Crack Paths 2012
Gc(x,z) = gc[l+é‘gc (61,2) +82 %(a,z)¢1(a;z)]+ 0(5).
X
One thus gets the following conditions:
G°=€ ,
$(a;k,)=-—g*“"§1)
.
l/qkLG (a)
Goda
.
A k)
2 ;k = ;k’k _ki — x W1) lkl—Lo L‘Qw 1 )\k\—"° ‘16)
1
Goda
Goda
><—g@(a’k1_k) dk+ +mai(a,k)—gc(a’kl_k) dk.
dG°
—w 8x
dG°
\k, — k j — % ( a )
lk1—kl—%(a)
The first condition determines the position a of the reference straight front as a
function of the loading applied, and the second and third conditions then determine the
first- and second-order perturbations of this front.
A P P A R ETNOTU G H N EOSFAS H E T E R O G E NMEAOTUESR I A L
In this section and the next one, we introduce the following hypotheses on the variation
of the unperturbed ERR:
o
2 0
d G ( a ) < 0 ;
d G
(a) > 0.
(17)
2
a
a
The first hypothesis is necessary for crack propagation to be strictly stable, and the
second is reasonable since G°(a) is then a positive decreasing function of a.
W e consider here a material with a randomly heterogeneous, but statistically
homogeneousdistribution of fracture toughness. The “apparent toughness” Giff (a) of
this material is defined as the value of the E R Rfor a fictitious straight crack having the
same average position as the real, curved one:
05%)= G0 (a+€<¢1(a;Z)>+€2 <¢2(a;z)>+ 0(5))
(18)
where <¢1(a;z)> and <¢2(a;z)> are the average values of ¢1(a;z) and ¢2(a;z).
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