Two-dimensional mechanical metamaterials with unusual Poisson ratio behavior*

We design two-dimensional mechanical metamaterials that may be deformed substantially at little or no energy cost. Examples of such deformable structures are assemblies of rigid isosceles triangles hinged in their corners on the macro-scale, or polymerized phenanthrene molecules forming porous graphene on the nano-scale. In these and in a large class of related structures, the Poisson ratio ν diverges for particular strain values. ν also changes its magnitude and sign, and displays a shape memory effect.