13th International Conference on Fracture June 16–21, 2013, Beijing, China —3— Fig.1 Calculation model predicting crack initiation based on damage mechanics theory From figure 1(3), some equations may be obtained: r r d− =arccos θ （2 ） θ θ ( )sin 2 A r r r d D Δ = − − （3） θ θ π θ sin D r A d r − Δ = − （4 ） From above analysis, ratio of reduction of the specimen area to original area ΔAD/AD should equal to ratio of attenuation of nominal stress to steady stress ΔS/S, namely: S S A A D D Δ = Δ (5) Based on above equations, when specimen radius r is determined (e.g. 5mm), the relation of equivalent length d of crack initiation and ΔS/S will be built. Through test data processed (see figure2), the following equation would be set up: （ ）1 -0.5006 4.1698d 1.7765d -0.1186d 2 3 = + + = Δ R S S (6) ΔS/S = -0.1186d3 + 1.7765d2 +4.1698d - 0.5006 R2 = 1 0 20 40 60 80 100 120 0.00 2.00 4.00 6.00 8.00 10.00 12.00 ΔS/S，% d/mm Fig.2 Relation curve and corresponding equation of S/S △ -d 3. Correction for prediction model for low cycle fatigue crack initiation Considering deviation of damage effect of one crack equivalent to several cracks in one section or several sections, damage coefficient λ (0～1) was introduced. Simultaneously, many factors including some measurement errors of attenuation of stress, areas swept by cracks and section distortion when specimen fractured and so on, which would directly influence calculation precision of equivalent crack length. Supposing calculating error is d0 (a constant relevant to material), then expression (6) can be further amended:
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