Fatigue Crack Paths 2003

kt

Nt t

) ( W

where

is the summation of the weights, and N is the number of time

∑ =

W

1

instants considered.

Four kinds of weight functions are examined in the following.

Weight I [3]

(2)

1)(W1=kt

According to such a weight function, each positions of the principal axes influences the

mean position of the principal axes to the same degree, irrespective of the stress values.

Application of this weight function gives us W = N, and leads to the arithmetic means.

Weight II [3]

k σ c σt ) ( 1

) ( W

0

for

⎧

af

<

⎪⎩⎪ ⎪ ⎨

σ σ

c t

m

for

k c t

(3)

σ

,

σ

⎜⎝⎛

⎟⎠⎞

af

t

) (

2

1

) (

k

afk

≥

σ

1

=

where: constant - c = 0.5. This weight function includes only those principal axes

positions for which the maximumprincipal stress σ1(tk) is greater than the product of c

and the limit σaf; the participation of such positions in averaging exponentially depends

on the parameter mσof the Wöhler curve.

Weight III

cW

) (

) ( W ⎧

< t W f o r k

0

t n 2 ) (

n

naf ,

⎪

) (

m

(4)

,

σ

,

cW

⎪

⎟⎠⎞

≥ t W f o r n k

⎨⎜⎝⎛

,

n a f

3

cWtW

k

nafk

⎪⎩

=

) ] ( ) ( ) ( 2 1 ) ( k n k n k n t t t W ε σ = . Weight III is a new proposition sgn[ ) ] ( t σ + sgn[ k n k n t ε

where

2

based on the parameter of normal strain energy density Wn [9]. This weight function

includes only those principal axes positions for which the parameter of normal strain

energy density Wn(tk) is greater than the product of c = 0.25 and the fatigue limit Waf,n

expressed by the energy density parameter:

af

2

σ

(5)

W

=

2

E

naf ,

The participation of such positions in averaging exponentially depends on the parameter

0.5mσ of the new fatigue curve (Wa,n – Nf), based not on stress amplitude σa but on

parameter of normal strain energy density amplitude Wa,n: