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CHAPTER 7

Effective, Intergranular, andTotal Stress

7.1 INTRODUCTION

The compressibility, deformation, and strength prop-erties of a soil mass depend on the effort required todistort or displace particles or groups of particles rel-ative to each other. In most engineering materials,resistance to deformation is provided by internalchemical and physicochemical forces of interactionthat bond the atoms, molecules, and particles together.Although such forces also play a role in the behaviorof soils, the compression and strength properties de-pend primarily on the effects of gravity through selfweight and on the stresses applied to the soil mass.The state of a given soil mass, as indicated, for ex-ample, by its water content, structure, density, or voidratio, reflects the influences of stresses applied in thepast, and this further distinguishes soils from mostother engineering materials, which, for practical pur-poses, do not change density when loaded or unloaded.

Because of the stress dependencies of the state, agiven soil can exhibit a wide range of properties. For-tunately, however, the stresses, the state, and the prop-erties are not independent, and the relationshipsbetween stress and volume change, stress and stiffness,and stress and strength can be expressed in terms ofdefinable soil parameters such as compressibility andfriction angle. In soils with properties that are influ-enced significantly by chemical and physicochemicalforces of interaction, other parameters such as cohe-sion may be needed.

Most problems involving volume change, deforma-tion, and strength require separate consideration of thestress that is carried by the grain assemblage and thatcarried by the fluid phases. This distinction is essentialbecause an assemblage of grains in contact can resistboth normal and shear stress, but the fluid and gas

phases (usually water and air) can carry normal stressbut not shear stress. Furthermore, whenever the totalhead in the fluid phases within the soil mass differsfrom that outside the soil mass, there will be fluid flowinto or out of the soil mass until total head equality isreached.

In this chapter, the relationships between stresses ina soil mass are examined with particular reference tostress carried by the assemblage of soil particles andstress carried by the pore fluid. Interparticle forces ofvarious types are examined, the nature of effectivestress is considered, and physicochemical effects onpore pressure are analyzed.

7.2 PRINCIPLE OF EFFECTIVE STRESS

The principle of effective stress is the keystone ofmodern soil mechanics. Development of this principlewas begun by Terzaghi about 1920 and extended forseveral years (Skempton, 1960a). Historical accountsof the development are described in Goodman (1999)and de Boer (2000). A lucid statement of the principlewas given by Terzaghi (1936) at the First InternationalConference on Soil Mechanics and Foundation Engi-neering. He wrote:

The stresses in any point of a section through a mass ofsoil can be computed from the total principal stresses, �1,�2, �3, which act in this point. If the voids of the soil arefilled with water under a stress u, the total principalstresses consist of two parts. One part, u, acts in the waterand in the solid in every direction with equal intensity. Itis called the neutral stress (or the pore water pressure).The balance � �1 � u, � �2 � u, and � �3 �� 1 2 3

u represents an excess over the neutral stress u, and it hasits seat exclusively in the solid phase of the soil.

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Copyright © 2005 John Wiley & Sons Retrieved from: www.knovel.com

174 7 EFFECTIVE, INTERGRANULAR, AND TOTAL STRESS

This fraction of the total principal stresses will be calledthe effective principal stresses . . . . A change in the neutralstress u produces practically no volume change and haspractically no influence on the stress conditions for failure. . . . Porous materials (such as sand, clay, and concrete)react to a change of u as if they were incompressible andas if their internal friction were equal to zero. All the meas-urable effects of a change of stress, such as compression,distortion and a change of shearing resistance are exclu-sively due to changes in the effective stresses and��, ��1 2

Hence every investigation of the stability of a saturated��.3

body of soil requires the knowledge of both the total andthe neutral stresses.

In simplest terms, the principle of effective stressasserts that (1) the effective stress controls stress–strain, volume change, and strength, independent of themagnitude of the pore pressure, and (2) the effectivestress is given by �� u for a saturated soil.1

There is ample experimental evidence to show thatthese statements are essentially correct for soils. Theprinciple is essential to describe the consolidation of aliquid-saturated deformable porous solid, as was donefor the one-dimensional case by Terzaghi and furtherdeveloped for the three-dimensional case by otherssuch as Biot (1941). It is also an essential concept forthe understanding of soil liquefaction behavior duringearthquakes.

The total stress � can be directly measured or com-puted using the external forces and the body force dueto weight of the soil–water mixture. A pore water pres-sure, denoted herein by u0, can be measured at a pointremote from the interparticle zone. The actual pore wa-ter pressure in the interparticle zone is u. We knowthat at equilibrium the total potential or head of thewater at the two points must be equal, but this doesnot mean that u � u0, as discussed in Section 7.7. Theeffective stress is a deduced quantity, which in practiceis taken as �� u0.

7.3 FORCE DISTRIBUTIONS IN APARTICULATE SYSTEM

The term intergranular stress has become synonymouswith effective stress. Whether or not the intergranularstress is indeed equal to � � u cannot be ascertained��iwithout more detailed examination of all the interpar-

1 The terms � and �� are the principal total and effective stresses.For general stress conditions, there are six stress components (�11,�22, �33, �12, �23, and �31), where the first three are the normal stressesand the latter three are the shear stresses. In this case, the effectivestresses are defined as � �11 � u, � �22 � u, � �33 �� 11 22 33

u, � , � and �� , � � � .12 12 23 23 31 31

ticle forces in a soil mass. Interparticle forces at themicroscale can be separated into the following threecategories (Santamarina, 2003):

1. Skeletal Forces Due to External Loading Theseforces are transmitted through particles from theforces applied externally [e.g., foundation load-ing) (Fig. 7.1a)].

2. Particle Level Forces These include particleweight force, buoyancy force when a particle issubmerged under fluid, and hydrodynamic forcesor seepage forces due to pore fluid movingthrough the interconnected pore network (Fig.7.1b).

3. Contact Level Forces These include electricalforces, capillary forces when the soil becomesunsaturated, and cementation-reactive forces (Fig.7.1c).

When external forces are applied, both normal andtangential forces develop at particle contacts. All par-ticles do not share the forces or stresses applied at theboundaries in equal manner. Each particle has differentskeletal forces depending on the position relative to theneighboring particles in contact. The transfer of forcesthrough particle contacts from external stresses wasshown in Fig. 5.15 using a photoelastic model. Strongparticle force chains form in the direction of majorprincipal stress. The evolution and distribution of in-terparticle skeletal forces in soils govern the macro-scopic stress–strain behavior, volume change, andstrength. As the soil approaches failure, buckling ofparticle force chains occurs and shear bands developdue to localization of deformation. Further discussionof microbehavior in relation to deformation andstrength is given in Chapter 11.

Particle weights act as body forces in dry soil andcontribute to skeletal forces, observed in the photo-elastic model shown in Fig. 5.15. When the pores arefilled with fluids, the weight of the fluids adds to thebody force of the soil–fluids mixture. However, hydro-static pressure results from the fluid weight, and theuplift force due to buoyancy reduces the effectiveweight of a fluid-filled soil. This leads to smaller skel-etal forces for submerged soil compared to dry soil.Seepage forces that result from additional fluid pres-sures applied externally produce hydrodynamic forceson particles and alter the skeletal forces.

7.4 INTERPARTICLE FORCES

Long-range particle interactions associated with elec-trical double layers and van der Waals forces are dis-

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INTERPARTICLE FORCES 175

External Load

InterparticleForces

InterparticleForces

(a)

Body Force

Buoyancy Forceif Saturated

Seepage

Viscous Drag bySeepage Flow

(b)

Capillary Force orCementation-reactiveForce

Electrical Forces

(c)

Figure 7.1 Interparticle forces at the particle level: (a) skeletal forces by external loading,(b) particle level forces, and (c) contact level forces (after Santamarina, 2003).

cussed in Chapter 6. These interactions control theflocculation–deflocculation behavior of clay particlesin suspension, and they are important in swelling soilsthat contain expanding lattice clay minerals. In densersoil masses, other forces of interaction become impor-tant as well since they may influence the intergranularstresses and control the strength at interparticle con-tacts, which in turn controls resistance to compressionand strength. In a soil mass at equilibrium, there mustbe a balance among all interparticle forces, the pres-sure in the water, and the applied boundary stresses.

Interparticle Repulsive Forces

Electrostatic Forces Very high repulsion, the Bornrepulsion, develops at contact points between particles.It results from the overlap between electron clouds,and it is sufficiently great to prevent the interpenetra-tion of matter.

At separation distances beyond the region of directphysical interference between adsorbed ions and hy-dration water molecules, double-layer interactions pro-vide the major source of interparticle repulsion. Thetheory of these forces is given in Chapter 6. As notedthere, this repulsion is very sensitive to cation valence,electrolyte concentration, and the dielectric propertiesof the pore fluid.

Surface and Ion Hydration The hydration energyof particle surfaces and interlayer cations causes largerepulsive forces at small separation distances betweenunit layers (clear distance between surfaces up to about2 nm). The net energy required to remove the last few

layers of water when clay plates are pressed togethermay be 0.05 to 0.1 J/m2. The corresponding pressurerequired to squeeze out one molecular layer of watermay be as much as 400 MPa (4000 atm) (van Olphen,1977).

Thus, pressure alone is not likely to be sufficient tosqueeze out all the water between parallel particle sur-faces in naturally occurring clays. Heat and/or highvacuum are needed to remove all the water from a fine-grained soil. This does not mean, however, that all thewater may not be squeezed from between interparticlecontacts. In the case of interacting particle corners,edges, and faces of interacting asperities, the contactstress may be several thousand atmospheres becausethe interparticle contact area is only a very small pro-portion (�� 1%) of the total soil cross-sectional areain most cases. The exact nature of an interparticle con-tact remains largely a matter for speculation; however,there is evidence (Chapter 12) that it is effectively solidto solid.

Hydration repulsions decay rapidly with separationdistance, varying inversely as the square of the dis-tance.

Interparticle Attractive Forces

Electrostatic Attractions When particle edges andsurfaces are oppositely charged, there is attraction dueto interactions between double layers of opposite sign.Fine soil particles are often observed to adhere whendry. Electrostatic attraction between surfaces at differ-ent potentials has been suggested as a cause. When the

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gap between parallel particle surfaces separated by dis-tance d at potentials V1 and V2 is conductive, there isan attractive force per unit area, or tensile strength,given by (Ingles, 1962)

�6 24.4 � 10 (V � V )1 2 2F � N/m (7.1)2d

where F is the tensile strength, d is in micrometers,and V1 and V2 are in millivolts. This force is indepen-dent of particle size and becomes significant (greaterthan 7 kN/m2 or 1 psi) for separation distances lessthan 2.5 nm.

Electromagnetic Attractions Electromagnetic at-tractions caused by frequency-dependent dipole inter-actions (van der Waals forces) are described in Section6.12. Anandarajah and Chen (1997) proposed a methodto quantify the van der Waals force between particlesspecifically for fine-grained soils with various geomet-ric parameters such as particle length, thickness, ori-entation, and spacing.

Primary Valence Bonding Chemical interactionsbetween particles and between the particles and adja-cent liquid phase can only develop at short range. Co-valent and ionic bonds occur at spacings less than 0.3nm. Cementation involves chemical bonding and canbe considered as a short-range attraction.

Whether primary valence bonds, or possibly hydro-gen bonds, can develop at interparticle contacts with-out the presence of cementing agents is largely amatter of speculation. Very high contact stresses be-tween particles could squeeze out adsorbed water andcations and cause mineral surfaces to come close to-gether, perhaps providing opportunity for cold weld-ing. The activation energy for soil deformation is high,in the range characteristic for rupture of chemicalbonds, and strength behavior appears in reasonableconformity with the adhesion theory of friction (Chap-ter 11). Thus, interatomic bonding between particlesseems possible. On the other hand, the absence of co-hesion in overconsolidated silts and sands arguesagainst such pressure-induced bonding.

Cementation Cementation may develop naturallyfrom precipitation of calcite, silica, alumina, iron ox-ides, and possibly other inorganic or organic com-pounds. The addition of stabilizers such as cement andlime to a soil also leads to interparticle cementation. Iftwo particles are not cemented, the interparticle forcecannot become tensile; they loose contact. However, ifa particle contact is cemented, it is possible for someinterparticle forces to become negative due to the ten-sile resistance (or strength) of the cemented bonds.

There is also an increase in resistance to tangentialforce at particle contacts. However, when the bondbreaks, the shear capacity at a contact reduces to thatof the uncemented contacts.

An analysis of the strength of cemented bondsshould consider three cases: (i) failure in the cement,(ii) failure in the particle and (iii) failure at the ce-ment–particle interface. The following equation can bederived (Ingles, 1962) for the tensile strength �T perunit area of soil cross section:

1 n� � Pk (7.2)� �T n1 � e

A� i1

where P is the bond strength per contact zone, k is themean coordination number of a grain, e is the voidratio, n is the number of grains in an ideal breakageplane at right angles to the direction of �T, and Ai isthe total surface area of the ith grain.

For a random and isotropic assembly of spheres ofdiameter d, Eq. (7.2) becomes

Pk� � (7.3)T 2�d (1 � e)

For a random and isotropic assembly of rods of lengthl and diameter d

Pk� � (7.4)T �d(l � d /2)(1 � e)

Bond strength P is evaluated in the following way (Fig.7.2) for two cemented spheres of radius R. It may beshown that

(R � cos �)� cosh � R sin � (7.5)

�

so for known �, � can be computed. Then, for cementfailure,

2P � � � �� (7.6)c

where �c is the tensile strength of the cement; forsphere failure,

2P� � � � �(��) (7.7)s

where �� R sin �, and �s is the tensile strength ofthe sphere, and for failure at the interface

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INTERPARTICLE FORCES 177

Figure 7.2 Contact zone failures for cemented spheres.

sin � 2P� � � � � 2�R (1 � cos �) (7.8)1 �

where �1 is the tensile strength of the interface bond.In principle, Eq. (7.6), (7.7), or (7.8) can be used toobtain a value for P in Eq. (7.2) enabling computationof the tensile strength �T of a cemented soil.

The behavior of cemented soils can depend on thetiming of cementation development. Artificially ce-mented soils are often loaded after cementation hasdeveloped, whereas cementation develops during or af-ter overburden loading in natural soils. In the formercase, the particles and cementation bonding are loadedtogether and contact forces can become negative de-pending on the tensile resistance of cementation bond-ing. The distribution and magnitude of skeletal forcesare therefore influenced by both geometric arrange-ment of particles and the cementation bonding at theparticle contacts. In the latter case, on the other hand,the contact forces induced by external loading are de-veloped before cementation coats the already loadedparticles. In this case, it is possible that cementationcreates extra forces at particle contacts. In some ce-

mented natural materials, if the soil is unloaded fromhigh overburden stress, elastic rebound may disrupt ce-mented bonds.

Cementation allows interparticle normal forces tobecome negative, and, therefore, the distribution andevolution of skeletal forces may be different than inuncemented soils, even though the applied externalstresses are the same. Thus, the stiffness and strengthproperties of a soil are likely to differ according towhen and how cementation was developed. How toaccount for this in terms of effective stress is not yetclear.

Capillary Stresses Because water is attracted tosoil particles and because water can develop surfacetension, suction develops inside the pore fluid when asaturated soil mass begins to dry. This suction acts likea vacuum and will directly contribute to the effectivestress or skeletal forces. The negative pore pressure isusually considered responsible for apparent and tem-porary cohesion in soils, whereas the other attractiveforces produce true cohesion.

When the soil continues to dry, air starts to invadeinto the pores. The air entry pressure is related to thepore size and can be estimate using the following equa-tion, assuming a capillary tube as shown in Fig. 7.3a:

2� cos �awP � (7.9)c rp

where is the capillary pressure at air entry, �aw isPc

the air–water interfacial tension, � is contact angle de-fined in Fig. 7.3, and rp is the tube radius. For purewater and air, �aw depends on temperature, for exam-ple, it is 0.0756 N/m at 0�C, 0.0728 N/m at 20�C, and0.0589 N/m at 100�C. If the capillary pressure Pc

(� ua � uw, where ua and uw are the air and waterpressures, respectively) is larger than then air in-P ,c

vades the pore.2 Since soil has pores with various sizes,the water in the largest pores is displaced first followedby smaller pores. This leads to a macroscopic modelof the soil–water characteristic curve (or the capillarypressure–saturation relationship), as discussed in Sec-tion 7.11.

If the water surrounding the soil particles remainscontinuous [termed the ‘‘funicular’’ regime by Bear(1972)], the interparticle force acting on a particle withradius r can be estimated from

2 It is often assumed that ua � 0 (for gauge pressure) or 1 atm (forabsolute pressure). However, this may not be true in cases such asrapid water infiltration when air in the pores cannot escape or the airboundary is completely blocked.

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Capillary TubeRepresenting a Pore

2rp

dc

θ

(a) (b)

ua

uwPc = ρw gdc =

rp

^ 2σawcosθ

Figure 7.3 Capillary tube concept for air entry estimation: (a) capillary tube and (b) bundleof capillary tubes to represent soil pores with different sizes.

22�r � cos �aw2 ˆF � �r P � (7.10)c c rp

where rp is the size of the pore into which the air hasentered. Since the fluid acts like a membrane with neg-ative pressure, this force contributes directly to theskeletal forces like the water pressure as shown in Fig.7.4a.

As the soil continues to dry, the water phase be-comes disconnected and remains in the form of me-nisci or liquid bridges at the interparticle contacts[termed the ‘‘pendular’’ regime by Bear (1972)]. Thecurved air–water interface produces a pore water ten-sion, which, in turn, generates interparticle compres-sive forces. The force only acts at particle contacts incontrast to the funicular regime, as shown in Fig. 7.4b.The interparticle force generally depends on the sep-aration between the two particles, the radius of the liq-uid bridge, interfacial tension, and contact angle (Lianet al., 1993). Once the water phase becomes discontin-uous, evaporation and condensation are the primarymechanisms of water transfer. Hence, the humidity ofthe gas phase and the temperature affect the water va-por pressure at the surface of water menisci, which inturn influences the air pressure ua.

7.5 INTERGRANULAR PRESSURE

Several different interparticle forces were described inthe previous section. Quantitative expression of the in-

teractions of all these forces in a soil is beyond thepresent state of knowledge. Nonetheless, their exis-tence bears directly on the magnitude of intergranularpressure and the relationship between intergranularpressure and effective stress as defined by �� u.

A simplified equation for the intergranular stress ina soil may be developed in the following way. Figure7.5 shows a horizontal surface through a soil at somedepth. Since the stress conditions at contact points,rather than within particles, are of primary concern, awavy surface that passes through contact points (Fig.7.5a) is of interest. The proportion of the total wavysurface area that is comprised of intergrain contact areais very small (Fig. 7.5c).

The two particles in Fig. 7.5 that contact at point Aare shown in Fig. 7.6, along with the forces that act ina vertical direction. Complete saturation is assumed.Vertical equilibrium across wavy surface x–x is con-sidered.3 The effective area of interparticle contact isac; its average value along the wavy surface equals thetotal mineral contact area along the surface divided bythe number of interparticle contacts. Define area a as

3 Note that only vertical forces at the contact are considered in thissimplified analysis. It is evident, however, that applied boundary nor-mal and shear stresses each induce both normal and shear forces atinterparticle contacts. These forces contribute both to the develop-ment of soil strength and resistance to compression and to the slip-ping and sliding of particles relative to each other. These interparticlemovements are central to compression, shear deformations, and creepas discussed in Chapters 10, 11, and 12.

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INTERGRANULAR PRESSURE 179

(a)

Soil Particles

ContinuousWater Film

Negative pore pressure acts allaround the particles

(b)

Suction forces act only at particlecontacts and the magnitude of theforces depends on the size of liquidbridges.

LiquidBridges

Soil Particles

InterparticleForces

Pores of Radiusrp Filled with Air

Air

Figure 7.4 Microscopic water–soil interaction in unsaturated soils: (a) funicular regime and(b) pendular regime.

Figure 7.6 Forces acting on interparticle contact A.

Figure 7.5 Surfaces through a soil mass.

the average total cross-sectional area along a horizontalplane served by the contact. It equals the total hori-zontal area divided by the number of interparticle con-tacts along the wavy surface. The forces acting on areaa in Fig. 7.6 are:

1. �a, the force transmitted by the applied stress �,which includes externally applied forces andbody weight from the soil above.

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2. u(a � ac), the force carried by the hydrostaticpressure u. Because a �� ac and ac is very small,the force may be taken as ua. Long-range,double-layer repulsions are included in ua.

3. A(a � ac) � Aa, the force caused by the long-range attractive stress A, that is, van der Waalsand electrostatic attractions.

4. A�ac, the force developed by the short-range at-tractive stress A�, resulting from primary valence(chemical) bonding and cementation.

5. Cac, the intergranular contact reaction that is gen-erated by hydration and Born repulsion.

Vertical equilibrium of forces requires that

�a � Aa � A�a � ua � Ca (7.11)c c

Division of all terms by a converts the forces tostresses per unit area of cross section,

ac� � (C � A�) � u � A (7.12)a

The term (C � A�)ac /a represents the net force acrossthe contact divided by the total cross-sectional area(soil plus water) that is served by the contact. In otherwords, it is the intergrain force divided by the grossarea or the intergranular pressure in common soil me-chanics usage. Designation of this term by gives��i

�� A � u (7.13)i

Equations analogous to Eqs. (7.11), (7.12), and (7.13)can be developed for the case of a partly saturated soil.To do so requires consideration of the pressures in thewater uw and in the air ua and the proportions of areaa contributed by water aw and by air aa with the con-dition that

a � a � a i.e., a → 0w a c

The resulting equation is

aw�� A � u � (u � u ) (7.14)i a w aa

In the absence of significant long-range attractions,this equation is similar to that proposed by Bishop(1960) for partially saturated soils

�� u � �(u � u ) (7.15)i a a w

where � � aw /a. Although it is clear that for a dry soil� � 0, and for a saturated soil � � 1.0, the usefulnessof Eq. (7.15) has been limited in practice because ofuncertainties about � for intermediate degrees of sat-uration. Further discussion of the effective stress con-cept for unsaturated soils is given in Section 7.12.

Limiting the discussion to saturated soils, two ques-tions arise:

1. How does the intergranular pressure relate to��ithe effective stress as defined for most analyses,that is, �� u?

2. How does the intergranular pressure relate to��ithe measured quantity, � � � u0, that is taken��mas the effective stress, recalling (Section 7.2) thatpore pressure can only be measured at points out-side the true interparticle zone?

Answers to these questions require a more detailedconsideration of the meaning of fluid pressures in soils.

7.6 WATER PRESSURES AND POTENTIALS

Pressures in the pore fluid of a soil can be expressedin several ways, and the total pressure may involveseveral contributions. In hydraulic engineering, prob-lems are analyzed using Bernoulli’s equation for thetotal heads and head losses associated with flow be-tween two points, that is,

2 2p v p v1 1 2 2Z � � � Z � � � h (7.16)1 2 1–2 2g 2gw w

where Z1 and Z2 are the elevations of points 1 and 2,p1 and p2 are the hydrostatic pressures at points 1 and2, v1 and v2 are the flow velocities at points 1 and 2,w is the unit weight of water, g is the acceleration dueto gravity, and h1–2 is the loss in head between points1 and 2. The total head H (dimension L) is

2p vH � Z � � (7.17)

2gw

Flow results only from differences in total head;conversely, if the total heads at two points are thesame, there can be no flow, even if Z1 Z2 and p1 p2. If there is no flow, there is no head loss and h1–2

� 0.The flow velocity through soils is low, and as a re-

sult v2 /2g → 0, and in most cases it may be neglected.Therefore, the relationship

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WATER PRESSURE EQUILIBRIUM IN SOIL 181

p p1 2Z � � Z � � h (7.18)1 2 1–2 w w

is the basis for evaluation of pore pressures and anal-ysis of seepage through soils and other porous media.

Although the absence of velocity terms is a factorthat seems to simplify the analysis of flows and pres-sures in soils, there are other considerations that tendto complicate the problem. These include:

1. The use of several terms to describe the status ofwater in soils, for example, potential, pressure,and head.

2. The possible existence of tensions in the pore wa-ter.

3. Compositional differences in the water frompoint-to-point and adsorptive force fields fromparticle surfaces.

4. Differences in interparticle forces and the energystate of the pore fluid from point to point owingto thermal, electrical, and chemical gradients.Such gradients can cause fluid flows, deforma-tions, and volume changes, as considered in moredetail in Chapter 9.

Some formalism in definition and terminology isnecessary to avoid confusion. The status of water in asoil can be expressed in terms of the free energy rel-ative to free, pure water (Aitchison, et al., 1965). Thefree energy can be (and is) expressed in different ways,including

1. Potential (dimensions—L2T�2: J/kg)2. Head (dimensions—L: m, cm, ft)3. Pressure (dimensions—ML�1 T�2: kN/m2, dyn/

cm2, tons/m2, atm, bar, psi, psf)

If the free energy is less than that of pure waterunder the ambient air pressure, the terms suction andnegative pore water pressure are used.

The total potential (head, pressure) of soil water isthe potential (head, pressure) in pure water that willcause the same free energy at the same temperature asin the soil water. An alternative definition of total po-tential is the work per unit quantity to transport re-versibly and isothermally an infinitesimal amount ofpure water from a pool at a specified elevation at at-mospheric pressure to the point in soil water underconsideration.

The selection of the components of the total poten-tial � (total head H, total pressure P) is somewhatarbitrary (Bolt and Miller, 1958); however, the follow-ing have gained acceptance for geotechnical work(Aitchison, et al., 1965):

1. Gravitational potential �g (head Z, pressure pz)corresponds to elevation head in normal hydrau-lic usage.

2. Matrix or capillary potential �m (head hm, pres-sure p) is the work per unit quantity of water totransport reversibly and isothermally an infinites-imal quantity of water to the soil from a poolcontaining a solution identical in composition tothe soil water at the same elevation and externalgas pressure as that of the point under consider-ation in the soil. This component corresponds tothe pressure head in normal hydraulic usage. Itresults from that part of the boundary stressesthat is transmitted to the water phase, from pres-sures generated by capillarity menisci, and fromwater adsorption forces exerted by particle sur-faces. A piezometer measures the matrix poten-tial if it contains fluid of the same compositionas the soil water.

3. Osmotic (or solute) potential �s (head hs, pres-sure ps) is the work per unit quantity of water totransport reversibly and isothermally an infinites-imal quantity of water from a pool of pure waterat a specified elevation and atmospheric pressureto a pool containing a solution identical in com-position to the soil water, but in all other respectsidentical to the reference pool. This componentis, in effect, the osmotic pressure of the soil wa-ter, and it depends on the composition and abilityof the soil particles to restrain the movement ofadsorbed cations. The osmotic potential is nega-tive, that is, water tends to flow in the directionof increasing concentration.

The total potential, head, and pressure then become

� � � � � � � (7.19)g m s

H � Z � h � h (7.20)m s

P � p � p � p (7.21)z s

At equilibrium and no flow there can be no varia-tions in �, H, or P within the soil.

7.7 WATER PRESSURE EQUILIBRIUM IN SOIL

Consider a saturated soil mass as shown in Fig. 7.7.Conditions at several points will be analyzed in termsof heads for simplicity, although potential or pressurecould also be used with the same result. The system isassumed at constant temperature throughout. At point0, a point inside a piezometer introduced to measure

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Figure 7.7 Schematic representation of a saturated soil for analysis of pressure conditions.

pore pressure, Z � 0, hm � hm0, and hs0 � 0 if purewater is used in the piezometer. Thus,

H � 0 � h � 0 � h0 m0 m0

It follows that

P � h � u (7.22)0 m0 w 0

the measured pore pressure.Point 1 is at the same elevation as point 0, except it

is inside the soil mass and midway between two clayparticles. At this point, Z1 � 0, but hs 0 because theelectrolyte concentration is not zero. Thus,

H � 0 � h � h1 m1 s1

If no water is flowing, H1 � H0, and

h � h � hm1 s1 m0

Also, because p1 � p0 � u0

u � h � h (7.23)0 m1 w s1 w

At point 2, which is between the same two clay par-ticles as point 1 but closer to a particle surface, therewill be a different ion concentration than at 1. Thus,at equilibrium, and assuming Z2 � 0,

u0h � h � h � h � h �m2 s2 m1 s1 m0 w

A similar analysis could be applied to any point in thesystem. If point 3 were midway between two clay par-ticles spaced the same distance apart as the particleson either side of point 1, then hs3 � hs1, but Z3 0.Thus,

u0 � Z � h � h � Z � h � h (7.24)3 m3 s3 3 m3 s1w

A partially saturated system can also be analyzed,but the influences of curved air–water interfaces mustbe taken into account in the development of the hm

terms.The conclusions that result from the above analysis

of component potentials are:

1. As the osmotic and gravitational componentsvary from point to point in a soil at equilibrium,

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MEASUREMENT OF PORE PRESSURES IN SOILS 183

the matrix or capillary component must also varyto maintain equal total potential. The concept thathydrostatic pressure must vary with elevation tomaintain equilibrium is intuitive; however, theidea that this pressure must vary also in responseto compositional differences is less easy to vi-sualize. Nonetheless, this underlies the wholeconcept of water flow by chemical osmosis.

2. The total potential, head, and pressure are meas-urable, and separation into components is possi-ble experimentally, although it is difficult.

3. A pore pressure measurement using a piezometercontaining pure water gives a pressure u0 � wh,where h is the pressure head at the piezometer.When referred back to points between soil par-ticles, u0 is seen to include contributions fromosmotic pressures as well as matrix pressures.Since osmotic pressures are the cause of long-range repulsions due to double-layer interactions,measured pore water pressures may include con-tributions from long-range interparticle repulsiveforces.

7.8 MEASUREMENT OF PORE PRESSURES INSOILS

Several techniques for the measurement of pore waterpressures are available. Some are best suited for lab-oratory use, whereas others are intended for use in thefield. Some yield the pore pressure or suction by directmeasurement, while others require deduction of thevalue using thermodynamic relationships.

1. Piezometers of Various Types Water in the pi-ezometer communicates with the soil through aporous stone or filter. Pressures are determinedfrom the water level in a standpipe, by a manom-eter, by a pressure gauge, or by an electronicpressure transducer. A piezometer used to mea-sure pressures less than atmospheric is usuallytermed a tensiometer.

2. Gypsum Block, Porous Ceramic, and FilterPaper The electrical properties across a spe-cially prepared gypsum block or porous ceramicblock are measured. The water held by the blockdetermines the resistance or permittivity, and themoisture tension in the surrounding soil deter-mines the amount of moisture in the block(Whalley et al., 2001). The same principle can beapplied by placing a dry filter paper on a soilspecimen and allowing the soil moisture to ab-sorb into the paper. When the suction in the filterpaper is equal to the suction in the soil, the two

reach equilibrium, and the suction can be deter-mined by the water content of the filter paper.These techniques are used for measurement ofpore pressures less than atmospheric.

3. Pressure-Membrane Devices An exposed soilsample is placed on a membrane in a sealedchamber. Air pressure in the chamber is used topush water from the pores of the soil through themembrane. The relationship between water con-tent and pressure is used to establish the relation-ship between soil suction and water content.

4. Consolidation Tests The consolidation pressureon a sample at equilibrium is the soil water suc-tion. If the consolidation pressure were instanta-neously removed, then a negative water pressureor suction of the same magnitude would beneeded to prevent water movement into the soil.

5. Vapor Pressure Methods The relationship be-tween relative humidity and water content is usedto establish the relationship between suction andwater content.

6. Osmotic Pressure Methods Soil samples areequilibrated with solutions of known osmoticpressure to give a relationship between watercontent and water suction.

7. Dielectric Sensors Such as Capacitance Probesand Time Domain Reflectometry Soil moisturecan be indirectly determined by measuring thedielectric properties of unsaturated soil samples.With the knowledge of soil water characteristicsrelationship (Section 7.11), the negative porepressure corresponding to the measured soilmoisture can be determined. The capacitanceprobe measures change in frequency response ofthe soil’s capacitance, which is related to dielec-tric constants of soil particle, water, and air. Thecapacitance is largely influenced by water con-tent, as the dielectric constant of water is largecompared to the dielectric constants of soilparticle and air. Time domain reflectrometrymeasures the travel time of a high-frequency,electromagnetic pulse. The presence of water inthe soil slows down the speed of the electromag-netic wave by the change in the dielectric prop-erties. Volumetric water content can therefore beindirectly measured from the travel time mea-surement.

Piezometer methods are used when positive porepressures are to be measured, as is usually the case indams, slopes, and foundations on soft clays. The othermethods are suitable for measurement of negative porepressures or suction. Pore pressures are often negativein expansive and partly saturated soils. More detailed

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descriptions and comparisons of these and other meth-ods are given by Croney et al. (1952), Aitchison et al.(1965), Richards and Peter (1987), and Ridley et al.(2003).

7.9 EFFECTIVE AND INTERGRANULARPRESSURE

In Section 7.5, it was shown that the intergranular pres-sure is given by

�� A � u (7.25)i

where u is the hydrostatic pressure between particles(or hmw in the terminology of Section 7.7). General-ized forms of Eq. (7.24) are

u � Z � h � h (7.26)0 w m w s w

and

u � h � u � Z � h (7.27)m w 0 w s w

Thus, Eq. (7.25) becomes, for the case of no elevationdifference between a piezometer and the point in ques-tion (i.e., Z � 0),

�� A � u � h (7.28)i 0 s w

Because the quantity hsw is an osmotic pressure andthe salt concentration between particles will invariablybe greater than at points away from the soil (such asin a piezometer), hsw will be negative. This pressurereflects double-layer repulsions. It has been termed Rin some previous studies (Lambe, 1960; Mitchell,1962). If hsw in Eq. (7.28) is replaced by the absolutevalue of R, we obtain

�� A � u � R (7.29)i 0

From Eq. (7.25), it was seen that the intergranularpressure was dependent on long-range interparticle at-tractions A as well as on the applied stress � and thepore water pressure between particles u. Equation(7.29) indicates that if intergranular pressure is to��ibe expressed in terms of a measured pore pressure u0,then the long-range repulsion R must also be taken intoaccount. The actual hydrostatic pressure between par-ticles u � u0 � R includes the effects of long-rangerepulsions as required by the condition of constant to-tal potential for equilibrium.

In the general case, therefore, the true intergranularpressure � � � A � u0 � R and the conventionally��i

defined effective stress �� u0 differ by the netinterparticle stress due to physicochemical contribu-tions,

�� A � R (7.30)i

When A and R are both small, as would be true ingranular soils, silts, and clays of low plasticity, or incases where A � R, the intergranular and effectivestress are approximately equal. Only in cases whereeither A or R is large, or both are large but of signifi-cantly different magnitude, would the intergranular andeffective stress be significantly different. Such a con-dition appears not to be common, although it might beof importance in a well-dispersed sodium montmoril-lonite, where compression behavior can be accountedfor reasonably well in terms of double-layer repulsions(Chapter 10).4

The derivation of Eq. (7.30) assumed vertical equi-librium, with contributing forces parallel to each other,that is, the intergranular stress is the sum of the��iskeletal forces (defined as �� u0) and the elec-trochemical stress (A � R), as illustrated in Fig. 7.8a.This implies that the deformation induced by the elec-trochemical stress (A � R) is equal to the deformationinduced by the skeletal forces at contacts [i.e., a ‘‘par-allel’’ model as described by Hueckel (1992)]. Thechange in pore fluid chemistry at constant confinement(��) leads to changes in intergranular stresses re-(��),i

sulting in changes in shear strength, for example.An alternative assumption can be made; the total

deformation of soil is the sum of the deformations ofthe particles and in the double layers as illustrated inFig. 7.8b. The effective stress �� is then equal to theelectrochemical stress (R � A):

�� R � A � �� u (7.31)i 0

This is called the ‘‘series’’ model (Hueckel, 1992), andthe model can be applicable for very fine soils at highwater content, in which particles are not actually incontact with each other but are aligned in a parallelarrangement. Increase in intergranular stress or ef-��ifective stress �� changes the interparticle spacing,which may contribute to changes in strength propertiesupon shearing.

4 A detailed analysis of effective stress in clays is presented by Chat-topadhyay (1972), which leads to similar conclusions, including Eq.(7.29). was termed the true effective stress and it governed the��ivolume change behavior of Na–montmorillonite.

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ASSESSMENT OF TERZAGHI’S EQUATION 185

Skeletal Forceσ� = σ _ u0

ElectrochemicalForce A _ R

σi�

σi�

Deformation atthe Contact

σi� = σ _ u0 + A _ R

(a)

Total Deformationat the Contactσ�i

σi�

σi� = σ _ u0 = A _ R

(b)

Particle Deformationby Skeletal Force

ElectrochemicalForce A _ R

Skeletal Forceσ� = σ _ u0

Electrochemical Force

Skeletal Force

Skeletal Force Electrochemical Force

Skeletal ForceElectrochemical Force Electrochemical Force

Skeletal Force

Figure 7.8 Contribution of skeletal force (� � u0) and electrochemical force (A � R) tointergranular force �i: (a) parallel model and (b) series model.

Since the particles are arranged in parallel as wellas nonparallel manner, the chemomechanical couplingbehavior of actual soils can be far from the predictionsmade by the above two models. In fact, Santamarina(2003) argues that the impact of skeletal forces by ex-ternal forces, particle-level forces, and contact-levelforces on soil behavior is different, and mixing bothtypes of forces in a single algebraic expression in termsof effective stress can lead to incorrect prediction [e.g.,Eq. (7.15) for unsaturated soils and Eq. (7.30) for soilswith measurable interparticle repulsive and attractiveforces].

7.10 ASSESSMENT OF TERZAGHI’S EQUATION

The preceding equations and discussion do not confirmthat Terzaghi’s simple equation is indeed the effectivestress that governs consolidation and strength behaviorof soils. However, its usefulness has been establishedfrom the experience of many years of successful ap-plication in practice. Skempton (1960b) showed thatthe Terzaghi equation does not give the true effectivestress but gives an excellent approximation for the case

of saturated soils. Skempton proposed three possiblerelationships for effective stress in saturated soils:

1. The true intergranular pressure for the case whenA � R � 0

�� (1 � a )u (7.32)c

in which ac is the ratio of contact area to totalcross-sectional area.

2. The solid phase is treated as a real solid that hascompressibility Cs and shear strength given by

� � k � � tan � (7.33)i

where � is an intrinsic friction angle and k is atrue cohesion. The following relationships werederived: For shear strength,

a tan �c�� 1 � u (7.34)� �tan ��

where �� is the effective stress angle of shearingresistance. For volume change,

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Table 7.1 Compressibility Values for Soil, Rock,and Concrete

Material

Compressibilitya

per kN/m2 � 10�6

C Cs Cs /C

Quartzitic sandstone 0.059 0.027 0.46Quincy granite (30 m deep) 0.076 0.019 0.25Vermont marble 0.18 0.014 0.08Concrete (approx.) 0.20 0.025 0.12Dense sand 18 0.028 0.0015Loose sand 92 0.028 0.0003London clay (over cons.) 75 0.020 0.00025Gosport clay (normally cons.) 600 0.020 0.00003

After Skempton (1960b).aCompressibilities at p � 98 kN/m2; water Cw � 0.49

� 10�6 per kN/m2.

Cs�� 1 � u (7.35)� �C

where C is the soil compressibility.3. The solid phase is a perfect solid, so that � � 0

and Cs � 0. This gives

�� u (7.36)

To test the three theories, available data were studiedto see which related to the volume change of a systemacted upon by both a total stress and a pore waterpressure according to

V� �C �� (7.37)

V

and also satisfied the Coulomb equation for drainedshear strength �d :

� � c� � �� tan �� (7.38)d

when both a total stress and a pore pressure are acting.It may be noted that this approach assumes that theCoulomb strength equation is valid a priori.

The results of Skempton’s analysis showed that Eq.(7.32) was not a valid representation of effective stress.Equations (7.34) and (7.35) give the correct results forsoils, concrete, and rocks. Equation (7.36) accountswell for the behavior of soils but not for concrete androck. The reason for this latter observation is that insoils Cs /C and ac tan � / tan �� approach zero, and,thus, Eqs. (7.34) and (7.35) reduce to Eq. (7.36). Inrock and concrete, however, Cs /C and ac tan � / tan ��are too large to be neglected. The value of tan � / tan�� may range from 0.1 to 0.3, ac clearly is not negli-gible, and Cs /C may range from 0.1 to 0.5 as indicatedin Table 7.1.

Effective stress equations of the form of Eqs. (7.32),(7.34), (7.35), and (7.36) can be generalized to the gen-eral form (Lade and de Boer, 1997):

�� u (7.39)

where � is the fraction of the pore pressure that givesthe effective stress.5 Different expressions for � pro-posed by several researchers are listed in Table 7.2.

5 A more general expression has been proposed as �ij � � �iju,��ijwhere �ij is the tensor that accounts for the constitutive characteristicsof the solid such as complex kinematics associated with anisotropicelastic materials (Carroll and Katsube, 1983; Coussy, 1995; Did-wania, 2002).

A more rigorous evaluation of the contribution ofsoil particle compressibility to effective stress wasmade by Lade and de Boer (1997) using a two-phasemixture theory. The volume change of the soil skeletoncan be separated into that due to pore pressure incre-ment u and that due to the change in confining pres-sure (� � u) (or � � u). The effective stressincrement �� is defined as the stress that produces thesame volume change,

CV �� V � V � CV ( � � u)0 sks sku 0

� C V u (7.40)u 0

where Vsks is the volume change of soil skeleton dueto change in confining pressure, Vsku is the volumechange of soil skeleton due to pore pressure change,V0 is the initial volume, C is the compressibility of thesoil skeleton by confining pressure change, and Cu isthe compressibility of the soil skeleton by pore pres-sure change. Rearranging Eq. (7.40) leads to

Cu �� 1 � u (7.41)� �C

Lade and de Boer (1997) used this equation to de-rive an effective stress equation for granular materialsunder drained conditions. Consider a condition inwhich the total confining pressure is constant [ (� �

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ASSESSMENT OF TERZAGHI’S EQUATION 187

Table 7.2 Expressions for � to Define Effective Stress

Pore Pressure Fraction � Note Reference

1 Terzaghi (1925b)n n � porosity Biot (1955)

1 � ac ac � grain contact area per unit area of plane Skempton and Bishop (1954)

1 � ac

tan �

tan ��

Equation (7.34) Skempton (1960b)

1 �Cs

CEquation (7.35); for isotropic elastic

deformation of a porous material; for solidrock with small interconnected pores andlow porosity (Lade and de Boer, 1997)

Biot and Willis (1957), Skempton(1960b), Nur and Byerlee (1971), Ladeand de Boer (1997)

1 � (1 � n)Cs

CEquation (7.43) Suklje (1969); Lade and de Boer (1997)

After Lade and de Boer (1997).

Figure 7.9 Variation of � with stress for quartz sand andgypsum sand (Lade and de Boer, 1997).

u) � 0], but the pore pressure changes by u.6 Thevolume change of soil skeleton caused by change inpore pressure ( Vsku) is attributed solely from the vol-umetric compression of the solid grains ( Vgu). Hence,

V C V u � C (1 � n)V u V orsku u 0 s 0 gu

C � C (1 � n)u s (7.42)

where Cs is the compressibility of soil grains due topore pressure change and n is the porosity. SubstitutingEq. (7.42) into (7.41) gives

Cs �� 1 � (1 � n) u or �C

Cs� � 1 � (1 � n) (7.43) �C

Figure 7.9 shows the variations of � with stress forquartz sand and gypsum sand (Lade and de Boer,1997). For a stress level less than 20 MPa, � is essen-tially one. Thus, Terzaghi’s effective stress equation,while not rigorously correct, is again shown to be anexcellent approximation in almost all cases for satu-rated soils (i.e., solid grains and pore fluid are consid-ered to be incompressible compared to soil skeletoncompressibility).

6 An example of this condition is a soil under a seabed, in which thesea depth varies. This condition is often called the ‘‘unjacked con-dition.’’

Can the effective stress concept also be applied forundrained conditions where drainage is prevented?That is, when an isotropic total stress load of �iso isapplied, is u equal to �iso? Using a two-phase mix-ture theory, the total stress increment ( �iso) is sepa-rated into partial stress increments for the solid phase( �s) and the fluid phase ( �ƒ) (Oka, 1996). Consid-ering that the macroscopic volumetric strains by twophases are equal but of opposite sign for undrained

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θ

θ

Solid Surface

AirWater(reference fluid)

Air

Solid surface

(a)

Water(reference fluid)

(b)

(c)

Air

Water

Solid

Figure 7.10 Wettability of two fluids (water and air) on asolid surface: (a) contact angle less than 90�, (b) contact an-gle more than 90�, and (c) unsaturated sand with water as thewetting fluid and air as the nonwetting fluid.

conditions, Oka (1996) showed that the partial stressesare related to the total stress as follows:

C � Cs � � �ƒ iso(C /n) � (1 � 1/n)C � Cs l (7.44)

[(1/n) � 1]C � (C /n) � Cs l � � �s iso(C /n) � (1 � 1/n)C � Cs l

where n is the porosity, C is the compressibility of soilskeleton, Cs is the compressibility of soil particles, andCl is the compressibility of pore fluid.

If the excess pore pressure generated by undrainedisotropic loading � is u, the partial stress incrementfor the fluid phase becomes (Oka, 1996)

� � n u (7.45)ƒ

Combining Eqs. (7.45) and (7.46),

C � Cs u � � (7.46)isoC � C � n(C � C )s l s

The multiplier in the right-hand side of the aboveequation is in fact Bishop’s pore water pressure coef-ficient B (Bishop and Eldin, 1950).7 For typical soils(Cs � 1.9 � 2.7 � 10�8 m2 /kN, Cl � 4.9 � 10�9

m2/kN, C � 10�5 � 10�4 m2/kN), so the values of Bare roughly equal to 1. Hence, it can be concluded thatTerzaghi’s effective stress equation is also applicablefor undrained conditions for most soils.

7.11 WATER–AIR INTERACTIONS IN SOILS

Wettability refers to the affinity of one fluid for a solidsurface in the presence of a second or third fluid orgas. A measure of wettability is the contact angle,which was introduced in Eq. (7.9). Figure 7.10 illus-trates a drop of the reference liquid (water for Fig.7.10a and air for Fig. 7.10b) resting on a solid surfacein the presence of another fluid (air for Fig. 7.10a andwater for 7.10b). The interface between the two fluidsmeets the solid surface at a contact angle �. If the angleis less than 90�, the reference fluid is referred to as thewetting fluid for a given solid surface. If the angle isgreater than 90�, the reference liquid is referred to asthe nonwetting phase. The figure shows that water and

7 A similar equation for B value has been proposed by Lade and deBoer (1997).

air are the wetting and nonwetting fluid, respectively.8

The environmental SEM photos in Fig. 5.27 showedthat water can be either wetting or nonwetting fluiddepending soil mineralogy.

The contact angle is a property related to interac-tions of solid and two fluids (water and air, in thiscase).

� � �as wscos � � (7.47)�aw

where �as is the interfacial tension between air andsolid, �ws is the interfacial tension between waterand solid, and �aw is the interfacial tension between

8 Some contaminated sites contain non-aqueous-phase liquids(NAPLs). In general, NAPLS can be assumed to be nonwetting withrespect to water since the soil particles are in general primarilystrongly water-wet. Above the water table, it is usually appropriateto assume that the water is the wetting fluid with respect to NAPLand that NAPL is a wetting fluid with respect to air, implying thatthe wettability order is water � NAPL � air. Below the water table,water is the wetting fluid and NAPL is the nonwetting fluid.

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WATER–AIR INTERACTIONS IN SOILS 189

Volumetric Water Content θ w

0.1 0.2 0.3 0.4 0.5 0.60.010-1

100

101

102

103

104

105

106

1

2

3

5

4

6

7

1 Dune Sand2 Loamy Sand3 Calcareous Fine Sandy Loam4 Calcareous Loam5 Silt Loam Derived from Loess6 Young Oligotrophous Peat Soil7 Marine Clay

Mat

ric s

uctio

n u a

– u

w (

kPa)

Figure 7.11 Soil–water characteristic curves for some Dutchsoils (from Koorevaar et al., 1983; copied from Fredlund andRahardjo, 1993).

air and water. The microscopic scale distribution ofwater and air is illustrated in Fig. 7.10c, whereby it isassumed that water is wetting the grain surfaces.

The aforementioned discussion on wettability andcontact angle assumes static water drops on solid sur-faces. It has been observed for movement of water rel-ative to soil that the ‘‘dynamic’’ contact angle formedby the receding edge of a water droplet is generallyless than the angle formed by its advancing edge.

Matric suction (or capillary pressure) refers to thepressure discontinuity across a curved interface sepa-rating two fluids. This pressure difference exists be-cause of the interfacial tension present in the fluid–fluid interface. Matric suction is a property that causesporous media to draw in the wetting fluid and repelthe nonwetting fluid and is defined as the differencebetween the nonwetting fluid pressure and the wettingfluid pressure. For a two-phase system consisting ofwater and air, the matric suction � is

� � u � u (7.48)n w

where un is the pressure of the nonwetting fluid (air)and uw is the pressure of the wetting fluid (water).

Assuming that the soil pores have a cylindricalshape, like a bundle of capillary tubes as illustrated inFig 7.3b, the interface between two liquids in each tubeforms a subsection of a sphere. The capillary pressureis then related to the tube radius, contact angle, andthe interfacial tension between the two liquids. Thepressure drop across the interface is directly propor-tional to the interfacial tension and inversely propor-tional to the radius of curvature. It follows that higherair pressure is required for air to enter water-saturatedfine-grained than coarse-grained materials.

Soil contains a range of different pore sizes, whichwill drain at different capillary pressure values. Thisleads to a soil–water characteristic relationship inwhich the matric suction is plotted against the volu-metric water content (or sometimes water saturationratio) such as shown in Fig. 7.11.9 The curves are oftendetermined during air invasion into a previously water-saturated soil. As the volumetric water content de-creases, as a result of drainage or evaporation, thematric suction increases. When water infiltrates intothe soil (wetting or imbibition), the conditions reverse,with the volumetric water content increasing and ma-tric suction decreasing. Usually drainage and wetting

9 The soil–water characteristic curve is referred to by a variety ofnames depending on different disciplines. They include moisture re-tention, soil–water retention, specific retention, and moisture char-acteristic.

processes do not follow the same curve and the volu-metric water content versus matric suction curves ex-hibit hysteresis during cycles of drainage and wettingas shown in Fig. 7.12a. One cause of hysteresis is theexistence of ‘‘ink bottle neck’’ pores at the microscopicscale as shown in Fig. 7.12b. Larger water-filled porescan remain owing to the inability of water to escapethrough smaller openings below in the case of drainageor above in the case of evaporation. Another cause isirreversible change in soil fabric and shrinkage duringdrying.

The curves in Fig. 7.11 have two characteristicpoints—the air entry pressure �a and residual volu-metric water content �r as defined in Fig. 7.12a. Theentry pressure is the matric suction at which the airbegins to enter the pores and the pores become inter-connected (Corey, 1994). At this point, the air per-meability becomes greater than zero. Corey (1994)also introduced the term ‘‘displacement pressure’’ (�d

in Fig. 7.12b) and defined it as the matric suction atwhich the first water desaturation occurs during adrainage cycle.10 The entry pressure is always slightly

10 For the Dense NAPL–water two-phase system (often Dense NAPLis the nonwetting fluid and water is the wetting fluid), the displace-ment pressure may be important to examine the potential of DNAPLinvading into a noncontaminated water-filled porous media.

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Water Content

Initial drainageCurve

Main WettingCurve

ScanningCurve

Main Drying Curve

Hysteresis ScanningCurve

(a)

θ r

ψa

θ r Residual Water Content

ψa Air Entry Value

Draining

Wetting

(b)

ψd

ψd Displacement pressure

Suc

tion

Figure 7.12 Hysteresis of a soil–water characteristic curve: (a) effect of hysteresis and (b)ink bottle effect: a possible physical explanation for the hysteresis.

greater than the displacement pressure because porethroats smaller than the maximum must be penetratedto establish air connectivity. The air entry pressure ismuch greater for fine-grained than for coarse-grainedsoils because of their smaller pore sizes.

Residual water content �r is defined as the watercontent that cannot be further reduced by the increasein matric suction. At this stage, the water phasebecomes essentially discontinuous and the regimechanges from the funicular to pendular state, as de-scribed in Section 7.4. However, this does not meanthat the soil cannot have a degree of saturation lessthat the residual saturation because residual water cancontinue to evaporate. Hence, it is important to notethat the residual saturation defined here is a mathe-matical fitting parameter without a specific quantitativevalue.

The shape of the soil–water characteristic curve de-pends on many factors, including the grain size distri-bution, soil fabric, the contact angle, and the interfacialtension [see Eq. (7.11)]. If the material is uniform witha narrow range of pore sizes, the curve has three dis-tinct parts: a straight part up to the air entry pressure,a relatively horizontal middle part, and an end part thatis almost vertical (soil 1 in Fig. 7.11). On the otherhand, if the material is well graded, the curve issmoother (soils 3, 4, and 5 in Fig. 7.11). The capillarypressure increases gradually as the water saturation de-creases and the middle part is not horizontal. Many

algebraic formulas have been proposed to fit the mea-sured soil-water characteristic relations. The most pop-ular ones are (a) the Brooks–Corey (1966) equation:

� � � when � � � (7.49)m d

�1/�� r� � � when � � � (7.50)� �d d� � �m r

where �m is the volumetric water content at fullsaturation and � is the curve-fitting parameter calledthe pore size distribution index and (b) the van Gen-uchten equation (1980):

�1 / m 1�m� � �r� � � � 1 (7.51) � � �0 � � �m r

where �0 and m are curve-fitting parameters.Various modifications have been proposed to these

equations to include behaviors such as hysteresis, non-wetting fluid trapping, and three-phase conditions.

7.12 EFFECTIVE STRESS IN UNSATURATEDSOILS

Although it seems clear that the volume change andstrength behavior of partly saturated soils are con-

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EFFECTIVE STRESS IN UNSATURATED SOILS 191

(ua_uw)/(ua

_uw)b

(ua_uw)b = Air Entry Value

(a) (b)

S (%)

1. CompactedBoulder Clay2. Compacted Shale3. Breadhead silt4. Silt5. Silty clay6. Sterrebeek silt7. White clay

Degree of Saturation

Coe

ffici

entχ

χ =(ua – uw)(ua – uw)

– 0.55

Coe

ffici

entχ

Figure 7.13 Variation of parameter � with the degree of water saturation Sr for differentsoils: (a) � versus water saturation (after Gens, 1996) and (b) � versus suction (after Khaliliand Khabbaz, 1998).

Applied Pressure (kPa )

Voi

d R

atio

e

10 100 10000.64

0.68

0.72

0.76

0.80

0.84

Air Dry (8 specimens)

Soaked at Constant Void Ratio

Soaked at Constant AppliedPressure

Initially Soaked Test

Figure 7.14 Oedometer compression curves of unsaturatedsilty soils (after Jennings and Burland, 1962 in Leroueil andHight, 2002).

trolled by an effective stress that is not the same as thetotal stress, the appropriate formulation for the effec-tive stress is less certain than for a fully saturated soil.As noted earlier, Bishop (1960) proposed Eq. (7.15)(assuming �� ):��i

�� u � �(u � u ) (7.52)a a w

The term � � ua is the net total stress. The termua � uw represents the soil water suction that adds tothe effective stress since uw is negative. Thus, theBishop equation is appealing intuitively because neg-ative pore pressures are known to increase strength anddecrease compressibility. Using Eq. (7.52), the shearstrength of unsaturated soil can be expressed as

� � {(� � u ) � �(u � u )}tan �� (7.53)a a w

where �� is the effective friction angle of the soil.However, difficulties in the evaluation of the parameter�, its dependence on saturation (� � 1 for saturatedsoils and � � 0 for dry soils), and that the relationshipbetween � and saturation is soil dependent, as shownin Fig. 7.13a, all introduce problems in the applicationof Eq. (7.53). Since water saturation is related to matricsuction as described in Section 7.11, it is possible that� depends on matric suction as shown in Fig. 7.13b.Nonetheless, because of the complexity in determining�, the attempt to couple total stress and suction to-gether into a single equivalent effective stress is un-certain (Toll, 1990).

Limitations in Bishop’s equation were highlightedby Jennings and Burland (1962) in their experimentsinvestigating the volume change characteristics of un-saturated soils. Figure 7.14 shows that the oedometercompression curve of air-dry silt falls above that ofsaturated silt. Also, as shown in the figure, some air-dry samples were consolidated at four different pres-sures (200, 400, 800, and 1600 kPa) and then soaked.

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Preconsolidationpressure

25 50 100 200 4000.95

1.00

1.05

1.10

1.15

1.20

1.25

σ _ua (kPa)

ua_ uw (kPa)

300 kPa

200 kPa

100 kPa0 kPa

Curves are Averages ofSeveral Tests

Voi

d R

atio

e

Figure 7.15 Isotropic compression tests of compacted kaolin(after Wheeler and Sivakumar, 1995 in Leroueil and Hight,2002).

The void ratio decreased upon soaking and the finalstate was very close to the compression curve of thesaturated silt. Additional tests in which constant vol-ume during soaking was maintained by adjusting theapplied load were also done. Again, after equilibrium,the state of soaked samples was close to the compres-sion curve of the saturated silt. Soaking reduces thesuction and, hence, Bishop’s effective stress decreases.This decrease in effective stress should be associatedwith an increase in void ratio. However, the experi-mental observations gave the opposite trend (i.e., a de-crease in void ratio is associated with irreversiblecompression). The presence of meniscus water lensesin the soil before wetting was stabilizing the soil struc-ture, which is not taken into account in Bishop’s equa-tion (7.52).

An alternative approach is to describe the shearstrength/deformation and volume change behavior ofunsaturated soil in terms of the two independent stressvariables � � ua and ua � uw (Coleman, 1962; Bishopand Blight, 1963; Fredlund and Morgenstern, 1977;Fredlund, 1985; Toll, 1990, Fredlund and Rahardjo,1993; Tarantino et al., 2000). Figure 7.15 shows theresults of isotropic compression tests of compacted ka-olin. Different compression curves are obtained forconstant suction conditions, and relative effects of � �ua and ua � uw on volume change behavior can beobserved. Furthermore, the preconsolidation pressure(or yield stress) increases with suction.

On this basis, the dependence of shear strength � onstress is given by equations of the form

� � a(� � u ) � b(u � u ) (7.54)a a w

in which a and b are material parameters that may alsodepend on degree of saturation and stress. For exam-ple, Fredlund et al. (1978) propose the following equa-tion:

b� � (� � u )tan �� (u � u )tan � (7.55)a a w

where �b is the angle defining the rate of increase inshear strength with respect to soil suction. An exampleof this parameter as a function of water content, fric-tion angle, and matric suction is given by Fredlund etal. (1995).

Similarly, the change in void ratio e of an unsat-urated soil can be given by (Fredlund, 1985)

� � a (� � u ) � a (u � u ) (7.56)t a m a w

where at is the coefficient of compressibility with re-spect to changes in � � ua and am is the coefficient ofcompressibility with respect to changes in capillarypressure. A similar equation, but with different coef-ficients, can be written for change in water content.For a partly saturated soil, change in water content andchange in void ratio are not directly proportional.

The two stress variables, or their modifications thatinclude porosity and water saturation, have been usedin the development of elasto-plastic-based constitutivemodels for unsaturated soils (e.g., Alonso et al., 1990;Wheeler and Sivakumar, 1995; Houlsby, 1997; Gallip-oli et al., 2003). The choice of stress variables is stillin debate; further details on this issue can be found inGens (1996), Wheeler and Karube (1996), Wheeler etal. (2003), and Jardine et al. (2004).

Bishop’s � parameter in Eq. (7.52) is a scalar quan-tity, but microscopic interpretation of water distributionin pores can lead to an argument that � is directionaldependent (Li, 2003; Molenkamp and Nazemi,2003).11 During the desaturation process, the numberof soil particles under a funicular condition decreases,and they change to a pendular condition with furtherdrying. For particles in the funicular region, the suctionpressure acts all around the soil particles like the waterpressure as illustrated in Fig. 7.4a. Hence, the effect isisotropic even at the microscopic level. However, oncethe microscopic water distribution of a particle changesto the pendular condition, the capillary forces only acton a particle at locations where water bridge forms andthe contribution to the interparticle forces becomes

11 A microstructural analysis by Li (2003) suggests the following ef-fective stress expression:

�� u � � � (u � u )ij ij a ij ij a w

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QUESTIONS AND PROBLEMS 193

more or less point wise, as shown in Fig. 7.4b. Asdescribed in Section 7.3, the magnitude of capillaryforce depends on the size of the water bridge and theseparation of the two particles, and hence, the contactforce distribution in the particle assembly becomes de-pendent not only on pore size location and distributionbut also on the relative locations of particles to oneanother (or soil fabric). It is therefore possible that thedistribution of the pendular-type capillary forces be-comes directional dependent.

In clayey soils, water is attracted to clay surface byelectrochemical forces, creating large matric suction.Although uw � u0 is used in practice, the actual porepressure u acting at interparticle contacts may be dif-ferent from u0, as discussed in Section 7.9. The con-tribution of the long-range interparticle forces tomechanical behavior of unsaturated clayey soils re-mains to be fully evaluated.

7.13 CONCLUDING COMMENTS

The concepts in this chapter provide insight into themeanings of intergranular pressure, effective stress,and pore water pressure and the factors controllingtheir values. Because soils behave as particulate ma-terials and not as continua, knowledge of these stressesand of the factors influencing them is a necessary pre-requisite to the understanding and quantification ofcompressibility, deformation, and strength in constitu-tive relationships for behavior. Various interparticleforces have been identified and their possible effectson soil behavior are highlighted.

The effective stress in a soil is a function of its state,which depends on the water content, density, and soilstructure. These factors are, in turn, influenced by thecomposition and ambient conditions. The relationshipsbetween soil structure and effective stress are devel-oped further in Chapter 8. Chemical, electrical, andthermal influences on effective pressures and fluidpressures in soils have not been considered in the de-velopments in this chapter. They may be significant,however, as regards soil structure stability fluid flow,volume change, and strength properties. They are an-alyzed in more detail in subsequent chapters.

An understanding of the components of pore waterpressure is important to the proper measurement ofpore pressure and interpretation of the results. Inclu-sion of the effect of pore water suction and air or gaspressure on the mechanical behavior of unsaturatedsoils requires modification of the effective stress equa-tion used for saturated soils. Complications arise fromthe difficulty in the choice of stress variables and intreatment of contact-level forces (i.e., capillary forces

in the pendular regime) in the macroscopic effectivestress equations.

QUESTIONS AND PROBLEMS

1. A sand in the ground has porosity n of 0.42 andspecific gravity Gs of 2.6. It is assumed that thesevalues remain constant throughout the depth. Thewater table is 4 m deep and the groundwater is un-der hydrostatic condition. The suction–volumetricwater content relation of the sand is given by soil1 in Fig. 7.11.a. Calculate the saturated unit weight and dry unit

weight.b. Evaluate the unit weights at different saturation

ratios Sw.c. Plot the hydrostatic pore pressures with depth

down to a depth of 10 m and evaluate the satu-ration ratios above the water table.

d. Along with the hydrostatic pore pressure plot,sketch the vertical total stress with depth usingthe unit weights calculated in parts (a) and (b).

e. Estimate the vertical effective stress with depth.Use Bishop’s equation (7.52) with � � Sw. Com-ment on the result.

2. Repeat the calculations done in Question 1 with soil5 in Fig. 7.11. The specific gravity of the soil is2.65. Comment on the results by comparing themto the results from Question 1.

3. Using Eq. (7.3), estimate the tensile strength of asoil with different values of tensile strengths of ce-ment, sphere, and interface. The soil has a particlediameter of 0.2 mm and the void ratio is 0.7. As-sume k / (1 � e) � 3.1. Consider the following twocases: (a) � � 0.0075 mm and � � 5� and (b)� � 0.025 and � � 30�. Comment on the results.

4. Compute the following contact forces at differentparticle diameters d ranging from 0.1 to 10 mm.Comment on the results in relation to the effectiveand intergranular pressure described in Section 7.9.a. Weight of the sphere, W � �Gswd3, where Gs

1–6is the specific gravity (say 2.65) and w is theunit weight of water.

b. Contact force by external load, N � d2��, where�� is the external confining pressures applied.The equation is approximate for a simple cubicpacking of equal size spheres (Santamarina,2003). Consider two cases, (i) �� 1 kPa (�depth of 0.1 m) and (ii) �� 100 kPa (� depthof 10 m).

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c. Long-range van der Waals attraction force, A �Ahd / (24t2), where Ah is the Hamaker constant(Section 6.12) and t is the separation betweenparticles (Israelachvili, 1992, from Santamarina,2003). Use Ah � 10�20 N-m and t � 30 A.

5. Discuss why it is difficult to measure suction usinga piezometer-type tensiometer for long-term moni-toring of pore pressures. Describe the advantages ofother indirect measurement techniques such as po-rous ceramic and dielectric sensors.

6. For the following cases, compare the effectivestresses calculated by the conventional Terzaghi’sequation and by the modified equation (7.39) withvalues presented in Fig. 7.8. Discuss the possibleerrors associated with effective stress estimation byTerzaghi’s equation.a. Pile foundation at a depth of 20 m.b. A depth of 5 km from the sea level where the

subsea soil surface is 1 km deep.

7. Give a microscopic interpretation for why an un-saturated soil can collapse and decrease its volumeupon wetting as shown in Fig. 7.14 even though theBishop’s effective stress decreases.

8. Clay particles in unsaturated soils often aggregatecreating matrix pores and intraaggregate pores. Airexists in the matrix pores, but the intraaggregatepores are often saturated by strong water attractionto clay surfaces. The total potential of unsaturatedsoil can be extended from Eq. (7.19) to � � �g ��m � �s � �p, where �p is the gas pressure poten-tial.12 Discuss the values of each component of theabove equation in the matrix pores and the intraag-gregate pores.

12 This was proposed by a Review Panel in the Symposium on Mois-ture Equilibrium and Moisture Changes in Soils Beneath CoveredAreas in 1965.

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