Why threshold stress intensity range is a function of the crack length: an explanation using fractals

It has long been recognized that the fatigue growth behaviour of cracks having a length comparable with the material microstructure size (the so-called short or small cracks) 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, the crack surface is firstly treated as a self-similar invasive fractal set (which is characterized by a uniform fractal dimension) and, owing to the fractional physical dimension of the fracture surface, the stress intensity factor is shown to be a function of the crack length. Consequently, the threshold stress intensity range is deduced to be a function of the crack length. Then the fractal dimensional increment is assumed to vary from 0 to 1 since, in the physical reality, the fractal dimension of the crack surface may change with the crack length. This allows us to put forward a new interpretation of the Kitagawa diagram within the framework of the fractal geometry.