The use of the effective stress principle led to a general theory for the strength and volumetric behavior of saturated soils. Presently, all constitutive models for saturated soils are based on this principle. In 1959, Bishop proposed an equation for the effective stress of unsaturated soils. However, it was severely criticized because it could not explain by itself the phenomenon of collapse upon wetting. Moreover, an analytical expression for the determination of its main parameter  was not provided, and in addition, its value could not be easily determined in the laboratory. Since then, several equations to determine the value of parameter  have been proposed. Sixty years later, it is acknowledged that Bishop’s effective stress equation can be employed to simulate the behavior of unsaturated soils when it is complemented with a proper elastoplastic framework.

Based on the analysis of the equilibrium of solid particles of an unsaturated sample subject to a certain suction, it is possible to establish an analytical expression for Bishop´s parameter X. With this parameter, the effective stress can be evaluated and used to predict the shear strength and volumetric behavior of unsaturated soils. For the determination of parameter X, three elements are required: the saturated fraction, the unsaturated fraction, and the degree of saturation of the unsaturated fraction of the sample. The equation established for parameter X clarifies some features of the strength of unsaturated soils that, up to now, had no apparent explanation. A drawback to this expression is that the three required parameters for the determination of X cannot be obtained from current experimental procedures.

Based on the analysis of the equilibrium of solid particles in a soil sample subject to a certain suction, an analytical expression for the value of Bishop´s parameter χ was established in the previous chapter. This parameter can be written as a function of the saturated fraction, the unsaturated fraction, and the degree of saturation of the unsaturated fraction of the soil. However, the determination of these three parameters cannot be made from current experimental procedures. Therefore, a porous-solid model simulating the structure of soils is proposed herein and used to determine these parameters. The data required to build up the porous-solid model are the void ratio of the sample and their grain and pore size distributions.

In the previous chapter, a computational network porous-solid model was developed to simulate the hydraulic behavior of unsaturated soils. However, important computational constraints make this model unpractical. In this chapter, a probabilistic porous-solid model is developed to overcome these constraints. The probabilistic model is an alternative to the use of computational network models and shows important advantages. This model is built from the probability of a certain pore to be filled or remain filled with water during a wetting or drying process, respectively. The numerical results of the probabilistic model are compared with those of the computational network model showing only slight differences. Then the model is validated by making some numerical and experimental comparisons. Finally, a parametric analysis is presented.

In the previous chapter, a probabilistic porous-solid model with the ability to simulate both branches of the soil-water retention curve, was developed. In this chapter, the model is used to interpret more realistically the results of mercury intrusion porosimetry tests. Moreover, it is used to obtain the pore size distribution of soils employing both branches of the soil-water retention curve as data. The numerical and experimental comparisons for different soils show that the model approximately reproduces the pore size distributions obtained from mercury intrusion porosimetry tests. Finally, a procedure to fit the numerical with the experimental soil-water retention curves in order to obtain the pore size distribution of soils is presented. 

 In this chapter, the probabilistic porous-solid model is used to determine the mean effective stress of soils at failure. The plots of the deviator stress against the mean effective stress show a unique failure line for a series of triaxial tests performed at different confining net stress and suctions for both wetting and drying paths. This result confirms that the proposed effective stress equation is adequate to predict the shear strength of unsaturated soils. It also results in different strengths for wetting and drying paths, as the experimental evidence indicates. 

In this chapter, the probabilistic porous-solid model is used to simulate the tensile strength of unsaturated soils tested at different water contents. The strength of unsaturated soils can be split into two parts: one related to the net stress and the other to suction stress. The strength generated by suction has its origin in the additional contact stresses induced to solid particles by water meniscus. This additional contact stress is called matric suction stress when it is solely related to matric suction. In such a case, the tensile strength of soils represents the matric suction stress of the material at that particular water content. The numerical and experimental comparisons of the tensile strength of unsaturated soils tested at different water contents show that the probabilistic porous-solid model can simulate this phenomenon quite accurately.

In this chapter, an equation to account for the volumetric behavior of unsaturated soils is proposed. This equation is based on the effective stress principle and results in a unifying framework for the volumetric behavior of both saturated and unsaturated soils. The numerical results of the proposed equation are compared with experimental results published by different researchers. These comparisons show that the equation is adequate to account for wetting-drying and net stress loadingunloading paths. This analysis confirms that the effective stress principle can be applied to the volumetric behavior of unsaturated soils.

This chapter presents the modeling of the phenomenon of collapse upon wetting using the effective stress approach established in Chapter 2 and the elastoplastic framework for the volumetric behavior of soils proposed in the previous chapter. Using the probabilistic porous-solid model, Bishop’s parameter  can be obtained to determine the current effective stress. The proposed framework includes the hysteresis of the SWRC and, to some extent, the hydro-mechanical coupling of unsaturated soils. This model is able to reproduce some particularities of the phenomenon of collapse upon wetting that other models cannot simulate. 

In this chapter, the elastoplastic framework for the volumetric behavior of soils developed in Chapters 8 and 9 is extended to account for the case of expansive soils. The hydraulic behavior of the soil is simulated using the porous-solid model developed in Chapter 4. The result is an elastoplastic framework where the value and sign of the expansion index depend on the density of the soil as well as the state of stresses and the direction of the increment of the effective stress with respect to the yield surfaces in the plane of effective mean stress against suction. Experimental and numerical comparisons show the ability of the model to simulate the behavior of expansive soils under different stress paths.

The phenomenon of hysteresis during wetting-drying cycles can be simulated using the porous-solid model developed in Chapter 3. This model employs the current pore-size distribution of the material. The term “current pore-size distribution” means that the size of pores can be updated as the soil deforms. In that sense, the porous-solid model can be used advantageously for the development of fully coupled hydro-mechanical constitutive models, as the influence of the volumetric deformation on the retention curves and effective stresses can be easily assessed. By including some experimental observations related to the behavior of the pore size distribution of soils subjected to loading or suction increase, volume change can be related to the reduction in the size of macropores. This methodology avoids the necessity of any additional parameter or calibration procedure for the hydromechanical coupling of unsaturated soils. 

In previous chapters, it has been shown that the principle of effective stresses can be applied to the shear strength, the tensile strength, and the volumetric behavior of unsaturated soils. This chapter shows that the critical state line for unsaturated soils shifts with respect to the saturated critical state line in a quantity that depends on the suction stress. Taking into account this phenomenon and the influence of hydro-mechanical coupling on the behavior of unsaturated soils, a fully coupled general constitutive model for soils is developed. This model is based on the modified Cam-Clay model but includes a yield surface with anisotropic hardening that takes into account the shift of the critical state line with suction. The result is a very simple model with symmetric stiffness matrix that can be used for the case of saturated, unsaturated, and compacted materials. 

During the determination of the main drying curve, the soil is subjected to high suctions, which induce important volumetric deformations. These volumetric deformations modify the pore size distribution of the sample affecting both the drying and the wetting branch of the retention curves. Although most deformation occurs at drying, this branch is only slightly affected by soil deformation. In contrast, the wetting branch shows important shifting when volume change is considered. A porous-solid model based on the grain and pore size distributions of the soil is coupled with a mechanical model to simulate the soil-water retention curves while the material is deforming.

 When undrained triaxial tests are performed, two main phenomena occur. First, the compression of the sample produces an increase in the degree of saturation and therefore, a reduction in the value of suction. Second, with the reduction in the sizes of pores, the retention curves shift on the axis of suction. Thereafter, the simulation of undrained triaxial tests requires the correct simulation of the hydromechanical coupling phenomenon. A fully coupled constitutive model for unsaturated soils is used herein to simulate the behavior of unsaturated soils subjected to undrained conditions. The mechanical model is based on the modified Critical State model and the effective stress concept. The hydraulic model uses the grain and pore size distributions to approximately reproduce the structure of soils. This model is able to simulate the soil-water retention curves during wetting-drying cycles. Plastic volumetric strains modify the pore size distribution of the soil, which in turn affects the retention curves and, therefore, the current effective stress. Some comparisons between numerical and experimental results of undrained triaxial tests show the adequacy of the model.

When unsaturated soils are subjected to drained or undrained compression tests, they approach the saturated compression line with different slopes. This difference in slopes is produced by the amount of collapse of each path. During compression, four main phenomena occur in these materials: first, with the reduction in volume, the degree of saturation increases; second, with the reduction in the size of pores, the soil-water retention curve shifts on the suction axis; third, these two phenomena produce an increase in the suction stress and, finally, this increase in suction stress produces a certain amount of collapse on the sample. In this chapter, a coupled model is employed to simulate the volumetric behavior of compacted soils under different stress paths. The comparison between experimental and numerical results shows the pertinence of the model. 

In this chapter, the probabilistic porous network model is employed to establish a fully analytical equation for the relative hydraulic conductivity of soils. From this approach, a parameter accounting for the proportion of pores of different sizes forming continuous paths of saturated elements between the boundaries is used to compute the hydraulic conductivity. This approach avoids the effect of the size of the network on the results and the necessity of the pore-scale model approach required by computational networks to obtain the hydraulic conductivity of soils. Similarly, constraints related to computing time and memory size are avoided. In addition, single, double, or triple structured soil can be considered for the network. The theoretical and experimental comparisons indicate that capillary flow can account for the hydraulic conductivity of soils for the full range of suction of sandy and silty soils. Finally, all parameters required in the relative conductivity equation can be obtained by fitting the numerical with the experimental retention curves.