Theorem on the Compatibility of Spherical Kirigami Tessellations

TL;DR

This paper establishes a theorem on the compatibility and bistability of spherical kirigami tessellations, revealing conditions for strain-free configurations and transitions between spherical and planar forms, with implications for designing morphable structures.

Contribution

The paper introduces a theorem characterizing the compatible states of spherical kirigami tessellations and conditions for their bistability, advancing the understanding of morphable non-Euclidean structures.

Findings

Spherical kirigami tessellations have at most two compatible states.

Bistability occurs when slits form parallelogram voids with zero Gaussian curvature.

Theorem enables design of deployable, dome-like structures.

Abstract

We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned slits forming freeform quadrilateral meshes. We show that the spherical kirigami tessellations have either one or two compatible states, i.e., there are at most two isolated strain-free configurations along the deployment path. The theorem further reveals that the rigid-to-floppy transition from spherical to planar kirigami tessellations is possible if and only if the slits form parallelogram voids along with vanishing Gaussian curvature, which is also confirmed by an energy analysis and simulations. On the application side, we show a design of bistable spherical dome-like structure based on the theorem. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.