N.: Fractal fragmentation theory for shape effects of quasi-brittle materials in compression

The influence of the slenderness (shape effects) of a specimen in compression and of the friction between it and the loading platens on the dissipated energy density and on the compressive strength (strain-softening response) is theoretically and experimentally analysed. The energy dissipated during the process is assumed to be proportional to the area of the free surface of the fragments created under compression. A very general law, describing the energy dissipation in natural as well as in man-made fragmentation phenomena, is herein presented, obtaining, as particular cases, the classical comminution laws (Surface, Volume and Third Comminution theories). As a consequence, the dissipated energy density and the strength for a structural element under compression are obtained, by varying its slenderness under different boundary friction conditions. Finally, a comparison between experimental data and theoretical predictions on shape effects is presented. The influences of the specimen slenderness and friction on dissipated energy density and compressive strength are captured by the proposed model in a satisfactory way. Notation A f total fracture surface area of fragments b specimen basis side D fractal exponent h specimen height Np total number of fragments P cumulative size-distribution function for fragments p probability size-distribution function for fragments r fragment size rmax size of the largest fragment rmin size of the smallest fragment V = l3 specimen volume under compression Vf = l 3 f fragmented volume W energy dissipated during fragmentation friction-exponent º = h b specimen slenderness C material strengt