PSI - Issue 25

Marco Maurizi et al. / Procedia Structural Integrity 25 (2020) 268–281 M. Maurizi and F. Berto / Structural Integrity Procedia 00 (2019) 000–000

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angles greater than ∼ 102 . 6 ◦ , for which the Mode II eigenvalue λ II > 1, meaning that a non-singular Mode II stress field is obtained, while the Mode 0 eigenvalue still λ 0 < 1 (out-of-plane singularity). This latter has been reported by Harding et al. (2010), as shown in Fig. 3, and extensively investigated by Berto et al. (2011c).

Fig. 3: Eigenvalues as function of the V-notch opening angle for Mode II and 0. Comparison between analytical and numerical (symbols) results by Harding et al. (2010)

The validity of the previous 3D analytical model (Eq. (9) and (10)) for the strength of the stress singularities (1 − λ ) has been confirmed through finite element simulations by Harding et al. (2010); the good match between the analytical and numerical solutions λ (2 α ) (2 α is the opening angle) is highlighted in Fig. 3. Besides, from Eq. (9) and (10), confirmed by 3D finite element models (Berto et al. (2013a)), it can be deduced that the singular Mode II induced by anti-plane loading has the same eigenvalues of the shear Mode II (in-plane loading); vice versa, the characteristic equation ( cos ( λ 3 a = 0)) for the singular Mode 0 induced by in-plane shear loading is the same as for Mode III, even if the two modes have a completely di ff erent nature (next paragraph). Also, non-singular anti-plane loading of cracks, i.e. K III = 0, can cause singular coupled Mode II, which shows a strong K C II variation along z and a dependence on the plate thickness. The possibility to have a singularity at the crack / notch tip surface even if the applied mode is not singular could strongly contribute to failure initiation. The Averaged Strain Energy Density (ASED) approach might well capture the failure location along the thickness of a cracked brittle plate due to a singular coupled Mode 0, despite the non-singular nature of the applied in-plane loading, as documented through finite element modeling by Berto et al. (2011a). One of the first three-dimensional finite element analyses dedicated to studying the 3D e ff ects in fracture mechanics has been conducted by Nakamura and Parks (1989), who investigated the out-of-plane induced mode (Mode 0) on a thin elastic cracked plate under Mode II loading. They noticed that the coupled Mode 0 K 0 is zero at the mid-plane and it increases along the thickness up to a maximum on the intersection of the crack front with the free surface (vertex singularity), assuming higher values as the Poisson’s ratio goes up. This was consistent with the results of Bažant and Estenssoro (1979), who predicted a K 0 approaching infinity, but in contradiction with the boundary condition of zero shear stress ( τ z θ ) at free surfaces. More recent numerical results, such as Harding et al. (2010), Berto et al. (2011b), Berto et al. (2011c) confirmed the latter conclusion, i.e. K 0 →− 0 at free surfaces, arguing that too coarse meshes have been utilized in previous studies. The discussion about the e ff ect and the theoretical description of the corner point singularity still survives in the scientific community, as later briefly reported. An opposite trend occurs for the applied Mode II close to free surfaces. These trends are schematically shown in Fig. 2b and the results of finite element simulations of Kotousov et al. (2013) are reported in Fig. 4b. In Fig. 4a the typical local out-of-plane deformation due to in-plane shear loading close to a crack surface front is highlighted; the displacement field shown is deliberately exaggerated. This behaviour and trend of (N)SIFs around cracks and notches have been observed in several works by FE analyses, citing only as example Harding et al. (2010), Berto et al. (2011b), Berto et al. (2012), Kotousov et al. (2013) for sharp and blunt V-shaped notches, Berto et al. 4.1. Mode 0 of Cracks and Notches Under Shear Loading