Fractality in the threshold condition of fatigue crack growth: an interpretation of the Kitagawa diagram

It has long been recognized that cracks having a length of the order of magnitude as that of the material microstructure size (the so-called short or small cracks) exhibit a fatigue growth behaviour which is remarkably different from that of long cracks. In particular, the threshold condition of fatigue crack growth is seen to be correlated to the crack length and the material microstructure. The well-known 'Kitagawa diagram' describes the variation of the threshold stress intensity range against the crack length, showing the existence of a transition value of length beyond which the threshold of fatigue crack growth is governed by linear elastic fracture mechanics. In the present paper, treating fracture surfaces as self-similar invasive fractal sets (which are characterized by a uniform fractal dimension), owing to their fractional physical dimensions, the stress intensity factor is shown to be a function of the crack length. Consequently, the threshold stress intensity range appears to be also a function of the crack length. In the physical reality, the fractal dimension of the fracture surfaces may change with the crack length and, thus, a varying fractal dimensional increment (with respect to the Euclidean domain where the fractal set is contained) from 0 to I is here assumed. This allows us to put forward a new interpretation of the Kitagawa diagram within the framework of the fractal geometry.