Fractal behaviour in fatigue of materials

Abstract

Following the pioneering work of Mandelbrot [1], the use of "fractal" methods have become popular and their range of application has increased dramatically. In the application of fractal geometry to problems in fatigue and fracture of materials, fractal behaviour has figured prominently in many investigations, either as an assumption, a deduction or an observation [2-6]; attention was focused primarily on simple self-similar fractals that can be characterized by a single fractal dimension D~. However, to our knowledge, so far there is no rigorously theoretical analysis concerned with the physical origin of the fractal behaviour in these processes. The principle aim of the present work is to develop a more rigorous theory of fatigue fracture in metals and ceramics by treating the fatigue crack growth as a stochastic process, and to find the necessary and sufficient conditions resulting in fractal behaviour in fatigue processes in the theoretical framework of non-equilibrium statistical mechanics. All the considerations originate from the well-recognized fact that fatigue crack growth is fundamentally a random phenomenon [7, 8]. The sources of this randomness are material inhomogenetity which causes the variability of the materials resistance to fatigue crack growth along the path of the crack, and the random nature of the fatiguing stresses which dominates the probabilistic aspects of the problem in most practical applications. In the development of this approach the deterministic equation for rate of change of a given fatigue microcrack will be made stochastic by addition of a random "noise" term, i.e. it is assumed that the macroscopic fatigue rate equation is only approximate; in reality individual microcracks depart from this behaviour in an unpredictable way. Therefore, the resulting rate of a given fatigue crack growth can be expressed in a unified form as