本文研究雙異質線彈性複合層板之暫態響應，解析含有限長裂紋之線彈性複合層板受反平面均佈動力負載時裂紋尖端之應力強度因子。文中將問題分為兩個部份，問題一先求解不含裂紋之複合層板於自由表面受反平面動力負載時之解，問題二則於拉普拉斯轉換域中在複合層板的裂紋面上反加問題一所產生的剪應力。解析時將利用奇異積分方程與切比雪夫多項式來求得轉換域中之解，最後再利用Durbin數值拉普拉斯逆轉換法獲得應力強度因子之時域解。數值計算時，將與文獻上施加應力在裂紋面上之解析解作比較，並作詳細之討論。

In this study, the transient response of a finite crack lying on the interface of a linearly elastic composite strip is investigated. The laminate is subjected to uniformly dynamic anti-plane loading on the free surfaces. This transient problem can be treated as the superposition of two problems. Problem 1 considers a crack-free laminate subjected to uniformly dynamic anti-plane loading on the free surfaces. While problem 2 concerns a linearly elastic laminate containing an interfacial crack with the crack faces subjected to the loads that cancel out the shear stress induced by problem 1. The solution procedures are based on the use of integral transforms, singular integral equations and Chebyshev polynomial expansions. Durbin’s method is used to carry out the numerical inversion of Laplace transform. The numerical results are evaluated and discussed in detail. Furthermore, the accuracy is examined through some specified cases in the literature.

目錄............................................I
圖目錄............................................III
第一章 緒論....................................1
1.1 研究動機...................................1
1.2文獻回顧....................................3
1.3內容簡介....................................5
第二章 理論基礎................................6
2.1線彈性材料之反平面控制方程與本構方程式......6
2.2拉普拉斯轉換及逆轉換........................7
2.3 傅立葉轉換及逆轉換 ........................7
2.4 Durbin 方法................................8
2.5奇異積分方程 (Singular Integral Equation)...8
2.6切比雪夫多項式..............................9
第三章 含有限長裂紋之複合層板之動力破壞........11
3.1 問題描述...................................11
3.2 無裂紋受反平面均佈應力之複合層板之問題描述..12
3.3 無裂紋之複合層板受反平面均佈應力之解析.....13
3.4含有限長裂紋於裂紋面承受負載複合層板之問題描述.............................................16
3.5 含有限長裂紋於裂紋面承受負載之複合層板之解析.............................................17
第四章  數值結果與討論.........................27
      4.1數值計算時應注意事項..................27
      4.2數值解結果比較........................30
第五章  結論與成果.............................35
      5.1本文結論...............................37
      5.2本文成果.............................................38
      5.3尚待研究的方向.......................................................38
參考文獻.................40
附錄一 論文簡要版..............................................................................................66














圖 目 錄
圖3-1含有限長裂紋之複合層板受反平面均佈應力之圖形..…44
圖3-2無裂紋之複合層板受反平面均佈應力之圖形…...…...……45
圖3-3含有限長裂紋於裂紋面承受負載之複合層板之圖形..…46
圖4-1 於不同的Durbin項數之比較...............................................47
圖4-2應力強度因子於不同積分上限.............................................................48
圖4-3應力強度因子在不同的 級數項數.............................................49
圖4-4應力強度因子在不同的積分精確度...................................................50
圖4-5應力強度因子在Durbin的不同加總項數.........................................51
圖4-6應力強度因子在不同的 值.............................................52
圖4-7應力強度因子於不同週期之比較........................................................53
圖4-8應力強度因子於裂紋長度 時不同厚度之比較...............54
圖4-9應力強度因子於裂紋長度 時不同厚度之比較...............55
圖4-10 應力強度因子於裂紋長度 時不同厚度之比較...........56
圖4-11	應力強度因子於長時間的影響.........................................................57
圖4-12	應力強度因子於 之值的影響...........................................58
圖4-13於裂紋上施加均佈負載之複合層板之圖形................................59
圖4-14於裂紋上施加均佈負載的複合材料之應力強度因子................60
圖4-15為計算直接施加均佈負載於裂紋面上情況之應力強度因子 61
圖4-16 負載函數f(t)之圖形     62
圖4-17 於裂紋上施加f(t)形式負載之複合層版之圖形 63
圖4-18 於裂紋上施加f(t)型式負載的複合材料之應力強度因子 64
圖4-19 單一材料於裂紋上施加f(t)型式負載情況之應力強度因子 65

Boby, D.B., (1972) “The plane soluteion for anisotropic elastic wedges under normal and shear loading,” J. Appl. Mech 39,1103–1109.

Chen, Z.T., and Karihaloo, B.L., (1999) “Dynamic response of a cracked piezoelectric ceramic under arbitrary electro-mechanical impact,” International Journal of Solids and Structures 36,5125–5133.

Chen, Z.T., and Meguid, S.A., (2000) “The transient response of a piezoelectric strip with a vertical crack under electromechanical impact load,” International Journal of Solids and Structures 37, 6051–6062.

Dubner, R., and Abate, J., (1968) “Numerical inversion of Laplace transforms by relating them to the finite Fourier Cosine transform,” JACM 15, 115–123.

Durbin, F., (1974) “Numerical inversion of Laplace transforms: an efficient improvement to Duber and Abate’s method,” The Computer Journal 17, 371–376.

England, A.T., (1965) “A crack between dissimilar media,” J. Appl. Mech. 32,400–402.

Erdogan, F., (1965) “Stress distribution in bonded dissimilar material with cracks,” J .Appl. Mech. 32,403–410.

Feng, W.J. and, Pan, E., (2008) “Dynamic fracture behavior of an internal interfacial crack between two dissimilar magneto-electro- elastic plates,. ” Engineering Fracture Mechanics 75, 1468–1487.

Hu, S., Shen, S., and Nishioka, T., (2007) “Numerical analysis for a crack in piezoelectric material under impact,” International Journal of Solids and Structures 44, 8457–8492.



Ing, Y. S., and Ma C. C., (1997) “Dynamic fracture analysis of a finite crack subjected to an incident horizontally polarized shear wave,” International Journal Solids Structures 34, 895–910.

Kuo, M.C., and Bogy, D.B., (1974) “Plane solution for the displacement and traction-displacement problems for anisotropic elastic wedges. ” J. Appl. Mech. 41, 197–202.

Li, Q., and Chen, Y., (2007) “Analysis of a permeable interface crack in elasticdielectric/piezoelectric bimaterials,” Acta Mech Sin 23, 681–687.

Li ,Y.D., and Lee, K.Y.,(2008) “Fracture analysis in micropolar elasticity: anti-plane crack, ” International Journal of Fracture 152, 163–168.

Meguid, S.A., and Chen, Z.T., (2001) “Transient response of a finite piezoelectric strip containing coplanar insulating cracks under electromechanical impact,” Mechanics of Materials 33, 85–96.

Meguid, S.A., and Zhao, X., (2002) “The interface crack problem of bonded piezoelectric and elastic half-space under transient electromechanical loads ,”Journal of Applied Mechanics, Transactions ASME 69, 244–253.

    Rice, J.R., and Sih, G.C., (1965) “Plane problems of cracks in dissimilar media,” J. Appl. Mech. 32, 418–423.

Shin, J.W., Kwon, S.M., and Lee, K.Y., (2001) “An eccentric crack in a piezoelectric strip under anti-plane shear impact loading,” International Journal of Solids and Structures 38, 1483–1494.

Ting, T. C. T.,(1986) “Explicit solution and invariance of the singularities in an interface crack in anisotropic composites,”  J. Appl. Mech.38, 505–513.



Ting, T. C. T.,(1995) “Generalized Dundurs constants for anisotropic materials,”  International Journal of Solid & Structures 32, 483-500.

Ueda, S.,(2003) “Diffraction of anti-plane shear waves in a piezoelectric laminate with a vertical crack,” European Journal of Mechanics A/Solids 22, 413–422.

Wang, B.L., and Han, J.C., (2007) “Multiple Surface Cracking of a Piezoelectric Layer Bonded to an Elastic Substrate Under Dynamic Anti-plane Electromechanical Impacts,” Journal of Intelligent Material Systems and Structures 18, 1203–1213.

Wang, B.L., and Han, J.C., (2007) “Fracture of a finite piezoelectric strip with a crack vertical to its borders – An exact analysis and its applications,” International Journal of Applied Electromagnetics and Mechanics 26 , 87–99.

Wang, B.L., Han, J.C., and Du, S.Y., (2000) “Electroelastic fracture dynamics for multilayered piezoelectric materials under dynamic anti-plane shearing,” International Journal of Solids and Structures 37, 5219–5231.

Wang, X., and Yu, S., (2000) “Transient response of a crack in piezoelectric strip subjected to the mechanical and electrical impacts,” International Journal of Solids and Structures 37, 5795–5808.

Wang, X.D., (2001) “On the dynamic behaviour of interacting interfacial cracks in piezoelectric media,”  International Journal of Solids and Structures 38, 815–831.

Zhao, X., (2004) “An efficient approach for the numerical inversion of Laplace transform and its application in dynamic fracture analysis of a piezoelectric laminate ,” International Journal of Solids and Structures 41, 3653-3674.

廖雪吩 (2007)，應用數值拉普拉斯逆轉換法於壓電材料動力破壞之研究，淡江大學航空太空工程學系碩士班碩士論文。

許吉勝 (2008)，含有限長裂紋之彈壓電複合層板動力破壞分析，淡江大學航空太空工程學系碩士班碩士論文。