非饱和土力学讲义(Fredlund在香港科大)chapter8ustmod

•The mechanical behavior of an unsaturated soil is directly affected by changes in the pore-air and pore-water pressures. Undrained loading of an unsaturated soil generates pore pressures in both the air and water phases.•This chapter presents the pore-pressures generated as a result of the application of total stresses to the soil.•The compressibility of air, water, and air-water mixtures is presented along with the compressibility of the soil structure which is summarized in the form of a constitutive relationship.•The pore pressure response is expressed in terms of pore pressure parameters, which relate the development of pore-pressures to a change in total pressure applied to a soil mass. These parameters perform a useful role in visualizing unsaturated soil behavior.•Estimated pore pressures are required at the start of a transient consolidation type analysis.•Comparisons between the predictions and measurements of pore pressures generated by applied loads are presented and discussed.•This chapter addresses pore pressures generated under various loading conditions.Additional notes:•During undrained compression of an unsaturated soil, volume change occurs as a result of the compression of air and to a lesser extent, compression of the water. Consequently, the pore-air and pore-water pressures increase.•Soil solids can be considered to be incompressible for the stress ranges commonly encountered in engineering practice.•The compressibility of a material at a point can be defined on the volume-pressure curve during compression.•Isothermal compressibility is defined as the volume change of a fixed mass with respect to a pressure change per unit volume at a constant temperature.Additional notes:•Isothermal compressibility of air is defined as the volume change of a fixed mass of air as the pressure is changed.•The volume versus pressure relationship for air during isothermal, undrained compression can be expressed using Boyle’s law, and the final air volume, V a ,is a function of the applied absolute air pressure,•Differentiating the volume of air, V a , with respect to the absolute air pressure, defines an expression for the infinitesimal volume change of air with respect to an infinitesimal change in the absolute air pressure.•Combining this equation with Boyle’s law permits expressing the volume of air derivative with respect to the absolute air pressure.•Air compressibility can then be written as the inverse of the absolute air pressure, since the incremental change in absolute air pressure is equal to the incremental change in the gauge pressure.•The air compressibility decreases as the absolute air pressure increases.Additional notes:a u•Water compressibility can be expressed as the product of the inverse of the water volume and the water volume change with respect to a change in the water pressure.•Water compressibility measurements (Dorsey, 1940) are also a function of temperature.•Dissolved air in water produces an insignificant difference between the compressibility of water.Additional notes:•The air phase, water phase, and solid phase volumetric relations in an unsaturated soil are as shown above.•The volumetric relations are used in the formulation of the compressibility of air-water mixtures found in an unsaturated soil mass.Additional notes:•The compressibility of an air-water mixture can be derived using directproportioning of the air and water compressibilities.•The air, water and solid volumetric relations can be described in terms of the degree of saturation, S , and a porosity, n,for an unsaturated soil.•The total volume of the air-water mixture is the sum of the individual components, V a + V w . The dissolved air, V d , is within the volume of water.•The pore-air and pore-water pressures are u a and u w , with u a > u w . The soil is subjected to a compressive total stress, σ.•Applying an infinitesimal increase in total stress, d σ, to the undrained soil results in increases in both pore-air and pore-water pressures, while the volumes of air and water decreases.•The compressibility of an air-water mixture for an infinitesimal increase in total stress can be written using total stress as a reference.Additional notes:•The compressibility of an air-water mixture presented in the previous slide is slightly different from the compressibility equation proposed by Fredlund (1976) in that the pore-water pressure change, du w , was used as the reference pressure in the 1976 compressibility equation.•The term [d(V w -V d )/d σ] is considered to be equal to dVw since the dissolved air is a fixed volume internal to the water.•The total volume of water, Vw , is therefore used in computing the compressibility ofwater [ i.e., C w = -(1/V w )(dV w /du w )].•The total air volume change can be obtained directly using Boyle’s law byconsidering the initial and final pressures and the volumetric conditions with respect to the air phase.•The free and dissolved air can be considered as one volume with uniform pressure, and although the volume of dissolved air is a fixed quantity, it is carried along in the formulation.•The chain rule of differentiation can be applied to the compressibility equation.Additional notes:•The compressibility of an air-water mixture equation can be rearranged in the form shown in the first equation in order to permit the use of the volume relations, S,and the expressions previously defined for air compressibility,C a ,and water compressibility,C w , to yields the second equation shown above.•The isothermal compressibility of air, C a ,is equal to the inverse of the absolute air pressure, and therefore a third equation can be written.•The ratio between the pore pressure and the total stress change,(du/d σ), is referred to as a pore pressure parameter (Skempton, 1954; and Bishop, 1954).•The pore pressures parameters for air and water are different and depend primarily upon the degree of saturation of the soil.•The pore pressure parameters can also be experimentally measured in laboratory.•For isotropic loading conditions, the parameter is called the B pore pressure parameter.•In the absence of soil solids, the pore pressures parameters, B a and Bw , are equal to 1.0. •In the presence of soil solids, the surface tension effects and the compressibility of air, will cause the B a and B w values to be less than 1.0.Additional notes:•Several equations for the compressibility of an air-water mixture have been proposed by researchers.•The above equation is obtained by ignoring the first term of the previous equation for air-water mixture compressibility (i.e., the water compressibility term) andsetting the Ba and Bwvalues to 1.0.•The result is an equation applicable to the case where the air phase constitutes a significant portion of the fluid, and is similar to the equation proposed by Bishop and Eldin(1950).•Bishop and Eldin(1950) assumed the compressibility of air with reference to the initial volume of air, Vao, as shown in the second equation above.•Such an assumption yields a slightly different equation for the air compressibility,C a , as expressed by the third equation above which gives the average aircompressibility, (uao /ua2) during an air absolute pressure change from uaoto ua.•Replacing the air compressibility term (i.e., 1 / ua ) with the average aircompressibility term [i.e., (uao /ua2)] yields the air-water compressibility equationproposed by Bishop and Eldin(1950), expressed as the fourth equation above.•The last equation was suggested by Koning(1963) by expressing the pore-air and pore-water pressure changes as a function of surface tension. The solubility of air in water and the effect of matric suction were neglected.Additional notes:•Kelvin’s equation [i.e., (u a -u w ) = 2T s /R s ] relates matric suction to surface tension and the radius of curvature.•Attempts have been made to use Kelvin’s equation in writing an equation for the compressibility of air-water mixtures (Schuurrman, 1966; Barends, 1979).•In particular, problems arise in the case of occluded air bubbles in a soil with adegree of saturation greater than 85 %.•Kelvin’s equation results in the incorporation of the radius curvature,R s , as avariable. However, R s ,is not measurable in a soil element.•Kelvin’s equation describes a microscopic phenomenon within the soil element.The radii of the occluded air bubbles should not be incorporated into amacroscopic type formulation for compressibility.•Figure 8.6b shows a soil that is almost saturated and has its macroscopic behavior governed by effective stress. At a microscopic level, there exists numerous inter-granular stresses acting at the contacts between the soil particles in the element.•The net results of attempts to apply Kelvin’s equation, togetherHenry’s laws, to the compressibility of an air-water mixture is that an anomalyarises from a theoretical point of view. Such a formulation predicts that anincrease in matric suction occurs as the total stress is increased under undrainedloading (Fredlund and Rahardjo, 1993).Additional notes:•Experimental results indicate that the pore-air and pore-water pressures gradually increase towards a single value as the matric suction approaches zero and the total stress is increased under undrained loading. The process is gradual and in response to several increments of total stress.•The above figure illustrates the development of air and water pore pressures as well as the changes in both the shape and volume of the air bubbles within an unsaturated soil during undrained loading, as the total stress is increased.•Non-spherical air bubbles in Zone 1 could provide an explanation to justify that the decrease in free air volume is not necessarily accompanied by a decrease in the controlling radius of curvature. The assumption is made that only the controlling minimum radius is of relevance in Kelvin’s equation.•The above figure shows that although the volume of the continuous air phase in Zone 1decreases due to an increase in the pore-air pressure from ua1to ua2,the controlling radiusmay increase from Rs1to Rs2,and therefore the matric suction decreases.•Nevertheless, the increase in total stress will eventually cause the air bubbles to take on a spherical form, as shown in Zone 2.•For spherical air bubbles, a decrease in volume must be followed by a decrease in the radius. In this case, the increase of matric suction postulated by Kelvin’s equation, cannot be resolved.•It would appear that the presence of air bubbles merely renders the pore-fluid compressible. Therefore, it is recommended that the pore-air and pore-water pressures be assumed to be equal in Zone 2.Additional notes:•The pore-pressure response for a change in total stress during undrained compression has been expressed in terms of pore-pressure parameters (i.e., B a and B w ), in previous sections.•In this section , derivations are presented for pore-pressure parameters corresponding to various loading conditions.•The pore pressure parameters for the air and water phases of an unsaturated soil can be defined either as tangent-type or secant-type parameters.•These definitions are similar in concept to the tangent and secant moduli used in the theory of elasticity.•Isotropic loading is a particular case of the more general triaxial loading and is used to express the definition of the secant pore pressure parameter for the air phase.•The secant-type pore-air pressure parameter (i.e., B a ’) is defined as the ratio between the increase in pore-air pressure (i.e., response) and the increase in isotropic pressure (i.e., σ3) from the initial condition.Additional notes:•The secant-type pore-water pressure parameter (i.e., B w ’) is defined as the ratio of the increase in pore-water pressure (i.e., response) to the increase in isotropic pressure (i.e., σ3) from the initial condition.•The secant-type parameter requires the definition of initial conditions for the soil specimen in terms of both pore-pressures and applied total stress.Additional notes:•If an infinitesimal increase in the isotropic confining pressure is considered at a point along the pore-air pressure versus isotropic confining stress, σ3, relationship,the pore-air pressure response at that point can be expressed as the tangent, B a ,pore-air pressure parameter.•Similarly, a tangent pore-water pressure parameter, B w , can be defined.•The concepts of secant and tangent-type pore-air and pore-water pressure parameters are illustrated on the next slide.Additional notes:•A linear equation for total volume changes within a localized region of the constitutive surface can be written as proposed by Fredlund and Morgenstern (1976).•The compressibility parameters, m 1s and m 2s ,correspond to changes in the stress state variables, (σ-u a ) and (u a -u w ), respectively.•Total volume changes can then be predicted by using the constitutive surfaces when changes in the stress state variables are known.Additional notes:•The linear equation for pore-air volume changes can be written as proposed by Fredlund and Morgenstern (1976).•The compressibility parameters, m 1a and m 2a ,correspond to changes in the stress state variables, (σ-u a ) and (u a -u w ), respectively.•Air volume changes can be predicted using the constitutive surfaces when changes in the stress state variables are known.Additional notes:•A linear equation for pore-water volume changes can be written as proposed by Fredlund and Morgenstern (1976).•The compressibility parameters, m 1w and m 2w , correspond to changes in the stress state variables (σ-u a ) and (u a -u w ), respectively.•Water volume changes can be predicted by using the constitutive surfaces when changes in the stress state variables are known.•The continuity requirement for a referential element of an unsaturated soil can be expressed by equating total volume change to the summation of the air volume and water volume changes.•This requirement leads to the conclusion that there are two closed-formrelationships between the compressibility.Additional notes:•The application of an all-around, positive (i.e., compressive) total stress, d σ, either in drained or undrained loading, can cause a change in volume.•In drained loading, air and water are allowed to drain from the soil subsequent to the application of a total stress increment. The stress state variables in the soil are altered and the volume of the soil changes.•The volume change can be computed from the stress state variable changes in accordance with the constitutive relationship for the soil structure.•In undrained loading, the air and water are not allowed to drain from the soil. The total stress increase causes the pore-air and pore-water pressures to increase, and consequently the stress state variables also change.•The increase in the pore fluid pressures occurs in response to a compression of the pore fluid.•The volume change equivalent to the pore fluid compression, dVv , can becomputed by multiplying the compressibility of air-water mixture, C aw , by thepore fluid volume V v (i.e., V w + V a = nV ) and the total stress increment, d σ.•The volume change can also be expressed in terms of the stress state variable [i.e., (σ-u a ) and (ua -u w )] changes in accordance with the constitutive relationships for the soil structure. The equation at the bottom of the slide is the volume change due to an increase in net total stress, d(σ-u a ).Additional notes:•An increase in the total stress results in an increase in pore fluid pressures (i.e., air and water) in response to compression of the pore fluids. Therefore,there is a reduction in matric suction since the increase in pore-water pressure is greater than the increase in pore-air pressure.•The volume change due to a decrease in matric suction can be expressed by the above equation.•The total volume change can then be written as the summation of the two previous volume changes due to the changes in the stress state variables (i.e., (dVv /V o )1and (dV v /V o )2].•The last equation expresses the volume change obtained from the constitutiverelationships and can be equated to the volume change due to pore fluid compression.•The concepts involving drained and undrained volume changes are illustrated on the next slide.Additional notes:•The increase in total stress results in an increase in net normal stress and a decrease in matric suction. The stress state variable changes can be used to define the volumetric changes in the unsaturated soil mass by using the constitutive relationships.•The above figure shows the comparison between undrained loading and drained loading by using the constitutive surfaces combined with changes in stress state variables due to an increase in total stress under undrained compression.Additional notes:•In a saturated soil, the pore voids are filled with water. The pore fluidcompressibility is equal to the compressibility of water.•At saturated conditions, a total stress increase, d σ, in undrained loading is almost entirely transferred to the water phase (i.e., du w ≈d σ) and the pore-water pressure parameter approaches 1.0.•The effective stress in undrained loading remains essentially constant, and as a consequence, the volume change computed from the constitutive relationship for the soil structure is extremely small as shown in Figure 8.11a.•The soil volume change obtained from the pore fluid compression is also small because of the low compressibility of water.•In a dry soil, the pore voids are primarily filled with air which is much more compressible than the soil structure.•A total stress change during undrained loading is almost entirely transferred to the soil structure.•Figure 8.11b above illustrates a dry soil under undrained loading which also illustrates that the pore-air pressure remains constant (i.e., the B parameter is essentially zero).Additional notes:•The K o loading conditions are illustrated in the above figure where the total stress increment is applied in the vertical direction.•The total stress increment is denoted as, d σ, and the vertical direction is assumed to be the major principal stress direction.Additional notes:•The volume change equation previously presented, associated the soil volume change of the soil structure to the volume change due to the pore fluid compression (i.e., under undrained loading).•Using K o loading conditions, the pore pressure parameter equation can be expressed by the above equation where the compressibility parameters correspond to K o loading.•The last two equations extend the compressibility of the air-water mixture,C aw , by substituting the change in total stress with the soil response in terms of pore pressures. •The first term of the compressibility equation (i.e., C aw equation) refers to thecompression of the pore-water phase and the second term refers to the compression of the pore-air phase.Additional notes:•The previous volume change equation can be rearranged to yield an expression for the change in the pore-water pressure, du w , in response to a total stress change, d σ.•In the last equation, the compressibility, m 2s ,has been written as a ratio of the compressibility with respect to a total stress change, m 1k s ,which gives rise to the parameter, R sk .Additional notes:•The previous equation can be simplified by introducing the compressibility parameters, R 1k and R 2k•There are two unknowns, du w and d σy , and therefore a second independent equation is required.•The second equation is derived by considering the change in the volume of air. •The change in volume is described by the constitutive relationship for the air phase which is equated to the volume change of air due to the compressibility of the air term multiplied by pore fluid volume (i.e., nV).•In this equation, the compressibility parameter, m 1k a ,is introduced.•In a similar procedure to that used for the soil structure, the last equation introduces the parameter R ak , that is defined as the ratio between the air phase compressibility, m2a ,and the air phase compressibility with respect to a total stress change, m 1k a .Additional notes:•The previous equation can be rewritten to give the change in pore-air pressure, du a , due to a total stress increment, d σy .•The expression can be simplified by introducing the ratio parameters, R 3k , and R 4k , as illustrated above.Additional notes:•The pore-air and pore-water parameters for Ko undrained loading can be written as B ak and B w k parameters, respectively. These pore-pressure parameters are defined as tangent-type parameters referenced to a particular stress point.•The expressions for du a and du w are from the previous equations where the pore-air and pore-water responses, at any stress point during Ko undrained loading, are a directfunction of the B ak and B w k pore pressure parameters.Additional notes:•Hilf (1948) outlined a procedure to compute the change in pore pressure in compacted earth fills as a result of an applied total stress. This method has been extensively used by the United States Bureau of Reclamation (i.e., U.S.B.R.), and has proven to be to be quite satisfactory in engineering practice (Gibbs et al, 1960).•The derivation is based on the results of a one-dimensional oedometer test on acompacted soil, Boyle’s law, and Henry’s law. A relationship between total stress and pore-water pressure is derived.•Hilf (1948) stated: “... consider a sample of moist earth compacted in a laboratory cylinder, as illustrated before. If a static load is applied by means of a tight-fitting piston , permitting neither air nor water to escape, it is found that there is a measurable reduction in volume of the soil mass”.•The reduction in volume was assumed to be the result of compression of free air and free air dissolving into water. Free and dissolved air are considered as a single phase.•The initial and final conditions considered in Hilf’s analysis are shown in the next slide. The total volume of air associated with the initial condition, Vao , can be writtenas shown above, with the first and second terms representing thefree and dissolved air volumes.•The air volume change can be written as a change in porosity, ∆n , times the initial volume of soil, V o ,. Therefore, the total volume of air under final conditions, V af , can be expressed by the above equation. •The final absolute air pressure is shown in the last equation.Additional notes:•Boyle’s law can be applied to the initial and final conditions of the free and dissolved air.•By substituting the initial and final air volumes into the above equation, andrearranging, it yields an expression for the change in the pore-air pressure as a function of the initial pore-air pressure.•The above equation for the change in absolute pore-air pressure is commonly referred to as Hilf’s equation, which provides a relationship between the change in pore-air pressure and the change in pore-air volume (i.e., ∆n ).•In order to reach saturation, the soil volume change, ∆Vv , must equal the volume offree air (i.e., (1-S o ) n o V o ). The change in porosity corresponding to this condition is expressed above (i.e., ∆n ).•Substituting the equation for a change in porosity, ∆n,into the equation for thechange in pore-air pressure, gives the absolute pore-air pressure change (i.e., increase) required to saturate a soil.Additional notes:•The previous equation for pore-air pressure change can be written in an alternate way form by replacing the change in porosity, ∆n,with (∆V v /V o ), which expresses the volume change due to the compression of air.•Assuming that the change in matric suction is negligible, the total volume change can also be obtained from the constitutive surfaces relationship [i.e., as a function of ∆(σv -u a )].•Near saturation, the soil compressibility parameter, m 1s ,can be assumed to be equal to the coefficient, m v ,measured under saturated conditions.Additional notes:•The pore-pressure parameter for K o undrained loading can be derived using Hilf’s analysis by equating the volume change of the soil structure to the volume change due to the compression of air.•The second equation in this slide shows the expression for the change in pore-air pressure, which is obtained by rearranging the first equation shown.•The secant pore pressure parameter, B áh , derived from Hilf’s analysis is thenexpressed as the third equation on the above slide.Additional notes:•For an isotropic soil under three-dimensional loading, the constitutive relationship for the soil structure is expressed by the above equation.•In the case of isotropic loading, the total stress increments are equal in all directions.Therefore, the constitutive relationship for an isotropic soil under isotropic loading can be written as the second equation.•The compressibility of the air-water mixture, C aw , is obtained from the previously shown equations by using the isotropic pressure increment, d σ3, for the total stress increment, d σ.•Volume change due to the pore fluid compression, dVv , is computed from thecompressibility, C aw equation multiplied by the pore fluid volume, V v (i.e., nV ), and the isotropic pressure increment, d σ3.Additional notes:•The change in volume of the soil structure must be equal to the change in volume due to pore fluid compression, as expressed by the first equation on this slide.•That equation can be rearranged to express the change in pore-water pressure, du w , as a function of the change in pore-air pressure and the change in total confining stress.•Another equation can be derived by considering the continuity of the air phase. The constitutive relationship for the air phase of an isotropic soil under three-dimensional loading can be written as the third equation above.•For isotropic loading, the constitutive relationship for the air phase of an isotropic soil is obtained by substituting the condition of ( d σ1 = d σ2 = d σ3). The result is the fourth equation.•Volume change due to the compression of air, dV a , is also represented by the fifth equation which is the second term of the pore fluid compression equation , dV v .•The last equation is obtained by equating the volume change due to thecompression of air to the volume change predicted by the constitutive relationship.Additional notes:•Solving the previous equation for the change in pore-air pressure, du a , results the first equation given above.•The compressibility of the soil structure, m 2s ,can be expressed as a ratio of m 1s , as shown by the R s parameter.•The expression for the pore-water pressure change can then be written as shown by the third equation, and simplified, by using the R 1and R 2parameters.Additional notes:•Solving the previous equation for the change in pore-air pressure, du a , results in the first equation given above.•In a similar manner, the compressibility, m 2a , can be related to the m1acompressibility coefficient by the ratio R a,as shown in the first equation.•The expression for the pore-air pressure change can then be written in the form presented in the second equation, and simplified by using the R 3and R 4parameters to give the form presented in the third equation on the slide.•Therefore, the pore pressure parameters B a (i.e., du a /ds 3) and B w (i.e., du a /ds 3) can be expressed as shown above.Additional notes:。