本論文研究利用應變能密度理論分析無窮功能梯度平面嵌入一中央裂縫，並改變裂縫面上的邊界負載，探討裂縫即將開裂時其裂出角度。材料的蒲松氏比為一定值，但功能梯度曲線可任意旋轉，材料性質沿任意方向做指數型函數變化。承接Konda and Erdogan (1994)的研究，利用Gauss-Chebyshev積分方法將積分方程式展開為一組線性的代數方程式進行數值求解，可求出裂縫尖端第I型與第II型的無因次應力強度因子，並將數值解結果與文獻相互驗證，證明所推導之公式其可用性。再將不同邊界負載下所求得的無因次應力強度因子代入應變能密度理論，討論當改變材料梯度方向以及非均質材料參數時對無因次化應變能密度因子之影響，並觀察無因次化應變能密度曲線，找出應變能密度因子極小值所對應的角度，即為所預測之裂縫可能開裂角度，並與利用最大周向應力理論所預測之可能開裂角度其之間的差異做一比較。由於材料的破壞韌性在非均質材料中仍是未知，因此裂縫是否真的會沿預測的角度裂出仍需要更多的實驗結果來輔助驗證。

In this research, the strain energy density theory is employed to predict the extension direction of a central crack embedded in a functionally graded material. The variation of material property, which vary along arbitrary direction, is assumed in an exponential form and the Poisson ratio is kept as a constant. Following the previous study of Konda and Erdogan (1994), the formulations will be derived from the beginning. The derived singular integral equations are solved numerically. The normalized stress intensity factors of two simple loading cases are compared and checked correctly to validate the formulations and numerical computations. The direction of minimum strain energy density factor Smin, which is used to indicate the crack extension direction, for different loading directions are obtained and compared with the results predicted by the maximum circumferential stress theory. The factors that affect the magnitude of Smin and direction include the non-homogeneous material parameter, crack length, and the variation direction of material properties. Since the fracture toughness of a non-homogeneous material is still unknown, the possibility of using the strain energy density theory for predicting the direction of crack extension needs more experimental evidence.

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