13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- where, B and n are material parameters determined by the fatigue crack propagation tests. Taking logarithm both sides of Eq. (15), Eq. (15) can be rewritten as ) lg( lg d d lg th B n K K N a ⎟ = + Δ −Δ ⎠ ⎞ ⎜ ⎝ ⎛ , (16) It can be found from Eq. (16) that the relation between logrithmic crack propagation rate lg(da/dN) and lg(ΔK−ΔKth) shows a linear one, and the slope and intercept are individually m and lgB. As aforementioned above, in the cases of stress ratio R=0.02, 0.1, 0.2 and 0.4, the threshold stress intensity factor ranges (ΔKth) measured in the tests are 11.842, 11.313, 9.773 and 7.440MPa·m1/2, respectively. The linear regression analysis on the experiment data is performed by using Eq. (16), and the crack propagation rate equations considering ΔKth are individually derived as 11.842) lg( 6.858 2.055 d d lg ⎟ =− + × Δ − ⎠ ⎞ ⎜ ⎝ ⎛ K N a , (17) 11.313) lg( 6.994 2.192 d d lg ⎟ =− + × Δ − ⎠ ⎞ ⎜ ⎝ ⎛ K N a , (18) 9.773) lg( 7.326 2.457 d d lg ⎟ =− + × Δ − ⎠ ⎞ ⎜ ⎝ ⎛ K N a , (19) 7.440) lg( 7.376 2.571 d d lg ⎟ =− + × Δ − ⎠ ⎞ ⎜ ⎝ ⎛ K N a , (20) Considering the threshold stress intensity factor range ΔKth, and in double-logrithmic coordinates, the theoretical curves of crack propagation rate da/dN versus stress intensity factor rang ΔK are plotted by using Eq. (17), Eq. (18), Eq. (19) and Eq. (20), as shown the solid lines in Fig. 5(b). 45 90 135 180 10-5 10-4 10-3 10-2 10-1 R=0.02 R=0.1 R=0.2 R=0.4 da/dN/(mm.cycle−1) ΔK/MPa.m1/2 (a) 45 90 135 180 10-5 10-4 10-3 10-2 10-1 R=0.02 R=0.1 R=0.2 R=0.4 dla/dN/(mm.cycle −1) ΔK/MPa.m1/2 (b) Figure 5. Curves of crack propagation rate: (a) by Paris formula; (b) by modified Paris formula It can be found from Fig. 5(b) that the theoretical results are well fitted with test data, especially in the case of small values of stress intensity factor range ΔK, comparing with Eq. (11) to Eq. (14), the precise theoretical results are obtained by using Eq. (17) to (20), which can be seen from the comparison between theoretical curve and experimental data in Fig. 5 and Fig. 6. Therefore, the modified Paris formula can not only be used to predict the value of threshold stress intensity factor at different stress ratio, but also can describe the fatigue crack propagation behavior accurately. 4.3. Fractography reverse model As aforementioned above, the fracture surface morphologies of standard compact tension samples of GH4133B superalloy are investigated using an EVO scanning electron microscope, and a series of fatigue striations are found in the crack steady propagation region. According to these fatigue striations on the fracture surface, the crack propagation rate equations and loads are derived using
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