Fatigue Crack Paths 2003

homogeneous and highly inhomogeneous stress fields and to predict both the crack

initiation and its propagation through the damaged material as a united process.

In this paper, we apply the local and non-local approaches to initiation and

propagation a periodic system of multiple fatigue cracks. Note that some other

particular non-local approaches were used for predicting fatigue life in [5-9].

L O C ABLRITTLSE T R E N G TAHN DDURABILITCYO N D I T I O N S

Let us consider a cyclic process in a body Ω represented as a cycle sequence {}..2,1),(=mcijxmσ of connected closed but generally non-coinciding loops (cycles)

] ) ; , ( [ ) , ( 1 m m ij cij x x m τ τ τ τ σ σ ≤ ≤ = − in the stress space, ) , ( = ij ij m τx

) , (

x

1− σ, where m

τ

m=1,2,... is the cycle number, x∈Ω.Then the cyclic fatigue can be described in terms of

the cycle number n (instant n) as a discrete or continuous time-like parameter.

To describe fatigue crack initiation and propagation, we will analyse the brittle

strength, that is strength at a particular point y along a particular infinitesimal plane with a normal vector ζr at that point. The local fatigue brittle strength condition for a plane

ζr at a point y ∈ Ω can be taken in the form [4], 1),,)};,(({<⋅Λζσ r y n Herye

c

r ) , , ) } ;ζ, ( ( { σ y n y c ⋅ Λ is a local brittle Cyclic Normalized Equivalent Stress Functional

...2,1 and is positively homogeneous in }{cijσ

{

} ) , (

c

(CNESF)defined on sequences

ij σ x m

= m

and non-decreasing in n . It is considered as a material characteristic.

For example, the CNESF associated with the power S-N diagram

R b n) / ( * *1Δ Δ =σ σ and the Palmgren-Miner linear accumulation rule has form [1,4],

r

v

} b /1

{

n

b

y n y

y m

dm

c

∫

1 ) , , ) } ; , ( ( = ⋅ Λ ζ σ

[

Δ

σ

) ] , , ( ζ

(1)

ζζ

*

Δ

0

σ

) , , ( ζ σζζ v y m Δ

R

1

is the normal stress range on the plane ζr

where

at the point y during the

cycle m; b is a non-negative material constant and

is a material parameter

Δ*1Rσ

y R =

y m

y m

r

v

v

.

ζζ

ζζ

) , (

/ ) , , (

σ ζ

depending only on the asymmetry ratio

min σ ζ

) , , ( ζ

max

Let a an open domain Ω(n) occupied by the body at an instant n has a boundary

)(nΓ consisting of an initial body boundary Γ(0) and of a new crack surface )(*nY

occurring and growing during the cyclic process. Then we have the following

conditions for the fatigue crack initiation and propagation,

r , 1 ) , , ) } ; ) , ( ; ( ( { < ⋅ Γ ⋅ Λ ζ σ y n y )(nyΩ∈c ζr ∀ , (2) ,1))(,,)};),(;(({*=⋅Γ⋅Λyrynycζσ n∈ )Y(* y (3)

(4)

, 0 ) ( ) ) , ( ; ( * = Γ y y n n j i j ζ σ )∈( * n Y y

where ζr)(*y

is the vector normal to the crack surface )(*nY.