非饱和土力学讲义(Fredlund在香港科大)chapter8ustmod.pdf

1 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 2 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 3 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 Va is a function of the applied absolute air pressure Differentiating the volume of air Va 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 4 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 5 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 6 The compressibility of an air water mixture can be derived using direct proportioning 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 Va Vw The dissolved air Vd is within the volume of water The pore air and pore water pressures are uaand uw with ua uw The soil is subjected to a compressive total stress Applying an infinitesimal increase in total stress d 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 7 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 duw was used as the reference pressure in the 1976 compressibility equation The term d Vw Vd d is considered to be equal to dVwsince the dissolved air is a fixed volume internal to the water The total volume of water Vw is therefore used in computing the compressibility of water i e Cw 1 Vw dVw duw The total air volume change can be obtained directly using Boyle s law by considering 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 8 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 Ca and water compressibility Cw to yields the second equation shown above The isothermal compressibility of air Ca 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 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 Baand Bw are equal to 1 0 In the presence of soil solids the surface tension effects and the compressibility of air will cause the Baand Bwvalues to be less than 1 0 Additional notes 9 The first term in the compressibility equation accounts for the compressibility of the water portion of the mixture while the second term accounts for the compressibility of the air portion The contribution of each compressibility component to the overall compressibility of the air water mixture is illustrated for various degrees of saturation The case considered has an initial absolute air pressure of 202 6 kPa i e 2 atm Values of Baand Bware assumed to be equal to 1 0 for all degree of saturation This assumption may be unrealistic for low degrees of saturation however it simplifies the comparison of the compressibility components The compressibility of an air water mixture is predominantly influenced by the compressibility of the free air portion When the degree of saturation is equal to zero the compressibility of the pore fluid is equal to the isothermal compressibility of air i e 4 94 x 10 3 1 kPa When the degree of saturation becomes equal to 1 0 the pore fluid compressibility becomes equal to that of water i e 4 58 x 10 7 1 kPa Additional notes 10 The solution of air in water gives the effect that the water is compressible The compressibility due to the solution of air in water is approximately two orders of magnitude greater than the compressibility of the water as saturation is approached Air dissolving in water significantly affects the compressibility of an air water mixture when the free air volume becomes less than approximately 20 of the volume of voids The effect of air solubility on the compressibility of an air water mixture is shown for several initial absolute air pressures The Baand Bwparameters are assumed to be equal to 0 8 and 0 9 respectively The effect of air solubility on the compressibility of an air water mixture is the same i e on a logarithmic scale for any initial air pressures However the effect of air dissolving in water does not result in a smooth transition in compressibility of an air water mixture as saturation is approached Consequently the second term in the compressibility of air water mixture must be dropped and the compressibility abruptly decreases to the compressibility of water If the free air does not have time to dissolve in water the solubility term must be set to zero and the compressibility of pore fluid has a smooth transition back to saturation Additional notes 11 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 and setting the Baand 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 Ca as expressed by the third equation above which gives the average air compressibility uao ua2 during an air absolute pressure change from uaoto ua Replacing the air compressibility term i e 1 ua with the average air compressibility term i e uao ua2 yields the air water compressibility equation proposed 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 12 Kelvin s equation i e ua uw 2Ts Rs 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 a degree of saturation greater than 85 Kelvin s equation results in the incorporation of the radius curvature Rs as a variable However Rs 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 a macroscopic 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 together Henry s laws to the compressibility of an air water mixture is that an anomaly arises from a theoretical point of view Such a formulation predicts that an increase in matric suction occurs as the total stress is increased under undrained loading Fredlund and Rahardjo 1993 Additional notes 13 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 1 decreases due to an increase in the pore air pressure from ua1to ua2 the controlling radius may 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 14 The pore pressure response for a change in total stress during undrained compression has been expressed in terms of pore pressure parameters i e Baand Bw 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 Ba is defined as the ratio between the increase in pore air pressure i e response and the increase in isotropic pressure i e 3 from the initial condition Additional notes 15 The secant type pore water pressure parameter i e Bw is defined as the ratio of the increase in pore water pressure i e response to the increase in isotropic pressure i e 3 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 16 If an infinitesimal increase in the isotropic confining pressure is considered at a point along the pore air pressure versus isotropic confining stress 3 relationship the pore air pressure response at that point can be expressed as the tangent Ba pore air pressure parameter Similarly a tangent pore water pressure parameter Bw 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 17 The above figure illustrates the development of pore air and pore water pressures during isotropic undrained compression The pore water pressure increases faster than the pore air pressure in response to an increase in total confining stress At a point during the loading of the soil e g Point 1 the tangent and the secant pore pressure parameters can be defined for both the air and water phases The concept of a tangent B pore pressure parameter has been used in previous sections where an infinitesimal increase in total increase is considered Secant B pore pressure parameters are particularly useful in computing the final pore air and pore water pressures following a large change in total stress As saturation is approached the pore water pressure approaches the pore air pressure i e uwapproaches uaor Bwapproaches Ba and the air bubbles dissolve in the water At saturation Ba Bw 1 0 and a change in total stress is equal to a change in pore water pressure Additional notes 18 The theoretical formulation of pore pressure parameters requires volume change constitutive relationships for an unsaturated soil These relationships describe the volume change that takes place under drained loading The volume changes later explained in Chapter 12 are expressed in terms of the stress state variables The above figure illustrates the nature of the constitutive surfaces for an unsaturated soil specimen that undergoes one dimensional drained compression The stress state variables ua and ua uw change as the soil is compressed Volume change is primarily the result of compression of the pore fluid since the soil solids are essentially incompressible The three dimensional surfaces presented above are the idealized and linearized constitutive surfaces for the soil structure air and water phases of an unsaturated soil specimen under one dimensional drained compression Additional notes 19 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 m1sand m2s correspond to changes in the stress state variables ua and ua uw respectively Total volume changes can then be predicted by using the constitutive surfaces when changes in the stress state variables are known Additional notes 20 The linear equation for pore air volume changes can be written as proposed by Fredlund and Morgenstern 1976 The compressibility parameters m1aand m2a correspond to changes in the stress state variables ua and ua uw respectively Air volume changes can be predicted using the constitutive surfaces when changes in the stress state variables are known Additional notes 21 A linear equation for pore water volume changes can be written as proposed by Fredlund and Morgenstern 1976 The compressibility parameters m1wand m2w correspond to changes in the stress state variables ua and ua uw 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 form relationships between the compressibility Additional notes 22 The application of an all around positive i e compressive tot