Some Notes on Granular Mixtures with Finite, Discrete Fractal Distribution

Why fractal distribution is so frequent? It is true that fractal dimension is always less than 3? Why fractal dimension of 2.5 to 2.9 seems to be steady-state or stable? Why the fractal distributions are the limit distributions of the degradation path? Is there an ultimate distribution? It is shown that the finite fractal grain size distributions occurring in the nature are identical to the optimal grading curves of the grading entropy theory and, the fractal dimension n varies between -infinity and infinity . It is shown that the fractal dimensions 2.2-2.9 may be situated in the transitional stability zone, verifying the internal stability criterion of the grading entropy theory. Micro computed tomography (mu CT) images and DEM (distinct element method) studies are presented to show the link between stable microstructure and internal stability. On the other hand, it is shown that the optimal grading curves are mean position grading curves that can be used to represent all possible grading curves.

Automated summary

Why are fractal distributions so common? Is fractal dimension always less than 3? Why does the fractal dimension between 2.5 and 2.9 seem to be stable? Fractal distributions are the limit distributions of degradation paths, but is there an ultimate distribution? It is shown that the finite fractal grain size distributions in nature are the same as the optimal grading curves of the grading entropy theory, with the fractal dimension n varying between negative infinity and positive infinity. Additionally, it is suggested that fractal dimensions between 2.2 and 2.9 may fall within a transitional stability zone, confirming the internal stability criterion of the grading entropy theory. Micro computed tomography (mu CT) images and distinct element method (DEM) studies demonstrate the connection between stable microstructure and internal stability. On the other hand, it is shown that the optimal grading curves represent all possible grading curves as mean position grading curves.