PSI - Issue 25

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

270

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

3

(a)

(b)

Fig. 1: (a) Notation for the stress field near the crack tip. (b) Notation for sharp V-notches. Small partial cylindrical zone around a V-notch represented in figure; the dashed red line represents the mid-plane.

Disclinations could be related to the behavior of elements of the crack / notch tip surface near free surfaces, the so-called corner point (vertex) singularity, investigated in detail by Bažant and Estenssoro (1979) and subsequently by Benthem (1980). They found out that in the vicinity of a corner, that is the intersection between a crack and a free surface, the singularity changes. A 3D analytical frame, which reduces the linear elastic problem to a system of one bi-harmonic equation, representing the in-plane problem, and one harmonic equation for the displacement field ( w ) along z axis, has been proposed by Lazzarin and Zappalorto (2012); Zappalorto and Lazzarin (2013). Although it well approximates the finite element solutions at the middle plane of a plate Berto et al. (2016), it is believed to fail on the free surfaces Zappalorto and Lazzarin (2013), where vertex singularities act. Despite this, the 3D frame predicts what have been in the last two decades numerically observed Harding et al. (2010); Berto et al. (2011c); Kotousov et al. (2013), i.e. the coupling between the anti-symmetric modes II and III. They are thought to be related to each other through the Poisson’s ratio and the thickness, even if these two parameters do not have the same influence on the coupled mode induced by in plane or anti-plane loading. Therefore, a shear (II or III) isolated mode cannot exist. Despite the e ff orts made by many researchers, a complete understanding and a theoretical description of 3D e ff ects on fracture mechanics still lacks. In light of this, to clarify what has been already done and to highlight contradictions and zones of no-consensus, we briefly review some 3D analytical formulations, the coupled modes and the vertex singularities.

2. 3D analytical formulations

Kane and Mindlin (1955) have laid the groundwork for the available analytical frameworks for the elastostatic 3D problem ahead of crack / notch tip. By studying the high-frequency extensional vibrations of plates, they proposed the following displacement field:

z h

u x = u ( x , y )

u y = v ( x , y )

u z =

w ( x , y ),

(1)

where 2 h is the plate’s thickness. Based on this assumption and on the Mindlin’s solutions in terms of stress resultants (Lazzarin and Zappalorto (2012)), Yang and Freund (1985) studied the 3D stress field in through-the thickness cracked thin plates. By means of the equilibrium equations, the constitutive model and the compatibility