MICROMECHANICS ANALYSIS FOR UNSATURATED GRANULAR SOILS

Acta Mechanica Solida Sinica, Vol. 24, No. 3, June, 2011ISSN 0894-9166Published by AMSS Press, Wuhan, ChinaMICROMECHANICS ANALYSIS FOR UNSATURATEDGRANULAR SOILS**Weihua Zhang*Chenggang Zhao(School of Civil Engineering, Being Jiaotong University , Beijing 100044, China)Received 25 March 2010; revision received 3 March 2011ABSTRACT This paper aims at establishing an anisotropic stress expression for unsaturatedpendular-state granular soils. Using the second-order fabric tensor, we formulate a micromechan-ics scheme of soils with statistically averaging method, and reveal that the macroscopic averagestress of unsaturated granular soils in pendular- state is not isotropic. Not only is the stress fromcontact forces anisotropic due to the fabric, but also the capillary stre8s is directional dependent,which is different from the common point that the capillary stress is isotropic. The capillary stressof unsaturated pendular -state granular soils is determined by the orientation distribution of con-tact normals, 8o it is closely related to the initial and induced anisotropy of soils. Finally, DEMnumerical simulations of triaxial compression tests of pendular state soils at different degrees ofsaturation are used to verify the existence of above anisotropy of stresses.KEY WORDS unsaturated granular soils, pendular state, average stress, fabric, capillary stress,anisotropyI. INTRODUCTIONGranular soils are particulate in nature and composed of distinct particles which vary over a verylarge size range and behave as independent units. The mechanical properties of granular particles dependon the geometrical packing and interparticle interactions and can be described through the stress-forcefabric relationship, which is a decomposition of the stress tensor into components related to interparticleforces and to parameters reflecting the microscopic geometry. Microscopic characteristics such as thearrangement of particles, contact distribution, the orientations 8nd magnitudes of contact forces, etc,determinate the macroscopic properties of granular soils. Since granular particles are distinct and theirlocations are random, micro- and macro- relationships are developed by averaging techniques over agroup of particles based on principle of statistics. Following the work of Dantn[1] and Weberl2, a numberof researchers, such as Christoffersen et alMehrabadi et al.l4), Cundall and Stracklo, Thorntun andBarnes[6), Rothenburg and Bathurst!7, Mehrabadi et al,[8], Chang et a1.l9), have conducted the researchon deducing explicit relationships between the stress tensors and parameters that describe interparticleforces and fabric.No matter dry, saturated or unsaturated granular soils, their macroscopic properties are governed byinterparticle contacts. For dry soils, the interparticle forces are directly related to the external loads. In thecase of wetted soils, different stages of saturation can be identifed. The fully saturated state correspondsto a two phase material with water completely flling the voids between particles. Terzaghilof defined* Corresponding author. E mail: zwh2530@gmail.com中国煤化工** Project supported by the National Natural Science Foundation ofMHCNMHGNational BasicResearch Program (973) of China (No. 2010CB732100)..274.ACTA MECHANICA SOLIDA SINICA2011effective stresses as the diference between the total stresses and pore water pressure. Physically, efectivestresses describe the stresses acting on the soil skeleton. For unsaturated soil, the physical meaningof efective stresses remains the same. However, two additional factors must be considered: (1) thestress acting through the air phase; (2) the diference between the pore air pressure and the pore waterpressure, or matric suction; Three different soil- water states at different degrees of saturation can bedistinguished for unsaturated soils, namely, the pendular, funicular and capillary statesl11]. During thependular state, the water phase is discontinuously distributed in menisci located at the interparticlecontacts, and the capillary forces are applied only at the contacts. As the water content increases, thewater phase begins to extend outward surrounding the soil particles and acts like a membrane withnegative pressure which is homogenized throughout the soil skeleton like the water pressure, then thewater phase and the air phase become continuous, which is called funicular state. When the watercontent continues to increase, the air-water menisci no longer overlap the soil particle and close as airbubbles. Only water phase is continuous when the soil is fully saturated, which is called the capillarystate.Bishopl2) applied efective stress approach to unsaturated soil and extended Terzaghi's classicalefftective stress equation as followsσ' =(σ-ua) + x(ua - uw)(1)whereσ- ua is the net total stress, ua - Uuo the soil water suction (matric suction) that adds to theeffective stress, and x(ua - Uw) the contribution of matric suction to the efective stress. x is a parameterto be experimentally determined that is generally considered to vary between zero (dry soil) and unity(fully saturated soil). Bishop's x parameter is a scalar quantity, but microscopic interpretation of waterdistribution in pores can lead to an argument that x is directional dependentl13, 14), which is our mainconcern of this paper.Focusing on the pendular state and based on micromechanics, this paper first analyzes the interpar-ticle forces of unsaturated granular soils, and then uses the statistically averaging principle to establishrelationship between macroscopic stresses and microscopic interparticle forces. Starting from interpar-ticle contact normal orientation and using the second order fabric tensor, we deduce the anisotropicstress expression of unsaturated granular soils, which shows the effects of the fabric on the anisotropyof the average stress of unsaturated granular soils. It reveals the following phenomenon: For the macrscopic stress of unsaturated pendular-state granular soils, not only the stress from particle contacts isanisotropic which is infuenced by the fabric, but also the capillary stress generated at particle con-tacts is directional dependent (it is different from the point that capillary stress is isotropic) which isdetermined by the orientation distribution of contact normals. So the capillary stress of unsaturatedpendular-state granular soils is closely related to the initial and induced anisotropy of soils, and finally,DEM numerical simulations of triaxial compression tests at different degrees of saturation are used toverify the revealed phenomenon are objectively existent. In order to simplify the analysis, all particlesare assumed to be spherical.II. CONTACT FORCES AND CAPILLARY FORCES .For unsaturated granular soils, the cohesion forces; come from different interparticle physicochemicalforces such as (i) van der Waals attraction, (i) electrical double-layer repulsion or attraction, (li)cementation due to solute precipitation, and (iv) capillary forces due to negative pore water pressureand 8urface tension. The first three categories of interparticle bonding forces can exist under saturated orunsaturated states which are most pronounced in the clays but could be negligible for granular particlesfor the bulky volume and small specific surfacel1s. The capillary force is of interest in nature and canapparently only exist in the unsaturated state. According to granular material mechanics theory, eachinteraction of wet granular soils can be described as the sum of a dry contact force and a capillaryforce resulting from the presence of an air-water meniscus at the contact. The dry contact interactionis described by an elastic-plastic relation between the force and the relative displacemnent between twointeracting particles. The normal contact forces f2 are modeled as 8中国煤化工parallelwith a dashpot. The tangential contact forces are idealized as a Kelv|YHCN M H Grictionalslider. The tangential and normal contact forces are related by a slip swtcaupy.Vol. 24, No. 3 Weihua Zhang et al: Micromechanics Analysis for Unsaturated Granular Soils275Capillary force refers to the net interparticle force generated within a matrix of unsaturated granularparticles. at the contacts due to the combined effects of negative pore pressure and surface tension.Fisher[16] proposed a simple spherical model with a toroidal approximation of the liquid bridge, whichassumes that the surface tension T。exists at the air-water meniscus, the air- water meniscus is concavetowards the water phase, and hence subjected to a pressure lowerthan air. He demonstrated thatmeniscus water induces only a force normal to the plane passing through the spheres' contact pointand orthogonal to the line connecting their centers. Here the interparticle capillary force can be givena8 fllowsl7,18l; .Rsin θfop = 2rRTsin0 |sin(0 +8)+-2-(片一)n2)particles); R is the mean particle radius, θ and δ are the flling angles and contact angles, respectively;T1 and r2 are the radi of the principal curvatures of the pendular bridge and are the function of 0and 8. The water content w (or the degree of saturation Sr) is a function of θ and δl19, so the fllingangle θ is determined by the water content w(or the degree of saturation S). It is worth noting thatmicromechanical studies(20] demonstrated tbat in the case of more complex particle shapes, ua - Uwinduces normal forces only at interparticle contact points without causing any tangential forces.III. MACROSCOPIC STRESSES VERSUS MICROSCOPICINTERPARTICLE FORCE-FABRIC .3.1. Macroscopic Stresses and Microscopic ForcesGranular soils carry external loads by distributing them between individual contacts. So there shouldbe a link between contact forces and the macroscopic stress in a representative elemental volume (REV)to meet the requirement of quastic equilibrium. Based on averaging principles and using continwumtheorywe analyzes the link between contact forces and macroscopic stresses in this subsection. Thebasic assumptions made in the analysis are listed as follows:(1) The soil skeleton is treated as an assembly of particles that are assumed to be rigid bodies incontact with each other.(2) Each particle is surrounded by pore fuid (partially air and partially water). The surface area ofeach particle (or overlap length) in contact with neighboring particles is negligible compared with thesurface area exposed to the pore fuid.(3) The interparticle bonding forces (including van der Waals attraction, electrical double-layerrepulsion or attraction and cementation due to solute precipitation) are negligible.Here we consider the soil skeleton at a scale much larger than the particle size in an REV of athree-phase granular soil, which conforms to all above assumptions. The boundary distributed forcesp:(xi) are applied to the continuum boundary S of REV or each particle, and the volume force 7i:(xi)are acting within it. Under a quasi-static condition, as is shown in Fig.1, the equilibrium equation foran infinitesimal continuum element withi each particle is、 Boundary of REV (S)REV|w42中国煤化工MHCNMHGFig. 1. Free body diagram of granular culuu.276ACTA MECHANICA SOLIDA SINICA2011Oσi,j+Yi=0(3And the boundary condition can be written asOijnj= Pi(4in which n is the normal orientation of the boundary surface at the contact area pointing inwardsfollowing the sign convention in soil mechanics. The convention is adopted hereinafter. The averagestress 码of an individual particle n is defined in correspondence with Hill's averaging principlel21]唱=lovd(5where un is the particle volume, the bar on top of the stress parameters indicates the average character.Applying the divergence theorem to Eq.(5) and combing the result with the equilibrium conditions (3)and (4) leads tooij =亦Grk65kdu=.__. oTxj.dvu== . (ox)h- xou,JdvonJOinhkz;dS -xzyhdv; =px;dS-xjnidu(6)with Sn representing the particle surface. In granular material, the forces on the boundary are actingon the discrete points, and the volume force is the total gravity force acting on the gravity center ofthe particle, the average stress tensor of the particle can be expressed in terms of the boundary forcefi exerted on the boundary point and the volume force W; at the gravity center Xj醋=-，H paxyds-.J.Tj7idu =in2,f?cxZC + X;W.(7)where superscript c notes the n-th particle contact point, fnc the interparticle force in the i-th orientationat contact point C, xnC the distance in the j-th orientation between the particle contact and the referenceorigin, and Nn the total contact number of particle n. For derivation of the macroscopic stress tensorσij, we consider a representative volume of granular soils with particles that are suficiently small wbencompared to the scale of a boundary value problem. Furthermore, the number of particles needs to besufficiently large, which can represent the overall soil in statistical sense. In correspondence with theseaspects, the macroscopic stress tensors may be determined by the individual average particle stresseswith volume averaging technique:s=六V吗=f*x”*+ 2 X;W.(8)n=1Ln=l c=1in which M represents the total number of particles in the representative volume V. The quasi-staticcondition ensures that the force equilibrium condition is satisfied not only for the whole assembly butalso for any micro-element inside .Thus for the arbitrary particle m, one hasErme +W?"=0(9)Substituting Eq(9) into Eq.(8) yields的=二唱=六2(10)ln=1c=1m∈λcEmDefine the contact vectoruj = Xj-xj as the vector from the contact point C on the particle boundaryto the gravity center of the particle, then Eq.(10) can be expressed as中国煤化工的=2V喝=-0Ln=1c=1f"eg- 2 EmYHCNMHG (11)mEM cEmVol. 24, No.3Weibua Zhang et al: Micromechanics Analysis for Unsaturated Granular Soils. 277The double summation over the number of particles M and the number of the contacts per particleNn can be replaced by a single summation over the total number of contacts Ne, and the point c is aninternal contact point between two contact particles m and n. The number of forces is halved whengoing from particle forces to contact forces; hence, these two terms compensate each other, resulting in2Smn= 2 Emxme(12)n∈v cEnm∈M cEmwhere f?c = -fm;, which are the force and counter-force, respectively; so the average stress could bewritten as==2f°8L°where the branch vector L° denotes the distance between the centers of two particles m and n, i.e, Lym =R}m - Rnn, Eq.(13) is the microscopic expression of stress tensor. For packing of spherical particles,the orientations of branch vectors and contact normal orientations coincide each other, therefore, wehaveLnm = II"|n = 2Rn(14)where R is the average radius of particles.The interparticle forces of unsaturated granular soils include the average normal contact force, theaverage tangential contact force and the average capillary forcef(n)= F"(n)+ f(n)+ fC"(n)(15)Substituting Eq.(14) and Eq.(15) into Eq.(13), and assuming p(n) is the contact probability in thenormal orientation n, the average stress could be rewritten as(16)o= V [xm/*"()+ F'()+ F叫8 2Rndin3.2. Fabric CharacterizationGeometrical properties of granular packing have been of interest in various areas of engineering.Studiesl22,23)] suggest that among the geometrical properties, the important factors infuencing themechanical behavior are the radii of particles, the contact number in the volume, the orientationdistribution of the branch vectors joining the centroids of two particles in contact, and the orientationdistribution of the interparticle contacts. For spherical particles, the orientations of branch vectors andcontact normal vectors coincide. Only the orientation distribution of contact normal is considered inthis paper.3.2.1. The orientation distribution probability of contact normale p(n)A second-order symmetric tensor, named as fabric tensor, has been used by some investigators(23- 25].Here we approximate the distribution of contact probability p(n) in the normal orientation n by asecond-order tensor d(dij = bij + Pij, φij is the cofficient tensor)p(n)= -d:(n⑧n)=(8j +φi): (n@n)(17)Here[ p(n)dn =:。jd:(n@n)=1The mechanical property of soils is transversely isotropic, 80 the second-order coefficient tensor φij is .given asPij =(18)中国煤化工where a is a second-order parameter indicating the degree of (YHCNMH G.278ACTA MECHANICA SOLIDA SINICA20113.2.2. Approximations for mean contact forceSince the average normal contact force fn(n) = f"(n) .n is an even function of orientation due toits definition as a dot product, i.e., f"(n)= fn(-n), it is a scalar function of orientation and couldbe represented as followsl26):F"(n) = F0[(δ + D"): (n 8 n)]n(19)Here the tensor D" describes the deviation of orientational dependent mean forces from the isotropicstate. In the isotropic state, fn(n)= fn(- -n) and the constant fn(n) is the mean force, i.e.,.[j(nz)dn;=P(1+2)=PThe average tangential contact force ft(n) = f(n)- f(n)n, ft(n) = -f(-n) only containsodd-order terms, thus can be represented as follows:F(n)= F°[D' .n- D' :(n & n)n}(20)where D' is the second order tensor of average tangential force distribution.The average capillary force is generated from the air-water menisci at contacts, and its orientationcoincides with the contact normal orientation. Assuming granular soils are well distributed rendersjCaP(n)= fcap(n) = feap(n)n(21)3.3. Stress- Force-FabricSubstituting Eq8.(17), (19), (20) and (21) into Eq.(16), then Eq.(16) can be rewritten asi=e . d:(n@n){7!(6 + D"):(n 8n)n+鬥D* :n- D* :(n@n)m|+fcap(n)n}凶[2Rn]dn(22)And Eq.(22) can be decomposed as the following:耐=唱+ogap = Fyjo° + REζ°Po°OP(23)d(n) = dmnmNn(m,n= 1,2,3)f[(n) = f'[Dfmp + (D- D知enn] (ij,k,l= 1,2,3)go 二2RN.J031F= |onnJm + ghDnlmo + dmJsHkm(Di- D咖.= bis+ g4xs + 74jkD森+ 2(4j* D喝+ D嘬&Pin) + priD&8&s(24)呢= D嗡- D}x([()]=I@nJ]2Rnjdn= FEgDoS2RNe jfcap .FEOP= dmnJmnij = 8ij+3V开[ nmninjdn = gJmij， Jmnij = (0mn8ij + 5msOnj + 5mj8nj)Jijklmn(oyjJkmm + 8sxJmn + AuJjkmn + omnJskm)In Eq.(23), 0ij is the macroscopic average stress tensor, the first item on the right indicates the netcontact stress stemming from dry contact force, σ means the net contact stress in isotropic state; Fijdenotes the anisotropic parameter tensor of net contact stress, depending on the orientation distributionof contact normal and contact force; the second item o "P which is induced by the capillary force andtermed as capillary stress denotes the average capillary stress tensor中国煤化工er menisciat contacts, and σδP denotes the isotropic capillary stress; FifaPden:ter tensorof capillary stress which is diretly related to the contact normalHCNMH G.Vol. 24, No. 3 Weihua Zhang et al: Micromechanics Analysis for Unsaturated Granular Soils279 .3.4. The Constitutive Framework for Unsaturated Granular SoilsThe stress-strain-strength response of granular soil, either dry, saturated or unsaturated, impliesthe response of the soil skeleton, which is governed by the contact force between particles as well as bycontact material properties. For dry or saturated soils, the average stress is directly determined by theapplied load or by the applied load and hydrostatic pressure. According to Eq.(23), the incrementalstress-strain relation is expressed bydoij = Himn(doij, r)demn(25)where k represents a group of internal variables of the soil skeleton and Hijmn the contact stifness. Forthe soil under saturated conditions, the constitutive relations can be obtained by substituting Eq.(23)into the above relation, as followsdor*= Hjmn(om + opgaP,x)demn - dPrf°PσaP - FECjPdoCaP(26)Unsaturated granular soils are in funicular or capillary state during the high degree of saturationwhere capillary pressure or negative hydrostatic pressure is unanimously applied on the soil skeleton andhas no shear effect. However, liquid bridges form at particle contacts where capillary force is generated.As we can see from Eq,(26), a change in Fij(just the change of contact normal orientation) under aconstant capillary stress could act as a loading increment to cause the deformation of soil skeleton. Alsowe can see from the second item in Eq.(26) that capillary stress is applied on the soil skeleton whichis not isotropic but has capillary-induced shear effects. Analysis is shown as follows:° (27)Then the capillary stress can be decomposed into two parts: a normal stress (isotropic) ogcap anda deviatoric (purely anisotropic) stress σqcap , which are represented asopcep =oiqa + o2ap + o3S8P= oaP(28a)3o°=°iSXP- osS = ((911 - 433)oSO = gaσ°(28b)where 411 and 433 are the major and minor principal value of the contact normal distribution, respec-tively; and the latter is seldom considered when analyzing the effects of capillary stress.IV. THE DIRECTIONALITY VERIFICATION TEST OF CAPILLARY STRESSThe idea of the verification test is simple. If the magnitude of capillary stress varies in diferentorientations during wetting on an initially isotropic soil specimen subjected to triaxial compression test,it means capillary stress is directional dependent.The DEM numerical simulations are under the condition of dry conditions, the sample is a polydisperseassembly composed of 10000 spherical soil particles, with a particle size distribution ranging from 0.035mm to 0.07 mm, and a porosity equals 0.39. The sample is large enough compared with the maximumlength of the particles to be structurally typical of the tested material. And the sample is representedas an assembly of circular particles. The input parameters are listed in Table 1.Table 1. Input parameters for numerical siomulation by DEMGlobal modulus E(MPa)_ a=Kt/rn Frictional angle中(°)_50.530First, the sample is prepared using an isotropic-compaction techniaue ronsistinσ nf twn main stages:(a) All particles are randomly positioned inside a cube made中国煤化工Ils such thatno overlap/ contact force develops between any two particles. TYHCNMHGgleissettobe a small value (the smaller the friction angle is, the denser the assetnoly 18 ana a vaue of 0.5 hasACTA MECHANICA SOLIDA SINICA2011been chosen here) and particle radi are then homogeneously increased, whereas the boundary wallsremain fixed. The process continues until the confining pressure (5 kPa in this study) is reached andthe static equilibrium between the internal stress state and the external load is satisfied.(b) The interparticle friction coeficient is then changed to a value which is common used in DEM sim-ulation to reproduce an acceptable shear strength and the boundary walls are controlled in displacementto keep the equilibrium state.Then, a set of triaxial compression tests has been carried out on the numerical specimen, which wasraised from the initial state of 5 kPa to upper confining pressures by isotropic compaction, consistingof three main stages:(a) A set of triaxial compression tests are carried out on the numerical specimen and the confiningpressure are 5 kPa, 10 kPa, 20 kPa.The specimen is progressively wetted by gradually increasing the water content as it occurs duringcapillary condensation. This progress is controlled by suction variation. To ensure the pendular stateassumption in the medium, only saturation degrees below 10% were considered (Checking whether themeniscus on each particle meet any neighboring particles shows that, for this particular assembly, 10%is the critical degree of saturation from which liquid bridges start to overlap).(c) When the wetting equilibrium is reached, the axial and lateral capillary stresses are calculated,and plot the principal capillary stress σ&P and σ2P. Assuming k = 2(oSaP- oSQP)/(σSaP +σSaP) whichdenotes a representative index of the sphericity tensor obtained from triaxial tests at different degreesof saturation. When K = 0, it means capillary stress is isotropic, and the greater the index K is, thegreater the anisotropic degree of capillary stress is.As indicated in Fig.2 (a), we can see, at the same degree of saturation, the axial capillary stress whichis just in force-chain orientation is always larger than the lateral capillary stress. And we can also seefrom Fig.2(b) that the index K approaches to be a constant as the axial strain increases, which meansthe difference between axial and lateral capillary stress gradually keeps balance. On the other hand, theindex K tends to decrease as the degree of saturation increases, namely, the difference between axial andlateral capillary stress gradually decreases indicating the fabric has less effects on the capillary stresswith the increasing degree of saturation. To sum up, the test result indeed confirms the directionalityof capillary stress..0r0.14-1.2.00.10- + -8-25%3.0 F+ S- 10%.xin0.08+&-1% .K 0.06.0 H0.020.00.0 1.0 2.0 3.0.0 5.0Axial strain (%)(b)Fig. 2. (a) Axial (oild lines) and lateral (dashed lines) capillary stresses versus axial strain ftting curves, (b) Variationindex k derived from triaxial tests at different degrees of saturation versus axial strain curves.V. CONCLUSIONUsing the continuum mechanics theory and integrating with statistically averaging techniques, thispaper attempts to relate macroscopic stresses to microscopic interparticle forces of unsaturated granularsoils in pendular state via a second-order fabric tensor, and analyzes the magnitude and directionalityof interparticle stresses. Some conclusions have been made based中国煤化工alysis:(1) In condition of low saturation, the macroscpic stress teCNMH Gular-stateonsorgranular soils include the net interparticle contact stress tensor σn (rrju) anu capiay ouless tensorVol. 24, No.3 Weihua Zhang et al: Micromechanics Analysis for Unsaturated Granular Soils281 .σ;cap (FGaPσaP). The former is infuenced by the orientations of contact normal and contact force; andthe latter is directional dependent on the contact normal orientation.(2) If the orientation distribution of contact normal is isotropic, then the capillary stress is isotropic,which happens when the external load is isotropically applied on the initially isotropic granular soils.However, the capillary stress is anisotropic when deviatoric load is applied or in natural depositionprocess which leads to the force chain orientation to change, i.e., the stress- induced anisotropy or thepreferred orientation of the contact normal orientation. So it shows that the capillary stress of unsaturatedpendular-state granular soils is closely related to the initial and the stress-induced anisotropy.References[1] Dantu,P., Contribution a 1etude mechanique et geometrique des milieux epulverulents. In: Proceedings ofthe Fourth Intermational Conference Soil Mech, 1957, 1: 144-148.2] Weber ,J, Recherche concernant les contraintes intergranulaires dans les milieux pulverulents. Buletin deLiaison des Ponts et Chassees, Paris, 1966, 20: 1-20.3] Christoffersen,J, Mehrabadi,M.M. and Nemat- Nasser,S., A micromechanical description on granular ma-terial behavior. Jourmnal of Applieded Mechanics, 1981, 48: 339 344.[4] Mehrabadi,M.M., Nemat-Nasser,S. and Oda,M., On statistical description of stress and fabric in granularmaterials. International Journal of Numerical and Analytical Method in Geomechanic, 1982, 6: 95 108.[5] Cundall,P.A. and Strack,O.D.L., Modeling of microecopic mechanisms in granular materials. In: Jenk-ins,J.T, Satake,M. (Eds.). Proceedings of the US-Japan serminar on new models and constitutive relationsin the mechanics of granular materials, 1983: 137-149.6] Thornton,C. and Barnes,D.J, Computer simulated deformation of compact granular assemblies. Acta Me-chanica, 1986, 64: 45-61.7] Bathurst,R.J. and Rothenberg,L, Micromechanical aspects of isotropic granular assemnblies with linearcontact interactions. ASME Journal of Applied Mechanics, 1988, 55: 17-23.8] Mehrabadi,M.M. and Loret,B., Incremental constitutive relations for granular materials based on microme-chanics. Proceedings of the Royal Society of London, 1993, 441: 443- 463.9] Chang,C.S. and Hicher,P.Y., An elastoplastic model for granular materials with microstructural consider-ation. International Jourmal of Solid and Structures, 2005, 42(14): 4258 4277.[10] Terzaghi,K., Principles of Soil Mechanics. New York: John Wiley and Sons, 1963.[11] Bear,J, Dynamics of Fluids in Porous Media, New York: Elsevier, 1972.[12] Bishop,A.W and Blight,G.E, Some aspects of effective stress of in saturated and partially saturated soils.Geotechnique, 1963, 13: 177-179.[13] Li,X.S., Efective stress in unsaturated soil: A microstructural analysis. Geotechnique, 2003, 53: 273-277.14] Molenkamp,F. and Nazemi,A.H, Micromechanical considerations of unsaturated pyramidal packing.Geotechnique , 2003, 53(2): 195-206.[15] James,K.Mitchell and Kenichi, Fundamentals of Soil Behavior. Hoboken, New Jersey: John Wiley & Sons,Inc., 2004.[16] Fisher,RA., On the capillary forces in an ideal soil; Corrections of formulae given by Haines,W.B. Jounalof Agricultural Science, 1926, 16: 492-505.17] Lu,N., Wu,B. and Tan,C.P, Tensile strength characteristics of unsaturated sands. Jourmal of Geotechnicaland Geoenviromental Engineering, ASCE, 133: 144 154. doi :10.1061/(ASCE) 1090-0241, 2007, 133: 2(144).[18] Lian,G., Thornton,C. and Adams,M.J., A Theoretical study of liquid bridge force between rigid sphericalbodies. Journal of Collide and Interface Science, 1993, 161: 138-147.[19] Pietsch,W .B., Tensile strength of granular materials. Nature, 1968, 217: 736-737.[20] Gili,Y.Y., Modelo microstructural para medios hranulares ni saturados. Doctoral Thesis, UniversitatPolitecnica de Catalunya, 1988.[21] Hill,R., The essential structure of constitutive laws for metal composite sand polycrystals. Joumnal of theMechanics and Physics of Solids, 1967, 15: 79-95.[22] Gray,W .A., The Packing of Solid Particles. London: Chapman and Hall, 1968.[23] Oda,M, Nemat-Nasser S. and Meharabad,M.M., A statistical study of fabric in random assembly of spheri-cal granulates. International Jourmnal of Nurmerical and Analytical Method in Geomechanic, 1982, 6(1): 77-94.[24] Satake,M., Fabric tensor in granular materials. In: proc. IUTAM Symp. On Deformation and Failure ofGranular materials (Edited by P.A. Vermeer and H.J.Luger), 1982, 63-68. Balkema,A.A, Delft, The Nether-lands.[25] Cowin,S.C, The relationship between the elastic tensor and the fabric tensor. Mechanics of Materials, 1985,4(2): 137-147.[26] Rothenberg,L. and Bathurst,.J, Analytical study of induced中国煤化工，ular materials.Geotechnique, 1989, 39(4), 601-614 .MHCNMHG