local minimizers that closely resemble experimental observations and have additional regions of stretching near lines of inflection. Furthermore, in the Kirchhoff approximation to the elastic energy, we show that there exists 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 in crochet models of the hyperbolic plane. The predicted morphologies however differ from what is observed in experiments on hydrogel disks highlighting the need for further theoretical studies.