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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10

where b n are the coe ffi cients for the William’s expansion for the in-plane displacements ( u x and u y ) around a crack tip, whereas f 0 n are analogous to F 0 of Eq. (19). Eq. (20) highlights the existence of the coupled Mode 0 even for non-singular ( K ∞ II = 0) in-plane loading, still depending on the thickness.

4.2. Coupled Mode associated with anti-plane loading

The coupling between Mode II and III is not valid only to one direction: anti-plane loading generates in-plane shear singular mode (Mode II) for cracks and notches, as highlighted in Fig. 5a. The boundary conditions negate the transverse shear stresses at the free surfaces, inducing a coupled in-plane shear mode.

(a)

(b)

Fig. 5: Coupled Mode II induced by anti-plane loading. (a) FE example of local in-plane shear mode generated by Mode III loading. Deflections are expressly exaggerated. (b) Induced singular Mode II and primary Mode III stress intensity factor (normalized) variation along z and Poisson’s ratio influence. This latter is shown only for the primary mode because very weak influence is exhibited for the induced mode. Berto et al. (2013a) and Kotousov et al. (2013) investigated this phenomenon on cracks and sharp V-notches, obtaining the trends on the primary and coupled mode (N)SIFs reported in Fig. 5b. As the free surface is approached, the primary SIF K III ( z ) tends to zero, whereas the coupled in-plane SIF K II increases. Moving towards planes near the mid-plane, the induced SIF K c II ( z ) drops down to zero. Indeed, close to the mid-plane the stress field converges to the plane stress condition and the far-field K ∞ III is recovered, while the coupled mode disappears, leading to K c II →− 0. Analogously to the previous coupling, also non-singular anti-plane loading can give rise to singular coupled in-plane mode Kotousov et al. (2013). Besides, in contrast with in-plane loading, whose coupled Mode 0 vanishes for ν = 0, anti-plane loading conditions lead to coupled in-plane shear mode also when ν = 0. The di ff erence arises from the nature of the coupling: the coupled Mode 0 is caused by the transversal deformation due to the Poisson’s ratio, while the coupled in-plane mode is generated by a mechanism of redistribution of the transversal shear stress components close to the free surfaces, where they assume null values, as imposed by the free-stress boundary conditions. 4.2.1. Scale e ff ect It has been noticed by Berto et al. (2013a) that notched plates loaded in Mode III have a deterministic scale e ff ect, which has a similar form of Eq. (19):

, ν ,

II

z h

K c

( λ 3 − λ 2 ) F

II ( z ) = K ∞ III h

(21)