Crack Paths 2012

0 2 4 6 8 10 12 14 16 18 20

Dimensionlessprojected coordinate, x/d

(a)

4th iteration

3rd iteration

2nd iteration

1st iteration

0

0.2

0.4

0.6

0.8

1

Dimensionless projected coordinate, x/d

(b)

0

1

2

3

4

Dimensionless projected coordinate, x/d

(c)

Figure 3. Illustrative sketches (not to scale) of: (a) A periodic zig-zag crack with kinking

angle decreasing with increasing crack length; (b) First four iterations in the generation of a self-similar zig-zag crack; (c) A self-similar zig-zag crack at the 4th iteration with

fractal dimension decreasing with increasing crack length.

Crack Growth Rate

Crack kinking induces a geometrical effect on crack growth rate. Since the fractal length

of the crack

is equal to

D l , the derivation chain rule yields a relation between the

*l

scale-invariant crack growth rate dNdl* and its nominal counterpart dNdl [8]:

dl

dl

D 1

dl

dl

Dl

*

*

dN

dl dN

dN

(7)

Fatigue Crack GrowthLaw

According to Ref. [8], the following modified Paris law can be used to describe the

fatigue crack growth for a fractal crack

(8)

m I K C d N d l * * '

where *l is the renormalized crack length having physical dimension D L .

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