13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- Table 1. The 7 different test cases Test Case Kt Notch radius, (mm) Notch depth, h (mm) R-ratio Average alternating HCF strength at 106 cycles (MPa) 1 2.78 0.330 0.729 -1 173.6 2 2.78 0.330 0.729 0.1 158.9 3 2.78 0.330 0.729 0.5 104.6 4 2.78 0.203 0.254 0.10 167.2 5 2.78 0.203 0.254 0.50 105.2 6 2.78 0.127 0.127 0.10 144.7 7 2.78 0.127 0.127 0.50 111.0 For a smooth specimen with defects having a fatigue damage process zone of volume V, the whole volume is divided into small volume elements, dV with probability of failure of a sufficiently small volume element given as: dP dV = λ (9) Where here, λ is the critical defect density defined as the expected number of defects per unit volume of the smooth specimen. Using weakest link theory, the probability of survival of the entire volume is obtained from the probability of survival of all “m” number of sub-volumes i.e. ( ) ( ) 1 1 1 1 m m s i i i P dP dV = = = − = − Π Π λ (10) This equations assumes that the defects are randomly distributed within the volume and thus do not interact, which is only reasonable when considering the formation of a fatigue crack(s) in high cycle (HCF) and very high cycle fatigue (VHCF) regimes. Following the framework presented in [14], as the volume of each small element tends to zero, equation (10) can be transformed into exp d s V P dV = − ∫ λ (11) Using the generalized extreme value distribution function, the distribution of defects, a, that are above the threshold, ath, is modeled by a power law of the form 1 0 0 1 1 ε λ ξ − − = + th a a V a (12) Figure 4. Domain decomposition of the cylindrical notched specimen geometry. Figure 5 Finite element mesh for 0.33 mm notch root radius and Kt = 2.78 consisting of four-node linear tetrahedron element type (C3D4)
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