本文以Reissner混合變分原理（Reissner’s mixed variational theorem, RMVT）推衍非局部（nonlocal）Timoshenko梁理論（Timoshenko beam theory, TBT），進行不同邊界條件下具彈性支承之單壁奈米碳管（single-walled carbon nanotube, SWCNT）之力學行為分析，並與基於虛位移原理（principle of virtual displacement, PVD）之TBT分析結果做相互比較，最後探討各參數對SWCNT力學行為分析結果之影響。文中基於Hamilton原理推導RMVT非局部TBT之強形式（strong form），其中奈米微尺度效應由Eringen非局部理論之組成關係予以考慮，而SWCNT與周圍彈性介質間之相互作用，則使用Winkler或Pasternak型式之彈性支承予以模擬，SWCNT力學行為則由無網格法（meshless method）進行求解，其中形狀函數由微分再生核（differential reproducing kernel, DRK）內插法或微分擬合（differential quadrature, DQ）法建構之。數值範例中首先分別以RMVT和PVD推衍之非局部TBT，分析不同邊界條件下，無彈性支承SWCNT之靜態與動態行為，其結果顯示無網格法之收斂快速，且其解與文獻中解析解結果一致，惟RMVT非局部TBT在精確度和收斂速度上均較PVD非局部TBT略勝一籌。其後再以RMVT非局部TBT分析具彈性支承SWCNT之力學行為，並討論各項參數（非局部參數、長細比、Winkler勁度及周圍介質之剪切勁度）對靜態與動態行為之影響。

A nonlocal Timoshenko beam theory (TBT), based on the Reissner mixed variational theorem (RMVT), is developed for the analysis of a single-walled carbon nanotube (SWCNT) embedded in an elastic medium and with various boundary conditions. The comparisons between the results obtained by using the RMVT-based nonlocal TBT and those of principle of virtual displacement (PVD)-based one. The strong formulations of the RMVT- and PVD-based nonlocal TBTs are derived by using Hamilton’s principle, in which Eringen’s nonlocal constitutive relations are used to account for the small-scale effect. The interaction between the SWCNT and its surrounding elastic medium is simulated using the Winkler and Pasternak foundation models. The static and free vibration of the embedded SWCNT are thus investigated by using these nonlocal TBT combined with the meshless collocation methods, in which the shape functions are constructed by either the differential reproducing kernel (DRK) interpolation method or the differential quadrature (DQ) one. In the implementation of these meshless colocation methods, the results show the performance of RMVT-based nonlocal TBT is superior to that of the PVD-based one. A parametric study with regard to some crucial effects on the static and free vibration characteristics of the embedded SWCNT is undertaken, such as different boundary conditions, nonlocal parameters, aspect ratios, spring constants and shear modulus of the foundation.

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