13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Fig. 3. Evolution of the ratio Ktf /Kt with respect to the relative length of the short crack After the above discussion of the stress gradient with the propagation of short cracks at the sharp notch root, the variation of ΔK which associates with the fatigue stress concentrations Kf can be better understood. The Kf is defined as the ratio of the fatigue limit of a smooth specimen, S0, to that of a notched specimen, S0’, under the same experimental conditions. Obviously, it is related to the size and geometry of the specimen and dimension of the notch. However, to determine experimentally the Kf with the grand variation of the notch shape (from the blunt notch to the critical condition “crack”) is too laborious. Thus the analytical method based on some experimental results to predict the Kf is rather necessary. Indeed, if the crack is regarded as the critical condition of “notch”, the nominal stress which can initiate the existing crack will be equal to the value of S0’. That implies that the Kf is not only the key point for the crack initiation life analysis but also for the short crack growth estimations. In 1979, El Haddad et al. associated the fatigue limit with the crack threshold through the Eq. (3) – Eq.(4) [7]. ( ) ( ) ( ) 0 0 0 0 K a a K a K a a a η σ π Δ = ⋅Δ + ≥Δ =Δ ⋅ + (3) ( )2 0 0 0 (1/ ) /( ) a K S = ⋅ Δ ⋅Δ π η (4) Following Bazant’s reasoning in [8], a more general threshold expression as Eq. (5) was presented by introducing an adjustable parameter γ to fit experimental data as described by Eq.(5) ( ) / 2 1 0 0 0 K (a) K 1 a a γ γ − ⎡ ⎤ Δ =Δ ⋅ + ⎢ ⎥ ⎣ ⎦ (5) For the case where the crack is much smaller or much larger than the notch dimensions, the stress intensity range of a semi-elliptical notch has been mentioned in [9] with the expression in Eq.(6) and Eq.(7). ( ) 0 t K K a a η σ π Δ = ⋅ ⋅Δ ⋅ + for a << b (6) ( ) K a b η σ π Δ = ⋅Δ ⋅ + for a >> b (7) For the case of very small crack with a << a0, Δ σ is the notch root stress range; for the long crack with a >> b, Δ σ tends to the nominal stress Δ σn. In most cases, the Kt is used to indicate the stress concentration factor of the notch root without the crack growth. Hence, the function ϕ (a) in Eq. (8)
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