PSI - Issue 28

Carlos D.S. Souto et al. / Procedia Structural Integrity 28 (2020) 146–154

148

Carlos D.S. Souto et al. / Structural Integrity Procedia 00 (2020) 000–000

3

Combining Equation 2 and Equation 3, the following relations are obtained:

1 2 ∆ a · t

G I = −

Y j · ∆ v i

(4)

1 2 ∆ a · t

G II = −

X j · ∆ u i

(5)

Once obtained the value of G , and considering a linear-elastic problem, the stress intensity factor, K , can be ob tained by the well-known relations:

K 2 E

(plane stress)

(6)

G =

K 2 (1 − ν 2 ) E

G =

(plane strain)

(7)

Where E is the Young’s modulus and ν is the Poisson’s ratio. Finally, it must be noted that the presented method works for a xy referential aligned with the crack, as shown in Figure 1. If an inclined crack is considered, and a global XY referential is used, all vector quantities must first be transformed using a rotational matrix ( x = T X ).

2.2. J-integral

In the context of Linear Elastic Fracture Mechanics (LEFM), the calculation of the J-integral is another way to calculate the strain energy release rate, G , which in turn can be used to determine the stress intensity factor. In LEFM, G = J . The J-integral in 2D was introduced by Rice (1968) and it is defined by Equation 8. It is the integral around the contour Γ belonging to a solid that contains a crack whose facets are traction free. As shown in Figure 2a, the contour Γ must start in one of the crack’s facets, go around the crack-tip counterclockwise, and end on the other facet. Rice also showed that the value of the J-integral is always the same even for di ff erent paths, meaning that the J-integral can be considered path-independent. J = Γ Wdy − T ∂ u ∂ x ds (8) contour element ds . Owen and Fawkes (1983), as well as Kuna (2013), present a direct procedure to numerically calculate J based on a finite element model. This procedure is e ff ective and does not add any approximations, since the contour Γ is defined along the finite element’s integration points. However, in order to guarantee that the contour Γ is continuous, this Where W = mn 0 σ i j d i j is the strain energy density; n and T are the normal and traction vectors, respectively, of a

(a) The J-integral (Rice, 1968).

(b) Defining Γ for the direct procedure (Kuna, 2013).

(c) J-integral as an EDI (Abaqus, 2017).

Fig. 2: The J-integral and its numerical approaches.