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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whose crack tip surface displacements are Mode I, and the antisymmetric mode, which is a combination of Mode II and III. Exploiting of the properties of J-integral in LEFM, recent numerical results of He et al. (2015) have confirmed the predictions of Bažant and Estenssoro (1979) and Benthem (1980) under the hypothesis of β = 90 ◦ for the local Mode I and II SIFs in the vicinity of vertex singularities, i.e. the behavior shown in Fig. 2. Bowed crack fronts on fatigue precracks in fracture toughness test specimens is an example of the e ff ect of corner point singularities, since a propagating crack front shape converges to β c . Despite the application of the analysis of Bažant and Estenssoro (1979) can be used to explain the behaviour of a crack front at a corner point, it needs to be revised. For instance, it predicts that, for a crack front intersection angle of 90 ◦ , K III (therefore also K 0 ) would tend to infinity close to the vertex singularity; however, this does not occur, as shown in Fig. 4, due to the free-transversal-shear-stress boundary conditions at free surfaces. On the other hand, also this statement could be confuted, arguing that Mode III (or Mode 0) is basically torsion, therefore, warp of sections leads to values of τ zy di ff erent from zero. However, this e ff ect does not justify high values (to infinity) as those predicted by Bažant and Estenssoro (1979). We think the question to be addressed would be, Can a unique stress intensity parameter completely describe the intensity of the singular stress field around a crack surface, including the corner point singularity ? Indeed, finite element results of Berto et al. (2017) on cracked plates and discs under anti-plane loading highlighted the non linear behavior of τ yz with the distance from the crack tip ( r ) on a log-log plot, leading to apparent values of K III (computed by the definition of SIF of Eq. (13)) dependent on r (Fig. 1). As a consequence, Berto et al. (2017) argued that the current knowledge seems to suggest that the singular stress field in the vicinity of a corner point may be sum of two singularities of di ff erent strength, not theoretically, numerically or experimentally proven yet. The most representative theoretical and numerical results on 3D e ff ects on LEFM have been provided and dis cussed. 3D analytical frameworks have been briefly presented, highlighting their key strengths and limits. These latter have been discussed referring to the finite element results available in literature, which prove the existence of coupled shear modes, as predicted by the theoretical frame. The good match between the analytical and numerical strength of the singularities has been underlined. At the same time, particular attention to the limit of the analytical formulation of (notch) stress intensity factors for Mode I and II, which does not imply a variation of K I and K II along the thickness, as numerically shown, has been payed. Besides, the theoretical problem of vertex singularities has been introduced, highlighting the main contradictions, and providing an interpretation, which might lead to future numerical, theoreti cal and experimental tests. What appears to be only theoretical speculation, it has actually practical implications. Design against brittle failure and fatigue crack growth of cracked and notched components is based on standards, which in turn are written fol lowing the scientific / engineering research. Lack of 3D applicable results has lead to leave the 2D analytical frame as base for tests and measurements. Coupled singular shear modes and their presence also for non-singular primary modes, and the variation of stress intensity factors along the thickness as well as the thickness scale e ff ect on the induced modes could shift the brittle failure location (from the mid-plane to the lateral surfaces or vice versa) or the expected critical point for fatigue crack initiation. Fatigue of welded lap joints is only a practical example for which these phenomena occur. Moreover, local e ff ects of corner point singularities may contribute to global change in the crack front geometry, which in turn provokes a variation in the stress field far away from free surfaces. We argue that one interesting approach to studying the 3D e ff ects on LEFM seems to be the average strain energy density, which has the ability to take into account the multiaxiality and the coupled modes. In scientific literature, it has been shown that this approach is also able to predict the shift in the failure location due to coupled modes. However, a theoretical breakthrough is needed to deeply understand coupled modes and vertex singularities and their e ff ects on the singular stress field ahead of crack tip surfaces. 6. Conclusions

Appendix A. Out-of-plane shear stress singularity

By means of Eq. (7) and (8), the anti-plane singular shear stresses and the correspondent displacement along the z-axis read as: