本文以 Reissner 混合變分原理(Reissner’s mixed variational theorem, RMVT)推衍非局部(nonlocal) Euler-Bernoulli樑理論(Euler-Bernoulli beam theory, EBT)，進行不同邊界條件下具彈性支承之單層奈米樑(single-layered nanobeam, SLNB)或單壁奈米碳管(single-walled carbon nanotube, SWCNT)之力學行為分析。SLNB/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。SLNB/SWCNT的運動方程式與其對應的邊界條件則由Hamilton’s principle結合Eringen的非局部理論推導出。文中SLNB/SWCNT之力行為問題皆由無網格法進行求解，其中形狀函數由微分再生核(differential reproducing kernel, DRK)內插法建構之。分析之力行為包括:受均佈載重且具彈性支承的SLNB/SWCNT其撓度及次要變量(可藉由微分主要變量之結果間接求得)之分析、具彈性支承的SLNB/SWCNT其自由振動的頻率參數之分析以及具彈性支承的SLNB/SWCNT其挫曲載重參數之分析。

A Reissner’s mixed variational theorem (RMVT)-based nonlocal Euler-Bernoulli beam theory (EBT) is developed for the bending, free vibration and buckling analyses of a single-layered nanobeam (SLNB) (or a single-walled carbon nanotube, SWCNT) embedded in an elastic medium and with combinations of simply-supported and clamped edges. The interaction effect between the SLNB/SWCNT and its surrounding elastic medium is simulated using either a Winkler or a Pasternak foundation model. The SLNB/SWCNT’s equations of motion and the associated possible boundary conditions are derived by using Hamilton’s principle combined with Eringen’s nonlocal constitutive relations. A meshless collocation method is applied to obtain the deflection and stress-resultant components induced in a loaded SLNB/SWCNT, frequency parameters of an unloaded SLNB/SWCNT, and critical load parameters of an axially-loaded one, in which a differential reproducing kernel interpolation method is used to construct the shape functions of each field variable.

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