13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Where a0 and ξ are the scale and shape parameters respectively. Substituting (12) into (11) yields 1 0 0 1 exp 1 d th s V a a P dV V a − − = − + ∫ ε ξ (13) If * 0 th a a= ξ , re-arranging equation (13) yields, 1 * 0 0 1 exp s a P dV V a − = − ∫ ε (14) Where a0 * is regarded here as the mean defect size. Equation (14) is valid only if 0≠ξ . The critical defect size is related to the microscopic stress (taking here as a random variable) through a power law relationship of the form z A a =σ (15) where A and z are materials constants. Similarly, the stress amplitude, σ0 corresponding to the mean defect size a0 * can be taken as the fatigue limit of the reference volume Vo for 50% failure probability. The two parameters can also be related by a power law of the form: 0 * 0 z A a = σ (16) Combining equations (15) and (16) we have 0 * 0 z a a a = σ σ (17) Substituting Equation (17) into Eq. (14) yields 0 0 1 exp b sP dV V = − ∫ σ σ (18) Where b=z/ξ. For ξ>0, b and σ0 represents a 2-parameter Weibull shape and scale parameters. The cumulative probability of HCF failure of the component, specifically defined can be obtained from Eq. (18) as 0 0 1 1 exp b fP dV V = − − ∫ σ σ (19) To facilitate development of the expression for fatigue notch factor from Eq. (18), the concept of stress homogeneity factor that have been used is introduced here. Thus equation (18) can be re-written as, max 0 1 exp b f o kV P V = − − σ σ (20) where max 1 b k dV V = ∫ σ σ (21) is regarded as the stress homogeneity factor. Conventionally, the fatigue notch factor is the ratio of
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