Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

在三维（3D）非局部弹性理论的框架内，作者开发了一种渐近理论，以研究非嵌入或嵌入到弹性介质中的简单支撑的单壁碳纳米管（SWCNT）的自由振动特性。时标法。采用艾林根（Eringen）的非局部本构关系来解释配方中的小长度尺度效应。SWCNT及其周围介质之间的相互作用被建模为两参数Pasternak基础模型。在执行了一系列数学过程（包括无量纲化，渐近展开和逐次积分等）之后，作者获得了针对各种阶次问题的运动方程组的递归集。非局部经典壳理论（CST）作为当前3D非局部弹性问题的一阶近似推导，高阶问题的运动方程与非局部CST的运动方程保持相同的微分算子，尽管具有不同的非均质性条款。获得了非嵌入式或嵌入式SWCNT的自然频率参数及其相应的贯穿厚度模态应力和位移分量分布的当前渐近解，以评估文献中可用的各种非局部壳和梁理论的准确性。