Issue 63

P. Livieri et alii, Frattura ed Integrità Strutturale, 63 (2023) 71-79; DOI: 10.3221/IGF-ESIS.63.07

In a previous paper [20] the authors investigated the failure of the O-integral in presence of high curvature for regular (  of class C 2 ) cracks. In terms of regularity, a corner Q’ means a singularity. This requires a new technique in order to correct the O-integral. In particular, in this paper we are interested in adjusting the O-integral for convex polygons. The analytical results will be compared with numerical ones obtained from an accurate three-dimensional FE analysis.

W EIGHT FUNCTION FOR A THREE - DIMENSIONAL CRACK : A NALYTICAL BACKGROUND

O

ore and Burns proposed a general equation for the evaluation of the SIF of a two-dimensional crack inside a three-dimensional body subjected to a nominal tensile stress σ n (Q). The nominal stress σ n (Q) is evaluated without the presence of the crack. Q is the inner point of the crack. The crack can be considered as an open bounded simply connected subset Ω of the plane as reported in Fig. 1. We define:

ds





f Q

( )

(1)

2

 Q P s

( )



where   ( , ) Q Q x y , s is the arch-length parameter and point P(s) runs over the boundary  . In reference [4] Oore Burns proposed an empirical formula for the evaluation of the mode I stress intensity factor at each point of the border crack of boundary  :

 n

( ) Q

2









, I OB K Q

d

Q

( ')

,

'

(2)

2

 

 f Q Q Q ( )

'

Under reasonable hypotheses on the function σ n (Q), the integral (2) is convergent and the proof is based on the asymptotic behaviour of f(Q) [21]. In different papers, the authors analysed the properties of the Oore-Burns integral where the accuracy of the equation was tested in the particular case of an elliptical crack [22, 23].

Figure 1: Inner crack.

C ONVEX POLYGON

I

n a previous paper [20] we investigated the proprieties of the O-integral in the presence of high curvature for regular (  of class C 2 ) cracks. Henceforth, we denote by  =  (Q’) the correction factor of Eqn. (2) at Q’ 

 

K Q

, I OB K Q

( ')

(Q')

( ')

(3)

I

In order to draw out the factor  , we need to take into account some hints:

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