The Finite Fractal Distributions as Mean Grain Size Distributions of Granular Matter

Abstract

The grading entropy is the statistical entropy of the finite discrete grain size distribution on N uniform statistical cells in terms of the N sieve cells, consisting of two terms, the base entropy and the entropy increment (depending on N), which have normalized forms as well (basically independent of N). Being the most adequate statistical variables, both physical phenomena and physical model parameters can be best described by their use. Among others, the normalized base entropy A can be used to measure internal stability, being related to erosion, piping and liquefaction phenomena. Its value classifies the grading curves. Each class (with a fixed value of A) has a mean grading curve with finite fractal distribution, the fractal dimension varies from minus to plus infinity. (These mean gradings indicate a unique relation between the four entropy coordinates and four central moments). The internally stable fractal dimensions - between 2 and 3 – are occurring in nature verifying the internal stability rule of grading entropy. The widespread fractal soils are formed by degradation, which has a directional grading entropy path, with different features in terms of non-normalized and normalized grading entropy coordinates.