PSI - Issue 28

Alberto Sapora et al. / Procedia Structural Integrity 28 (2020) 446–451 Author name / Structural Integrity Procedia 00 (2019) 000–000

448

3

 representing the well-known William’s eigenvalue. The shape function  related to the present geometry was evaluated by (Dunn et al. 1997): it varies from 1.12  for the crack case (  = 0°) to 1 for the plane geometry (  = 180°). In order to implement the FFM criterion expressed by Eq. (1), the stress field and the SIF functions are needed. The former can be approximated by the asymptotic relationship

V I

K



(4)

( )    y x



 1 x 



2



whereas the latter through the expression proposed by (Hasebe and Iida 1978):

0.5

( ) ( )      K c

V K c





(5)

I

I

The parameter  increases from unity, when  = 0°, up to 1.12   , when Eq. (5) coincides with the formula for the SIF of an edge crack. Accurate values can be found in tabulated form in (Livieri and Tovo 2009). The substitution of Eqs. (4) and (5) into system (1) yields :

2(1 )       ( ) l

, V I th   K

(6)

0

th

where

1 2 1 



  2 



   

 

(7)

( )   

  

2 

/ 2

 

and

2

0      th K   

th l

 

(8)

The crack advance l c takes the following form:

  2(1 ) 2 2 2 

l

l 



(9)

c

th

 

Indeed, for approaches based on a critical distance it has been proved (Lazzarin and Zambardi 2001; Atzori et al. 2005; Carpinteri et al. 2010) that Eq. (6) still keeps true, but for a different definition of the function  (Eq. (7)), which depends on the adopted criterion. Substituting Eqs. (3) and (6) into the fatigue limit condition (2) yields:

 



(10)

f



1 a 



0

where / th a a l  is the dimensionless notch depth. Predictions according to Eqs. (10) are reported in Fig. 2a for increasing notch amplitudes  , leading to the so-called generalized Kitagawa diagram for V-notches.