Crack Paths 2012

$(a;k,)=-—g*“"§1)

.

l/qkLG (a)

Goda

.

A k)

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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