Fractal nature of material microstructure and size effects on apparent mechanical properties

Abstract

The problems of the size effects on tensile strength and fracture energy of brittle and disordered materials (concrete, rocks, ceramics, etc.) are reconsidered under a new and unifying light cast on by fractal geometry. It is physically impossible to measure constant material properties unless we depart from integer dimensions of the material ligament at peak stress and of the fracture surface at final rupture. In this way we can define new tensile properties with physical dimensions depending on the fractal dimension of the damaged microstructure, which turn out to be scale-invariant material constants. This represents the so-called renormalization procedure, already proposed in the statistical physics of random processes. Variations in the fractal dimension of fracture surfaces produce variations in the physical dimension of toughness, and not, as asserted by some authors, only in the measure of toughness. In disordered materials an attenuation of the size effects due to the dimensional disparity between strength and toughness is found. As a limit case, any size effect vanishes when both tensile strength and fracture energy present the physical dimension characteristic of the stress-intensity factor, [ F] [ L] -3/2. It is very likely that this critical situation is achieved only inside a very disorderly damaged microstructure, e.g., in the vicinity of the crack tip. In the case of tensile strength, the dimensional decrement represents self-similar weakening of the material ligament, due to pores, voids, defects, cracks, aggregates, inclusions, etc. Analogously, in the case of fracture energy, the dimensional increment represents self-similar tortuosity of the fracture surface, as well as self-similar overlapping and distribution of microcracks in the direction orthogonal to that of the forming macrocrack.