13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Having established the equations, the plastic zone sizes and stress distribution along the elastic ligament σ(x) can be determined by solving the above equation using the weight function method and Newton-iteration. Then, the corresponding crack opening profile for the strip yield model can be obtained by superposition. 2.5. Examples and validations In this section, two and three collinear cracks in finite and infinite sheets are solved using the above methods. Some existing results and FEM results are also presented for comparison. 2.5.1 Three collinear cracks with plastic zones coalesced in a finite sheet For a given three crack configuration a1=0.3, a2=0.1 and d=0.2 in a finite sheet shown in Fig.1, the critical applied stress σc/ σs and fictitious crack length a are obtained based on the method described in section 2.2, which are 0.3777 and 0.7417 respectively. Figure 4 shows the corresponding CODs determined by WFM and FEM, and very good agreement is observed. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.1 0.2 0.3 0.4 x/a u(a,x)E‘/(σa) Line: Finite element analysis Symbols: Closed−form weight function Fig.4. Crack opening profile for the fictitious crack of three un-equal length cracks with coalesced plastic zones. (In Fig.1, a1 = 0.3, d = 0.2, 2a2 = 0.2, the half-length of fictitious crack a = 0.7417) [7]. 2.5.2 Two equal-length collinear cracks in finite and infinite sheets The strip yield models for two equal-length collinear cracks infinite sheets can be obtained by using the WFM and the unified method. However, for the cracks in finite sheet, the unified method is applied. Figure 5 shows some typical results of the inner plastic zone sizes and CTOD as a function of the applied stress. These results are normalized by the plastic zone size r0 of a single Dugdale crack of the same length, r0=a[sec(0.5πσ/σs)-1], a=(c-b)/2. Also shown in these figures are the results for the plastic zones critical coalescence. To verify the solution accuracy of the present weight function approach, the results are compared to those given by Collins and Cartwright [5] by using complex stress function method. It is observed that the results for two collinear cracks in an infinite sheet obtained from the weight function method, complex stress function method and unified method agree very well. Fig.6 shows the CODs for the strip yield model of the two cracks in finite sheet subjected various applied loads. Also shown in this figure are the FEM results, very good agreement is observed. However, the unified method (UM) is much more efficient than FEM for solving the strip yield model. 2.5.3 Three collinear cracks in an infinite sheet Figure 5 and 6 show the strip yield model for three symmetric collinear cracks in an infinite sheet. The plastic zone sizes and CODs were obtained by using the WFM and unified method.
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