Crack Paths 2012

where

th c c a* a,

m2 / am) 2. (Fro*m the last expression in Eqs (7), the

relationship between the number of loading cycles n and the current damage )(nD can

be obtained:

0 D n D

0

) ( f o r

)( N n e n D ; ˜

¯ ® ­

21

) ( 2

n D N

n D

1 ) (

N n

for

o

;

) ( l n

(12)

Finally, by equating the number of load cycles n (see Eq. (11) and Eq. (12)), we get a

relationship between

P D based on the Paris law and ) ( n D aDccording to the damage

model :

* a a D * 1 * 2 mP mc D a N ; m t h c P * 0 ) *

e D

> @ * * / 1 0 * * 1 * 2 m m t h a a D a » º ª ˜ ˜ ; ˜ (13)

( 1

or

D

P

« ¬

c

P

¼

and the crack size

D a corresponding to the damage D becomes (see Eq. (9)):

0

*

* / 1 * 1 * 2 l n m

1 ) ˜

m t h P c a a D a a a a D a a a c

˜

(14)

˜ ˜ ; ˜

>

(

)

(

«¬ª

D

th

c

c th P t h

th

@

»¼º

*

Then, by substituting the expression ln ( N2 n D ˜ ; (obtained from Eq. (12)) in Eq. (14), the crack growth rate d n d/ a Dcan be determined: > @ ) * ( 1 a a D )

*

> @ * 0 * )* 1 ( 0 ) * ( m t h c P a a D

˜

»¼º

mm m t h c P

da

a

a N N n «¬ª ˜ ˜

D

c

(15)

* ) ( 1

* * * * mc a m N dn ˜ ˜

mc

*

depend on the stress amplitude

a V , the crack growth

Since the terms ,N ; and

ca

rate based on the present damage model is dependent on

a V . Such a dependence is

shown to not affect significantly the results.

A P P L I C A T I O N

The above damage model and the crack propagation-based assessment are employed to

examine the fatigue behaviour of a real metallic material. The damage model

parameters connected to the Wöhler regime (referred to the crack initiation-dominated

regime) are firstly determined through the damage increment defined in Eq. (3). Then,

the parameters related to the crack propagation-dominated life (Eq. (4)) are obtained,

and the corresponding crack growth rate (Eq. (15)) is graphically represented.

The material to be examined is the Aluminium Alloy Al 2024-T3 [18] which is

characterized by the fatigue-fracture parameters reported in Tab. 1.

0 , , , , , , V q V h C B), dAetermined through

The damage model parameters (521,...,,aaa ,

the damage increments in Eqs (4, 6) neglecting the stress gradient effect ( 0 J ) and

assuming

q h1 – corresponding to thecrack initiation-dominated fatigue (Wöhler

300