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In this paper, the vibration of a stacked multilayered graphene/black phosphorus (G/BP) heterostructure is investigated via the mesh-free method. The shape function and its derivatives are addressed by the moving least squares (MLS) approach. Optimization of the sequential quadratic programming method is adopted to calculate the distance between the arbitrary layers. Therefore, coefficients of the van der Waals (vdW) interaction between arbitrary layers of heterostructures are obtained. Then the frequencies and mode shapes of the multilayered G/BP heterostructure, considering the vdW interaction between arbitrary layers, are compared with considering only the vdW interaction among adjacent layers. The effects of the number of layers and aspect ratio of the G/BP heterostructure on the frequencies are investigated. The results demonstrate that coefficients of the vdW interaction, considering the arbitrary layers, are larger than those considering only adjacent layers. The difference between natural frequencies considering arbitrary layers and those considering adjacent layers is not clear for the low-order cases. Alternatively, the difference between natural frequencies obtained considering arbitrary layers and those considering adjacent layers are obvious for high-order cases. This paper provides a useful method to optimize the vdW interaction between multilayered G/BP heterostructures and can adequately simulate their vibration behaviors.

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The effect of the stacking order of bilayer black phosphorus (BLBP) on the vibration properties is investigated by using molecular dynamics (MD) simulation and continuum modelling. The results show that the interlayer shear effect plays a very important role in the vibration behaviours of BLBPs. In MD simulation, a negative shear modulus is found in BLBP nanoresonators with stacking orders AA and AA′, which can decrease the natural frequency of the nanoresonators. The prediction of the natural frequency of the BLBP from the orthotropic sandwich plate model (OSPM) agrees with the MD simulation results very well. Moreover, for the vibration of the high-order mode, the interlayer shear modulus of the OSPM along one direction can mainly influence the natural frequency of the vibration mode, with the antinodes strictly aligning the direction for the BLBP. The results are thus important for understanding the dynamic behaviour of multilayer orthotropic nanostructures and designing multilayer orthotropic nanostructure resonators.