基於二維等向性反平面之彈線性理論及複變理論的框架下，本文主旨在求解塗層多邊形孔洞受到無窮剪應力之應力強度因子解析，並對基材上的裂紋產生何種影響及變化進行討論。透過保角映射法將塗層孔洞轉換成同心圓孔洞，可將物理平面轉換至為數學平面，再利用解析連續以及交替法等方法計算出所需函數。研究中藉由界面上應力連續以及位移連續之條件，使用交替法反覆疊代求得完整之應力場級數解。再利用未定係數插植公式、疊加法滿足其沿著裂紋之差排分佈及表面無曳引力條件下，對應力勢能函數透過Gauss-Chebyshev數值積分求得差排密度，繼而求得應力強度因子。最後探討不同剪力模數、形狀因子及塗層形狀對應力強度因子之影響以了解塗層特性對含裂紋孔洞彈性體之整體穩定度評。

Based on two-dimensional isotropic anti-plane elasticity and complex variable theory, a general analytical solution to a coated arbitrary shape hole interacted with a crack embedded in an infinite matrix under a remote uniform shear load is provided in this article. According to the approach of conformal mapping and the method of analytical continuation theorem in conjunction with alternating technique, a convergent series solution in the reinforcement layer and the matrix for both the displacement and stress functions is expressed in terms of the corresponding homogeneous complex potential. By using the superposition method, a singular integral equation (SIE) satisfying the traction-free condition along the crack surface is then established. The effects of material properties and geometric configurations of a coating layer on the mode-III stress intensity factors (SIFs) are discussed in detail and shown in graphic form.

[1] Irwin, G. R. (1997). Analysis of stresses and strains near the end of a crack traversing a plate.
[2] Westergaard, H.M., Bearing pressures and cracks, Journal of Applied Mechanics, Vol. 6, pp. A49-53, 1939
[3] Atkinson, C. (1972). The interaction between a crack and an inclusion. International Journal of Engineering Science, 10(2), 127-136.
[4] Erdogan, F., & Gupta, G. (1971). The stress analysis of multi-layered composites with a flaw. International Journal of Solids and Structures, 7(1), 39-61.
[5] Erdogan, F., Gupta, G. D., & Ratwani, M. (1974). Interaction between a circular inclusion and an arbitrarily oriented crack.
[6] Muskhelishvili, N. I. (1963). Some basic problems of the mathematical theory of elasticity. Noordhoff, Groningen, 17404.
[7] Chen, Y. Z., & Cheung, Y. K. (1990). New integral equation approach for the crack problem in elastic half-plane. International Journal of Fracture, 46(1), 57-69.
[8] Cheung, Y. K., & Chen, Y. Z. (1987). Solutions of branch crack problems in plane elasticity by using a new integral equation approach. Engineering fracture mechanics, 28(1), 31-41.
[9] Chao, C. K., & Shen, M. H. (1995). Solutions of thermoelastic crack problems in bonded dissimilar media or half-plane medium. International Journal of Solids and Structures, 32(24), 3537-3554.
[10] Chao, C. K., and Wu, S. P., 1995, "Explicit Solutions for the Anti-plane Problem of Bonded Dissimilar Materials with Two Concentric Circular-Arc Inclusions", Journal of Energy Resources Technology, Transactions of the ASME, 117(1), pp. 1-6.
[11] Chao, C. K., & Kao, B. (1997). A thin cracked layer bonded to an elastic half-space under an anti-plane concentrated load. International journal of fracture, 83(3), 223-241.
[12] Lekhnitskii, S. G. Theory of Elasticity of an Anisotropic Elastic Body (Holden‐Day, San Francisco, 1963).
[13] Chao, C. K., & Young, C. W. (1998). On the general treatment of multiple inclusions in anti-plane elastostatics. International journal of solids and structures, 35(26-27), 3573-3593.
[14] Chao, C. K., & Chiang, T. F. (1996). Anti-plane interaction of an anisotropic elliptic inclusion with an arbitrarily oriented crack. International journal of fracture, 75(3), 229-245.
[15] Chao, C. K., Chen, F. M., & Lin, T. H. (2017). Stress intensity factors for fibrous composite with a crack embedded in an infinite matrix under a remote uniform load. Engineering Fracture Mechanics, 179, 294-313.
[16] Gong, S. X., & Meguid, S. A. (1994). A screw dislocation interacting with an elastic elliptical inhomogeneity. International Journal of Engineering Science, 32(8), 1221-1228.
[17] Wang, X., & Zhong, Z. (2003). A circular inclusion with a nonuniform interphase layer in anti-plane shear. International journal of solids and structures, 40(4), 881-897.
[18] Gong, S. X. (1994). Anti-plane interaction of line crack with an arbitrarily located elliptical inclusion. Theoretical and applied fracture mechanics, 20(3), 193-205.
[19] Chen, F. M., Chao, C. K., & Chen, C. K. (2011). Interaction of an edge dislocation with a coated elliptic inclusion. International Journal of Solids and Structures, 48(10), 1451-1465.
[20] Xiao, Z. M., & Chen, B. J. (2001). On the interaction between an edge dislocation and a coated inclusion. International Journal of Solids and Structures, 38(15), 2533-2548.
[21] Chao, C. K., & Wikarta, A. (2012). Mode-iii stress intensity factors of a three-phase composite with an eccentric circular inclusion. Computer Modeling in Engineering and Sciences, 84(5), 439.
[22] Chao, C. K., Tseng, S. C., & Chen, F. M. (2018). Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load. Journal of the Chinese Institute of Engineers, 41(7), 590-602.
[23] Yoshikawa, K., & Hasebe, N. (1999). Heat source in infinite plane with elliptic rigid inclusion and hole. Journal of engineering mechanics, 125(6), 684-691.
[24] Chao, C. K., Tseng, S. C., & Chen, F. M. (2018). Analytical solutions to a coated triangle hole embedded in an infinite plate under a remote uniform heat flow. Journal of Thermal Stresses, 41(10-12), 1259-1275.
[25] Tseng, S. C., Chao, C. K., & Chen, F. M. (2020). Stress Field for a Coated Triangle-Like Hole Problem in Plane Elasticity. Journal of Mechanics, 36(1), 55-72.
[26] Yang, Q. Q., Zhu, W. G., & Li, Y. (2017, October). Stress concentration analysis of an arbitrary shape hole coated by a functionally graded layer. In 2017 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications (SPAWDA) (pp. 293-297). IEEE.
 
[27] Wang, S., Xing, S., Chen, Z., & Gao, C. (2019). A nanoscale hole of arbitrary shape with surface elasticity. Journal of Elasticity, 136(2), 123-135.
[28] Tseng, S. C., Chao, C. K., Chen, F. M., & Chiu, W. C. (2020). Interfacial stresses of a coated polygonal hole subject to a point heat source. Journal of Thermal Stresses, 1-26.