PSI - Issue 25

Marco Maurizi et al. / Procedia Structural Integrity 25 (2020) 268–281

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

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where F II ( z / h , ν ) is a dimensionless function dependent on z and Poisson’s ratio, although this latter parameter does not have a great influence as for the Mode 0, as well as the local primary mode K II change with ν , in case of in-plane shear loading, is small compared to the correspondent induced mode K 0 . λ 3 corresponds to λ 3 a of Eq. (10), whereas λ 2 is the eigenvalue of the coupled mode, extending the definition of (N)SIF of Eq. (15) to the induced mode. It is worth noticing that the exponent of h is negative for notch opening angles greater than zero, meaning that brittle failure of thinner plates is more influenced by the in-plane coupled mode. For the crack case, the dependence on h disappears. The eigenvalue problem is indeed the same as before; therefore, the strength of singularity of primary and coupled mode is still governed by Eq. (9) and (10), whose behavior is shown in Fig. 3. The 3D analytical frameworks before mentioned are supposed to be valid in the mid-plane of plates, as proven by finite element calculations (Berto et al. (2016)). However, at the intersection of crack / (sharp)notch front with free surfaces a phenomenon occurs, the so called corner point (or vertex) singularity. The analytical formulation still lacks the understanding of this latter phenomenon. Bažant and Estenssoro (1979) were the first authors to introduce the concept of vertex singularities in crack fronts, later also studied by Benthem (1980) and Pook (1994). In the works of Bažant and Estenssoro (1979) and Pook (1994), the stress field near a vertex singularity is described by a stress intensity measure K λ , defined analogously to the (notch) stress intensity factor; nonetheless, spherical coordinates ( r , θ, Φ ) are used instead of polar coordinates, measuring the angle Φ from the crack front. Additionally, no explicit expressions are available for K λ , which is generically related to the characteristic crack dimension a (e.g. crack length) and the remote applied stress σ ∞ through an unknown dimensionless parameter Y , as K λ = Y σ ∞ ( π a ) λ . Stress and displacement field are proportional to K λ / r λ and K λ r 1 − λ , respectively, where r is the radial distance measured from the corner point in spherical coordinates. The strength λ of the corner point singularity has been found to be function of Poisson’s ratio ν and the crack front intersection angle β , shown in Fig. 6a (Pook (1994)); so, these three parameters are related, and knowing two of them, the third is deduced. 5. Corner Point Singularities

(a)

(b)

Fig. 6: (a) Crack front intersection angle. (b) Crack surface intersection angle.

The basic idea of Bažant and Estenssoro (1979) is that, from energetic and other considerations, a crack front tends to intersect the surface at a critical angle β c , such that λ = 0 . 5. Indeed, for small values of β the SIFs tend to zero at the corner point, whereas for large values of β they tend to infinity. Therefore, at the critical angle, SIFs are recovered, i.e. the singularity along the crack front is constant. As reported in the work of Pook (1994), for β < β c , λ < 0 . 5, hence, K I in plane stress conditions tends to zero at the corner point (deduced by considerations on the relation between crack profile and SIF, see Pook (1994)), while for β > β c , λ > 0 . 5 and K I tends to infinity as the vertex singularity is approached. As shown in Fig. 2a from numerical results, K I ( z ) drops down as the corner point is approached, meaning that a strength singularity at the vertex λ < 0 . 5 is suggested, from the previous point of view. Pook (1994) argued that even though the crack front tends to the critical angle, as experimentally proven for bowed crack fronts in plates of constant thickness, the transition between zero and infinity in the behavior of K I at the corner point is not sharp as theoretically predicted, and problems in the experimental measurement of β have been found to be crucial in the validation of the theoretical formulation. Besides, when β = γ = 90 ◦ , with γ defined in Fig. 6b as the crack surface intersection angle, two modes of the stress intensity measure are exhibited: the symmetric mode,