本文基於Eringen 非局部彈性力學理論與多重時間尺度法發展平面應變漸近非局部彈性力學理論，應用於具簡支承邊界之多層均向石墨烯薄板系統嵌入彈性介質中的圓柱型撓曲振動分析。頂層和底層石墨烯板與其周圍環境彈性介質及相鄰兩石墨烯板彼此間的相互作用均以不同勁度係數的單參數Wrinkler 形式之彈性支承模擬。演算過程利用非局部參數導入微小尺度效應於非局部本構方程式，並應用無因次化、漸近展開及連續積分之數學運算過程於獨立單層石墨烯板，再將各層石墨烯板之運動方程式組合成多層石墨烯板系統，形成一套遞迴循環的各階問題之運動方程式。非局部平面應變問題之首階近似解可解得非局部多層古典板理論之解，高階問題之運動方程式具有與非局部多層古典板理論相同之微分運算子，惟控制方程式之非齊性項不同。本文將以嵌入或非嵌入彈性介質的多層石墨烯板系統之自然頻率參數及其相應振動模態佐證漸近非局部彈性力學理論之精確與收斂快速。

An asymptotic nonlocal elasticity theory for cylindrical bending vibration analysis of simply-supported, Nl-layered, and uniformly- or nonuniformly-spaced, graphene sheet (GS) systems embedded in an elastic medium is developed by using the Eringen nonlocal elasticity theory and multiple time scale method. Both the interactions between the topmost and bottommost GSs and their surrounding medium and the interactions between each pair of adjacent GSs are modelled as one-parameter Winkler models with different stiffness coefficients. In the formulation, the small length scale effect is introduced to nonlocal constitutive equations using a nonlocal parameter, and then the nondimensionalization, asymptotic expansion, and successive integration mathematical processes are performed for a typical GS. After assembling the motion equations for each individual GS to form those of the multiple GS system, recurrent sets of motion equations can be obtained for various order problems. Nonlocal multiple classical plate theory (CPT) is derived as a first-order approximation of the current nonlocal plane strain problem, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal multiple CPT, although with different nonhomogeneous terms. Some nonlocal plane strain solutions for the natural frequency parameters of the multiple GS system with and without being embedded in the elastic medium and their corresponding mode shapes are presented to demonstrate the performance of the asymptotic nonlocal elasticity theory.

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