Fatigue Crack Paths 2003

range 0th K Δ for long cracks (which is crack-size independent), and the material length

represents the intrinsic crack length

0a .

0l

E X P E R I M E N TA PAPLL I C A T I O N

The multifractal law of Eq. (6) is here applied to interpret

th K Δ against a results of

some experimental tests carried out by Tanaka and coworkers [7]. The material tested

was a ferritic and pearlitic mild steel with the carbon content of 0.20%. The grain size of

the ferritic phase was changed by a heat treatment from 7.8 μ m(Material A) into 55 μ m

(Material B). Fatigue tests were conducted on plate specimens at room temperature

under fully reversed bending. K-decreasing tests were performed to obtain the threshold

stress intensity range on plates containing either a centre crack, or a surface crack or a

corner crack. The threshold condition was conventionally determined for a crack growth

rate equal to 10-11 m/cycle. The threshold stress intensity range

th K Δ was

experimentally evaluated (7 values for Material A, 12 values for Material B) for the

crack length ranging from 6 μ mto 1383 μm.

thKΔ against a curves (see Eq. (6)) are shown in Fig. 4 together

The best-fitting

with the experimental data reported in Ref. [7]. The best-fitting procedure allows us to

determine the parameters

and

0l of the present model (Table 1). Note that the

∞ΔthK

correlation coefficient R is approaching the unity (corresponding to a perfect

correlation) for both materials being examined.

It is self-evident that the tendency of the experimental

thKΔ against a data can be

well described also according to ElHaddad model [6], knowing the two parameters

0 0 a= l ).

0thKΔ and

0a (see Eq. (1), and Eq. (6) with

0th th K Δ = ΔK∞ and

Note that, in

the tests by Tanaka and coworkers [7], the value of

0th K Δ was experimentally

0a (see Table 1) was obtained from Eq. (2) by

determined, while the value of

considering the fatigue limit for smooth specimens (0thσΔ) computed through an

empirical expression depending on the grain size.

and

Therefore, the reason for determining the parameters

from a fitting of

∞ΔthK

0l

thKΔ against a data is to show a general way of application of the proposed

multifractal law, without knowing a priori the physical meaning of ∞ Δ

th K and 0l. Such

a meaning is then revealed by comparing Eq. (6) with Eq. (1). In other words, Equation

(6) does not provide a new expression for the threshold stress intensity range as a

function of the crack length, but it only demonstrates that the relationship describing the

Kitagawa diagram can be obtained following a non-conventional approach based on the

(multi)fractal geometry.