Confined disclinations: exterior vs material constraints in developable thin elastic sheets

TL;DR

This paper investigates how thin elastic sheets with inserted wedges deform when pressed against a plane, revealing universal buckling behavior and a law of shape determination based on geometric constraints.

Contribution

It introduces a theoretical framework linking material and exterior constraints to the shape of developable thin sheets, including a law of corresponding states for shallow cones and wedges.

Findings

Unbuckled sector is always semicircular, regardless of wedge angle.

Shape determined by a single parameter elta/psilon^2 in the shallow cone regime.

Experimental observations show slow convergence to the predicted semicircular buckling.

Abstract

We examine the shape change of a thin disk with an inserted wedge of material when it is pushed against a plane, using analytical, numerical and experimental methods. Such sheets occur in packaging, surgery and nanotechnology. We approximate the sheet as having vanishing strain, so that it takes a conical form in which straight generators converge to a disclination singularity. Then its shape is that which minimizes elastic bending energy alone. Real sheets are expected to approach this limiting shape as their thickness approaches zero. The planar constraint forces a sector of the sheet to buckle into the third dimension. We find that the unbuckled sector is precisely semicircular, independent of the angle $\delta$ of the inserted wedge. We generalize the analysis to include conical as well as planar constraints and thereby establish a law of corresponding states for shallow cones of slope $\epsilon$ and thin wedges. In this regime the single parameter $\delta/\epsilon^2$ determines the shape. We discuss the singular limit in which the cone becomes a plane. We discuss the unexpected slow convergence to the semicircular buckling seen experimentally.