Auxetic structures, effusion-cooling auxetic sheets, systems and devices with auxetic structures, and methods of using and methods of making auxetic structures are disclosed. An auxetic structure is disclosed which includes an elastically rigid body with opposing top and bottom surfaces. First and second pluralities of elongated apertures extend through the elastically rigid body from the top surface to the bottom surface. The first plurality of elongated apertures extends transversely with respect to the second plurality of elongated apertures. The first and/or second pluralities of elongated apertures have distorted shapes projected through the elastically rigid body at an oblique angle. The elongated apertures are cooperatively configured to provide a desired stress performance while exhibiting negative Poisson's Ratio (NPR) behavior under macroscopic planar loading conditions. By way of example, the auxetic structure may exhibit a reduction in stress concentration proximate the elongated apertures and a Poisson's Ratio of approximately -0.0001 to -0.9%.