M-積分是研究具有裂縫之物體破壞行為之重要參數。本論文結合有限元素法研究針對具有多裂縫之三維線彈性材料物體，受混合載重作用下，計算其M-積分的數值分析。首先依序針對具有任意形狀之二維裂縫問題以及具有任意形狀之三維裂縫問題，進行M-積分式的理論介紹，其次証明M-積分具有與積分曲面無關的性質。在三維問題，對單裂縫的M-積分計算結果顯示與積分曲面無關和具有與原點無關的特性；關於多裂縫問題的M-積分計算，考慮一特殊狀況，在一個無限域的線彈性結構體內包含多裂縫，受到遠端均勻載重作用，在此特殊的邊界情況下，選取任意積分原點且任意包含所有裂縫的積分區域，計算出的M-積分值皆會相同。接著利用不同裂縫間距去探討裂縫間距對於M-積分的影響，而結果呈現出，隨裂縫間距縮小的情況下所得到的M積分值也越小。二維情況下M-積分的物理意義為裂縫面形成時能量變化的兩倍，三維中M-積分的物理意義則為裂縫面形成時能量變化的三倍。M-integral is an important parameter for fracture analysis. In this research, a numerical procedure, incorporated with the finite element method, that is developed for calculation of the 3D linear elastic solid subjected to mixed-mode load with cracks. First, we verify M-integral for arbitrary-shaped cracks in 2D problems and the arbitrary-shaped cracks in 3D problems. Second,we verify the property of surface independence.In 3D single crack problems, the M-integral is verified to be surface independent and origin independent. In 3D multiple cracks problems, the result of M-integral also has the property of surface independence and origin independence on specific boundary conditions. We also discuss the relation between cracking space and the results of the M-integral.In 2-D situation, M-integral is equal to twice the surface energy required for the formation of the whole cracks. In 3-D situation, due to the different geometry of the cracks, the M-integral appears to be equal to triple the surface energy required for the formation of the whole cracks.
