Freeform Auxetic Mechanisms Based on Corner-Connected Tiles

Abstract

Auxetic mechanisms based on corner-connected polygonal tiles have been used to design deployable structures and are currently applied to programmable surfaces. However, existing surface structures are realized by compliant kirigami, and the realization with rigid-body mechanism, in particular with thick panels, is still limited to configurations with global symmetries due to the mechanism's overconstraining nature. In this study, we generalize the auxetic mechanisms into freeform surfaces by imposing local symmetries on polyhedral surfaces. From the discussion of kinematics, we show that polyhedral surfaces whose edges coincide with a Voronoi diagram of points on the surface can be converted to kinematics systems of corner-connected kinematic tiles. We propose hard constraints to ensure the Voronoi property required for the kinematics and soft constraints to attain a large expansion ratio. Then, we provide an optimization-based scheme using the proposed constraints to achieve a mechanism from a given target surface. We also propose methods for accommodating the thickness of the tiles and show different variations of joints. As a result, we obtained deployable surfaces of positive and negative Gaussian curvature that can deploy and contract with a one-DOF mechanism. If the structure is viewed as a cellular material, it has an auxetic property with Poisson's ratio of -1. It is also potentially scalable to architectural applications because our mechanism is composed of rigid bodies and hinges.