CONSOLIDATION HEORY OF UNSATURATED SOIL BASED ON THE THEORY OF MIXTURE(Ⅰ)

Unsaturated soil is a three-phase media and is composed of soil grain, water and gas. In this paper, the consolidation problem of unsaturated soil is investigated based on the theory of mixture. A theoretical formula of effective stress on anisotropic porous media and unsaturated soil is derived. The principle of effective stress and the principle of Curie symmetry are taken as two fundamental constitutive principles of unsaturated soil. A mathematical model of consolidation of unsaturated soil is proposed, which consists of 25 partial differenfial equations with 25 unknowns. With the help of increament linearizing method, the model is reduced to 5 governing equations with 5 unknowns, i.e., the three displacement components of solid phase, the pore water pressure and the pore gas pressure. 7 material parameters are involved in the model and all of them can he measured using soil tests. It is convenient to use the model to engineering practice. The well known Biot’s theory is a special case of the model.     

CONSOLIDATION HEORY OF UNSATURATED SOIL BASED ON THE THEORY OF MIXTURE(Ⅰ)

Unsaturated soil is a three-phase media and is composed of soil grain,water andgas.In this paper,the consolidation problem of unsaturated soil is investigated basedon the theory of mixture.A theoretical formula of effective stress on anisotropicporous media and unsaturated soil is derived.The principle of effective stress and theprinciple of Curie symmetry are taken as two fundamental constitutive principles ofunsaturated soil.A mathematical model of consolidation of unsaturated soil isproposed,which consists of25 partial differenfial equations with25 unknowns.Withthe help of increament linearizing method,the model is reduced to5 governingequations with5 unknowns,i.e.,the three displacement components of solid phase,thepore water pressure and the pore gas pressure.7 material parameters are involved inthe model and all of them can be measured using soil tests.It is convenient to use themodel to engineering practice.The well known Biot’s theory is a special case of themodel.     

A micropolar mixture theory of multi-component porous media

A mixture theory is developed for multi-component micropolar porous media with a combination of the hybrid mixture theory and the micropolar continuum theory. The system is modeled as multi-component micropolar elastic solids saturated with multi- component micropolar viscous fluids. Balance equations are given through the mixture theory. Constitutive equations are developed based on the second law of thermodynamics and constitutive assumptions. Taking account of compressibility of solid phases, the volume fraction of fluid as an independent state variable is introduced in the free energy function, and the dynamic compatibility condition is obtained to restrict the change of pressure difference on the solid-fluid interface. The constructed constitutive equations are used to close the field equations. The linear field equations are obtained using a linearization procedure, and the micropolar thermo-hydro-mechanical component transport model is established. This model can be applied to practical problems, such as contaminant, drug, and pesticide transport. When the proposed model is supposed to be porous media, and both fluid and solid are single-component, it will almost agree with Eringen＇s model.     

THE APPLICATION OF THE FINITE ELEMENT METHOD TO SOLVING BIOT’S CONSOLIDATION EQUATION

Biot’s theory of consolidation of saturated soil regards the con-solidation process as a coupling problem between stress of elas-tic body and flow of fluid existing in pores.It can moreprecisely reflect the mechanism of consolidation than Terzhi-gi’s theory.In this article,we obtain the general Biot’sfinite element equations of consolidation with classical varia-tional principles.The equations have clear physical meaningand have been applied to analysing the consolidation of Bajia-zui earth dam.The computational results are in accord withengineering practice.     

VERTICAL VIBRATION ANALYSIS OF AXISYMMETRIC SATURATED SOIL

Based on Biot’s dynamic consolidation equations,by means of Laplace-Hankel transform technology,the integral solutions of stress and displacement in saturated soil with subjacent rock-stratum under axisymmetric arbitrary excitations were derived.In- fluence of the reflected wave generated by the boundary was revealed.Numerical results indicate that the vibration frequency has some effect on the vertical displacement of satu- rated soil.The vertical displacement at the surface of saturated soil lags in phase with the load.Furthermore,the dynamic permeability coefficient of saturated soil has significant effect on the vertical displacement at the initial stage of load applied,but when the load becomes stable,the effect is inapparent.     

Lamb＇s integral formulas of two-phase saturated medium for soil dynamic with drainage

When dynamic force is applied to a saturated porous soil, drainage is common. In this paper, the saturated porous soil with a two-phase saturated medium is simulated, and Lamb＇s integral formulas with drainage and stress formulas for a two-phase saturated medium are given based on Biot＇s equation and Betti＇s theorem （the reciprocal theorem）. According to the basic solution to Biot＇s equation, Green＇s function Gij and three terms of Green＇s function G4i, Gi4, and G44 of a two-phase saturated medium subject to a concentrated force on a spherical coordinate are presented. The displacement field with drainage, the magnitude of drainage, and the pore pressure of the center explosion source are obtained in computation. The results of the classical Sharpe＇s solutions and the solutions of the two-phase saturated medium that decays to a single-phase medium are compared. Good agreement is observed.     

AN ANALYTICAL SOLUTION FOR THREE-DIMENSIONAL DIFFRACTION OF PLANE P-WAVES BY A HEMISPHERICAL ALLUVIAL VALLEY WITH SATURATED SOIL DEPOSITS

An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot's dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot's dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.     

Exact solutions to one-dimensional transient response of incompressible fluid-saturated single-layer porous media

Based on the Biot theory of porous media,the exact solutions to onedimensional transient response of incompressible saturated single-layer porous media under four types of boundary conditions are developed.In the procedure,a relation between the solid displacement u and the relative displacement w is derived,and the well-posed initial conditions and boundary conditions are proposed.The derivation of the solution for one type of boundary condition is then illustrated in detail.The exact solutions for the other three types of boundary conditions are given directly.The propagation of the compressional wave is investigated through numerical examples.It is verified that only one type of compressional wave exists in the incompressible saturated porous media.     

WAVE PROPAGATION WITH MASS-COUPLING EFFECT IN FLUID-SATURATED POROUS MEDIA

This article utilizes the theory of mixtures to formulate a general theory of wavepropagation with mass-coupling effect in fluid-saturated porous media.An attempt is madeto discuss the physical interpretation and the thermodynamic restriction of the coefficientsappearing in the equations obtained.By the comparison it is shown that Biot’s classicaltheory and the present one are essentially consistent.Also,wave velocities in some specialcases are calculated,from which it is concluded that mass-coupling and permeability ofmedia greatly affect wave propagation behavior.     

Characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media

According to generalized characteristic theory, a characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media was performed. The characteristic differential equations and compatibility relations along bicharacteristics were deduced and the analytical expressions for wave surfaces were obtained. The characteristic and shapes of the velocity surfaces and wave surfaces in the transversely isotropic fluid-saturated porous media were discussed in detail. The results also show that the characteristic equations for stress waves in pure solids are particular cases of the characteristic equations for fluid-saturated porous media.     

GIBBS-APPELL’S EQUATIONS OF VARIABLE MASS NONLINEAR NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Gibbs-Appell’s equations of motion are extended to the most generalvariable mass nonholonomic mechanical systems.Then the Gibbs-Appell’s equations ofmotion in terms of generalized coordinates or quasi-coordinates and an integral variationalprinciple of variable mass nonlinear nonholonomic mechanical systems are obtained.Finally,an example is given.     

DOUBLE-MEDIUM CONSTITUTIVE MODEL OF GEOLOGICAL MATERIAL IN UNIAXIAL TENSION AND COMPRESSION

Based on elasto-plasticity and damage mechanics, a double-medium constitutive model of geological material under uniaxial tension and compression was presented, on the assumption that rock and soil materials are the pore-fracture double-medium, and porous medium has no damage occurring, while fracture medium has damage occurring with load. To the implicit equation of the model, iterative method was adopted to obtain the complete stress-strain curve of the material. The result shows that many different distributions (uniform distribution, concentrated distribution and random distribution) of fractures in rock and soil material are the essential reasons of the daedal constitutive relations. By the reason that the double-medium constitutive model separates the material to be porous medium part, which is the main body of elasticity, and fracture medium part, which is the main body of damage, it is of important practical values and theoretical meanings to the study on failure of rock and soil or materials containing damage.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

KANE’S EQUATIONS FOR PERCUSSION MOTION OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEMS

In this paper,the Kane’s equations for the Routh’s form of variable massnonholonomic systems are established.and the Kane’s equations for percussion motionof variable mass holonomic and nonholonomic systems are deduced from them. Secondly,the equivalence to Lagrange’s equations for percussion motion and Kane’sequations is obtained,and the application of the new equation is illustrated by anexample.     

Dynamics of a spacecraft with large flexible appendage constrained by multi-strut passive damper

This paper is concerned with the dynamics of a spacecraft with multi-strut passive damper for large flexible appendage.The damper platform is connected to the spacecraft by a spheric hinge,multiple damping struts and a rigid strut.The damping struts provide damping forces while the rigid strut produces a motion constraint of the multibody system.The exact nonlinear dynamical equations in reducedorder form are firstly derived by using Kane’s equation in matrix form.Based on the assumptions of small velocity and small displacement,the nonlinear equations are reduced to a set of linear second-order differential equations in terms of independent generalized displacements with constant stiffness matrix and damping matrix related to the damping strut parameters.Numerical simulation results demonstrate the damping effectiveness of the damper for both the motion of the spacecraft and the vibration of the flexible appendage,and verify the accuracy of the linear equations against the exact nonlinear ones.     

THERMAL CONSOLIDATION OF LAYERED POROUS HALF-SPACE TO VARIABLE THERMAL LOADING

An analytical method was derived for the thermal consolidation of layered, saturated porous half-space to variable thermal loading with time. In the coupled governing equations of linear thermoelastic media, the influences of thermo-osmosis effect and thermal filtration effect were introduced. Solutions in Laplace transform space were first obtained and then numerically inverted. The responses of a double-layered porous space subjected to exponential decaying thermal loading were studied. The influences of the differences between the properties of the two layers (e.g., the coefficient of thermal consolidation, elastic modulus) on thermal consolidation were discussed. The studies show that the coupling effects of displacement and stress fields on temperature field can be completely neglected, however, the thermo-osmosis effect has an obvious influence on thermal responses.     

Structural bonding-breakage constitutive model for natural unsaturated clayey soils

The natural clayey soils are usually structural and unsaturated,which makes their mechanical properties quite different from the remolded saturated soils.A structural constitutive model is proposed to simulate the bonding-breakage micro-mechanism.In this model,the unsaturated soil element is divided into a cementation element and a friction element according to the binary medium theory,and the stress-strain coordination for these two elements is obtained. The cementation element is regarded as elastic,whereas the friction element is regarded as elastoplastic which can be described with the Gallipoli＇s model.The theoretical formulation is verified with the comparative experiments of isotropic compressions on the saturated and unsaturated structural soils.Parametric analyses of the effects of damage variables on the model predictions are further carried out,which show that breakage deformation of natural clayey soils increases with the rising amount of initial defects.     

THE COMPLEMENTARY PROPERTY OF LINDELF’S WORK AND CHAPLYGIN’S WORK

Lindelf’s equation is derived by using the Vakonomic model,which shows that Lindelf’s work coincides with Vakonomic model. Chaplygin’s equation is derived by using Chetaev’s model, which shows that Chaplygin’s work coincides with Chetaev’s model. On basis of these, by improving the expressions of Chaplygin’s equation and Lindelf’s equation, the reasonable transition from Chaplygin’s equation to Lindelf’s equation is realized, the reasonable transition from Lindelf’s equation to Chaplygin’s equation is realized too. Finally, a typical example is given. The work of this paper shows that, just as the Vakonomic model and Chetaev’s model are complementary to each other, Lindelf’s work and Chaplygin’s work are complementary to each other too.     

Axisymmetrical analytical solution for vertical vibration of end-bearing pile in saturated viscoelastic soil layer

Based on elasticity and the theory of saturated porous media, and regarding the pile and the soil as a single phase elastic and a saturated viscoelastic media, respectively, the dynamical behavior of vertical vibration of an end-bearing pile in a saturated viscoelastic soil layer is investigated in the frequency domain using the Helmholtz decomposition and variable separation method. The axisymmetrical analytical solutions for vertical vibrations of the pile in a saturated viscoelastic soil layer are obtained, and the analytical expression of the dynamical complex stiffness of the pile top is presented. Responses of dynamic stiffness factor and equivalent damping of pile top with respect to the frequency are shown in figures using a numerical method. Effects of the saturated soil parameters, modulus ratio of the pile to soil, slenderness ratio of pile and pile＇s Poisson ratio, etc. on the stiffness factor and damping are examined. It is shown that, due to the effect of the transversal deformation of the pile and the radial force of the saturated viscoelastic soil acting on the pile, the dynamic stiffness factor and the damping derived from the axisymmetrical solution are greatly different from those derived from the classical Euler-Bernoulli rod model, especially at some specific excitation frequencies. Therefore, there are limitations on applicability of the Euler-Bernoulli rod model in analyzing verticai vibration of the pile. More accurate analysis should be based on a three-dimensional model.     

Formulation of thermo-hydro-mechanical coupling behavior of unsaturated soils based on hybrid mixture theory

Thermo-Hydro-Mechanical （THM） coupling pro- cesses in unsaturated soils are very important in both theoretical researches and engineering applications. A coupled formulation based on hybrid mixture theory is derived to model the THM coupling behavior of unsaturated soils. The free-energy and dissipative functions for different phases are derived from Taylor＇s series expansions. Constitutive relations for THM coupled behaviors of unsaturated soils, which include deformation, entropy change, fluid flow, heat conduction, and dynamic compatibility conditions on the interfaces, are then established. The number of field equations is shown to be equal to the number of unknown variables; thus, a closure of this coupling problem is established. In addition to modifications of the physical conservation equations with coupling effect terms, the constitutive equations, which consider the coupling between elastoplastic deformation of the soil skeleton, fluid flow, and heat transfer, are also derived.