Abstract This study addresses the dynamic stability of an Euler–Bernoulli nanobeam under time-dependent axial loading based on the nonlocal strain gradient theory (NSGT) and considering the surface stress effects. The studied nanobeam cross-section was rectangular, and simply-supported boundary conditions were assumed. Moreover, a uniform thermal gradient was applied to the nanobeam. The elastic medium was modeled based on the Pasternak theory. The strain–displacement relations were derived using the Von Karman equations. The governing equations were obtained by the energy method and applying the Hamilton's principle. Furthermore, the Bolotin and Incremental Harmonic Balance (IHB) methods were used to solve the differential equations. This study investigates the impact of such parameters as the small-scale parameter, the material length scale, surface effects, elastic medium parameters, temperature variations, geometry, and the static loading factor on the Dynamic Instability Region (DIR). The results are suggestive of the shift of the DIR to lower frequency zone by increasing the small-scale Eringen's nonlocal theory parameter, whereas an increase in the material length scale from the strain gradient theory moves the region to higher frequencies. In case the said parameters are equal, the result conforms to the classical beam theory. In addition, assuming a Pasternak medium and taking into account the effects of surface stress (Young's modulus and the residual stress of the surface) shifts the DIR to higher frequencies, whereas applying a compressive static load moves the region to lower frequencies. Moreover, depending on the thermal expansion coefficient of the medium, temperature variations can also displace the DIR.

中文翻译：

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基于非局部应变梯度理论和表面效应的 Euler-Bernoulli 纳米梁动态稳定性分析中 Bolotin 和增量谐波平衡方法的比较

摘要 本研究基于非局部应变梯度理论 (NSGT) 并考虑表面应力效应，研究了欧拉-伯努利纳米梁在瞬态轴向载荷下的动态稳定性。所研究的纳米梁横截面为矩形，并假定为简支边界条件。此外，对纳米束施加了均匀的热梯度。弹性介质是基于帕斯捷尔纳克理论建模的。应变-位移关系是使用 Von Karman 方程导出的。控制方程是通过能量法并应用哈密顿原理得到的。此外，使用 Bolotin 和增量谐波平衡 (IHB) 方法来求解微分方程。本研究调查了诸如小尺度参数、材料长度尺度、表面效应、弹性介质参数、温度变化、几何形状和动态不稳定区域 (DIR) 上的静态载荷因子。结果表明，通过增加小尺度 Eringen 的非局部理论参数，DIR 向低频区域移动，而应变梯度理论中材料长度尺度的增加将该区域移动到更高频率。如果上述参数相等，则结果符合经典梁理论。此外，假设帕斯捷尔纳克介质并考虑到表面应力（杨氏模量和表面残余应力）的影响，会将 DIR 移至较高频率，而施加压缩静载荷会将区域移至较低频率。此外，根据介质的热膨胀系数，