本研究使用有限元素法作為分析工具，利用有限元素法計算的結果，搭配自行撰寫的計算程式，來分析含有裂縫之功能梯度材料平板與圓管在不同的材料與體積分率函數的物理特性，包含熱傳導特性與破壞分析。其中探討功能梯度材料圓管承受內壓力與熱應力之破壞問題，比較五種不同體積分率對於應力強度因子與溫度變化趨勢的影響，以及使用Fe/SiC與Ti-6Al-4V/ZrO2兩種組成的差異。從本研究所模擬的結果能發現，金屬與陶瓷的材料性質將會導致整體結果有很大差異。金屬材料增加會使材料的導熱性提升，因此溫度上升的幅度較大，金屬含量最高與最低的組合，溫差最大約120度。在破壞的分析中，金屬材料的破壞韌性比陶瓷更高，因此能夠提高功能梯度材料的破壞抵抗性，加入50%的金屬最多能使韌性提高4.4倍。本文所分析的兩種材料組成，應力強度因子會隨著陶瓷材料的比例提高而增加，將金屬材料含量提高20%，能使應力強度因子下降約10%。最後將功能梯度材料應用到實際工程構件，並評估其可行性，以及找出適當的材料與體積分率函數。

In industry, many high strength or high-temperature resistance materials are used to satisfy the requirement of some working conditions, sometimes more than two materials would be chosen. In traditional, welding or plating is a common method to combine them. However, the interface between these two materials was not continuous in that way, some defects or stress concentration would appear in the material interface, it may weaken a structure, decreasing working life or strength. Functionally graded materials (FGMs) has a characteristic of gradual changing of material properties. However, it’s inevitable that there are some defects and cracks inside structures because of working for a long time or manufacturing process. Crack resistance is an important problem to analysis. Functionally graded material is an inhomogeneous material, it’s very difficult to get analytical solutions of an inhomogeneous material with cracks or complex boundary. In this research use finite element method software which is a common and well-developed method, and combined with some calculating programs, to analyze the fracture properties of FGM plates and pipes with crack. Simulating two types of FGMs, Fe/SiC and Ti-6Al-4V/ZrO2, 5 different volume fraction, comparing the results while using them in high pressure and thermal stress environment. According to the results, temperature distribution affected by the mechanical properties of metal and ceramics. Higher content metal will increase thermal conductivity, so the temperature rises more quickly. There is about 120℃ difference between the highest and lowest metal content FGM. Fracture toughness of metal is higher than ceramic, so it can provide more fracture resistance, 50% of metal could increase about 4.4 times higher than pure ceramic material. Higher volume of ceramic may cause higher stress intensity factor, increasing 20% of metal material, value of stress intensity factor could be decreased about 10%. Finally, we use FGM to design a pressure vessel, to find some proper parameters and evaluating the feasibility.

[1] Gururaja Udupa, S. Shrikantha Rao, K.V. Gangadharan, “Functionally Graded Composite Materials: An Overview,” Procedia Materials Science 5, pp. 1291-1299 ,2014.
[2] Valmik Bhavar1, Prakash Kattire1, Sandeep Thakare1, Sachin patil and Dr. RKP Singh, “A Review on Functionally Gradient Materials (FGMs) and Their Applications,” Mater. Sci. Eng., 229, 2017.
[3] A.J. Marlworth, K.S. Ramesh, W.P. Parks Jr., “Review: Modelling Studies Applied to Functionally Graded Materials,” Journal of Materials Science 30, pp. 2183-2193, 1995.
[4] D.T. Sarathchandra, S.Kanmani Subbu, N. Venkaiah, “Functionally graded materials and processing techniques: An art of review,” Materials Today: Proceedings 5 pp. 21328-21334, 2018.
[5] M.S.EL-Wazery, A.R.EL-Desouky, “A review on Funcitonally Graded Ceramic-Metal Materials,” Materials Environment Sci. 6 pp. 1369-1376, 2015.
[6] Naotake Noda, “Thermal Stresses in Functionally Graded Material,” J. Therm. Stresses, 22, pp. 477–512, 1999.
[7] Z.H. Jin and R. C. Batra, “Some basic fracture mechanics concepts in functionally graded materials,” J. Mech. Phys. Solids, Vol.44 No.8, pp. 1221-1235, 1996.
[8] Drake, J. T., Williamson, R. L. and Rabin, B.H. “Finite Element Analysis of Thermal Residual Stresses at Graded Ceramic-Metal Interfaces, Part II: Interface Optimization for Residual Stress Reduction,” J. Appl. Phys. 74, 1321-1326, 1993.
[9] Yanyan Zhang, Licheng Guo, Xiaoli Wang, Rilin Shen, Kai Huang, “Thermal Shock Resistance of Functionally Graded Materials with Mixed-Mode Cracks,” International journal of solids and structures 164, pp. 202-211, 2019.
[10] James R. Rice, “Mathematical Analysis in the Mechanics of Fracture ,” Fracture-An Advance Treatise, Vol. II, H. Liebowitz (Ed.), Academic, New York, pp. 191-308, 1968.
[11]S.J. Chu, C.S. Hong, “Application of the Jk-Integral to Mixed Mode Crack Problems for Anisotropic Composite Laminates,” Engineering Fracture Mechanics, Vol 35, Issue 6, pp. 1093-1103, 1990.
[12]J.W. Eischen, “An Improved Method for Computing the J2-Integral,” Engineering Fracture Mechanics, Vol.26, No.5, pp. 691-700, 1987.
[13]Serkan Dag, K. Ayse Ilhan, “Mixed-mode Fracture Analysis of Orthotropic Functionally Graded Material Coating Using Analytical and Computational Methods,” Journal of Applied Mechanics, Vol. 75, 2008.
[14]Serkan Dag, “Mixed-Mode Fracture Analysis of Functionally Graded Materials Under Thermal Stresses: A New Approach Using Jk-Integral,” Journal of Thermal Stresses, pp. 269-296, 2007.
[15]W.K. Wilson, I.W. Yu, “The Use of the J-Integral in the Thermal Stress Crack Problems,” International Journal of Fracture, Vol.15, No. 4 ,1979.
[16]Serkan Dag, E. Erhan Arman, Bora Yildirim, “Computation of Thermal Fracture Parameters for Orthotropic Functionally Graded Materials Using Jk-Integral,” International Journal of Solids and Structures, pp. 3480-3488, 2010.
[17] Kieback B., Neubrand A., Riedel H. “Processing Techniques for Functionally Graded Materials,” Materials Science and Engineering A362, pp. 81-105, 2003.
[18] Erdogan F., Y-D Lee “Residual/Thermal Stresses in FGM and Laminated Thermal Barrier Coatings,” Vol.69, Issue 2, pp. 145-165, 1994.
[19] J.R. Cho, D.Y. Ha, “Averaging and Finite-Element Discretization Approaches in the Numerical Analysis of Functionally Graded Materials,” Materials Science and Engineering, pp. 187-196, 2001.
[20] Hatta H, Taya M. “Effective Thermal Conductivity of a Misoriented Short Fiber Composite,” J. Appl. Phys. 58(7): pp. 2478–2486, 1985.
[21] Rosen BW, Hashin Z. “Effective Thermal Expansion Coefficients and Specific Heats of Composite Materials,” Int. J. Eng. Sci. 8: pp. 157–173, 1970.
[22] T. L. Anderson, “Fracture Mechanics: Fundamentals and Applications,” CRC press, 2017.
[23]Logan, D.L., “A First Course in the Finite Element Method Fourth Eidition,” pp. 617-622, CL-Engineering, January 2011.
[24] Peng Liu, Tiantang Yu, Tinh Quoc Bui, Chuanzeng Zhang, Yepeng Xu , Chee Wah Lim, “ Transient Thermal Shock Fracture Analysis of Functionally Graded Piezoelectric Materials by the Extended Finite Element Method,” International Journal of Solids and Structures, 51, pp. 2167-2182, 2014.
[25] 劉軒瑋, “高溫功能材料熱破壞的RPIM分析, ” 碩士論文, 國立成功大學, 2016.
[26] 林子杰, “壓電橢圓管受管內壓與熱負載之破壞分析,’’ 碩士論文,”國立成功大學, 2019.
[27] Thanyaporn Yodakew, “Sintered Fe-Al2O3 and Fe-SiC Composites,” Journal of Metals, Materials and Minerals, Vol.18 No.1, pp.57-61, 2008.