非饱和土力学讲义(Fredlund在香港科大)chapter2ustmod

非饱和土力学1. 简介非饱和土力学是土力学中的一个重要分支，研究非饱和土的力学性质和行为规律。

非饱和土是指含有一定空隙度和部分或全部未饱和的水分的土壤。

相比于饱和土，非饱和土具有一些特殊的力学性质和变形特性，因此对其力学行为的研究具有重要的实际意义。

2. 非饱和土特性非饱和土的特性主要包括以下几个方面：2.1 吸力吸力是非饱和土中水分存在的特殊状态所引起的一种力。

在非饱和土中，由于存在着未饱和水分，土颗粒表面会形成一种吸附力，即吸力。

吸力的大小与土壤的孔隙结构密切相关。

2.2 干湿收缩性非饱和土在干燥过程中会发生干缩现象，而在被湿润后会发生湿润膨胀。

这是因为非饱和土中的水分含量影响着土颗粒之间的接触状态和土壤体的结构。

2.3 孔隙气压非饱和土中的气体存在一定的孔隙气压，该气压与土壤孔隙水的张力有关。

在非饱和土力学中，孔隙气压的变化对土体的力学行为有重要影响。

3. 非饱和土力学实验为了研究非饱和土的力学性质和行为规律，人们进行了大量的实验研究。

常用的非饱和土力学实验包括以下几种：3.1 吸力试验吸力试验是用来测试非饱和土吸力大小的实验。

在吸力试验中，通常采用吸力仪器对土样进行测量，得到吸力与土壤含水量之间的关系。

3.2 干湿循环试验干湿循环试验是用来模拟非饱和土在干燥和湿润过程中的变形行为的实验。

通过反复进行干燥和湿润过程，可以观察并记录土样的收缩和膨胀行为。

3.3 压缩试验压缩试验是用来研究非饱和土的压缩变形特性的实验。

实验中通常使用压缩装置对土样施加压力，并记录土样的变形和力学参数的变化。

4. 非饱和土的工程应用非饱和土力学的研究对于土木工程的设计和施工具有重要的指导意义。

非饱和土的一些特性和行为规律在以下方面有着广泛的应用：4.1 坡面稳定性分析非饱和土在坡面稳定性分析中发挥着重要作用。

由于非饱和土具有较好的抗侵蚀和抗冲刷能力，因此在坡面设计中通常采用非饱和土力学原理。

4.2 基础工程在基础工程中，非饱和土的力学行为对基底承载力和变形进行了特别的研究。

•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:。

文章编号:1000-582x(2000)S0-0197-04非饱和土的力学理论Ξ陈正汉,王权民,李　刚,孙树国(后勤工程学院土木工程系,重庆　400041) 摘　要:介绍了笔者多年来在非饱和土力学领域研究的主要理论成果。

内容包括非饱和土的建模理论、应力理论、渗流理论、本构理论和固结理论,其中本构理论和固结理论都涉及线性、非线性、弹塑性、结构性、动力特性和热力特性等6个方面。

关键词:非饱和土;力学理论;渗流理论;本构模型;固结模型 中图分类号:TU43;O357.3 文献标识码:A 非饱和土的力学理论包括渗流理论、本构理论、强度理论和固结理论,其中非饱和土的固结涉及水气渗流和土骨架的变形两个方面,是一个综合问题。

非饱和土中渗流包括水流和气流,变形有线性、非线性、弹塑性、结构性、动力特性和热力特性之分,从而组成了非饱和土的本构模型谱系和固结模型谱系。

通过笔者的多年努力,这两个谱系已基本形成,笔者简要介绍有关研究成果。

1　非饱和土建模的公理化理论体系渗流理论、本构模型和固结模型都是数学模型,建模需要理论指南。

1994年,笔者融理性力学、不可逆过程热力学和土力学的精华于一体,建立了岩土力学的公理化理论体系[1]。

该理论体系包括5个基本定律和8个本构原理,对饱和土与非饱和土都适用。

5个基本定律是:质量守恒定律、动量守恒定律、动量矩守恒定律、能量守恒定律和热力学第二定律。

8个本构原理是:等存性原理、相容性原理、客观性原理、Curie对称原理、Onsager原理、压硬剪胀原理、有效应力原理和记忆原理。

2　非饱和土的应力理论[2,3]非饱和土是固-液-气三相复合介质,描述应力状态一般需要三个应力张量。

在不计土粒的压缩性时,则可用两个应力状态变量σij-u aδij和(u a-u w)δij描述,分别称为净总应力张量和吸力张量;也可用下式表达的有效应力刻划之:σ′ij=σij-u aδij+χ(u a-u w)δij称为Bishop公式,其中σij、σ′ij、u a、u w分别是总应力、有效应力、孔隙气压力和孔隙水压力,χ是有效应力参数,与土的饱和度及应力路径有关,且0≤χ≤1。

非饱和土力学
非饱和土力学是研究非饱和土的力学性质和行为的学科。

非饱和土是指土壤中含有水分但不是完全饱和状态的土壤。

在非饱和状态下，土壤的力学性质和行为与饱和状态下有很大的不同。

非饱和土力学主要研究以下几个方面：1.非饱和土的吸力特性：非饱和土中的水分存在于土颗粒之间的微小孔隙中，这些孔隙中的水分会受到吸力的作用。

吸力是非饱和土力学中的一个重要参数，它对土壤的力学性质和行为有很大的影响。

2.非饱和土的渗透特性：非饱和土的渗透特性与饱和状态下有很大的不同。

在非饱和状态下，土壤中的水分会受到吸力的作用，因此渗透速度会比饱和状态下慢很多。

3.非饱和土的力学性质：非饱和土的力学性质与饱和状态下也有很大的不同。

在非饱和状态下，土壤中的水分会受到吸力的作用，因此土壤的强度和变形特性会受到吸力的影响。

4.非饱和土的稳定性：非饱和土的稳定性也是非饱和土力学研究的一个重要方面。

在非饱和状态下，土壤中的水分会受到吸力的作用，因此土壤的稳定性会受到吸力的影响。

总之，非饱和土力学是一个非常重要的学科，它对于土壤工程和地下水工程的设计和施工都有着重要的意义。

非饱和土力学引言非饱和土力学是土力学的一个分支，研究非饱和土壤的力学性质和行为。

饱和土壤是指土壤中的孔隙完全充满水分，而非饱和土壤是指土壤中的孔隙中同时存在气体和水分。

非饱和土力学的研究对于土壤工程、农业、环境科学等领域具有重要的意义。

本文将介绍非饱和土力学的几个重要概念和应用。

概念非饱和土壤的吸力非饱和土壤中存在着气体和水分。

由于毛细现象的存在，非饱和土壤中的水分会受到一定的吸力作用。

吸力是指土壤颗粒表面附近的环境中存在的气体与土壤孔隙中的水分之间的一种力。

吸力是非饱和土力学研究的基础，它与土壤的水分含量、孔隙结构等因素密切相关。

孔隙水压力非饱和土壤中的水分受到吸力作用，会产生一定的压力，称为孔隙水压力。

孔隙水压力是非饱和土力学中的一个重要概念，它描述了土壤中水分的分布和移动情况。

孔隙水压力的变化会影响非饱和土壤的力学性质和行为。

非饱和土壤的力学性质非饱和土力学研究的一个重要目标是揭示非饱和土壤的力学性质。

非饱和土壤的力学性质与饱和土壤有一些明显的差异。

例如，非饱和土壤的抗剪强度和变形特性会受到吸力和孔隙水压力的影响。

非饱和土壤的力学性质的研究对于土壤工程的设计和施工具有重要的指导意义。

应用土壤含水量测定非饱和土力学的研究需要准确测定土壤中的水分含量。

常用的方法有重量法、电容法、压力传感器法等。

这些方法可以测定不同吸力下土壤中的水分含量，从而揭示土壤的吸力特性和水分传输规律。

非饱和土壤的稳定性分析非饱和土壤的力学性质和行为与饱和土壤存在一定的差异。

因此，非饱和土壤的稳定性分析需要考虑吸力和孔隙水压力等因素对土壤的影响。

非饱和土壤的稳定性分析可以用于土壤边坡、挡土墙、基础等工程的设计和施工。

土壤水分调控非饱和土力学的研究成果可以应用于农业、环境科学等领域。

例如，在农业生产中，了解土壤中的水分分布和移动规律，可以合理调控土壤水分，提高植物的生长和产量。

此外，在环境科学研究中，非饱和土力学的研究成果可以用于土壤污染防治和土壤水源涵养等方面。

Professor Lee Barbour•The modeling of unsaturated flow through soils is the most commonly used application of unsaturated soil mechanics.•Emphasis upon the environmental and geo-environmental engineering areas, in addition to classical soil mechanics problems, has resulted in the development of severalcomprehensive software computer codes.•This chapter deals with the flow of the fluid phases (i.e., air and water) through an unsaturated soil.•The analysis of fluid flow through an unsaturated soil requires the definition of the driving potential and a flow law associated to each fluid phase.•The diffusion of air through water is also discussed as a phenomenon associated to unsaturated soils.Additional notes:•The analysis of fluid flow requires a law to relate the flow rate to the driving potential through the use of appropriate coefficients.•Air in an unsaturated soil can occur either in an occluded form or a continuous form depending upon the degree of saturation of the soil.•A knowledge of the driving potentials that causes air and water to flow through an unsaturated soil is necessary for solving geotechnical engineering problems.•The movement of air through the water phase is referred to as air diffusion and its driving potential needs to be defined.•Water flow is caused by a hydraulic head gradient, where the hydraulic head consists of an elevation head plus a pressure head.•The flow of air in the continuous phase form is governed by a concentration or pressure gradient. The elevation gradient has a negligible effect.Additional notes:•What is the driving potential for the flow of water through an unsaturated soil?•Several concepts, such as water content gradient, pressure head gradient, soil suction gradients, and hydraulic head gradient have been used to explain the flow of water through an unsaturated soil.•The above figure illustrates some hypothetical cases which demonstrate that matric suction gradient is not the driving potential that causes water to flow through an unsaturated soil.•Case 1 illustrates that water can flow through an unsaturated soil from a region of low matric suction to another region at a relatively high matric suction.•Case 2 illustrates that the flow of water through an unsaturated soil can occur between regions with equal matric suctions.•Case 3 illustrates that water can flow through an unsaturated soil even when the left-hand side (i.e., departing region) is at a matric suction higher than the right-hand side (i.e., target region).•Water flow can be defined most appropriately in terms of a hydraulic head gradient. •The special case where the air pressure is zero, which is the common situation in nature, the matric suction gradient is numerically equal to the pressure gradient in the water.•The elevation head can be omitted then the matric suction gradient is equal to the hydraulic head gradient and all points in the soil mass are at the same elevation (i.e., horizontal flow).Additional notes:No suctiongradient•A point (e.g., an elemental volume) in the water phase has three primary components of energy; namely, gravitational, pressure, and velocity.•Point A has a gravitational potential energy, Eg , which is a function of the water element mass and the distance, y , relative to a datum.•When the water density, ρw , is constant, the pressure energy can be expressed as a function of the pore-water pressure, the water volume, and the water density at point A .•The flow rate of water at point A gives rise to a velocity energy that can be expressed as a function of the mass of water and the velocity of water at point A .•The total energy (i.e., mechanical energy) at point A is the summation of thegravitational, pressure, and velocity components.•Potential or hydraulic head is defined as energy per unit of weight. Therefore, the hydraulic head consists of three components; namely, the gravitational head, the pressure head and the velocity head.•The velocity head in a soil is commonly negligible in comparison to the gravitational and the pressure heads.Additional notes:•The heads used in the definition of hydraulic potential have the dimensions of length. •Hydraulic head is a measurable quantity, the gradient of which causes flow in saturated and unsaturated soils.•Piezometers and tensiometers can be used to measure the in situ pore-water pressure at any point in a soil mass.•A piezometer can be used to measure positive pore-water pressures at a point in a soil mass.•A tensiometer can be used to measure negative pore-water pressures at a point in a soil mass.•In the above figure, point A has a higher total head than point B [i.e., hw (A) > h w (B)] and therefore water will flow from point A to point B due to the hydraulic head gradient.•The driving potential causing flow in the water phase through a soil mass has the same form for both saturated (i.e., point B ) and unsaturated (i.e., point A ) soils (Freeze & Cherry, 1979).Additional notes:•Darcy’s law describes the flow of water through an unsaturated soil. However, the coefficient of permeability in an unsaturated soil need not be a constant.•The coefficient of permeability in an unsaturated soil is a variable that is predominantly a function of the water content or the matric suction in the unsaturated soil.•Water can be visualized as flowing only through the pore space filled with water. The air filled pore spaces are nonconductive channels to the flow of water.•To verify Darcy’s law for unsaturated soils, experiments were conducted using acolumn of unsaturated soil, with a uniform water content and a constant water pressure head subjected to various gradients of gravitational heads.•The results indicated that at a specific water content, the coefficient of permeability, k w , is constant for various hydraulic head gradients applied to the unsaturated soil.•The rate of water flow through an unsaturated soil is linearly proportional to the hydraulic head gradient, with the coefficient of permeability being a constant, similar to the situation for a saturated soil.•Therefore, Darcy’s law can also be applied to unsaturated soils.However, themagnitude of the coefficient of permeability is a function of volumetric water content, θw .Additional notes:•The coefficient of permeability depends upon the properties of the fluid and the properties of the porous medium. Different types of fluids (e.g., water and oil) ordifferent types of soil (e.g., sand and clay) produce different values for the coefficient of permeability, k w .•The intrinsic permeability of a soil, K , represents the characteristics of the porous medium and is independent of the fluid properties.•In geotechnical engineering, the term coefficient of permeability, kw , is a commonly used term.•In an unsaturated soil, the coefficient of permeability, k w , is affected by changes in void ratio and more significantly by the degree of saturation (or water content).•As a soil becomes unsaturated, air first replaces some of the water in the large pores, and this causes water to flow through the smaller pores, with an increased tortuosity to the flow path.•Additional increases in matric suction leads to further decreases in the the pore volume occupied by water, and the air-water interface is drawn closer and closer to the soil particles.•As a result, the coefficient of permeability with respect to the water phase decreases rapidly as the space available for water flow reduces.Additional notes:•Change in void ratio in an unsaturated soil may be small. However, the effect of a change in degree of saturation may be highly significant.•As a result, the coefficient of permeability is often described as a singular function of the degree of saturation, S,or the volumetric water content, θw.•A change in matric suction can produce a more significant change in the degree of saturation than can be produced by a change in net normal stress.•Degree of saturation has been commonly described as a function of matric suction. •Numerous semi-empirical equations for the coefficient of permeability have been derived using the matric suction versus versus degree of saturation curve.•In these equations, the pore-size distribution forms the basis for predicting the coefficient of permeability.•The residual degree of saturation,Sr , is defined as the degree of saturation at whichan increase in matric suction does not produce a significant change in the degree of saturation.Additional notes:•The matric suction versus degree of saturation curve exhibits hysteresis and thedrainage curve is most commonly used. In addition, the soil structure is assumed to be incompressible.•The air entry value of the soil, (u a -u w )b , residual degree of saturation, S r , and the pore size distribution index, λ, are three parameters commonly used when deriving equations for predicting the coefficient of permeability, k w .•Using the concept of effective degree of saturation, S e , the above figure illustrates a procedure for determining the air entry value, the residual degree of saturation, and the pore size distribution index.•The effective degree of saturation can be computed by first estimating the residual degree of saturation.•Plotting the effective degree of saturation versus matric suction and adjusting a horizontal and a sloping line to the calculated data permit the use of a new assumed residual degree of saturation.•The procedure is repeated until all points on the sloping line constitute a straight line. The pore size distribution index, λ,is defined as the negative slope of the effective degree of saturation, S e , versus the matric suction, (u a -u w ), curve.•The intersection point between the straight sloping line and the saturation ordinate (i.e., S e = 1.0) defines the air entry value of the soil.Additional notes:•The above figure shows an equation for effective degree of saturation as a function of the matric suction for values of matric suction greater than the air entry value of the soil.•The figure also illustrates typical matric suction versus degree of saturation curves for various soils.Additional notes:•The above figure illustrates typical λvalues for various soils that have been obtained from matric suction versus degree of saturation curves.•Soils with a wide range of pore sizes have a small value for λ. The more uniform the distribution of pore sizes in a soil, the larger the value for λ.Additional notes:•Brooks and Corey (1964) proposed an equation for the coefficient of permeability, k w , by using the matric suction versus the degree of saturation curve and the concepts of effective degree of saturation and pore-size distribution index , λ.•The coefficient of permeability, kw , for a sandstone is expressed in terms of the relative permeability as compared to the coefficient of permeability of the soil at saturated conditions, k s .•The figure also presents the air permeability of the soil to illustrate its decreasing value as the soil approaches a degree of saturation equal to 100%.Additional notes:•The above table presents several empirical constant, δvalues (i.e., Brooks and Corey, 1964 parameter) and their corresponding pore size distribution indices,λ, for various soil types.(Note that δ= (2+3λ) / λ)•The above values can be used in practice as a first estimation for different soils.Additional notes:•The above figure presents the equation proposed by Gardner (1958) for predicting the coefficient of permeability for an unsaturated soil as function of matric suction.•The equation provides as flexible permeability function that is defined by two constants (a and n ) where “a”is related to the breaking point (i.e., air entry value) and “n ” is defined by the slope of the function.•Four typical functions, with differing values of “a”and “n”illustrates the forms produced by Gardner’s (1958) equation.Additional notes:Gardner’s equation(1958)•The above table summarizes several of the proposed relationships between the coefficient of permeability and matric suction.•The equations illustrate the relationship between coefficient of permeability and matric suction by using soil constants that can be evaluated from the matric suction versus degree of saturation curves.Additional notes:•A coefficient of permeability function, k w (θw ), was proposed using the configurations of the pore space filled with water (Childs and Collis-George, 1950).•The soil-water characteristic curve can also be presented in terms of volumetric water content, θw , versus the matric suction. Therefore, k w can also be expressed directly as a function of matric suction.•The permeability function, k w (θw ), is derived based on Poiseuille’s equation.•The above permeability function has a similar form to the function presented by Kunze et al., (1968) and is slightly modified to use SI units and matric suction is used in the equation instead of pore-water pressure head.•The calculation of the permeability function, k w (θw ),at a specific volumetric water content involves the summation of the matric suction values that correspond to the volumetric water content at and below the value under consideration, (θw )i .Additional notes:Childs andCollis-George(1950)•The calculation of the permeability function, k w (θw ),at a specific volumetric water content involves the summation of the matric suction values that correspond to the volumetric water content at and below the water content under consideration, (θw )i .•A matching factor, (k s /k sc ), based on the saturated coefficient of permeability is necessary in order to provide a more accurate representation of the unsaturated permeability function.•The shape of the permeability function is determined by the terms inside thesummation sign portion of the equation as obtained from the soil-water characteristic curve.•The magnitude of the permeability function needs to be adjusted with reference to the measured saturated coefficient of permeability, k s , by using the matching factor.•If the saturated coefficient of permeability is measured, the permeability function can be predicted directly from the soil-water characteristic curve because all of the terms in front of the summation sign in the equation can be considered as an adjusting factor.•Therefore, the permeability function, k w (θw ), can be written as shown at the bottom of the slide.Additional notes:Childs andCollis-George(1950)•The soil-water characteristic curve can be visualized as an indication of theconfiguration of water-filled pores in the soil mass.•The coefficient of permeability is obtained by dividing the soil-water characteristic curve into “m” equal intervals along the volumetric water content axis.•The matric suction corresponding to the midpoint of each interval is used to calculate the coefficient of permeability.•This computational procedure for obtaining the permeability function appears to be most successful for sandy soils having a narrow pore size distribution (Nielsen et al., 1972).•The soil-water characteristic curve and the prediction of the permeability function for a fine sand (Lakeland sand) are shown on the next slides.Additional notes:•The soil-water characteristic curve for the Lakeland fine sand is shown above. •The figure shows a low air entry value for the soil specimens tested [i.e., (u a -u w )b around 3 kPa] and an estimated residual degree of saturation around 10 kPa.Additional notes:•A comparison between the permeability function, kw (θw ),computed from theproposed equation and experimental data is shown for a fine sand.•There is a satisfactory match between the measured and calculated values.Additional notes:•The coefficient of permeability, k w , is generally assumed to be uniquely related to the degree of saturation, S , or the volumetric water content, θw .•This assumption is reasonable since water flows through the pore-water phase.•The degree of saturation or volumetric water content shows significant hysteresis when plotted versus matric suction.Additional notes:•The relationships between the water content (i.e., S or θw ) and thecoefficient of permeability appear to exhibit little hysteresis (Nielsen and Biggar, 1961).•The permeability function, k w (or θw ), for coarse-textured soils isapproximately the same for both wetting and drying (Nielsen et al., 1972.)Additional notes:No hysteresis•Since there is hysteresis in the relationship between water content and matricsuction, there is also hysteresis in the relationship between the coefficient ofpermeability and matric suction.•The hysteresis in the relationship between the coefficient of permeability andmatric suction depends upon the distribution of pores in the soil.Additional notes:•Air in an unsaturated soil can be present in two forms; namely, a continuous form or as occluded air bubbles.•Air phase is generally continuous when the degree of saturation is below 85%.•Air phase becomes occluded bubbles as the degree of saturation is higher than 90% and the air flow is reduced to a process of diffusion through the pore-water.•Concentration or pressure gradient can be used as the driving potential for the flow of the continuous air phase through an unsaturated soil.•Fick’s law is often used to describe the diffusion of gases through liquids.•A modified Fick’s law, as shown above, relates the mass rate of flow, Ja , to theconcentration gradient, (dC /dy ), by using the transmission constant for air flow through a soil, D a .•The negative sign in the relationship indicates that air flows in the direction of a decreasing concentration gradient.•The concentration of the air, C , in a soil can be expressed as a function of the air pressure by using the air density which is related to the absolute air pressure in accordance with the gas law.Additional notes:•The modified Fick’s law, as shown above, can be used to express the mass rate of flow, Ja , to the pore-air pressure gradient, (d ua /d y ), by using a modified coefficient oftransmission for air flow through soils,D a *.•It can be observed that D a *is a function of the volume-mass properties of the soil.•This modified form of Fick’s law has been used in geotechnical engineering to describe air flow through soils (Blight, 1971).•Air flow, under steady-state conditions, can be measured through a soil mass. At the exit point, the mass rate of the air flow is measured at a constant air density, ρma .•This particular steady-steady condition permits the establishment of a relationship between the flow rate, v a , and the air pressure gradient, d u a /d y .Additional notes:•The pore-air pressure, u a , can be expressed in terms of pore-air pressure head, h a , using a constant air density (i.e., by neglecting the elevation head gradient).•Therefore, the equation for air flow through an unsaturated soil mass has the same form as Darcy’s equation for air phase, when the term k a is used for D a * g , and the hydraulic head gradient consists of the pore-air pressure head gradient as the driving potential.•The coefficient of permeability for the air phase is a function of the fluid (i.e., air) and the soil volume-mass properties.•Air flows through the air phase and as the matric suction increases, or the degree of saturation increases and the air coefficient of permeability increases.•Relationships between air coefficient of permeability and matric suction or degree ofsaturation reflect the pore size distribution and the matric suction versus degree of saturation curve of the soil.•The relationship at the bottom is obtained when the effective degree of saturation, S e , is expressed in terms of matric suction.•The air coefficient of permeability, k a , is essentially the inverse of the water coefficient of permeability function, k w .(Brooks and Corey, 1964)Additional notes:•The relationships between ka and degree of saturation or matric suction are also validfor the air coefficient of transmissivity, Da *, since the two coefficients are related bythe constant, “g”.•Experimental verifications have been performed for Fick’s and Da1971) and some results are presented.•The tests were performed by establishing steady-state air flows through dry soils.The soils were assumed to have a rigid structure because no measurable volume change occurred during the tests.•In order to use the bulk soil as a reference, the mass rate of air flow must be multiplied by the air porosity, na.•The above figure illustrates the applicability of Fick’s law to air flow. For a small change in the pore-air pressure gradient, the mass flow rate,J, is almost linearlyproportional to the pore-air pressure gradient, with Da * being the coefficient ofproportionality.•The air pressure gradients used in the experiments were high, and the pressures applied could affect the soil structure (i.e., altering the pore-soil distribution) and themagnitudes for both the Da * and kacoefficients.Additional notes:Fick’s Law•The above figure illustrates the applicability of Darcy’s law to air flow. For a smallchange in the pore-air head gradient, the flow rate of air, va , is almost linearlyproportional to the pore-air head gradient, with ka being the coefficient ofproportionality.•The above results were calculated using a constant air density, ρa , at a pressure of 101.3kPa and a temperature of 20 o C (i.e., 1.20 kg/m3). Additional notes:Darcy’s Law•The above figure illustrates the agreement between measured data and the theoretical air coefficient of permeability function by using the air permeability equation defined with effective degree of saturation expressed in terms of matric suction.•The results are presented in terms of relative permeability with the maximum air coefficient of permeability corresponding to the dry soil conditions.Additional notes:•The above figure presents air and water coefficients of permeability for a soilcompacted at different water contents or matric suction values.•The air coefficient of permeability, ka , decreases as the soil water content ordegree of saturation decreases (Ladd, 1960; Barden and Pavlakis,1971).•The air and the water coefficients of permeability were measured on the same soil specimens under steady-state flow conditions induced by small pressuregradients applied to each phase.•The air coefficient of permeability , ka , decreases rapidly as optimum watercontent is approached. At this point, the air phase becomes occluded and the flow of air takes place as diffusion of air through water.•The coefficient of permeability decreases and the water coefficient ofpermeability increases with increasing water content, however, the aircoefficients of permeability remain significantly greater than the watercoefficients of permeability.•The difference in air and water viscosities (i.e., water viscosity is about 56 times the air viscosity at atmospheric conditions) is one of the reasons for the aircoefficient of permeability being greater than the water coefficient ofpermeability.Additional notes:•The diffusion process occurs in response to a concentration gradient. Ionic or molecular movement take place from regions of higher concentration to regions of lower concentration.•There are two common diffusion mechanisms of interest in unsaturated soil mechanics, the first is the flow of air through the pore-water in a saturated or unsaturated soil, the second process is related to air flow through the water in a high air entry ceramic disk. •Fick’s law, which is shown above, is often used to describe the first diffusion process. The concentration gradient provides the driving potential for the diffusion process and is expressed with respect to the soil voids (i.e., air and water phases).•Fick’s law equation can also be written in terms of the partial pressure of a constituent gas diffusing through the pore-water in a soil, by expressing the concentration of the gas constituents in terms of partial pressures.Additional notes:•The mass rate of the gas constituents diffusing across a unit area of the soil voids (i.e., J a ) can be determined by measuring the volume of the diffused constituents under constant pressure conditions.•The ideal gas law is applied to the diffusing constituents.•A change in concentration of the diffusing constituent relative to a change in partial pressure is obtained by using the change in density of the dissolved constituent in the pore-water.•Using the ideal gas law yields the equation at the bottom of the slide which expresses the change in concentration of the diffusing gas in terms of the partial pressure of the diffusing constituent and the volume of the dissolved constituent, V d i , through the pore-water.Additional notes:•The ratio of the volume of dissolved constituent to the volume of water, (i.e., V di /V w ) is referred to as the volumetric coefficient of solubility, h .•The flow rate of the diffusing constituent across a unit area of the soil voids, v f i , can beexpressed as a function of the partial pressure gradient, (i.e.,in this case in the y-direction) by combining the previous equations.•The equation for the diffusing constituent, vf i , can be applied to air or gas diffusion through the pore-water in a soil or through free water or some other material such as a rubber membrane (Poulos, 1964).•The equation for the diffusing constituent, v fi , can also be developed in terms of the partial pressure head, h fi , with respect to the constituent density ρfi , which corresponds to the absolute constant pressure,u fi , used in the measurement of the diffusing constituent volume.•By defining the diffusion coefficient of permeability for air through an unsaturated soil with occluded air bubbles, the equation for the flow rate of the diffusing gas can be expressed as a modified form of Darcy’s law, as shown at the bottom of the slide.Additional notes:。