Crack Paths 2006

2 / 1 2

2/1

2

¨ ¨ © §

¸ ¸ ¹ ·

, , 1 2 , 1 , 2 x a S I I A I W S r x h r x h A I I S I W V a x a x r x h r xa h ¨ © § ˜ ˜ ¸ ¹ ·

r

o 0

V

˜

¸¸¹·¨¨©§¸¹·¨©§ ˜

(2)

0 , V S Therefore, for solving the integral equations (1) the following expressions were , , , a A I W A I I V S I W S I I

r x h r x h r x h r x

assumed, by which, adopting the notations M = I or II (mode of facture), P=V or W (nominal

stress component) and C=S or A (symmetrical or antisymmetrical), the W Fcomponents are

expressed either:

) , , (

21

i

2

ª

n

a r x h

2

1 ¨ © § ˜ «

¸ ¹ ·

S

ax

(3)

MC

P

P

M

˜

r c

¦

C

i

˜

a

i

0

« ¬

»¼º

or

12

2 »¼º « « ¬ ª ¸ ¹ · ¨ © § ˜ ˜ ˜ ˜ ¦ i n i i M C M C a x r c a x a a r x h P P (4) 0 1 2 )

Equation (3) holds for (,,)IShxraV, (,,)IIShxraV, (,,)IAhxraW and (,,)IIAhxraW, whereas

eqn. (4) holds for (,,)IAhxraV, (,,)IIAhxraV,

(,,)IShxraW and (,,)IIShxraW. In the present

analysis the number of terms of the power expansion was limited to 3, so that n=2 in the

summations (3) and (4). In order to fulfil the general asymptotic conditions, expressed by

(2), the first coefficients corresponding to i=0 for the diagonal and off-diagonal terms of the

W Fare respectively:

P

or M I I I P P V W (5a)

1 for M M S i A i c r c r i

0 and

P

or M I I I P P V W (5b)

0 for M M S i A i c r c r i

0 and

The other coefficients corresponding to i>0 depend on the r ratio. In order represent the

dependence over a wide range of r the following function was considered:

i

Ai

(6)

Ai r r c )()/(P i M A S

i r E r C ˜ ˜ ˜ i i F i D

Ai

r B

The constant parameters Ai, Bi, Ci, Di, Ei, Fi were determined by least square fitting the

SIF values calculated for reference loading conditions at different r ratios. In particular,

having assumed n=2, eight linearly independent nominal stress distributions applied on the

crack edges were considered, as described in [8]. A data base of SIFs values was built up by

performing several FE analyses with r ratio varying over a wide range [0.005, 40], in order

to encompass the two limit conditions for the crack: an embedded Griffith crack and a

shallow delamination immediately beneath the free surface. The SIFs calculated by using

the W Ffor the reference cases were compared with the original values obtained by the FE

analysis. The relative differences between the W Fand the FE SIFs for both Modes showed

a satisfactory agreement being generally within 1.0% in the whole considered range of r.