Fractals in microstructure of metals

Abstract

Microstructures range between highly ordered and disordered. Fractal geometry deals with highly disordered morphologies, for example, rough interfaces or porous structures. They can in part be characterised by non-integer fractal dimensions D> d. Fractal analysis is applied to microstructural elements, which are usually described by their integer Euclidean dimension 0 ≤ d ≤ 3, for example, vacancies, dislocations, grain boundaries, dispersoid particles. In addition to their dimensions D > d, a fractal structure must be geometrically scaling. Self-similarity implies that a similar morphology appears in a wide range of magnifications in metallographic analysis. Examples are given for fractal analysis of dislocations, grain boundaries, particle distributions, and surfaces. In dendrites and martensitic microstructures, different degrees of fragmentation can be recognised. There exist attempts to describe slip step spacings and distribution of residual austenite by dimensions D < 1. Fractured and worn surfaces require careful metallographic analysis to determine not only a d value, but also to establish their self-similarity. Fractal microstructures exist far from thermodynamical equilibria. They are of various origins, but their formation always requires a high input of energy. A base for more than empirical fractal microstructure–property relations is in an early stage of development.