Auxetic properties and anisotropy of elastic material constants of 2D crystalline media

Anisotropies of Young's modulus E, the shear modulus G, and Poisson's ratio v of all 2D symmetry systems are studied. The shear modulus and the ratio v/E of 2D crystals have fourfold symmetry. Simple necessary and sufficient conditions on their elastic compliances are derived to identify if any of these crystals is completely auxetic, non-auxetic or auxetic. Examples of all types of auxetic properties of crystals of oblique and rectangular symmetry are presented. Particular attention is paid to 2D crystals of quadratic symmetry. All mechanically stable quadratic crystals are characterized by three parameters belonging to a prism with the stability triangle in the base. Regions in the stability triangle in which quadratic materials are completely auxetic, non-auxetic, and auxetic are established.