Issue34

A. Campagnolo et alii, Frattura ed Integrità Strutturale, 34 (2015) 190-199; DOI: 10.3221/IGF-ESIS.34.20 196 the region where realistic values of K II cannot be calculated. The extent of the linear portion, in terms of plate thickness, decreases as t/a increases but is still present when t/a = 3 [3]. This is in contrast with the discs results [2] where linear portions are less extensive and K II becomes essentially zero for s > 40 mm. Maximum values of K II are at the surface. This is within the region where calculated K II values are not realistic so caution is needed in the interpretation of results. D ISCUSSION here has been a lot of discussion on whether K III tends to zero or infinity as a corner point is approached [2,3]. When apparent K III values are calculated from stresses at a constant distance from the crack tip then K III appears to tend to zero as the model surface is approached (Fig. 5-6), in accordance with the linear elastic prediction. However, apparent values of K III at the surface (Fig. 4) increase strongly as the distance from the crack tip at which they are calculated decreases. These results can be interpreted as indicating that K III tends to infinity at a corner point in accordance with Bažant and Estenssoro’s prediction. The results in Figs. 5-6 also show that K II does appear to tend to infinity as the surface is approached, in accordance with Bažant and Estenssoro’s prediction. The discussion is futile because, as pointed out by Benthem [9], K III is meaningless at a corner point and there is no paradox. For s ≥ 0.25 mm the value of λ calculated from τ xy is close to the theoretical value of 0.5 for a stress intensity factor singularity so K II provides a reasonable description of the crack tip stress field. Similarly, K III provides a reasonable description of the crack tip stress field for s ≥ 1 mm. At the surface, values of λ obtained from τ xy are always less than the theoretical value for a corner point singularity. The distribution of τ yz at the surface (Fig. 2) cannot be accounted for on the basis of Bažant and Estenssoro’s analysis. There is clear evidence of a boundary layer effect whose extent is independent of the thickness. The only available characteristic dimension controlling the boundary layer thickness is the crack length, a . S TRAIN ENERGY DENSITY THROUGH THE DISC AND PLATE THICKNESS he intensity of the local stress and strain state through the disc and plate thickness can be easily evaluated by using the strain energy density (SED) averaged over a control volume embracing the crack tip (see Ref. [13] for a review of the SED approach). The main advantage with respect to the local stress-based parameters is that it does not need very refined meshes in the close neighbourhood of the stress singularity [19]. Furthermore the SED has been considered as a parameter able to control fracture and fatigue in some previous contributions [14-18, 22] and can easily take into account also coupled three-dimensional effects [2, 3, 23, 24]. With the aim to provide some numerical assessment of the contribution of the three-dimensional effects, specifically the coupled fracture mode, K II , the local energy density through the disc and plate thickness has been evaluated and discussed in this section. Figure 7 : Discs case: through the thickness SED distribution for t/a = 0.50, 1, 2, 3. Control radius R 0 = 1.00 mm. T T