Auxetic Metamaterial and Flagstone Tessellation Patterns Via Convex Airy Stress Functions

Auxetic metamaterials are gaining popularity as deployable architectural design systems due to their inherent flexibility, adaptability and programmable nature. The efficiency and constructability of these systems – which comprise of rigid panels and hinges - is interlinked to their geometry and corresponding kinematic behaviour. As such they can be considered as structural design systems in which the structural performance directly defines their architectural conceptual design. This research paper presents a purely geometrical methodology for developing Auxetic metamaterials based on graphic statics. Specifically, it is shown how by employing reciprocal convex Airy Stress Functions (ASF) and Minkowski sums it is possible to derive in a direct (and noniterative) way non-standard 2D Auxetic patterns with negative Poisson’s ratio. Furthermore, this methodology can be applied to cases of convex 3D polyhedral surfaces by combining said reciprocal ASFs into cantellated polyhedral objects. Lastly, it is discussed how this construction relates to origami flagstone tessellations.