PSI - Issue 26

Marco Maurizi et al. / Procedia Structural Integrity 26 (2020) 336–347 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

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performing the average of the obtained values along the distance from the notch tip r , in the range 0 to R / 10. The stress component to use in Eq. (4) depends on the mode of loading: yy -component for mode 1, and yz -component for anti-plane mode. It is worth noting that the stress components were extrapolated from integration points to nodes along the bisector line, giving rise to further numerical approximations. The results were therefore confirmed by performing the log-log regression also on the displacement field, which behaves like r λ and it is directly extracted at the nodes. This latter procedure resulted to be useful at the plate surface, where the corner point singularity influence might create numerical errors, as mentioned.

3. Results and Discussion

After mesh convergence tests on the FE model of Fig. 3b, the numerical simulations under mode 1 allowed to characterize the near-notch-front stress field and to understand the mechanisms behind the causes of the 3D corner point singularity and its e ff ect on brittle failure of materials subjected to in-plane tensile loading. The simulations of the plate (Fig. 3b) subjected to anti-plane loading (mode 3) were performed to observe the vertex singularity e ff ect on the out-of-plane shear stress component.

3.1. In-plane Mode 1 loading

In Fig. 4a the generic stress component σ yy as a function of the notch front distance r , along distinct planes z / h = constant , is shown; whereas, in Fig. 4b the same variable is plot along the notch front at three di ff erent radii.

Fig. 4: Stress component on a sharply-notched plate with thickness 2 h = 20 mm, notch opening angle 2 α = 60 ◦ and Poisson’s ratio ν = 0 . 3, under mode 1. (a) Stress component σ y y along the bisector line vs. distance from the notch tip, for di ff erent z -planes. (b) Stress component σ y y along the notch front at three distinct radii. Unambiguously the stress σ yy intensity drops down as the free surface is approached, in agreement with what reported by Kotousov (2010) for the NSIF, with higher gradient as the notch front distance decreases (Fig. 4b). In Fig. 4a a slight bending of the line log( σ yy ) vs. log( r ) at the free surface occurs, probably due to the numerical approximation before mentioned. The stress singularity, i.e. the power-law exponent 1 − λ 1 of Eq. (3), for the opening angles 2 α = 30 ◦ and 60 ◦ , is plotted along the notch front ( z / h ), highlighting the e ff ect of the Poisson’s ratio, in Fig. 5. The range 0 . 8 < z / h < 1, in the proximity of the free surface, in Fig. 5 was selected to highlight the singularity behaviour due to the corner point, while the upper value of the Poisson’s ratio range was limited to 0.3 for the opening angle of 30 ◦ (extended plot in Fig. A.9 for 2 α = 60 ◦ ) due to numerical "instabilities" of the computed stress near the free surface due probably to the stronger e ff ect of the vertex singularity (which reflects on the extrapolations from the integration points of the FE model), symptom of a powerful Poisson’s ratio influence. The asymptotic trend of 1 − λ 1 towards the 2D eigenvalue solution (Williams (1952)) for z / h < 0 . 8 is clearly visible (values down to z / h = 0 not shown for clarity) for both the notch opening angles, recovering at the mid-plane the 2D singularity of the stress