13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- 3.1 Effective crack extension length and residual Young's modulus The linear asymptotic superposition assumption is considered in the analytical method presented by Xu and Reinhardt [7, 8] to introduce the concept of linear elastic fracture mechanics for calculating the double-K fracture parameters. Detailed explanation of the above assumption can be found elsewhere [7]. Based on this assumption, the value of the equivalent-elastic crack length for WS specimen is expressed as: 0 1/2 0 ) } 9.16 13.18 ){1 ( ( h E b c a h h − ⋅ ⋅ + = + − (4) Where c=CMOD/P is the compliance of specimens, b is specimens thickness; h is specimens height and h0 is the thickness of the clip gauge holder. For calculation of critical value of equivalent-elastic crack length ac, the value of crack mouth opening displacement (CMOD) and P is taken as CMODc and Pu respectively. The residual Young's modulus E is calculated using the P-CMOD curve as: [13.18 (1 ) 9.16] 1 2 × − − = α i bc E (5) Where ci=CMODini/Pini, is the initial compliance before cracking, α= (a0+h0)/ (h+h0). The value of critical equivalent-elastic crack length ac and residual Young's modulus E are listed in Table 2. 3.2 Crack opening displacement along the fracture process zone Since the cohesive stress distribution along the fracture process zone depends on the crack opening displacement and the specified softening law, it is important to know the value of crack opening displacement along the fracture line. It is difficult to measure directly the value of COD along the fracture process zone, for practical purposes the value of COD(x) at the crack length x is computed using the following expression [3]: 2 1/2 2 {(1 ) (1.018 1.149 )[ ( ) ]} ( ) a x a x h a a x COD x CMOD − − − + = (6) For calculation of critical value of crack tip opening displacement CTODc, the value of x and a (see in Fig.4) in Eq. (6) is taken to be ao and ac, respectively. The value of cohesive stress along the fictitious fracture zone to the corresponding crack opening displacement is evaluated using bilinear stress-displacement softening law as given in Eq. 3. 3.3 Determination of stress intensity factor caused by cohesive force The standard Green’s function [27] for the edge cracks with finite width of plate subjected to a pair of normal forces is used to evaluate the value of cohesive toughness. The general expression for the crack extension resistance for complete fracture associated with cohesive stress distribution in the fictitious fracture zone for Mode I fracture is given as below: adx h a a x x F K a a c I π σ2 ( ) ( , ) / 0 ∫ = (7) Where
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