Issue 44

M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 58 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 <a 0 S < a*< a S * ) In the original case of Fig.5 we recognize case (b) which is also typical for metals for high stress concentrations whereas also case (a) is possible since F K /F R =2 -10 . In any event, 5 regions can be seen in the Atzori-Lazzarin diagram. Fig.6 also show the construction of the generalized Wöhler curves corresponding to the key sizes (in other words, we have up to 4 distinct Wöhler lines corresponding to the sizes a 0 ,a 0 S , a*, a S * ). With reference to Fig.5 (also reproduced in the details of the construction in Fig.7), the 5 regions correspond to different trend of the resulting slope in the ( a 0 <a 0 S < a*< a S * ). In the first region, for a < a 0 , k’=k remains unvaried to the value of the unnotched specimen. In the second region a 0 <a<a 0 S , we obtain a decrease of k’ up to a limit value, then remaining constant in the entire third region S a 0 <a<a* , and which we obtain easily from writing a Wohler-like power-law between the static strength value to the fatigue limit divided by the K t factor,             k k R t N N K 0 0 Δ (16) If we divide the original Wöhler curve (1) by (16) term by term, we obtain          k k k R t k k R N K 0 0 Δ Δ (17) i.e.         R min t R k F k K F Log Log (18) 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 k R f N N K ' 0 0 Δ (19) and dividing again for (1)                    R min t R N N k F N N k K F 0 ' ' 0 / Log Log / Log (20) 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.