Crack Paths 2006

between the different W Fterms in the matrix product of eqn. (11), the GF can be obtained

by solving 3.(n+1)2 integrals of one variable. Having assumed n=2 for the WF, the number

of integrals to be evaluated is 27. However, by considering that due to the asymptotic properties of the WF, some of the W Fcoefficients are zero, the evaluation of only 22 of the

27 integrals is necessary. In particular, the following three classes of integrals have to be

determined:

k 21 ¸¹·¨©§

1

¨ © §

¸ ¹ ·

j

x x I a 1

2

b ¨ © § ˜ x 1

b x

1

'

db ¸ ¹ · ¨©§ ˜ ¸ ¹ · 1

(13a)

³

n j k

, 0 , ,

kj

b

)',(

x x )',ma (

1 1

' 1

21

¸¹·¨©§

¸¹·¨©§

21

x x I )',(

k

j

(13b)

n j k , 1 , ,

bx b x b ¸¹·¨©§ ˜¸ ¹ · ¨ © § ˜ ³ axx ) ' , m a x ( 2

db

kj

2

1 1

' 1

¨ © §

21

¸¹·¨©§

21

x x I )',(

k

j

¸¹·

(13c)

n j k , 0 , ,

bx b x b ¸¹·¨©§ ˜¸ ¹ · ¨ © § ˜ ³ axx ) ' , m a x ( 3

db

kj

3

Integrals of type I1 and I3 were analytically solved by using a recursive strategy, whereas

I 2 integrals were reduced to the solution of elliptic integrals. By knowing the analytical GFs,

the C O Dcomponents can be determined at any location of the crack for any loading

condition by eqn. (12) when the nominal stress components V x and Wx are known on the

crack edge. The problem of crack closure can therefore be faced in an efficient way by

using the procedure explained in [11] for non symmetrical problems, that accounts for

couplings effects, active between normal stresses and tangential displacement and between

normal displacement and tangential stress. An example of the C O Dv component calculated

by the proposed GF approach and by the FE modelling is shown in figure 5, where the

conditions of partial crack closure produced by a load P inclined by an angle of 45° with

respect to the normal at the free surface and pointing inward the semiplane are plotted.

3.0E-02 456

2.0E-02

W F

1.0E-02

r=2 L=0

F E

0.0E+00

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

x/a

Fig. 5: v components of C O Dcalculated by the GFs method and by the FE analysis. The

v values are normalised by the following characteristic parameter v0=a.Vo/E (V0=2P/Sa)