Atomistic simulation of the bending stiffness of sp2 and sp3 hybridized carbon sheets

The bending stiffness of two-dimensional materials is a fundamental property governing their mechanical stability and performance in flexible electronic devices. In the case of carbon-based two-dimensional materials, the bending stiffness of single-layer graphene is now well established. However, experimental studies on few-layer graphene report a wide range of behaviors depending on the measurement techniques and mechanical constraints applied to the samples. In addition, while sp3-hybridized carbon sheets produced through the hydrogenation of graphene offer an attractive route for opening an electronic bandgap, the bending properties of graphane and diamane—the sp3 analogues of graphene and bi-layer graphene—remain largely unexplored. In this work, we present a comprehensive atomistic simulation study of the bending properties of both sp2- and sp3-bonded carbon sheets. Our results show that the bending stiffness of graphane and diamane follows remarkably well the predictions of classical elastic thin-plate theory over a wide range of thicknesses. In contrast, bending in few-layer graphene is strongly influenced by non-linear structural relaxation mechanisms, including interlayer shear, interlayer slip, and the formation of ripplocations. The activation of these mechanisms depends sensitively on the mechanical constraints imposed on the system and leads to significant deviations from the cubic thickness dependence predicted by plate theory. These findings clarify the origin of the discrepancies between theoretical predictions and experimental measurements reported for multilayer graphene and provide a consistent framework for understanding the bending mechanics of both sp2- and sp3-based two-dimensional carbon membranes.