PSI - Issue 26

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

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

Fig. 6: (a) 3D corner point stress singularity vs. Poisson’s ratio ( ν ) for notch opening angles of 30 ◦ and 60 ◦ , and 24 ◦ and 96 ◦ extracted from the work of Bažant and Estenssoro (1979), indicated as [Previous work] in the plot. (b) Size of non-uniform longitudinal strain ( ε zz ) in terms of % of the plate thickness (2 / h ). The inset shows the strain component ε zz around the corner point for 2 α = 60 ◦ and ν = 0 . 3, highlighting the region aroud the corner point where the non-uniformity occurs.

ments are directly computed at the nodes, their magnitude is not a ff ected by the extrapolation from the integration points, as for the stresses, whose values at the free surface resulted to be abundantly distorted. Therefore, the slope of u y at the free surface as determined is the 3D corner point singularity. This latter strongly depends on the Poisson’s ratio, especially for positive values. In fact, as this material parameter decreases, down to ν = − 0 . 3, the gradient of 1 − λ 1 c does so. To have a benchmark, values extracted from the work of Bažant and Estenssoro (1979) are plotted in Fig. 6a. Despite di ff erent values of notch angle are available in literature, Fig. 6a shows that the singularity monoton ically decreases for rising values of Poisson’s ratio, in a similar fashion, also in agreement with the results obtained by Bažant and Estenssoro (1979); Benthem (1980) for cracks under mode 1, collected by Pook (1994). Moreover, looking for example at the data for ν = 0 . 3, the nonlinear increment of the singularity as the angle changes stands out. Catastrophic brittle failure of materials caused by stress raisers, such as notches, occurs suddenly when a certain ex ternal load limit is reached, provoking a crack initiation and rapid propagation. Therefore, neglecting the conditions of crack propagation, which necessarily force one to taking into account the e ff ects of the intersection angles (Fig. 1), the only parameter involved in 3D corner point singularities is the Poisson’s ratio. This latter material constant can hence be considered the cause behind the mechanism of vertex singularity, at least under mode 1. The inset in Fig. 6b features the contour plot of the strain component ε zz around the corner point, which unambiguously produces a non-uniform region in terms of lateral contraction (in z -direction, Fig. 2a) of the plate, undermining the hypothe sis of generalized plane strain (Lazzarin and Zappalorto (2012)), almost always adopted to formulate 3D analytical closed solutions for elastic problems. The depth of such region along the notch front was deduced by identifying the z -coordinate at the notch tip for which the function ε zz ( z ) started changing, compared to the uniform far-field value (at the mid-plane). The estimation of the relative non-uniform zone size with respect to the plate thickness, displayed in Fig. 6b, steeply rises for positive values of ν , remaining nearly 0 % in the negative range of Poisson’s ratio, leading to an asymmetric behavior, also confirmed by the singularity at the vertex (Fig. 6). The region of non uniformity can be considered as the boundary layer previously defined; in fact, the order of magnitude for positive Poisson’s ratio resembles that obtained by Shivakumar and Raju (1990); De Matos and Nowell (2008). It is further worth noting that at the higher opening angle, 60 ◦ , corresponds a greater region size, especially at ν = 0 . 45, and at ν = 0 the non uniformity disappears, confirming the Poisson’s ratio-leading nature of 3D corner point singularity around V-notches under mode 1. Finally, the normalized NSIF ( K I / K ∞ I ), whose critical value activates the brittle fracture, is featured in Fig. 7. Whilst inside the boundary layer region the validity of the single power-law is still unclear, computing the NSIF by means of Eq. (4) provides interesting information. The trend of σ yy for ν = 0 . 3 in Fig. 4b demonstrates a continuous decreasing in the proximity of the free surface for every radius r ; the opposite occurs for negative values (not reported here for brevity). Therefore, if at the extremes of the boundary layer region the model represented by Eq. (3) is