PSI - Issue 37

Jesús Toribio et al. / Procedia Structural Integrity 37 (2022) 1029–1036 Jesús Toribio / Procedia Structural Integrity 00 (2021) 000 – 000

1031 3

(

) ( 1 2

) ) − 2

1 4

(1)

2 a D a D a D ) ( −

0.473 3.286 −

14.797(

Y

a D

=

+

The second SIF solution was numerically obtained by the finite element method using the stiffness derivative technique on the basis of a virtual crack extension to calculate the energy release rate, it yielding the following expression for the dimensionless SIF Y :

(2)

2 3 4 1.4408 3.6364( ) 19.3500( ) 34.7849( ) 36.8446( ) = − + − + Y a D a D a D a D

and the latter is a local K -solution valid for the center of a straight-fronted edge crack. Astiz (1986) obtained the energy release rate under plane strain conditions using also the stiffness derivative technique on the basis of a virtual crack extension. The author performed a polynomial fitting on the numerical results using the least squares method, so that the following expression was obtained for the dimensionless SIF Y :

4 3

( i 0 j 0 i 1 = =  =  Y C a D a b ) ( ) i ij

j

(3)

and the coefficients C ij are given in Table 1. Therefore, the equation (3) provided by Astiz is a two-parameter K solution dependent on both the relative crack depth a/D and the crack aspect ratio a/b , cf. Fig. 1. Table 1. Coefficients C ij of equation (3) i \ j 0 1 2 3 0 1.118 -0.171 -0.339 0.130 2 1.405 5.902 -9.057 3.032 3 3.891 -20.370 23.217 -7.555 4 8.328 21.895 -36.992 12.676 Carpinteri (1992a, 1992b) computed the dimensionless SIF for straight-front and semi-elliptical cracks using the finite element method and 3D isoparametric elements of 20 nodes. The stress r – 1/2 -singularity was modelled by using quarter-point singular elements close to the crack tip, i.e., by moving the mid-side nodes of the brick elements ahead to the crack tip to the quarter-point position. Levan and Royer (1993) calculated the dimensionless SIF in round bars with transverse circular cracks, using the boundary integral equation method. The crack was modelled with isoparametric 2D elements of 6 and 8 nodes, and the stress singularity was introduced using quarter-point nodes along the crack front. Couroneau and Royer (1998) performed a finite element analysis of a quarter of the cracked cylinder using 3D isoparametric elements of 15 and 20 nodes. Singular elements were used in the close vicinity of the crack tip by moving the mid-side nodes to the quarter-point position, so as to introduce the r – 1/2 -stress singularity. Shih and Chen (1997, 2002) built a 3D finite element model of a quarter of the round bar with a semielliptic crack, using 20 node brick elements and collapsed singular elements to model the r – 1/2 -stress singularity along the crack front. The SIF was obtained form the nodal displacements in the vicinity of the crack tip. Several curves were fitted to the numerical results in the matter of dimensionless SIF in the points A and B but, unfortunately, in some cases the fitting procedure produces negative K -values, as discussed by Cai and Shin (2004). Shin and Cai (2004) computed the dimensionless SIF in a round bar with a transverse semielliptical crack using a finite element numerical procedure to obtain the stress-strain state and a virtual crack extension to compute the SIF. Due to the symmetry of the problem, only a quarter of the round bar was considered. Collapsed singular elements (with the mid-side nodes in the quarter-point position) were used to model the stress r – 1/2 -singularity. From the mechanical point of view, the analysis was carried out under two different boundary: (i) free sample ends, i.e., unrestrained bending; (ii) constrained sample ends, i.e., restrained bending.