Crack Paths 2009

KO E%J€

(6)

From(5), by meansof a Taylor expansion, we have

axo

2 Ko(s,ot) : — + s (0,ot)+O(s2)

(7)

J;

.

.

5 K0

.

.

Thedetailed calculation of

(0, 0t) is reported in reference [9].

68

Finally, after some calculations, we obtained the following approximation for Oore

Burns integral (1) as a function of angle 0t:

K O(8,0t) : % { l + sEin eci“°n‘}+O(s2)

(8)

W h e r e cn are the Fourier coefficients of the first order crack front position

6 R . . 0t —>S(ot):—(0,ot)1n the sense that $(q):20nem“ . +00

6e

cw

The En coefficients, are independent from the homotopyR(s,ot) and are reported in table

1.

In general, by taking into account that an a-dilatation of Q under uniform normal

tension cs produces factor J 3 in the expression of K0, from (8) we are able to state the

following final equation:

KO(8,0t) : 2 ? ?{1+sibflEn ei“°‘}+O(s2)

(9)

where bn are the Fourier coefficients of or —>lg(0,ot)and S(8,0t) describes the

a

boundary of 69(8)

Table 1. En coefficients

n

En

n

En

0

1/2

6

—l.58042

l

0

7

— l . 8 1911

2

—0.4

8

—2.04377

3

—0.74286

9

—2.2566

4

—l.04762

10

—2.45929

5

—l.32468

ll

—2.653l8

lll5