Fatigue Crack Paths 2003

d 2 K A ω σ σ

ω

ω

⎢⎣⎡

pA − −

⎥⎦⎤

d

l

l

d

d

I

n

n

p n

n

(

η ω

−

+ − =

d

2

or

1 1 1

)

*

0

0

1

.

)

( )

(39)

d

d

d

ω

(

= π

−

−

* c 0

o

n

Integrating Eq. (39), we obtain

pA

+ − = Δ

dl

( ) . d 1 n n p n

1 1 p 0 ∫ − − n ω ⎢⎣⎡

⎥⎦⎤

−

1

(40)

η

ω ω ω

( )

ω

n

k

k n ω and

where

p n ω denote the damage values at the beginning and the end of the

propagation stage III.

The relation (40) specifies the crack growth during one cycle, so that Δl = dl/dN. The

consecutive stage IV corresponds to elastic unloading, so that

(41)

( ) . 0 d , 0 d , 0 d 0 d I = = < < l K n n ω σ

Using the double logarithmic scale, the crack propagation curves are shown in Figs 4a

and 4b, for varying exponent values n and for varying values of damage growth pa

rameter A. The curves can be compared with the usual diagrams dl/dN = f(ΔKI) avail

able in literature. It is seen that the crack propagation curves correspond qualitatively

well to experimental curves. WhenKI tends to

*Ic K , the crack propagation rate tends to

infinity, when KI tends to

Ith K , the propagation rate tends to zero (or the logarithmic

measure to minus infinity).

a)

b)

(

)*Ic

log

Figure 4. Crack propagation curves log(Δl/d0) versus

/ K Ka) dependen:ce on Imax

the exponent n, b) dependence on the parameterA.