Issue 44

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

By looking at fixed dimension of the notch, and interpolating between static and infinite-life strength, we can obtain the “generalized Wöhler curve”, as well as a “generalized Wöhler coefficient, k’(a) ”. We shall start with a simplified version of the Atzori-Lazzarin schematically represented in Fig.5, i.e. the criterion with line segments, to make the easiest possible estimate of the generalized slope k’ --- we shall return to the more accurate El Haddad version in the last paragraph. Since a 0 S /a 0 =(F K /F R ) 2 and a * /a 0 = K t 2 we also obtain a 0 S / a * =(F K /F R ) 2 / K t 2 . Hence if K t < F K /F R then a 0 S > a * whereas if K t >F K /F R then a 0 S < a *. . We shall only consider F K >F R or as a limit case, F K =F R hence we have 3 cases: 1. Case (a) F K >F R and K t < F K /F R (top of Fig.6 where we see a 0 < a* < a 0 S < a S * ) 2. Case (b) F K =F R and K t >F K /F R =1 (bottom of Fig.6 where we see a 0 =a 0 S < a*= a S * ) 3. Case (c) F K >F R but K t >F K /F R (Fig.7 where we see a 0

k

 

  

Δ

k R

0

 N N

  

(16)

0

K

t

If we divide the original Wöhler curve (1) by (16) term by term, we obtain

k

k R k

 N K 0 0 Δ Δ    k

k

(17)

t

R

i.e.

  R

k F Log

  k

  Log

(18)

min

K F

t R

In other words, k’ decreases from the unnotched specimen case up to a limit value (depending on Kt) given by Eq.18. Notice that this equation has been obtained without any need to specify N 0 and N ∞ , except of course that these values are assumed to remain constant independently on the size of the notch. If a more general choice had been made, i.e. using new values N’ 0 and N’ ∞ , and not the original N’ 0 and N’ ∞ of the Wöhler curve in Eq.1,

k

  

   

Δ

k R

'

0

 N N

  

(19)

0

K

f

and dividing again for (1)

  

   

  ' N N N N / /

  R

0 ' 0

  Log

k F Log

  k

 

(20)

min

K F

Log

t R

and the decrease of k’ depends now on the variation of N’ 0

and N’ ∞

, as a function of the notch size, and not just on K t .

However, we shall neglect this possibility.

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