Crack Paths 2009

o-OIo,(rIR)I2E+p

(34)

where as before 6 is the externally applied biaxial stress and p is the contact pressure.

Then: 0', I@ I 2I[1_ 2cGa—éffj%

(35)

I‘

where l/Gefl I l/G + (25' _ 1)/G‘

(36)

Now,for the case of no misfit between the inclusion and exterior body, and bonded

together for plane strain or smooth frictionless contact for plane stress between them,

the stress intensity factor is of the form:

K, I 0_'\/7r—aF,,(a/R,

yo, /G)

(37)

where F0 is defined by Eq. (38) and where 7/ I G‘/ G which applies for a / R << 1.

a G- _ A _ Gig FU[E,%%]_2[[1+0122E][1p ] 7[[1+0.0731+7][1 2e G JR] (38) Gefl 1 _ _Z

G

Now,for large cracks the values of F0, approach asymptotically to constants which are:

F,,( )Il

(for cracks on both sides of the inclusion)

F ( )I1/\/2 (for a crack on one side of the inclusion)

U

Further for the full range of crack sizes 0 < a / R < co the cubic asymptotic fit of the end

2 3 curves will be applied. It is: FU(x,)/,)/,fi.)IA+B[ x ] + C [ x ] + D [ x j

(39)

1+x 1+x 1+x

where: A()/, yeff ) I 2(1+ 0-122/(1+ 7))(1_ 378,7)

B(y, ),,f)I _%(1+ 0.073/(1+ )))(1_ 2pm,)

(40)

Forcrackso n bothsides of the incusion:

c(y,y,fi.)=3_3A_2B ,

D(y,y,fi.)=_2+2A+B

(41)

where: x I a / R,

y/IG‘lG,y e fi I G e fl / G

However,for a crack on one side of the inclusion Eqs. (41) should be replaced by Eqs.

(42). This completes the discussion for the externally applied stresses with differing

elastic properties of the inclusion.

C I 2 . 1 2 1 — 3 A — ,2 BD I — 2 + 2 A + B

(42)

Next the case of a radial difference, A , interference betweenthe inclusion and its nest

in the exterior body will be considered. For small cracks a/ R <<1 on one side or both

sides of the inclusion the result is:

K, I 20,, Aw/naFA(£, 7]

(43)

R

R

where: FA(a/R,)/) I 1+ 0.122/(1+ 7) _i(1+0.073/(1+ 7))a/R

(44)

d’

For large cracks, a/ R >> 1, the result for a crack on one side of the inclusion is:

501