在岩石破壞力學的工程問題中，數值技術已成為不可或缺的工具。而許多的研究也著重在發展新的數值方法來求得應力強度因子，加上現實中具有多變的工程條件，使得發展一套新的數值方法或替代技術能夠有效以及準確的解決較具複雜的問題。而現今學者研究各種試驗方法中，目前較多圍繞在模態I、II型和I+II複合型裂縫破壞展開，對於模態III型裂縫破壞的研究還處於初級階段，僅有少量的文獻報導。但是岩體地下工程和邊坡工程中往往會發生模態III型加載下的岩石破壞，故發展模態III型破壞之應力強度因子的研究有利於暸解岩石破壞行為。
本研究提出一套單域邊界元素法(或稱對偶邊界元素法)，結合異向性線彈性理論，藉以Fortran語言撰寫成程式，針對異向性岩石於模態III型之破壞，探討岩石中裂縫前緣之應力強度因子。主要討論橫向等向性材料在不同裂縫長度、材料寬度、材料層面傾角以及上述情況於不同之異向性程度的影響下，其應力強度因子KI、KII、KIII的變化，並以KIII值做進一步的討論。
先利用加載裂縫位於中央的試體，發現與加載方向垂直以及平行的兩裂縫前緣，其應力強度因子KII、KIII的分佈為一拋物線。而加載裂縫位於邊緣的試體，發現當裂縫開裂的越深，或是長寬比較小的試體其應力強度因子的值(KII、KIII)較高，而結果也證明應力強度因子是一種抵抗的能力，越容易達到破壞的試體其應力強度因子越大。
最後，本研究提供各種試驗方法，其特色在於將模態I型及II型的破壞控制住，並且KI、KII值亦趨近於零，以達到接近純模態III型破壞之試驗，可供學者在進行III型試驗中做參考。

Numerical technique in rock fracture mechanics have become indispensable tools for solving all kinds of science and engineering problems. Extensive research has been carried out for the development of new numerical methods to determine the stress intensity factors (SIFs). Due to the varied engineering conditions in the practical area, it is imperative to develop new numerical methods or to explore alternative techniques for the purpose of solving the complicated problems and to improve the efficiency and accuracy of the existing or new numerical methods.

In recent years, scholars investigate methods of the various tests and focus on the mode-I, mode-II and mixed-mode I-II. The mode III fracture is still in the initial stage. However, the rock mass often occur in mode-III fracture under load in underground engineering and slope engineering. Thus, the development of mode-III SIF will help to study the behavior of rock fracture.

This thesis presents the dual boundary element methods (dual-BEM) or single-domain BEM to analyze anisotropic rocks in mode-III fracture and adopts the Fortran language to develop the numerical program. For anisotropic rocks of mode-III fracture, the SIF along the crack front can be discussed. The aim of these discussions are to compare the variation of SIFs (KI, KII and KIII) for different crack length, material width, material inclined angle and the anisotropic influence in transversely isotropic material. Further, the value of KIII can be discussed.

The results show that the SIFs (KII, KIII) is a parabolic distribution when the loading directs the vertical and parallel crack fronts crack in the rocks within a center crack. For the edge crack, the higher values of SIFs (KII, KIII) occur at the deeper crack or smaller ratios of length and width. The results also proved that the SIF is an ability to resist and the greater SIF occur at the easier arrives failure.

This study provides three-dimensional test method, and its characteristics is that control mode-I and mode-II fractures (i.e. KI, KII near to zero), then approximate the pure mode-III fracture test. Finally, this study can provide the reference for the mode-III test.

1. 王志、饒秋華、謝海峰 “脆性岩石反平面剪切(Ⅲ型)斷裂機理的有限元分析”，湖南理工學院學報，第20卷2期，75-77頁，2007。
2. 黎立云、寧海龍、許風光、王見強、李海雲、侯蘭英 “III型裂紋斷裂韌性測試及數值分析”，岩石力學與工程學報，第25卷12期，2523-2528，2006。
3. 許鳳光 “岩石裂紋的兩類剪切斷裂實驗與有限元研究”，碩士論文，北京科技大學，2003。
4. 陳家豪 “邊界元素法於異向性岩石破壞力學之研究”，博士論文，成功大學，2008。
5. 涂家輝 “異向性雙合成材料之破壞力學性質分析”，碩士論文，成功大學，2002。
6. 柯建仲 “利用邊界元素法探討異向性岩石之裂縫傳播路徑”，博士論文，成功大學，2007。
7. 帥玉康 “以平板彎曲試驗求取石材III型破裂韌度之研究”，碩士論文，成功大學，2008。
8. Abdul, M.M.J. and Fenner, R.T. Some boundary integral equation solutions for thrss-dimensional stress concentration problems, J. Strain Anal., 18(4), 207-215, 1983
9. Agrawal, A.K. Free surface effect on moving crack under impact loading by BEM, Engng. Anal. Boundary Elements, 26, 253-264, 1984.
10. Albuquerque, E.L., Sollero, P. and Fedelinski, P. Dual reciprocity boundary elemant method in Laplace domain applied to anisotropic dynamic crack problems, Comput. Struct., 81, 1703-1713, 2003.
11. Aliabadi, M. H., Fracture of Rock, WITpress, Boston, 1999.
12. Aliabadi, M.H. Boundary element formulations in fracture mechanics, Appl. Mech. Rev., 50, 83-96, 1997.
13. Al-Shayea, N.A., Khan, K. and Abduljauwad, S.N. Effects of confining pressure and temperature on mixed mode(I+II) fracture toughness of a limestone rock, Int. J. Rock Mech. Min. Sci., 37, 629-643, 2000.
14. Antonio, J. and Tadeu, A. 3D seimic response of a lmited valley vis BEM using 2.5D analytical Green’s function for an infinite free-rigid layer, Soil Dyn. Earthq. Engng., 22, 659-673, 2002.
15. Ariza, M.P. and Dominguez, J. Dynamic BE analysis of 3-D cracks in transversely isotropic solids, Comput. Meth. Appl. Mech. Engng., 193, 756-779, 2004.
16. Ariza, M.P., saez, A. and Dominguez, J. A singular element for three-dimensional fracture mechanics analysis, Engng. Anal. Boundary Elem., 20, 275-285, 1997.
17. Atkinson, C., Smelser, R.E. and Sanchez, J. Combined mode fracture via the cracked Brazilian disk test, Int. J. Frac., 18(4), 279-291, 1982.
18. Awaji, H. and Sato, S. Combined mode fracture toughness measurement by the disk tect, J. ENGNG. Materials Tech., 100, 175-182, 1978.
19. Ayatollahi, M.R. and Hashemi, R. Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching, Comput. Struct., 78, 602-609, 2007.
20. Blackburn, W.S. Three dimensional calculation of growth of cracks starting in parallel planes by boundary elements, Int. J. Fatigue, 21, 933-939, 1999.
21. Brebbia, C.A. The boundary element method for engineers, Pentech Press, Plymouth and Londton, 1978.
22. Carini, A. and Gioda, G. A boundary integral equation technique for visco-elastic stress analysis, Int. J. Num. Anal. Meth. Geom., 10, 585-608, 1986.
23. Chatterjee, J., Ma, F., Henry, D.P. and Banerjee, P.K. Two- and three-dimensional transient heat conduction and thermoelastic analyses by BEM via efficient time convolution, Comput. Meth. Appl. Mech. Engng., 196, 2828-2838, 2007.
24. Chen, C. S., Pan, E., and Amadei, B., Fracture mechanics analysis of cracked discs of anisotropic rock using the boundary element method. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 35, No.2, 195-218, 1998.
25. Chen, C.H., Chen, C.S. and Wu, J.H., Fracture toughness analysis on cracked ring disks of anisotropic rock, Rock Mech Rock Engng, 41(4), 539-562, 2007.
26. Chen, C.S., and Chen, C.H., and Pan, E. Three-dimensional stress intensity factors of a central square crack in a transversely isotropic cuboid with arbitrary material orientations, Engng. Anal. Boundary Elem., 33, 128-136, 2008.
27. Chen, C.S., Pan, E. and Amadei, B. Determination of deformability and tensile strength of anisotropic rock using Brazilian tests, Int. J. Rock Mech. Min. Sci., 35(1), 43-61, 1998.
28. Cisilino, A.P. and Ortiz, J. Boundary element analysis of three-dimensional mixed-mode cracks vie the interaction integral, Comput. Meth. Appl. Mech. Engng., 194, 935-956, 2005.
29. Cruse, T.A. Application of the boundary integral equation method to three-dimensional stress analysis, Comput. Struct., 3, 509-527, 1973.
30. Cruse, T.A. Numerical solutions in three dimensional elastostatics, Int. J. Solids Structures, 5, 1259-1274, 1969.
31. dell’Erba, D.N. and Aliabadi, M.H. On the solution of three-dimensional thermoelastic mixed-mode edge crack problems by the dual boundary element method, Engng. Fract. Mech., 66, 269-285, 2000.
32. Edward, W., Lee, J.Li., Evaluation of the Edge Crack Torsion(ECT) Test for Mode III Interlaminated Fracture Toughness of Laminated Composites, NASA Technical Memorandum 110264 U. S. Army Research Laboratory Technical Report, 1210, 1996.
33. Ehart, R.J.A., Stanzl-Tschegg, S.E., Tschegg, E.K., Crack face interaction and mixed mode fracture of wood composites during mode III loading, Engng Frac Mech, 61, 253-278, 1998.
34. Farshad, M., Wildenberg, M. W. and Flueler, P., Determination of shear modulus and Poisson's ratio of polymers and foams by the anticlastic plate-bending method, Materials and Structures Constructions, 30, 377-382, 1997.
35. Fenner, R.T. The boundary integral equation(boundary element) method in engineering stress analysis, J. Strain Anal., 18(4), 199-205, 1983.
36. Fredholm, E.I. Sur une class d’equations functionelles, Acta Math., 365-390, 1903.
37. Griffith, A.A., The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London, Series A., 221, 163-198, 1920.
38. Gual, L. Schanz, M. A comparative study of three boundary element approachs to calculate the transient response of viscoelastic solids with unbounded domains, Comput. Meth. Appl. Mech. Engng., 179, 11-123, 1999.
39. Guzina, B.B., Pak, R.Y.S. and Martinez-Castro, A.E. Singular boundary elements for thrss-dimensional elasticity problems, Engng. Anal. Boundary Elem., 30, 623-639, 2006.
40. Hatzigeorgiou, G.D. snd Beskos, D.E. Static analysis of 3D damage solids and stuctures by BEM, Engng. Anal. Boundary. Elem., 26, 521-526, 2002.
41. He, W.J., Lin, Y. and Ding, H.J. A three-dimensional formula for determing stress intensity factors in finite element analysis of cracked bodies, Engng. Fract. Meth., 57(4), 409-415, 1997.
42. Hiroshi Suemasu, An experimental method to measure the mode-III interlaminar fracture toughness of composite laminates, Composites Science and Technology, 59, 1015- 1021, 1998.
43. Inglis, C., Stresses in a plate due to the presence of cracks and sharp corners. Trans. Inst. Naval Architects., 55, 219-241, 1913.
44. Irwin, G. R., Analysis of stresses and strains near the end of a crack. Trans. ASME, J. Appl. Mech., 24, 361-364, 1957.
45. Jaswon, M.A. Integral equation methods in potential theory –I, Proc. Roy. Soc. Lond., A275, 23-32, 1963.
46. Kirsch, G., Die theorie der elastizitat und die bedurfnisse derfestigkeitslehre, Veit. Ver. Deut. Ing., 42, 797-807, 1898.
47. Kirsch, G., Z. Verein Deutscher Ing. (VDI)., 42, 113, 1898..
48. Kupradze, O.D. Potential methods in the theory of elasticity, Daniel Davy, New York, 1965.
49. Lazarus, V., Leblond, J.B. and Mouchrif, S.E. Crack front rotation and segmentation in mixed mode I+III or I+II+III. Part I: Calculation of stress intensity factors, J. Mech. Physics Solids, 49, 1399-1420, 2001.
50. Lazarus, V., Leblond, J.B. and Mouchrif, S.E. Crack front rotation and segmentation in mixed mode I+III or I+II+III. PartI: Calculation of stress intensity factors, J. Mech. Physic Solids, 49, 1399-1420, 2001.
51. Lekhnitskii, S.G. Theory of elasticity of an aniaotropic elastic body, translated by P. Fern, Holden-Day, San Francisco, 1963.
52. Leung, A.Y.T. and Su, R.K.L. A numberical study of singular stress field of 3D cracks, Finite Elem. Anal. Design, 18, 389-401, 1995.
53. Liang, L.H., Liu, Y, Jia, G.S. and Yang, J. Elastoplastic and creep line spring models for surface flaws by BEM, Int J. Pres. Ves. Piping, 76, 781-787, 1999.
54. Lim, I.L., Johnston, I.W. and Choi, S.K. Assessment of mixed mode fracture toughness testing methods for rock, Int. J. Rock. Nech. Min. Sci. Geom. Abs., 31(3), 265-272, 1994.
55. Liu, Y., Liang, L.H., Hong , Q.C. and Antes, H. Non-linear sirface crack analysis by thrss dimensional boundary element with mixed boundary conditions, Engng. Fract. Mech., 63, 413-424, 1999
56. Lo, S.H., Dong. C.Y. and Cheung, Y.K. Integral equation approach for 3D Mutiple-crack problems, Engng. Fract. Mech., 72, 1830-1840, 2005.
57. Marrin, L. Lesnic, D. Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition. Comput. Meth. Appl. Meth. Engng., 191, 3257-3270, 2002.
58. Mezrhab, A. and Bouzidi, M. Computation of view factors for surfaces of complex shape including screening effects and using a boundary element approximation, Engng. Comput., 22(2), 132-148, 2005.
59. Mukherjee, S. Shi. X. and Mukherjee, Y.X. Integral variables and their sensitivities in three- dimensional linear elasticity by the boundary contour method, Comput. Meth. Appl. Mech. Engng., 187, 289-306, 2000.
60. Muskhelishvili, N.I. Some basic problems of the mathematical theory of elasticity, Noordhoff, Holland, 1953.
61. Niu, Y. and Dravinski, M. Direct 3D BEM for scattering of elastic waves in a homogeneous anisotropic half-space, Wave Motion, 38, 165-175, 2003
62. Ong, E.T. and Lim, K.M. Three-dimensional singular boundary element for corner and edge singularities in potential problems, Engng. Anal. Boundary Elem., 29, 175-189, 2005.
63. Ouchterlony, F. Suggested methods for determining the fracture toughness of rock, Int. J. Rock. Mech. Min. Sci. Geom. Abs., 25(2), 69-71, 1988.
64. Pan and Yuan, F.G. Boundary element analysis of three-dimensional cracks in anisotropic solids, Int. J. Numer. Meth. Engng., 48, 211-237, 2000.
65. Pan, E. and Amadei, B. A 3-D boundary element formulation of anisotropic elasticity with gravity, Appl. Math. Model., 20, 114-120, 1996.
66. Pan, E. and Amadei, B., Boundary element analysis of fracture mechanics in anisotropic bimaterials. Engineering Analysis with Boundary Elements, 23, 683-691, 1999.
67. Pan, E. and Young, B. Three-dimensional interfacial Green’s functions in anisotropic biomaterials, Appl. Math. Model., 27, 307-326, 2003.
68. Pan, E., Sassolas, C., Amedei, B. and Pfeffer, W.T. A 3-D boundary element formulation of viscoelastic media with gravity, Comput. Mech., 19, 308-316, 1997.
69. Pan, E.and Amedei, B. A 3-D boundary element formulation of anisotropic elasticity with gravity, Appl. Math. Model., 20, 114-120, 1996.
70. Partheymuller, P. Haas, M. and Kuhn, G. Comparison of the basic and discontinuity formulation of the 3D-dual Boundary element method, Engng. Anal, Boundary Elem., 24, 777-788, 2000.
71. Polizzotto, C. A symmetric Galerkin boundary/domain element method for finite elastic deformations, Comput. Meth. Appl. Mech. Engng., 189, 481-514, 2000.
72. Popov, V., Power, H. and Walker, S.P. Numercial comparison between two possible mutipole alternative for the BEM solution of 3D elasticity problems based upon Taylor series expansions, Engng. Anal. Boundary Elem., 27, 521-531, 2003.
73. Qin, T.Y., Zhu, B.J. and Noda. N.A. Mixed-mode stress intensity factors of a three-dimensional crack in s bonded biomaterial, Engng. Comput., 25(3), 251-267, 2008.
74. Quinlan, P.M., Grannall, J.J., Atluri, S.N. and Fitzgerald, J.E. Boundary discretization using the edge-function method in three dimensional elasticity, Appl. Math. Model., 3, 18-24, 1979.
75. Rizzo F.J. and Shippy, D.J. An advanced boundary integral equation method for three-dimensional thermoelasticity, Int. J. Num. Meth. Engng., 11, 1753-1768, 1977.
76. Rizzo, F.J. An integral equation approach to boundary value problems of classical elasticitics, Q. Appl. Math., 25(1), 83-95, 1967.
77. Singh, D. and Shetty, D., Fracture toughness of polycrystalline ceramics in combined mode I and mode II loading. J. Am. Ceram. Soc., 72 No.1, 78-84, 1989.
78. Somigliana, C. Sopra l’equilibrio di un corpo elastico isotropo. I1 Nuovo Ciementi, 17-19, 1886.
79. Symm, G.T. Integral equation methods in potential theory –II, Proc. Roy. Soc. Lond. A275, 33-46, 1963.
80. Tan, C.L. and Fenner R.T. Three-dimensional stress analysis by the boundary integral equation method, J. Strain. Anal., 13, 213-219, 1978.
81. Tohgo. K., Otsuka. A. and Yuuki. R., Fatigue Crack Growth of a Mixed Mode Three-Dimensional Crack (1st Report, Analysis of Stress Intensity Factors of Mixed Mode Three-Dimensional Crack Based on the J-Integral Concept), Trans. Japan Soc. Mech. Engrs., 52, No.476 A, 909-918, 1986.
82. Tonon, F., Pan, E. and Amadei, B. Green’s functions and boundary element method formulation for 3D anisotropic media, Comp. Stuct., 79, 469-482, 2001.
83. Vogel, S.M. and Rizzo, F.J. An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems, J. Elasticity, 3(3), 203-216, 1973.
84. Wang, C.B. Chatterjee, J. Banerjee, P.K. An efficient implementation of BEM for two-and three-dimensional multi-region elastoplastic analyses, Comput. Meth. Appl. Meth. Engng., 196, 829-842, 2007.
85. Weaver, J. Three-dimensional crack analysis, Int. J. Solid. Struct., 13, 321-330, 1997.
86. Westergaard, H., Bearing pressures and cracks. Journal of Applied Mechanics., 61, 49-53, 1939.
87. Whittaker, B. N., Singh, R. N. and Sun, G., Rock Fracture Mechanics：Principles, Design and Applications., Elsevier, New York, 1992.
88. Whittaker, B.N., Singh, R.N. and Sun, G. Rock fracture mechanics: principles, design and applications, Elsevier, New York, 1992.
89. Williams, M. L., The stresses around a fault or crack in dissimilar media. Bulletin of Seismological Society of America., 49, No.2, 199-204, 1959.
90. Yue, Z.Q., Xiao, H.T. and Pan, E. Stress intensity factors of square crack inclined to interface of transversely isotropic bi-meterial, Engng. Anal. Boundary Elem., 31, 50-65, 2007.
91. Zhao, M.H., Fan, C.Y., Liu, T. and Yang, F. Extended displacement discontinuity Green’s functions for three-dimensional transversely isotropic magneto-electro-elastic media and applicatios, Engng. Anal. Boundary Elem., 31, 547-558, 2007.
92. Zhou, W., Lim, K.M., Lee, K.H. and Tay, A.A.O. A new variable-order singular boundary element for calculating stress intensity factors in three-dimensional elasticity problem, Int. J. Solids Struct., 42, 159-185, 2005.