▎ 摘　　要

Single-layer graphene sheets (SLGSs) with dimensions of the order of a few nanometres are relatively new, but expected to have several applications. When SLGSs experience displacements that are large in comparison with their extremely small thickness, the membrane forces that develop lead to non-linear behaviour. Knowing the modes of vibration of SLSGs is important, because these modes provide a picture of what one may expect not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of flat single-layer graphene sheets are investigated. For that purpose, a Galerkin type formulation, based on classic plate theory with Von Karman non-linear terms and resorting to Airy' s stress function, is implemented. The formulation takes into account non-local effects, which are thought to be important in very small structural elements. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM), with several harmonics, and are subsequently solved by an arc-length continuation method. The combined importance of non-local effects and of the geometrical non-linearity on the non-linear modes of vibration is analysed. They result in alterations of the natural frequencies, variations in the degrees of hardening, changes in the frequency content of the free vibrations, and alterations in shapes assumed along a period of vibration. The main outcome of this work is the finding that the small scale has a major effect on interactions between the first and higher order modes, interactions which are induced by the geometrical non-linearities. It turns out to be possible, e.g., for non-local effects to considerably change the frequencies at which internal resonances occur, or even to eliminate those internal resonances.