Crack Paths 2012

n

1

(2)

¦

N n

i N

i

1

¦ i n

2

1 n

2

(3)

σ

i N N

1

The chosen frequency function f(N), used for the determination of interval [N1, N2] in

which crack will be initiated with the probability P(N), should satisfy the following

conditions [9]:

0 t N f

for every N

(4.a)

³f f

1 d N N f

(4.b)

2 1 2 N N N P d N N f N

³

d d

(4.c)

N

1

Practical considerations of the fatigue analyses show that the Weibull and lognormal

distributions are best suited for this type of problems. In this paper the lognormal

distribution is utilised. In this case, the variable (the number of loading cycles N) can

take values in the interval [0, ∞), which is applicable to the fatigue problem. The

frequency function is defined as [9]:

ª 21 e x p π 2 σ 1 *

º

¨©§

¸¹·

N N

ln σ

*

(5)

«

»

N N f

*

« ¬

» ¼

where N * and σ* are defined as follows:

2

N

*

N

ln

(6)

2

2

σ

N

(7)

σ

¸¸¹·¨¨©§ 22 σ 1 l n N

*

The interval [N1, N2] in which the crack is initiated with the probability P(N) can

then be estimated by using Eqs 4 and 5. Example for probability P = 0.9 is shown in the

Fig. 6.

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