Fatigue Crack Paths 2003

(6)

m2 σ

N W

⎜ ⎜ ⎝ ⎛

⎟ ⎟ ⎠ ⎞

f

, naf

W

n a

0

,

= , where N0 – a number of cycles corresponding to the fatigue limit Waf,n. Parameter of

normal strain energy density is calculated in each time instant on the plane where

product of normal stress σn(tk) and normal strain εn(tk) reaches maximumvalue.

Weight IV

ctW W

cW W

ns

ns

t

< t W f o r

⎧

nsaf ,

k

n ska f

k n s k

⎪

m

t W f o r

4

τ

⎪ ) ( W

) (

(7)

,

=

20 ) (

⎜ ⎜ ⎝ ⎜ ⎛

⎟ ⎟ ⎟ ⎞

⎨

,

⎪

≥

n s a f

⎪

⎠

,

⎩

) ] ( s g n [ k n s t ε ) ( ) ( 2 1 ) ( k n s k n s k n s t t t W ε σ = . Weight III was formed on the ) ] ( t σ + sgn[ k n s

where

2

same theoretical grounds as weight function W3 but is based on the parameter of shear

strain energy density Wns. This weight function includes only those principal axes

positions for which the parameter of shear strain energy density Wns(tk) is greater than

the product of c = 0.25 and the fatigue limit Waf,ns expressed by the energy density

parameter.

af 2

τ

,

(8)

W nsaf

=

4

,

G

where: τaf– shear stress fatigue limit, G – modulus of shear elasticity. The participation

of such positions in averaging exponentially depends on the parameter 0.5mτof the new

fatigue curve (Wa,ns – Nf), based not on stress amplitude τa but on parameter of shear

strain energy density amplitude Wa,ns:

τ m

(9)

.

nsaf f N W ⎟⎠⎞ = ⎜⎝⎛

2

,

0 nsa N W The parameter of shear strain energy density is calculated in each instant time on the

,

plane where the product of shear stress σns(tk) and shear strain εns(tk) reaches maximum

value.

After the averaging procedure of Euler angles the components of cosines matrix of

averaged principal stress directions are calculated [4].