First, the expensiveness of the modern building materials makes people think more and more to the use of local materials in the fabrication of bricks for the housing. These types of construction have been known since the age-old times through the Nubian arch of the area of the Nil, the “sakh” and the “hut” of Sine, Senegal, the “banco boumg “of Casamance. These earthen constructions or in fiber ameliorated earth were spread in Senegal and through the world. The banco (in Africa) and adobe (in Europe) are a technology used since millennia all over the world. It is a raw earth brick molded and dried under the sun, sometimes added with binding fiber or aggregates. The raw material, of so clayey nature (up to 30% of thin portions), but very sandy, is added with water to obtain a semi firm paste (15% to 30% of water). The mixture is put into a mold so that to make the brick [1] [7] . The blocks are assembled by a mud mortar made from the same earth. It is largely used in the majority of the Senegal areas. The adobe bricks are cast in an open parallelepiped frame.

In the frame of this study two types of soils are used. Two samples will be submitted to that study. It’s about the Ndouloumadjie Dembe quarry hammer ( Figure 1(a) ) and the Tatttaguine termite ( Figure 1(b) ) whose grain size distribution is shown in Figure 2 . A sample of raw earth for the construction derived from the Ndouloumadjie quarry (Matam, Senegal), and another sample stemming from an exploited termite mount in Tattaguine, Fatick region ( Figure 3 ). The described experimental procedure in Figure 4 will be continued.

These raw samples are used for experimental analysis in order to have a geotechnical characterization by determining their identification and classification parameters. This characterization is done in performing identification tests at geotechnical laboratory of Thies University. Among the geotechnical tests we have, the granulometric analysis, the VBS value, the Atterberg limits and Proctor test.

The granulometric analysis ( Figure 2 ) shows the percentage of the loops to 0.075 micron is 38% for the Ndouloumadjie sample and 19% for the Tattaguine sample. These two percentages are inferior to 50%, we have a granny soil. The loops in the sieve 4.75 mm are 99% for the Ndouloumadjie sample and 100% for the Tattaguine sample. Since these percentages are superior to 50%, we have sands. The coefficient of uniformity of 4.5 (inferior to 6) and the coefficient of curvature of 0.42 is out of the range 1 - 3 hence, a graduated soil (SP). The loops to 0.075 micron are over 12% for the both samples. Thus, the Ndouloumadjie sample which the liquidity limit is 23 and the plasticity index is 13, is a soil of (SP-SM) type whereas for Tattaguine which has 32 as liquid limit and 15 as plasticity index, the soil is SP type ( Table 1 ).

A chemical analysis through spectrometry technique by X fluorescence to wave length dispersion of these samples has been carried out ( Table 2 ).

These components allowed determining two modules ( Table 3 ): the silica module given through equation 1 that inform on the ratio between the quantity of silicate (stemming from SiO₂) and that of aluminates (stemming from Al₂O₃ and Fe₂O₃), and the alumina module given in Equation (2) and which is the ratio between the quantity of the alumina and iron oxide. For Ndouloumadjie we have an alumina module (AM) of 0.47 (inferior to 0.638) whereas for the Tattaguine termite it is 1.97 (superior to 0.638). The Ndouloumadjie quarry material is then rich in iron whereas the Tattaguine termite normal. The alumina content is higher for the Tattaguine termite. Physically that is shown through the brown colour of the Ndouloumadjie sample (rich in iron) and yellow for the Tattaguine termite (rich in alumina).

$\text{MS}={\text{SiO}}_{\text{2}}/\left({\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}+{\text{Fe}}_{\text{2}}{\text{O}}_3\right)$ (1)

$\text{MA}={\text{Al}}_{\text{2}}{\text{O}}_{\text{3}}/{\text{Fe}}_{\text{2}}{\text{O}}_{\text{3}}$ (2)

An improvement with a small percentage 1%, 2% and 3% of the involucre of the millet collected has been made also. The millet involucre has been sieved at 5 mm to eliminate coarse elements. Figure 3 shows the tests execution procedure.

In order to improve the thermomechanical properties raw earth samples from Ndouloumadjie and of Tattaguine, millet involucre ( Figure 4(a) and Figure 4(b) ) has been used.

The compressive strength is determined on adobe bricks of size (15 cm × 15 cm × 40 cm) for raw and improved samples. The anchorage ( Figure 5(a) ) and resampling for making of adobes is done on the basis of the parameters of Proctor optimum (Apparent dry density and optimal water content). Mixing is done by shovels, hands or feet until obtaining a homogeneous and manageable paw ( Figure 6(a) ). The mold (15 cm × 15 cm × 40 cm) is then filled into three layers ( Figure 5(b ). Each layer filled undergoes 3 consecutive falls of 40 cm height in order to expel voids and have good compaction. After filling of three layers, a surplus of earth is deposited on the surface of the mould. It’s finally subjected to the blows of a dune metal plate of 72.5 g in size (15 cm, 30 cm, 4 cm). The number of strokes of the plate metal is between 10 and 15 up to cancellation of settlements. The samples are dried in the shade for 14 days ( Figure 5(c) ). These are briquettes will be crushed ( Figure 5(d) ) by the way.

$\text{Rc}=10\ast F/S$ (3)

where Rc = corrosion resistance in MPa, F = strength in MN, S = mm².

For bricks made from raw sample and ameliorated of Tattaguine and that of Ndouloumadjie improved, the mold (10 × 10 × 5) was used ( Figure 6 ). The filling of the mold is done in one go and suffers five consecutive blows until what is observed any settlement. Equation (3) is used to determine the compressive strength of those adobe bricks.

Thermic characterization consists to determine thermic parameters such as conductivity (λ) effusivity (E) and thermic Resistance (Rth) of raw and improved earth briquettes with involucre of millet. The thermic properties are determined using the plate method kept warm. This latest is a device which makes it possible to measure the thermic conductivity (λ) and thermal Resistance (Rh). A heat flow is generated by Joule effect at the center of a plate kept warm. The perimeter of the heating zone is surrounded by an electrical resistance, the guard. A regulation maintains a difference of null temperature between the guard and the heating zone. The flux which crosses a pair of test specimens, placed on either both sides of the hot plate, is then unidimensional. Two cold plates placed against the outer faces of the specimens allow them to set their temperature. Because of the two samples are identical the flow crosses at equal share each eprouvette. Measurement of two sides of each temperature on sample, once permanent regime reached then allows to calculate the thermic conductivity and deduct the thermal resistance.

$\lambda =Q\ast L/\Delta T$ en W/m·K (4)

QL: steady state heat flow;

ΔT: T_a − T_b = thermal gradient;

L: sample thickness.

The tests carried out in the laboratory make it possible to have results on very challenging changes. Then, use the techniques of interpolation and approximation are used [11] [12] . These are two techniques which make it possible to represent by one simple function, but in an approached way, an unknown function which we acknowledge the value in a certain number of points. Hence, it comes to the thermo-mechanical properties on the whole extension interval of the mixture. Two approximation methods will be used in the frame of this work, namely the Lagrange method and the Newton method (divided differences). The two methods will allow to do an approximation of compressive strength by polynomial: First of all, through Equation (5) and Equation (6), then Newton polynomial (divided differences) through Equations (7)-(9).

$P_n\left(x\right)={\displaystyle {\sum }_{i=0}^n{y}_i{L}_i\left(x\right)}$ (5)

L_i is Lagrange polynomial define by:

$L_i\left(x\right)={\displaystyle {\prod }_{j\ne 1}^n\left[\left(x-x_j\right)/\left(x_i-x_j\right)\right]}$ $i=0,1,\cdots ,n$ (6)

Newton polynomial (divided differences method) by function:

$P_n\left(x\right)=f\left[x_0\right]+f\left[x_0,x_1\right]\left(x-x_0\right)+\cdots +f\left[x_0,\cdots ,x_n\right]{\displaystyle {\prod }_{k=0}^{n-1}\left(x-x_k\right)}$ (7)

with

$f\left[x_{i-1},x_i\right]=\left(y_i-y_{i-1}\right)/\left(x_i-x_{i-1}\right)$ (8)

$f\left[x_{i-2},x_{i-1},x_i\right]=\left[f\left(x_{i-1},x_i\right)-f\left(x_{i-2},x_{i-1}\right)\right]/\left(x_i-x_{i-2}\right)$ (9)

These approximation methods are going to be couples with some laboratory data in order to extend the field of the study results. It is then to extend the results in numeric integrant of the points. This worked is executed by using the laboratory data in Table 4 for the compressive strength and data in Table 5 for the thermal variables. The tests are taken 14 days after dredging under shade on the raw test tubes or improved with millet involucre.

For the Ndouloumadjie earth bricks case, the uniaxial compressive strength is given through the following polynomial in relying on the recapitulated results in Table 4 .

For Ndouloumadjie, we have:

Hence, we have for Lagrange:

$P_2\left(x\right)=1.42\left(x-2\right)\left(x-3\right)/2-1.65\left(x-1\right)\left(x-3\right)+1.53\left(x-1\right)\left(x-2\right)/2$

For Newton method, in accordance with Equation (7) mentioned above, we have Table 6 .

Hence, the following Newton polynomial:

$P_n\left(x\right)=1.42+0.23\left(x-1\right)-0.1750\left(x-1\right)\left(x-2\right)$

For Tattaguine, we have the same $L_i\left(x\right)$ $i=0,1,2$, as for Ndouloumadjie, we have the following Lagrange polynomial:

$P_2\left(x\right)=1.38\left(x-2\right)\left(x-3\right)/2-1.54(x-1\left(x-3\right)+1.39\left(x-1\right)\left(x-2\right)/2$

Due to n = 2, the Newton polynomial writes:

$P_n\left(x\right)=f\left[x_0\right]+f\left[x_0,x_1\right]\left(x-x_0\right)+f\left[x_0,x_1,x_n\right]\left(x-x_0\right)\left(x-x_1\right)$

$P_2\left(x\right)=1.38+0.16\left(x-1\right)-0.155\left(x-1\right)\left(x-2\right)$

according to the presented results in Table 7 .

For these two materials, the shape of the resistance curves as a function of the percentage of millet involucre is identical even if the maximum values reached are different ( Figure 7 ).

By presenting the thermal capacity education in reference to the millet involucre percentage ( Table 5 and Figure 6 ), its evolution is analyzed.

For Ndouloumadjie,

The Lagrange polynomial is the following:

$P_2\left(x\right)=0.787\left(x-2\right)\left(x-3\right)/2-0.690\left(x-1\right)\left(x-3\right)+0.561\left(x-1\right)\left(x-2\right)/2$

For the Tattaguine samples, we have:

$P_2\left(x\right)=0.817\left(x-2\right)\left(x-3\right)/2-0.751\left(x-1\right)\left(x-3\right)+0.67\left(x-1\right)\left(x-2\right)/2$

With Newton’s method, we have Table 8 for Ndouloumadjie.

$P_n\left(x\right)=0.787-0.097\left(x-1\right)-0.016\left(x-1\right)\left(x-2\right)$

For Tattaguine,

$P_n\left(x\right)=0.817-0.066\left(x-1\right)-0.0085\left(x-1\right)\left(x-2\right)$

With Newton’s method, we have Table 9 for Tattaguine.

The obtained tends for the Newton method in the case of the uniaxial compressive strength, and the thermal capacity are the same as those obtained through Lagrange method ( Figure 8 ).

In this part, we study the correlation between the compressive strength and the thermal capacity. It has been noticed that between 1% and approximately 2.125% of involucre, the raison of the resistance to the compression leader to a decrease of the thermal capacity ( Figure 9 ). In contrast, between 2.125% and 3% of millet involucre capacity ( Figure 10 ).

We are going to calculate the linear correlation coefficient r, which given us a measurement of the intensity and of the meaning of the linear relation between two variables ( Table 10 ), thermal capacity and the uniaxial compressive strength. This coefficient is given through the following relation:

$r=\left[{\displaystyle {\sum }_{i=1}^n\left(x_i-\bar{x}\right)\left(y_i-\bar{y}\right)}\right]/\left[{\displaystyle {\sum }_{i=1}^n{\left(x_i-\bar{x}\right)}^2}\ast {\displaystyle {\sum }_{i=1}^n{\left(y_i-\bar{y}\right)}^2}\right]$

where x_i corresponds to uniaxial compressive strength; y_i corresponds to the thermal capacity.

Let’s be the averages $\bar{x}=1.57$ and $\bar{y}=0.73$.

Between 1% and 2.125% of millet involucre, we have for Ndouloumadjie a correlation coefficient r = −0.997 ( Figure 11 , Table 7 ) While for Tattaguine r = −0.987 between 1% and 2.05% and 3% that shows that the more the resistance to the compression raides on this interval, the conductivity increases meaning a bad thermal capacity ( Figure 12 ).

Test results for mud bricks (Ndouloumadjie and Tataguine samples) improved with millet involucre show that with the 1% and 2.125% improvement for Ndouloumadjie and 1% to 2.05% for Tattaguine, compressive strength gradually increases before gradually decreasing after these peaks to 1.67 MPa and 1.54 MPa respectively. These strength values obtained in are satisfactory for construction with local soil Materials (Rc > 0.6 MPa). The different silica ratios (higher for Tattaguine) are linked to their genesis, corresponding to simple transport for the termite mound and chemical alteration for the Matam sample. This genesis controls the values of iron and alumina moduli. At peak, the higher strengths for Ndouloumadjie than for Tattaguine are due to the phyllosilicate, iron and silica content of these materials. The ratios found in the Ndouloumadjie sample are more favorable to the manufacture of earthen eco-construction bricks and to the inactivity noted in the Matam sample, where Ca = 0.16 (≤0.75). Thermal test results show that the Ndouloumadjie Dembé sample has better thermal performance than the termite mound sample of the Tattaguine termite mound. Both samples have very good thermal performance. They are well suited to construction in intertropical countries like ours. These thermal qualities become even better with improvements using millet involucre. Millet involucre is an effective thermal regulator for ecological construction and saves energy. The materials with maximum particle size of 0.080 mm ( Figure 3 , Table 1 ) and with high plasticity are characterized by their aluminous phyllosilicate content. These minerals are able to regulate the temperature.