Fatigue Crack Paths 2003

N O N - L O C BARLITTLSE T R E N G TC OHN D I T I O N S

General Description W e will suppose that strength at a point y ∈ Ωon a plane ζr

depends not only on the

{

} ...2,1 ) , ( = m c

stress history at that point,

ij σ y m but also on the stress history in its

neighborhood and generally, in the whole of the body, {

} ) , (

c

= m

ij σ x m , x∈Ω. A non ...2,1 ) , , , }ζ ; ( { σ r y n c Γ, whΛich Θ

local brittle cyclic normalized equivalent stress functional

is positively homogeneous in σ and non-decreasing in n, can be introduced [1,4]. It is

considered as a material characteristics implicitly reflecting influence of material r microstructure. Then the non-local strength condition for a planeζ at a point y ∈ Ω takes the form 1 ) , , , } ; ( { < Γ Λ Θ ζ σ r y n c.

The simplest examples of the non-local brittle CNESFsand strength conditions are

) , (

by its non-local counterpart

obtained by replacing the local stress

x i j τ σ

) , , ; ( ζ τ σ r y ij Γ Θ

in the corresponding local brittle CNESFsdescribed in Section 1,

r r ) . , , , ) } ; , ) , ( , ( ( { ) , , , } ; ( { ζ ζ σ ζ yσ n y y n c c Γ ⋅ Γ ⋅ Λ Γ Λ Θ Θ (11) r

) , ) , ( ; ( ζ τ τ σ y r ij Γ Θ

can be taken particularly as a weighted average

The non-local stress

of ) ,x ( ij τ σ (see [12] and also [1-2, 4-5],

dxx ) ; ( ) ; , , ( ) , ) , ( ; ( ) ; , ( τ ζ τ τ σ ζ = Γ ∫ Γ Ω Θ Θ r r x y w y k l y i j k ij

(12)

r

where the weight function w and the non-locality zone

Θ Ω (some neighborhood of y)

are characteristics of material point y, planeζr and generally of the body shape Γ, such

j l i k y i j k l x y w δ δ ζ ζ r . For example, ) ; Γ, ( ΩΘ ζr = Γ ∫ΓΩΘ);,,();,( r

as

y

can be taken as a 2D disc of a diameter 2δ in a 3Dbody

Ω(n) or as a 1D segment of a length 2δ for a 2D body Ω(n), in the plane ζr

with the

centre at y, where δ is considered as a material parameter. Near the boundary Γ(n), ) ; , ( Γ ΩΘ y ζr should be taken as an intersection of the disc/segment with Ω(n).

r Using the introduced brittle non-local CNESF ) , , , ζ} ; ( { y σn c Γ, thΛe fatiΘgue

fracture process (the fatigue crack initiation and its propagation through the damaged

material) can be described as in Section 1 after replacement there the stress tensor σ b y

its non-local counterpartΘσ.

However it is sometimes more convenient to employ for that purpose an equation for

the crack rate vector instead of Eqs (2)-(4). Let us consider a 2D case for homogeneous

isotropic body under a cyclic process as an example. Suppose there exist k cracks with

). W ecan take the total derivative of non

χ * y (

K moving tips

1

≤ ≤ χ

k K2 ;

≤ ≤

k K

local counterpart of Eq. (3) at a crack tip

χ * y with respect to n and arrive for CNESF