本文基於Eringen非局部彈性力學理論探討單層奈米石墨烯板之三維撓曲、自然振動、挫屈行為分析。Eringen的非局部彈性力學理論與傳統局部彈性力學理論之主要差別在於虎克彈性體描述之組成關係不同，不在於廣義應力平衡方程式。基於局部彈性理論，受到載重作用的彈性體其特定點的應力分量只取決於該點應變分量；但基於非局部彈性理論，受到載重作用的彈性體其特定點的應力分量與該連續體所有點的應變分量有關。奈米石墨烯板與其周圍環境的彈性介質之相互作用以Winkler與Pasternak型式之彈性支承基礎模擬。文中首先提出基於Eringen非局部彈性力學理論，應用微擾方法於奈米石墨烯板之三維非局部靜態行為分析，再者，基於Eringen非局部彈性力學理論與多重時間尺度法應用於彈性介質中且具完全簡支承邊界之單層均向性奈米石墨烯板的自然振動行為分析。最後，基於Eringen非局部彈性力學理論，應用漸近展開分析具完全簡支承邊界之嵌入式單層均向性石墨烯與奈米板的挫屈行為。演算過程基於Eringen的非局部彈性理論於計算微小尺度效應且引入非局部彈性力學理論的本構方程式，並進行無因次化、漸近展開與連續積分之數學運算過程，並藉由運動方程推衍各階之控制方程式。理論推衍因微擾階數不同，各階問題有其相對應之控制方程式，其首階控制方程可解得非局部古典板理論之解，再據以逐階修正得精確之非局部三維彈性力學解。本文將以三維漸近非局部彈性理論探討奈米石墨烯板嵌入於彈性介質中，其撓曲、自然振動、挫屈等三維結構力學問題之解析。

A three-dimensional (3D) asymptotic theory is reformulated for the structural analysis of simply-supported, isotropic and orthotropic single-layered nanoplates and graphene sheets (GSs). Eringen’s nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these. The interactions between the nanoplates (or GSs) and their surrounding medium are modelled as a two-parameter Pasternak foundation. The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional (2D) nonlocal plate problems, the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory (CPT), although with different nonhomogeneous terms. Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions, we can obtain the Navier solutions of the leading-order problem, and the higher-order modifications can then be determined in a hierarchic and consistent manner. Therefore, some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory. The nonlocal elasticity solutions of the natural frequency parameters of nanoplates and GSs with and without being embedded in the elastic medium and their corresponding through-thickness distributions of modal field variables are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory. The nonlocal elasticity solutions of the critical load parameters of simply-supported, biaxially-loaded single-layered nanoplates and graphene sheets with and without being embedded in the elastic medium are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory.

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