本文研究主題為應用數值拉普拉斯逆轉換法於無窮域內含不可誘電邊界靜止裂紋之壓電材料動力破壞問題，求解一含有限長靜止裂紋之壓電材料於裂紋面上受均佈反平面動態應力及平面動態電位移負載的暫態問題。本文運用了三種數值拉普拉斯逆轉換法分別為Durbin方法、趙氏方法ㄧ及趙氏方法二，首先利用簡單函數範例來比較這些數值法的準確性，並由計算經驗分別建議三種數值法的參數設定範圍；最後以數值拉普拉斯逆轉換法配合積分轉換技巧，解析含裂紋之壓電材料動力破壞問題，並將所求得之應力強度因子數值解做詳細的計算與討論。

In this study, the dynamic fracture analysis of an infinite piezoelectric material containing a finite crack with impermeable boundary conditions is investigated by using numerical inversion of Laplace transform. The finite crack is subjected to uniformly anti-plane stress and in-plane electric displacement impacts. Three numerical methods, Durbin’s method, Zhao’s method 1 and Zhao’s method 2 are used in this study. The accuracy is examined through some specified functions and the applicable numerical parameters are suggested respectively by the experience of calculation. Finally, the dynamic fracture analysis of a piezoelectric crack is carried out by numerical inversion of Laplace transform and the technique of integral transformation. Numerical solutions for dynamic stress intensity factors are evaluated and discussed in detail.

目     錄
中文摘要 ……………………………………………………………… I
英文摘要 …………………………………………………………… II
目錄 ………………………………………………………………… IV
表目錄 …………………………………………………………… VI
圖目錄 …………………………………………………………… VIII
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 ………………………………………………6
第二章 數值拉普拉斯逆轉換………………………………………8
2.1  拉普拉斯轉換及逆轉換……………………………8
2.2  Durbin方法…………………………………………9
2.3  趙氏方法一 (Zhao’s method1)………………………19
2.4  趙氏方法二 (Zhao’s method2)………………………22
2.5  簡單數值範例…………………………………………27
第三章 壓電材料動力破壞之應用…………………………35
3.1  問題描述…………………………………………35
3.2  理論解析…………………………………………37
第四章 數值結果與討論………………………………………55
4.1  求解弗萊德積分方程式之數值法…………55
4.1.1  高斯積分法…………………………………55
4.1.2  L.U.法………………………………………56
4.1.3  求解弗萊德積分方程式………………………58
4.2  數值結果比較…………………………………63
第五章 結論與展望…………………………………………72
5.1  本文結論…………………………………………72
5.2  本文成果……………………………………………72
5.3  尚待研究的方向……………………………………73
參考文獻………………………………………………………………75

表 目 錄

表2-1 文獻範例  運用三種數值方法之結果………………………81
表2-2 文獻範例  運用三種數值方法之結果………………………82
表2-3 文獻範例 運用三種數值方法之結果………………………83
表2-4 文獻範例  運用三種數值方法於較佳設定參數之結果……84
表2-5 文獻範例 運用三種數值方法於較佳設定參數之結果………85
表2-6 文獻範例 運用三種數值方法於較佳設定參數之結果………86
表2-7   Durbin數值方法比較範例結果……………………………87
表2-8   Durbin方法比較 範例結果………………………………88
表2-9   趙氏數值方法一比較 範例結果……………………………89
表2-10   趙氏數值方法一比較 範例結果…………………………90
表2-11   趙氏數值方法二比較 範例結果…………………………91

圖 目 錄

圖2-1  及 、 、 之圖形…………………………92
圖2-2  及 、 、 之圖形…………………………93
圖2-3  發散及 、 、 之圖形……………………94
圖2-4  收斂及 、 、 之圖形……………………95
圖2-5   於Durbin方法逆轉換在不同週期 之比較圖……………96
圖2-6   於Durbin方法逆轉換在不同週期 之比較圖……………97
圖2-7   於Durbin方法逆轉換在不同 之比較圖…………………98
圖2-8   於Durbin方法逆轉換在不同 之比較圖…………………99
圖2-9   於Durbin方法逆轉換在不同項數 之比較……………100
圖2-10   於Durbin方法逆轉換在不同項數 之比較……………101
圖2-11   於趙氏方法一逆轉換在不同週期 之比較……………102
圖2-12   於趙氏方法一逆轉換在不同週期 之比較……………103
圖2-13   於趙氏方法一逆轉換在不同項數 之比較……………104
圖2-14   於趙氏方法一逆轉換在不同項數 之比較……………105
圖2-15   於趙氏方法二逆轉換在不同週期 之比較……………106
圖2-16   於趙氏方法二逆轉換在不同週期 之比較……………107
圖2-17   於趙氏方法二逆轉換在不同項數 之比較……………108
圖2-18   於趙氏方法二逆轉換在不同項數 之比較……………109
圖2-19   於三種數值法逆轉換之比較圖………………………110
圖2-20   於三種數值法逆轉換之比較圖………………………111
圖2-21   於三種數值法逆轉換之比較圖………………………112
圖2-22 三種數值法逆轉換結果之比較…………………………113
圖3-1 無窮域含裂紋壓電材料受反平面均佈應力及平面電位移負載之結構示意圖………………………………………………………114
圖4-1 無窮域含裂紋之彈性材料受反平面均佈應力運用不同數值法
之結果比較圖………………………………………………115
圖4-2 無窮域含裂紋之壓電材料運用Durbin方法於不同項數 時之比較圖………………………………………………116
圖4-3 無窮域含裂紋之壓電材料運用Durbin方法於不同項數 時之短時間比較圖………………………………………117
圖4-4 無窮域含裂紋之壓電材料運用趙氏方法一於不同項數 時之比較圖………………………………………………118
圖4-5 無窮域含裂紋之壓電材料運用趙氏方法一於不同項數 時之短時間比較圖………………………………………119
圖4-6 無窮域含裂紋之壓電材料運用趙氏方法二於不同項數 時之比較圖………………………………………………120
圖4-7 無窮域含裂紋之壓電材料運用趙氏方法二於不同項數 時之短時間比較圖………………………………………121
圖4-8 無窮域含裂紋之壓電材料運用Durbin方法於 不同時之比較圖………………………………………………………122
圖4-9 無窮域含裂紋之壓電材料運用趙氏方法一於 不同時之比較圖……………………………………………………123
圖4-10 無窮域含裂紋之壓電材料運用趙氏方法二於 不同時之比較圖……………………………………………………124
圖4-11 無窮域含裂紋之壓電材料運用不同數值法之長時間數值解比較圖…………………………………………………………125
圖4-12 無窮域含裂紋之壓電材料運用Durbin方法於不同電位移負載大小之比較圖……………………………………126
圖4-13無窮域含裂紋之壓電材料運用趙氏方法一於不同電位移負載大小之比較圖……………………………………127
圖4-14無窮域含裂紋之壓電材料運用趙氏方法二於不同電位移負載大小之比較圖……………………………………128
圖4-15無窮域含裂紋之壓電材料運用三種數值法之比較……129
圖4-16 Chen and Karihaloo(1999)之結果圖………………………130
圖4-17無窮域含裂紋之壓電材料受之電位移強度因子關係圖
……………………………………………………………131
圖4-18第一種負載函數之圖形……………………………………132
圖4-19無窮域含裂紋之壓電材料之三種數值法比較圖………133
圖4-20第二種負載函數之圖形……………………………………134
圖4-21 無窮域含裂紋之壓電材料受運用Durbin方法於不同加
載上升時間之比較圖…………………………………135
圖4-22無窮域含裂紋之壓電材料受運用趙氏方法一於不同加載上升時間之比較圖……………………………………136
圖4-23無窮域含裂紋之壓電材料運用趙氏方法二於不同加載上升時間之比較圖………………………………………137

參考文獻

Albrecht, P. and Honig, G., (1977) “Numerical inversion of Laplace transforms: [ Numerische inversion der Laplace transformierten ],” Angew Inf Appl Inf , Vol. 19, pp. 336-345.

Arai, M., Sato, Y. and Adachi, T., (2003) “Elastodynamic crack analysis by boundary element method using numerical inversion of Laplace transform ,” JSME International Journal, Series A: Solid Mechanics and Material Engineering , Vol. 46, pp. 131-139.

Bellman, R., Kalaba, R. E. and Lockett, J. A., (1966) “Numerical inversion of Laplace transform,” American Elsevier, New York.

Bleustein, J. L., (1968) “A new surface wave in piezoelectric materials,” Applied Physics Letters, Vol. 13, pp. 412-413.

Chen, Z.T. and Yu, S.W., (1997a) “Crack tip fields of piezoelectric materials under antiplane impact,” Chinese Science Bulletin, Vol. 42, pp.  1615–1619.

Chen, Z.T. and Yu, S.W., (1997b) “Anti-plane dynamic fracture mechanics in piezoelectric materials,” International Journal of Fracture, Vol. 85, L3-L12.

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, Vol. 36, pp. 5125–5133.

Cheney, W. and Kincaid, D., (1999) Numerical mathematics and computing, 4th ed. Brooks/Cole Publishing .

Crump K. S., (1976) “Numerical inversion of the Laplace transforms using a Fourier series approximation,” J Assoc Comput Mach, Vol. 23, pp. 89-96.

Dubner, H. and Abate, J., (1968) “Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform,” Journal of the Association for Computing Machinery, Vol. 15, pp.115-123.

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

Evans, G. A., (1993) “Numerical inversion of Laplace transforms using contour methods,” International Journal of Computer Mathematics, Vol. 49, pp. 93-105.

Honig, G. and Hirdes, U., (1984) “A method for the numerical inversion of Laplace transforms,” Journal of Computational and Applied Mathematics, Vol. 10, pp. 113-132.

Hosono, T., (1979) “Numerical inversion of the Laplace transform,” Electrical Engineering in Japan (English translation of Denki Gakkai Ronbunshi) , Vol. 99, pp. 43-49.

Hosono, T., (1981) “Numerical inversion of the Laplace transform and some applications to wave optics,” Radio Science, Vol. 16, pp. 1015-1019.

Hüpper, B., and Pollak, E., (1999) “Numerical inversion of the Laplace transform,” Journal of Chemical Physics, Vol. 110, pp. 11176-11186.

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, Vol. 34, pp. 895-910.

Inoue, H., Kamibayashi, M., Kishimoto, K., Shibuya, T. and Koizumi, T., (1992) “Numerical Laplace transformation and inversion using fast Fourier transform,” JSME International Journal, Series 1: Solid Mechanics, Strength of Materials, Vol. 35, pp. 319-324.

Kitahara, N., Nagahara, D. and Yano, H., (1988) “Numerical inversion of Laplace transform and its application,” Journal of the Franklin Institute, Vol. 325, pp. 221-233.

Kwok, Y. K. and Barthez, D., (1989) “An algorithm for the numerical inversion of Laplace transforms,” Inverse Problems, Vol. 5, pp. 1089-1095.

Li, X. F. and Duan, X. Y., (2003) “Comparison of dynamic response of a piezoelectric ceramic containing two parallel cracks via two methods of Laplace inversion,” International Journal of Fracture, Vol.  122, pp. L131-L136.

Li, X. F. and Lee, K. Y., (2004) “Dynamic behavior of a piezoelectric ceramic layer with two surface cracks ,” International Journal of Solids and Structures, Vol. 41, pp. 3193-3209.

Li, X. F. and Lee, K.Y., (2006) “Transient response of a semi-infinite piezoelectric layer with a surface permeable crack ,” Zeitschrift fur Angewandte Mathematik und Physik, Vol. 57, pp. 636-651.

Li, X. F. and Yang, J., (2005) “Electromagnetoelastic behavior induced by a crack under antiplane mechanical and inplane electric impacts,” International Journal of Fracture, Vol. 132, pp. 49-64.

Loeber, J. F. and Sih, G. C., (1968) “Diffraction of antiplane shear waves by a finite crack,” The Journal of the Acoustical Society of America, Vol. 44, pp. 90-98.

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, Vol. 33, pp. 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 of the ASME, Vol. 69, pp. 244-253.

Miller, M. K. and Guy, W. T.,(1966) “Numerical inversion of the Laplace transform by use of Jacobi polynomials,” SIAM Journal on Numerical Analysis, Vol. 3, pp.624-635.

Milovanović, G. V. and Cvetković, A. S.,(2005) “Numerical inversion of the Laplace transform,” Electronics and Energetics, Vol. 18, pp.515-530.

Nakhla, M., Singhal, K. and Vlach, J., (1973) “Numerical Inversion of the Laplace transform,” Electronics Letters, Vol. 9, pp.  313-314.

Narayanan, G. V. and Beskos, D. E., (1982) “Numerical operational methods for time-dependent linear problems,” International Journal for Numerical Methods in Engineering, Vol. 18 , pp. 1829-1854.

Papoulis, A., (1956) “A new method of inversion of Laplace transform,” Quart. Appl. Math., Vol.14, pp.405-414.

Schmittroth, L. A., (1960) “Numerical inversion of Laplace transforms,” Communications of the ACM, Vol. 3, pp.171-173.

Singhal, K., Vlach, J. and Vlach, M., (1975) “Numerical Inversion of multidimensional Laplace transform,” Proceedings of the IEEE, Vol. 63, pp. 1627-1628.

Shih, D. H., Shen, R. C. and Shiau, T. C., (1987) “Numerical inversion of multidimensional Laplace transforms,” International Journal of Systems Science, Vol. 18, pp. 739-742.

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, Vol. 38, pp. 1483–1494.

Therapos, C. P. and Diamessis, J. E., (1982) “Numerical Inversion of a class of Laplace transforms,” Electronics Letters, Vol. 18, pp. 620-622.

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, Vol.  37, pp. 5219-5231.

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

Week, W. T., (1966) “Numerical inversion of Laplace transforms using Laguerre functions,” J. ACM, Vol. 13, pp.419-429.

Wen, P. H., Aliabadi, M. H. and Rooke, D. P., (1996a) “Dynamic analysis of mode III cracks in rectangular sheets,” International Journal of Fracture, Vol. 80, R37–R41.

Wen, P. H., Aliabadi, M. H. and Rooke, D. P., (1996b) “The influence of elastic waves on dynamic stress intensity factors (two dimensional problem),” Archive of Applied Mechanics, Vol. 66, pp.  326–335.

Wu, J. L., Chen, C. H. and Chen, C. F., (2001) “Numerical inversion of Laplace transform using Haar wavelet operational matrices,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 48, pp. 120-122.

Zakian V. and Coleman R., (1971) “Numerical inversion of rational Laplace transforms,” Electronics Letters, Vol. 7, pp. 777-778.

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, Vol. 41, pp. 3653-3674.