PSI - Issue 26

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347

342

M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

7

Fig. 5: Stress singularity under mode 1 along the notch front, varying the notch opening angle and the Poisson’s ratio. (a) Stress singularity 1 − λ 1 vs. z / h , notch opening angle 2 α = 30 ◦ . (b) Stress singularity 1 − λ 1 vs. z / h , notch opening angle 2 α = 60 ◦ ; extended plot including nu = 0 . 45 is reported in Fig. A.9. (a) and (b) show a zoom in the range 0 . 8 < z / h < 1.

field corresponding to the applied (plane stress) displacement field of Eq. (1aa-b), hence confirming the correctness of the assumption R = h . The first notable trend corresponds to ν = 0; in fact, for both the opening angles the stress singularity remains constant along the notch front, corresponding to the 2D stress solution (Williams (1952)). As the Poisson’s ratio becomes positive, 1 − λ 1 starts increasing (relative di ff erence greater than 0.1 %) at about z / h = 0 . 95 − 0 . 90 for ν = 0 . 1 and 0.3, respectively, i.e. at ∼ 2 . 5 − 5 % of the plate thickness (2 h ) of distance from the free surface, reaching a maximum just before ( < 1 % of the plate thickness) the free surface and falling to values smaller than the far-field singularity at z / h = 1. This e ff ect, i.e. maximum and minimum in the proximity of the free surface, is amplified as the Poisson’s ratio increases, as also evident from Fig. A.9, in agreement with previous results for cracks by Shivakumar and Raju (1990); De Matos and Nowell (2008). The region where the stress singularity at the crack front deviates from the far-field 2D solution and oscillates was called boundary layer zone by Shivakumar and Raju (1990), and it represents a small area close to the free surface, whose size (between 0 and 5 % of the plate thickness) depends on ν , where Eq. (3), applied to cracks and with λ 1 not being function of z , might loose its meaning, and a sum of two singularities, i.e. the asymptotic and vertex singularity, considered instead. The mentioned authors proved that inside such region the separation of variables assumption fails, meaning that the stress singularity and the displacement power-law exponent, computed by the log-log regression analysis of σ yy and u y , respectively, depend on the angle θ . However, at the inner and outer boundary of this region, i.e. at the beginning of the deviation from the far-field value and at the free surface, the single power-law equation was demonstrated (Shivakumar and Raju (1990); De Matos and Nowell (2008)) to be suitable to describe the near-crack-front stress field. Despite the evidence that shows the limit of a single power-law model for cracks, a clear proof of the suggested model (sum of two singularities) is still not available. Extending the reasoning also to sharp v-notches, object of the present work, and for lack of other information, we limited our analysis to assuming that Eq. (3) as well as the separation of variables is valid at the extreme of the boundary layer, as done by Shivakumar and Raju (1990) and De Matos and Nowell (2008); further investigations are left for future works. On the other hand, merely as a speculative approach, it could be argued that Eq. (3) may be corrected by including the dependence of λ 1 on θ , without having anyway a physical proof of the validity. The stress singularity seems to have an opposite trend for negative values of Poisson’s ratio. Indeed, 1 − λ 1 has a minimum just before the free surface, where it suddenly goes up. A weaker change of the singularity at the free surface with respect to the far-field solution, compared to that corresponding to the positive Poisson’s ratio counterpart (e.g. ν = − 0 . 3 and ν = 0 . 3), appears to be evident for both the opening angles. Fig. 6a can help to deeply understand this latter consideration and to highlight the 3D corner point singularity dependence on the Poisson’s ratio for sharp V-notches. The vertex singularity 1 − λ 1 c was obtained from the slope of the near-notch-front displacement component u y at the free surface in the log( u y )-log( r ) space (see Fig. A.10), evaluated at θ = γ (Fig. 2b for notation). Since the displace-