本文基於協合應力偶理論(Consistent Couple Stress Theory , CCST)，發展了Hermitian C2有限層狀元素法(Finite Layer Method, FLM)，用於對簡支承邊界下的功能性(Functionally Graded, FG)壓電圓柱微殼進行三維尺度相關的應力與變形分析。該圓柱微殼處於開放迴路表面條件，並在內外表面承受外力與電位移作用。首先，我們利用位能駐值原理推導三維弱形式，並將圓柱微殼人為地劃分為nl層，其中每層的主要變數為彈性位移分量與電位勢。接著，我們將層狀變位模型納入弱形式理論表述中，進一步發展半解析有限層狀元素方法，以求解圓柱微殼內誘發的彈性與電場變數。其中，每個主要變數均在面內方向使用雙重傅立葉級數展開，在厚度方向則採用Hermitian C2多項式函數進行內插擬合。為驗證本方法的準確性，我們將材料尺度參數設為零，並將所得解與文獻中FG壓電圓柱宏觀殼的三維精確解進行比較。此外，我們亦探討重要影響因素對圓柱微殼域內電場與彈性場變數的影響，其中包括半徑與厚度比、非均勻指數、材料尺度參數及壓電效應。研究結果顯示，隨著厚度、材料尺度參數與壓電效應的增加，微殼剛性提升，進而使得位移、面內應力與橫向應力顯著減小；而非均勻性指數愈高，則所誘發之彈性場與電場變數在厚度方向上變化愈劇烈。

Based on the consistent couple stress theory (CCST), this study develops a Hermitian C² finite layer method (FLM) for conducting a three-dimensional, size-dependent stress and deformation analysis of a simply supported functionally graded (FG) piezoelectric cylindrical microshell. The cylindrical microshell is subjected to mechanical loads and electric displacements on its inner and outer surfaces under open-circuit boundary conditions. Using the stationary principle of potential energy, we first derive a 3D weak formulation, in which the shell is artificially divided into nl layers, with the elastic displacement components and the electric potential of each layer being selected as the primary variables. By integrating a layer-wise generalized displacement model into the weak formulation, a semi-analytical finite layer method is developed to solve for the elastic and electric field variables induced in the cylindrical microshell. Each primary variable is expanded in the in-surface directions using a double Fourier series, while Hermitian C² polynomial functions are employed for interpolation along the thickness direction. To verify the accuracy of the proposed method, we set the material length-scale parameter to zero and compare the obtained results with existing 3D exact solutions for FG piezoelectric cylindrical macroscale shells in the literature. Additionally, we investigate the effects of several essential factors on the elastic and electric field variables within the cylindrical microshell, including the radius-to-thickness ratio, the inhomogeneity index, the material length-scale parameter, and piezoelectricity.

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