Crack Paths 2012

modulus and Poisson’s ratio, respectively. For a general mixedmodeI+II problem, the two

analytic functions gz5(z) and X(z) can be chosen as series of complexeigenvalue Goursat

functions (Sih and Liebowitz [5])

¢ l ( z:)Z A T V Z A:RZ A T L T A H Q i A H Q , = Z B n Z A n:+Zl B n T A n + l e i ( ) \ n + l ) 9

n = 0

n = 0

n = 0

n = 0

Q M Z ): 2 Gnzx,: Z Gnrxneixne, X2(Z) : Z H n z x n n: Z H n r x n n e u x n n p(6)

n=0

n=0

n=0

n=0

'1

'

' -

- - - Knrihaloo and Xiao (2007)

.

-

Y

,7.

Polynominlfl)

Malennl"l"

’

.

A 0,,

k. a.

A y

I

" I

E 0.6 .Q 2

E 0.4

G

C

D

Z

traction-free i

cohesive

0.2

crack

_

10113

Material “2" real crack tip fictitious crack tip 1;. p.

It“

‘

'

° 0

0.2

0.4

"-6

'18

1

Nondimensional opening(-)

Figure 1. A traction free-crack

Figure 2, Cohesive law

at a bi-material interface.

comparison[4].

Eq.(5) is applied to material 1 in Fig. 1 (0 g 6 3 it) while Eq.(6) is applied to

material 2 (-7r g 6 g 0), wherethe complexcoefficients are A n I a m+ iazmBnI

bln + ib2mGn= g m+ iggn and H ”= h m+ ihgn. The eingenvalues, A” and coefficients

a1", a2n,b1n,b2n,g1n,g2n,h1n and hzn are real. B y substituting the complexfunctions (5)

in Eqs(2),(3) and (4), the completeseries expansionof the displacements and stresses near

the tip of the crack can be written exactly as in Karilahoo and Xiao [1].The coefficients

almazmblnand bzn are used in the case of material 1. Thecoefficients g1n,g2n,h1n and hzn

are used for material 2.For moredetails see Alberto A.,Barpi F. and Valente S. [6].

T h econditionsat the bi-material interface

The opening displacement (COD)of the crack faces can be written as w = o‘ — a‘

0=7r

6=I7r.

A n 0) I 2 ’"2

k+)\,, A n + 1 k+)\,, > \ + 1 _ 1 a m‘1' 01” + 2 91” + n h1n:| S111 AnTl' M1 #1 #2 H2

n = 0

and the sliding displacement (CSD) can be written as 5 = iii

— u‘

9=7r

9=I7r.

671