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and and are the void ratio at respectively air entry (in the intra-aggregate pores) and the swelling limit.
2.2.6. The Model of Olsen and Haugen (1998)
Olsen and Haugen (1998) proposed a second order hyperbolic equation, using in its positive solution to describe the shrinkage curve between the zero and the normal shrinkage, and its negative solution to describe the shrinkage curve from normal to structural shrinkage. This model contains six parameters.
where reflects the curvature at the transition zones between residual and normal shrinkage, reflects curvature at the transition zones between normal and structural shrinkage; is a coefficient depending on the upper asymptote; is the moisture ratio where the two domains of the shrinkage curve join.
2.2.7. The Model of Braudeau et al. (1999)
Braudeau et al. suggested a seven-parameter-model similar to the Tariq and Durnford (1993b) model. They divided the structural zone into a linear and curvilinear zone, including a point of friability:
The slopes of the linear curves are:
2.2.8. The Model of Chertkov (2000, 2003)
The author proposed an expression based on the statistical analogy between crack networks and the probabilistic microstructure of a matrix consisting only of clay particles:
μ is a model coefficient; is the density of water; is the density of the solid particles; is the liquid limit, which is the maximum moisture ratio in the solid state of the clay, or at which the shear strength approaches that of a liquid.
3. Material and Method
To reproduce the soil’s shrinkage curve experimentally, we must measure the change of the volume and the weight during all the test process simultaneously. To perform this experiment, we use the measurement device basically used to carry out the desiccation test according to  (Figure 4).
This measurement device is usually used to measure the axial deformation during the drying process, but in our study we use it to measure the axial deformation in both wetting and drying processes. The intact sample submitted for testing was a clayey soil with a little carbonate nodules (7%) from the village of Moulay el Bergui near the city of Safi (Morocco). The intact samples were taken from 1.8 - 2.4 m depth.
The tests were performed as follows:
At first, undisturbed samples were taken from field using a sampling box, in the view to preserve the initial structure of the soil. Then, test tubes of 3.6 cm diameter were carefully cut from the undisturbed bloc, and placed in the testing apparatus. Once the test tube was fixed in the receptacle, we place all the mechanism over a balance in order to measure the weight and the volume change both at the same time. After a first reading at its natural state, we begin supplying water by stages (2 g of water at each stage) and at each stage the weight and the axial deformation were taken after the stabilization of the axial deformation. During the wetting process, we protected the upper plane of the test tube by a thin plastic film to avoid water evaporation, and all the mechanism was placed in a box whose the temperature and the humidity were controlled.
After saturation and total stabilization of the axial deformations, we begin the drying process. We start to take measurements along the free air dehydration, then when the axial deformations were stabilized, we place the sample in the oven (105˚C) for 72 hours, taking its weight and deformations values every 6 hours.
The temperature of the testing room was 20˚C and its humidity was 50%.
4. The Shrinkage Curve Modeling
In our testing approach, we study the unidimensional
Figure 4. Measurement device of the volume’s changes. (a) Le bâti; (b) Plaque amovible vue en coupe; (c) Plaque amovible vue de dessue.
volume variation of three test-tubes, considering that the tested soil is non-rigid and homogeneous and that there is no shearing between the soil particles. The choice of the physical parameters for our model was based on the fact that the value of the soil’s deformation is the result of the spacing between the particles following the thickness variations of the diffuses layer. This is the variation of the void ratio according to the water content of the medium.
The shrinkage curve model integrates only intrinsic physical parameters of the soil, and the model is described by a third degree polynomial equation as follow:
The values will be deduced from the boundary conditions of the process as follow:
When the soil is dry: so
When the soil is saturated: so
By derivation of the Equation (1):
When the soil is dry:, so
When the soil is saturated: so
We obtains the Equation (3) as follow:
Because the results obtained by the Equation (3) was not too accurate, we opted for a new water coefficient, where we deduced the shrinkage limit from both the maximal water content and the considered water content as follows (changing by)The analytical model of the soil’s behavior during the desaturation phase:
where is the void ratio at the shrinkage limit; is the maximal void ration in a saturated state; is the maximal water content, and is the shrinkage limit.
We also try to adapt this model to the saturation curve, according to the following formulation:
where is the natural water content.
5. Results and Discussion
The experimental data and the corresponding soil’s shrinkage curve are represented in Tables 1(a) and (b) and Figure 5. Note that the data represented below are the average of three tests conducted on the same clayey soil.
For the desaturation curve, we observe that the measured shrinkage of the samples cover practically the complete water content range, from the shrinkage curve’s wet side to its dry one.
The comparison between the shrinkage curve experimentally performed and the one calculated by the previous model shows a good correlation between the two methods, and proves that this model is functional for this type of soil. The advantages of this model during the desorption process are:
• The use of a single equation which covers all the phases of the shrinkage curve;
• A reduced number of physical parameters;
• A good correlation between the analytical and the experimental results during the drying process.
In addition, we try to evaluate the adsorption curve for the same soil with the same model, except that we change by. For the adsorption curve, the model does not follow the experimental curve perfectly; it did not give a perfect correlation between the experimental results and the analytical model.
The current paper proposes a new model of the shrinkage curve on the basis of the soil’s water content and its structural evolution. This current model is able to cover
Figure 5. Curve of adsorption and desorption of undisturbed clay samples.
Table 1. (a) Experimental results of water content and void ratio variations; (b) Calculated results of water content and void ratio variations.
the fourth parts of the shrinkage curve (structural, normal, residual and zero shrinkage) by using only a third degree polynomial equation according to the limits of its hydro-structural boundaries. For the testing undisturbed soil, the comparison between the experimental tests and the analytical model gives a good correlation between the two methods during the drying process.
In addition, we try to evaluate the adsorption curve for the same soil with the same model, except that we changed by, but it did not give a perfect correlation between the experimental results and the analytical model.
This work is part of a research project “Clay Soil Behavior during the Drying Process”, conducted in collaboration by the “Centre Expérimental des Sol-Laboratoire Public d’Essais et d’Etudes” (CES-LPEE) CasablancaMaroc and “Ecole Mohammadia des Ingénieurs” (EMIUM5) Rabat-Maroc.