Crack Paths 2006

dl *

m

K' C

(12)

dN

*

I *

where

*l is the renormalized crack length having physical dimensions

D L , while the

1 3 2D

˜

material parameter

* C has the following dimensions:

2 L F .

Since D l l * , the derivation chain rule yields a relationship between the renormalized

fatigue crack growth dNdl* and its nominal counterpart dNdl, namely

dl

D 1

dl

dl

d l D l

*

*

dN

dl dN

dN

(13)

By substituting Eqs 11 and 13 in Eq. 12, the following fatigue crack growth law in

terms of the nominal quantities dNdl and

I K ' can be obtained :

ª

º

C cos22ln *

l

K

2 1

D m ¸¹·¨©§ * 1 2 1

m ¸¹·¨©§ ¸ ¸ ¹ · ¨ ¨ © § cos22ln4ln1

mI

mI

ª

º

«

»

dNdl

lDC

K

»'

«

»

-

«

»

(14)

'

4ln

« ¬

» ¼

«

»

« ¬

¼

-

Note that, conversely to the fatigue crack growth law in Eq. 7 for periodically-kinked

cracks, Equation 14 for continuously-kinked cracks explicitly depends on the crack

length l and, hence, it accounts for crack size effects on the fatigue crack growth rate.

C O M P A R I SWOINT HE X P E R I M EANNT DDISCUSSION

It is instructive to consider here some experimental fatigue crack growth results

exhibiting crack size effects, since such results might be regarded as a counter-example

for the validity of the crack growth law in Eq. 7. The experimental data are related to

fatigue crack propagation in three-point bend high-strength plain-concrete specimens

[11]. One series of three two-dimensional geometrically similar cracked beams (A, B and

C) with height equal to hA = 38 mm,hB = 108 m mand hC = 304 mm,initial length of the

crack of 0.16h (lA = 6.3 mm,lB = 18.0 m mand lC = 50.4 mm), span of2.5h and thickness

of 38 m mwas tested. The maximumsize of the aggregate was equal to 9.5 m mand the

mean compression strength was equal to 90.3 MPa.

The nominal values of the crack growth rate against SI range [11] are reported as a

bilogarithmic plot in Fig. 4 (17, 16 and 12 experimental points for beams A, B and C,

respectively). The value of the fractal dimension D can be calculated by applying Eq. 14

to the data in Fig. 4 through a best-fit procedure (see Ref. [6] for details). It turns out that

the fractal dimension D is equal to 1.27 (being the slope of the Paris-Erdogan law m =

8.2) and hence, by applying Eq. 10, the kinking angle -

results to be equal to 54°. The

value of -

is deemed to be correlated to the material microstructure, but further work is

needed in order to determine quantitative relationships.