Theorem on the Compatibility of Spherical Kirigami Tessellations

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 domelike structure based on the theorem. Our study provides new insights into the rational design of morphable structures based on Euclidean and non-Euclidean geometries.

Figure 1

$5\times 5$ SQK tessellation. (a) The kirigami pattern. (b) The compatible configuration at $\omega =\pi /2$. (c),(d) The physical model made of rubber.

Figure 2

(a) The geometric notations of a deployed $3\times 3$ SQK tessellation. (b) A $3\times 3$ SQK tessellation at $\cos {\beta }_1=-1$ (left), 0 (middle), and 1 (right). We replace the hinge at the top-central vertex with a spring and permit the overlap of the rigid panels, such that the incompatible configuration (middle) is determined by minimizing the spring elongation. (c) The optimized $3\times 3$ SQK tessellation at $\cos {\beta }_1=-1$ (left), $-0.5$ (middle), and 0 (right). The compatible configuration is shifted from $\cos {\beta }_1=1$ to 0. (d) Curves of the loop functions.

Figure 3

(a) The evolutions of elastic energy $E$ of $5\times 5$ SQK and PQK shells with equal opposite side lengths of slits for different values of Gaussian curvature $K=1$, 0.7, 0.4, and 0. The energy curve of a PQK tessellation ($K=0$) is constantly zero. (b) Undeployed patterns of the tessellations. The aspect ratios of slits are fixed as 0.4.

Figure 4

Reconfigurable SQK domelike shell structure. (a) The kirigami pattern is perforated on a spherical square of side length $s=0.465\pi$ and Gaussian curvature $K=1$. (b) The deployed configuration covers a spherical dome of height $h=1.2$. (c),(d) The physical model made of resin.