Fatigue Crack Paths 2003

Ao'lj (m,y) IAqO(m) |(5',j(a(m),y)|.

The stress field 6'1]. (a(m),y) induced by the

loading @(x1) is supposed to be available analytically or numerically.

Let us take C N E S Fin the form (1), then the equation for the crack initiation moment

according to Eq. (3) is

j|Aqo (m)|bdm : (621A )b,

(6)

b é-zzwo’y’fi)

0

where a0 I 0 if there is no crack initially in the body, y* is the tip of an already

existing crack or the stress concentration point where the crack will initiate. If there

exists an initial crack system with a(0) I a0 I 0, then Eq. (6) implies n; I 0 due to the

(5'2 (aO,aO)|I 00, ie the cracks start to propagate

stress singularity at the crack tip,

without any delay after the load application.

Let the origin of the coordinate system be in the middle of a crack. Then the

coordinate of the crack tip is y: I a(n) and the dependence a(n) for the developing

crack length is to be obtained from (3), which is reduced to the following non-linear

Volterra integral equation of the first kind for a(n) s l ,

- (020” — enowarnnl” jlAqmlbdm.

(7)

622(a(m),a(n))|b|Aq0(m)|bdm

I

W e can change variables in (7) similar to Zobnin and Rabotnov (see [10] where a

solution of a corresponding creep crack problem is presented for b=1) and arrive at the

following non-convolution linear Volterra equation of the first kind to be solved for g(a) : [Aq0(m(a))/o';1A ] hdm(a) /da,

a(n)

ezzwmaonl” A

ezzwonlbgwna :1

I

b ,

aO£a(n)£l.

(8)

G22(a0:a0)|

“0

Periodic Collinear Cracks in an Infinite Plane UnderUniformLoading

Consider nowa more particular example for 2l - periodic collinear straight cracks with a

length 2a(m) in an infinite plate. Let a uniform cyclic traction with a range

Aq(m,x) I Aq0(m) be applied at infinity normal to the crack line, Fig.2. For an elastic

body, the normal stress range Ao'22(m, x1) ahead of the crack has the form [11],

AK1

A622(m;a(m),y1) I

(9)

\/2ltg(&))[sin2 (7371) - sin2 ( K b )

2l

2l

AK1(m;a(m)) : Aq0(m)‘/2ltg%;n),

a(m) g l.