Constitutive Model of Unsaturated Soils Considering the Effect of Intergranular Physicochemical Forces
Publication: Journal of Engineering Mechanics
Volume 142, Issue 11
Abstract
A clayey soil is an electrically charged porous medium whose behavior is sensitive to the variations in the composition and concentration of pore water. Pronounced physicochemical interaction can occur between the solid particle and the pore water so that the clayey soil shows strong chemomechanical coupling effects and complex mechanical behaviors. If the degree of saturation varies the pore-water composition and concentration can also vary, resulting in intensive physicochemical effects in the soil. In this paper, a conceptual constitutive model for unsaturated soils is proposed to explain the influence of pore-fluid chemistry on the chemomechanical behavior of unsaturated clayey soils. A new intergranular stress which can effectively account for the physicochemical effects including osmosis, capillarity, and adsorption is introduced as the stress-state variable. The formulation of the proposed model can lead to a remarkable unification of the experimental results obtained under complex chemomechanical loading conditions. The proposed model is validated by comparing the theoretical calculations with the experimental results. It is shown that the proposed model is capable of addressing the effect of water content, concentration and species variation on the mechanical behavior of the clayey soil.
Introduction
In recent years, geotechnical practice has increasingly encountered situations where the pore-fluid chemistry of soil varies under environmental impacts (Barbour et al. 1992). As a consequence, the chemomechanical behavior of clayey soils is of increasing interest in many areas of geoengineering, such as geological disposal of nuclear waste, natural gas hydrate exploitation, oil and gas wellbore-stability analysis, municipal waste isolation, landform stability assessment, and so on. The importance of chemomechanical coupling effect in geomaterials has been clearly demonstrated by many engineering practices. For instance, in the North Sea chalk reservoir, enhanced reservoir compaction and seabed subsidence has been found during the replacement of oil in chalk pores by seawater, which can be attributed to the weakening effect of the injected seawater on the chalk (Homand and Shao 2000). In petroleum engineering, it is well recognized that wellbore stability can be improved by circulating a mud, which is believed to create a membrane that is at least partially impermeable to salts (Loret et al. 2002).
The chemomechanical behavior of soil can be considered as the response of the soil to the variation of the chemical compositions and/or concentrations of the pore water (solution). When the pore water composition and concentration vary, chemical–mechanical interactions occur between the soil matrix and the pore water (Di Maio 1996; Sridharan 2014). These interactions determine the physical and mechanical properties of fine-grain soils including plasticity, swelling, compression, strength, and fluid conductivity, and strongly influence the hydromechanical behavior of the soils. Recently, the chemomechanical behavior of clay has received considerable attention. Many experimental studies were carried out on the shear strength and volumetric compression behaviors of the clayey soils exposed to different salt solutions (Di Maio 1996; Witteveen et al. 2013; Wahid et al. 2011). Based on the experimental results, many chemomechanical models for fully saturated soils have been developed (e.g., Hueckel 2002; Loret et al. 2004; Guimarães et al. 2007; Gens 2010; Witteveen et al. 2013). Despite these efforts, some issues remain to be resolved, including those related to the physicochemical interactions between soil grains and pore water.
Over the last two decades intensive effort has been made to develop the coupled hydromechanical constitutive models for unsaturated soils (e.g., Alonso et al. 1990; Bolzon et al. 1996; Loret and Khalili 2002; Gallipoli et al. 2003; Wheeler et al. 2003; Sheng et al. 2004; Ehlers et al. 2004; Santagiuliana and Schrefler 2006). Although significant advances have been achieved in unsaturated soil mechanics, the physicochemical effects are seldom addressed. Baker and Frydman (2009) pointed out that in many constitutive models for unsaturated soils, the measured matric potential had traditionally been identified as a stress variable (or suction), which is questionable since these quantities have different physical characteristics despite their common dimensions. In addition, in the current geotechnical practice, it is not uncommon that the matric suction measured by a mechanical device (e.g., a pressure plate) is erroneously considered as the negative pore-water pressure.
To address the chemomechanical behavior of geomaterials, it is crucial to analytically define the effective stresses so that the chemo-mechanical coupling effect can be properly described. Since Karl Terzaghi proposed the effective stress formulation for the saturated soils, intensive research has been performed to extend the effective stress concept to other situations. Most of the research efforts in this regard are devoted to unsaturated soils (e.g., Bishop 1959; Kohgo et al. 1993; Khalili et al. 2004; Laloui and Nuth 2009). However, it has been well recognized that Terzaghi’s effective stress and its extensions are insufficient in describing the chemomechanical behavior of clayey soils (Barbour et al. 1992; Mitchell and Soga 2005). As noted by Lu and Likos (2006), the microscopic intergranular stresses such as those arising from physicochemical and capillary forces are not explicitly included in the effective stress formulation, so that the chemomechanical effect cannot be properly addressed.
The intergranular physicochemical forces stem from the electrically charged nature of clay minerals, including the long-ranged attractive and repulsive forces developing between charged soil particles. Several options have been available to characterize these forces (e.g., Mašín and Khalili 2016). One is to adopt the intergranular (or mineral) stress (Sridharan and Venkatappa Rao 1973), defined by , where is the total stress; is the measured pore water pressure; and and collectively represent the microscopic intergranular attractive and repulsive forces, respectively. Another option is to assume that the intergranular stress, the effective stress, and the intergranular physicochemical stress are all equal (Hueckel 1992), i.e., . Alternatively, Mašín and Khalili (2016) have recently suggested that the effect of the intergranular physicochemical forces is constitutive in nature and can be properly characterized in a constitutive model using the original Terzaghi effective stress. Noticeably, in the intergranular stress equation, must be considered as the pressure of the equilibrium solution contained in a reservoir, which is in contact and in equilibrium with the soil water (Mitchell and Soga 2005: Chapter 7). Thus far, however, explicit expressions for and have not been available.
In a similar, intuitive way, Lu and Likos (2006) have developed an effective stress equation for unsaturated soils, which includes a stress term called the suction stress (Lu and Likos 2006; Lu et al. 2010). The suction stress conceptually combines all the effects (other than air pressure and external loading) into a macroscopic stress variable, and can account for physicochemical effects, cementation, surface tension, and the force arising from negative pore-water pressure. Remarkably, the suction stress variable can be represented by a function of the degree of saturation (or moisture content), which is independent of the soil-water characteristic relationship. Although the suction stress concept has been very well applied to analyze the unsaturated soil issues related to failure, a general constitutive model based on this concept has yet to be developed.
To properly address the effect of physicochemical forces in unsaturated soils, it is important to characterize the true pore-water pressure. In general, the true pore-water pressure is significantly different than the equilibrium solution pressure that is measurable using a pore-water pressure transducer. Such a difference can be clearly illustrated by referring to the concept of disjoining pressure (e.g., Gonçalvès et al. 2010). The disjoining pressure represents the pressure difference between a thin fluid film present between two platelets of a clay mineral and the bulk solution where these possibly charged platelets are embedded. In a rough sense, when this concept is applied to a fully-saturated soil, the disjoining pressure can be considered as the microscopic counterpart of the pressure difference between the true pore-water pressure and the equilibrium solution pressure.
The diffuse double layer concept can also clearly point to the difference between the pore water and the equilibrium solution pressure. In clayey soils, a diffuse double layer can develop in the vicinity of the pore water around clay particles (Barbour et al. 1992). Consequently, some ions are retained in the soil pores so that the clay behaves as a nonideal, semipermeable membrane (Witteveen et al. 2013). Due to the existence of the double layer, the pore-water concentration is generally different from that of the equilibrium solution (Gajo and Loret 2007), resulting in a pressure difference between the pore water and the equilibrium solution in the reservoir, which is called the Donnan osmotic pressure (Dominijanni et al. 2013). Although the Donnan osmotic pressure can be analytically evaluated, it excludes the effect of sorption, which is significant for clayey soils (Nitao and Bear 1996; Baker and Frydman 2009).
Recently, Wei (2014) developed a theoretical framework for modeling the chemomechanical behavior of unsaturated soils with saturation varying from 100% to a low extreme value, which could address skeletal deformation, fluid flow, heat conduction, solute diffusion, chemical reaction, and phase transition in a consistent and systematic way. Based on the equilibrium properties of porous media, a relationship between the true pore-water pressure and the equilibrium solution pressure is established, and an explicit formulation of effective stresses is proposed for unsaturated soils. In addition, true pore-water pressure can be additively decomposed into two components: the equilibrium solution pressure, which can be measured through traditional methods, and the generalized-osmotic pressure, representing the physicochemical effects. The generalized-osmotic pressure has two contributions: one is from the Donnan osmotic pressure accounting for the repulsion between the double layers around the electrically charged soil grains, and the other is due to the adsorptive effect between the soil grains and the pore water.
In this paper, a conceptual constitutive model is first developed for modeling the chemomechanical behavior of unsaturated soils using the intergranular stress formulation proposed by Wei (2014); the proposed model is then used to analyze the chemomechanical behavior of unsaturated soils with varying pore-water saturation and composition. The new model combines the true pore-water pressure with the mechanical stress, which can eliminate the physical inconsistency.
Theoretical Background
For simplicity, it is assumed that (1) any chemical reaction is excluded; (2) there is no mass or ion exchange between the pore water and the soil particles; and (3) the soil water (i.e., the pore water) is ideal. In addition, it is assumed that the fixed charge density, i.e., the total number of the electric charges fixed in a unit volume of the soil, is constant. Based on Assumption 3, the chemical activity of a solute species is equal to its molar fraction, which is generally valid for a dilute solution.
Pore-Water Pressure
Consider a representative volume of the soil of concern that is embedded in (or in contact with) a reservoir containing a solution of the same compositions as the soil water. If the whole system (including the soil and the reservoir) is in thermodynamic equilibrium, then the solution in the reservoir is called the equilibrium solution of the soil. In general, the true porewater pressure, , in the soil can be expressed as (Wei 2014)where is the pressure of the equilibrium solution; and is the generalized-osmotic pressure stemming from the physicochemical interactions between the soil particles and pore solution, including Donnan osmosis, capillarity, and adsorption, given bywhere is the mass density of pure water, and equals under the standard atmospheric pressure; and is Donnan’s osmotic pressure, given bywhere is the universal gas constant, ; the absolute temperature (K); the molar mass of pure water and equals ; and and are the activities of the water species (i.e., the solvent) in the equilibrium solution and the soil water, respectively, which are functions of , , and the molar fraction of water . For the ideal dilute solution, and are approximately equal to the molar fraction of pure water and , respectively. Noticeably due to Donnan’s effect (Mitchell and Soga 2005; Wei 2014), is generally different than .
(1)
(2)
(3)
is the surface energy potential that is associated with the surface forces stemming from the interaction between pore solution and soil particles. In general, is a function of and the volume fraction of liquid phase, , andwhere = mass density of the pore solution, and approximately equals for a dilute solution; is the matric suction; is the pore air pressure; and = volume fraction of liquid phase at full saturation. Eq. (4) implies that can be determined using the soil water characteristic curve, provided that function is given. Remarkably, Coussy (2011) had proposed a formulation for the so-called interfacial energy, which is formally similar to Eq. (4), except that the term of is excluded. It is noted that inclusion of in Eq. (4) highlights the difference between the soil water and the equilibrium solution. Traditionally, such a difference has been overlooked.
(4)
For clayey soils with a high-fixed negative charge density, the magnitude of could be very large. As the pore-water concentration varies, may change significantly. In current practice, pressures and are commonly used interchangeably. Apparently, if the physicochemical effect comes into play, such a practice must be exercised with caution.
Donnan’s osmotic pressure, , has yet to be evaluated. However, it can be theoretically calculated in some particular cases. Consider a clayey soil, which has a fixed charge density of [moles per cubic meter ()] representing the total number of electric charges fixed in a unit volume of the soil. For this example, it is assumed that the soil saturated with a NaCl solution. The soil is in contact and in equilibrium with an equilibrium solution with a concentration of . By enforcing the equilibrium conditions, can be calculated as a function of , , and (Wei 2014), and the results are depicted in Fig. 1. It is clear from Fig. 1(a) that the largest variations of occurs when is lower than . As further increases, varies at a lesser rate; this is consistent with the results inferred from the measurements of residual strength (Di Maio 1996). When reaches at the saturated concentration, becomes vanishing and Donnan’s osmotic effect disappears. Hence, in characterizing the Donnan effect, the soil with a saturated salt solution can be used as a reference.
EM.1943-7889.0001146/asset/f48c6cb3-129b-4649-9c32-d7863e6c3155/assets/images/large/figure1.jpg)
By its very definition [i.e., Eq. (1)], can be quantitatively considered as the swelling pressure. Sufficient experimental results show that the swelling pressure depends largely on the soil dry density, the nature of exchangeable cations, and the pore fluid salinity (Sun et al. 2015; Castellanos et al. 2008; Alawaji 1999). For a soil with a fixed porosity of [Fig. 1(a)], is constant, and thus the variation of is simply equal to that of . Fig. 1(a) shows that the swelling pressure decreases with the increase of the pore-fluid salinity. Fig. 1(b) indicates that increases with the decrease of porosity (equivalently, the increase of the soil dry density), i.e., the swelling pressure increases with the increase of the dry density.
From Fig. 1(b), it is clear that if the saturation varies considerably, can also change significantly, implying that the physicochemical effect could be significant in modelling the behavior of unsaturated soils with varying saturation.
Intergranular Stress
Based on the concept of true-pore pressure, an explicit formulation has been developed for the intergranular stress of unsaturated soils, which can address the physicochemical effect (Wei 2014), i.e.where is referred to as the intergranular stress tensor; = total stress tensor; = second-order unit tensor; and = so-called suction stress (Lu and Likos 2006), which is an intergranular stress component accounting for the effects of capillarity and adsorption as well as Donnan’s osmosis.
(5)
At full saturation, and . One haswhere ends up with Terzaghi’s effective stress tensor for the fully saturated soils. Hence, the as-defined intergranular stress can smoothly switch between a partially saturated state and the fully saturated state.
(6)
Intergranular Stress versus Bishop-Type Effective Stress
Bishop’s effective stress is given bywhere = coefficient, which is a function of saturation and assumes a value between 0 and 1. In application, it is usually assumed that (Houlsby 1997). For convenience, we shall adopt this assumption in the following text.
(7)
To demonstrate the advantages of the proposed intergranular stress over Bishop’s type effective stress, we compare the simulated results for the deformation of a porous-glass rod induced by the change of saturation using the two previous stress variables. Amberg and Mclntosh (1952) studied the water adsorption/desorption of a porous-glass rod and measured the change of the rod length. They found that during adsorption/desorption processes, the rod length change was fully reversible, suggesting that all the measured strains were fully elastic. The relationship between relative humidity (RH) with water content () was shown in Fig. 2, in which two adsorption-scanning curves were also given. A plot of the length variations corresponding with the isotherm was given in Fig. 3 (represented by the datum points).
EM.1943-7889.0001146/asset/197ccd3d-2495-49a7-8fba-9086832487d6/assets/images/large/figure2.jpg)
EM.1943-7889.0001146/asset/71349eba-5c73-403d-93da-b6239990a7a1/assets/images/large/figure3.jpg)
The intergranular stress induced by drying of the material is purely hydrostatic. For a homogeneous and isotropic material, the deformations can translate to uniform strains in all directions. Linear deformation, i.e., drying shrinkage, in any one direction is equal to of the total volumetric deformation. Assuming that the deformation is linearly elastic, the strain of the rod can be calculated as follows:
The porous-glass rod did not contain any fixed charges at the particle surfaces, and thus there no Donnan osmotic effect existed, so that , , and can be calculated by Eq. (4). The porous glass has an extremely narrow distribution of pore sizes with an overwhelming majority of the pores, with sizes of the order of nanometers and the pores of micron sizes are nearly absent (Nordberg 1944; Scherer 1986). Due to its morphology, the porous glass tends to retain the saturating liquid during the initial stages of drying while precipitously losing a large amount of pore water in a narrow range of relative humidity values, as evident by the water retention data displayed in Fig. 2. In the simulation, the air entry value is assumed to be 7,600 kPa, corresponding to the pore sizes. At full saturation, the generalized osmotic pressure , and the compression modulus .
The calculated results of Eqs. (8) and (9) are given in Figs. 3(a and b), respectively. Eq. (8) yields much better results compared to Eq. (9), especially in the regime of low saturation.
Uniqueness of the Failure and Critical Lines
In the following text some fundamental concepts in the constitutive modeling of soils are evaluated by using the proposed intergranular stress [i.e., Eq. (5)], and particularly the uniqueness of the failure and critical lines is examined based on the experimental data available in the literature.
Fig. 4(a) shows the experimental results obtained through oedometer tests on a fully saturated bentonite clay (Di Maio 1996). The results are presented in the form of the residual shear strength () against the applied axial stress () for different pore-water concentrations. The effect of pore-water concentration on the failure line is quite pronounced in the concentration range of 0–0.5 moles per liter (), and diminishes when the concentrations . This can be explained by the proposed intergranular stress. Indeed, as the concentration increases, decreases [as shown in Fig. 1(a)] and decreases, inducing an increase in the intergranular stress. Consequently, the residual shear strength increases with the pore-water concentration. Because the variation of (or ) is more severe at low concentration ranges and tends to become stable at high concentration, the effect of concentration on the residual shear strength is more significant when the concentration is low and becomes vanishing as the concentration increases.
EM.1943-7889.0001146/asset/76b1488e-1345-4074-9690-aca5152157c8/assets/images/large/figure4.jpg)
Pore-fluid concentration has a noticeable influence not only on the magnitude of the residual shear strength, but also on the slope of the failure lines. The specimens were prepared by mixing the powdered clay with an electrolyte of specified concentration to a slurry at about the liquid limit. Depending upon the electrolyte concentration, the void ratio, , of a specimen can vary significantly as the axial stress increases. According to Di Maio (1996), if the specimen is saturated with distilled water, at and at ; while if the specimen is saturated with the saturated NaCl solution, at and at . Clearly, at the same axial stress the higher the concentration of the saturating fluid, the denser the prepared sample and the higher the residual shear strength. In addition, due to the nonproportional increase of the residual shear strength with the axial stress, the slopes of the failure lines are different for various pore-fluid concentrations, as shown in Fig. 4(a).
The residual shear strength of soil is generally measured when a failure surface has clearly formed through the soil so that in the vicinity of the failure surface most of soil grains move in a mode of slipping instead of rolling with each other. Because of this, the effect of the soil fabric is minimized and the physicochemical effect becomes dominant in measuring the residual shear strength. Hence, the authors suggest herein that if Eqs. (5) or (6) properly address the physicochemical effect, it should be possible to lump all the failure lines into a single line using the proposed intergranular stress. To validate this idea, the failure line is replotted in Fig. 4(b) using the intergranular axial stress (). To that end, Eq. (6) is used and is assumed to be for the bentonite. It is quite clear that all the data points can be approximately lumped into a curve. Indeed, there exists some inconsistency between the failure line of the sample saturated with the pure water and those with NaCl solutions. Such a discrepancy is probably due to the fact that in calculating for the sample saturated with the pure water, the concentration of the saturating fluid is arbitrarily chosen as (since the soil skeleton contains some dissoluble ions).
Fig. 5(a) shows the experimental results on the critical line of a Kaolin clay (Sivakumar 1993). The datum points are quite scattered for different suctions. However, when the data is replotted in the mean intergranular stress-deviatoric stress space, the critical state lines become unique, as shown in Fig. 5(b). As shown in Figs. 6 and 7, the critical state line is also unique for silts and sands if the proposed intergranular stress [i.e., Eq. (5)] is adopted to interpret the experimental data. In applying Eq. (5), it is assumed that and , since the plasticity of the three soils previously described are low and Donnan’s osmotic effect is negligible.
EM.1943-7889.0001146/asset/f6be2094-86cf-4c04-9ce6-c067921e8688/assets/images/large/figure5.jpg)
EM.1943-7889.0001146/asset/2263cfba-4d6b-4817-add1-4ca49448fb11/assets/images/large/figure6.jpg)
EM.1943-7889.0001146/asset/d135ee01-bced-4097-b4a3-3c97bcd17d35/assets/images/large/figure7.jpg)
Constitutive Equations
Based on the proposed intergranular stress equation [i.e., Eq. (5)], a conceptual constitutive model of unsaturated soils with chemomechanical coupling is presented in the following text.
Yield Function
The stress-strain relationship is developed within the framework of the modified Cam-Clay model (Roscoe and Burland 1968), in which the yield function is given bywhere = deviatoric stress; = mean intergranular stress; = slope of the critical state line in the plane, which is unique and independent of the saturation and the pore solution concentration; and is the preconsolidation pressure.
(10)
To account for the physicochemical effect on soil hardening, we introduce the internal pressure, , defined by , as a new hardening variable. According to Lu and Likos (2006), is indeed the suction stress accounting for the effect of the intergranular physicochemical forces. Clearly, the yield stress of the soil increases as increases. Hence it is proposed here thatwhere = initial preconsolidation pressure of the fully saturated soil, and a function of plastic volumetric strain and the saturated concentration of pore water; is a function of , accounting for all the physicochemical effects associated with unsaturation; and = initial generalized osmotic stress of the saturated soil, which is used to describe the physicochemical effect on the mechanical behavior of the fully saturated soil. It is assumed that
(11)
(12)
Here, is the initial yield pressure of the saturated soil when and ; is a hardening parameter, accounting for the physicochemical effect; is the specific volume, , and the void ratio; and and are the slopes of the normal consolidation line and the unloading-reloading line in the plane, respectively. As a first approximation, and are assumed to be constants independent of the saturation and physicochemical effect. Fig. 8 shows the relationship between and the yield pressure for a fully saturated soil, showing that the mechanical yield pressure, , decreases as increases. This is consistent with experimental results (Witteveen et al. 2013). When the pore-water concentration is increasing, decreases and the corresponding yield pressure increases.
EM.1943-7889.0001146/asset/22bb077d-b8f1-4570-a0f1-9cb55d3aad0b/assets/images/large/figure8.jpg)
Function is the hardening function attributed to the physicochemical effects associated with unsaturation. As a first approximation, we proposed that
(13)
Noticeably, when the soil becomes saturated, and so that the proposed model degenerates into its saturated counterpart. As shown in Fig. 9, Eq. (11) provides a good fit to the initial yield stresses measured under constant-suction loading conditions.
EM.1943-7889.0001146/asset/164015cb-b8f9-4e1c-9d7e-5bf2e96679db/assets/images/large/figure9.jpg)
The yield surface is moving with the evolution of , which is characterized by the double-hardening mechanisms described in Eq. (11). As previously discussed, based on the intergranular stress concept, the critical state line is unique and independent of the saturation and the physicochemical effect. Hence one can simply assume
(14)
Stress-Strain Relationship
The incremental elastic volumetric and shear strains can be generally expressed aswhere is the elastic shear modulus.
(15)
An associated flow rule is adopted herein, i.e., the plastic potential coincides with the yield function. The incremental plastic volumetric and deviatoric strains are given bywhere is the plastic multiplier that can be determined from the consistency condition.
(16)
Description of Soil-Water Characteristic Curve (SWCC)
A hysteretic SWCC model, developed by Wei and Dewoolkar (2006), is adopted here to describe the relationship between the degree of saturation, , and the measured suction, , are not unique and exhibit hysteresis. Although this SWCC model is capable of predicting all types of primary, secondary, and higher-order scanning curves within the boundary loop, it does not address the effect of skeletal deformation. For a deforming soil, one has (Ma et al. 2015)
(17)
Clearly, the variation of saturation generally has two contributions: one is solely due to seepage or dissipation, and the other is due to the change in the pore volume, which are presented by the first and second terms of the right-hand side, respectively. According to Wei and Dewoolkar (2006)where denotes the hydraulic loading direction and assumes a value of 1 (or ) for drying (or wetting); and is the negative slope of the current soil-water characteristic curve (either scanning or boundary), which is a function of , , and .
(18)
The air entry value of soil can increase with the decrease of void ratio. As a consequence, the soil-water characteristic curve may shift upward on the plane with the development of volumetric strain, . To address this issue, it is proposed herein thatwhere function assumes the same form as the original one given in Wei and Dewoolkar (2006), representing the negative slope of the corresponding main boundary (flagged by ); for the main drying boundary, , and for the main wetting boundary, ; is a positive material parameter which is used to describe the scanning behavior, and for simplicity it is assumed that is constant; is the matric suction value on the corresponding main boundary, i.e., for drying and for wetting, where subscripts DR and WT stands for drying and wetting, respectively; and are the main drying and wetting boundaries, respectively; and is the current size of the bounding zone, i.e., . Because elastic deformation is generally much smaller than plastic deformation, in Eq. (19) can be approximately replaced by the total plastic volumetric strain, .
(19)
Van Genuchten (1980) model is chosen to describe the bounding curveswhere the residual degree of saturation is assumed zero; and and are two material parameters accounting for the air entry value and the slope of SWCC, respectively. assumes values for wetting and drying are different.
(20)
Experimental results (Vanapalli et al. 1999) suggest that the skeletal deformation changes the position of the SWCC only and leaves the shape of the curve almost unchanged. Thus, for simplicity, we propose thatwhere and = curving-fitting parameters, .
(21)
Evaluation of Constitutive Parameters
The proposed model is based on the modified Cam-Clay model, including an additional parameter to describe the intergranular physicochemical effects associated with unsaturation. The constitutive parameters can be divided into the following three groups:
1.
Conventional: , , , and ;
2.
Soil-water characteristic: , , , , , and ; and
3.
Physicochemical: , , and .
Those conventional mechanical parameters are independent of the saturation and the saturating fluid concentration, and can be determined in the same way as those in the modified Cam-Clay model. The soil-water characteristic parameters can be determined as proposed by Ma et al. (2015). is related to the cation exchange capacity (CEC) of a soil, and is equal to the air entry value of the soil; can be determined by fitting the experimental results obtained from the relationship between the preconsolidation pressure and the intergranular stress.
Model Performance
In this section, the experimental data available in the literature (Di Maio 1996; Sharma 1998; D’onza et al. 2011) are used to qualitatively illustrate the performance of the proposed model in describing the chemical–mechanical behavior of saturated and unsaturated soils.
Chemical–Mechanical Coupling of a Fully Saturated Soil
Di Maio (1996) carried out a series of oedometer tests on the reconstituted samples of Ponza bentonite, which is composed mainly of Na-montmorillonite. The material was found to have 80% clayey fraction, with a plasticity index of 320%. Reconstituted specimens were prepared by mixing powdered clay with distilled water to a slurry at approximately the liquid limit. The specimens were first consolidated to several axial stresses, and were subsequently exposed to the saturated solutions by replacing the water in the cell. Experimental results are presented in Figs. 10(a and c), where the compression and the swelling curves for specimens embedded in the distilled water and in a saturated NaCl solution are shown. Very different volumetric strains were measured between samples in the distilled water and in a saturated NaCl solution.
EM.1943-7889.0001146/asset/a138056a-c710-40b5-9f86-f9caa4a064a7/assets/images/large/figure10.jpg)
In addition, Figs. 10(a and c) illustrate that the specimens undergo apparent compressive volumetric strain when exposed to the saturated NaCl solution, and the volume decrease depends on the applied axial stress. The phenomenon of solution-induced volume decrease cannot be explained using Terzaghi’s effective stress since the effective stress is constant during the impact of saturated NaCl solution. Based on the concept of the intergranular stress, however, when the distilled water was replaced by the saturated NaCl solution, the intergranular stress was increased due to the decrease of Donnan’s osmotic pressure, resulting in additional volumetric strain. Remarkably, when the specimen was exposed to water again, noticeable swelling behavior appeared.
The simulated results of the proposed model are shown in Figs. 10(b and d). In simulation, is assumed to be . Although there exists some inconsistency between the prediction and experimental results, the proposed model captures the main features of the deformation of the fully saturated bentonite sample under chemomechanical loading. The discrepancy is probably because of other chemical effects (e.g., the nonideality of the saturating fluid), which the proposed model has not taken into account.
Isotropic Loading under Constant Suction
Sharma (1998) conducted a series of isotropic compression tests on a compacted mixture of bentonite and kaolin. The parameters are determined based on the experimental results, and listed in Table 1. In addition, is assumed to equal .
| Parameter | Value |
|---|---|
| 0.219 | |
| 0.035 | |
| 0.016 | |
| (kPa) | 32.1 |
| (kPa) | 66.2 |
| 0.13 | |
| (kPa) | 8.99 |
| (kPa) | 10 |
| (kPa) | 135 |
| (kPa) | 135 |
Fig. 11 shows the comparison between experimental and simulated behaviors for the isotropic loading tests under constant-suction conditions (100, 200, and 300 kPa, respectively), and the corresponding stress paths and yield surfaces are also shown. Clearly the model can effectively capture the main features of the mechanical behavior of the unsaturated soil experiencing loading-unloading cycle. In addition, the proposed model can also effectively describe the increase of the degree of saturation with the volume deformation.
EM.1943-7889.0001146/asset/266853a1-5a2d-4b8f-9ebf-a3b723c8fb97/assets/images/large/figure11.jpg)
The yield stresses are well predicted by the proposed model [Fig. 11(a)]. Fig. 11(c) shows that the intergranular stress increases during a loading process. In the elastic phase, the internal pressure, , increases slightly due to the increase of saturation. When the intergranular stress touches the yield curve, irreversible change of void ratio occurs, and increases considerably. The yield curve expands during loading and the final yield curves are shown as dotted lines in the figures. During unloading processes the soil experiences elastic deformation. In general, model simulations agree with the experimental data.
Fig. 12 compares the simulations to the experimental results obtained for a mixed loading condition in which a wetting/drying cycle was first applied to the soil under a constant mean net stress of 10 kPa, and then an isotropic loading under a constant suction of 300 kPa was applied.
EM.1943-7889.0001146/asset/04a20832-b35f-4c1e-a809-f42397168d80/assets/images/large/figure12.jpg)
Although the proposed model yields a more rigid response of the soil, it can reasonably capture the trend of skeletal deformation in either the wetting/drying or the loading-unloading stage. Fig. 12(c) illustrates that some amount of plastic deformation had occurred in the soil during the wetting/drying cycle, which cannot be addressed by the model. Fig. 13 shows that the stress path during the wetting/drying cycle is totally located within the yield surface, implying that the soil deformation is fully elastic. As a consequence, the model underestimates the volumetric strain after the wetting/drying cycle, resulting in an overestimation of the soil-specific volume during the coming loading phase [Fig. 12(a)]. Apparently, the discrepancy is mainly due to the oversimplification of the proposed model. Nonetheless, the model correctly predicts the occurrence of capillary hysteresis in a deforming soil and the irreversible change of void ratio under the constant-suction loading condition.
EM.1943-7889.0001146/asset/1dc1cc12-6481-42dd-b5c3-decd204e3039/assets/images/large/figure13.jpg)
Fig. 14 depicts the experimental and simulated results of the unsaturated soil sample subjected to isotropic loading-unloading cycles under a constant suction of 200 kPa. During the experiment, when the first unloading to was finished, a wetting/drying cycle was applied to the soil under constant mean net stress. Then the soil was isotropically reloaded under a constant suction of 200 kPa to , where the soil was unloaded again. From Fig. 14, several features of the soil response can be clearly identified, including: (1) the soil was softened by the applied wetting/drying cycle [Fig. 14(a)]; (2) the change of saturation during wetting/drying was not reversible [Fig. 14(b and d)]; and (3) a small amount of deformation occurred after the wetting/drying cycle [Fig. 14(c)].
EM.1943-7889.0001146/asset/d6312974-ad0c-46f2-a2ca-daebcb639e38/assets/images/large/figure14.jpg)
In general, the proposed model reasonably describes the soil response under such a complicated loading condition. In particular, the model effectively predicts the trend of the saturation variation during the deforming process of the unsaturated soil [Fig. 14(b)]. Some discrepancies occur between the simulated and measured volume changes [Fig. 14(a)]. Although the volume change during the first loading/unloading cycle is effectively described, it is significantly underestimated by the proposed model after the wetting/drying cycle. In addition, the model predicts that the soil becomes hardened after the wetting/drying cycle, which is inconsistent with the experimental observation.
To explore the logic behind the afore mentioned inconsistency, one can refer to Eqs. (11) and (13). After the wetting/drying cycle, the matric suction remains as 200 kPa, but the degree of saturation increases by some amount. In the whole process, remains practically constant. According to Eq. (4), the increase of saturation results in a decrease of , and thus an increase of as implied by Eq. (2). Hence, the hardening stress variable, , may increase or decrease depending upon the relative magnitudes of the variation of and (noted that is constant and ). Because the experimental data are lacking, the SWCC adopted in simulation may not exactly represent the true soil-water characteristics. This is implied by Fig. 14(d), in which one can see that the variation of saturation with matric suction deviates considerably from the measurement at low matric suction. Because of this, the model predicts an increase in the yield stress after the wetting/drying cycle.
Fig. 15 illustrates the predicted stress path that the soil experiences. During the whole wetting/drying cycle, the soil deformation is fully elastic and irreversible deformation occurs during the isotropic loading processes.
EM.1943-7889.0001146/asset/ee0f6d21-01f3-458a-a210-1c208d410b00/assets/images/large/figure15.jpg)
Based on the previous discussions, one can see that with its very nature of simplicity the proposed model can well capture the main features of the mechanical behavior of the unsaturated soils under complicated loading conditions. The model performance can be improved by introducing a more realistic soil-water characteristic model.
Triaxial Compression under Constant Suction
The results of a series of suction-controlled triaxial experiments on compacted samples of Jossigny silt by Casini (2008) are used to further illustrate the performance of the proposed model. The experimental data are available on the MUSE website (http://muse.dur.ac.uk). The tested soil includes 5% sand, 70% silt, and 25% clay. In simulation, is assumed to be , and . The material parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| 0.123 | |
| 0.0057 | |
| 0.034 | |
| G (GPa) | 36 |
| 1.28 | |
| (kPa) | 9.94 |
| 0.18 | |
| (kPa) | 4.57 |
| (kPa) | 10 |
| (kPa) | 135 |
| (kPa) | 135 |
Figs. 16 and 17 compare the simulations and experimental results of two triaxial compression tests performed under a constant suction of 200 kPa. One test involved anisotropic loading (denoted by TX08), while the other involved anisotropic loading prior to shearing (TX06). The loading anisotropy is represented by , which is defined as the ratio between the increments of deviator and mean net stress during loading.
EM.1943-7889.0001146/asset/ae7bf2b3-def9-47e6-a1cc-16cfa118f99c/assets/images/large/figure16.jpg)
EM.1943-7889.0001146/asset/8c4da44b-7be0-42a8-8daf-6a445802cff4/assets/images/large/figure17.jpg)
It can be seen from the comparison that the proposed model reasonably predicts the soil response in both tests. In addition, the predicted yield stresses agree well with the measurements, though the new model predicts a sharp transition from elastic to plastic behavior. The predicted variations of the volumetric strain with the axial strain are reasonably close to the observations. Comparison between the predicted and measured relationships also confirms that the model is capable of describing the dependency of degree of saturation on the volumetric strain. Despite of some predicted discrepancies, due to its very nature of simplicity, one can still conclude that the proposed model is capable of capturing the main features of the shearing behavior of unsaturated soils under complex loading conditions.
Conclusions
A conceptual elastoplastic constitutive model for unsaturated soils is developed based on the concept of intergranular stress, which explicitly accounts for the effects of osmosis, adsorption, and capillarity. The applicability of the proposed intergranular stress formulation is first illustrated using the experimental results available in the literature. In describing the elastic deformation of a porous rod the new stress variable performs much better than Bishop’s type effective stress, especially when the saturation is low. In addition, based on the proposed intergranular stress, a unique failure line exists for a soil with different saturations and pore-water concentrations.
Using the new intergranular stress, the constitutive equations are developed with the framework of the modified Cam Clay model, in which a hardening stress variable, i.e., , is introduced to lump all the physicochemical effects into an individual variable. The constitutive model is thus formulated in terms of a single yield curve so that the chemomechanical behavior of soils under chemical or mechanical loadings can be characterized in a consistent and systematic way.
A series of experiments on various type of soils subjected to a wide variety of stress paths are analyzed and simulated, demonstrating the performance of the proposed model. Comparison between predicted and experimental results confirms the potential of the proposed model in capturing the important features of the chemical-mechanical behavior of unsaturated soils.
Acknowledgments
This research was funded by one of the National Basic Research Programs of China under Grant 2012CB026102, and the National Science Foundation of China (NSFC) under Grants 51239010, 11502276, and 11372078.
References
Alawaji, H. A. (1999). “Swell and compressibility characteristics of sand-bentonite mixtures inundated with liquids.” Appl. Clay Sci., 15(3-4), 411–430.
Alonso, E. E., Gens, A., and Josa, A. (1990). “A constitutive model for partially saturated soils.” Geotechnique, 40(3), 405–430.
Amberg, C. H., and McIntosh, R. (1952). “A study of adsorption hysteresis by means of length changes of a rod of porous glass.” Can. J. Chem., 30(12), 1012–1032.
Baker, R., and Frydman, S. (2009). “Unsaturated soil mechanics: Critical review of physical foundations.” Eng. Geol., 106(1-2), 26–39.
Barbour, S. L., Fredlund, D. G., and Pufahl, D. E. (1992). “The osmotic role in the behavior of swelling clay soils.” Proc., NATO Advances Research Workshop, Springer, Berlin, 97–139.
Bishop, A. W. (1959). “The principle of effective stress.” Tek. Ukebl., 106(39), 113–143.
Bolzon, G., Schrefler, B. A., and Zienkiewicz, O. C. (1996). “Elastoplastic soil constitutive laws generalized to partially saturated states.” Geotechnique, 46(2), 279–289.
Casini, F. (2008). “Effetti del grado di saturazione sul comportamento meccanico del limo.” Ph.D. thesis, Univertita degli Studi di Roma La Sapienza, Rome.
Castellanos, E., Villar, M. V., Romero, E., Lloret, A., and Gens, A. (2008). “Chemical impact on the hydro-mechanical behaviour of high-density FEBEX bentonite.” Phys. Chem. Earth, 33(Suppl. 1), S516–S526.
Coussy, O. (2011). Mechanics and physics of porous solids, Wiley, West Sussex, U.K.
Di Maio, C. (1996). “Exposure of bentonite to salt solution: Osmotic and mechanical effects.” Geotechnique, 46(4), 695–707.
Dominijanni, A., Manassero, M., and Puma, S. (2013). “Coupled chemical-hydraulic-mechanical behaviour of bentonites.” Geotechnique, 63(3), 191–205.
D’Onza, F., et al. (2011). “Benchmark of constitutive models for unsaturated soils.” Geotechnique, 61(4), 283–302.
Ehlers, W., Graf, T., and Ammann, M. (2004). “Deformation and localization analysis of partially saturated soil.” Comput. Methods Appl. Math., 193(27–29), 2885–2910.
Gajo, A., and Loret, B. (2007). “The mechanics of active clays circulated by salts, acids and bases.” J. Mech. Phys. Solids, 55(8), 1762–1801.
Gallipoli, D., Gens, A., Sharma, R., and Vaunat, J. (2003). “An elasto-plastic model for unsaturated soil incorporating the effects of suction and degree of saturation on mechanical behavior.” Geotechnique, 53(1), 123–135.
Gens, A. (2010). “Soil-environment interactions in geotechnical engineering.” Geotechnique, 60(1), 3–74.
Gonçalvès, J., Rousseau-Gueutin, P., de Marsily, G., Cosenza, P., and Violette, S. (2010). “What is the significance of pore pressure in a saturated shale layer?” Water Resour. Res., 46(4), W04514.
Guimarães, L. D., Gens, A., and Olivella, S. (2007). “Coupled thermo-hydro-mechanical and chemical analysis of expansive clay subjected to heating and hydration.” Transp. Porous Media, 66(3), 341–372.
Homand, S., and Shao, J. F. (2000). “Mechanical behavior of a porous chalk and water/chalk interaction. Part I: Experimental study.” Oil Gas Sci. Technol., 55(6), 591–598.
Houlsby, G. T. (1997). “The work input to an unsaturated granular material.” Geotechnique, 47(1), 193–196.
Hueckel, T. (1992). “Water-mineral interaction in hygromechanics of clays exposed to environmental loads: A mixture theory approach.” Can. Geotech. J., 29(6), 1071–1086.
Hueckel, T. (2002). “Reactive plasticity for clays during dehydration and rehydration. Part 1: Concepts and options.” Int. J. Plast., 18(3), 281–312.
Khalili, N., Geiser, F., and Blight, G. E. (2004). “Effective stress in unsaturated soils: Review with new evidence.” Int. J. Geomech., 115–126.
Kohgo, Y., Nakano, M., and Miyazaki, T. (1993). “Theoretical aspects of constitutive modelling for unsaturated soils.” Soils Found., 33(4), 49–63.
Laloui, L., and Nuth, M. (2009). “On the use of the generalized effective stress in the constitutive modelling of unsaturated soils.” Comput. Geotech., 36(1-2), 20–23.
Loret, B., Gajo, A., and Simões, F. M. F. (2004). “A note on the dissipation due to generalized diffusion with electro-chemo-mechanical couplings in heteroionic clays.” Eur. J. Mech. Solids, 23(5), 763–782.
Loret, B., Hueckel, T., and Gajo, A. (2002). “Chemo-mechanical coupling in saturated porous media: Elastic-plastic behavior of homoionic expansive clays.” Int. J. Solids Struct., 39(10), 2773–2806.
Loret, B., and Khalili, N. (2002). “An effective stress elastic-plastic model for unsaturated porous media.” Mech. Mater., 34(2), 97–116.
Lu, N., Godt, J. W., and Wu, D. T. (2010). “A closed-form equation for effective stress in variably saturated soil.” Water Resour. Res., 46(5), W05515.
Lu, N., and Likos, W. J. (2006). “Suction stress characteristic curve for unsaturated soil.” J. Geotech. Geoenviron. Eng., 131–142.
Ma, T., Wei, C, Wei, H., and Li, W. (2015). “Hydraulic and mechanical behavior of unsaturated silt: Experimental and theoretical characterization.” Int. J. Geomech., D4015007.
Maatouk, A., Leroueil, S., and Rochelle, P. L. (1995). “Yield and critical state of a collapsible unsaturated silty soil.” Geotechnique, 45(3), 465–477.
Mašín, D., and Khalili, N. (2016). “Swelling phenomena and effective stress in compacted expansive clays.” Can. Geotech. J., 53(1), 134–147.
Mitchell, J. K., and Soga, K. (2005). Fundamentals of soil behavior, Wiley, New York.
Nitao, J. J., and Bear, J. (1996). “Potentials and their role in transport in porous media.” Water Resour. Res., 32(2), 225–250.
Nordberg, M. E. (1944). “Properties of some vycor-brand glasses.” J. Am. Ceram. Soc., 27(10), 299–305.
Rampino, C., Mancuso, C., and Vinale, F. (2000). “Experimental behavior and modelling of an unsaturated compacted soil.” Can. Geotech. J., 37(4), 748–763.
Roscoe, K. H., and Burland, J. B. (1968). “On the generalized stress-strain behavior of ‘wet’ clay.” Engineering plasticity, J. Heyman and F. A. Leckie, eds., Cambridge University Press, Cambridge, 535–609.
Santagiuliana, R., and Schrefler, B. A. (2006). “Enhancing the Bolzon-Schrefler–Zienkiewicz constitutive model for partially saturated soil.” Transp. Porous Med., 65(1), 1–30.
Scherer, G. W. (1986). “Dilatation of porous glass.” J. Am. Ceram. Soc., 69(6), 473–480.
Sharma, R. S. (1998). “Mechanical behavior of an unsaturated highly expansive clays.” Ph.D. dissertation, Univ. of Oxford, Oxford, U.K.
Sheng, D., Sloan, S. W., and Gens, A. (2004). “A constitutive model for unsaturated soils: Thermomechanical and computational aspects.” Comput. Mech., 33(6), 453–465.
Sivakumar, V. (1993). “A critical state framework for unsaturated soil.” Ph.D. dissertation, Univ. of Sheffield, Sheffield, U.K.
Sridharan, A. (2014). “Fourth IGS-Ferroco Terzaghi oration: Soil clay mineralogy and physico-chemical mechanisms governing the fine-grained soil behaviour.” Indian Geotech. J., 44(4), 371–399.
Sridharan, A., and Venkatappa Rao, G. (1973). “Mechanisms controlling volume change of saturated clays and the role of the effective stress concept.” Géotechnique, 23(3), 359–382.
Sun, D. A., Zhang, L., Li, J., and Zhang, B. (2015). “Evaluation and prediction of the swelling pressures of GMZ bentonites saturated with saline solution.” Appl. Clay Sci., 105, 207–216.
Vanapalli, S. K., Fredlund, D. G., and Pufahl, D. E. (1999). “The influence of soil structure and stress history on the soil-water characteristics of a compacted till.” Geotechnique, 49(2), 143–159.
van Genuchten, M. T. (1980). “A closed-form equation for predicting the hydraulic conductivity of unsatutated soils.” Soil Sci. Soc. Am. J., 44(5), 892–898.
Wahid, A. S., Gajo, A., and Di Maggio, R. (2011). “Chemo-mechanical effects in kaolinite. Part 1: Prepared samples.” Geotechnique, 61(6), 439–447.
Wei, C. (2014). “A theoretical framework for modeling the chemomechanical behavior of unsaturated soils.” Vadose Zone J., 13(9), 1–21.
Wei, C., and Dewoolkar, M. M. (2006). “Formulation of capillary hysteresis with internal state variables.” Water Resour. Res., 42(7), W07405.
Wheeler, S. J., Sharma, R. S., and Buisson, M. S. R. (2003). “Coupling of hydraulic hysteresis and stress-strain behavior in unsaturated soils.” Geotechnique, 53(1), 41–54.
Witteveen, P. J., Ferrari, A., and Laloui, L. (2013). “An experimental and constitutive investigation on the chemo-mechanical behavior of a clay.” Geotechnique, 63(3), 244–255.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 142 • Issue 11 • November 2016
Copyright
This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.
History
Received: Dec 30, 2015
Accepted: May 25, 2016
Published online: Jul 26, 2016
Published in print: Nov 1, 2016
Discussion open until: Dec 26, 2016
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.