Abstract In the present study, it is shown that the fully nonlocal elasticity is failed to analyze the dynamic stability of Timoshenko nanobeams subjected to an axial load and, to solve this problem, for the first time, the two-phase local/nonlocal elasticity, as a paradox free form of nonlocal elasticity, is utilized to study the size-dependent dynamic stability and damping vibration of Viscoelastic Functionally Graded Porous (VFGP) Timoshenko nanobeams incorporating surface effects. Firstly, the governing equations, in presence of the axial and transverse displacements, are obtained through the Hamilton's principle. Next, to investigate the vibration and dynamic stability of nanobeams with different boundary conditions, the Generalized Differential Quadrature Method (GDQM) as well as Bolotin's method are utilized. After validation of the present results and formulation, to examine the influences of different parameters such as local phase fraction factor, nonlocal parameter, damping factor, FG index, volume fraction of porosities and surface effects in different boundary conditions, various benchmark results are presented. Furthermore, it is indicated that, against the fully nonlocal theory, using two-phase elasticity makes it possible to study the size dependent vibration and stability for several boundary conditions of Timoshenko nanobeams which are subjected to axial load.

中文翻译：

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结合表面效应和不同边界条件的两相局部/非局部 VFGP 纳米梁的动态稳定性和振动

摘要 在目前的研究中，表明完全非局部弹性无法分析铁木辛科纳米梁在轴向载荷下的动态稳定性，为了解决这个问题，首次使用两相局部/非局部弹性，作为非局部弹性的无悖论形式，用于研究粘弹性功能梯度多孔 (VFGP) 铁木辛柯纳米梁的尺寸相关动态稳定性和阻尼振动，其中包含表面效应。首先，通过哈密顿原理得到存在轴向和横向位移的控制方程。接下来，为了研究具有不同边界条件的纳米梁的振动和动态稳定性，利用广义微分正交法 (GDQM) 以及 Bolotin 方法。在对现有结果和公式进行验证后，为了检验不同参数（如局部相分数因子、非局部参数、阻尼因子、FG 指数、孔隙体积分数和表面效应）在不同边界条件下的影响，提出了各种基准结果。此外，表明与完全非局部理论相反，使用两相弹性可以研究受到轴向载荷的 Timoshenko 纳米梁的几种边界条件的尺寸相关振动和稳定性。提供了各种基准测试结果。此外，表明与完全非局部理论相反，使用两相弹性可以研究受到轴向载荷的 Timoshenko 纳米梁的几种边界条件的尺寸相关振动和稳定性。提供了各种基准测试结果。此外，表明与完全非局部理论相反，使用两相弹性可以研究受到轴向载荷的 Timoshenko 纳米梁的几种边界条件的尺寸相关振动和稳定性。