Continuum thin-shell model of the anisotropic two-dimensional materials: Single-layer black phosphorus

Using the geometric parameters and the elastic moduli determined by the density functional theory (DFT) calculations as the benchmarks, the original Stillinger-Weber (SW) potential of the single layer black phosphorus (SLBP) is revised. The revised SW potential can effectively describe the elastic behavior of SLBP, including both the elastic moduli and the stress-strain relationship under large deformation. Using the molecular dynamics (MD) simulations (based on the revised SW potential), it is found that the continuum thin-shell theory can effectively describe the bending and buckling behavior of SLBP: although the elastic behavior of SLBP is highly anisotropic, the thin-shell thicknesses along different directions determined from the continuum theory are very close to each other. In addition, the thin-shell thickness of SLBP is very close to its physical thickness, and the thin-shell thickness determined from the bending theory is also very close to that determined from its buckling behavior, which are highly different from the corresponding behaviors of graphene. Thus, the lower limit for effectively applying the continuum thin-shell theory to investigating the mechanical behavior of two-dimensional (2D) materials is that 2D-materials include at least two atomic layers.