M-積分是研究物體具有裂縫之破壞行為的重要參數。本論文結合有限元素法並利用M-積分分析三維多裂縫線彈性材料在非均勻分佈載重作用下的表現。本研究針對具有任意形狀之二維多裂縫及三維圓盤形裂縫問題，進行M-積分式的理論及物理意義的推導，其次証明M-積分具有與積分曲面無關的特性。
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與積分原點無關的特性；多裂縫問題的M-積分計算，則需對所有裂縫的幾何形狀中心做計算，並且裂縫的幾何位置亦會影響M-積分。
二維M-積分的物理意義為兩倍裂縫面形成時所需要的能量變化，三維M-積分的物理意義則為三倍裂縫面形成時所需要的能量變化。
此外，利用不同裂縫間距的有限元素網格探討裂縫間距對於M-積分的影響。裂縫間距比較小的情況下所得到的M積分值較小。
關鍵詞：M-積分、有限元素法、積分曲面無關、積分原點無關、幾何形狀中心、裂縫間距。

In this research, we analyze 2D and 3D crack problems by using the M-integral. First, we derive the M-integral for multiple arbitrary-shaped cracks in 2-D and 3D situations. Then, we verify the physical intepretation and path independent property of the M-integral.
In 3D single crack situations, the result of M-integral has the property of surface independence and origin independence. In 3D multiple cracked situation, the result of M-integral associated with geometric center is calculated, and the origin of the coordinate system affects the computation result.
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.
Furthermore, we also study the relation between periodic cracks and the M-integral.
Keywords : M-integral, surface independent, origin independence, geometric center, surface energy, periodic crack

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