PSI - Issue 7

G. Härkegård / Procedia Structural Integrity 7 (2017) 343–350 F / Structural Integrity Procedia 00 (2017) 000–000 G. Härkegård 0

345

3

El Haddad et al. (1979) suggested a simple analytical model of the Kitagawa-Takahashi observations by introducing an equivalent large-crack stress-intensity range ( ) eq 0 π K F a a σ ∆ = ∆ + , (1)

where the ‘intrinsic’ crack depth, 0 a , is given by 2 th 0 2 A 2 th 1 , π 1 K L a F F K σ  ∆  = =  ∆     ∆ 

(2)

.

L

= 

π

A ∆    σ

The crack-geometry factor, F , depends on the shape and location of the crack. For the surface through-crack of Fig. 1, F = 1.1215. Letting eq th K K ∆ = ∆ , and solving eq. (1) for σ ∆ , yields the following expression for the fatigue limit of the cracked member:

(3)

A 1 a a σ ∆ +

σ ∆ =

.

0

This equation has been used for plotting the El Haddad curve in Fig. 2. In the following, it will be demonstrated how eq. (3) can be related to power laws of the type

(4)

, m a m

constant

0 < ,

σ ∆ =

⋅

1 3 m = − , and Murakami (2002),

1 6 m = − . Taking logarithms of

which have been proposed by Frost et al. (1974),

both members of eq. (4) and differentiating with respect to ln a yields d ln d ln m a σ ∆ = ,

(5)

the slope of a graph of eq. (4) in a log-log diagram. Similarly, taking logarithms of both members of eq. (3) and differentiating with respect to ln a yields

(6)

d ln

1

a a

σ ∆

= −

.

0

d ln

2 1

a

a a

+

0

The slopes according to eqs. (5) and (6) coincide if

(7)

1 2 1

a a

a

m

m

= −

⇒ = −

.

0

a a

a

m

+

+

1

2

0

0

0 a a and

A σ σ ∆ ∆ have been compiled in Table 2.

Associated values of m ,