13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Boundary Condition b: 0 2 2 ( ) pz dr pz SIF dr I dr dr K K G x dx . (6) By applying the superposition to stress filed, the total SIF for the semi-infinite crack loaded as shown on the left side of Figure 2 is: 2 2 tot dr dr pz K K K L . (7) 2.2. COD Formula The COD (x) is convenient to express in the following integral form using an axillary fictitious force Q (x) perpendicular crack face [14]: 0 1 2 ( ) ( , ) ( , ) ' L I I Q x K K Q d E . (8) We select the origin of coordinate at the beginning of distributed load , i.e., on the distance L from the crack tip (see Figure 2a). For calculations of COD due to distributed load two cases should be considered: Case I: 0 x L , x L and Case II: 0 x , 0 L . In the Case I, 1 2 , ( , ) ( , ) ' L I I Q x x K K Q d E (9) Substituting Eqs. 4 and 5 into Eq. 9, one can find the COD formula. It is noted that x in Eq. 4 should be replaced by x due to the change of the origin. Finally, the COD is: Case I: 8 , ' L x x d E x ; and Case II: 0 8 , ' L x d E x (10) Performing the integration in (10), and shifting the origin to the crack tip one finds the COD: 4 8 , ln , 0 L x x L x Lx x E E x L . (11) The reduction of COD due to the closing force dr is determined by the same procedure. It takes form: 4 8 , ln , 0 pz dr pz dr dr dr pz pz x x x x x E E x . (12) The analytical expression for total COD for the semi-infinite crack loaded as shown in Fig.2 is: 4 8 ( , ) ( , ) ln 4 8 ln tot dr pz dr pz dr pz pz L x x L x x x Lx E E x L x x x E E x . (13) Figure 3 shows an example of COD profile for normalized stress 0.4 dr and CL sizes
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