A closed form for the Stress Intensity Factor of a small embedded square-like flaw

In the present work, the stress intensity factor (SIF) of a small embedded square-like flaw is calculated by means of a procedure based on the Oore-Burns integral. An explicit equation is given to evaluate the SIF along the two axes of symmetry that correspond to the points where the SIF takes its maximum and minimum value on the contour crack. The SIF is calculated in accordance with FE numerical results.


INTRODUCTION
he analytical evaluation of the stress intensity factor (SIF) of three-dimensional cracks is more complicated than cases where the crack is sketched as a straight line in a two-dimensional boundary. Few examples of convex threedimensional cracks can be found in textbooks [1][2][3]. In order to give a general formulation for the SIF, Bueckner [4] and Rice [5] performed the weight function technique. In this way, the calculation is reduced to an integral exactly over the region covered by the crack and this becomes particularly useful in the case of three-dimensional cracks. The exact weight function is known only in few cases.. Galin evaluated the exact weight function for the circle but the equation is not simple to manage due to the presence of a singularity typical of the weight function itself (see reference [3]). However, the authors, by means of the Oore-Burns (OB) weight function, give a simplified formulation of the weight function of the analytical circle where the singularity disappears by means of an appropriate change of variable [6]. This was performed because the OB integral is exact on circles. In others cases, the OB integral is a first level of approximation for the SIF in the case of circle-like cracks [7]. Usually, for irregular contours, a crack-like elliptical crack can be adopted [8][9]. For example, in flaw characterisation adopted in the fitness-for-service procedure, the flaw is modelled by a simpler geometry such as a trough crack with a straight crack front or with an elliptical or semi-elliptical shape. In this cases, the OB integral should be adopted because the characterisation of a semi-axial ellipse (1, b), when eccentricity e tends to zero, the main contribution of the OB integral T differs from Irwin's analytical solution [11] for a small amount equal to 2 20 e  [12], where e is the eccentricity of the ellipse. The OB integral could also be useful in the fatigue life assessment of materials with small internal defects. In order to obtain an acceptable approximation of the maximum SIF KImax, Murakami took into account the square root of the crack area where Y is a coefficient that was evaluated as best fitting for numerical and analytical results; see also [13]) of a general crack shape. The internal defect often has rounded corners and an analytical equation for a squarelike flaw could be useful for design purposes. In this paper, we use the Oore-Burns integral to obtain a closed form solution for a square-like flaw that is intermediate from a circular flaw and a square crack. The results are then compared with the FE results obtained with a fine mesh.

WEIGHT FUNCTION FOR A THREE-DIMENSIONAL CRACK: ANALYTICAL BACKGROUND
ig. 1 shows a two-dimensional crack inside a three-dimensional body subjected to a nominal tensile loading σ n (Q) that 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. We define: s is the arch-length parameter and point P(s) runs over the boundary  . In their pioneering work in 1980, Oore-Burns [10] proposed the following expression for the mode I stress intensity factor (SIF) for a twodimensional crack of boundary  : Under reasonable hypothesis on the function σn(Q), the integral (2) is convergent and the proof is based on the asymptotic behaviour of f(Q) [14].

STRESS INTENSITY FACTOR FOR A SQUARE LIKE-FLAW
n the case of a square-like flaw, as indicated in Fig. 2, the Oore-Burns integral can be analytically expressed in simplified form at the two points A and B. The Oore-Burns integral will be approximated by means of Riemann sums plus a F I suitable asymptotic (in terms of mesh size) correction. The equation of a unitary square-like flaw in a Cartesian coordinate system x,y is given in the form: Let us consider Q' as the point where the SIF is calculated. As a first calculation, we consider Q' as overlapping point A.
In order to evaluate the O-B integral, we take into account a new convenient Cartesian orthogonal reference system u,v with its origin in Q' with a u axis tangent to  (see Fig. 3).
The relation between the two reference planes can be summarised as follows: A mesh of size δ on  can be considered, where δ divides the length of  . P m in Fig. 3 is a point of coordinate mδ with respect to the initial point PO. On the x,y plane, the coordinates of Pm are equal to (xm, ym)= R(mδ) (cos(mδ), sin(mδ)) with m from 0 to M and 2Mδ=2π. From Eq. (3) we have: If we know the polar equation of the crack border ( ) R  , it is convenient to discretise the angle α instead of the arc length s. The angle α is equal to mδ. Now we consider: The coordinates of generic point P m = (x m , y m ) on the u,v plane become: we denote by Q jk =k  (cos(j )  ,sin(jδ)) the generic point on the plane (u,v) of the semi-circular mesh (see Fig. 4). In order to establish whether point Qjk of the coordinates (u,v) is inside the crack, the following inequality must be verified: In this paper, Eq. (11) is very simple to use with respect to the general equations proposed in references [15 and 16] for a star domain crack. From Eqns. (1) and (2), the SIF K IA at point A results: The sum (12) is made for 1 k N   and 0 /2 j M   . The value of N is calculated in order to obtain a crack inside the mesh as appears in Fig. 4. The asymptotic correction term, according to reference [16] is given by: where     Γ max , 0, 2        . Now we can evaluate the SIF also for point B as in Fig. 5. Eqns. (10)- (14) are replaced with Eqns. (15)- (19) and the mesh for Riemann sums is shown in Fig. 6.
The sum (17) In Eqns. (14) and (19), the value of  can be numerically evaluated from Fig. 7.  is close to 1.265.

NUMERICAL VALIDATION
qns. (12) and (17) allow us to numerically evaluate the SIF at the two characteristic points A and B of the squarelike crack without any specific computer strategies [17]. In this case, any generic mathematical code can be used to calculate the SIF. Tabs. 1 and 2 report the value obtained with Eq. (12) and (17). The stability of the equation is clear as well as the fundamental rule of asymptotic term C. Mastrojannis et al. [18] numerically solved the case of square crack with a small rounded apex. They found that the maximum SIF was assumed to be at the middle point of the edge and the maximum shape factor max ,max /( ) was equal to 0.76. Furthermore, for a sharp square crack by means of numerical methods, Helsing el al. [19] calculated a value for Y max equal to 0.75, Weaver [20] gave 0.74, whereas Isida et al. [21] gave 0.76. The results relating to a square-like crack are compared with those obtained by means of the FE code. The percentage of errors are around a few per cent for point B, while they increase up to 10% for point A. Figs. 8 and 9 show the model and the mesh used in the numerical FE analysis. The mesh used in this work is accurate as proposed in previous works [22][23][24][25]. The dimensions of smaller elements at the tip of the crack were in the order of 10 -5 mm. Fig. 10 highlights that the slope of stress σ z in a double logarithm scale is close to the theoretical value of 0.5 for point B [26]. The y' axis has the origin at point B.
Regarding the engineering applications for this topic, the fatigue behaviour of welded joints with internal defects such as pores could be appropriate. For example, the morphology and size distribution of gas pores within the entire weld joints, were examined in references [27][28]. By observing the typical spatial shape of the defects, sometime, the square-like flaw E u v kꞏδ jꞏδ B Qjk appears more appropriate than others shapes when a two-dimensional simplified representation of the defects is requested [13]. The pores show a rounded shape sometime away any ellipse typically used for simplified fatigue assessment [8].    Fig. 4

CONCLUSIONS
he Oore-Burns weight function gives us a closed formula for the estimation of the stress intensity factor of a squarelike flaw with a rounded corner. The errors, with respect to the FE results, are around a few per cent in the middle of the side, while they increase up to 10% at the corner. This suggests the need for a corrective procedure in order to significantly improve the Oore-Burns integral. After careful investigations, it is now clear, without any doubt, that the accuracy of the integral is not satisfactory at the high curvature points of the crack.