Issue 50

P. Livieri et alii, Frattura ed Integrità Strutturale, 50 (2019) 613-622; DOI: 10.3221/IGF-ESIS.50.52 614 simplifications in the form of a defect is common and it is comparable to relatively simple geometric shapes of which the solution is already known. Usually, by simplifying, the defect is compared to an elliptical equivalent crack [7], while in order to simulate the propagation phase, the defect is continuously assimilated to a succession of ellipses [8–9]. The weight function introduced by Oore-Burns (OB) allows the value of the SIF of three-dimensional cracks under mode I loading with a generic shape to be calculated [10]. Although not exact, this weight function, for the three-dimensional case [11], is an excellent approximation that retains all the typical characteristics of the weighted functions in [12]. In particular, in the case of a semi-axial ellipse (1, b), when eccentricity e tends to zero, the main contribution of the Oore-Burns integral differs from Irwin’s analytical solution [13] for an amount equal to 2 20 e  [14]. The advantage offered by the Oore-Burns integral is to use a generic shape of the defect and the evaluation of SIF at each point of the creak contour without the need to consider the elliptical shape. Unfortunately, the difficulties associated with the analytical calculation of the integral result in the user choosing a particular grid [12, 15]. The resolution strategy proposed in [15] is not always easy to reach the convergence condition. However, it should be emphasised that the Oore-Burns integral is recommended by ASM standards in the case of isolated three-dimensional cracks to have an estimate of the SIF value throughout the profile of a real crack. The designer is therefore motivated to find a solution in closed form for a rapid evaluation of the SIF of a crack with generic form [16]. In this paper, we first summarise an analytical technique to get the Oore-Burns integral as an exact solution. Then, for a circular crack, a simple general equation for the SIF estimation is proposed and, as an example, it is used in the case of welded structures. Finally, the Oore-Burns integral is very well approximated by means of an explicit form for an irregular crack shape similar to a star domain under uniform tensile loading. A NALYTICAL BACKGROUND et  be an open bounded simply connected subset of the plane as in Fig. 1. We define: 2 ( ) ( ) ds f Q Q P s     (1) where ( , ) Q Q x y   , s is the arch-length parameter and the point P(s) runs over the boundary  . In their famous work in 1980, Oore-Burns proposed the following expression for the mode I stress intensity factor for a crack subjected to a nominal tensile loading σ n (Q) evaluated without the presence of the crack: 2 ( ) 2 ( ') , ' ( ) ' n I Q K Q d Q f Q Q Q         (2) Under reasonable hypothesis on the function σ n (Q), the integral (2) is convergent and the proof is based on the asymptotic behaviour of f(Q) [17]. Figure 1 : Inner crack L