基於協合應力偶理論(consistent couple stress theory, CCST)，本文發展Hermite-Family C1與C2有限層狀元素法(finite layer methods, FLMs)，並針對在開放或封閉迴路(open-/closed-circuit)表面條件下具簡支承邊界之功能性(functionally graded, FG)壓電材料微米板進行三維撓曲和自由振動分析。在撓曲分析中，假設微米板置放在Winkler-Pasternak彈性基礎上並施加正弦形式或均勻分布之電彈(electro-mechanical)荷載。文中之FLMs是將微米板切割為多個層狀元素，每層中彈性場及電場的主變數以傅立葉函數及Hermite多項式分別對內、外表面進行內插擬合。本文透過將材料尺度參數假設為零以及忽略壓電與撓電效應，分別使原FG壓電材料微米板之理論表述簡化為適用於FG壓電材料宏觀板及 FG彈性材料微米板之理論表述，並藉由與文獻中FG壓電材料宏觀板及FG彈性材料微米板之參數解進行比較，以此評估Hermite-Family C1與C2 FLMs之精確度及收斂情形。結果顯示使用Hermite-Family C1與C2 FLMs能更快收斂且收斂解與文獻之精確解高度契合。文中亦探討關於FG壓電材料微米板之撓曲變形和自由振動特性的影響因素，其中包含材料尺度參數、板之寬厚比、壓電效應、撓電效應、材料特性梯度指數、基礎介質的剪切模數及不同表面條件等參數。

Based on the consistent couple stress theory (CCST), the authors develop Hermite-family C1 and C2 finite layer methods (FLMs) for the three-dimensional (3D) static bending and free vibration analyses of a simply-supported, functionally graded (FG) piezoelectric microplate which is placed under open-circuit and closed-circuit surface conditions. In the bending analysis, the microplate of interest is assumed to be resting on a Winkler-Pasternak foundation and to be subjected to either sinusoidal or uniformly distributed electro-mechanical loads. In the formulation of the FLMs, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The FLMs for analyzing EG piezoelectric microplates are reduced to the FLMs for analyzing FG piezoelectric macroplates and FG elastic microplates by setting the material length scale parameter at zero and by ignoring the piezoelectric and flexoelectric effects, respectively. The accuracy and convergence rate of the FLMs are assessed by comparing the solutions it produces with the exact and approximate 3D solutions of FG piezoelectric macroplates and FG elastic microplates which have been reported in the literature. The authors examine and discuss some key effects on the bending and free vibration characteristics of an FG piezoelectric microplate, including the impact of the material length scale parameter, the length-to-thickness ratio, the piezoelectric effect, the flexoelectric effect, the material-property gradient index, and the different surface conditions.

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