Ultimate particle size distribution of granular materials based on fractal scaling: Definition and its application

The concept of relative breakage has been increasingly used for the quantitative analysis of particle breakage in granular materials, and its calculation relies significantly on the ultimate particle size distribution. However, a precise characterization of this distribution remains unresolved. This study introduced a novel concept of the ultimate particle size distribution, defined by the ultimate space-filling capacity of the particles, employing fractal theory. High-pressure confined compression tests were conducted on quartz sand and gravel to investigate the evolution of the particle size distribution due to particle breakage. Fractal scaling served as a modeling tool to track the change in the particle number during particle breakage. The findings revealed a transition from diverse initial distributions to a fractal distribution with increasing breakage. A finite particle size scale exhibited an ultimate fractal distribution, where the ultimate fractal dimension was a function calculating the ratio of the minimum to maximum particle size, with values in the range of 1.5 to 3. The ultimate fractal distribution characteristics depended on the measurement scale of the maximum and minimum particle sizes, regardless of the uniformity of the initial distribution. The transition from a particle size distribution to a fractal distribution in the confined compression tests was characterized by a constant ratio between volumetric strain and relative breakage. This ratio held promise for accurately estimating relative breakage. This study provides insights into the ultimate particle size distribution and enhances relative breakage measurement in granular materials.