An extended (fractal) Overlapping Crack Model to describe crushing size-scale effects in compression

The inherent microstructural disorder strongly influences the mechanical behaviour of heterogeneous materials such as concrete and rocks. Tensile and compression tests, in fact, evidenced a localization of strain and dissipated energy in the post-peak softening branch, with a consequent scale dependence of the stress–strain response. For this reason, the well-known Cohesive Crack Model and the recently proposed Overlapping Crack Model are useful tools for describing the size effects in tension and compression, respectively. In general, strain localization, damage and fracture, which are phenomena affecting the failure of concrete, are not rigorously interpretable in the framework of continuum mechanics. On the other hand, since the flaw and the aggregate distributions in quasibrittle materials are often self-similar (i.e. they look the same at different magnification levels), the microstructure may be correctly modelled by fractal sets. In this paper, the approach based on fractal geometry, that has profitably been applied for the tensile behaviour, is applied to obtain a fractal overlapping law from uniaxial compression tests. According to this approach, it is assumed that energy dissipation, stress and strain are not defined with respect to the canonical physical dimensions, though on fractal sets presenting noninteger physical dimensions. As a consequence, these classical parameters should be substituted by fractal quantities, which become the true material properties.