Bending and twisting rigidities of 2D materials

A novel torus-based surface parametrization is proposed and used to evaluate the bending and twisting rigidities of 2D materials of arbitrary symmetries. A generalized constitutive model is used to capture the effects of simultaneous bending and twisting curvatures on the strain energy density of a 2D material. The flexural rigidities are evaluated by subjecting the 2D material specimens to different curvature states, including simultaneous bending and twisting, evaluating the corresponding strain energy densities and curve fitting the constitutive model to the data. The proposed methodology is independent of the method used to evaluate the strain energy of the 2D materials, e.g., density functional theory or classical interatomic potentials. The flexural rigidities of graphene, molybdenum disulfide, and black phosphorus are evaluated and compared to those published in the literature. The directional dependence of the flexural rigidities is investigated. Dimensionless coefficients are presented for quantifying the bidirectional bending and bending-twisting coupling effects in 2D materials. Lastly, this study includes an analysis of the flexural anisotropy and flexure-thickness distension coefficients of black phosphorus.