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Why Threshold Stress Intensity Range is a Function of the Crack Length: an Explanation using Fractals
Last modified: 2013-12-02
Abstract
It has long been recognized that the fatigue growth behaviour of crackshaving a length comparable with the material microstructure size (the so-called shortor small cracks) is remarkably different from that of long cracks. In particular, thethreshold condition of fatigue crack growth is seen to be correlated to the crack lengthand the material microstructure. The well-known “Kitagawa diagram” describes thevariation of the threshold stress intensity range against the crack length, showing theexistence of a transition value of length beyond which the threshold of fatigue crackgrowth is governed by linear elastic fracture mechanics. In the present paper, the cracksurface is firstly treated as a self-similar invasive fractal set (which is characterized bya uniform fractal dimension) and, owing to the fractional physical dimension of thefracture 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 thecrack length. Then the fractal dimensional increment is assumed to vary from 0 to 1since, in the physical reality, the fractal dimension of the crack surface may changewith the crack length. This allows us to put forward a new interpretation of theKitagawa diagram within the framework of the fractal geometry.
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