**3.2 Fractal dimension (FD)**

Marmi [24], in 2019 he expressed the fractal dimension as a parameter exists in classical geometry, and is a line is a one-dimensional object a surface a two-dimensional object, a volume a dimensional object. We are therefore used to objects whose dimension (D) is an integer 1,2 or 3. But it is not specified, what would be the dimension of a series of points on a line, an irregular and plane curve, a surface full of convolutions. For this purpose, the term fractal dimension was introduced by B. Mandelbrot in 1975 the fractal dimension is therefore a number which measures the degree of irregularity or fragmentation of an object or which measures the roughness of a surface.

The fractal dimension is the fraction or an irrational number (; 1.23; etc.) or an integer.

This notion of fractal dimension applies to scale-invariant objects: there are parts which are similar to the object itself up to an expansion (enlargement).

When we change the observation scale of a scale invariant object, we keep the shapes.

The particle size distribution curves of the cumulative sieve percentages as a function of the grain dimensions can be transformed to a straight line representing cumulative numbers as a function of the grain dimensions.

We can do this by assuming that the shapes of the grains have the same oval shape, and this, if we adopt the same hypothesis proposed by Lecomte and Thomas [25] in his work, which first touched on the analysis fractal and through which he approached the application of the fractal dimension in the determination of three types of granular mixtures for high-performance concrete. And in 1992, he achieved his study results by applying fractal analysis to a granular mixture of concrete related to the definition of granular analysis of granular mixture of concrete, which consists of several granular types. These results indicate at the time that he adopted the hypothesis of the dimension of a spherical grain of aggregate of main and standard dimension G, and the relation (8) below summarizes the determination of the volume of the spherical grain.

We can estimate the mass of the grains, called **M** mass, as well as their true density **ρ** and the average size of the grains **V** for each initial class of grains, n being the partial number of grains refused on the opening screen **G** [25] as a relationship (9) below.

$$\boldsymbol{\nu} = \frac{\pi}{6} \mathbf{G}^3 \tag{8}$$

$$N = \frac{\mathbf{M}}{\rho \left(\frac{\pi}{6}\right) \mathbf{G}^3} \tag{9}$$

It is also possible to express the cumulative number Nc of aggregate grains whose dimension is greater than or equal to the size of the opening of the sieve G, and Relation No. (10) shows the determination of the cumulative number of grains of aggregate. Thus, Relation (11) allows us to express the number of grains of aggregate rejected in a sieve, in terms of the cumulative numbers of all the granular components.

$$\mathbf{N}\_{\mathbf{c}} = \sum\_{i=1}^{n} \frac{\mathbf{M}i}{\rho\left(\frac{\pi}{6}\right) G i^3} \tag{10}$$

*Application of a Granular Model to Identify the Particle Size of the Granular Mixtures… DOI: http://dx.doi.org/10.5772/intechopen.99969*

$$\mathbf{N}\_{\mathbf{c}} = \sum\_{i=1}^{n} \frac{\mathbf{M}\dot{\mathbf{r}}\mathbf{P}}{\rho \frac{\pi}{6} \mathbf{G} \dot{\mathbf{t}}^{3}} \mathbf{N}\_{\text{cumule}} = \sum\_{i=1}^{n} \frac{\mathbf{M}\dot{\mathbf{r}}\mathbf{P}}{\rho \frac{\pi}{6} \mathbf{G} \dot{\mathbf{t}}^{3}} \tag{11}$$

The granulometric analysis of cement is done by laser, "Laser granulometry" this technique is based on the diffraction of light and was proposed by Fraunhofer under the application of their theory of Fraunhofer ". We have **Table 1** below which shows by a sub-detail of the fractal analysis which will identify the particle size of an example of CPA cement by the fractal dimension (FD).

FD fractal dimension is, therefore, an approximation of the granular distribution curve. If this approximation is good over almost the entire grain size measurement field, the granular distribution line is said to be the fractal or quasi-fractal dimension. If the curve obviously tends towards a limit when the dimension of the seeds tends towards zero, then this curve is said to be semi-fractal.

We show without difficulty that any physical measurement on a granular structure, even purely fractal, results in a granular curve on a logarithmic scale (quasifractal) due to the smaller dimension of d mm, an empirically necessary procedure. It turns out that only successive zooms, and logarithmic scale transformations


#### **Table 1.**

*Particle size analysis and fractal analysis of CPA.*

(d, D), probably reveal (on the slope of the lower higher convergence line) the effective quasi-fractal drift of the studied process.

If this drift has several changes in the slope, then in some cases it will be referred to as "multi-fractal."

**Figure 1** Presented the granular distributions of four types of cement identified by the fractal line, the cements are:

Portland cement compound class 42.5 MPa CPJ 42.5.

Cement sulphate resistant class 42.5 MPa CRS 42.5.

Artificial Portland cement class 52.5 MPa CPA 52.5.

Portland cement compound class 42.5 MPa type P6 CPJ P 42.5.

We present in **Figure 1** - the different fractal distribution with correlation coefficients of the fractal lines of the granular distributions, and the minimum correlation coefficient value is R<sup>2</sup> = 0.96. Appears in the granular distribution between three closely related types of cement, and another is different.

We followed the same method according to the results **Figure 2** of Lecomt [25] presented in the **Figure 2** which has ideally defined an example of the granular mixture containing a spread granular for a high-performance concrete, as well as all the granular classes of this concrete including the active mineral additions were used.

**Figure 1.** *Fractal lines of granular distributions of four types of cements alone.*

#### **Figure 2.**

*Transformation from a particle size distribution to a fractal distribution for a concrete mixture and its components from the Lecompt [25]. a) Particle size curves, b) fractal line.*

*Application of a Granular Model to Identify the Particle Size of the Granular Mixtures… DOI: http://dx.doi.org/10.5772/intechopen.99969*

All concrete formulation methods, old or new, are based on particle size for determining the different dosages of the granular constituents. The results obtained by Lecompte [25] and Chouicha [2], show that these methods which use a granular distribution, indirectly use a fractal distribution.

In **Figure 3** of Chouicha [2], the particle size curves for different granular mixtures that he identified with a uniform particle size range, we can determine the particle distribution of the granular mixtures with a fractal dimension from FD = 0.5 to FD = 7, knowing that this field is for the granular mixtures in general, which is much larger than the field of the granular concrete mix, so it is outside the concrete field, because the fractal dimension FD does not exceed the value of 3 .

We applied one of the three high quality BHP concrete mixes that he adopted by Lecomt in his research (**Figure 2**), we clearly show through **Figure 4** the curves of the granular distribution of the component classes of the concrete, as well as its curve of the granular mixture, and this gives some similarities between Lecomt's work and what we got despite using different components in terms of density and type of aggregate ect, and what we got despite using different components in terms of density and origin of gravels, and this is due to our relying on the granule size assumption of the spherical-shaped relation (8) to obtain the fractal distribution of this granular mixture with its components as shown in **Figure 4 (a), (b)** which gives us the results of converting granular curves to fractal lines. Lines.

**Figure 3.** *Particle size curves of the different granular mixes identified by FD Chouicha [2].*

**Figure 4.**

*Transformation from a particle size distribution to a fractal distribution for a concrete mixture and its components (example BHP) a) particle size curves, b) fractal line.*

**Figure 5.** *Transformation of a particle size curve of a granular mixture 0,63/25 to a) fractal line particle size curves, b) fractal line.*

The process of transforming the particle size curve into a fractal line has a direct relation to the granular variety of a granular class or of a granular mixture regardless of the mass taken for the granular variety. **Figure 5 (a)** shows that the grain size curve remains the same, regardless of the mass for the same grain class. **Figure 5 (b)** shows the transformation into a fractal line which gives us two lines of the fractal distribution, for each mass gives a fractal line for the same granular class, but with the same slope value, so it is the same value of the fractal dimension, and this is what he had confirmed by Chouicha [2] in his work in 2006.
