臺灣博碩士論文加值系統

本文係求解一簡易支撐含壓電之彈性凸緣對稱疊層圓型板，在有預拉伸力且受均勻側面壓下之非線性大撓曲問題做探討，其方法係以馮卡門大撓曲理論為基礎，並延伸至一含壓電對稱疊層凸緣之情況，來建構其非線性大撓曲掌控方程。求解過程並先在小撓曲考量下省略其非線性項，來探討其簡化式線性問題，且運用特殊函數之重現關係以求得其解析解，並探討其極限條件(純板件、純薄膜)之情況。而其非線性問題則利用有限差分數值法，配合一數值迭代技巧，並以前述線性解析解作為其迭代初始值之方式，來尋求其非線性數值解，所提出之方法將針對典型矽質類半導體材料疊層構件來求解。惟為驗證所提出之方法，數值求解過程將先考慮一淺寬凸緣、低電壓條件下之擬單層對稱疊層板件，以與文獻既有之結果作比對。此外，數值實例還包含不同凸緣與環板厚度比、彈性凸緣體之徑向尺寸大小、各式施予壓電層之電壓、面壓與預拉伸力大小等，以探討各式幾何回應，包含中心撓度與曲率、全域式斜率、撓度、曲率、機械敏性(無因次最大徑向應力)等，並探討其轉折行為。而各式物理參數如凸緣體尺寸大小(含厚度比與徑向尺寸)、電壓、側向面壓與預拉伸力大小等之影響，也將作一系列密集式調查，來完成其參數化研究。
針對擬單層板件在淺寬凸緣體與低電壓下所得之結果發現脗合文獻所提之擬單層板件純機械負荷下之解，也驗證了所提方法之正確性。此外，壓電效應在低機械負荷下有其明顯影響，且會隨著電壓與凸緣體寬度變大，而更趨顯著，惟提高機械負荷後，則逐漸消失。其中，在低預拉伸力條件下，含壓電層為淺寬凸緣時，其壓電效應之影響最為明顯，但提高凸緣體厚度時，其影響相較略不顯著，但會增強在凸緣體與環板介面處之介面效應。

Problem of nonlinear large-deflection of a simply supported and symmetrically layered circular piezoelectric plate with an elastic boss under pre-tension due to uniform lateral load is studied. Von-Karman plate theory for large deflection is employed and extended to a symmetrically layered plate including a piezoelectric layer with an elastic boss, to derive the nonlinear governing equation for large deflection. The small deflection condition is considered first to study a simplified linear problem. The associated analytical solution is developed following recurrence relations of special functions, and the limited cases of pure membrane and pure plate are studied. For the nonlinear problem, on the other hand, numerical solution is sought using a finite difference method incorporated with an iteration scheme, by taking the simplified linear solution as the initial guess. The developed approach is implemented with typical silicon-based semi-conducting materials. To verify the proposed method, however, a nearly monolithic plate with a shallow and wide boss under a very low applied voltage will be considered, first. The results are checked against those available in literature. In addition, the numerical examples also include the cases of different thickness ratios between center boss and annular plate, various radial sizes of the elastic boss, different applied voltages across the piezoelectric layer and a wide range of lateral loads and pre-tensions. The objective is to explore all kinds of geometric responses including the center deflection and curvature, lateral deflection, slope and the mechanical sensitivity (dimensionless maximum radial stress) and curvature as well as the transition behavior of the bossed piezoelectric plates. The influence of a variety of physical parameters such as the size of the elastic boss (including the thickness ratio and the radial size), the applied voltage, lateral loads and pre-tension will be intensively investigated for a thorough parametric study.
For a nearly monolithic plate with a shallow and wide boss under a low applied voltage, the results correlate well with those available for a single-layer plate with pure mechanical load in literature. Thus, the presented approach is checked. In addition, the numerical results show that piezoelectric effect may arise at a low mechanical loading condition and becomes more apparent when the center boss is widened and the applied voltage is increased. It will gradually diminish when the mechanical load is raised, however. Under a low initial tension condition, specifically, the influence of piezoelectric effect will be more apparent when a shallow but wide elastic boss is considered. Yet, thickening the center boss will moderate the influence of the piezoelectric effect but induces a more serious interface effect between the center boss and the annular plate.

中文摘要 i
Abstract ii
致謝 iv
目錄 v
表目錄 vii
圖目錄 viii
符號說明 xii
第一章 前言 1
1-1研究動機與目的 1
1-2文獻回顧 4
1-3本文架構 7
第二章 物理問題與基礎理論描述 9
2-1問題描述 9
2-2對稱疊層圓型板大撓曲掌控方程 9
2-2-1力平衡方程 9
2-2-2力矩平衡方程 10
2-3含壓電層之對稱疊層圓型板之大撓曲掌控方程 12
2-3-1增量型式 12
2-3-2無因次化 13
2-4含壓電彈性凸緣對稱疊層圓型板之大撓曲無因次掌控方程 14
2-4-1外環板部分 14
2-4-2含壓電層彈性凸緣部分 15
2-5 無因次邊界條件之設定 15
第三章 解析與數值方法推導 16
3-1線性小撓曲問題 16
3-1-1當機械(力)效應大於電效應 16
3-1-2當電效應大於機械(力)效應 20
3-1-3極限條件下之線性解析解 24
3-2非線性大撓曲問題 26
3-2-1有限差分法之應用 26
3-2-2非線性數值解程式化技巧 30
3-2-3非線性數值解加快收斂技巧 30
第四章 結果與討論 31
4-1壓電效應下大撓曲問題之線性解析結果 31
4-1-1中心撓度對預拉伸力之結果 32
4-1-2全域式側向斜率之結果 33
4-1-3全域式側向撓度之結果 34
4-1-4全域式側向曲率之結果 35
4-1-5全域式最大徑向應力(機械敏性)之結果 36
4-2大撓曲非線性問題之數值結果 37
4-2-1中心撓度對面壓大小之結果 37
4-2-2中心曲率對面壓大小之結果 38
4-2-3全域式側向斜率之結果 39
4-2-4全域式側向撓度之結果 40
4-2-5全域式側向曲率之結果 40
4-2-6全域式最大徑向應力(機械敏性)之結果 41
4-3真實幾何與結構回應之數值結果討論 42
4-3-1無因次預拉伸參數(k)與真實預拉伸力之檢視 43
4-3-2無因次面壓(P)與真實面壓之檢視 43
第五章 結論 44
參考文獻 45
圖表整彙 49

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