Crack Paths 2012

The calculation of these average values is madeeasier by assuming the distribution of

toughness to be periodic in the direction z , so that (,)cgxz is of the form

+∞

0 ( , ) ()imkz c m m g x z c x e =−∞ = ∑

(19)

may be assumed

where

0 k is a positive wavenumber and the () m c x coefficients;

0 ( ) c x

to be zero for every x since the degree of arbitrariness left on the definition of c G

permits to consider it as identical to the exact average value of the local toughness.

Calculating then

1 ( ; ) a z φ and

2 ( ; ) a z φ and using the definition (18), one gets

d G

dG

0

2 0

()

c

c

a m k

a

ε

G a G

−

+∞

()

()

0

2

() c a

da

da

eff

2

( )

(20)

O ε +

∑

m

2

3

= +

0 0

2

0

() d G m k a G d a ⎡ ⎤ − ⎢ ⎥ ⎣ ⎦

1

m

=

where the

m c2 ( ) x have been averaged over a distance much larger than the typical

distance of fluctuation of the local toughness, though still much smaller than the

distance over which

0 ( ) G a v ries significantly. By inequalities (17), each term in the

()

series is negative, so that

eff c G a is slightly less than the exact average toughness

c G ;

upon the position a

the effect is tied to the dependence of the unperturbed E R R 0 ( ) G a

0

() 0 dG da a → .

of the straight crack front and disappears in the limit

E Q U I L I B R I SU HM A P EO FA C R A CFKR O NMTE E T I NAGNO B S T A C L E

W efinally consider a matrix of toughness

M c G containing an obstacle of width 2d in

the direction z , infinite length in the direction x and toughness

cO cM G > G . The

toughness distribution maybe represented by Eq. (14) with

( , ) g x z

G G

. x d x d < >

G G

O M

⎧ ⎪ = ⎨

ε

10 if

−

c c

;

;

M

c c

c

(21)

=

=

cM G

⎪⎩

W eare especially interested here in the first- and second-order perturbations of the

0

( ) 0

dG da a → (meaning that the characteristic distance of

crack front in the limit

variation of the unperturbed E R Ris much larger than that of toughness fluctuations).

757