Closed-form Molecular Mechanics Formulations for the 3D Local Buckling and 2D Effective Young’s Modulus of the Nanosheets

A closed form three-dimensional solution is presented for determination of the local buckling (cell buckling) load of the nanosheets. Moreover, an expression is proposed for the effective 2D Young’s modulus of the unit cell of the nanosheet. In this regard, a three-dimensional efficient space-frame-like geometrical model with angular and extensional compliances is considered to investigate stability and effective Young’s modulus of the nanosheet in terms of the generally possible relative movements of the atoms of the unit cell, in the Cartesian coordinates. The molecular dynamics approach is employed in development of the formulation, taking into account the force constants and bond characteristics. The governing equations are derived based on the principle of minimum total potential energy. Results of the special cases of each of the proposed expressions are verified by the results available in literature or results of the traditional approaches. Comparisons are made with various buckling results reported for different nanosheets, based on different approaches of determination of the stiffness parameters, and a good agreement is noticed.