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

Abstract

Anisotropies of Young's modulus E, the shear modulus G, and Poisson's ratio ν of all 2D symmetry systems are studied. The shear modulus and Poisson's ratio 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.

Auxetic properties of polycrystals

“…For quadratic materials more reasonable choice of independent parameters was proposed in our paper 3, and for cubic materials by Every 12: …”

“…2 and 3). Acoustic (12–14) and elastic properties 3, 4 of a particular initially unstrained cubic and quadratic material are characterized by s 1 , which has dimensionality of stiffnesses, and two dimensionless parameters s 2 and s 3 belonging to the appropriate STs. Isotropic materials are represented by points of the interval of the line s 3 = 0 belonging to the suitable ST. For acoustic and elastic characteristics the parameter s 1 is a scaling parameter.…”

“…In our previous papers we presented maps of E , G , and ν for quadratic materials 3 and cubic 4. Since for monocrystalline media the mentioned characteristics are anisotropic, these maps refer to n and m directed along high symmetry crystalline directions.…”

“…Our particular attention was paid to 2D 3 and 3D cubic 4 structures. Auxetic properties of cubic materials were also studied by Brańka et.…”

“…In Refs. 3, 4 we established regions of ST in which crystalline quadratic and cubic materials are completely auxetic, auxetic, and non‐auxetic. One may expect that complete auxetics are particulary promising from the standpoint of technology.…”