Nonlinear elastic analysis of 2D materials of arbitrary symmetries with application to black phosphorus

Murnaghan’s polynomial based nonlinear elastic constitutive model has been previously applied to 2D materials of hexagonal symmetry. We present a general approach for determining the nonlinear elastic constants of 2D materials of arbitrary symmetries and any constitutive polynomial order. The methodology, which is based on ray sampling of the strain energy density in strain space, is independent of the energy calculation method. The ray based methodology is verified by evaluating the elastic constants of graphene which is a hexagonally symmetric 2D material previously considered in the literature. The methodology is then applied to determine the elastic constants of black phosphorus, an orthorhombic 2D material whose comprehensive nonlinear elastic behavior has not been previously considered in the literature. The energy calculations are carried out using plane-wave density functional theory. Detailed convergence analyses are performed to assess the accuracy of the nonlinear elastic constants. The linearized mechanical properties of black phosphorus are obtained from the elastic constants for comparison with the results published in the literature.