Three-dimensional fractal analysis of concrete fracture at the meso-level

The fracture process in concrete-like materials cannot be properly modelled in an Euclidean framework, due to its complex morphology at the micro- and meso-level. The inherent flaws interact through a multi-scale process, leading to self-affine fracture surfaces. Moreover, the self-organized network of microcracks displays fractal properties prior to the formation of the final fracture surface. At the same time, due to the presence of pores and voids, the stress-carrying cross section is a rarefied fractal domain, even from the beginning of the loading process. A new experimental equipment has been developed which allows the entire fracture surface, or any plane cross section, to be digitised and analysed. This represents an important progress with respect to the study of one-dimensional profiles. In this paper, the three-dimensional algorithms for evaluating the fractal dimension of invasive surfaces and lacunar sections are described. The invasive fractal character of the fracture surfaces is confirmed. Moreover, the lacunar fractal character of the stress-carrying cross sections, a priori assumed by Carpinteri [A. Carpinteri, Mechanics of Materials 18 (1994) 259–266], is now proven experimentally.