Multifractal scaling laws in the breaking behaviour of disordered materials

Abstract

A strong relationship between critical phenomena and fractals can be recognized, implying a deep analogy between geometry and physics: the common underlying rule is the scale invariance, which forms the basis of the renormalization group theory, successfully applied to the analysis of phase transitions. In the case of fracture of heterogeneous media, the macroscopic effect of the disordered microstructure turns into the incomplete self-similarity of the phenomenon with respect to the main macroscopical quantities, which have to be renormalized and assume non-integer (anomalous) dimensions. The dimensional fractional increment of the dissipation space in the case of fracture energy, as well as the dimensional fractional decrement of the material ligament in the case of ultimate strength, are shown to explain the experimental size-dependent trends of these mechanical quantities. On the other hand, the continuous vanishing of fractality with increasing the observation scale seems to be peculiar of all natural fractals (multifractals). Extrapolating to physics, this implies that the effect of microstructural disorder on the mechanical behaviour becomes progressively less important for larger structures (i.e. large when compared with the microstructural characteristic size) where the disordered microstructure is somehow homogenized. Therefore, the scale effect should vanish in the limit of structural size tending to infinity, where asymptotic values of the physical quantities can be determined. Two multifractal scaling laws are proposed, for fracture energy and tensile strength, respectively, where the dimensional transition (with increasing structural size) from a microscopic Brownian disorder, to a macroscopic Euclidean behaviour, is controlled by the slope in the bilogarithmic diagram. Best-fitting of relevant experimental results has confirmed the soundness of this new approach.