An approximation in closed form for the integral of Oore–Burns for cracks similar to a star domain

Abstract

A B S T R A C T In this paper, we give an explicit new formulation for the three-dimensional mode I weight function of Oore-Burns in the case where the crack border agrees with a star domain. Analysis in the complex field allows us to establish the asymptotic behaviour of the Riemann sums of the Oore-Burns integral in terms of the Fourier expansion of the crack border. The new approach gives remarkable accuracy in the computation of the Oore-Burns integral with the advantage of reducing the size of the mesh. Furthermore, the asymptotic behaviour of the stress intensity factor at the tip of an elliptical crack subjected to uniform tensile stress is carefully evaluated. The obtained analytical equation shows that the error of the Oore-Burns integral tends to zero when the ratio between the ellipse axes tends to zero as further confirmation of its goodness of fit.Keywords 3D weigh function; fracture mechanics; stress intensity factor. N O M E N C L A T U R Ea,b = dimensionless semi-axis of an elliptical crack a; b = actual semi-axis of an elliptical crack e = eccentricity of ellipse k I = mode I stress intensity factor for a dimensionless domain u , v = auxiliary dimensionless coordinate system x , y = dimensionless Cartesian coordinate system x; y = actual Cartesian coordinate system E(e) = elliptical integral of second kind K I2 = Taylor expansion up to second order of K I for an ellipse K(e) = elliptical integral of first kind K I = mode I stress intensity factor K Irw = mode I stress intensity factor from Irwin's equation Q = point of Ω Q ' = point of crack border δ = size of mesh over crack Δ = distance between Q and ∂Ω σ n = nominal tensile stress in x; y actual Cartesian coordinate system σ = nominal tensile stress in x; y dimensionless Cartesian coordinate system Ω = crack shape ∂Ω = crack border I N T R O D U C T I O NThe advantages of the use of weight functions for the assessment of stress intensity factors (SIFs) are well known in the literature, especially when many loads act on the component. For each geometry, we have to estimate the correct weight function related to the location where the crack nucleates and then propagates under fatigue loading. For a correct evaluation of the SIF, the proper weight function should be calculated;Correspondence: P. Livieri.

The Stress Intensity Factor of convex embedded polygonal cracks

“…In the case of a square-like flaw, the Oore-Burns integral can be analytically expressed in simplified form in the middle of the side and in the middle of the rounded corner [24]. The Oore-Burns integral will be approximated by means of Riemann sums plus a suitable asymptotic correction in terms of mesh size [25,26].…”

“…( 12) appears precise and modifies the value of the SIF only near the corner. The value of K I,OB was evaluated by means of the procedure valid for a star domain proposed in reference [26] where a mesh with a step of 0.0157 radiant in the tangential direction and 80 terms in the Fourier series have been considered.…”

Stress intensity factor for small embedded cracks in weldments

“…In Fig. 4, a disc is put at the weld toe of the welded T-joint and the stress intensity factor is evaluated by considering the asymptotic stress field as reported in [22][23]. The crack lies along the bisector at variable distance d from the weld corner.…”