Issue34

A. Campagnolo et alii, Frattura ed Integrità Strutturale, 34 (2015) 190-199; DOI: 10.3221/IGF-ESIS.34.20 197 Figs. 7-8 show the local SED variation respectively across the disc and the plate averaged over a cylindrical volume having radius R 0 and height h , with h about equal to R 0 . In Refs [13, 14-16, 17-18] R 0 was thought of as a material property which varies under static and fatigue loading but here, for the sake of simplicity, R 0 and h are simply set equal to 1.0 mm, only to quantify the three-dimensional effects through the disc and plate thickness. The influence of the applied mode III loading combined with the induced singular mode II loading is shown in Figs. 7-8. For the discs results the ratio between the maximum values of K II and K III is quite low (about 2) as can be seen from Fig. 5, so that the maximum contributions are not significantly different and the position of the maximum SED results to be a function of disc thickness [2]. For thin discs it is close to the lateral surface, but for thick discs the maximum SED is at the mid-plane and its value is about 1.5 times the value at the lateral surface. For the plate results (Fig. 6), indeed, the maximum contribution of the coupled mode II is significantly higher (about 4 times) compared to the maximum contribution of the applied mode III. It is evident that the position of the maximum SED is the same in all cases. It is close to the lateral surface, where the maximum intensity of the coupled mode II takes place, both for thin plates, t / a = 0.5 and 1.0, and for thick ones, t / a = 2 and 3. Figure 8 : Plates case: through the thickness SED distribution for t/a = 0.50, 1, 2, 3. Control radius R 0 = 1.00 mm. C ONCLUSIONS (1) The results obtained from the highly accurate finite element analyses have improved understanding of the behaviour of through cracked discs and plates under anti-plane loading. In particular, it is confirmed that mode III does induce coupled mode II c . (2) The influence of plate bending is increasingly important as thickness decreases. The anti-plane loading used is a nominal mode III loading. For thin discs and plates it is a mixed modes III and II loading, in which mode III induces mode II c and vice versa. At the present state of the art it is not possible to separate the coupled modes from the applied modes. (3) Bažant and Estenssoro’s analysis works well for the symmetric mode (mode I), but it is incomplete for the asymmetric mode (a combination of modes II and III). (4) Discussion on whether K III tends to zero or infinity as a corner point is approached is futile because, as pointed out by Benthem, K III is meaningless at a corner point. (5) The present results do not confirm the existence of a corner point singularity dominated region within a K -dominated region. It appears that a new field parameter, probably a singularity, is needed to describe the stresses at the free surfaces. (6) Calculation of the strain energy density (SED) in a control volume at the crack tip shows that the position of the maximum SED in the discs case is a function of the thickness. In the plates case instead the position of the maximum SED is independent of plate thickness, contrary to discs results.