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.

中文翻译：

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二维材料的弯曲和扭转刚度


提出了一种新颖的基于环面的表面参数化，并用于评估任意对称性二维材料的弯曲和扭转刚度。广义本构模型用于捕获同时弯曲和扭转曲率对二维材料应变能密度的影响。通过使二维材料样本处于不同的曲率状态（包括同时弯曲和扭曲）、评估相应的应变能密度以及将本构模型与数据进行曲线拟合来评估弯曲刚度。所提出的方法独立于用于评估二维材料应变能的方法，例如密度泛函理论或经典原子间势。对石墨烯、二硫化钼和黑磷的弯曲刚度进行了评估，并与文献中发表的结果进行比较。研究了弯曲刚度的方向依赖性。无量纲系数用于量化二维材料中的双向弯曲和弯曲扭转​​耦合效应。最后，本研究还分析了黑磷的弯曲各向异性和弯曲厚度膨胀系数。