In this work, a unified high-order nanobeam model considering various high-order shear deformation beam theories is established to investigate the vibration response of nanobeam on the basis of two-phase local/nonlocal strain and stress gradient theory, as well as surface elasticity theory. The unified model also includes the Euler-Bernoulli beam model and Timoshenko beam model. Herein, the elastic dynamics governing equations and boundary conditions are derived using Hamilton's principle, and the analytical solutions, such as exact formulas for natural frequencies, are obtained by employing the Navier method for simply supported boundary conditions. The effects of local volume fraction, nonlocal parameter, material length scale parameter, shear deformation and surface energy in stress and strain-driven models are analyzed in detail, respectively. The parametric studies reveal that the two scale parameters (nonlocal parameters and material length characteristic parameters) have opposite effects on the stiffness of the nanobeams in the two driving models, while the surface parameters have the same effect on the stiffness of the two driving models. The influence of the slenderness ratio on the surface effect and scale effect is opposite, meaning that the increase of the slenderness ratio deepens the influence of the surface effect but weakens the influence of the scale effect. There are also differences in the effects of higher-order modes on the two effects. Higher modes lead to more significant scale effects, but the effect of higher modes on surface effects depends on the surface elastic properties of the material. We also find that the introduction of surface elasticity increases the gap between the TBT and other higher-order beams, which indicates the prediction results of the higher-order beam model are more accurate when both surface and nonlocal effects are considered. In addition, it is represented that the surface elasticity makes aluminum nanobeams exhibit a stiffness softening effect, while the effect of surface elasticity on the stiffness of silicon nanobeams is significantly dependent on the slenderness ratio and the number of modes.

在这项工作中，建立了一个统一的高阶纳米梁模型，考虑了各种高阶剪切变形梁理论，以两相局部/非局部应变和应力梯度理论以及表面弹性理论为基础，研究纳米梁的振动响应。理论。统一模型还包括 Euler-Bernoulli 梁模型和 Timoshenko 梁模型。在此，弹性动力学控制方程和边界条件是利用Hamilton原理推导出来的，而对于简支边界条件，则采用Navier方法得到解析解，例如固有频率的精确公式。详细分析了应力应变驱动模型中局部体积分数、非局部参数、材料长度尺度参数、剪切变形和表面能的影响，分别。参数研究表明，两个尺度参数（非局部参数和材料长度特征参数）对两种驱动模型中纳米束的刚度具有相反的影响，而表面参数对两种驱动模型的刚度具有相同的影响。长细比对表面效应和尺度效应的影响是相反的，即长细比的增加加深了表面效应的影响，但减弱了尺度效应的影响。高阶模式对这两种效应的影响也存在差异。更高的模式会导致更显着的尺度效应，但更高模式对表面效应的影响取决于材料的表面弹性特性。我们还发现，表面弹性的引入增加了 TBT 与其他高阶梁之间的差距，这表明当考虑表面和非局部效应时，高阶梁模型的预测结果更准确。此外，表明表面弹性使铝纳米梁表现出刚度软化作用，而表面弹性对硅纳米梁刚度的影响显着取决于长细比和模数。