Elastic anisotropy measure for two-dimensional crystals

Similar to bulk crystals, many two-dimensional crystals exhibit elastic anisotropy. However, the measures of the elastic anisotropy in two-dimensional crystals are rarely explored. In this work, we propose a two-dimensional elastic anisotropy index (ASU) which is shown to be applicable for all two-dimensional crystals. Meanwhile, from the existing ’bulk’ elastic anisotropy indices, we derive the specific forms of their two-dimensional counterparts. In addition, physically, ASU has a clear meaning; Technically, it has a simple and explicit expression that is easy to be used in applications. Thus we suggest ASU as a simple yet universal elastic anisotropy measure. Finally, an elastic anisotropy diagram is constructed for 328 two-dimensional crystals that are grouped into four crystal systems: oblique, rectangular, square, and hexagonal crystals. We find that although the lower symmetry lattices prefer to endow two-dimensional crystals with higher elastic anisotropy, the bounds of elastic anisotropy degree are lattice-independent except the highest symmetry lattice (i.e. hexagonal lattice). This work provides a fundamental metric in characterizing elastic anisotropy of two-dimensional crystals.