**3. New models (mathematical analysis)**

#### **3.1 Fractal analysis**

The fractal model for determining the granular distribution is a conclusion drawn from fractal analysis and is a new model for the mathematical description of everything used and found in nature, in which its truncated shapes reveal patterns similar to increasingly precise scales. And irregular and recurring shapes can be described using mathematical models. The term "fractal" is a mathematical term coined by Benoit Mandelbrot [22] from the Latin root fraction. It was originally used as an adjective (fractal line) and today is a noun meaning broken or irregular.

Sebsadji and Chouicha in 2012 [5–10, 23] showed that Fractals can be defined as disordered systems that are self-similar independent of scale of observation. Their fundamental property is a non-integer dimension called fractal dimension, which can measure.

Its result builds on the findings of previous researches (Lecomte and Thomas, 1992; Chouicha, 2006), according to which ideal grading curves of concrete can be transformed into straight-lines power-law of the form given in Eq. (11).
