On the possible generalizations of the Kitagawa-Takahashi diagram and of the El Haddad equation to finite life

The celebrated Kitagawa-Takahashi (KT) diagram, and the El Haddad (EH) equation, have received great attention since they define quite successfully the region of non-propagation (or the condition of self-arrest) for short to long cracks. The EH equation can be also seen as an "asymptotic matching" between the fatigue limit and the threshold of crack propagation. Above this curve, finite life is expected, since cracks propagate and eventually lead to final failure. In this paper, possible extensions of the EH equation to give the life of a specimen with a given initial crack as a function of the applied stress range, using only "asymptotic matching" equation between known regimes, namely the Wohler SN curve (or some simplified form, like Basquin law), and the crack propagation rate curve (or just the Paris' law). This permits an extension of the so-called "intrinsic crack" size concept in the EH equation for infinite life. The generalized El Haddad equation permits to take into account approximately of some of the known deviations from the Paris regimes, for short cracks, near the fatigue threshold or fatigue limit, or to the static failure envelope. The new equations are also plotted as SN curves, showing that power-law regimes seem very limited with many possible deviations and truncations, even when the crack propagation law has a significant power-law regime. The diagram remains partly qualitative (for example, we neglect geometric factors), and can be considered a first attempt towards more realistic maps. Particularly interesting are the cases with the Paris exponent m < 2, in which propagation tends to be very slow until very close to the toughness failure, making the maps qualitatively different.