本論文主要目的在於分析探討含有單一嵌入式裂縫功能梯度材料條板面外問題的破壞分析。依據彈性材料相關之理論基礎，運用傅立葉轉換法，將此混合邊界值問題推導出一組奇異積分方程式，再藉由Gauss- Chebyshev多項式技術化為代數聯立方程組，以求得應力強度因子之數值解。從數值結果分別討論條板邊界條件、裂縫長度與位置及材料非均質參數對應力強度因子的影響。另外，本論文推導所得之力場解可以簡化至數個文獻上已經發表之案例，藉此證明本論文結果之正確性。

This thesis deals with the fracture behavior of a functionally graded material strip with an embedded crack under anti-plane shear loads. Using the theory of Elasticity and Fourier transform, the field equations with mixed boundary condition can be transformed into a system of integral equation and then solved numerically by employing the Gauss-Chebyshev polynomials. The results show the effects of boundary conditions, crack geometry and nonhomogeneous parameters on the stress intensity factors. In additions, the derived elastic field solutions can be reduced to several simple problems and compared well with the results of previous studies.

[1] Lee, W. Y., Bae, Y. W., Berndt, C. C., Erdogan, F, Lee, Y. D. and Mutasim, Z., The concept of functionally gradient materials for advanced thermal barrier coating applications: a review. Journal of the American Ceramic Society 79, 1996.
[2] Cherradi, N., Kawasaki, A. and Gasik, M., Worldwide trends in functionals research and development. Composites Engineering 4, 883-894, 1994.
[3] 王保林, 杰才, 幸. 非均材料力. 科出版社, 北京市, 2003.
[4] Chue, C. H. and Ou, Y. L., Mode III crack problems for two bonded functionally graded piezoelectric materials. International Journal of Solids and Structures 42, 3321-3337, 2005.
[5] Chue, C. H. and Ou, Y. L., Mode III eccentric crack in a functionally graded piezoelectric strip. International Journal of Solids and Structures 43, 6148-6164, 2006.
[6] Chue, C. H. and Ou, Y. L., Two mode III internal crack located within two bonded functionally graded piezoelectric half planes respectively. Arch Appl Mech 75, 364-378, 2006.
[7] Erdogan, F., The crack problem for bonded nonhomogeneous materials under antiplane shear loading. Transactions of the ASME, Journal of Applied Mechanics 52, 823-828, 1985.
[8] Jin, Z. H. and Noda, N., Crack tip singular fields in nohomogeneous materials. ASME Journal of Applied Mechanics 61, 738-740, 1994.
[9] Chen, Y.F. and Erdogan, F., The interface crack problem for a nonhomogeneous coating bounded to a homogeneous substrate. Journal of the Mechanics and Physics of Solids 44, 771-787, 1996.
[10] Gu, P. and Asaro, R. J., Cracks in functionally graded materials. International Journal of Solids and Structures 34, 1-17, 1997.
[11] Honein, T. and Herrmann, G., Conservation laws in nonhomogeneous plane elastostatics. Journal of the Mechanics and Physics of Solids 45, 789-805, 1997.
[12] Ueda, S. and Shindo, Y., Crack kinking in functionally graded materials due to an initial strain resulting from stress relaxation. Journal of Thermal Stresses 23, 285-290, 2000.
[13] Prabhakar, R. M. and Tippur, H. V., Numerical analysis of crack-tip fields in functionally graded materials with a crack normal to the elastic gradient. International Journal of Solids and Structures 37, 5353-5370, 2000.
[14] Shul, C. W. and Lee, K. Y., A subsurface eccentric crack in a functionally graded coating layer on the layered half-space under an anti-plane shear impact load. International Journal of Solids and Structures 39, 2019-2029, 2002.
[15] Wang, B. L., Mai, Y. W. and Sun, Y. G., Anti-plane fracture of a functionally graded material strip. European Journal of Mechanics A/Solids 22, 357-368, 2003.
[16] Wang, B. L., Mai, Y. W. and Noda, N., Fracture mechanics analysis models for functionally graded materials with arbitrarily distributed properties (Modes II and III problems). International Journal of Fracture 126, 307–320, 2004.
[17] Dag, S., Mixed-Mode Fracture Analysis of Functionally Graded Materials Under Thermal Stresses: A New Approach Using Jk-Integral. Journal of Thermal Stresses 30, 269–296, 2007.
[18] Zhong, Z., and Cheng, Z., Fracture analysis of a functionally graded strip with arbitrary distributed material properties. International Journal of Solids and Structures 45, 3711-3725, 2008.
[19] Erdogan, F., and Ozturk, M., Diffusion problems in bonded nonhomogeneous materials with an interface cut. International Journal of Engineering Science 30, 1507–1523, 1992.
[20] Muskhelishvili, N. I., Singular Integral Equations. Noordhoff International Publishing, Groningen, The Netherlands, 1953.
[21] Erdogan, F, Gupta, G. D. and Cook, T. S., Numerical solution of singular integral equations. In Mechanics of Fracture 1: Method of analysis and solution of crack problem, edited by G. C. Sih, Chapter 7, Noordhoff International Publishing, Leyden, The Netherlands, 1973.
[22] Erdogan, F. and Gupta, G. D., On the numerical solution of singular integral equations. Quarterly of Applied Mathematics 30, 525-534, 1972.
[23] Rivlin, T. J., The Chebyshev polynomials. Wiley, New York, 1974.