Fractal Dimensions of Granular Materials Based on Grading Curves

A grading curve is an important measure for illustrating the grain size distribution of granular materials. In general, there exists a statistical fractal relationship between the cumulative number of particles and the grain size for granular materials in nature. The objective of this paper is to calculate fractal dimensions for different types of grading curves of granular materials. The common grading curves are classified into four categories: concave, convex, combinational, and gapped. The formulas of the concave and convex grading curves are derived from the same statistical fractal relation. The fractal dimensions can then be obtained from the known grading curves by a linear fitting approach. For combinational and gapped grading curves, two different fractal dimensions are used to describe the fine and coarse grain parts. In addition, a criterion to determine the optimal demarcation point of a combinational grading curve is suggested. All types of grading curves of common granular materials in civil engineering are considered in this study. Their statistical fractal dimension can be used to study the physical and mechanical properties of materials containing granular ingredients to facilitate the consideration of the complexity of grain distribution.