PSI - Issue 25

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

274

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

7

the-thickness cracks or Berto et al. (2016) for notched plates, have proved that K N I and K N

II depend on the z-coordinate;

this behavior is shown in Fig. 2 for cracks.

(a)

(b)

Fig. 2: Primary Mode I and II and induced Mode 0 normalized stress intensity factors variation along the thickness for di ff erent Poisson’s ratios ν from He et al. (2016). (a) Mode I SIF variation for ν = 0 . 15 (1), 0 . 30 (2), 0 . 50 (3). (b) Primary Mode II and induced Mode 0 SIF dependence on z for ν = 0 . 1 (1), 0 . 30 (2), 0 . 50 (3) Primary modes term indicates the principal fracture modes obtained directly by the imposed loading conditions, whereas the term coupled modes emphasises that the fracture modes, with the corresponding SIF, are induced by the primary modes. In Fig. 2 the SIFs are normalized with respect to the far-field (asymptotic) stress intensity factors K ∞ I and K ∞ II , which have been adopted to apply the 2D far-field displacement (plane stress) solution (William’s solution) as boundary condition; the distinction between far-field and local modes is necessary, in light of the dependence of these latter from the z-coordinate. Besides, in Fig. 2b the induced coupled Mode 0 K 0 is shown varying with the Poisson’s ratio and achieving the maximum near the free plate surface, approaching zero when this latter is reached (more details in the next paragraphs), as well as K I depends also on the Poisson’s ratio. Defining the local in-plane (notch) stress intensity factors as in Eq. (14) and (15), but introducing the dependence of z in σ θθ ( r , θ, z ) and τ r θ ( r , θ, z ), and considering the linearity of the elastic problem (LEFM), the functional relationship between the local and applied in-plane (N)SIFs reads as: K I ( z ) = K ∞ I F I z h , ν (17) K II ( z ) = K ∞ II F II z h , ν (18) where F I and F II are dimensionless functions dependent on the z-position and Poisson’s ratio, as clear from Fig. 2. These local in-plane (N)SIFs do not depend on the thickness.

4. Coupled Modes

In the last three decades the existence of shear coupled modes, i.e. Mode II and III, has been proven (e.g. Nakamura and Parks (1989),Harding et al. (2010),Berto et al. (2011a),Kotousov et al. (2013)). Additionally, even non-singular shear modes can induce singular coupled modes. Specifically, non-singular terms in the William’s expansion (higher order terms) of the far-field in-plane shear stress can induce the singular out-of-plane mode (Mode 0), as numerically proven by Berto et al. (2011a) for a cracked plate. The same situation also occurs for sharp V-notches with opening