Crack Paths 2012

suitable for stable crack propagation in the L E F Mregime, or by the Donahueet al. law

[16], suitable also for short cracks.

For a cyclic loading with zero meanvalue and constant stress amplitude (R I -1) and

assuming absence of stress gradient effect, the stress invariants and the deviatoric stress

invariants can be easily determined, and the damageincrement given by Eq. (4) becomes

[8]:

dD=Dq-lA-EB-dE)= Dq-A-[W-o—o0]B-dE=

B

(6)

The integration of the damageincrement from the initial damage DO up to the generic

damage D, occurring after n < N cycles, can be performed [9]:

d D _

_ A

3+1

3+1 _

E)?_lnD(n)—lnD0_2nB+l- [Wop-0'0] —[W-o'a—o'0]

_

E(o'a):0

A

+1

A

3+1

-

(7)

I:

fl 0 1 B

lI

a

0]

: 2 n W—-o'-—0' : 2 n ~W-0' —o'

thatls,

lnD( )—lnD +2 i - [ W - _ P“ or D(n)—fi—e2'E(”_N) n — 0 " 3 + 1 G a 6 0 _ 65.210 _

where N is the numberof loading cycles up to the final collapse (i.e. whenD =1), and

o'fl is equal to the constant stress amplitude 0'” of the cyclic loading. Fromthe above

relationship, we can note that D(n) I DO for n I 0, and D(n) I l for n I N .

C R A CGKR O W LTAHW S

As is well-known, the Paris law [11] describes the stable crack propagation regime for

long cracks whenthe L E F Mhypothesis holds (Fig. 1). O n the other hand, the Donahue

et al. [16] equation is suitable also for the near-threshold regime. Such laws are expressed

by:

§ 2 C'Mm’ % = C "(AK —AK.)'" - Cum/E— 4123'" - Grim/4o — an]: (8)

n

n

where a is the crack length, C',m are constants depending on the material, A K is the

effective Stress-Intensity Factor (SIF) range, and AKth is the threshold SIF. Let us

define a crack size-based damage DP for the material :

ac _ath

298