Crack Paths 2006

0 3 3 l { 3 l G

(2)

2

G K IC c

for plane strain.

and

2 1 Q E

In order to calculate this change in potential energy between the two states one has to

specify the behavior of the displacement field for the V-notched cracked domain.

This can be done by an asymptotic analysis of the displacement field when a small finite

crack is introduced at the V-notch tip.

Using a coordinate transformation

/ i i y x l ,

/r l U , we observe the very close

neighborhood of the V-notch tip and match the terms of the solution of this “inner

expansion” with the terms of the “outer expansion” (as

r o 0 and U f o).

Matching the solutions it can be shown that:

T U

T U

. . . ˆ 2 v

0,0

lA

(3)

, u l l u y y l 2 1

u

ˆ u l A v D D 2 1 2 2

1

1 D D

1

2

1

Where

as U f o vv0ˆ,ˆ21 now need to olve for

and

2ˆv elasticity problems with prescribed vanishing

1ˆv

behavior at infinity:

(4)

ˆ v V | i

0

ˆ or v

|

0

i

o f

o f

U

U

Using Betti’s theorem one mayshow that:

u u T f g T g

0 l l l u u where 3 3 3 { (5) 0 1 , 2 0 , G \ l

*

¬

Inserting (3) in to (5) one obtains:

H l A H H l A A H l A D D D D G ) ( 22 22 21 12 2 1 1 1 2 2 1 2 2 1 1 3

u v H u v H u v U \ U \ U \ ,ˆ ,ˆ ,ˆ { { {

(6)

H

ij i i

ji j i D j

i j

ii

D

D

i

i

It is important to note that the

ij H 's are only a function of the notch geometry and

material elastic parameters. See [11] for details on the different functions

ij H obtained

by FE analyses.

Following the initial analysis proposed in [13], inserting (6) in to (1) we can obtain a

lower limit for l :