Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.

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基于非局部应变梯度理论的轴向运动振动功能梯度纳米板的稳定性

考虑到非局部尺度和材料长度尺度效应，我们研究了具有轴向运动的功能梯度纳米板的振动和稳定性行为，以便对纳米工程中的二维纳米带进行建模。应用非局部应变梯度理论，采用微分非局部应变梯度本构关系。使用功能梯度薄板的物理中性平面，我们通过汉密尔顿原理推导出具有轴向运动的功能梯度纳米板的运动控制方程，其中运动学特性被引入动态行为。使用 Galerkin 方法对控制方程进行数值求解。非局部尺度和材料长度尺度参数、轴向速度、梯度指数、讨论了双轴预张力和纵横比。结果表明，具有轴向运动的功能梯度纳米板的复杂频率随着亚临界区域轴向速度的增加而降低，而运动纳米板在超临界区域经历发散不稳定性或颤振不稳定性。固有频率和临界速度随着非局部尺度参数的增加而减小，而随着材料长度尺度参数的增加而增加，分别反映了非局部软化和应变梯度硬化机制。此外，固有频率和临界速度随着双轴预张力和纵横比的增加而增加，但随着梯度指数的增加而减小。特别是，