本文基於Reissner混合變分原理(Reissner’s mixed variational theorem, RMVT)發展幾何非線性非局部Timoshenko梁理論(Timoshenko beam theory, TBT)，應用於嵌入式多層奈米碳管(multi-walled carbon nanotube, MWCNT)，在考慮凡得瓦爾力作用下，該結構之幾何非線性靜力分析。嵌入式MWCNT的最外層受到外部載重作用，配合自由端、簡支承及固端支承的組合作為邊界條件，除此之外，本文亦考慮MWCNT任兩層間的凡得瓦爾力互制效應，並使用Pasternak基礎模型來模擬MWCNT與周圍介質之相互作用。文中以Eringen非局部彈性理論解釋小尺度效應，推衍相應之幾何非線性TBT控制方程式和可能的邊界條件，MWCNT的非線性位移則由微分擬合法以及直接迭代法求解。在數值範例中，本文證實了基於RMVT非局部TBT在求解上能夠快速收斂，且其線性收斂解與文獻中使用各式虛位移原理(principle of virtual displacement, PVD)非局部梁理論得到的解相當一致。

Based on Reissner’s mixed variational theorem (RMVT), rather than the principle of virtual displacement (PVD), we present a nonlocal Timoshenko beam theory (TBT) for the geometrically nonlinear static analysis of multi-walled carbon nanotubes (MWCNT) embedded in an elastic medium. The embedded MWCNT is subjected to mechanical loads on its outer-most surface, with combinations of simply-supported and clamped edge conditions. The van der Waals interaction between any pair of walls constituting the MWCNT is considered, and the interaction between the MWCNT and its surrounding medium is simulated using the Pasternak-type foundation model. In the formulation, the governing equations of a typical wall and the associated boundary conditions are derived, in which von Karman geometrical nonlinearity is considered. Eringen’s nonlocal elasticity theory is used to account for the small length scale effect. The deformations induced in the embedded MWCNT are obtained using the differential quadrature method and a direct iteration approach.

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