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.

中文翻译：

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范德华力对堆叠多层石墨烯/黑磷异质结构振动的影响

在本文中，通过无网格方法研究了堆叠的多层石墨烯/黑磷 (G/BP) 异质结构的振动。形状函数及其导数通过移动最小二乘 (MLS) 方法来处理。采用顺序二次规划方法的优化计算任意层之间的距离。因此，获得了任意异质结构层之间的范德华（vdW）相互作用的系数。然后将考虑任意层之间的 vdW 相互作用的多层 G/BP 异质结构的频率和模式形状与仅考虑相邻层之间的 vdW 相互作用进行比较。研究了G/BP异质结构的层数和纵横比对频率的影响。结果表明，考虑任意层的 vdW 相互作用的系数大于仅考虑相邻层的系数。对于低阶情况，考虑任意层的固有频率与考虑相邻层的固有频率之间的差异并不明显。或者，对于高阶情况，考虑任意层和考虑相邻层获得的固有频率之间的差异是显而易见的。本文提供了一种有用的方法来优化多层 G/BP 异质结构之间的 vdW 相互作用，并可以充分模拟它们的振动行为。对于低阶情况，考虑任意层的固有频率与考虑相邻层的固有频率之间的差异并不明显。或者，对于高阶情况，考虑任意层和考虑相邻层获得的固有频率之间的差异是显而易见的。本文提供了一种有用的方法来优化多层 G/BP 异质结构之间的 vdW 相互作用，并可以充分模拟它们的振动行为。对于低阶情况，考虑任意层的固有频率与考虑相邻层的固有频率之间的差异并不明显。或者，对于高阶情况，考虑任意层和考虑相邻层获得的固有频率之间的差异是显而易见的。本文提供了一种有用的方法来优化多层 G/BP 异质结构之间的 vdW 相互作用，并可以充分模拟它们的振动行为。