13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- H H HH H H H H HH H HH HH H H H H H H H H H H Nominal strain (%) Nominal stress (MPa) H-precharged Fracture origin: Al2O3・(CaO)x = 27.6m TiN Al2O3·(CaO)x Non-charged Fracture origin: Al2O3・(CaO)x = 23.6 m TiN Al2O3·(CaO)x 10 m 10 m 0 500 1000 1500 2000 2500 3000 0 1 2 3 4 5 6 7 8 9 10 Reduced variate, yʹ = ‒ln (‒ln (F)) Fracture origin: Al2O3·(CaO)x = 27.6 m Fracture origin: TiC = 11.6 m yʹ = 0.245 ‒ 4.373 Vs = 477 mm3 n= 14 Al2O3·(CaO)x TiN TiC 20 50 100 200 1000 2000 10 1 10 50 80 90 95 98 99 99.5 99.8 99.9 99.95 0 1020304050 Cumulative distribution function, F % -2 -1 0 1 2 3 4 5 6 7 8 max (m) Tʹ = 1/(1‒F) 10 m 10 m 500 where σ0 is the remote tensile stress, a' is the large semi-axis of the ellipsoid and λ is the crack length measured from the equator of the ellipsoid [15]. As shown in Fig. 10, KI for a crack with λ/a' of 0.2 is approximately equal to the stress intensity factor for a penny shaped crack with the radius of a'+λ. By considering such a steep growth in KI value together with the presence of the interfacial cracking or cracked inclusion in the early stage of the fracture process (cf. Fig. 8), it can be deemed that the ellipsoidal nonmetallic inclusions are mechanically equivalent to a penny-shaped crack. Such an equivalence has experimentally been confirmed for the small fatigue cracks and defects [7, 8, 10]. Accordingly, we approximate KTH in terms of σf and the inclusion size as follows: KTH = (2/π) σf ' πa (1) As illustrated in Fig. 8, most of the Al2O3·(CaO)x type inclusions yield a spherical-like cavity [6]. The shape of the cavity is determined by the inclusion shape, which is not perfectly spherical but is irregular in nature. Also in the case of the TiN type inclusion, the shape of the original crack generated λ b aʹ λ b aʹ ρ b=aʹ, b=2aʹ, = λ / aʹ 0 0.5 1.0 1.0 0.5 0 FI for penny shaped crack b= 0.5 aʹ b= aʹ b= 2 aʹ area x z y O σ0 σ0 KImax 0.5 σ0 KImax 0.65 σ0 x z y area σ0 σ0 Figure 8. Illustration of crack initiation processes from two kinds of non-metallic inclusions [6] Figure 9. Statistics of extremes distribution of inclusions at the fracture origin [12] Figure 10. Normalized stress intensity factor for an annular crack emanating from an ellipsoidal cavity [15] (a) Internal crack (b) Surface crack Figure 11. Stress intensity factor for planer cracks with arbitrary shapes [7, 8, 10]
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