PSI - Issue 37

Jesús Toribio et al. / Procedia Structural Integrity 37 (2022) 1029–1036 Jesús Toribio / Procedia Structural Integrity 00 (2021) 000 – 000

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Finally, to have a complete picture (proper of a review paper) of SIF solutions applicable to surface cracks in round bars, the British Standard BSI 7910 (1999) should be analyzed. This Standard includes SIF solutions for straight and semicircular crack fronts taken from the review paper by James and Mills (1988). Such solutions (in the form of polynomial equations) were obtained by curve fitting. For round bars in tension with straight-fronted edge cracks, the fitting by James and Mills was made from the previous results calculated by the experimental compliance method (Daoud et al . (1978), Bush (1981)) and by FEM techniques (Blackburn (1976), Daoud et al . (1978), Salah el Din and Lovegrove (1981), Mattheck et al . (1984)). In round bars in tension with circular surface cracks, James and Mills describe a fitting equation obtained by Forman and Shivakumar (1986) from numerical results using different methods: FEM (Astiz and Elices (1980), Salah el Din and Lovegrove (1981), Nezu et al . (1982), Trantina et al . (1983), Daoud and Cartwright (1985), Raju and Newman (1986), Nord and Chung (1986), Caspers et al . (1986)), BIEM (Johnson (1972), Athanassiadis et al . (1981)) and experimental techniques (Härkegård (1974), Wilhem et al . (1981), Mackay and Alperin (1985)). One problem of the analysis performed by James and Mills (1988) is the mixture of results at the crack center (A) and the cylinder surface (B), cf. Fig. 1, that increases the scattering due to the fact that the SIF is maximum at the crack center or at the cylinder surface depending on the crack aspect ratio. In addition, the evolution of SIF values with the crack depth in the British Standard BSI 7910 (1999) is made by interpolation of previous numerical results and changing from semicircular to straight crack front as the crack depth increases. The review and analysis provided by James and Mills tries to cover not only smooth round bars with surface cracks, but also cracked bolts, although most of the cited results are for (smooth) round bars and only the pioneering work by Nord and Chung (1986) is for cracked bolts. With regard to this, other pioneering works are those by Reibaldi (1986), Toribio et al. (1991a, 1991b, 1991c) and Toribio (1992a, 1992b, 1994). Paper by Toribio (1992a) is cited by the British Standard BSI 7910 (1999) as a source of SIF solutions applicable to crack in bolts, i.e., such a paper provides a reference solution for the aforesaid Standard. This paper clearly demonstrates the important effect of the crack aspect ratio on the SIF values at the crack center and at the cylinder surface and thus the serious flaw of the review by James and Mills (1988) when it mixes the SIF solutions at both locations. 3. Comparison between SIF solutions Most of the K -solutions for elliptical cracks reviewed in this paper represent two-parameter approaches depending on the relative crack depth a/D and the crack aspect ratio a/b , with two exceptions: (i) the early results for straight fronted edge cracks in round bars obtained by Valiente as a function of only the relative crack depth a/D ( one parameter approach); (ii) the most recent results for semi-elliptical edge cracks in round bars computed by Shin and Cai as a function of the relative crack depth a/D , the crack aspect ratio a/b and the specific position x/h of the considered point at the crack front ( three-parameter approach). Results are given in Figs. 2 to 5. All the researchers assumed free ends in the bar (i.e., unrestrained bending), with the exception of Shin and Cai who included also the case of a round bar with constrained ends (i.e., restrained bending). Plots associated with Carpinteri and Shih-Chen come directly form the numerical results (prior to fitting), while results by Valiente, Astiz, Levan-Royer, Couroneau-Royer and Shin-Cai are plotted after curve fitting to obtain analytical expressions. Fig. 2a plots the K -results at the crack center of straight-fronted cracks ( a/b =0). It shows that the global solution given by Valiente underestimates K , especially for the longest cracks. On the other hand, numerical results by Levan and Royer clearly overestimate K . In addition, results by Couroneau and Royer clearly contradict those by Levan and Royer. Other K -solutions such as those by Carpinteri or Shih and Chen are slightly above the average trend, the local solutions given by Valiente and Astiz appear as the best ones, since they plot just the average trend. The recent K solutions by Shin and Cai provide the engineer with an analysis of the effect of geometrical constraint of the anchorage, since both the cases of unrestrained bending (free ends: F symbol in the plots) and restrained bending (constrained ends: C symbol in the plots) are analyzed. Results show a decrease of stress intensity in the vicinity of the crack tip when bending of the sample is not permitted.