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研究生:莊明達
研究生(外文):Ming-TaChuang
論文名稱:以應變能密度理論分析含單一任意方向裂縫之功能梯度條板問題
論文名稱(外文):Fracture analysis of a FGM strip containing an arbitrarily oriented crack by using strain energy density theory
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:碩士
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:2012
畢業學年度:100
語文別:中文
論文頁數:53
中文關鍵詞:裂縫功能梯度材料面內問題應力強度因子應變能密度因子
外文關鍵詞:crackFGMinplane problemstress intensity factorstrain energy density factor
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  本論文主要目的在於探討含單一嵌入式裂縫功能梯度材料條版面內問題的破壞分析。依據彈性力學之基礎理論,利用傅立葉轉換法,將混合邊界值問題推導出一組奇異積分方程式,再藉由Chebyshev多項式技術化為代數聯立方程組以求數值解,由於其中核函數相當複雜並未如預期順利解出其數值解,而為了進一步利用應變能密度理論觀察其破壞行為,本論文將引用Long及Delale研究中所得應力強度因子之結果代入應變能密度因子公式,因此從應變能密度因子曲線圖中所能討論的破壞行為將受到Long及Delale之研究結果所侷限。我們可從S曲線中找到最小應變能密度因子Smin及裂縫開裂角度Θ0,再配合S理論討論其破壞行為,並探討裂縫長度及裂縫旋轉角度對破壞行為的影響。結果顯示無論裂縫旋轉角度為0°或45°,裂縫開裂的驅動力Smin都將隨裂縫長度增長而增加;當裂縫旋轉角度為0°時,必須藉由實驗得到臨界值Scr才能預測裂縫將從哪端開裂;而當裂縫旋轉角度為45°之情況下,由於在裂縫a端(材料較軟處)的Smin值較b端的Smin值高,因此可判斷裂縫將優先從a端開裂。
  This thesis deals with the fracture behavior of a functionally graded material strip with an embedded crack under inplane loads. A system of Cauchy-type singular integral equations is formulated by employing the Fourier transforms. Although the numerical technique of using the Chebyshev polynomials is a powerful tool to solve the singular equations, we face some difficulty and fail to get numerical results. In order to apply the strain energy density theory on the fracture problems of nomhomogeneous material, all numerical results of strain energy density factor S are calculated directly from the stress intensity factors plotted in the study of Long and Delale. Therefore, the discussion on the fracture behavior of the problem becomes very limited.
  From the variations of S with the local coordinate system Θ, the minimum strain energy density factor Smin and crack extension angle Θ0 can be obtained to discuss the effects of crack length and crack angle on the fracture behavior. Two cases of crack angle with θ=0° and 45° are adopted in the discussion. As it is expected, the crack driving force for longer crack length will be higher at both crack tips. Since the critical strain energy density factor Scr for nonhomogeneous materials has to be obtained by testing, there are no sufficient evidences to get the conclusion that the crack will extend at which crack tips for the case θ=0°. However, for θ=45°, the Smin at the crack tip within softer material (i.e. crack tip a) is greater than that at crack tip b and the crack can be predicted to propagate at crack tip a.

摘要 I
Abstract                   II
致謝                        IV
目錄                        VI
表目錄                        VIII
圖目錄                        IX
符號說明                        XI
第一章 緒論                   1
1-1 前言                        1
1-2 文獻回顧                   2
1-3 研究動機 4
第二章 問題分析與推導              5
2-1 問題描述                   5
2-2 疊加原理與邊界條件 6
2-3 求解過程 8
2-4 應變能密度理論 24
第三章 數值運算方法 27
3-1 第一型奇異積分方程式 27
3-2 Gauss-Chebyshev的數值積分法 28
第四章 結果與討論 33
4-1 裂縫長度及裂縫角度對應力強度因子的影響 33
4-2 裂縫長度及裂縫角度對S曲線的影響 36
第五章 結論 45
附錄A 47
附錄B 48
附錄C 50
參考文獻 51

[1] Chue, C. H. and Ou, Y. L., Mode III crack problems for two bonded functionally graded piezoelectric materials. International Journal of Solids and Structures 42, 3321-3337, 2005.
[2] 王保林, 韓杰才, 張幸紅, 非均勻材料力學, 科學出版社, 北京市, 2003.
[3] Erdogan, F. and Delale, F., The crack problem for a nonhomogeneous plane. Transaction of the ASME, Journal of Applied Mechanics 50, 609-614, 1983.
[4] Jin, Z. H. and Noda, N., Crack tip singular fields in nonhomogeneous materials. ASME Journal of Applied Mechanics 61, 738-740, 1994.
[5] Erdogan, F. and Wu, B. H., The surface crack problem for a plate with functionally graded properties. Transactions of the ASME, Journal of Applied Mechanics 64, 449-456, 1997.
[6] Ozturk, M. and Erdogan, F., The axisymmetrical crack problem in a nonhomogeneous medium. Transaction of the ASME, Journal of Applied Mechanics 60, 406-413, 1993.

[7] Konda, N. and Erdogan F., The mixed mode crack problem in a nonhomogeneous elastic medium. Engineering Fracture Mechanics 47, 533-545, 1994.
[8] 廖經元, 以應變能密度理論分析含中央裂縫之功能梯度平面問題, 國立成功大學, 2011.
[9] Long X. and Delale F., The general problem for an arbitrarily oriented crack in a FGM layer. International Jounnal of Fracture 129, 221-238, 2004.
[10] Muskhelishvili, N. I., Singular Integral Equations. Noordhoff Internaltional Publishing, Groningen, The Netherlands, 1953.
[11] Sih. G. C., Mechanics of Fracture Initiation and Propagation. Kluwer Academic Publishing, Dordrecht, Boston, Chapter 1, 1991.
[12] Erdogan, F., Gupta, G. D. and Cook, T. S., Numerical solution of singular integral equations. In Mechanics of Fracture 1: Method of analysis and solution of crack problem, edited by G. C. Sih, Chapter 7, Noordhoff Internaltional Publishing, Leyden, The Netherlands, 1973.
[13] Rivilin, T. J., The Chebyshev Polynomials. Wiley, New York, 1974.
[14] Erdogan, F. and Gupta, G. D., Numerical solution of singular integral equations. Quarterly of Applied Mathematics 29, 525-534, 1972.
[15] Gdoutos, E. E., Fracture Mechanics, Springer, Xanthi, Greece, 2005.

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