本論文針對壓電圓形及環形板的靜態與振動分析，發展一種混合型有限元素列式，此列式結合傳統的位移-電位列式及piezoelectric Reissner`s principle，並給予相對權重；權重則透過與實驗值比較來調整。藉由這種列式，應力及電位移會如同位移及電位一樣是主要的變數，不只在元素內部而且在元素間也可以保證連續；在分析時，包含位移、應力、電位和電位移邊界條件也可以正確的加上。使用此種方法，本文分析一些典型圓形及環形壓電板的靜態變形及自由振動頻率，並且和其他方法的結果相互比較。研究結果發現，若以本文所比較的實驗結果為基準，則使用本文的混合型列式，並取權重因子為1時，一般可以得到比其他列式更準確的結果。

The present study developes a mixed finite element formulation for the analysis of piezoelectric circular and annular plates. This formulation combines the conventional displacement-electric potential type variational principle and the piezoelectric Reissner`s principle with a weighting factor which represents ratio of weights imposed on the above two variational principles, and which can be adjusted by comparing with experiment results. With this formulation, stresses and electric displacements, like displacements and electric potential, are primary variables and are continuous across elements and element interfaces. Also, all displacement, stress, electric displacement, and electric potential boundary conditions can be easily and exactly imposed. Static deformations and vibration frequencies of some typical piezoelectric circular and annular plates are then obtained with the present approach and are compared with those by other methods. Based on experiment results in the literature, it is found that better results could be obtained in general by the present mixed finite element formulation than the others when 1 is chosen as the weighting factor.

摘要 i
ABSTRACT ii
目錄 iii
表目錄 vi
圖目錄 vii
符號說明 ix

第一章 緒論 1

1.1 研究目的與源起 1
1.2 相關研究背景 2
1.2.1 壓電現象[1] 2
1.2.2 線彈性壓電理論 3
1.2.3 固體力學變分原理[2] 5
1.3 文獻回顧 6
1.3.1 圓形及環形壓電板的靜態變形問題 7
1.3.2 圓形及環形壓電板的自由振動問題 10
1.3.3 變分理論應用於壓電材料 13
1.4 全文架構 16

第二章 壓電板的自由振動分析 19

2.1 簡介 19
2.2 軸對稱元素 19
2.3 場變數 20
2.4 相容方程式 (compatibility equations) 23
2.5 物性方程式 (constitutive equations) 24
2.6 變分列式 27
2.7 圓形壓電板自由振動分析(一) 38
2.7.1 問題描述 38
2.7.2 結果與討論 39
2.8 圓形壓電板自由振動分析(二) 47
2.8.1 問題描述 47
2.8.2 結果與討論 48
2.9 環形壓電板自由振動分析 49
2.9.1 問題描述 49
2.9.2 結果與討論 50

第三章 壓電板的靜態變形分析 70

3.1 簡介 70
3.2 軸對稱元素 70
3.3 場變數 71
3.4 物性方程式及相容方程式 72
3.5 變分列式 72
3.6 部份電極圓形壓電板電極邊緣的電彈場集中現象 76
3.6.1 問題描述 76
3.6.2 結果與討論 78

第四章 結論與建議 88

4.1結論 88
4.2建議 89

參考文獻 90
附錄一：無阻尼自由振動方程式元素矩陣細部 98
附錄二：靜態變形方程式元素矩陣細部 102

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