In this paper, a size dependent numerical model for free vibration and bending analyses of hexagonal beryllium crystal (HBC) nanoplates is presented. Based on the nonlocal strain gradient theory (NSGT), both nonlocal and strain gradient effects are considered in the present model. In addition, the behavior of HBC material can be considered as an anisotropic one. By employing principle of virtual displacement (PVD), the governing equations of motion for a Galerkin weak for the higher-order shear deformation theory (HSDT) are deduced. Thereafter, the natural frequency and deflection of the HBC nanoplates are determined by solving those governing equations using an isogeometric analysis (IGA). Numerical results show that bending and vibration behaviors of the anisotropic nanoplates are affected by the geometry, boundary conditions, length-to-thickness ratios, exponential factor, nonlocal parameter and strain gradient parameter. On the other hand, when the material length scale parameter is smaller than or equal to the nonlocal parameter then the natural frequency predicted by nonlocal strain gradient theory is lower than that evaluated by classical continuum theory. Conversely, obtained results are higher than the referenced ones by classical continuum theory when the material length scale parameter is larger than or equal to the nonlocal parameter. Whilst, a contrary case occurs for the displacements. Furthermore, by respectively ignoring the strain gradient and nonlocal parameters, the pure nonlocal and strain gradient models can be recovered from the present model.

在本文中，提出了六方铍晶体 (HBC) 纳米板的自由振动和弯曲分析的尺寸相关数值模型。基于非局部应变梯度理论（NSGT），本模型同时考虑了非局部和应变梯度效应。此外，HBC 材料的行为可以被认为是各向异性的。利用虚位移(PVD)原理，推导出高阶剪切变形理论(HSDT)下Galerkin弱运动的控制方程。此后，通过使用等几何分析 (IGA) 求解这些控制方程来确定 HBC 纳米板的固有频率和偏转。数值结果表明各向异性纳米板的弯曲和振动行为受几何形状、边界条件、长厚比、指数因子、非局部参数和应变梯度参数。另一方面，当材料长度尺度参数小于或等于非局部参数时，非局部应变梯度理论预测的固有频率低于经典连续介质理论估计的固有频率。相反，当材料长度尺度参数大于或等于非局部参数时，获得的结果高于经典连续介质理论的参考结果。同时，位移发生相反的情况。此外，通过分别忽略应变梯度和非局部参数，可以从现有模型中恢复纯非局部和应变梯度模型。当材料长度尺度参数小于或等于非局部参数时，非局部应变梯度理论预测的固有频率低于经典连续介质理论估计的固有频率。相反，当材料长度尺度参数大于或等于非局部参数时，获得的结果高于经典连续介质理论的参考结果。同时，位移发生相反的情况。此外，通过分别忽略应变梯度和非局部参数，可以从现有模型中恢复纯非局部和应变梯度模型。当材料长度尺度参数小于或等于非局部参数时，非局部应变梯度理论预测的固有频率低于经典连续介质理论估计的固有频率。相反，当材料长度尺度参数大于或等于非局部参数时，获得的结果高于经典连续介质理论的参考结果。同时，位移发生相反的情况。此外，通过分别忽略应变梯度和非局部参数，可以从现有模型中恢复纯非局部和应变梯度模型。当材料长度尺度参数大于或等于非局部参数时，获得的结果高于经典连续介质理论的参考结果。同时，位移发生相反的情况。此外，通过分别忽略应变梯度和非局部参数，可以从现有模型中恢复纯非局部和应变梯度模型。当材料长度尺度参数大于或等于非局部参数时，获得的结果高于经典连续介质理论的参考结果。同时，位移发生相反的情况。此外，通过分别忽略应变梯度和非局部参数，可以从现有模型中恢复纯非局部和应变梯度模型。