Nonlocal vibration of embedded double-layer graphene nanoribbons in in-phase and anti-phase modes

Graphene nanoribbons (GNRs), the finite-wide counterparts of crystalline graphene sheets, have been potential materials used in nano-devices because of their excellent electronic, thermal and mechanical properties. In this work, a theoretical analysis of nonlocal elasticity theory for the free vibrational characteristics of embedded double-layer GNRs (DLGNRs) is proposed based on continuum and Winkler spring models. We find two types of vibrational modes, in-phase mode (IPM) and anti-phase mode (APM). The results show that the vibrational properties of DLGNRs show different behaviors in IPM and APM. The natural frequencies of DLGNR embedded in an elastic matrix are significantly influenced by nonlocal effects, the aspect ratio of DLGNRs and the Winkler foundation modulus