Thin-Shell Thickness of Two-Dimensional Materials

In applying the elastic shell models to monolayer or few-layer two-dimensional (2D) materials, an effective thickness has to be defined to capture their tensile and out-of-plane mechanical behaviors. This thin-shell thickness differs from the interlayer distance of their layer-by-layer assembly in the bulk and is directly related to the Foppl-von Karman number that characterizes the mechanism of nonlinear structural deformation. In this work, we assess such a definition for a wide spectrum of 2D crystals of current interest. Based on first-principles calculations, we report that the discrepancy between the thin-shell thickness and interlayer distance is weakened for 2D materials with lower tensile stiffness, higher bending stiffness, or more number of atomic layers. For multilayer assembly of 2D materials, the tensile and bending stiffness have different scaling relations with the number of layers, and the thin-shell thickness per layer approaches the interlayer distance as the number of layers increases. These findings lay the ground for constructing continuum models of 2D materials with both tensile and bending deformation.