Issue 44

M. Ciavarella et alii, Frattura ed Integrità Strutturale, 44 (2018) 49-63; DOI: 10.3221/IGF-ESIS.44.05 52         k k R f N N N 0 0 ,    f N N N 0 (1) where ∆σ is the stress range (we assume at the moment for simplicity that amplitude and range coincide i.e. the load ratio R=0, although it is clear that in general it would perhaps be appropriate to rewrite Eq.(1) in terms of amplitude of the cycle σ) and the N 0 , and  N are the number of cycles as defined in Fig.1. Clearly, Eq.1 also implies           r N k F k N 0 Log Log (2) and typically for steels considering N ∞ =10 7 and N 0 =10 3 , for F R  R  0 = 2 we would have k=13.3 , while for F R =3 , k=8.4 , in the typical range k= 6-14 for Al or ferrous alloys. In strain-controlled fatigue, the fatigue curve is replaced by a sum of two power/law functions assuming the fatigue life to be dominated by plastic strain in the LCF regime, and elastic strains or stresses in the HCF. The resulting well know equation (Coffin/Manson) is expected to be more accurate (if anything because it has more degrees of freedom to reproduce the experimental SN curve) although there is still a need to introduce the cut-off thresholds on very low and very high number of cycles, particularly on the low number of cycles where it tends to have the wrong concavity. N  R 0 N oo  0 tan( ) = k  Figure 1 : The simplified Wohler curve. Paris’ law The second important power law in fatigue is Paris’ law, giving the advancement of fatigue crack per cycle, v a , as a function of the amplitude of stress intensity factor ΔK (see Fig.2)    m a da v C K dN ;     th Ic K K K (3) where ΔK th is the “fatigue threshold”, and K Ic the “fracture toughness” of the material. There is therefore no dependence on absolute dimension of the crack. The law is mostly valid in the range 10 -5 —10 -3 mm/cycle, and in a simplified form it can be considered intersecting ΔK th and K Ic at 10 -6 , 10 -4 mm/cycle, respectively . This means that the constant C is not really arbitrary, since by writing the condition at the intersections,      m m th Ic C K K 6 4 10 10 . An alternative form can be obtained considering that Paris’ law is in general valid in the range 10 -5 —10 -3 mm/cycle and hence instead of the constant C it is perhaps more elegant to define a constant ΔK -4 , i.e. the range corresponding to a speed of propagation of 10 -4 mm/cycle