13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Figures 3a shows the P-Δ relationships for the experimental specimen of a0/W = 0.222 with a growing crack indicated by the discrete circular symbols. The good agreement between the test data and the FE results at the zero crack extension, Δa = 0, confirms the boundary conditions of the FE models. In addition, the numerical P-Δ curve computed from the FE model with the same crack size as the experimental specimen at the end of the fracture test intersects with the test P-Δ curve at the end of the measured history. Figure 3b shows the strain energy U versus the crack extension evaluated from the P-Δ curves for the shallow cracked SE(B) specimen. The U value is computed at varied CMOD levels (Δi), which correspond to the intersections between the experimental and numerical P-Δ curves. The discrete symbols in Figure 3b represent the strain energy value computed from the FE model with the corresponding initial crack lengths a0+Δai. The solid lines in Figure 3b correspond to the second-order polynomial functions derived from the regression analysis. The J-value corresponding to each crack extension, Δai, can be derived from Eq. (4), using the first-order derivative of the fitted polynomial for the strain energy U (at the corresponding Δi) with respect to the crack depth. This leads to a J-R curve schematically shown in Figure 1c. The determination of the J-R curve by the CMOD-based hybrid approach for the SE(B) specimen with a0/W = 0.511 follows the same procedures described in Figure 3a and 3b. Figure 3c compares the experimental J-R curves with those determined from the CMOD-based hybrid method for SE(B) specimens with two crack depth ratios, a0/W = 0.222 and 0.511. The J-R curves obtained from the CMOD-based methods agree closely with the test results for both the shallow-crack and the deep-crack SE(B) specimens. This validates both the applicability and the accuracy of the CMOD-based hybrid method in determining the J-R curves for SE(B) specimens. 4. Validation on the Mixed-Mode Specimens The verification of the hybrid method on the determination of the J-R curves utilizes two mixed-mode Al-alloy 5083 H-112 specimens reported by the authors [5, 11], the mode I dominant specimen AM1 (mode-mixity βeq = tan-1(KI/KII) = 75 o ) and the mode II dominant specimen AM5 (βeq = tan -1(KI/KII) = 20 o) [5]. Figure 4a and 4b shows the typical FE models for the two mixed-mode specimens, AM1 and AM5. The geometrical configurations and the orientation of the crack planes follow exactly the test procedures described by the authors [5, 11]. Figure 4c and 4d illustrates the close-up view of the region around the crack tips for AM1 and AM5, respectively. The crack path deviates by 20o from the original crack plane in the mode I dominant specimen AM1 and by 9o from the original crack plane in the mode II dominant specimen AM5. The multiple FE models have varied crack lengths of a0+Δai along the crack directions observed from the tests, which have been summarized in Table 2. The verification of the CMOD-based hybrid approach on the mixed-mode specimens includes eight and six FE models with different crack sizes for the mode I dominant AM1 and the mode II dominant specimen AM5, respectively, as shown in Table 2. The crack tips for both specimens are simulated with an initial root radius of 25 μm to facilitate the convergence of the large-deformation analysis, similar to the method shown in the Figure 2c. The element type, boundary conditions and the calculation procedures follow similarly the methods described by Qian and Yang [7]. The material properties of the elements remain the same as those shown in Figure 2a. Figure 5a and 5b show the M-θ curves and the FV-δV relationships computed from the FE models listed in Table 2. The circles in Figure 5a and 5b represent the test results for AM1. Figure 5c and 5d illustrate the mode I strain energy UI versus Δa curves and the mode II strain energy UII versus Δa curves evaluated from the FE results at varied deformation levels listed in Table 2. The symbols
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