13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- unmatched to notched fatigue strength at the same probability of failure (usually 50%). Using equation 20, the probability of failure of unmatched specimen and a notched specimen will be the same when max, max, 0 0 exp exp b b s n s s n n o o kV k V V V − = − σ σ σ σ (22) where the subscripts n and s represent the respective value of the variable for notched and smooth (unmatched) specimens. The ratio of the smooth to notch fatigue driving force parameters (i.e., the stress amplitude) is used to define a new fatigue notch factor given as 1 1 max, max, b b s n n f n s s k V k k V = = σ σ (23) For smooth specimen that is loaded at a very low stress or strain amplitude in the HCF regime, the number of critically stressed grains (or elements) is very small. Thus for the life limiting case in which only one grain or element is critically stressed above the threshold, Vs = Ve (i.e. volume of element or grain) and Ks = 1; thus Equation (23) becomes ( ) 1 1 1 1 max, max, max, 1 b b b b s n n b f n n e n e V V k k dV V V V = = = ∫ σ σ σ σ (24) However, if the materials contain some pores or inclusions, equation (23) must be used. It is important to state that Eqs. (23) and (24) can be used only if subsurface crack initiation is the failure process, if crack originates from the surface, then the volume parameter in this equation should be replace with the surface area. 5. Closed Form Solution for Fatigue Notch Factor To resolve inelastic deformation at the scale of microstructure to facilitate next generation microstructure-sensitive notch root analyses inherently requires mesh refinement to the scale of microstructure, which is often several orders of magnitude finer than the scale of the component. Moreover, the kind of constitutive equations that must be used are often of advanced form and requiring rather sophisticated and time-consuming computational strategies to perform concurrent analyses at the component and notch root microstructure scales. Accordingly, direct application of multiscale finite element analysis is simply too computationally time consuming for practical microstructure-sensitive fatigue damage assessment of notched components under multiaxial loads. Thus, for practical engineering application, a more simplified and approximate model for fatigue notch factor is presented here based on closed form solution for stress distribution at the notch developed by Glinka using the Creager-Paris solutions of the stress field ahead of a crack. For a notch component with notch root radius ρ and stress concentration factor, kt, the axial stress distribution along the notch root centre line is given as: 3 3 1 1 2 2 2 2 max, 1 1 1 1 2 2 2 2 2 2 2 2 t n k S x x x x = + = + + + + + ρ ρ ρ ρ σ σ ρ ρ ρ ρ (25) Finding the ratio of the stress amplitude to the maximum stress and substituting into equation (24) at x
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