Issue 41

F. Berto, Frattura ed Integrità Strutturale, 41 (2017) 475-483; DOI: 10.3221/IGF-ESIS.41.59 481 Through-the-thickness distributions of K II and K III for t/a = 1 is shown in Fig. 7. From Tabs. 1 and 2 the values of K II are not realistic for s < 0.25 mm and the values of K III are not realistic for s < 1 mm. The distribution of K III presents a maximum at the centre line, but K III then remains nearly constant for about half the distance to the plate surface. The influence of plate bending means that maxima steadily decrease as t/a decreases. As a surface is approached K III first decreases slightly then increases to a maximum at about 0.15 mm from the surface. There is then an abrupt drop which is within the region where realistic values of K III cannot be calculated. Plate bending theory [2] suggests that K II should be zero on the centre line, with a linear increase towards a surface. K II does indeed increase linearly for much of the thickness with a greater increase as the surface is approached. This is within 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. 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 [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. 7), in accordance with the linear elastic prediction. However, apparent values of K III at the surface, linked to finite values of τ yz (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 Fig. 7 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 ≥ 0.2 mm 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, and decrease with increasing plate thickness. The distribution of τ yz at the surface (Fig. 4) 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 plate thickness. The only available characteristic dimension controlling the boundary layer thickness is the crack length, a . S TRAIN ENERGY DENSITY THROUGH THE PLATE THICKNESS he intensity of the local stress and strain state through the 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 [14-16, 34-35] and can easily take into account also coupled three-dimensional effects [4,22]. 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 plate thickness is evaluated and discussed in this section. Fig. 8 shows the local SED variation across 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 thickness. The influence of the applied mode III loading combined with the induced singular mode II loading is shown in Fig. 8. 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. In fact, as can be seen from Fig. 7, the maximum contribution of the coupled mode II, at the lateral surface, is significantly higher (about 4 times) compared to the maximum contribution of the applied mode III, at the mid plane. T T