本文根據三維電彈性力學理論，藉由微擾法推導出含壓電材料雙曲率複合層殼撓曲與自由振動之三維漸近解析理論。首先在撓曲問題上，將三維電彈性力學之22條基本方程式重新組合，消去曲面應力場量，並以位移場、橫向應力及電位移等8個場量為主要變數，化簡成8條微分方程式；再藉由適當的無因次化處理，使用漸近展開法，將各場量展開成與一微小參數相關之冪級數型式，得以使原三維控制方程式分離成該微小參數所對應之不同階數的控制方程式。循序將各階方程式沿厚度方向積分，導得由低階至高階具遞迴形式各階問題對應之控制方程式。其中，首階方程式即為古典層殼理論之二維控制方程式，而高階修正場量以低階場量之解為依據，有系統地逐階循環修正，求得收斂之精確解。在含壓電材料雙曲率複合層殼振動反應之三維漸近解析上，則採用了多重尺度法，在基本運動方程式中引入多重時間尺度，求得無因次三維電彈性力學之運動方程式；再透過微擾法使原三維電彈性力學之運動方程式被分離成相應於某微小參數各個不同階數之運動方程式，同樣地循序將各階方程式沿厚度方向積分，可導得由低階至高階遞迴形式的運動方程式。其中，首階方程式亦為古典層殼理論之二維運動方程式，但與靜態問題不同的是高階修正場量則需透過可解與正規化條件求得，以確定此解有界且不含時間尺度上之非同階項。文中應用本三維漸近解析理論，解析壓電材料雙曲率複合層殼撓曲與自由振動問題之數值範例，其數值驗證，不論在收斂性和精確度上均顯示出理想之結果。

An asymptotic theory for the static and dynamic analyses of doubly curved laminated piezoelectric shells is developed on the basis of three-dimensional (3D) linear piezoelectricity. The twenty-two basic equations of 3D piezoelectricity are firstly reduced to eight differential equations in terms of eight primary variables of elastic and electric fields. By means of nondimensionalization, asymptotic expansion and successive integration, we can obtain recurrent sets of governing equations for various order problems. The two-dimensional equations in the classical laminated piezoelectric shell theory (CST) are derived as a first-order approximation to the 3D piezoelectricity. Higher-order corrections as well as the first-order solution can be determined by treating the CST equations at multiple levels in a systematic and consistent way. In the cases of dynamic analysis, multiple time scales are introduced into the present formulation to eliminate the secular terms so that the asymptotic expansion is uniform and feasible. Again, the leading order solution can be obtained by solving the CST-type motion equations. Higher-order corrections can then be determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Several benchmark solutions for various piezoelectric laminates are given to demonstrate the performance of the theory.

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