臺灣博碩士論文加值系統

　　本論文包含兩部分，第一部份是分析含界面裂紋雙磁電彈楔形之反平面破壞力學問題。經由梅林轉換及殘值定理，可以推導出一組奇異積分方程式，並求解楔形結構裂紋尖端間環形區域內之反平面磁電彈場。結果顯示，所獲得的應力場、電位移場與磁感場之間並無耦合現象，而且均與材料性質無關。這個現象來自於裂紋在電與磁雙方面均假設為非滲透型，以及對稱的幾何外形與邊界條件。

　　本文以含裂紋之單材料磁電彈楔形結構為例，證明能量密度理論在預測裂紋成長方面優於能量釋放率理論。研究結果發現在裂紋尖端附近的能量釋放率為負值，無法評估裂紋的破裂行為，因此本文採用能量密度理論。接著，依據裂紋尖端附近反平面磁電彈近場解，可以得到磁電彈場之各式強度因子，再總和而成單一之能量密度因子。結果發現該因子與裂紋尖端的局部圓柱座標q1無關。依據能量密度理論，此界面裂紋可能沿任意角度成長，而與裂紋尖端附近材料的破壞韌性有關。

　　為了瞭解界面裂紋的破壞行為，本文進一步探討楔形角度以及作用於楔形邊界之外在負荷等因素，與能量密度因子的關係。結果顯示，電場和磁場方向的逆轉在裂紋尖端造成不同的能量分佈。當二者方向相反時，將提高裂紋成長的趨動力。

　　本論文第二部份以雙磁電彈材料接合而成的楔形結構為例，利用梅林轉換，探討磁電彈場在楔形頂點之奇異性。經由特徵方程式，可以得到各種不同材料的組合與接合角度下之奇異性階數。結果顯示，兩材料BaTiO3體積百分比的差距越大，奇異性階數就越高，不利於該結構的可靠度。另外，從材料的選擇與接合角度，均可尋求奇異性消失的條件，以確保結構的安全。

　　本論文所有結果，均可以退化至壓電問題或單純彈性問題。

　This paper contains two parts. The first part is to study the fracture problem of a bimaterial magneto-electro-elastic (MEE) composite wedge containing an interface impermeable crack under antiplane shear and inplane electric and magnetic loads. After employing the Mellin transform and residue theorem, the governing equations are reduced to a system of singular integral equations. The antiplane magneto-electro-elastic field inside a circular region between the crack tips is obtained. It is found that the fields of stress, electric displacement and magnetic induction are uncoupled and are independent of material properties. This result is based on the assumption of the impermeable crack and the symmetry of structure geometry and boundary conditions.

　To prove the superior of the energy density theory to the energy release rate theory, the problem of a single MEE wedge containing a radial internal crack located at the symmetric axis is considered. A negative energy release rate near the crack tip is obtained and cannot be used to predict the fracture propagation.

　In order to apply the energy density theory, the energy density factors at the crack tips in each material are computed from the intensity factors of stress, strain, electric displacement, electric field, magnetic induction and magnetic field. The energy density factors are independent of the local coordinate q1 defined at the crack tips. It means that the crack may extend along any direction depending on the fracture toughness of the materials near the crack tip. For further investigations, the variations of the energy density factors with the wedge angles and the external loads applied on the wedge edges are plotted graphically. The results show that the change in directions of the magnetic field and electric field will significantly affect the energy distribution near the crack tips. If these two directions are opposite, the crack driving forces are enhanced to extend the cracks.

　The second part of this paper is to analyze the singularity behavior of a bimaterial MEE composite wedge. Using the Mellin transform, the characteristic equations of eigenvalues are derived analytically. The antiplane singularity orders are then obtained numerically under the combinations of different material properties and wedge angles. The results show that the singularities become stronger when the difference between the BaTiO3 volume fractions of both materials is larger. Also, the conditions for vanishing singularities can be determined if the material properties and wedge angles are selected properly.

　All results of this paper can be degenerated to the piezoelectric or elastic problems.

目錄

摘要 i
Abstract ii
誌謝 iii
目錄 iv
表目錄 vii
圖目錄 viii
符號表 xi


第1章 緒論 1
1.1 前言 1
1.2 壓電材料 2
1.3 磁電彈材料與壓磁材料 9
1.4 本研究相關文獻回顧 12
1.4.1 楔形問題 12
1.4.2 壓電材料裂紋問題 17
1.4.3 磁電彈材料裂紋問題 23
1.5 本論文分析內容 25

第2章 含界面裂紋之雙磁電彈楔形分析 27
2.1 問題定義 27
2.2 基本公式 31
2.3 求解 34
2.3.1 位移w、電位f 與磁位y 34��
2.3.2 f(r)、g(r)、l(r)函數之求解 52
2.3.3 應力、電位移與磁感之求解 57
2.4 強度因子 59
2.5 能量密度因子 65
2.6 數值結果與討論 67
2.6.1 BaTiO3-CoFe2O4的材料係數 67
2.6.2 強度因子的討論: 楔形角a 的影響 69
2.6.3 強度因子的討論: 界面裂紋位置的影響 71
2.6.4 強度因子的討論: 界面裂紋長度的影響 74
2.6.5 能量密度因子的討論: 楔形角a 的影響 75
2.6.6 能量密度因子的討論: 電負荷的影響 77
2.6.7 能量密度因子的討論: 磁負荷的影響 78
2.6.8 CoFe2O4導磁係數G11之影響 79
2.6.9 界面裂紋面上的位移、電位、磁位 82
2.6.10 能量密度因子與能量釋放率的比較 85
2.7 其他討論 87
2.7.1 無限域楔形之反平面問題 87
2.7.2 點集中負荷問題 88
2.7.3 壓電材料-空氣界面邊界條件與裂紋的滲透性問題 91
2.7.4 磁電彈材料-空氣界面邊界條件與裂紋的滲透性問題 93

第3章 含界面裂紋之雙壓電楔形分析 97
3.1 問題定義與求解 97
3.2 數值結果與討論 102
3.2.1 a=p/2且a®0特例的強度因子 102
3.2.2 能量密度因子 104

第4章 雙磁電彈楔形頂點之奇異性分析 106
4.1 問題定義 106
4.2 基本公式與求解 108
4.3 退化例子 114
4.4 數值結果與討論 115
4.4.1 材料係數 115
4.4.2 單材料磁電彈楔形 116
4.4.3 雙磁電彈楔形(邊界條件A-A組合) 118
4.4.4 磁電彈/壓電楔形(邊界條件A-A組合) 121
4.4.5 磁電彈/壓磁楔形(邊界條件A-A組合) 122
4.4.6 磁電彈/彈性楔形(邊界條件A-A組合) 124
4.4.7 邊界條件A-B組合下之雙磁電彈楔形 125

第5章 結論 127

參考文獻 129

附錄A 140
附錄B 143
附錄C 154
附錄D 158
附錄E 161

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