Fatigue Crack Paths 2003

(1) after some manipulations using the non-local counterpart of expression (5), at the

following crack growth rate equation,

(Mg (m, y,(f)b

dm ] (13)

I _[Mg.o1ba

5M1.

dy?‘

6

n

d n

ayi

y=y*x(n)

Here is no sum in )5, and if‘ (n) is the outward unit vector tangent to the crack. The

fracture plane is determined from the non-local counterpart of Eq. (5). Due to the

integral term, Eq. (13) accounts for all damage history at the point y *9‘ . Note that for an

arbitrary CNESF,the crack growth rate equation is obtained after solving a system of K

linear algebraic equation with respect to K scalar unknownvalues I y ”(n) |, where

dy:"(n)/dn I $ 0 1 M ? (”'1) |~

ExampleofNon-LocalDurability Analysis

Let us consider the 2D problem from Section 2.1 using the non-local durability analysis

with particular non-local C N E S F(11), (12) where the crack propagation plane 5* is

prescribed by the problem symmetry, Q®(y,@;1") is

the interval

(y, — 5 . 0 0 %+ 5.01)) for y ahead Of the Crack a(m), 5.01) I min(5, yr — a(m)|)

y1 —l|) and 6 is a material constant. Let w,,,,(y,x,§) I w(y, 005,15”,

5.(y1) :min(5,

wherew(y,x) is a boundedfunction, whichis considered as a material characteristics to

be identified. As possible approximations, one can choose e.g. w(y,x) constant w.r.t.

xe§2(y) (thus arriving at the Neuber stress averaging, cf. [12]), a piece-wise linear or a

more smooth hat-shaped dependence on x.

Repeating the same reasoning as in Section 2 but now for the non-local stress

Ao'lf’ (m;a, y,) , we arrive at the same Eqs (6)-(8) where (5'22 (a, y,)must be replaced by

1 - y I y) w(y1 ,x, )0‘22 (a, y,)dxl . For a problem with initially existing crack, 5 + ( 1 ) A . . . . .

6§;(a,y1):j

y1’5I(Y1

the crack propagation start instant n; obtained from the non-local counterpart of (6) is

non-zero since lo'g(ao,ao)l

A (9 _ b G22(a0 aao - example, the start delay for a constant loading qo is n; I n :,

Differentiating the non-local counterpart of (7) w.r.t. a(n) (cf. Eq. (13)), we arrive at

the following linear non-convolution Volterra equation of the second kind for the

unknownfunction g(a),

g(a(n)) + (n)K(a(n),a)g(a)da

: Y(a(n)),

a0 g a(n) s l,

(14)

K(a(n),a) I |6§’2(a(n),a(n)[b

fiwpgwrnnl‘b.

Y(a(n)) : -|a§; (a0, aO)[bK(a(n),aO).