Deployable Surfaces Featuring Bistable Auxetics Kirigami

Ancient motifs have inspired families of kirigami surface structures that are mechanically bistable under tension. In-plane mechanics of periodic tessellations have been investigated in literature. In this work, we aim to understand the effect of aperiodic tessellation and the resulting out-of-plane transformation due to geometric frustration. We show that when such unit cells are uniaxially stretched, the Poisson's ratio of is -1 throughout. By parametrically varying the unit cell geometry, we are able to bistably transform unit cells with a theoretical maximum scale factor of 2. Such expansion can be captured by conformal maps, which locally preserve angles and shapes, but not size or curvature. Indeed, the Gaussian curvature of a surface is proportional to the Laplacian of the conformal scale factor. If we specify a target 3D shape, we can obtain a scalar field of scale factors required to flatten it. By discretizing the shape and tessellate spatially varying unit cells based on the scale factors, we demonstrate families of unit cells that can all stably deploy into 3D shape at various length scales. Lastly, by abstracting the in-plane deformation, we propose a continuum model to capture the deployment process as well as the mechanical characteristics when deployed.