Crack Paths 2012

Ic f m m K N a Y N K ' ˜ ' ˜ ' ) ( ) ( S V allowing us to express the final crack size ratio,

m f w f a a , , / as:

1

2

VV

(5)

, mw f a

¸¸¹·¨¨©§''wm

!

.

Assuming a constant geometry factor Y during the crack propagation, the current

crack size ratio r

2

(6)

) ( ¸ ¸ ¹ · ¨ ¨ © § ' ' w m V

mw r N a

remains constant along the whole propagation process (see Fig. 3(b)), in particular, from

the beginning of the fatigue process, i.e., for the initial crack sizes

m i m a a ,

up to the final failure state, proving that on a logarithmic scale, the vertical

and

w i w a a , ,

distance c=log r between the curves related to

wia, and

mia, remains constant all along

the propagation process (see Fig. 3(b)).

(a)

(b)

Figure 3. Crack growth curves for the best and worst initial defect cases for a given

material (a) plotted in natural scale, and (b) plotted in logarithmic scale.

While the initial worst crack size a , related to p=0, is determined from Eq. (2), w, i

another crack size, a m, i , related to a given probability p (particularly the best one bia,

related to p=1, remaining, in principle, unknown) can be obtained by integration of the

) ( / K G d' N,ddue tao its uniqueness, for a generic number

crack growth rate law,

L N of cycles to failure , i.e.:

a

da

L N d N

N

m if

,

L

N a a a

³

) , , ( , m L m i m f V' . (7)

o

) ) ( ( a K G ' ³ 0

m f

, ,

,

Since Eq. (6) applies irrespective of the number of cycles considered, the above

quantities can be used to write a system of two equations:

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