Auxetic behavior on demand: a three steps recipe for new designs

Despite their outstanding mechanical properties, with many industrial applications, a rational and systematic design of new and controlled auxetic materials remains poorly developed. Here a unified framework is established to describe bidimensional perfect auxetics with potential use in the design of new materials. Perfect auxetics are characterized by a Poisson's ratio $\nu=-1$ over a finite strain range and can be modeled as materials composed of rotating rigid units. Inspired by a natural connection between these rotating rigid units with an antiferromagnetic spin system, here are unveiled the conditions for the emergence of a non-trivial floppy mode responsible for the auxetic behavior. Furthermore, this model paves a simple pathway for the design of new auxetic materials, based on three simple steps, which set the sufficient connectivity and geometrical constraints for perfect auxetics. In particular, a new exotic crystal, a Penrose quasi-crystal and the long desired isotropic auxetic material are designed and constructed for the first time. Using 3D printed materials, finite element methods and this rigid unit model, the auxetic behavior of these designs is shown to be robust under small disturbances in the structure, though the Poisson's ratio value relies on system's details, approaching $-1$ close to the ideal case.