本文發展基於協合應力偶理論(Consistent couple stress theory , CCST)的三維弱形式表述，將其應用於對具簡支撐和冪次法則/指數型的功能性(Functionally graded, FG)微米板之位移、應力及自由振動分析。在理論推衍中，微米板被人為地劃分為許多有限數目的微米尺度層，其中傅立葉函數(Fourier functions)和拉格朗日/埃爾米特多項式(Lagrange/Hermite Polynomials)分別用於描述該微米板每個獨立層中位移變量的面內和面外變化。基於本弱形式表述，本文亦發展Lagrangian C0二次、三次及Hermitian C1的兩節點及三節點的有限元素方法(Finite element methods, FEMs)。透過與文獻中相關的三維精確解進行比較，驗證本Lagrangian C0及Hermitian C1的有限元素方法的準確性和收斂性。基於比較目的，本文也發展了基於修正應力偶理論(Modified couple stress theory, MCST)的三維弱形式表述，並進行研究和討論使用CCST和MCST的弱形式表述兩者獲得的結果之間的相似性與差異。另探討對功能性微米板的應力、變形和自由振動特性的一些關鍵影響，包括材料尺度參數、材料特性梯度指數和長厚比效應，結果顯示這些影響對於功能性微米板極為重要。

A three-dimensional (3D) weak formulation based on the consistent couple stress theory (CCST) is developed for displacement, stress, and free vibration analyses of simply-supported, power-law- /exponential-type functionally graded (FG) microscale plates. In the formulation, the microscale plate is artificially divided into numerous finite microscale layers, where the Fourier functions and the Lagrange/Hermite polynomials are used to interpolate the in- and out-of-plane variations in the displacement variables for each individual layer, respectively. Based on the weak formulation, the authors develop layer-wise C0 quadratic and cubic finite element methods (FEMs), as well as layer-wise C1 two- and three-node FEMs to address the current issue. The accuracy and convergence of these layer-wise C0 and C1 FEMs are validated by comparing their solutions with the 3D exact and the quasi-3D results available in the literature. A 3D weak formulation based on the modified couple stress theory (MCST) is also developed for comparison purposes. The similarity and the differences between the results obtained using the CCST- and the MCST-based weak formulations are examined and discussed. Some key effects on stress, deformation, and free vibration characteristics of the FG microscale plate are examined, including the material length scale, the material-property gradient index, and the length-to-thickness ratio effects, which are shown to be significant for the FG microscale plate.

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