On notch and crack size effects in fatigue, Paris’ law and implications for Wöhler curves

As often done in design practice, the Wöhler curve of a specimen, in the absence of more direct information, can be crudely retrieved by interpolating with a power-law curve between static strength at a given conventional low number of cycles N₀ (of the order of 10-10³), and the fatigue limit at a “infinite life”, also conventional, typically N_∞=2·10⁶ or N_∞=10⁷ cycles. These assumptions introduce some uncertainty, but otherwise both the static regime and the infinite life are relatively well known. Specifically, by elaborating on recent unified treatments of notch and crack effects on infinite life, and using similar concepts to the static failure cases, an interpolation procedure is suggested for the finite life region. Considering two ratios, i.e. toughness to fatigue threshold F_K=K_Ic/DK_th, and static strength to endurance limit, F_R =s_R /Ds₀, qualitative trends are obtained for the finite life region. Paris’ and Wöhler’s coefficients fundamentally depend on these two ratios, which can be also defined “sensitivities” of materials to fatigue when cracked and uncracked, respectively: higher sensitivity means stringent need for design for fatigue. A generalized Wöhler coefficient, k’, is found as a function of the intrinsic Wöhler coefficient k of the material and the size of the crack or notch. We find that for a notched structure, k<k’<m, as a function of size of the notch: in particular, k’=k holds for small notches, then k’ decreases up to a limiting value (which depends upon K_t for mildly notched structures, or on F_K and F_R only for severe notch or crack). A perhaps surprising return to the original slope k is found for very large blunt notches. Finally, Paris’ law should hold for a distinctly cracked structure, i.e. having a long-crack; indeed, Paris’ coefficient m is coincident with the limiting value of k’_lim. The scope of this note is only qualitative and aims at a discussion over unified treatments in fatigue.