Crack Paths 2009

(mode I plus T-stress) in a homogeneousmaterial. This topic has been recently revisited

[13,14] leading to define a positive threshold of the T-stress below which no branching

can occur as observedin someexperiments [15].

In the case of an interfacial crack, Williams’ expansion takes the form

g(x1,x2) : g(0) + rel/Mme)+ E W M Q w+ n)ae)+

(5)

Where rt(6) is the first non singular term of the expansion, the associated stress field

does not grow to infinity but remains constant as r —>0. The coefficient T is the

corresponding intensity factor. Introducing a new mixed modeparameter M (r) allows

writing

MR6‘) :Q(0)+Kr1’2”€(a(6)+m(r) E(6‘)+M(r)t(6‘)+-~)

with MU)_1r1/24g

0:169) I Kr’1'2”€(g(6)+m(r) §(6)+M(r) 560+")

_ K

(6)

The energy condition takes the following form

G : KI? X(a,m(l),M(l)) +... 2 G,

(7)

With

X(a,m(l),M(l)) : 2Re[A(a) +m(l)A‘(a) +M(£)B(a)] w h oC(a) m+...

The new complex term B and the real one C play roles similar to A and A‘ (Eq. 4).

The stress condition derived from Eq. 62 leads to

K1? r(e,m(r),M(r)) z r of

(8)

With

Y(e,m(r),M(r)) : [s,,(e) + ma) s,,(e) +Ma)1,,(0) +42

The compatibility between the two inequalities 7 and 8 gives an equation for the crack

initiation length is as a function of a

<9>

C Y(a,m(ic),M(fc)) 012

It was not possible to extract such a length from Eq. 4 where i appears only through the

oscillating term in m “ ) .It is n o wfeasible thanks to the additional T-stress term which

involves an exponent larger than 1/2.

Finally Eq. 7 with t : t6 gives a condition on ‘K i for crack initiation in the directiona :

858