13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- high strength steel. During the formation of FGA, localized and irreversible plastic flows are blocked within the twin martensite, and fatigue crack occur with the fracture of martensite when the accumulated energy reaches the critical value. We assume that the dislocations are piled up along the slip band in the martensite, and introduce the influence of the strength of the material and the law of FGA formation on the VHCF crack initiation life, thus we have the modified Tanaka-Mura model for life prediction of FGA: ( )2 2 , / = FGA m b FGA net N l G C area π τ κ σ ⋅ − ⋅ ⋅ (1) Where lm is the width of martensite, σb is tensile strength of the material, areaFGA,net is the area of FGA minus the area of inclusion, and C is a correction coefficient which shows the relationship between the strength of the martensite and tensile stress of material. Other parameters are same as Tanaka-Mura model[18]. With Von Misses yield criteria, Eq. [1] can be expressed as: ( ) 2 2 2 4 2 9 16 = 2 - b FGA FGA Inc m D CG K N area l σ π σ π σ σ ⎛ ⎞ Δ − ⎜ ⎟ ⎝ ⎠ (2) In Eq. [2], σD is the fatigue limit. With Eq. [2], sampling-based sensitivity measures are performed with direct Monte-Carlo simulation[19; 20]. The bulk shear modulus G, correction coefficient C, martensite width lm and fatigue limit σD are assumed to be deterministic variables with values of 79 GPa, 1, 575 nm and 600MPa respectively. local stress σ, ΔKFGA, and inclusion size Inc area are assumed to be random variables. The stress is normally distributed with a mean value of 750 MPa and a COV of 0.03. ΔKFGA is a normal distribution with a mean value of 5.5 MPam0.5 and a COV of 0.03. Inclusion size is Gumbel distributed with a mean value of 30 μm and a COV of 0.4. The calculated results of sensitivity measures are shown in Fig. 6. Both sensitivity of mean value Sμ and sensitivity of standard deviation Sσare dimensionless and appropriate for variable ranking, and they are defined as[21]: i i f fP P S i σ μ μ / / ∂ ∂ = , i i f fP P S i σ σ σ / / ∂ ∂ = (3) Pf , μi and σi are respectively the probability of failure, mean value and standard deviation of a variable Xi. Fig. 6. (a) sensitivity of mean value ( Sμ ); (b) sensitivity of standard deviation ( Sσ). 0.0 0.2 0.4 0.6 0.8 1.0 -2 -1 0 1 2 3 4 5 Sensitivity of CDF w.r.t. Std. Dev. (Sσ) Probability of Fatigue Failure, Pf Inclusion size Stress ΔK FGA 0.0 0.2 0.4 0.6 0.8 1.0 -1 0 1 2 Inclusion size Stress ΔKFGA Sensitivity of CDF w.r.t. Mean (Sµ) Probability of Fatigue Failure, Pf
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