本文在三維非局部彈性力學理論體系下，針對具雙邊簡支承之嵌入式單壁奈米碳管(Single-walled carbon nanotubes, SWCNTs)，應用多重時間尺度法(The method of multiple time scales)研究其自由振動行為和受圍壓及軸壓作用下之挫屈行為，文中以Eringen非局部材料本構關係將微小尺度效應納入考量。單壁奈米碳管與周圍環境介質的交互作用以Pasternak或Winkler基礎模型模擬，將半管厚與中曲面半徑比之均方根作為微擾參數，透過無因次化、漸近展開及連續積分等數學運算過程得到各階問題的控制方程組。結果顯示，三維非局部彈性力學理論的首階控制方程式即為非局部古典殼理論(Classical shell theory, CST)之控制方程式，且高階問題與首階問題具有相同之微分運算子，惟控制方程式之非齊性項不同。文中應用本三維非局部彈性力學漸近理論，求解單壁奈米碳管之自然頻率及臨界挫屈載重參數，並據以檢核各類一維非局部梁及二維非局部殼理論的準確度。

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics and buckling behavior under combined hydrostatic pressure and axial compression 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 modelled as a two-parameter Pasternak foundation model or Winlder 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 and critical load parameters of non-embedded or embedded SWCNTs are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

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