Crack Paths 2009

Where r and θ are the polar coordinates emanating from O (figure 1) and where holds fo the complex stress intensity factor (SIF). It can be rewr tten, introducing the K

mixed modeparameter

()mr

[4,5]

( ) ( ) ( ) m r u

1 2 ( ) U x xU O , = + ( )

1/2i

2i

m r r

(

)

θ

θ

with () K =

Kr u ε +

+

+...

(3)

−

ε

Figure 1. The interface crack growth, (a) delamination, (b) kink out of the interface.

The incremental energy release rate for a crack increment A in the direction α now

writes [5,6]

( R e A m A α α + A )+ ( ) ( ) ' ( ) ...

2 G K K =

(4)

Here the complex terms A and 'A play the role of the single real scaling coefficient A

0 α = and A'(0) = 0 , then it is clear that the energy

in Eq. 1. For a delamination crack

(using either the incremental or the differential form) is meaningful

condition

c G ≥ G

c G

and can be used to predict the delamination growth. Note that

denotes here the

interface toughness, a “material” parameter which is difficult both to define (because of

the mixity of modes) and to determine experimentally [7,8].

On the other hand, for a crack kinking out of the interface A'(α) ≠ 0 . Then, due to the

term (Eq. 3), the limit as A → oes not exists, the differential energy release rate ()mA 0 d

is ill-defined. Different authors have tried to overcome this difficulty by assuming either

ε of the exponent, which is usually small, can be neglected or

that the imaginary part

by prescribing a given characteristic distance [9,10,11]. In the next section we show that

the next term of Williams’ expansion (Eqs. 2 and 3) can be used to avoid this problem.

T H ER O L EO FT H ET-STRESS

he pioneering work in this domain is due to Cotterell and Rice [12] who investigated

T

the influence of the T-stress on the kinking of a crack submitted to a biaxial loading

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