在二維熱彈性理論和複變理論的框架下，本文主要在求解嵌於無窮平板的對稱翼型裂紋承受遠端均勻拉伸和熱通量負載下之應力強度因子。先利用保角映射法將對稱翼型裂紋轉換成單位圓孔洞，再藉由解析連續法和介面上的絕熱和無曳引力之邊界條件計算出所需的溫度和應力函數。首先，分別求解對稱翼型裂紋僅承受遠程均勻拉伸或熱通量負載下之應力函數。接著，透過疊加原理將分別兩種不同形式負載的應力函數和應力強度因子疊加。在求得對稱翼型裂紋同時承受遠端均勻拉伸和熱通量之應力強度因子後，便可透過應變能密度準則計算出裂紋的破裂角度。最後，將探討不同方向、大小之拉伸負載和熱通量對於應力強度因子和破裂角度之影響。本文會逐一討論各組合的應力強度因子和破壞機制並藉由應變能密度等高線來做破裂角度的驗證。

Based on the two-dimensional thermoelasticity and complex variable theory, the stress intensity factors (SIFs) for a symmetrical airfoil crack embedded in an infinite matrix under a remote uniform tensile load and a remote uniform heat flow are primarily solved in this study. The conformal mapping method is applied to convert a symmetrical airfoil crack into a unit circular hole and the analytical continuation theorem is then used to obtain the undetermined temperature and stress functions with the adiabatic and traction-free condition. The stress function of a symmetrical airfoil crack subject to a remote uniform tensile load and a remote uniform heat flow is obtained respectively. The principle of superposition is subsequently utilized to superimpose the stress functions and SIFs of the two different loading conditions. After the SIFs are calculated, the strain energy density criterion is employed to determine the fracture angle of the crack. Finally, the effect of different orientations and magnitudes of applied tensile load and heat flux on the SIFs and fracture angle will be investigated. This study will comprehensively discuss the SIFs and failure mechanism of each case and the fracture angle will be verified using the strain energy density contour.

[1] Shan, D., Zhao, M., Shao, H., & Fang, C. (2019). Fracture behavior of notched TC21 alloy observed by In-situ SEM. Results in Physics, 15, 102604
[2] Muskhelishvili, N. I. (1953). Some basic problems of the mathematical theory of elasticity (Vol. 15). Groningen, Holland.
[3] Inglis CE. (1913). Stresses in a plate due to the presence of cracks and sharp corners. Transactions of the Royal Institute of Naval Architects , 60, 219–241.
[4] Florence, A. L., & Goodier, J. N. (1960). Thermal stresses due to disturbance of uniform heat flow by an insulated ovaloid hole. J. Appl. Mech. 27, pp.635-639.
[5] Yi-Zhou, C. (1984). Elastic analysis of an infinite plate containing hole with cusps and applied by concentrated forces. Engineering Fracture Mechanics, 20(4), 573-582.
[6] Chen, Y. Z. (2021). Thermal stress analysis for a hypocycloid-type crack problem under remote thermal loading. Journal of Thermal Stresses, 44(5), 634-641.
[7] Chao, C. K., Lu, L. M., Chen, C. K., & Chen, F. M. (2009). Analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load. International Journal of Solids and Structures, 46(14-15), 2959-2965.
[8] Chao, C. K., Tseng, S. C., and Chen, F. M. (2018). Analytical solution to a coated triangular hole embedded in an infinite plate under a remote uniform heat flow. Journal of Thermal Stresses, 41, 1259-1275.
[9] Tseng, S. C., Chao, C. K., & Chen, F. M. (2020). Interfacial stresses of a coated square hole induced by a remote uniform heat flow. International Journal of Applied Mechanics, 12(06), 2050063.
[10] 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, 43(12), 1487-1512.
[11] Sih, G. C. (1962). On the singular character of thermal stresses near a crack tip.
[12] Chao, C. K., & Kuo, L. Y. (1994). Thermoelastic problem of curvilinear crack with a point heat source. Journal of Thermal Stresses, 17(2), 271-273.
[13] Chen, Y. Z. (1984). Multiple crack problems of antiplane elasticity in an infinite body. Engineering Fracture Mechanics, 20(5-6), 767-775.
[14] Shahzad, S., Dal Corso, F., & Bigoni, D. (2017). Hypocycloidal inclusions in nonuniform out-of-plane elasticity: stress singularity vs. stress reduction. Journal of Elasticity, 126(2), 215-229.
[15] Zhu, N., & Oterkus, E. (2020). Calculation of stress intensity factor using displacement extrapolation method in peridynamic framework. Journal of Mechanics, 36(2), 235-243.
[16] 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.
[17] You-Sheng, D., Long, L., Li-Qing, M., & Cheng-Pu, P. (2021). Mode-I stress intensity factors for cracked R-fluted shells under complex loads. International Journal of Pressure Vessels and Piping, 194, 104490.
[18] Chao, C. K., & Chen, S. J. (1996). Stress intensity factors of two bonded half-plane problem with a point heat source. Nuclear Engineering and Design, 160(1-2), 97-109.
[19] Chen, Y. Z. (2020). Fracture analysis for a cusp type crack problem. Acta Mechanica, 231(8). 3123-3128.
[20] Sharma, D. S., & Dave, J. M. (2015). Stress intensity factors for hypocycloidal hole with cusps in infinite anisotropic plate. Theoretical and Applied Fracture Mechanics, 75, 44-52.
[21] Nik Long, N. M. A., & Yaghobifar, M. (2011). General analytical solution for stress intensity factor of a hypocycloid hole with many cusps in an infinite plate. Philosophical Magazine Letters, 91(4), 256-263.
[22] Chen, F. M., Chao, C. K., Chiu, C. C., & Noda, N. A. (2021). Stress intensity factors for cusp-type crack problem under mechanical and thermal loading. Journal of Mechanics, 37, 327-332.
[23] Lee, K. Y., & Choi, H. S. (1988). Determination of thermal stress intensity factors for traction free cusp cracks under uniform heat flow. Engineering Fracture Mechanics, 31(4), 661-672.
[24] Chauhan, M. M., Sharma, D. S., & Dave, J. M. (2016). Stress intensity factor for hypocycloidal hole in finite plate. Theoretical and Applied Fracture Mechanics, 82, 59-68.
[25] Sharma, D. S. (2016). Stresses around hypotrochoidal hole in infinite isotropic plate. International Journal of Mechanical Sciences, 105, 32-40.
[26] 潘睿澤 (2022)。含對稱翼型和對稱唇型裂紋平板承受機械和熱流負載下之應力強度因子分析與破裂角度預測。［碩士論文。國立台灣科技大學］台灣博碩士論文知識加值系統。
[27] Sekine, H. (1975). Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow. Engineering Fracture Mechanics, 7(4), 713-729.
[28] Nehari, Z. (2012). Conformal Mapping. Courier Corporation.
[29] England, A. H. (1971). Complex Variable Methods in Elasticity. John Wiley & Sons Ltd.
[30] Sih, G. C., & Macdonald, B. (1974). Fracture mechanics applied to engineering problems-strain energy density fracture criterion. Engineering Fracture Mechanics, 6(2), 361-386.
[31] Stewart, G. W. (1993). On the early history of the singular value decomposition.
SIAM review, 35(4), 551-566.