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研究生:歐怡良
研究生(外文):Yi-Liang Ou
論文名稱:含嵌入式裂縫之功能梯度壓電材料面外問題破壞分析
論文名稱(外文):Antiplane Fracture Analysis of Embedded Cracks in Functionally Graded Piezoelectric Materials
指導教授:褚晴暉褚晴暉引用關係
指導教授(外文):Ching-Hwei Chue
學位類別:博士
校院名稱:國立成功大學
系所名稱:機械工程學系碩博士班
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2006
畢業學年度:94
語文別:中文
論文頁數:158
中文關鍵詞:應力強度因子嵌入式裂縫功能梯度壓電材料電位移強度因子奇異積分方程式
外文關鍵詞:singular integral equationsfunctionally graded piezoelectric materialsstress intensity factorsembedded crackselectric displacement intensity factors
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本論文主要目的在於分析探討含有嵌入式裂縫之功能梯度壓電材料的破壞問題。壓電材料之極化方式為六方對稱型,依照其幾何外型依序分為單一梯度壓電材料全平面、單一梯度壓電材料半平面、雙梯度壓電材料半平面結合、單一梯度壓電材料條板及其與半平面結合等之各個子問題。
本文運用Fourier積分轉換法,分別依據各幾何形狀及混合邊界條件問題化成兩組奇異積分方程式,再藉由Gauss- Chebyshev定理及Chebyshev多項式定理將其再度化為代數聯立方程組,以求得應力強度因子與電位移強度因子之數值解。
研究結果顯示,應力強度因子與電位移強度因子分別僅與其外加之機械負載與電負載、裂縫長度及位置有關,而與材料特性無關,且彼此為非耦合,同時只要外加機械負載為常數且彼此相同,則其應力強度因子值在不可滲透型裂縫與可滲透型裂縫無異,與既有文獻結論一致。強度因子將隨著裂縫所在位置處的材料性質增強而提高,且邊界條件對於強度因子的影響甚劇,隨著離邊界的距離漸增,邊界的影響愈趨緩,另一方面,強度因子也會因為相鄰材料的梯度變化而有不同的結果,當其非均質參數分別趨近於兩個極限值時,可以分別退化成為簡單的不同邊界問題。
本文使用能量密度因子及能量釋放率作為裂縫成長之驅動力,分析結果可知能量密度因子皆為正值函數且與局部座標q1無關,能量釋放率與能量密度因子不同,在僅有外加電負載存在或電負載與機械負載的比值在某範圍外時將出現負值現象,預期裂縫在該情形下將不會成長。
能量密度因子與能量釋放率會受到材料性質的強烈影響,在材料性質較弱的一側由於其應變與電場值均較大,因此在均勻應力與電位移場作用之下兩種能量法則所得到的較大值都發生材料性質較弱的一方,然而因為梯度材料的材料參數SC及GC仍屬未知,故目前尚無法直接判斷裂縫開始成長的方向,待將來相關材料實驗作出進一步測試,可供作比較之用。
The fracture problems of embedded cracks within functionally graded piezoelectric materials (FGPM) are discussed in this thesis. The poling type of the piezoelectric material is in the form of 6mm symmetry. Several sub-problems with different geometries such as a FGPM full plane, a FGPM half plane, two bonded FGPM half planes, a FGPM strip and a FGPM strip bonded to a FGPM half plane are analyzed separately.
By applying Fourier integral transform, the field equations with mixed boundary conditions can be transformed into two singular integral equations. The Chebyshev polynomials are then employed to reduce the singular integral equations into one set of algebraic equations. The stress and electric displacement intensity factors are obtained numerically.
The results show that both stress and electric displacement intensity factors are uncoupled and depend on the applied mechanical and electrical loadings, crack length and crack location. They are independent of the material properties. If the applied mechanical loadings were the same for both impermeable and permeable crack case, the magnitudes of the stress intensity factors are completely identical. This conclusion agrees with the existence literatures. The magnitude of normalized intensity factors is higher for stronger material coefficients. The effects of boundary conditions and gradient variations of the adjacent bonded materials on the intensity factors are also significant.
In this article both the energy density factors and energy release rates are used as the driving forces of crack growth. The results show that the energy density factors are positive definite and independent of the local coordinate q1. Differing from energy density factors, the magnitude of energy release rates may be negative if only electrical loading existed or the ratio of mechanical and electrical loadings lay outside certain range. Crack growth will be impeded if the energy release rate is negative. Both energy criteria are strongly affected by the material nonhomogeneous parameters. From the expression of strain and electric fields, higher energy density factors and energy release rates occur at crack tip where the material is weaker. However, because the material parameters SC and GC are still unknown, it is unable to predict the crack propagation direction.
摘要 i
Abstract ii
誌謝 iii
目錄 iv
表目錄 vi
圖目錄 vi
符號說明 x
第一章 緒論 1
1-1 前言 1
1-2 文獻回顧 3
1-3 本論文分析問題簡介 16
1-4 本文架構 19
第二章 基本公式推導 20
2-1 壓電材料本構方程式基本理論 20
2-2 壓電材料裂縫面邊界條件假設 24
2-3 含單一裂縫梯度壓電材料全平面問題 27
2-4 含單一嵌入式裂縫梯度壓電材料半平面問題 35
2-5 含單一嵌入式裂縫雙接合梯度壓電材料半平面問題 40
2-6 含兩個嵌入式裂縫雙接合梯度壓電材料半平面問題 45
2-7 含單一嵌入式裂縫梯度壓電材料條板問題 51
2-8 含單一嵌入式裂縫梯度壓電材料半平面與條板接合問題 57
第三章 數值運算法與相關破壞準則預測 61
3-1 Gauss-Chebyshev積分式與Chebyshev多項式 61
3-2 能量密度因子與能量釋放率 69
第四章 結果與討論 76
4-1 含單一裂縫梯度壓電材料全平面問題 76
4-2 含單一嵌入式裂縫梯度壓電材料半平面問題 83
4-3 含單一嵌入式裂縫雙接合梯度壓電材料半平面問題 89
4-4 含兩個嵌入式裂縫雙接合梯度壓電材料半平面問題 97
4-5 含單一嵌入式裂縫梯度壓電材料條板問題 111
4-6 含單一嵌入式裂縫梯度壓電材料半平面與條板接合問題 118
第五章 結論 126
參考文獻 132
附錄A 144
附錄B 145
附錄C 147
附錄D 149
附錄E 152
附錄F 153
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