PSI - Issue 43

Jelena M. Djoković et al. / Procedia Structural Integrity 43 (2023) 95– 100 Author name / StructuralIntegrity Procedia 00 (2022) 000 – 000

98 4

The crack approaching the interface can behave in one of the two ways, presented in Figures 2(a) and 2(b). The unusual characteristics of cracks shown in Figure 2, is the fact that there is only one dominant mode of deformation at the crack tip, when the material across the interface is different. Thus, the asymptotic behavior at the crack tip depends on the distant loading only through one stress intensity factor. Existence of one dominant mode of deformation has a consequence the fact that whether the crack would penetrate the interface and continue to propagate across it (in material 1, Figure 2a) or would deflect into the interface (Figure 2b), or even bounce away from the interface (back in material 2), depends only on relative stiffness of the two materials constituting the interface. It is assumed that the length of the crack branch a is arbitrarily small with respect to the main (impinging) crack.

Fig. 2. (a) the crack crossing the interface; (b) the crack deflecting into the interface.

The competition between the crack crossing the interface and deflecting into it is being considered by comparing the energy release rates for these two events, G p for the former and G d for the latter. The crack would deflect into the interface if the following is valid, He and Hutchinson (1989):

G G



.

(5)



d

int



p

where: Γ – the fracture toughness of the material and Γ int – is the toughness of the interface. If the sign in (5) is reversed, the crack would penetrate the interface. For the case when the crack is crossing the interface, direction of the crack propagation, defined by angle  , has to be determined. That direction is being chosen in the direction of the maximum energy release rate. The stress intensity factors at the tip of the penetrating crack, depend on k 1 and a as, He and Hutchinson (1989): ( ) 1 2 k II I K iK c , , , k a      − + = , (6) where c 1 ( α , β ,  , ω ) is a dimensionless function of α , β ,  , ω , while  k is the real variable that depends on Dundurs parameters  and  ; factor k I is proportional to loading and it is not necessary to know it explicitly. The energy release rate for the crack that is crossing the interface is: 1 I

1

−



1 2 2 2 1 1 1  − I c k a

 k

G

=

.

(7)

p

2

The stress intensity factors for the crack deflecting into the interface (Figure 2b) are being calculated as:

1 2

( i I K iK k a d , a e , a        − −   + = +   . ) ( ) 1 2 1 1 k i

(8)

Thus, the ratio of the two energy release rates is independent of a and is determined as: ( ) 2 2 1 1 1 1 2 1 2 1 1 d max p d e Re d e G G c    + +   = − .

(9)