On the limitations of linear vibrations of membrane-like multi-layered graphene sheets carrying bi-directional electric currents: An insight into the nonlinear dynamical analyses

This study aims to present an original comprehensive investigation into the linear and nonlinear free vibration behaviors of pre-stretched multi-layered graphene nanomembranes in the presence of bi-directional electric currents. Using the Biot–Savart law, the applied nonlinear electromagnetic forces on each layer are calculated. By adopting the von Kármán large deflection theory and Eringen’s nonlocal differential/integral models, the nonlinear partial differential/integral equations of motion are methodically derived via Hamilton’s principle for the first time. These governing equations are then converted into a set of nonlinear ordinary differential equations via the Galerkin formulation, which are subsequently solved by applying the incremental harmonic balance (IHB) approach. Through numerous numerical analyses, the impacts of crucial factors, such as electric current, pre-tensioning forces, aspect ratio, number of layers, and nonlocality, on both the linear and nonlinear frequencies are comprehensively examined, further revealing the limitations of the linear analysis in capturing the free dynamic response. Such vital investigations also lead to the determination of the critical pre-tensioning forces and the critical electric currents for a special case of linear analysis, which will be of grave significance in the design and analysis of these nano-electro-mechanical systems. The results indicate that bi-directional electric currents and pre-tensioning forces exert opposing effects on the nonlinear dynamics of the nanosystem. Specifically, increasing the applied electric currents reduces the nonlinear frequencies, while increasing the pre-tensioning forces increases the nonlinear frequencies.