摘 要

本研究以異向性線彈性理論配合雙合成材料之基本解、邊界積分方程式、裂縫尖端模式及最大張應力準則為理論基礎，藉以Fortran語言撰寫成BEM分析程式，其可成功的分析均向性及異向性雙合成材料，裂縫尖端之應力強度子、初始開裂角度與裂縫傳播路徑等裂縫開裂問題。為檢驗數值分析結果之可靠度，特別設計一雙合成材料巴西圓盤試體(水泥-石膏巴西圓盤，簡稱CG-Disk)，進行混合模態載重之巴西試驗，並藉以高速攝影設備拍攝試體詳細破壞過程，以了解裂縫尖端動態行為及探討其破壞機制與裂縫傳播過程。其間發現實驗與BEM分析之比較結果非常吻合，且可成功地求得混合模態載重下裂縫尖端之應力強度因子與初始開裂角度，並證實本分析程式可準確預測裂縫實際之開裂與傳播行為。分析不同異向性程度之雙合成材料巴西圓盤，發現應力強度因子亦受材料異向性程度不同而有顯著的影響。

ABSTRACT

This study presents a single-domain boundary element method (BEM) for linear elastic fracture mechanics analysis in the 2-D anisotropic bi-material. In this formulation, the displacement integral equation is collocated on the uncracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. The complete fundamental solution (Green’s function) for anisotropic bi-materials was also derived and implemented into the boundary integral formulation so the discretization along the interface can be avoided except for the interfacial crack part. A special crack-tip element was introduced to capture exactly the crack-tip behavior. A computer program with the FORTRAN language has been developed to effectively calculate the stress intensity factors, crack initiation angle, and propagation path of an anisotropic bi-material. This BEM program has been verified having a good accuracy with the previous researches. In addition, a gypsum-cement disk (G-C disc) specimen with a central crack was made to conduct the Brazilian test under diametrical loading. The result shows that the numerical analysis can predict relatively well the direction of crack initiation and the path of crack propagation.

參考文獻
1. 柯建仲，異向性大理岩之破壞力學性質分析，國立成功大學資源工程學系碩士論文，(2000)。
2. Aliabadi, M. H., Fracture of Rock, WITpress, Boston, (1999)
3. Abdi, R. E., A special finite element for the analysis of the singularity in a biomaterial containing a crack perpendicular to the interface. Engineering Fracture Mechanics, Vol. 39, No. 6, pp. 1061-1065, (1991).
4. Atkinson, B. K., Fracture Mechanics of Rocks : Experimental fracture mechanics data of rock and minerals., Academic press, London, (1987).
5. Amadei, B., Rock Anisotropy and the Theory of Stress Measurements, Springer-Verlag, New York, (1983).
6. Atkinson, C., Smelser, R. E. and Sanchez, J., Combined mode fracture via the cracked Brazilian disk test. Int. J. Fracture, Vol. 18, No. 4, pp. 279-291, (1982).
7. Awaji, H. and Sato, S., Combined mode fracture toughness measurement by the disk test. J. Eng. Mater. Tech. Trans. ASME, Vol. 100, No. 4, pp. 175-182, (1978).
8. Bahar, L. Y., On an elastostatic problem in nonhomogeneous media leading to coupled dual integral equation. Ph.D dissertation, Department of Civil Eng., University of Lehigh, (1996).
9. Bhattacharya, J. P. and Walker, R. G., Facies Models Response to sea level change. Geol. Assoc. Canada., pp. 157-178, (1992).
10. Banerjee, P. K. and Butterfield, R., Boundary element methods in engineering science., McGraw-Hill, New York, (1981).
11. Brown, E. T., Rock characterization testing and monitoring : ISRM suggested methods, Pergamon press, Oxford, (1981).

12. Bieniawski, Z. T. and Hawkes, I., Suggested methods for determining tensile strength of rock materials. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 15, pp. 99-103, (1978)
13. 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., Vol. 35, No. 2, pp. 195-218, (1998).
14. Chen, C. S., Characterization of deformability, strength, and fracturing of anisotropic rocks using Brazilian tests. Ph.D. thesis, Department of Civil Eng., University of Colorado,(1996).
15. Chen, C. S. and Hsu, J., The stress intensity factors of slightly undulating interface cracks of bimaterials. Int. J. Fracture, Vol. 80, pp. 277-293, (1996).
16. Cerrolaza, M. and Alarcon, E., A bi-cubic transformation for the numerical evaluation of the Cauchy principal value integrals in boundary methods. Int. J. Numer. Methods Engng., Vol. 28, pp. 987-999, (1989).
17. Christensen, R.M. and Wu, E.M., A theory of crack growth in viscoelastic materials. Engineering Fracture Mechanics, Vol.14, pp. 215–225., (1981).
18. Comninou, M. and Dundurs, J., On the behaviour of interface crack, Research Mechanica, pp. 249-264, (1980).
19. Christensen, R.M., A rate-dependent criterion for crack growth. International Journal of Fracture, Vol. 15(1), pp. 3–21. (1979).
20. Comninou, M. and Schmueser, D., The interface crack in a combined tension-compression and shear field. Trans. ASME, J. Appl. Mech. Vol. 46, pp. 345-348, (1979)
21. Cherepanov, G. P., Mechanics of Brittle Fracture, McGraw-Hill, New York, (1979)
22. Comninou, M., The interface crack. Trans. ASME, J. Appl. Mech. Vol. 44, pp. 631-636, (1977).
23. Cook, T. S. and Erdogan, F., Stresses in bonded materials with a crack perpendicular to the interface. Int. J. Engng Sci., Vol. 10, pp. 677-697, (1972).
24. Elliott, T., Sedimentary environments and facies. Blackwell, pp. 113-154, (1986).
25. England, A. H., A crack between dissimilar media. Trans. ASME, J. Appl. Mech. Vol. 32, pp. 400-402, (1965).
26. Erdogan, F. and Sih, G. C., On the crack extension in plates under loading and transverse shear. J. Basic Eng., Vol. 85, pp. 519-527, (1963).
27. Erdogan, F., Stress distribution in bonded dissimilar materials with crack. Trans. ASME, J. Appl. Mech., Vol. 32, pp. 403-410, (1959).
28. Fischer M. P., Elsworth, D., Alley, R. B. and Engelder, T., Finite element analysis of the modified ring test for determining mode I fracture toughness. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 33, No. 1, pp. 1-15, (1996).
29. Fowell, R. J. and Xu, C., The use of the cracked Brazilian disc geometry for rock fracture investigations. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., Vol. 31, No. 6, pp. 571-579, (1994).
30. Guo, H., Aziz, N. I. and Schmidt, L. C., Rock fracture-toughness determination by the Brazilian test. Engineering Geology., Vol. 33, pp. 177-188, (1993).
31. Gao, H., Abbudi, M. and Barnett D. M., Interfacial crack-tip field in anisotropic elastic solids. J. Mech. Phys. Solids, Vol. 40, pp. 393-416, (1992).
32. Gramberg, J., A Non-conventional View on Rock Mechanics and Fracture Mechanics, Published of the Commission of the European Communities by A. A. Balkema, Rotterdam, (1989).
33. Gandhi, K. R., Analysis of inclined crack centrally placed in an orthotropic rectangular plate. J. Strain analysis, Vol. 7, No. 3, pp. 157-162, (1972).
34. Griffith, A. A., The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. London, Series A, Vol. 221, pp. 163-198, (1920).
35. Hamoush, S. A. and Ahmad, S. H., Mode I and mode II stress intensity factors for interfacial cracks in bi-material media. Engineering Fracture Mechanics, Vol. 33, No. 3, pp. 421-427, (1989).
36. Hutchinson, J. W., Mear, M. E., and Rice, J. R., Crack paralleling an interface between dissimilar materials. Trans. ASME, J. Appl. Mech., Vol. 54, pp. 828-832, (1987).
37. Huang, C. Y. and Cheng, Y. M., Oligocene and Miocene planktic foraminiferal biostratigraphy of northern Taiwan. Proc. Geol. Soc. China., Vol. 26, pp. 21-56, (1983).
38. Hondros, G., The evaluation of Poisson’s ratio and the modulus of materials of a low tensile resistance by the Brazilian (Indirect tensile) test with particular reference to concrete. Aust. J. App. Sci., Vol. 10, pp. 428-434, (1959).
39. Isida, M. and Noguchi, H., Formulae of stress intensity factors of branched cracks in plane problems. Trans. Japan Soc. Mech. Engrs., Vol. 49, No. 440, pp. 469-479, (1983).
40. Isida, M. and Noguchi, H., Plane elastostatic problems of bonded dissimilar materials with an interface crack and arbitrarily located cracks. Trans. Japan Soc. Mech. Engrs., Vol. 49, pp. 137-146, (1983).
41. Isida, M. and Noguchi, H., Plane problems of arbitrarily oriented cracks in bonded dissimilar materials. Trans. Japan Soc. Mech. Engrs., Vol. 49, pp. 36-45, (1983).
42. Isida, M., Analysis of stress intensity factors of plate with a arbitrary array crack and bent crack. Trans. Japan Soc. Mech. Engrs., Vol. 44, No. 380, pp. 1122-1133, (1978).

43. Isida, M., Arbitrary loading problems of doubly symmetric regions containing a central crack. Engineering Fracture Mechanics, Vol. 7, pp. 505-514, (1975).
44. Irwin, G. R., Analysis of stresses and strains near the end of a crack. Trans. ASME, J. Appl. Mech., Vol. 24, pp. 361-364, (1957).
45. Inglis, C., Stresses in a plate due to the presence of cracks and sharp corners. Trans. Inst. Naval Architects, Vol. 55, pp. 219-241, (1913).
46. Kirsch, G., Die theorie der elastizitat und die bedurfnisse der festigkeitslehre, Veit. Ver. Deut. Ing., Vol. 42, pp. 797-807, (1898).
47. Kirsch, G., Z. Verein Deutscher Ing. (VDI), Vol. 42, pp.113 (1898).
48. Kasano, H., Watanabe, T., Matsumoto, H., and Nakahara, I., Singular stress fields at the tips of a crack normal to the bimaterial interface of isotropic and anisotropic half planes. Trans. Japan Soc. Mech. Engrs., Vol. 52, No. 476, pp. 933-939, (1986).
49. Kaya, A. C. and Erdogan, F., Stress intensity factors and COD in an orthotropic strip. Int. J. Fracture, Vol. 16, No. 2, pp. 171-190, (1980).
50. Lahiri, S., Sankar, B. V., and Mataga, P. A., Evaluation of biomaterial stress intensity factors using a finite element-boundary element alternating method. Engineering Fracture Mechanics, Vol. 53, No. 2, pp. 289-302, (1996).
51. Lee, K. Y. and Choi, H. J., Boundary element analysis of stress intensity factors for biomaterial interface cracks. Engineering Fracture Mechanics, Vol. 29, No. 4, pp. 461-472, (1988).
52. Libatskii, L. L. and Kovchik, S. E., Fracture of discs containing cracks. Soviet Materials Science, Vol. 3, No. 4, pp. 334-339, (1967)
53. Lekhnitskii, S. G., Theory of elasticity of an anisotropic elastic body. translated by P. Fern, Holden-Day Inc., San Francisco, (1963).
54. Lekhnitskii, S. G., Anisotropic plates, translated by S.W. Tsai, Gordon and Breath, (1957).
55. Murakami, Y., A simple procedure for the accurate determination of stress intensity factors by finite element method. Engineering Fracture Mechanics, Vol. 8, pp. 643-655, (1976).
56. Malyshev, B. M. and Salganik, R. L., The strength of adhesive joints using the theory fracture. Int. J. Fracture, Vol. 1, pp. 114-128, (1965).
57. Muskhelishvili, N. I., Some basic problems of the mathematical theory of elasticity. Translated by J.R.M. Redok, Noordhoff (1953)
58. Nengquan, L., Nicholas, J., Altiero, N. J., Sur, U., An alternative integral equation approach applied to kinked cracked in finite plane bodies. Comp. Methods in Appl. Mech. and Eng., Vol. 84, pp. 211-226,(1990).
59. Prabhakar, R. M. and Hareesh, V. T., Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. International Journal of Solids and Structures, Vol. 37, pp. 5353-5370, (2000)
60. Pan, E. and Amadei, B., Boundary element analysis of fracture mechanics in anisotropic bimaterials. Engineering Analysis with Boundary Elements, Vol. 23, pp. 683-691, (1999).
61. Palaniswamy, K. and Knauss, W. G., Propagation of a crack under general, in-plane tension. Int. J. Fracture, Vol. 8, pp. 114-117, (1972).
62. Raveendra, S. T. and Banerjee, P. K., Computation of stress intensity factors for interfacial cracks. Engineering Fracture Mechanics, Vol. 40, No. 1, pp. 89-103, (1991).
63. Rice, J. R., Elastic fracture mechanics concepts for interfacial crack. Trans. ASME, J. Appl. Mech., Vol. 55, pp. 98-103 (1988).
64. Richard, H. A., Examination of brittle fractured criteria for overloapping mode I and mode II loading applied to cracks. In Application of Fracture Mechanics to Materials and Structures. ed. G. C. Sih et al. Martinus Nijhoff Pub., The Hague, pp. 309-316, (1984).

65. Rooke, D. P. and Tweed, J., The stress intensity factors of a radial crack in a point loaded disc. Int. J. Engng. Sci., Vol. 11, pp. 285-290, (1973).
66. Rice, J. R., A path independent integral and the approximate analysis of strain concentration by notches and cracks. Trans. ASME, J. Appl. Mech., Vol. 35, pp. 379-386, (1968).
67. Rizzo, F. J., An integral approach to boundary value problems of classical elastostatics. Q. appl. Math., Vol. 25, pp. 83-95, (1967).
68. Rice, J. R. and Sih, G., Plane problems of cracks in dissimilar media. Trans. ASME, J. Appl. Mech., Vol. 32, pp. 418-423, (1965).
69. Soh, A. K., An improved photo-elastic technique for determining the mixed-mode stress-intensity factors for interfacial cracks in a bi-material. Composites Science and Technology, Vol. 59, pp. 1033-1039, (1999).
70. Sollero, P., Aliabadi, M. H. and Rooke, D. P., Anisotropic analysis of crack emanating from circular holes in composite laminates using the boundary element method. Engineering Fracture Mechanics, Vol. 49, No. 2, pp. 213-224, (1994).
71. Suo, Z., Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. R. Soc. Lond. A, Vol. 427, pp. 331-358, (1990).
72. Singh, D. and Shetty, D., Fracture toughness of polycrystalline ceramics in combined mode I and mode II loading. J. Am. Ceram. Soc., Vol. 72 No. 1, pp. 78-84, (1989).
73. Schapery, R.A. A theory of crack initiation and growth in viscoelastic media – Parts I, II, III. International Journal of Fracture. Vol.11, pp.141-159, 369-388, 549-562. (1975).
74. Sih, G. C., Strain-energy density factor applied to mixed mode crack problems. Int. J. Fracture, Vol. 10, No. 3, pp. 305-321, (1974).

75. Sih, G. C., Handbook of Stress Intensity Factors, Inst. Of Fracture and Solid Mechanics, Lehigh University, Bethlehem, Pennsylvania, (1973).
76. Sih, G. C., Paris, P. C. and Irwin, G. R., On cracks in rectilinearly anisotropic bodies. Int. J. Fracture, Vol. 3, pp. 189-203, (1965).
77. Sih, G. C., Paris, P. C. and Erdogan, F., Crack tip stress-intensity factors for plane bending problems. Trans. ASME, J. Appl. Mech, Vol. 29, pp. 306-312, (1962).
78. Ting, T. C. T., Anisotropic elasticity:theory and application, University Press, New York: Oxford, (1996).
79. Tsamasphyros, G. and Dimou, G., Gauss quadrature rules for finite part integrals. Int. J. Numer. Methods Engng, Vol. 30, pp. 13-26, (1990).
80. Tan, C. L. and Gao, Y. L., Treatment of biomaterial interface crack problems using the boundary element method. Engineering Fracture Mechanics, Vol. 36, No. 6, pp. 919-932, (1990).
81. Tada, H., Paris, P. C. and Irwin, G. R., The Stress Analysis of Cracks, Handbook, Del Research Corporation., (1987).
82. Timoshenko, S. and Goodier, J. N., Theory of Elasticity, McGraw-Hill, New York, N. Y., (1970).
83. Vallejo, L. E., The brittle and ductile behavior of a material containing a crack under mixed-mode loading. Proc. 28th U.S. Symp. Rock Mech., University of Arizona, Tucson, pp.383-390, (1987)
84. Vutukuri, V. S., Lama, R. D. and Saluja, S. S., Handbook on mechanical properties of rocks. Vol. I, (1974).
85. Whittaker, B. N., Singh, R. N. and Sun, G., Rock Fracture Mechanics ：Principles, Design and Applications., Elsevier, New York, (1992).
86. Woo, C. W. and Wang, Y. H., Analysis of an edge crack in a finite bi-material plate. Engineering Fracture Mechanics, Vol. 42, No. 2, pp. 289-297, (1992).

87. Wu, K. C., Stress intensity factor and energy release rate for interfacial cracks between dissimilar anisotropic materials. Trans. ASME, J. Appl. Mech, Vol. 57, pp. 882-886, (1990).
88. Woo, C. W. and Ling, H. L., On angle crack initiation under biaxial loading. J. Strain Anal., Vol. 19, pp. 51-59, (1984).
89. Williams, M. L., The stresses around a faule or crack in dissimilar media. Bulletin of Seismological Society of America, Vol. 49, No. 2, pp. 199-204, (1959).
90. Williams, M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech., Vol. 19, pp. 526-528, (1952)
91. Westergaard, H., Bearing pressures and cracks. Journal of Applied Mechanics, Vol. 61, pp. 49-53, (1939).
92. Yuuki, R. and Cho, S. B., Efficient boundary element analysis of stress intensity factors for interface cracks in dissimilar materials. Engineering Fracture Mechanics, Vol. 34, pp. 179-188, (1989).
93. Yuuki, R. and Kitagawa, H., Stress intensity factors for branched cracks in a two-dimensional stress state. Trans. Japan Soc. Mech. Engrs., Vol. 41, No. 346, pp. 1641-1649, (1975).