PSI - Issue 7

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

347

5

4. Comparison between El Haddad and Murakami predictions for specimen with embedded elliptical crack Based on comprehensive fatigue testing of specimens with small defects, Murakami (2002) established empirical relationships between the fatigue limit and the size of the defect expressed as area , where area denotes the area of the defect projected in the direction of maximum normal stress. For an embedded defect (away from a free surface), Murakami suggested the following equation for the fatigue limit at 1 R = − of the defective specimen expressed in terms of the stress range:

(

)

(

)

(10)

2 1.56 HV 120 2 1.56 HV 120 ⋅ + ⋅ +

C

[

]

.

MPa

∆

=

=

=

σ

M

(

)

(

)

(

)

1

1

1

[ ] m

[ ] m

[ ] μm 10 6

6

6

area

area

area

Although Murakami’s empirical model is reported to be relatively insensitive to the shape of the defect, only an embedded elliptical crack will be considered here, see Fig. A.1. As shown in Appendix A, Fig. A.2, the stress intensity factor of an elliptical crack is mainly controlled by the area of the crack for axis ratios a c > 0.2. For an elliptical crack with a given area and axis ratio a c α = , one obtains

(11)

π area ac a π α = = .

A σ σ ∆ ∆ and

0 a a , and inserting eqs. (2) and (11),

Rewriting eq. (10) in terms of the dimensionless variables

yields

(12)

( ) 1 α 

1

1

−

−

   

2 6

F

a    

0   a    

6

6

1

C

C

σ

∆

.

  

=

=

M σ

M

(

)

1

1

A 0 ∆   a

σ

∆

π

α σ

6 L a

∆ ⋅

6

π

a

α

A

A

0

The ratio between the fatigue limit for an elliptical crack and that for a circular crack now becomes

(13)

1

1

( )

   

6    =   I a

2

F

α α

3

K K

∆ ∆

a σ σ r

.

  

=

( ) 1

2

F

I   r

Thus, the dependence of the ‘Murakami’ curve of the axis ratio becomes even weaker than what is shown in Fig. A.2. With fatigue data for 1 R = − and Vickers hardness from Table 1, one obtains 6 18.8 10 m L − = ⋅ and M 115.4 C = . Fig. 4 shows a comparison between the predictions due to El Haddad, eq. (3), and Murakami, eq. (12). For axis ratios in the range a c = 0.2–1 and a given 0 a a , ‘Murakami’ predictions vary < 3%. Moreover, these deviate 15% < from the El Haddad curve in the range 0 a a = 0.03–4, i.e ., over more than two orders of magnitude.