**4. The fractal line of sands/gravel and cement**

First, on the one hand, we deal with the transformation of the granulometric curves of local materials concerning the granular classes of sand and then of gravel. On the other hand, we show the transformation of the particle curves for the cementitious materials of CPJ 42.5 and CRS 42.5.

The determination of the granular distribution is one of the important physical properties for the definition of aggregates and as is known in the context of concrete formulation methods, and it is important to control and determine the proportions of the appropriate aggregate components for concrete.

#### **4.1 The fractal line of sands**

**Figure 6 (a)** is an example showing the grain size curve of dune sand with a grain range of 0/5, **Figure 6 (b)** is its transformation into a fractal line, and this sand is one of the 10 sand classes of dunes shown in **Table 2** and its smallest, granular extent is 0.005/0.63 and it is a very fine dune sand. As for the granular extent of coarse dune sand is 0.005/5. Thus, a quarry sand with its granular extent is 0.063/63.

The identification results shown in **Table 2** were obtained by conducting laboratory experiments to determine the varieties of sand dunes at the Building Materials Laboratory of the University of Adrar in cooperation with the Regional Workshop Laboratory. On desert techniques (ARTS) [26]. The sand studied is sand from sites approved for the use of dune sands from different sites in the Saharan region in Adrar, Algeria.

The graphical fit of the fractal distribution is a linear fit by the equation y = b + ax. And constant "a ", it is the slope which represents the fractal dimension. **Figure 6** is *Application of a Granular Model to Identify the Particle Size of the Granular Mixtures… DOI: http://dx.doi.org/10.5772/intechopen.99969*

**Figure 6.**

*Transformation of a grain size curve from a sand from Tilouline dune to a fractal line (example) [1]. a) Particle size curves, b) fractal line.*

an example showing the methodology for determining fractal dimensions of all granular classes, including dune sand.

$$\mathbf{Y} = 4, \mathbf{59-2, 69 (x)}\tag{12}$$

In this case, the slope is 2.69 and the correlation coefficient is R2 = 0.96. The slope obtained by cumulative number of grains according to grain dimensions is the fractal dimension DF, which determines the class of the distribution of grains of a granular class or of the granular mixture.

### **4.2 Physical properties of dune sands**

**Table 2** summarizes the conventional physical properties of dune sands in the Adrar region, namely granular extent, fineness modulus and sand equivalent. And the unconventional properties are the fractal dimension of the granular distribution, whose value varies from 1.14 to 4.20 with a correlation coefficient whose value varies from 0.82 to 0.98, which is a value close to one therefore is accepted. In addition to the total area of the grains of sand calculated by the fractal dimension.


**Table 2.**

*Conventional and unconventional parameters for certain sands of the Adrar-Algeria sites [1].*

**Figure 7.** *Transformation of the particle size curves of Adrar Dune Sands to fractal lines a) particle size curves, b) fractal line.*

Note: Physically, the value of the fractal dimension does not exceed the value of 3, because the study is within the framework of the three dimensions. But mathematically, that is to say according to the calculations of the hypothesis adopted, certain granular varieties can exceed the value of their fractal dimension the value 3.

But keep the optimal value that he adopted by chouicha [2], in his work, which does not exceed the value of 3 until the contrary is physically proven.

In **Figure 7 (a)** below, we show the grain size curves for all classes of dune sands as well as the quarry sand which we presented in **Table 2**, **Figure 7 (b)** we show the resulting fractal lines fractal distributions after transformation of their grain size curves for the different grain classes of dune sands. We note that the grain distribution of these sands is different and also gives us a different fractal distribution, which is proved by the obtained values of the fractal dimension.


#### **Table 3.**

*Identification by DF for granular mixtures of concrete at Adrar [1].*

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