PSI - Issue 37

Jesús Toribio et al. / Procedia Structural Integrity 37 (2022) 995–1000 Jesús Toribio / Procedia Structural Integrity 00 (2021) 000 – 000

996

2

Stress intensity factor (SIF) in a circular inner crack located inside a bar subjected to tensile loading has been obtained for the symmetrical case by Tada, Paris and Irwin (2000) and by Eshraghi and Soltani (2015). For the case of a circular crack exhibiting eccentricity in relation to the bar axis, the maximum SIF value has been calculated by Guinea, Rojo and Elices (2004). From the SIF calculation at several points of the crack front and using the Paris law, Nguyen, Gallimard and Bathias (2015) modeled the fish-eye fatigue crack growth in very high cycle fatigue. 2. Numerical modeling The finite element method (FEM) together with the MSC.Marc code was employed to obtain the SIF in an inner crack localized in a cylindrical bar subjected to tensile loading (Fig. 1a). The inner crack, characterized as a circle, is not centered in relation to the bar axis, so a quarter of the solid was modeled (Fig. 1b) with the adequate boundary conditions.

(a)

(b)

Fig. 1. Eccentric circular inner crack in a bar under tensile loading (a); 3D mesh of the bar (b).

Isoparametric hexahedral elements with 20 nodes were used in the computations, and the middle nodes of the first core surrounding the crack tip were shifted to the quarter-point position to reproduce the r – 1/2 singularity. In addition, a mesh sensitivity analysis was performed. The mode I (opening) SIF K I was computed from the J integral by using the expression (for plane strain conditions) obtained by Irwin (1957),

 − 2 1 EJ

(1)

=

K

I

where E is the Young’s modulus and ν the Poisson’s ratio. The geometry of the cracked bar (Fig. 2) was characterized by bar diameter D , inner crack diameter d and inner crack eccentricity ε (distance from the center of the inner crack to the center of the bar). The point A on the crack front is associated with the minimum distance to the bar surface and the point B with the maximum distance to the bar surface. The angle θ (measured from point A) is used to characterize each point on the crack front.



A

B



d

D

Fig. 2. Geometric parameters of the cracked bar.