Issue 44

M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05

These two curves (or better the inverse of these two curves) are reproduced in Fig.8,9, as a function of a/a 0 for some example cases (typical steel and typical ceramic, where F K =15.5, F R =2.4, and F K =2, F R =1.5, respectively). Since the plots are given as a function of a/a 0 , the 1/K f curve “bends” around x=1, whereas the corresponding 1/K S curve “bends” around x= a 0 S / a 0 which in fact scales with the square of the ratio F K /F R = 15.5/2.4=6.5, and F K =2/1.5=1.33, and hence a 0 S / a 0 =41.7 and 1.7 respectively, since a 0 S /a 0 =(F K /F R ) 2 . This El Haddad form is apparently more complicated, but in fact by repeating the same reasoning of the previous paragraph, we only need to distinguish 2 possible ranges: for a



k

F

Log

R



a

(26)

;

k



       

K F a a a a s 0

0

Log



i.e.



F

k

Log

R



a

(27)

;

k

       



            K R a F a F a F 0 2 1

 

K

Log



a

0

This slope has a limiting value of

        k k

F

Log

R

a=a*

(28)

;

      

 

lim

2

 

K t F K

1

Log



2

           K R

F K F

2

t



For a>a*, the slope increases again,



k

F

Log /

R



a*

*

(29)





k

    

 

s 0



K

a a K

   

Ic

Log

 Δ /

t

0

and finally we return to the original slope k , for a>a s *. The resulting slopes are also indicated as ratio k’/k<1 in the Fig.8,9 for 3 example stress concentration factors K t =2,5,10 , showing how for steel the generalized Wöhler slope is already about 60% of the original one for notches slightly larger than a 0 and with stress concentration factor only of about 2. The slope continues to decrease to about 40% when the notch is now significantly larger than a 0 (specifically about 20 times larger than a 0 ) and recollecting Eq. 6,2 for the estimate of the Paris and Wöhler slopes, respectively, we have about m=3.4, k=10.5 with the conclusion that the limit reduction of the generalized Wöhler slope is k’/k =32%, and hence with a stress concentration factor of about 5 we’re already very close to the limit slope. For the case of ceramic material in Fig.9, the estimates with eqts. 6,2 give m=13.3 and k=22.7 with the conclusion that the limit reduction of the generalized Wöhler slope is 59%. However, with the same concentration factors as the previous cases, i.e. K t =2,5,10 we obtain that the decrease of the slope is already almost complete with a notch of the order of 2 a 0 and with

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