On the mechanics of quasi-brittle materials with a fractal microstructure

The fractal approach to the mechanics of materials with a multiscale microstructure provides an elegant and unified explanation of the size effects the cohesive crack model parameters are subjected to. Aim of this paper is to collect the most recent developments of the fractal approach by Carpinteri and co-workers to damage and fracture of quasi-brittle materials. The first part shows how fractal patterns in the tensile failure of concrete specimens can be derived from the grain size distribution of the aggregates constituting the material. Once the concrete microstructure is analyzed, the geometrical and mechanical fractal quantities are highlighted. In terms of these quantities, a fractal cohesive crack model is proposed and its size-independence is proven. In the middle part of the paper, new mathematical operators from fractional calculus are introduced and applied to write the differential equations valid for fractal mechanical quantities and domains. The static and kinematic equations as well as the principle of virtual work for media with a fractal microstructure are obtained. The new mathematical formalism is applied to simple cases in order to obtain analytical results, such as the size effect upon strength in bending tests. The last part of the paper is devoted to the geometrical multifractal extensions of the scaling laws previously obtained. A multifractal scaling law (MFSL) for the critical displacement is proposed: together with the MFSLs for tensile strength and fracture energy, it completes the description of the size effects upon the cohesive crack model parameters.