13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- The crack growth rate can be correlated with the corrected stress intensify factor �KI by a power-law relationship (Paris relationship), which is a notable scaling law [16], between the fatigue-crack growth rate per cycle, da/dN, and the stress intensity factor range, �K. This relationship has been shown to apply over a wide range of cyclic crack growth rates for metallic, polymeric and ceramic materials. For the limited number of foil-thickness materials that have been studied, the m value of is higher than that commonly found for bulk (thicker) samples [5,6]. The Paris relationship (Eq. 1) is an empirical equation that provides little information of the effect that other parameters might influence cyclic crack growth behavior. Obviously, the average crack growth depends on (i) fatigue loading parameters, i.e. loading ratio R and frequency f, (ii) material properties, such as yield strength, �y and fracture toughness, Kc and (iii) specimen size. To examine the effect of various parameters on fatigue crack growth behavior, following the mathematical similarity approach from the work of Barenblatt et al [17] and Ritchie [18] crack growth rate can be expressed as: 2 , , , y c da K K R z f t dN K σ ∆ ∆ = Φ (4) z is the basic similarity parameter, given in terms of material constants and a characteristic specimen length scale h: y c h z K σ = (5) If incomplete similarity is assumed in the parameter �K << Kc, then ( ) ( ) ( ) 2 1 2 1 , , , c y c K K da R z R z K dN K α α α σ + ∆ ∆ Φ= Φ = Φ (6) Comparison with the Paris relation (Eq. 1) gives, ( ) 1 2 , , 2 ( , ) y C R z C m R z Kα α σ Φ = = + (7) where � is a function of loading ratio R and similarity parameter z. From Eqs. 6 and 7, the incomplete self-similarity infers that the constants C and m in the Paris power law relationship depend not only experimental parameters, such as �K, and material parameters, but also (through the similarity parameter z) on a specimen length scale h. For a set of loading parameters, the exponent � (and therefore m in Eq. 7) depends only on the parameter z. It has been first reported by proposed et al [17] and interpreted by Ritchie [18] that there is a trend of increasing Paris exponents m with increasing z corresponding to the transition from ductile (low m value) to brittle fracture (high m value). For the fatigue crack growth behavior of the Ti foils investigated, the experimentally measured Paris constant m varied between 4 to 5, which is slightly higher than the m value typically associated with bulk metal materials. The result implies accelerated fatigue crack growth rate due to foil thickness effect. According to incomplete self-similarity method, higher m value occurs with increasing z value and it corresponds to a transition from ductile fatigue fracture (m = 2~4) to more brittle fatigue fracture (m > 4). The higher m value induced by higher z value results from the competing mechanism between decreasing sample characteristic dimension h, fracture toughness Kc and increasing yield strength �y in the similarity parameter of z. Taking specimen thickness as the characteristic length scale h, would lead
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