Crack Paths 2012

2

(2)

G :G(1)+G(2)+...:(,/Gs)+,/G§1))

+(Gfj) +G§2))+...

->0 for 8 _>0.

(4)

It is worth mentioning that there are two sources of stress, i.e. mechanical (m) and

residual (r) which, separately applied, give the crack extension forces GIn and Gr. W h e n

both sources of stresses are applied together, then the crack extension force G is

calculated according to Eq. (4) (more details in [31]). Hence, in our case, the first term

of the total crack extension force due to the combined loading (flexural and thermal) is

given by:

I 1+ 2 Ga) :_I/Vs2bl 2W} H12K1P(b)(¢17)g261 l.\Pl_WS§l 5 12W 2 HIHZSZS d 1' _ + 2 5 + 0 1

S

25

S

'

(5)

t 2 1 262-1 '(K1p(b)(¢p)’\P1+K2p(b)(¢p)’qJ2)_WZsVVZ H 2 K2p(b)(<011)8 262

.\I12 2 0

S

The second term of the crack extension force due to the combinedloading is for the case

of the crack bifurcation given by

1/2

1/2

G(2) : Wsua, 51 +1

H l c i i m s sCl082(PP( J'Hyl ( y ' ) d y ;+ . I H y 1 " ( y ” ) d y g ] _

0

0

S

1/2

1/2

_ WSW,‘ 5, +1

P11603085‘

Sin (P. @0804 [14, (y')dy£ — [V1.5 (y")dy£']+

(6)

W

S

0

0

1/2

1/2

+ W.

Hptiflkr c082

1,52 5, +1

S

0

0

1/2

1/2

_ WS1+5Z 62 +1

11263350852

Sin (P, COS

S

0

0

For the case of the single crack deflection G0) adopts the same form as Eq. (6),

where only one integral (from 0 to 1 - for one branch) is considered within the brackets.

Note that GSIF or a T-stress is generally the sum of two contributions:

H,=H,’”+H,’; H2=H,'”+H2’; T=T’”+T’,

(7)

where Him is due to pure flexural loading and Hir is due to pure thermal loading

respectively. These parameters characterize the stress state in the crack tip vicinity. In

case, whensome of the GSIF (H1 or H2), are close (or equal) to 0 (eg case of the crack

perpendicular to the interface), then the Eqs. (5) and (6) wouldsignificantly simplify.

The factors Kim) ((01,) and the opening of the crack extension l/zyl. (y') , l/zyé (y') , etc.,

are calculated by means of F E Mon the inner domain once for all, since they depend

only on the local geometry and material properties — for details see [25, 28].

93