Soil Mechanics for Unsaturated Soils Professor Delwyn G. Fredlund•The study of a multi-phase system such as an unsaturated soil, requires an understanding of the properties of each component phase.rties of each phase of an unsaturated soil; ii.) an understanding of the interaction between air and water; iii.) establishment of useful volume-mass relations for solving engineering problems.Additional notes:Soil particles &contractile skinbehave as solidsAir & waterbehave as fluids•An unsaturated soil is commonly referred to as a 3-phase system composed of: i.) solids (soil particles); ii.) water; iii.) iii.) air.•The air-water interface (i.e., the contractile skin) warrants inclusion as an additional phase due to its unique and specific properties.•The contractile skin interacts with the soil particles in an independent manner and can significantly change the mechanical behavior of an unsaturated soil.fore can be neglected in volume-mass relationships.•The air phase might be present in an unsaturated soil either in a continuous or an occluded (bubbles) form.Additional notes:Required tounderstandthe stress stateanalysisSufficient forMapping phasechanges•Gravimetric and volumetric relationships for unsaturated soil are based on various mass and volume ratios of the component phases.•A rigorous microscopic analysis of the volume-mass properties of an unsaturated soil mass requires the inclusion of the contractile skin.•Using a macroscopic (phenomenological) approach, an unsaturated soil can be considered as a three-phase continuum system.•The most important property of the contractile skin is its ability to exert a tensile pull which is a consequence of the surface tension generated between the two pore-fluid phases (i.e., air and water).Additional notes:•Air is a mixture of several gases and varying amounts of water vapor. water vapor is present.•Air is compressible and its density requires additional considerations in terms of pressure and temperature.Additional notes:Air is a mixture of several gases that act in an independent manner•Air is considered to behave as an “ideal gas” under pressures and temperatures commonly encountered in geotechnical engineering. •The molecular mass of air , ωa , depends on the mixture of dry air and water vapor.lar mass ofwater vapor is 18.016 kg/kmol.from0.000002% to 5% (Harrison, 1965). Additional notes :Density is equal to mass per unit volume Density of air is directlyproportional to the absolute air pressure•Amounts of Nitrogen (N 2) and Oxygen (O 2) are essentially constant.•Carbon dioxide (CO 2) content in air may vary depending on the rate of consumption of fossil fuels. Additional notes:•Water is essentially a homogeneous substance the world over, except for variations due to salts and isotopes of hydrogen and oxygen (Dorsey, 1940).•Distilled water under the pressure of its saturated vapor is called pure, saturated water.•The density of water must be measured experimentally.•Variations in the density of water due to temperature differences are more significant than variations due to applied pressure.Additional notes:SURFACE TENSION •A molecule in the interior of the water experiences equal forces in all directions whilea water molecule in the contractile skin experiences an unbalanced force towards the interior of the water.actile skin.alled its surface tension,T s (i.e., units of N/m 2).•Contractile skin behaves like an elastic membrane.-water surface can be related to surface tensionand the radius of curvature by considering equilibrium.Additional notes:Surface Tension Phenomena•Surface tension is measured as the tensile force per unit length along the contractile skin (i.e., units of N/m).•Surface tension is tangential to the contractile skin surface. Its magnitude decreases as temperatures decreases.u a , that is greater than thewater pressure, u w .•The pressure difference, (u a -u w ) is referred to as matric suction.Additional notes:•Contractile skin forms a warped or saddle-shaped surface (i.e., 3-D membrane) that has different curvatures along two orthogonal (i.e., R 1and R 2for example).o C.-water interface.Additional notes:State Diagram for Water(from van Haveren and Brown, 1972) •Water phase can exist in one of three states: i.) solid state as ice; ii.) liquid state aswater; iii.) gaseous state as water vapor.•The vaporization curve is also called the vapor pressure curve of water.•The fusion curve separates the solid and liquid states.•The sublimation curve separates the solid and vapor states.•Triple point of water is achieved at a temperature of 0o C and a pressure of 0.61 kPa.Additional notes:•The vaporization curve represents an equilibrium condition between the liquid and vapor states of water.•At equilibrium, evaporation and condensation processes occur simultaneously at the same rate.•Rate of condensation depends on the water vapor pressure it reaches at itssaturation value on the vaporization value.•Evaporation rate depends only on temperature.•Water vapor is mixed with air in the atmosphere, and has no effect on thebehavior of the water vapor.•Water vapor may not be in equilibrium with adjacent water. Water vapor in air can be under-saturated, saturated or super-saturated depending on the partialpressure of water vapor.•The degree of saturation with respect to water vapor is referred to as relative humidity, RH.Additional notes:•Water molecules form a lattice structure with openings referred to as a “cage” that can be occupied by gas.•Air dissolves water and fills the “cages” which comprise approximately 2% by total volume.Additional notes:•The volumetric coefficient of solubility, h,varies slightly with temperature.•The rate at which mass is transferred across a unit area is equal to the product of the coefficient of diffusion, D,and the concentration gradient.•In the diffusion of air through water, the concentration difference is equal to the difference in density between the free air and the dissolved air in the water.•The pressure difference between the free air and the air dissolved in the water becomes the driving potential for the free air to diffuse into the water.•The coefficient of diffusion for air through the pore-water in a soil appears to differ by several orders of magnitude from that for air through free water.Additional notes:•Volume of air dissolved in water is essentially independent of air or water pressures.•Absolute pressure of the dissolved air is equal to the absolute pressure of the free air under equilibrium conditions.•Volume of air dissolved in water is constant for different pressures.•Ratio between the mass of each gas that can be dissolved in a liquid and the mass of the liquid is called the Coefficient of Solubility , H.•Coefficients of Solubility are referenced to a standard pressure of 101.3 kPa.•Ratio of the volume of dissolved gas, V d ,in a liquid to the volume of the liquid is the Volumetric Coefficient of Solubility , h.Additional notes :•A cylinder with a piston and porous stone analogy is useful in visualizing and analyzing the behavior of air-water mixtures.•The “porous stone” has pores equaling 2% of its volume. The “porto simulate the behavior of water.•An initial pressure is applied equally to the free air and the “•An imaginary valve is placed at the boundary between the free air and the “porous stone”.•When the piston is loaded, it compresses the free air above theporous stone inaccordance with Henry’s law.•Henry’s law states that the mass of gas dissolved in a fixed quantity of liquid, at constant temperature, is directly proportional to the absolute pressure of the gas above the solution (Sisler et al., 1953).•After several increases of the applied loading, all free air will have moved into the “porous stone”, and any additional applied load will be carriedAdditional notes:•The water lattice is relatively rigid and stable (Dorsey, 1940) and the density of water changes little as a consequence of the presence of the dissolved air.•Dissolved air has no measurable effect to the density of water at a temperature of about 22o C.Additional notes:•An unsaturated soil has air and water pressures at different magnitudes.•The mass of air going into or coming out of the water is time dependent.•The amount of gas dissolved in water is referred to as its Solubility .•The rate of solution of a gas is referred to as its Diffusivity .Additional notes:•The volume-mass relations of the soil particles, water, and air phases are useful properties in engineering practice, particularly for an unsaturated soil.•Both mechanical and hydraulic behaviors of an unsaturated soil mass are highly dependent on the relative proportion of the three component phases (i.e., solid, water and air).•Soil properties such as degree of saturation, porosity and void ratio express volumetric proportions involving the three phases of an unsaturated soil mass.•The above volumetric properties associated with the density of each phase allows the definition of all the index properties commonly utilized in engineering practice to characterize the state of a soil mass.Additional notes:•Each phase of an unsaturated can be expresses as a percentage of the totalvolume by using the definition of porosity.•The volume of the contractile skin can be assumed to be negligible.•The porosity with respect to the water phase, n w ,is equivalent to the volumetric water content, θw .•A soil mass might change its total volume in response to changes in the stress state applied. However, the volume of solid particles is constant.•Void ratio of a soil mass expresses the ratio of the void volume to the volume of the solid particles.Additional notes:•The degree of saturation expresses the percentage of void space filled by the liquid phase (i.e., water).dry (i.e., S = 0%),saturated (i.e., S = 100%) or unsaturated ( 0% < S < 100%).•The transition zone between a continuous air phase and occluded air bubbles occurs when the degree of saturation is between approximately 80-90%.•The water content can also be measured either in a gravimetric manner (i.e., ratio of water to solids mass) or in a volumetric manner (i.e., ratio of water volume to total volume).Additional notes:•Two commonly used soil density definitions are the total density and the dry density. Total density is called the bulk density.•Dry density of a soil mass is dependent upon the mineralogy of the solidphase, in terms of specific gravity,G s , and the porosity of the soil structure.•The saturated density of the soil is the total density of the soil for the casewhere the voids are filled with water (i.e., S = 100%).•The buoyant density of a soil is the difference between the saturated density of the soil and the density of water.•A “basic volume-mass relationship” for a soil mass can be derived by using the idealized diagram where each soil phase is defined by its volume and its mass.Additional notes:•Volume-mass relationships in engineering practice neglect the mass of airphase, but consider the volume of the air phase.•The expression (Se = wG s )is the basic volume-mass relationshipcommonly used in engineering practice and it represents the continuityequation for a soil mass.•Additional notes:•Density, ρis the ratio of mass to volume. Each soil phase has its own density.•Specific gravity of the soil particles, Gs ,is defined as the dimensionless ratio between thedensity of the soil particles and the density of water at a temperature of 4o C underatmospheric pressure conditions (i..e, 101.3 kPa).•Density of water at 4o C and at 101.3 kPa is 1000 kg/m 3.•Specific volume, v o ,is the inverse of density (i.e., ratio of volume to mass).•Unit weight, γis the product of density, ρ, and gravitational acceleration, g,(i.e., 9.81 m/s 2).Additional notes:•In engineering practice, the total density is often increased by imposing a reduction in the void volume of a soil mass.•Volume of pore-water in a soil mass is highly dependent uponenvironmental conditions and therefore is often a variable to be considered.•The dry density is independent of the volume of the pore-water and is often used in semi-empirical relationships for the soil.Additional notes:•The dry density curve corresponding to a degree of saturation of 100 % is called the “zero air voids”, ZAV, curve.•The dry density curves curves for different degrees of saturation should be presented in connection with soil compaction data.•Compaction is a mechanical process used to increase the dry density of soils (i.e., densification).Additional notes:-Start of process:•The basic volume-mass relationship (i.e., Se = w Gs ) applies to anycombination of S, e and w , and therefore it can be used to obtain changes in any of these properties when the other two properties are defined.•The basic volume-mass relationship can therefore be used to deriveequations associating changes of S, e and w .Additional notes:•Soil mixtures can occur in various forms in nature.•Mixture of soil particles and air constitutes a dry soil while a mixture of soil particles and water constitutes a saturated soil.•Unsaturated soils lie between dry and saturated soil conditions and constitute a mixture of soil particles, air and water in differing percentages.•The density of a soil mixture can also be formulated for situations where there is a change in the volume of air due to compression under undrained conditions. •The analogy of the piston-porous stone is a useful tool to derive the formulation of the density of the soil by considering the air compression.•In the above formulation, Boyle’s law is applied to the free and dissolved air under isothermal conditions.Additional notes:•Under isothermal conditions, the ratio of the density to the absolute pressure of the air phase (i.e., free and dissolved air) in a soil mixture is constant.•The final density of water is computed as the ratio of the total mass of water and dissolved air to the total volume of water and dissolved air under final equilibrium conditions.Additional notes:•The conservation of mass is herein applied to a system (i.e., soil mixture) under isothermal conditions, therefore, all the previous equations can be applied. •The total density of the soil mixture is therefore a function of the constant densities and volumes of water and solid phases as well as of the variable density and volume of the air phase.•The total density formulation can be applied to the definition of particular cases where one of the three phases (i.e., solid, water and air) in a soil mixture is not present.Additional notes:•The first equation above is a general mixture equation for density after the soil has been subjected to a change in air pressure.•The volume of dissolved air , V d ,is related to the volume of water, Vw ,by thevolumetric coefficient of solubility, h (i.e., V d = hV w ).•The equation incorporates the effect of air going into solution due to the change in the air pressure.•Where air does not have time to dissolve in water, the volumetric coefficient of solubility, h,might be set to zero.•The air-water mixture always has a density between that of air and water.•The density of an air-water mixture can be obtained from the general equation is obtained by setting the volume of the soil particles, Vs ,to zero, and the initial voidratio, e i ,becomes infinity.Additional notes:。