臺灣博碩士論文加值系統

本文透過應力強度因子來探討無窮均質平板含兩相異圓形異質與裂紋交互作
用之平面彈性問題。首先利用保角映射法將兩相異圓形異質問題轉換成兩同心圓
異質，將實際之物理平面轉換成數學平面，搭配解析連續與交替法得到差排作用下
無窮平板含兩相異圓形異質平面彈性場之全場應力級數解。接著利用此差排沿著
裂紋邊界積分配合疊加法求得奇異積分方程式，藉由Muskhelishvili 等向性二維合
力公式結合疊加法與奇異積分方程法，以矩陣形式計算差排密度之數值繼而得到
應力強度因子。本研究為無窮均質平板含兩相異圓形異質與一平行於x 軸之裂紋
受到垂直均佈拉力之影響，純一型應力強度因子數值結果，可清楚解釋兩相異圓形
異質剛性變化，以及裂紋尖端與圓形異質界面距離改變之間影響性。

In this study, the solution of a crack interacting with two circular inclusions under a
remote uniform load is provided. First, based on the technique of conformal mapping and
the method of analytical continuation in conjunction with the alternating technique, the
complex potential functions of dislocation interacting with two circular inclusions are
obtained. Second, by using Muskhelishvili’s complex potentials together with
superposition technique, the derivation of logarithmic singular integral equations by
introducing the complex potential functions of dislocation along the crack border is
established. The stress intensity factors are then determined to investigate the interaction
of a crack with two circular inclusions based on linear elastic fracture mechanics.
Numerical results of the mode-I stress intensity factors are studied in detail which are
dependent on the mismatch in the material constants, the distance between the circular
interface and a crack.

摘要 ................................................................................................................................ i
ABSTRACT .................................................................................................................ii
誌謝 ............................................................................................................................. iii
目錄 .............................................................................................................................. iv
圖目錄 .......................................................................................................................... vi
符號索引 ...................................................................................................................... ix
第一章緒論 ................................................................................................................... 1
1.1 研究動機 ........................................................................................................... 1
1.2 文獻回顧 ........................................................................................................... 2
1.3 本文做法 ........................................................................................................... 4
第二章理論基礎 ........................................................................................................... 6
2.1 等向性二維彈性力學公式 ............................................................................... 6
2.2 均質解 ............................................................................................................... 6
2.3 疊加法 ............................................................................................................... 7
2.4 保角映射法 ....................................................................................................... 7
2.5 解析連續與交替法 ........................................................................................... 9
2.5.1 解析函數 ....................................................................................... 9
2.5.2 連續定理 ..................................................................................... 10
2.5.3 交替法過程 ................................................................................. 11
2.6 數值求解方法 ................................................................................................. 12
2.6.1 插值公式 ...................................................................................... 12
2.6.2 奇異積分方程正解 ...................................................................... 12
2.6.3 Gauss-Chebyshev 積分法則 ........................................................ 13
2.7 應力強度因子 ................................................................................................. 13
第三章 兩相異圓形異質之平面彈性場通解 ........................................................... 18
3.1 奇異點與雙異質圓介質問題描述 ................................................................. 18
3.2 兩相異圓形異質應力函數推導 ..................................................................... 19
第四章數值求解方法與過程 ..................................................................................... 30
4.1 問題描述 ......................................................................................................... 30
4.2 奇異積分方程 ................................................................................................. 31
第五章數值結果與討論 ............................................................................................. 39
第六章結論與未來展望 ............................................................................................. 51
6.1 結論 ................................................................................................................. 51
6.2 未來展望 ......................................................................................................... 52
參考文獻 ..................................................................................................................... 53

[1] G. R. Irwin, Analysis of stresses and strains near the end of a crack traversing a plate, Journal of Applied Mechanics, Vol. 24, pp. 361-364, 1957.
[2] Y. Wang, SIF calculation of an internal crack problem under anti-plane shear,Computers & Structures, Vol. 48, No. 2, pp. 291-295, 1993.
[3] Y. Z. Chen and N. Hasebe, Dependence of stress intensity factor on loading condition, Engineering Fracture Mechanics, Vol. 55, No. 3, pp. 355-361, 1996.
[4] Z. M. Xiao and B. J. Chen, Stress intensity factor for a Griffith crack interacting with a coated inclusion, International Journal of Fracture, Vol. 108, pp. 193-205,2001.
[5] N. L. Muskhelishvili, Some Basic Problems of the Mathematical Theory of
Elasticity,Noordhoff,Groningen, 1953.
[6] Y. Z. Chen and Y. K. Cheung, Solutions of branch crack problems in plane
elasticity by using a new integral equation approach, Engineering Fracture
Mechanics, Vol. 28, pp.31-41, 1987b.
[7] Y. Z. Chen and Y. K. Cheung, New integral equation for plane elasticity crack problems, Theoretical and Applied Fracture Mechanics, Vol. 7, pp. 177-184, 1987a.
[8] Y. Z. Chen and N. Hasebe, Stress-intensity factors for curved circular crack in bonded dissimilar materials, Theoretical and Applied Fracture Mechanics, Vol. 17,pp. 189-196, 1992.
[9] Y. Z. Chen, New integral equation for curve crack problem in plane elasticity with arbitrary loading condition, International Journal of Fracture, Vol. 46, pp. R43-R46, 1990.
[10] Y. Z. Chen and R. S. Chen, Interaction between curved crack and elastic inclusion in an infinite plate, Archive of Applied Mechanics, Vol. 67, pp. 566-575, 1997.
[11] K. Y. Lam, P. P. Ong and N. Wude, Interaction between a circular inclusion and a symmetrically branched crack, Theoretical and Applied Fracture Mechanics, Vol.28, pp. 197-211, 1998.
[12] S. T. Shiue and S. Lee, The elastic interaction between screw dislocations and crack emanating from an elliptic hole, Journal of Applied Physics, Vol. 64, No. 1, pp.129-139, 1988.
[13] S. D. Wang, C. T. Hu and S. Lee, Screw dislocations nears a cross crack, Physica Status Solidi(a),Vol. 132, pp. 281-294, 1992.
[14] G. Anlas and M. H. Santare, Arbitrarily oriented crack Inside an elliptical inclusion,Journal of Applied Mechanics, Vol. 60, pp. 598-594, 1993.
[15] C. Y. Dong and K. Y. Lee, A new integral equation formulation of two-dimensional inclusion–crack problems, International Journal of Solids and Structures, Vol.42,pp. 5010–5020, 2005.
[16] C. K. Chao and M. H. Shen, Solutions of thermoelastic crack problems in bonded dissimilar media or half-plane medium, International Journal of Solids and Structures, Vol.32, pp. 3537-3554, 1995.
[17] C. K. Chao and J. Y. Lee, Interaction between a crack and a circular elastic inclusion under remote uniform heat flow, International Journal of Solids and Structures, Vol.33, pp. 3865-3880, 1996
[18] C. K. Chao and A. Wikarta, Mode-III stress intensity factors of a three-phase composite with an eccentric circular inclusion, Theoretical and Applied Fracture Mechanics, Vol. 53, pp. 205–210, 2010.
[19] C. K. Chao and A. Wikarta, Mode-III stress intensity factors of a three-phase composite with an eccentric circular inclusion, Computer Modeling in Engineering and Sciences, Vol.84, No.5, pp.439-458, 2012.
[20] C. K. Chao and A. Wikarta, Anti-plane interaction between a crack and anelliptically cylindrical layered media, Journal of Mechanics, Vol. 29, No. 1, pp.85-93, 2013.
[21] C. K. Chao and A. Wikarta, Solutions of a crack interacting with a three-phase composite inplane elasticity, Applied Mathematical Modelling, Vol. 40, pp.2454–2472, 2016.
[22] J. Li, Q. Fang and Y. Liu, Crack interaction with a second phase nanoscale circular inclusion in an elastic matrix, International Journal of Engineering Science, Vol.72, pp.89-97, 2013.
[23] P. B. N. Prasad, N. Hasebe, X. F. Wang and Y. Shirai, Green’s functions for a bimaterial problem with interfacial elliptical rigid inclusion and applications to crack and thin rigid line problems, International Journal of Solids and Structures, Vol.42, pp. 1513-1535, 2005.
[24] N. Hasebe, X. F. Wang, T. Saito and W. Sheng, Interaction between a rigid inclusion and a line crack under uniform heat flux, International Journal of Solids and Structures, Vol. 44, pp. 2426-2441, 2007.
[25] C. Y. Dong and K. Y. Lee, Boundary element analysis of an infinite anisotropic elastic medium containing inclusions and cracks, Engineering Analysis withBoundary Elements, Vol. 29, pp. 562–569, 2005.
[26] 李朝祥，「差排於異向彈性楔型體之分析」，碩士論文，國立台灣大學，台北，
1997
[27] 呂欣泰，「差排於異向性材料層域之全場解析與映射力硏究」，碩士論文，國
立台灣大學，台北，2001
[28] 陳富謀，「含直線或圓形界面異質之熱彈性問題解析」，博士論文，國立台灣
科技大學，台北，2005
[29] 陳錦坤，「橢圓形異質界面之彈性問題解析」，博士論文，國立台灣科技大學，
台北，2010
[30] Alief Wikarta，「含直線或圓形界面多層異質與裂紋交互作用之彈性問題解
析」，博士論文，國立台灣科技大學，台北，2013
[31] 侯金廷，「兩相異圓形異質之彈性問題解析」，碩士論文，國立台灣科技大學，
台北，2014
[32] 楊士玄，「兩相異圓形異質之點熱源彈性問題解析」，碩士論文，國立台灣科
技大學，台北，2015