The soil used for this study is taken from a production site in Yagoua (Djenseng), Far North Region of Cameroon and near the border with Chad (Bongor) ( Figure 1 ). This soil was chosen because of its abundant use in the production of adobe bricks by the local population. Moreover, this locality is chosen because of the frequent housing problems underlined in this locality due to the collapse of the houses linked to the severe weather. It’s dark brown earth. A previous study carried out on this soil indicated that it is suitable for the formulation of mud bricks [11] .

Fonio is a so-called minor cereal, which, after harvesting in the countryside, undergoes a threshing process that separates it from its lemellae. Fonio, which is grown on sandy or stony soils, is resistant to drought and heavy rainfall. It is mainly found everywhere in the Sahelian regions of Africa, particularly in Guinea, Mali, Benin, Burkina Faso, Togo, Niger, Cameroon… Fonio is much smaller than other commonly grown cereals. The paddy grain is ovoid in shape and is only 1 to 1.5 mm long. The mass of 1000 fonio grains is about 0.5 g. The bran obtained after slaughter is generally considered waste. They are very rarely eaten by some anime and are very often abandoned in the wild ( Figure 2 ). Very recently, some craftsmen have started to value them in mud bricks in order to improve their properties in order to have more comfortable buildings [14] .

The physical properties of sound fonio used for this study are presented in Table 1 .

The clayey soil used to produce adobes is dried up first and crushed afterward to obtain particles with a maximum grain size of 5 mm. Five compositions from the combination of dried powder of raw material with 0%, 1%, 2%, 3% and 4% by weight of fonio fibers content. Soil is mixed with plant fibers for fifteen minutes with 22% of water by dry weight of soil until it becomes a homogeneous mixture. The clayey paste is covered with plastics at room temperature for three days to promote the fermentation of fibers. The mixture (soil + fibers) is laid in 4 × 4 × 16 cm³ prismatic and 10 × 10 × 3 cm³ cubic mold in three layers and each layer is pressed manually with twenty shocks and progressively another layer is added until the mold is full after the third layer. The specimens are first kept in the shade (22˚C ± 5˚C) in ambient air for 24 h with a humidity of 60% before being demolded to dry. Before carrying out the various tests, the samples are first left in the shade for at least 21 days. Shade drying has been adopted to have samples without any cracks. After drying, the specimens are subjected to mechanical, durability and thermal characterizations.

The experimental device schematized in Figure 3 includes: A thin flat heating element on which is fixed a type K thermocouple (diameter wires 0.08 mm) inserted between two samples of identical dimensions of the material to be characterized, two isothermal aluminum blocks 4 cm thick with a section identical to that of the samples, a clamping device to control the clamping pressure and the thickness of the inserted device between the aluminum blocks.

The heating element is subjected to a flux step and the temperatures $T_s\left(t\right)$ are recorded at the center of the heating element and $T_0\left(t\right)$ aluminum blocks. The exploitation of the recording of $T_s\left(t\right)$ carried out on a time interval during which the transfer to the center of the heating element is 1D. 3D modeling makes it possible to define this time interval, a simplified 1D model is then used to make an estimate parameters over this time interval.

1D simplified models

The following assumptions are considered: The contact resistances and the thermal resistance of the heating element are negligible compared to the sample resistance , the transfer is 1D at the center of the system for the duration of the experiment, the following temperature gradient Oz is zero in the heating element (thin) and the temperature T₀(t) remains constant.

A probe with a maximum thickness of 2e_s = 0.3 mm and a thermal conductivity of the order of 0.15 W∙m⁻¹∙K⁻¹ has thermal resistance:

$\frac{e_s}{{\lambda }_s}={10}^{-3}{\text{m}}^2\text{}\cdot \text{K}\cdot {\text{W}}^{-2}$ (1)

To satisfy the first hypothesis, we will consider that the cases where $\frac{e_s}{{\lambda }_s}\gg {10}^{-3}{\text{m}}^2\text{}\cdot \text{K}\cdot {\text{W}}^{-2}$.

A 1D quadripolar modeling of the system makes it possible to write [13] - [17] :

$\left[\begin{array}{c}{\theta }_S\\ \frac{{\phi }_0}{p}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ C_{S}p& 1\end{array}\right]\begin{array}{cc}A& B\\ C& B\end{array}\left[\begin{array}{c}0\\ {\text{Φ}}_1\end{array}\right]$ (2)

When: $A=D=\cosh \left(\sqrt{\frac{p}{a}}e\right)$; $B=\frac{\sinh \left(\sqrt{\frac{p}{a}}e\right)}{\lambda S\sqrt{\frac{p}{a}}}$; $C=\lambda S\sqrt{\frac{p}{a}}\sinh \left(\sqrt{\frac{p}{a}}e\right)$

With: ${\theta }_s$ is a Laplace transform of the difference $T_s\left(t\right)-T_0\left(t\right)$, ${\rho }_s$ is a Density of the heating probe, $c_s$ is a mass heat capacity of the heating probe, $\lambda$ is a thermal conductivity of the material to be characterized, a the thermal diffusivity of the material to be characterized, p Laplace variable and S is the surface of sample and heating element. $C_s={\rho }_s{c}_s{e}_{s}S$.

This allows to write [18] [19] :

${\theta }_S\left(p\right)={\phi }_0\frac{B}{B{C}_{S}p+D}$ (3)

The temperature $T_s\left(t\right)$ in real space is obtained by inverse Laplace transformation which is carried out by De Hoog’s algorithm or Stehfest’s method:

$\text{Δ}T\left(t\right)=T_S\left(t\right)-T_0\left(t\right)=L^{-1}\left[{\theta }_S\left(p\right)\right]$ (4)

At very long times we get:

$T_S\left(t=\infty \right)-T_0\left(t\right)=\frac{{\text{Φ}}_0}{\frac{\lambda }{e}}$ (5)

This relation makes it possible to obtain in a very simple a value of the thermal conductivity $\lambda$ from the value of $\text{Δ}{T}_{\exp }$ obtained when the regime permanent is reached.

The principle of the proposed method is to estimate the values of the parameters $\lambda$ and incidentally ρc and ${\rho }_s{c}_s$ which minimize the sum of squared deviations:

$\psi ={\displaystyle {\sum }_{i=0}^n{\left[\text{Δ}{T}_{\exp }\left(t_i\right)-\text{Δ}{T}_{\mathrm{mod}}\left(t_i\right)\right]}^2}$ (6)

between the experimental curve and the theoretical 1D curve over a time interval over which the transfer is 1D at the center of the heating element [18] . The minimization of the sum $\psi$ is carried out using the Levenberg-Marquart algorithm.

For thermal properties measurement, samples are first put in an oven for 48 hours and kept in plastic for 48 hours before carrying out the tests. These bricks are made of cubic pieces of 10 × 10 × 3 cm³, each.

For each formulation, three samples, of 4 × 4 × 16 cm³ each, are used for the uniaxial compression test according to standards NF P18-406 and the six half specimens obtained after the uniaxial compression test are considered to determine the 3-point bending [20] [21] . The compressive and flexural strengths are measured, using a hydraulic press equipped with a 200 kN load, at a controlled displacement rate of 0.5 mm/min.

The compressive strength is calculated by [22] :

$R_c=\frac{F}{S}$ (7)

where F is the maximum loading force (N), S is the section (mm²).

The 3-point bending resistance is determined as given [23] :

$R_f=\frac{1.5F_{f}l}{b^3}$ (8)

where $F_f$ is the maximum loading force applied in the middle of the sample (N), $l$ is the length (distance between supports, mm), b is the side of the prism square section (mm).

The test to determine the capillarity water absorption coefficient (A) is carried out on cubic samples (4 × 4 × 16 cm³), dried up in an oven, at 105˚C for 24 hours. The water absorption coefficient (A) is determined as the drop of mass absorbed water by the sample plotted versus the square root of the time. The adobe samples are weighted using an electronic balance with 0.01 g precision. The value of the water absorption coefficient (A) is determined by the relation [24] :

$A=\frac{m_1-m_0}{S\sqrt{t}}$ (9)

where $m_0$ is the mass of the dry sample, $m_1$ is the mass of the sample immersed in water, t is time of immersion equal to 600 seconds, and S is the base surface area (4 × 16 cm²) of the sample. The capillarity water absorption was determined according to the literature [25] and the view of the samples as show in Figure 4 .

For erosion tests, three prismatic adobe samples of 4 × 4 × 16 cm³ of each formulation are used to simulate the erosion impact of rain for 10 min with a flow rate of 5 l/min. The samples are tilted at an angle of 30˚ to the horizontal. The spray is positioned around 120 mm in height above the sample. The coefficient of erosion test expressed as a percentage of loss material after the erosion test and is given in equation below [26] :

$CE=\frac{m_0-m_s}{m_0}\ast 100$ (10)

where $m_0$ is the mass of dried adobes and $m_s$ the mass of the adobes exposed to the erosion test and dried at 105˚C for a duration of 24 h.

The specific heat $C_{\exp }$ is determined from the heat capacity ${\left(\rho C\right)}_{\exp }$ measured using the hot plate device and the density.

Figure 5 shows different values of bricks as in terms of fonio. The results show that the effective temperature increases due to the increase in fonio in the bricks. These values are 1034; 1070; 1115; 1168 and 1217 J∙kg⁻¹∙K⁻¹ with 0%, 1%, 2%, 3%, and 4% fonio added. The specific heat of an object is the energy given to raise the temperature of one part of the object by 1 Kelvin, and we see a 17% difference in specific heat between actual bricks and bricks containing 4% agricultural residues.

These results are consistent with the authors’ view that the more an object is heated, the more it burns [27] [28] . The specific heat capacity of adobes increases when their density decreases, since the fibers plants have a higher capacity than mineral elements.

Figure 6 shows that the temperature change follows the temperature difference and decreases with increasing addition rate. In addition to low temperature, insulating materials must also have good ability to reduce thermal diffusion [29] .

In Figure 6 , it can be seen that the thermal diffusivity decreases with the addition of fonio fibers. A drop of 10% in thermal diffusivity between the reference brick (3.80 × 10⁻⁵ m²∙s⁻¹) and that having 4% fonio (1.98 × 10⁻⁵ m²∙s⁻¹) is obtained. This reduction is due to the pores in the fonio and their shape acting on the temperature within the group, helping to reduce the vulnerability of the compost and heat conduction. Table 2 reported differents values of thermal properties.

The porosity rate, the percentages of fiber addition, the composition of materials are all factors that influence the thermal performance of composite. In addition to low heat, thermal insulation should also have a good ability to reduce heat transmission [30] [31] .

One of the most important aspects for buildings in hot climate environments is the assessment of the behaviour of its building materials in the face of heat. This behavior is evaluated by thermal conductivity, which is the most important thermophysical parameter. It provides information on the ability of the material used in the building envelope, particularly in the load-bearing wall, to be easily or not penetrated by heat.

The following Figure 7 shows the evolution of the thermal conductivity of adobe bricks reinforced with 1%, 2%, 3% and 4% of fonio fiber waste.

Figure 7 illustrates the effect of adding fonio fibres to the mud bricks. The addition of 0%, 1%, 2%, 3%, and 4% in the soil matrix contributes to a gradual decrease in its thematic conductivity. Indeed, the new thermal conductivity values after adding 0%, 1%, 2%, 3% and 4% are 0.96 respectively; 0.96; 0.89; 0.81 (W∙m⁻¹∙K⁻¹).

The variation in thermal conductivity is closely related to the thermal properties of the plant fibers. Indeed, several authors reveal in their studies that plant fibers have a thermal insulating power [8] [14] - [16] . Therefore, the abundance of plant fibers within the matrix soil makes it difficult to access heat inside the building through the wall made of earth + plant fiber composite. As a result, the gradual increase of plant fibers in the soil matrix gradually strengthens the insulating properties of the composite [8] [14] - [16] . This behavior of slowing down the movement of heat through the load-bearing wall is of paramount importance for hot regions such as the Sudano-Sahelian region of Cameroon which requires a less hot indoor climate for occupants.

Moreover, this decrease in thermal conductivity is not perceptible at 1%, which would probably be related to the low content of the vegetable fibres and the low insulating properties of the plant fibres used. However, this increase starts to be noticeable from 2% and becomes a bit more satisfying up to 4%. Overall, we realize that fonio fibers make a certain positive contribution in terms of thermal insulation in the soil matrix but this improvement is relatively small; it is 16% corresponding to the addition of 4% of plant fibers. The same authors recently found an improvement of 23% and 54% respectively with the incorporation of millet and neem fibers in the same proportions ( Table 3 ). This confirms that fonio fibers have limited thermal insulation properties compared to mille, straw and neem leaf fibers [11] [12] .

The agreement obtained between the experimental and theoretical curves is then very good as shown by the curves of the estimation of covariance, sensibility and residuals in Figure 8 . That is a proof that the thickness/width ratio must always respect the limits set by the chart in Figure 8 to ensure that the transfer is 1D in the center and that the contact resistances are negligible [18] .

Ultimately, adobe bricks reinforced with fonio fibers from 2% limit heat transfer and are therefore better suited for buildings compared to adobe bricks without the addition of plant fibers [14] [26] .

However, many other authors have shown that the less plant fibers improve thermal properties, the more they improve mechanical properties.

The effect of fonio on the Compressive and flexural strength the soil appears on Figure 9 .

As illustrated on Figure 9 , a reading of this curve shows that compressive and flexural strengths vary according to the fonio contents incorporated into the soil matrix. We noticed an improvement of the mechanical characteristics with the addition of fonio waste.

It is quickly observed that the Compressive strength of the soil gradually increases with the fonio addition. Whatever the case, the Compresive strength values are improved, compared to soils without reinforcement. For this purpose, we obtained a resistance gain of 1.71%; 58.86%; 59.91% and 76.75%, respectively when we used 1%, 2%, 3% and 4% of fonio. As the fonio content increases, the compressive strength increases proportionately. The maximum compressive strength value is 8.29 MPa and corresponds to 4% fonio addition. At this percentage, the fonio fibers adhere better to the clay matrix, the composite obtained is more compact and resists better mechanical loads [20] [32] [33] . These results clearly show that fonio has a considerable effect on the improvement of the soil studied. The results obtained in this study show that fonio fiber waste improves compressive strength better in all proportions compared to the compressive strength values of some authors in the literature [34] [35] ( Table 4 ).

Many authors tried to explain the increase of the compressive strength of soils stabilized with fibers [26] [33] , this increase of the Compressive strength values is due to the reduction in the size of shrinkage cracks.

As illustrated in Figure 9 , it can be seen that the flexural strength decreased after 1% of fonio addition before increasing with 3, 4 and 4% of fonio. Similar results were obtained by Millogo et al., 2014 [26] and Danso et al., 2015 [34] . In fact, some authors point out that fibers are generally known rich in cellulose (tensile strength between 300 and 500 Mpa), responsible for the increase of the the soil + fibers mixture [36] [37] .

Fonio fibre-reinforced adobes have shown exceptional capabilities in terms of mechanical strength. They can therefore be used in the building without any damage to the occupant according to the recommendations in force in the building sector [38] . However, it is also important to evaluate the behaviour of these adobes in the face of water in order to enjoy not only the thermophysical and mechanical capacities, but also a certain guarantee of performance.

Figure 10 shows the evolution of the weigth loss before and after stabilization.

The results show that the values of weight loss of soils decrease with the addition of fonio content. These results clearly show the beneficial effect of stabilization. The same results have been obtained by [34] . For the authors, this decrease could be explained by the high adhesion of fibers with clayey matrix.

These results sufficiently prove that soils with low mass loss are more resistant to water erosion.

This is important because it justifies that adobe bricks reinforced with fonio fibers will resist tearing due to spraying from rain. For all percentages, adobe bricks reinforced with fonio fibers gave better stability compared to millet fibers, straws and neem leaves as shown in Table 5 .

Since moisture is the main enemy of earthworks, the understanding of the water stability of adobes was completed by the study of the water absorption of the samples in order to assess whether or not the formulated adobes promote water absorption by capillary action.

Figure 10 shows the effect of adding fonio waste on adobe bricks. Careful observation of this curve shows a decrease in the water absorption rate for all additions 1%, 2%, 3% and 4%. This reveals that the incorporation of the different fonio fibre contents in the adobe bricks helps to improve the behaviour of the mud bricks in the face of water. Indeed, when the absorption coefficient becomes lower and lower, as is the case in Figure 10 , the resulting composite is more resistant to water penetration and destruction [20] [25] [26] . Therefore, the addition of fonio fibers in the mud bricks enhances the durability of the mud bricks. Compared to millet and neem fibers previously studied with the same soil and with the same conditions, fonio fiber waste showed an improvement [11] [12] ( Table 6 ).

Other authors have obtained similar results with similar fibers and point out that certain thin, fine and short plant fibers promote the stability of mud bricks thanks to their physical properties.

Compared to millet and neem fibers previously studied with the same soil and with the same proportions fonio fiber waste showed an improvement [11] [12] .