13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- Some results for the inner crack tip A of three collinear cracks are given in Fig.8. It is observed that the results obtained from the unified method agree well with those from weight function method. It is also found that for σ/σs→0, the crack tip plastic zone size and opening displacement for the crack tip A approach the non-dimensional stress intensity factor squares [fA(a0,b0,c0)] 2·r 0 and [fA(a0,b0,c0)] 2·δ 0 shown in Figs.8a and b, respectively. The variables r0 and δ0 are the crack tip plastic zone size and opening displacement for a center crack in an infinite sheet. 3. Residual strength prediction and validation for sheets with MSD 3.1 CTOA criterion based on strip yield model The crack tip opening displacement in combination with the strip yield model had been used by several researchers to predict the stable crack growth behavior [10,15]. In the method, the crack growth was controlled by two parameters. One is critical crack opening displacement δ0 which is used to describe the crack initiation, Fig.9a; the other is used to characterize the stable crack growth by a constant critical CTOA α, Fig.9b. The crack growth equation for the whole process is ( ) ( ) ( ) ( ) 0 0 0 , ;crack initiation , 2 tan 2 , ;crackpropagation c c a a a a d d a d a d δ δ δ α δ = ⎧⎪ ⎨ − = + − − ⎪⎩ (8) where, δ(a, a-d) is the crack opening displacement at a distance d behind the crack tip, the first variable in the bracket is the crack length, and the second is the x location. δc(a-d, a-d) is the plastic wake height, which is equal to the crack tip opening displacement. αc is the critical CTOA. In practice, a pair of ‘optimal’ δ0 and αc is selected as the critical values. Using the ‘optimal’ values, the predicted crack growth behaviors of coupon specimens agree very well with the corresponding experiment observations. Here, the C(T) specimen is used to determine these critical values. The weight function method is adopted here to determine the COD for crack growth analysis. Figure 10 shows the predicted load-crack extension curves obtained by three different pairs of parameters. Also in the figure are the results measured from experiment [10]. It is observed that the predicted result obtained by the parameters δ0=0.10mm and αc=4° agrees well with test data. It is assumed that the criterion for single crack is also applicable to sheets with MSD. And these critical values will be used as material properties to predict crack growth for MSD specimens. x y o Plastic zone r δ0 a0 (a) Plastic wake Plastic zone r d α δ0 /2 x y o a0 Δa (b) δ0 /2 Fig.9. Crack opening profile of modified Dugdale strip yield model at (a) initiation and (b) at propagation with definition of crack growth parameters α and δ0
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