13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- ,Δ 1 1 $exp & "'( ! "Δ! ln lnΔ " δ 1 exp 3!"Δ! ln ln Δ " & "'( (5) For a specimen with circular cross section and variable diameter d(x) over its length, being d0 the minimal diameter in the section loaded by the maximum stress level ∆σ0, the summation can be extended to an integral. With Si= π d(x) dx, we get ,Δ 1 1 exp4 2 Δ Π ! 7 8 ln )ln)Δ 1 9:; 9 < ; * * = >? 1 @A B AB. (6) Due to symmetry, the integration is carried out over half the specimen length starting in the centre of the specimen, being the upper integration bound UB the x-coordinate, where (lnN−B)(ln(∆σ(x))−C)=λ. 2.3. Effective surface area For a specimen under variable stress state an effective specimen surface Seff can be defined having the same failure probability as the whole specimen but subjected to a constant stress range ∆σ. The normalizing variable for the nominal maximum stress Δσ0 acting in the central section of the specimen with diameter d0 is represented by 1 ln ln Δσ1 . For different specimen sections with diameter d(x) we have B ln ln Δσ1 ∙ A1D A B D ⁄ . An analytical expression for Seff is obtained equating Eqs. (5) (with Si =Seff) and (6): !E 2 ΠF B λ HA B dx 1 JK 1 λ H . (7) As can be observed from Eq. (7) Seff is independent of δ but depends on the parameters B, C, λ and β of the Weibull model and also on the number of cycles N and the stress range ∆σ. For given values of N and ∆σ0 and known material parameters B, C, λ and β the effective surface and the failure probability can be computed. However, for a specific specimen Seff cannot be calculated directly from the failure data, since the Weibull parameters are still unknown. Thus, an iterative process, as explained in [8], is used for the parameter estimation. Firstly, the n test data are fitted to the model given by Eq. (1), then the normalized values V0 are assigned their accumulated failure probabilities by L 0.3 . O 0.4 ⁄ . To refer the data to the surface element ΔS, those failure probabilities are shifted by using ,",∆ 1 )1 ,", QRR* ∆ QRR,# ⁄ . The 1," and their corresponding ,",∆ are fitted to a three-parameter Weibull distribution. The obtained values for λ and β are used to update the effective surface given by Eq. (7) in each iteration loop. Those steps are repeated until the Weibull parameters converge.
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