13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- Figure 1. Quadratic element for the higher order displacement discontinuity variation To eliminate the singularity of the displacements and stress calculation near the micro crack ends and increase the accuracy of order higher displacement discontinuity method around the original micro crack tip, a special treatment of the micro crack at the tip is necessary [24-27]. In this research three special crack tip elements at the end and initiation of each micro crack are used in the general higher order displacement discontinuity method. As shown in Fig. 2, using a special crack tip element with the length of 2c.The displacement discontinuity variations along this element can be written in the following form [24]: given as: () [ ()] () [ ()] () [ ()] () 3 3 2 2 1 1 D c D c D c D j T j T j T j (4) The crack tip element has a length 2 1 3 c c c c . Considering 3 2 1c c c , the shape functions ( ) 1 T , ( ) 2 T and ( ) 3 T can be written as: 2 5 1 2 5 2 3 1 2 3 2 1 1 2 1 3 2 5 1 2 5 2 3 1 2 3 2 1 1 2 1 2 2 5 1 2 5 2 3 1 2 3 2 1 1 2 1 1 8 5 2 5 8 5 3 ( ) and 4 3 2 3 3 4 3 5 ( ) , 8 - 8 15 ( ) c c c c c c c c c T T T (5) Figure 2. Quadratic element for the higher order displacement discontinuity variation 2 2c1 3 3 1 Dj 3 Dj 1 ζ One displacement discontinuity element are divided into three Sub elements 2c 2c 2 y 2b 2c2 Dj 2 2c3 y u 1 yD r θ v x 2 yD 3 yD 2c 2c1 2c2 2c3
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