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

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stress intensity factor (SIF or NSIF), i.e. the stress field, along the crack / notch front edge, including the corner point, is needed; specifically, the failure location can change as these parameters vary. Assuming the values of the angles in Fig. 1 be equal to 90 ◦ , a few researchers, after Benthem (1977, 1980) and Bažant and Estenssoro (1979), have studied the 3D corner point singularities for stationary cracks. Shivakumar and Raju (1990) performed simulations by using 3D finite element models of middle-crack tension and bend specimens under mode 1 loading, studying the variation of the stress singularity along the crack front. These authors have shown that the near-tip stress field component can be expressed as σ yy = C ( θ, z ) r − 1 / 2 + D ( θ, Φ ) r λ 1 C − 1 , being sum of two components. The first one represents the classical stress field in polar coordinates ( r , θ , z ), vanishing at the corner point, whereas the second component identifies the vertex singularity, which dominates the field at the surface, and it goes to zero away from a region called boundary layer. This latter has a thickness depending on the Poisson’s ratio, reaching the maximum value at about 5 % of the specimen thickness for ν = 0 . 45. Despite the small region size of influence, the corner point singularity is able to modify the (N)SIF along the crack / notch front, moving therefore the failure location. With the goal of determining the e ff ect of the 3D vertex singularity and the Poisson’s ratio (despite they are strictly related to each other) on the plasticity-induced fatigue crack closure, De Matos and Nowell (2008) studied the stress singularity and SIF change along the crack front of a cracked plate under mode 1 through a log-log linear regression analysis of the near-tip stress and displacement fields and by means of the J-integral, obtained by three-dimensional finite element simulations. Although e ff ort has been put on the identification of the e ff ects of 3D corner point singularities on stationary cracks, a lack of studies on their inherent causes, extending the concept to sharp V-notches (highlighted only by Bažant and Estenssoro (1979); Kotousov (2010))), seen as the theoretical generalization of cracks Williams (1952), is evident. In the present paper, we address the problem of determining quantitatively and qualitatively the causes of 3D corner point singularities on sharp V-notches loaded in mode 1 in the framework of LEFM, through three-dimensional accurate finite element simulations. Moreover, Poisson’s ratio values from -0.3 to 0.45 are considered, investigating the e ff ect of a negative ν compared to the positive counterpart. A clear representation of the 3D vertex singularity in terms of notch stress intensity factor is hence provided. Finally, a few observations on the e ff ect of the surface singularity on sharp V-notches under anti-plane loading are given, discussing previous explanations provided by some authors for cracks. To make the work clear and repeatable, we report in Fig. 2 the adopted notation for the stress field near the sharp V-notch tip and the relative geometrical features.

Fig. 2: (a) Notation for the stress field close to the V-notch and corner point in cylindrical coordinates ( r , θ , z ). Cartesian local coordinate system ( x , y , z ) centered at the V-notch tip on the mid-plane. (b) Geometrical features of the sharp V-notch.

2. Numerical experiments

In the theoretical framework of LEFM, to understand the causes and e ff ects of 3D corner point singularities on the stress state in the region close to the intersection between the sharp V-notch front and the surface (Fig. 3b), numerical experiments were performed by exploiting of three-dimensional accurate finite element models. Regarding to Fig. 3, in a generic body for which the hypotheses of LEFM hold, i.e. small scale plasticity in the process region, the 3D stress state region, whose size was quantified (Harding et al. (2010); Nakamura and Parks (1989) to be ∼ h , being 2 h the plate thickness, only for crack / notch under in-plane shear loading, is assumed to be included in a K-dominance