Crack Paths 2012

Expression ofthe energy-release-rate at order 2 in Fourier ’s space

From there, one m a y calculate the second-order expression of the energy-release rate

(ERR) G(a,e;z,). The result is best expressed in Fourier’s space. The definition of the

Fourier transform l/2(k) of an arbitrary function y/(z) adopted in this workis

+66 A —i 1.1/(z) = i, 1.1/roe" dk I> 1.1/(k) = aL, 1.1/(2)6 he iz A l + M

(10)

With these notations, one gets, G°(a) denoting the unperturbed ERR:

Gn(a,e;k,)=G°(a)5(k,)+eGI(a;k,)+e2G3(a;k,)+O(e3),

(11)

A

dG°

A

G1(a;k1)G:0(a)|: 0

( a ) _kl :|¢(kl)

G d d i

( 1 2 )

$00k.) - 5°00]: Q(a;k,k1 — kiitkiitk.

—k)dk.

.

I _ l d _2 lG d0 G 0l _ _ l

l

I

l2

Q(a,k,k)- ziGoda, 4—Goda(]k+k ]k]

k )+8{sgn(kk )(k+k) (13)

+ [sgn(k) — sgn(k ')] ]k + k ‘ (k—k‘)—(]k] —]k'])2}.

EQUILIBRIUCM R A C FKR O N TSHAPEIN A H E T E R O G E N E OMUESD I U M

W e now consider crack propagation governed by Griffith’s criterion in a material

having a heterogeneous fracture toughness Gc(x, 2) given by

6.0.2) = G_.[1+8g.(x,Z)]

(14)

where G, is a “mean fracture toughness”, e a small parameter and g, (x, z) a given

function.

Assuming G to be equal to GC at every point of the crack front, the distribution of

toughness determines the shape of this front in the form

x =a+8¢1(a;z)+€2¢2(a;z)+C(83)

(15)

where a, ¢1(6l;Z) and (i2(a;z) are a parameter and functions to be determined. To do

so, it suffices to equate the Fourier transform of the expansion of the ERR, deduced

from Eqs. (11), (12) and (13) with ¢=¢1+e¢2, to the Fourier transform of the

expansion of the local toughness,

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