63027 11a

369

CHAPTER 11

Strength and DeformationBehavior

11.1 INTRODUCTION

All aspects of soil stability—bearing capacity, slopestability, the supporting capacity of deep foundations,and penetration resistance, to name a few—depend onsoil strength. The stress–deformation and stress–deformation–time behavior of soils are important inany problem where ground movements are of interest.Most relationships for the characterization of thestress–deformation and strength properties of soils areempirical and based on phenomenological descriptionsof soil behavior. The Mohr–Coulomb equation is byfar the most widely used for strength. It states that

� � c � � tan � (11.1)ff ff

� � c� � �� tan �� (11.2)ff ff

where �ff is shear stress at failure on the failure plane,c is a cohesion intercept, �ff is the normal stress on thefailure plane, and � is a friction angle. Equation (11.1)applies for �ff defined as a total stress, and c and � arereferred to as total stress parameters. Equation (11.2)applies for defined as an effective stress, and c� and��ff�� are effective stress parameters. As the shear resis-tance of soil originates mainly from actions at inter-particle contacts, the second equation is the morefundamental.

In reality, the shearing resistance of a soil dependson many factors, and a complete equation might be ofthe form

Shearing resistance � F(e, c�, ��, ��, C, H, T, �, S)�,

(11.3)

in which e is the void ratio, C is the composition, His the stress history, T is the temperature, � is the strain,

is the strain rate, and S is the structure. All param-�eters in these equations may not be independent, andthe functional forms of all of them are not known.Consequently, the shear resistance values (including c�and ��) are determined using specified test type (i.e.,direct shear, triaxial compression, simple shear), drain-age conditions, rate of loading, range of confiningpressures, and stress history. As a result, different fric-tion angles and cohesion values have been defined, in-cluding parameters for total stress, effective stress,drained, undrained, peak strength, and residualstrength. The shear resistance values applicable inpractice depend on factors such as whether or not theproblem is one of loading or unloading, whether or notshort-term or long-term stability is of interest, andstress orientations.

Emphasis in this chapter is on the fundamental fac-tors controlling the strength and stress–deformationbehavior of soils. Following a review of the generalcharacteristics of strength and deformation, some re-

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

370 11 STRENGTH AND DEFORMATION BEHAVIOR

(a)

Strain

Peak

Critical State

Residual

ShearStress τ

Normal effective stress σ�

φresidual

φcritical state

φpeak

Tangent PeakStrength Envelope

Critical stateStrength Envelope

Residual Strength Envelope

(b)

Peak Strength

Secant PeakStrength Envelope

At Large Strains

a

b

c

da

b

c

d

Dense orOverconsolidated

a�

d�

Loose or NormallyConsolidated

ShearStress τor StressRatio τ/σ�

b�, c�

Figure 11.1 Peak, critical, and residual strength and associated friction angle: (a) a typicalstress–strain curve and (b) stress states.

lationships among fabric, structure, and strength areexamined. The fundamentals of bonding, friction, par-ticulate behavior, and cohesion are treated in some de-tail in order to relate them to soil strength properties.Micromechanical interactions of particles in an assem-blage and the relationships between interparticle fric-tion and macroscopic friction angle are examined fromdiscrete particle simulations. Typical values of strengthparameters are listed. The concept of yielding is intro-duced, and the deformation behavior in both the pre-yield (including small strain stiffness) and post-yieldregions is summarized. Time-dependent deformationsand aging effects are discussed separately in Chapter12. The details of strength determination by means oflaboratory and in situ tests and the detailed constitutivemodeling of soil deformation and strength for use innumerical analyses are outside the scope of this book.

11.2 GENERAL CHARACTERISTICS OFSTRENGTH AND DEFORMATION

Strength

1. In the absence of chemical cementation betweengrains, the strength (stress state at failure or theultimate stress state) of sand and clay is ap-proximated by a linear relationship with stress:

� � �� tan �� (11.4)ff ff

or

(�� ) � (�� )sin �� (11.5)1ff 3ff 1ff 3ff

where the primes designate effective stressesand are the major and minor principal�� 1ff 3ff

effective stresses at failure, respectively.2. The basic contributions to soil strength are fric-

tional resistance between soil particles in con-tact and internal kinematic constraints of soilparticles associated with changes in the soil fab-ric. The magnitude of these contributions de-pends on the effective stress and the volumechange tendencies of the soil. For such materialsthe stress–strain curve from a shearing test istypically of the form shown in Fig. 11.1a. Themaximum or peak strength of a soil (point b)may be greater than the critical state strength,in which the soil deforms under sustained load-ing at constant volume (point c). For some soils,the particles align along a localized failure planeafter large shear strain or shear displacement,and the strength decreases even further to theresidual strength (point d). The correspondingthree failure envelopes can be defined as shownin Fig. 11.1b, with peak, critical, and residualfriction angles (or states) as indicated.

3. Peak failure envelopes are usually curved in themanner shown in Fig. 4.16 and schematically inFig. 11.1b. This behavior is caused by dilatancysuppression and grain crushing at higherstresses. Curved failure envelopes are also ob-served for many clays at residual state. When

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GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION 371

Figure 11.2 Variation of residual strength with stress level (after Bishop et al., 1971): (a)Brown London clay and (b) Weald clay.

expressed in terms of the shear strength nor-malized by the effective normal stress as a func-tion of effective normal stress, curves of thetype shown in Fig. 11.2 for two clays are ob-tained.

4. The peak strength of cohesionless soils is influ-enced most by density, effective confiningpressures, test type, and sample preparationmethods. For dense sand, the secant peak fric-tion angle (point b in Fig. 11.1b) consists in part

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c�e

0Normal Effective Stress σ�

τ

eff

φ�e

φ�crit

A� A

AA�

σ�eσ�ff

σ�ff

Hvorslev Envelope

Peak Strength Envelope

Normally Consolidated

Virgin Compression

Rebound

Overconsolidated

Overconsolidated


She

ar S

tres

s τ

Voi

d R

atio

eW

ater

Con

tent

w

Figure 11.3 Effect of overconsolidation on effective stressstrength envelope.

of internal rolling and sliding friction betweengrains and in part of interlocking of particles(Taylor, 1948). The interlocking necessitateseither volume expansion (dilatancy) or grainfracture and/or crushing if there is to bedeformation. For loose sand, the peak frictionangle (point b� in Fig. 11.1b) normally coincideswith the critical-state friction angle (point c�),and there is no peak in the stress–strain curve.

5. The peak strength of saturated clay is influencedmost by overconsolidation ratio, drainage con-ditions, effective confining pressures, originalstructure, disturbance (which causes a change ineffective stress and a loss of cementation), andcreep or deformation rate effects. Overconsoli-dated clays usually have higher peak strength ata given effective stress than normally consoli-dated clays, as shown in Fig. 11.3. The differ-ences in strength result from both the differentstress histories and the different water contentsat peak. For comparisons at the same water con-tent but different effective stress, as for pointsA and A�, the Hvorslev strength parameters ce

and �e are obtained (Hvorslev, 1937, 1960).Further details are given in Section 11.9.

6. During critical state deformation a soil is com-pletely destructured. As illustrated in Fig. 11.1b,the critical state friction angle values are inde-pendent of stress history and original structure;for a given set of testing conditions the shearing

resistance depends only on composition and ef-fective stress. The basic concept of the criticalstate is that under sustained uniform shearing atfailure, there exists a unique combination ofvoid ratio e, mean pressure p�, and deviatorstress q.1 The critical states of reconstitutedWeald clay and Toyoura sand are shown in Fig.11.4. The critical state line on the p�–q plane islinear,2 whereas that on an e-ln p� (or e-log p�)plane tends to be linear for clays and nonlinearfor sands.

7. At failure, dense sands and heavily overconsol-idated clays have a greater volume after drainedshear or a higher effective stress after undrainedshear than at the start of deformation. This isdue to its dilative tendency upon shearing. Atfailure, loose sands and normally consolidatedto moderately overconsolidated clays (OCR upto about 4) have a smaller volume after drainedshear or a lower effective stress after undrainedshear than they had initially. This is due to itscontractive tendency upon shearing.

8. Under further deformation, platy clay particlesbegin to align along the failure plane and theshear resistance may further decrease from thecritical state condition. The angle of shear re-sistance at this condition is called the residualfriction angle, as illustrated in Fig. 11.1b. Thepostpeak shearing displacement required tocause a reduction in friction angle from the crit-ical state value to the residual value varies withthe soil type, normal stress on the shear plane,and test conditions. For example, for shale my-lonite3 in contact with smooth steel or other pol-ished hard surfaces, a shearing displacement ofonly 1 or 2 mm is sufficient to give residualstrength.4 For soil against soil, a slip along the

1 In three-dimensional stress space , �xy, �yz, �zx) or�� (��, ��, ��x y z

the equivalent principal stresses ( ), the mean effective��, ��, ��1 2 3

stress p�, and the deviator stress q is defined as

p� � (�� ) /3 � (�� ) /3x y z 1 2 3

q � (1 /2)

2 2 2 2 2 2(�� ) � (�� ) � (�� ) � 6� � 6� � 6�x y y z z x xy yz zx

2 2 2� (1 /2)(�� ) � (�� ) � (�� )1 2 2 3 3 1

For triaxial compression condition (�� ), p� � (�� 1 2 3 1

2��) /3, q � �� 2 1 22 The critical state failure slope on p�–q plane is related to frictionangle ��, as described in Section 11.10.3 A rock that has undergone differential movements at high temper-ature and pressure in which the mineral grains are crushed againstone another. The rock shows a series of lamination planes.4 D. U. Deere, personal communication (1974).

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(a-1) p� versus q

(a) (b)

Mean Pressure p�(kPa)

0 100 200 300 400 500 6000

100

200

300

400

500

0.7

0.6

0.5

0.4

0.3

100 200 300 400 500

Mean Pressure p� (kPa)

0 1 2 3 40

1

2

3

4

Mean Pressure p�(MPa)

(b-1) p� versus q

Initial State

Mean Pressure p�(MPa)

0.050.02 0.1 0.5 1 5

0.95

0.90

0.85

0.80

0.75

Overconsolidated


Overconsolidated


Isotropic NormalCompression Line

Critical State Line

Critical State Line Critical State Line

Critical State Line

Dev

iato

r S

tres

s q

(MP

a)

Voi

d ra

tio e

(a-2) e versus lnp�(b-2) e versus logp�

Dev

iato

r S

tres

s q

(kP

a)

Voi

d ra

tio e

Figure 11.4 Critical states of clay and sand: (a) Critical state of Weald clay obtained bydrained triaxial compression tests of normally consolidated (�) and overconsolidated (●)specimens: (a-1) q–p� plane and (a-2) e–ln p� plane (after Roscoe et al., 1958). (b) Criticalstate of Toyoura sand obtained by undrained triaxial compression tests of loose and densespecimens consolidated initially at different effective stresses, (b-1) q–p� plane and (b-2) e–log p� plane (after Verdugo and Ishihara, 1996).

shear plane of several tens of millimeters maybe required, as shown by Fig. 11.5. However,significant softening can be caused by strainlocalization and development of shear bands,especially for dense samples under low confine-ment.

9. Strength anisotropy may result from both stressand fabric anisotropy. In the absence of chemi-cal cementation, the differences in the strength

of two samples of the same soil at the same voidratio but with different fabrics are accountablein terms of different effective stresses as dis-cussed in Chapter 8.

10. Undrained strength in triaxial compression maydiffer significantly from the strength in triaxialextension. However, the influence of type of test(triaxial compression versus extension) on theeffective stress parameters c� and �� is relatively

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Figure 11.5 Development of residual strength with increasing shear displacement (afterBishop et al., 1971).

Figure 11.6 Effect of temperature on undrained strength ofkaolinite in unconfined compression (after Sherif and Bur-rous, 1969).

small. Effective stress friction angles measuredin plane strain are typically about 10 percentgreater than those determined by triaxial com-pression.

11. A change in temperature causes either a changein void ratio or a change in effective stress (ora combination of both) in saturated clay, as dis-cussed in Chapter 10. Thus, a change in tem-perature can cause a strength increase or astrength decrease, depending on the circum-stances, as illustrated by Fig. 11.6. For the testson kaolinite shown in Fig. 11.6, all sampleswere prepared by isotropic triaxial consolidationat 75�F. Then, with no further drainage allowed,temperatures were increased to the values indi-cated, and the samples were tested in uncon-fined compression. Substantial reductions instrength accompanied the increases in temper-ature.

Stress–Strain Behavior

1. Stress–strain behavior ranges from very brittlefor some quick clays, cemented soils, heavilyoverconsolidated clays, and dense sands to duc-tile for insensitive and remolded clays and loosesands, as illustrated by Fig. 11.7. An increase in

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Figure 11.7 Types of stress–strain behavior.10-4 10-3 10-2 10-1 100 101

Strain %Dynamic Methods

Local Gauges

Conventional Soil Testing

Retaining Walls

Foundations

TunnelsLinear Elastic

Nonlinear Elastic

Preyield Plastic

Full Plastic

(a) Typical Strain Ranges in the Field

(b) Typical Strain Ranges for Laboratory Tests

Stif

fnes

sG

or

E

Figure 11.9 Stiffness degradation curve: stiffness plottedagainst logarithm of strains. Also shown are (a) the strainlevels observed during construction of typical geotechnicalstructures (after Mair, 1993) and (b) the strain levels that canbe measured by various techniques (after Atkinson, 2000).

Figure 11.8 Effect of confining pressure on the consoli-dated-drained stress–strain behavior of soils.

confining pressure causes an increase in the de-formation modulus as well as an increase instrength, as shown by Fig. 11.8.

2. Stress–strain relationships are usually nonlin-ear; soil stiffness (often expressed in terms oftangent or secant modulus) generally decreaseswith increasing shear strain or stress level up topeak failure stress. Figure 11.9 shows a typicalstiffness degradation curve, in terms of shearmodulus G and Young’s modulus E, along withtypical strain levels developed in geotechnicalconstruction (Mair, 1993) and as associated withdifferent laboratory testing techniques used tomeasure the stiffness (Atkinson, 2000). For ex-ample, Fig. 11.10 shows the stiffness degrada-tion of sands and clay subjected to increase inshear strain. As illustrated in Fig. 11.9, the stiff-ness degradation curve can be separated into

four zones: (1) linear elastic zone, (2) nonlinearelastic zone, (3) pre-yield plastic zone, and (4)full plastic zone.

3. In the linear elastic zone, soil particles do notslide relative to each other under a small stressincrement, and the stiffness is at its maximum.The soil stiffness depends on contact interac-tions, particle packing arrangement, and elasticstiffness of the solids. Low strain stiffness val-ues can be determined using elastic wave veloc-ity measurements, resonant column testing, orlocal strain transducer measurements. The mag-nitudes of the small strain shear modulus (Gmax)and Young’s modulus (Emax) depend on appliedconfining pressure and the packing conditionsof soil particles. The following empirical equa-tions are often employed to express these de-pendencies:

nGG � A F (e)p� (11.6)max G G

nEE � A F (e)�� (11.7)i(max) E E i

where FG(e) and FE(e) are functions of voidratio, p� is the mean effective confining pres-sure, is the effective stress in the i direction,��iand the other parameters are material constants.

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Toyoura Sand

Ticino Sand

TC PSC

20

40

60

80

100

120

140

10010-110-210-310-4

Shear Strain (%)

(a)

10010-110-210-310-410-5

Shear Strain (%)

20

40

60

80

100

120

σc = 30 kPa

σc = 100 kPa

σc = 200 kPa

σc = 400 kPa

Confining Pressures

ConfiningPressure

78.4 kPa

49 kPa

(b)

Sec

ant S

hear

Mod

ulus

G (M

Pa)

Sec

ant S

hear

Mod

ulus

G (M

Pa)

Figure 11.10 Stiffness degradation curve at different confining pressures: (a) Toyoura andTicino sands (TC: triaxial compression tests, PSC: plain strain compression tests) (afterTatsuoka et al., 1997) and (b) reconstituted Kaolin clay (after Soga et al., 1996).

104103102101100100

101

102

103

104

nG = 0.13

nG = 0.65

nG = 0.63

Confining pressure, p� (kPa)

UndisturbedRemoldedRemolded with CaCO3

100 150 200 250 300

500

400

300

250

350

450

Vertical Effective Stress, σv� (kPa)

(b)(a)

nE = 0.49

At each vertical effective stress,horizontal effective stress σh� (kPa)was varied between 98 kPa and196 kPa

She

ar M

odul

us,G

max

MP

a

Ver

tical

You

ng's

Mod

ulus

Evm

ax/

FE(e

) (M

Pa)

Figure 11.11 Small strain stiffness versus confining pressure: (a) Shear modulus Gmax ofcemented silty sand measured by resonant column tests (from Stokoe et al. 1995) and (b)vertical Young’s modulus of sands measured by triaxial tests (after Tatsuoka and Kohata,1995).

Figure 11.11 shows examples of the fitting ofthe above equations to experimental data.

4. The stiffness begins to decrease from the linearelastic value as the applied strains or stressesincrease, and the deformation moves into thenonlinear elastic zone. However, a complete cy-cle of loading, unloading, and reloading withinthis zone shows full recovery of strains. Thestrain at the onset of the nonlinear elastic zoneranges from less than 5 � 10�4 percent for non-

plastic soils at low confining pressure conditionsto greater than 5 � 10�2 percent at high confin-ing pressure or in soils with high plasticity (San-tamarina et al., 2001).

5. Irrecoverable strains develop in the pre-yieldplastic zone. The initiation of plastic strains canbe determined by examining the onset of per-manent volumetric strain in drained conditionsor residual excess pore pressures in undrainedconditions after unloading. Available experi-

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1.00.80.60.40.2

q = σ�a-σ�r

0.2

0.4

0.6

0.8

0.0

-0.2

-0.4

MPa

MPa

Initial Condition

Yield State

Stress PathFailure Line

Failure Line

Yield Envelope

0.60.40.2

MPa0.0

-0.2

0.2

0.4

0.6

q = σ�a-σ�r

MPa

Yield State

Yield Envelope

Preyield Boundary

Pre-yield State

Initial State Surrounded byLinear Elastic Boundary

Linear ElasticBoundary

(a) (b)

p� = (σ�a + 2σ�r)/3 p� = (σ�a + 2σ�r)/3

Figure 11.12 Yield envelopes: (a) Aoi sand (Yasufuku et al., 1991) and (b) Bothkennar clay(from Smith et al., 1992).

mental data suggest that the strain level that in-itiates plastic strains ranges between 7 � 10�3

and 7 � 10�2 percent, with the lower limit foruncemented normally consolidated sands andthe upper limit for high plasticity clays and ce-mented sands.

6. A distinctive kink in the stress–strain relation-ship defines yielding, beyond which full plasticstrains are generated. A locus of stress statesthat initiate yielding defines the yield envelope.Typical yield envelopes for sand and naturalclay are shown in Fig. 11.12. The yield envelopeexpands, shrinks, and rotates as plastic strainsdevelop. It is usually considered that expansionis related to plastic volumetric strains; the sur-face expands when the soil compresses andshrinks when the soil dilates. The two inner en-velopes shown in Fig. 11.12b define the bound-aries between linear elastic, nonlinear elastic,and pre-yield zones. When the stress statemoves in the pre-yield zone, the inner envelopesmove with the stress state. This multienvelopeconcept allows modeling of complex deforma-tions observed for different stress paths (Mroz,1967; Prevost, 1977; Dafalias and Herrman,1982; Atkinson et al., 1990; Jardine, 1992).

7. Plastic irrecoverable shear deformations ofsaturated soils are accompanied by volume

changes when drainage is allowed or changes inpore water pressure and effective stress whendrainage is prevented. The general nature of thisbehavior is shown in Figs. 11.13a and 11.13bfor drained and undrained conditions, respec-tively. The volume and pore water pressurechanges depend on interactions between fabricand stress state and the ease with which sheardeformations can develop without overallchanges in volume or transfer of normal stressfrom the soil structure to the pore water.

8. The stress–strain relation of clays dependslargely on overconsolidation ratio, effectiveconfining pressures, and drainage conditions.Figure 11.14 shows triaxial compression behav-ior of clay specimens that are first normally con-solidated and then isotropically unloaded todifferent overconsolidation ratios before shear-ing. The specimens are consolidated at the sameconfining pressure but have different voidp�,0

ratios due to the different stress history (Fig.11.14a). Drained tests on normally consolidatedclays and lightly overconsolidated clays showductile behavior with volume contraction (Fig.11.14b). Heavily overconsolidated clays exhibita stiff response initially until the stress statereaches the yield envelope giving the peakstrength and volume dilation. The state of the

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Same Initial Confining Pressure

Loose Soil

Dense Soil

Critical State

DeviatorStress

Metastable Fabric

Loose Soil

Dense Soil

Metastable Fabric

0

+ΔV/V0

-ΔV/V0

Axial or Deviator Strain

(a)

Same Initial Confining Pressure

Loose Soil

Dense Soil

Critical State

DeviatorStress

Metastable Fabric

Loose soil

Dense Soil

Metastable Fabric

0

-Δu

+Δu

Axial or Deviator Strain

(b)

Critical State

Cavitation

Cavitation

Figure 11.13 Volume and pore pressure changes during shear: (a) drained conditions and(b) undrained conditions.

1 Normally consolidated

2 Lightly Overconsolidated

3 HeavilyOverconsolidated

U1

U2

U3D

Virgin Compression Line

CriticalState Line

1 Normally Consolidated

2 LightlyOverconsolidated

3 HeavilyOverconsolidated



3 Heavily Overconsolidated

Axial or Deviatoric Strain

DeviatorStress

+ΔV/V0

-ΔV/V0







Axial or Deviatoric Strain

DeviatorStress

-Δu

+Δu

VoidRatio

log p�

D Critical State

(a) (b) (c)

U1

U2

U3

p0�

Initial StateFailure at Critical State(D: Drained, U: Undrained)

Figure 11.14 Stress–strain relationship of normally consolidated, lightly overconsolidated,and heavily overconsolidated clays: (a) void ratio versus mean effective stress, (b) drainedtests, and (c) undrained tests.

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FABRIC, STRUCTURE, AND STRENGTH 379

Figure 11.16 Effect of temperature on the stiffness of Osakaclay in undrained triaxial compression (Murayama, 1969).

0.00.40.30.20.1

0.1

0.2

0.3

-0.1

-0.2

-0.3

(MPa)

(MPa)

Mean Pressurep� = (σ�a+ 2σ�r)/3

σ�r/σ�a = 0.54

σ�r/σ�a = 1.84

Failure Line inTriaxial Extension

Failure Line inTriaxial Compression

Initial At Failure

Anisotropically Consolidated σ�r/ σ�a = 0.54

Isotropically Consolidated

Dev

iato

r S

tres

s q

= σ

� a+

σ�rσ

Anisotropically Consolidated σ�r/σ�a = 1.84

Figure 11.15 Undrained effective stress paths of anisotrop-ically and isotropically consolidated specimens (after Laddand Varallyay, 1965).

soil then progressively moves toward the criticalstate exhibiting softening behavior. Undrainedshearing of normally consolidated and lightlyoverconsolidated clays generates positive excesspore pressures, whereas shear of heavily over-consolidated clays generates negative excesspore pressures (Fig. 11.14c).

9. The magnitudes of pore pressure that are de-veloped in undrained loading depend on initialconsolidation stresses, overconsolidation ratio,density, and soil fabric. Figure 11.15 shows theundrained effective stress paths of anisotropi-cally and isotropically consolidated specimens(Ladd and Varallyay, 1965). The difference inundrained shear strength is primarily due to dif-ferent excess pore pressure development asso-ciated with the change in soil fabric. At largestrains, the stress paths correspond to the samefriction angle.

10. A temperature increase causes a decrease in un-drained modulus; that is, a softening of the soil.As an example, initial strain as a function ofstress is shown in Fig. 11.16 for Osaka clay

tested in undrained triaxial compression at dif-ferent temperatures. Increase in temperaturecauses consolidation under drained conditionsand softening under undrained conditions.

11.3 FABRIC, STRUCTURE, AND STRENGTH

Fabric Changes During Shear of CohesionlessMaterials

The deformation of sands, gravels, and rockfills is in-fluenced by the initial fabric, as discussed and illus-trated in Chapter 8. As an illustration, fabric changesassociated with the sliding and rolling of grains duringtriaxial compression were determined using a uniformsand composed of rounded to subrounded grains withsizes in the range of 0.84 to 1.19 mm and a mean axiallength ratio of 1.45 (Oda, 1972, 1972a, 1972b, 1972c).Samples were prepared to a void ratio of 0.64 by tamp-ing and by tapping the side of the forming mold. Adelayed setting water–resin solution was used as thepore fluid. Samples prepared by each method weretested to successively higher strains. The resin wasthen allowed to set, and thin sections were prepared.The differences in initial fabrics gave the markedly dif-ferent stress–strain and volumetric strain curves shownin Fig. 11.17, where the plunging method refers to

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Figure 11.17 Stress–strain and volumetric strain relationships for sand at a void ratio of0.64 but with different initial fabrics (after Oda, 1972a). (a) Sample saturated with waterand (b) sample saturated with water–resin solution.

tamping. There is similarity between these curves andthose for Monterey No. 0 sand shown in Fig. 8.23. Astatistical analysis of the changes in particle orientationwith increase in axial strain showed:

1. For samples prepared by tapping, the initial fab-ric tended toward some preferred orientation oflong axes parallel to the horizontal plane, and theintensity of orientation increased slightly duringdeformation.

2. For samples prepared by tamping, there was veryweak preferred orientation in the vertical direc-tion initially, but this disappeared with deforma-tion.

Shear deformations break down particle and aggre-gate assemblages. Shear planes or zones did not appearuntil after peak stress had been reached; however, thedistribution of normals to the interparticle contactplanes E(�) (a measure of fabric anisotropy) didchange with strain, as may be seen in Fig. 11.18. Thisfigure shows different initial distributions for samplesprepared by the two methods and a concentration ofcontact plane normals within 50� of the vertical as de-formation progresses. Thus, the fabric tended towardgreater anisotropy in each case in terms of contactplane orientations. There was little additional changein E(�) after the peak stress had been reached, whichimplies that particle rearrangement was proceedingwithout significant change in the overall fabric.

As the stress state approaches failure, a direct shear-induced fabric forms that is generally composed ofregions of homogeneous fabric separated by discon-tinuities. No discontinuities develop before peakstrength is reached, although there is some particle ro-tation in the direction of motion. Near-perfect preferredorientation develops during yield after peak strength isreached, but large deformations may be required toreach this state.

Compaction Versus Overconsolidation of Sand

Specimens at the same void ratio and stress state be-fore shearing, but having different fabrics, can exhibitdifferent stress–strain behavior. For example, considera case in which one specimen is overconsolidated,whereas the other is compacted. The two specimensare prepared in such a way that the initial void ratio isthe same for a given initial isotropic confining pres-sure. Coop (1990) performed undrained triaxial com-pression tests of carbonate sand specimens that wereeither overconsolidated or compacted, as illustrated inFig. 11.19a. The undrained stress paths and stress–strain curves for the two specimens are shown in Figs.11.19b and 11.19c, respectively. The overconsolidatedsample was initially stiffer than the compacted speci-men. The difference can be attributed to (i) differentsoil fabrics developed by different stress paths prior toshearing and (ii) different degrees of particle crushingprior to shearing (i.e., some breakage has occurred dur-

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FABRIC, STRUCTURE, AND STRENGTH 381

Figure 11.18 Distribution of interparticle contact normals as a function of axial strain forsand samples prepared in two ways (after Oda, 1972a): (a) specimens prepared by tappingand (b) specimens prepared by tamping.

ing the preconsolidation stage for the overconsolidatedspecimen). Therefore, overconsolidation and compac-tion produced materials with different mechanicalproperties. However, at large deformations, both spec-imens exhibited similar strengths because the initialfabrics were destroyed.

Effect of Clay Structure on Deformations

The high sensitivity of quick clays illustrates the prin-ciple that flocculated, open microfabrics are more rigidbut more unstable than deflocculated fabrics. Similarbehavior may be observed in compacted fine-grainedsoils, and the results of a series of tests on structure-sensitive kaolinite are illustrative of the differences(Mitchell and McConnell, 1965). Compaction condi-tions and stress–strain curves for samples of kaolinitecompacted using kneading and static methods areshown in Fig. 11.20. The high shear strain associatedwith kneading compaction wet of optimum breaksdown flocculated structures, and this accounts for the

much lower peak strength for the sample prepared bykneading compaction.

The recoverable deformation of compacted kaolinitewith flocculent structure ranges between 60 and 90percent, whereas the recovery of samples with dis-persed structures is only of the order of 15 to 30 per-cent of the total deformation, as may be seen in Fig.11.21. This illustrates the much greater ability of thebraced-box type of fabric that remains after static com-paction to withstand stress without permanent defor-mation than is possible with the broken-down fabricassociated with kneading compaction.

Different macrofabric features can affect the defor-mation behavior as illustrated in Fig. 11.22 for the un-drained triaxial compression testing of Bothkennarclay, Scotland (Paul et al., 1992; Clayton et al., 1992).Samples with mottled facies, in which the bedding fea-tures had been disrupted and mixed by burrowingmollusks and worms (bioturbation), gave the stiffestresponse, whereas samples with distinct laminated fea-tures showed the softest response.

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2

1.5

1

0.1 1Mean Pressure p�

VoidRatio

Compacted Sample

OverconsolidatedSample

Normal Compression Line

(MPa)

0.60.40.200

0.2

0.4

0.6

0.8

1.0

p� (MPa)

Overconsolidated

Compacted

0 4 8 12 16 20

1.0

0.75

0.5

0.25

0

q (MPa)

Axial strain εa(%)

Compacted

Overconsolidated

(a)

(b)

(c)

q (M

Pa)

Figure 11.19 Undrained response of compacted specimen and overconsolidated specimenof carbonate sand: (a) stress path before shearing, (b) undrained stress paths during shearing,and (c) stress–strain relationships (after Coop, 1990).

If slip planes develop at failure, platy and elongatedparticles align with their long axes in the direction ofslip. By then, the basal planes of the platy clay parti-cles are enclosed between two highly oriented bandsof particles on opposite sides of the shear plane. Thedominant mechanism of deformation in the displace-ment shear zone is basal plane slip, and the overallthickness of the shear zone is on the order of 50 �m.Fabrics associated with shear planes and zones havebeen studied using thin sections and the polarizing mi-croscope and by using the electron microscope (Mor-genstern and Tchalenko, 1967a, b and c; Tchalenko,1968; McKyes and Yong, 1971). The residual strengthassociated with these fabrics is treated in more detailin Section 11.11.

Structure, Effective Stresses, and Strength

The effective stress strength parameters such as c� and�� are isotropic properties, with anisotropy in un-drained strength explainable in terms of excess porepressures developed during shear. The undrainedstrength loss associated with remolding undisturbed

clay can also be accounted for in terms of differencesin effective stress, provided part of the undisturbedstrength does not result from cementation. Remoldingbreaks down the structure and causes a transfer of ef-fective stress to the pore water.

An example of this is shown in Fig. 11.23, whichshows the results of incremental loading triaxial com-pression tests on two samples of undisturbed and re-molded San Francisco Bay mud. In these tests, theundisturbed sample was first brought to equilibriumunder an isotropic consolidation pressure of 80 kPa.After undrained loading to failure, the triaxial cell wasdisassembled, and the sample was remolded in place.The apparatus was reassembled, and pore pressure wasmeasured. Thus, the effective stress at the start of com-pression of the remolded clay at the same water con-tent as the original undisturbed clay was known.Stress–strain and pore pressure–strain curves for twosamples are shown in Figs. 11.23a and 11.23b, andstress paths for test 1 are shown in Fig. 11.23c.

Differences in strength that result from fabric dif-ferences caused by thixotropic hardening or by differ-ent compaction methods can be explained in the same

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FRICTION BETWEEN SOLID SURFACES 383

Figure 11.20 Stress–strain behavior of kaolinite compactedby two methods.

Figure 11.21 Ratio of recoverable to total strain for samplesof kaolinite with different structure.

0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

Axial Strain (%)0 2 4

0.0

0.2

0.4

0.6

Stress-Strain Relationships Stress Paths

MottledFacies

BeddedLaminated

MottledFacies

BeddedLaminated

(σ�a + σ�r)/2σ�ao

(σ� a

– σ

� r)/

2σ�

ao

(σ� a

– σ

� r)/

2σ�

ao

Figure 11.22 Effect of macrofabric on undrained responseof Bothkennar clay in Scotland (after Hight and Leroueil,2003).

way. Thus, in the absence of chemical or mineralogicalchanges, different strengths in two samples of the samesoil at the same void ratio can be accounted for interms of different effective stress.

11.4 FRICTION BETWEEN SOLID SURFACES

The friction angle used in equations such as (11.1),(11.2), (11.4), and (11.5) contains resistance contri-butions from several sources, including sliding ofgrains in contact, resistance to volume change (dila-tancy), grain rearrangement, and grain crushing. The

true friction coefficient is shown in Fig. 11.24 and isrepresented by

T� � � tan � (11.8)�N

where N is the normal load on the shear surface, T isthe shear force, and ��, the intergrain sliding frictionangle, is a compositional property that is determinedby the type of soil minerals.

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Figure 11.23 (a) and (b) Effect of remolding on undrained strength and pore water pressurein San Francisco Bay mud. (c) Stress paths for triaxial compression tests on undisturbed andremolded samples of San Francisco Bay mud.

Basic ‘‘Laws’’ of Friction

Two laws of friction are recognized, beginning withLeonardo da Vinci in about 1500. They were restatedby Amontons in 1699 and are frequently referred to asAmontons’ laws. They are:

1. The frictional force is directly proportional to thenormal force, as illustrated by Eq. (11.8) and Fig.11.24.

2. The frictional resistance between two bodies isindependent of the size of the bodies. In Fig.

11.24, the value of T is the same for a given valueof N regardless of the size of the sliding block.

Although these principles of frictional resistancehave long been known, suitable explanations camemuch later. It was at one time thought that interlockingbetween irregular surfaces could account for the be-havior. On this basis, � would be given by the tangentof the average inclination of surface irregularities onthe sliding plane. This cannot be the case, however,because such an explanation would require that � de-

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Figure 11.23(Continued )

Figure 11.24 Coefficient of friction for surfaces in contact.

crease as surfaces become smoother and be zero forperfectly smooth surfaces. In fact, the coefficient offriction can be constant over a range of surface rough-ness. Hardy (1936) suggested instead that staticfriction originates from cohesive forces between

contacting surfaces. He observed that the actual areaof contact is very small because of surface irregulari-ties, and thus the cohesive forces must be large.

The foundation for the present understanding of themobilization of friction between surfaces in contactwas laid by Terzaghi (1920). He hypothesized that thenormal load N acting between two bodies in contactcauses yielding at asperities, which are local ‘‘hills’’on the surface, where the actual interbody solid contactdevelops. The actual contact area Ac is given by

NA � (11.9)c �y

where �y is the yield strength of the material. Theshearing strength of the material in the yielded zone isassumed to have a value �m. The maximum shearingforce that can be resisted by the contact is then

T � A � (11.10)c m

The coefficient of friction is given by T /N,

T A � �c m m� � � � (11.11)N A � �c y y

This concept of frictional resistance was subse-quently further developed by Bowden and Tabor (1950,

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Figure 11.25 Contact between two smooth surfaces.Figure 11.26 Monolayer formation time as a function of at-mospheric pressure.

1964). The Terzaghi–Bowden and Tabor hypothesis,commonly referred to as the adhesion theory of fric-tion, is the basis for most modern studies of friction.Two characteristics of surfaces play key roles in theadhesion theory of friction: roughness and surface ad-sorption.

Surface Roughness

The surfaces of most solids are rough on a molecularscale, with successions of asperities and depressionsranging from 10 nm to over 100 nm in height. Theslopes of the nanoscale asperities are rather flat, withindividual angles ranging from about 120� to 175� asshown in Fig. 11.25. The average slope of asperitieson metal surfaces is an included angle of 150�; onrough quartz it may be over 175� (Bromwell, 1966).When two surfaces are brought together, contact is es-tablished at the asperities, and the actual contact areais only a small fraction of the total surface area.

Quartz surfaces polished to mirror smoothness mayconsist of peaks and valleys with an average height ofabout 500 nm. The asperities on rougher quartz sur-faces may be about 10 times higher (Lambe and Whit-man, 1969). Even these surfaces are probably smootherthan most soil particles composed of bulky minerals.The actual surface texture of sand particles depends ongeologic history as well as mineralogy, as shown inFig. 2.12.

The cleavage faces of mica flakes are among thesmoothest naturally occurring mineral surfaces. Evenin mica, however, there is some waviness due to ro-tation of tetrahedra in the silica layer, and surfaces usu-ally contain steps ranging in height from 1 to 100 nm,reflecting different numbers of unit layers across theparticle.

Thus, large areas of solid contact between grains arenot probable in soils. Solid-to-solid contact is throughasperities, and the corresponding interparticle contactstresses are high. The molecular structure and com-position in the contacting asperities determine the mag-nitude of �m in Eq. (11.11).

Surface Adsorption

Because of unsatisfied force fields at the surfaces ofsolids, the surface structure may differ from that in theinterior, and material may be adsorbed from adjacentphases. Even ‘‘clean’’ surfaces, prepared by fracture ofa solid or by evacuation at high temperature, are rap-idly contaminated when reexposed to normal atmos-pheric conditions.

According to the kinetic theory of gases, the timefor adsorption of a monolayer tm is given by

1t � (11.12)m �SZ

where � is the area occupied per molecule, S is thefraction of molecules striking the surface that stick toit, and Z is the number of molecules per second strik-ing a square centimeter of surface. For a value of Sequal to 1, which is reasonable for a high-energy sur-face, the relationship between tm and gas pressure isshown in Fig. 11.26. The conclusion to be drawn fromthis figure is that adsorbed layers are present on thesurface of soil particles in the terrestrial environment,

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Figure 11.27 Plastic junction between asperities with ad-sorbed surface films.

and contacts through asperities involve adsorbed ma-terial, unless it is extruded under the high pressure.5

Adhesion Theory of Friction

The basis for the adhesion theory of friction is in Eq.(11.10), that is, the tangential force that causes slid-ing depends on the solid contact area and the shearstrength of the contact. Plastic and/or elastic defor-mations determine the contact area at asperities.

Plastic Junctions If asperities yield and undergoplastic deformation, then the contact area is propor-tional to the normal load on the asperity as shown byEq. (11.9). Because surfaces are not clean, but are cov-ered by adsorbed films, actual solid contact may de-velop only over a fraction � of the contact area asshown in Fig. 11.27. If the contaminant film strengthis �c, the strength of the contact will be

T � A [�� (1 � �)� ] (11.13)c m c

Equation (11.13) cannot be applied in practice be-cause � and �c are unknown. However, it does providea possible explanation for why measured values of fric-tion angle for bulky minerals such as quartz and feld-spar are greater than values for the clay minerals andother platy minerals such as mica, even though thesurface structure is similar for all the silicate minerals.The small particle size of clays means that the loadper particle, for a given effective stress, will be smallrelative to that in silts and sands composed of the bulkyminerals. The surfaces of platy silt and sand size par-ticles are smoother than those of bulky mineral parti-

5 Conditions may be different on the Moon, where ultrahigh vacuumexists. This vacuum produces cleaner surfaces. In the absence ofsuitable adsorbate, clean surfaces can reduce their surface energy bycohering with like surfaces. This could account for the higher co-hesion of lunar soils than terrestrial soils of comparable gradation.

cles. The asperities, caused by surface waviness, aremore regular but not as high as those for the bulkyminerals.

Thus, it can be postulated that for a given numberof contacts per particle, the load per asperity decreaseswith decreasing particle size and, for particles of thesame size, is less for platy minerals than for bulkyminerals. Because � should increase as the normal loadper asperity increases, and it is reasonable to assumethat the adsorbed film strength is less than the strengthof the solid material (�c � �m), it follows that the truefriction angle (��) is less for small and platy particlesthan for large and bulky particles. In the event that twoplaty particles are in face-to-face contact and the sur-face waviness is insufficient to cause direct solid-to-solid contact, shear will be through the adsorbed films,and the effective value of � will be zero, again givinga lower value of ��.

In reality, the behavior of plastic junctions is morecomplex. Under combined compression and shearstresses, deformation follows the von Mises–Henkycriterion, which, for two dimensions, is

2 2 2� � 3� � � (11.14)y

For asperities loaded initially to � � �y, the appli-cation of a shear stress requires that � become lessthan �y. The only way that this can happen is for thecontact area to increase. Continued increase in � leadsto continued increase in contact area. This phenome-non is called junction growth and is responsible forcold welding in some materials (Bowden and Tabor,1964). If the shear strength of the junction equals thatof the bulk solid, then gross seizure occurs. For thecase where the ratio of junction strength to bulk ma-terial strength is less than 0.9, the amount of junctiongrowth is small. This is the probable situation in soils.

Elastic Junctions The contact area between parti-cles of a perfectly elastic material is not defined interms of plastic yield. For two smooth spheres in con-tact, application of the Hertz theory leads to

1 / 3d � (�NR) (11.15)

where d is the diameter of a plane circular area ofcontact; � is a function of geometry, Poisson’s ratio,and Young’s modulus6; and R is the sphere radius. Thecontact area is

6 For a sphere in contact with a plane surface � � 12(1 � � 2) /E.

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� 2 / 3A � (�NR) (11.16)c 4

If the shear strength of the contact is �i, then

T � � A (11.17)i c

and

T � 2 / 3 �1 / 3� � � � (�R) N (11.18)iN 4

According to these relationships, the friction coef-ficient for two elastic asperities in contact should de-crease with increasing load. Nonetheless, the adhesiontheory would still apply to the strength of the junction,with the frictional force proportional to the area of realcontact.

If it is assumed that the number of contacting as-perities in a soil mass is independent of particle sizeand effective stress, then the influences of particle sizeand effective stress on the frictional resistance of a soilwith asperities deforming elastically may be analyzed.For uniform spheres arranged in a regular packing, thegross area covered by one sphere along a potentialplane of sliding is 4R2. The normal load per contactingasperity, assuming one asperity per contact, is

2N � 4R �� (11.19)

Using Eq. (11.16), the area per contact becomes

� 3 2 / 3A � (4�R ��) (11.20)c 4

and the total contact area per unit gross area is

1 � �2 2 / 3 2 / 3(A ) � R (4��) � (4��)� �c T 24R 4 16 (11.21)

The total shearing resistance of �� is equal to thecontact area times �i, so

� �i2 / 3 �1 / 3� � (4��) � � K(��) (11.22)i16 ��

where K � �(4�)2 / 3 /16. On this basis, the coefficientof friction should decrease with increasing ��, but itshould be independent of sphere radius (particle size).

Data have been obtained that both support and con-tradict these predictions. A 50-fold variation in the nor-mal load on assemblages of quartz particles in contactwith a quartz block was found to have no effect on

frictional resistance (Rowe, 1962). The residual fric-tion angles of quartz, feldspar, and calcite are indepen-dent of normal stress as shown in Fig. 11.28.

On the other hand, a decreasing friction angle withincreasing normal load up to some limiting value ofnormal stress is evident for mica and the clay mineralsin Fig. 11.28 and has been found also for several claysand clay shales (Bishop et al., 1971), for diamond(Bowden and Tabor, 1964), and for solid lubricantssuch as graphite and molybdenum disulfide (Campbell,1969). Additional data for clay minerals show that fric-tional resistance varies as (��)�1 / 3 as predicted by Eq.(11.22) up to a normal stress of the order of 200 kPa(30 psi), that is, the friction angle decreases with in-creasing normal stress (Chattopadhyay, 1972).

There are at least two possible explanations of thenormal stress independence of the frictional resistanceof quartz, feldspar, and calcite:

1. As the load per particle increases, the number ofasperities in contact increases proportionally, andthe deformation of each asperity remains essen-tially constant. In this case, the assumption of oneasperity per contact for the development of Eq.(11.22) is not valid. Some theoretical considera-tions of multiple asperities in contact are availa-ble (Johnson, 1985). They show that the area ofcontact is approximately proportional to the ap-plied load and hence the coefficient of friction isconstant with load.

2. As the load per asperity increases, the value of �in Eq. (11.13) increases, reflecting a greater pro-portion of solid contact relative to adsorbed filmcontact. Thus, the average strength per contactincreases more than proportionally with the load,while the contact area increases less than pro-portionally, with the net result being an essen-tially constant frictional resistance.

Quartz is a hard, brittle material that can exhibit bothelastic and plastic deformation. A normal pressure of11 GPa (1,500,000 psi) is required to produce plasticdeformation, and brittle failure usually occurs beforeplastic deformation. Plastic deformations are evidentlyrestricted to small, highly confined asperities, and elas-tic deformations control at least part of the behavior(Bromwell, 1965). Either of the previous two expla-nations might be applicable, depending on details ofsurface texture on a microscale and characteristics ofthe adsorbed films.

With the exception of some data for quartz, thereappears to be little information concerning possiblevariations of the true friction angle with particle size.Rowe (1962) found that the value of �� for assem-blages of quartz particles on a flat quartz surface de-

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FRICTIONAL BEHAVIOR OF MINERALS 389

Figure 11.28 Variation in friction angle with normal stress for different minerals (afterKenney, 1967).

creased from 31� for coarse silt to 22� for coarse sand.This is an apparent contradiction to the independenceof particle size on frictional resistance predicted by Eq.(11.22). On the other hand, the assumption of one as-perity per contact may not have been valid for all par-ticle sizes, and additionally, particle surface textures ona microscale could have been size dependent. Further-more, there could have been different amounts of par-ticle rearrangement and rolling in the tests on thedifferent size fractions.

Sliding Friction

The frictional resistance, once sliding has been initi-ated, may be equal to or less than the resistance thathad to be overcome to initiate movement; that is, thecoefficient of sliding friction can be less than the co-efficient of static friction. A higher value of staticfriction than sliding friction is explainable by time-dependent bond formations at asperity junctions.Stick–slip motion, wherein � varies more or less er-ratically as two surfaces in contact are displaced, ap-pears common to all friction measurements of mineralsinvolving single contacts (Procter and Barton, 1974).Stick–slip is not observed during shear of assemblagesof large numbers of particles because the slip of indi-vidual contacts is masked by the behavior of the massas a whole. However, it may be an important mecha-nism of energy dissipation for cyclic loading at verysmall strains when particles are not moving relative toeach other.

11.5 FRICTIONAL BEHAVIOR OF MINERALS

Evaluation of the true coefficient of friction � and fric-tion angle �� is difficult because it is very difficult to

do tests on two very small particles that are slidingrelative to each other, and test results for particle as-semblages are influenced by particle rearrangements,volume changes, surface preparation factors, and thelike. Some values are available, however, and they arepresented and discussed in this section.

Nonclay Minerals

Values of the true friction angle �� for several mineralsare listed in Table 11.1, along with the type of test andconditions used for their determination. A pronouncedantilubricating effect of water is evident for polishedsurfaces of the bulky minerals quartz, feldspar, and cal-cite. This apparently results from a disruptive effect ofwater on adsorbed films that may have acted as a lu-bricant for dry surfaces. Evidence for this is shown inFig. 11.29, where it may be seen that the presence ofwater had no effect on the frictional resistance ofquartz surfaces that had been chemically cleaned priorto the measurement of the friction coefficient. Thesamples tested by Horn and Deere (1962) in Table 11.1had not been chemically cleaned.

An apparent antilubrication effect by water mightalso arise from attack of the silica surface (quartz andfeldspar) or carbonate surface (calcite) and the for-mation of silica and carbonate cement at interparticlecontacts. Many sand deposits exhibit ‘‘aging’’ effectswherein their strength and stiffness increase noticeablywithin periods of weeks to months after deposition,disturbance, or densification, as described, for exam-ple, by Mitchell and Solymar (1984), Mitchell (1986),Mesri et al. (1990), and Schmertmann (1991). In-creases in penetration resistance of up to 100 percenthave been measured in some cases. The relative im-portance of chemical factors, such as precipitation at

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Table 11.1 Values of Friction Angle (��) Between Mineral Surfaces

Mineral Type of Test Conditions �� (deg) Comments Reference

Quartz Block over particleset in mortar

Dry 6 Dried over CaCl2 beforetesting

Tschebotarioff andWelch (1948)

Moist 24.5Water saturated 24.5

Quartz Three fixed particlesover block

Water saturated 21.7 Normal load per particleincreasing from 1 to100 g

Hafiz (1950)

Quartz Block on block Dry 7.4 Polished surfaces Horn and Deere (1962)Water saturated 24.2

Quartz Particles onpolished block

Water saturated 22–31 � decreasing with in-creasing particle size

Rowe (1962)

Quartz Block on block Variable 0–45 Depends on roughnessand cleanliness

Bromwell (1966)

Quartz Particle–particle Saturated 26 Single-point contact Procter and Barton(1974)

Particle–plane Saturated 22.2Particle–plane Dry 17.4

Feldspar Block on block Dry 6.8 Polished surfaces Horn and Deere (1962)Water saturated 37.6

Feldspar Free particles on flatsurface

Water saturated 37 25–500 sieve Lee (1966)

Feldspar Particle–plane Saturated 28.9 Single-point contact Procter and Barton(1974)

Calcite Block on block Dry 8.0 Polished surfaces Horn and Deere (1962)Water saturated 34.2

Muscovite Along cleavagefaces

Dry 23.3 Oven dry Horn and Deere (1962)

Dry 16.7 Air equilibratedSaturated 13.0

Phlogopite Along cleavagefaces



Biotite Along cleavagefaces



Chlorite Along cleavagefaces



interparticle contacts, changes in surface characteris-tics, and mechanical factors, such as time-dependentstress redistribution and particle reorientations, in caus-ing the observed behavior is not known. Further detailsof aging effects are given in Chapter 12.

As surface roughness increases, the apparent anti-lubricating effect of water decreases. This is shownin Fig. 11.29 for quartz surfaces that had not beencleaned. Chemically cleaned quartz surfaces, whichgive the same value of friction when both dry and wet,

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FRICTIONAL BEHAVIOR OF MINERALS 391

Figure 11.29 Friction of quartz (data from Bromwell, 1966 and Dickey, 1966).

show a loss in frictional resistance with increasing sur-face roughness. Evidently, increased roughness makesit easier for asperities to break through surface films,resulting in an increase in � [Eq. (11.13) and Fig.11.27]. The decrease in friction with increased rough-ness is not readily explainable. One possibility is thatthe cleaning process was not effective on the roughsurfaces.

For soils in nature, the surfaces of bulky mineralparticles are most probably rough relative to the scalein Fig. 11.29, and they will not be chemically clean.Thus, values of � � 0.5 and �� 26� are reasonablefor quartz, both wet and dry.

On the other hand, water apparently acts as a lubri-cant in sheet minerals, as shown by the values for mus-covite, phlogopite, biotite, and chlorite in Table 11.1.This is because in air the adsorbed film is thin, andsurface ions are not fully hydrated. Thus, the adsorbedlayer is not easily disrupted. Observations have shownthat the surfaces of the sheet minerals are scratchedwhen tested in air (Horn and Deere, 1962). When thesurfaces of the layer silicates are wetted, the mobilityof the surface films is increased because of their in-creased thickness and because of greater surface ionhydration and dissociation. Thus, the values of ��

listed in Table 11.1 for the sheet minerals under satu-rated conditions (7�–13�) are probably appropriate forsheet mineral particles in soils.

Clay Minerals

Few, if any, directly measured values of �� for the clayminerals are available. However, because their surfacestructures are similar to those of the layer silicates dis-cussed previously, approximately the same valueswould be anticipated, and the ranges of residual fric-tion angles measured for highly plastic clays and clayminerals support this. In very active colloidal pureclays, such as montmorillonite, even lower friction an-gles have been measured. Residual values as low as 4�for sodium montmorillonite are indicated by the datain Fig. 11.28.

The effective stress failure envelopes for calciumand sodium montmorillonite are different, as shown byFig. 11.30, and the friction angles are stress dependent.For each material the effective stress failure envelopewas the same in drained and undrained triaxial com-pression and unaffected by electrolyte concentrationover the range investigated, which was 0.001 N to 0.1N. The water content at any effective stress was inde-pendent of electrolyte concentration for calcium mont-morillonite, but varied in the manner shown in Fig.11.31 for sodium montmorillonite.

This consolidation behavior is consistent with thatdescribed in Chapter 10. Interlayer expansion in cal-cium montmorillonite is restricted to a c-axis spacingof 1.9 nm, leading to formation of domains or layeraggregates of several unit layers. The interlayer spac-

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Figure 11.30 Effective stress failure diagrams for calcium and sodium montmorillonite (af-ter Mesri and Olson, 1970).

Figure 11.31 Shear and consolidation behavior of sodiummontmorillonite (after Mesri and Olson, 1970).

ing of sodium montmorillonite is sensitive to double-layer repulsions, which, in turn, depend on theelectrolyte concentration. The influence of the electro-lyte concentration on the behavior of sodium mont-morillonite is to change the water content, but not thestrength, at any effective consolidation pressure. Thissuggests that the strength generating mechanism is in-dependent of the system chemistry.

The platelets of sodium montmorillonite act as thinfilms held apart by high repulsive forces that carry theeffective stress. For this case, if it is assumed that thereis essentially no intergranular contact, then Eq. (7.29)becomes

�� A � u � R � 0 (11.23)i 0

Since � � u0 is the conventionally defined effectivestress ��, and assuming negligible long-range attrac-tions, Eq. (11.23) becomes

�� R (11.24)

This accounts for the increase in consolidation pres-sure required to decrease the water content, while at

the same time there is little increase in shear strengthbecause the shearing strength of water and solutions isessentially independent of hydrostatic pressure. Thesmall friction angle that is observed for sodium mont-morillonite at low effective stresses can be ascribedmainly to the few interparticle contacts that resist par-ticle rearrangement. Resistance from this source evi-dently approaches a constant value at the highereffective stresses, as evidenced by the nearly horizontalfailure envelope at values of average effective stressgreater than about 50 psi (350 kPa), as shown in Fig.11.30. The viscous resistance of the pore fluid maycontribute a small proportion of the strength at all ef-fective stresses.

An hypothesis of friction between fine-grained par-ticles in the absence of interparticle contacts is givenby Santamarina et al. (2001) using the concept of‘‘electrical’’ surface roughness as shown in Fig. 11.32.Consider two clay surfaces with interparticle fluid asshown in Fig. 11.32b. The clay surfaces have a numberof discrete charges, so a series of potential energywells exists along the clay surfaces. Two cases can beconsidered:

1. When the particle separation is less than severalnanometers, there are multiple wells of minimumenergy between nearby surfaces and a force isrequired to overcome the energy barrier betweenthe wells when the particles move relative to eachother. Shearing involves interaction of the mole-cules of the interparticle fluid. Due to the multi-ple energy wells, the interparticle fluid moleculesgo through successive solidlike pinned states.This stick–slip motion contributes to frictionalresistance and energy dissipation.

2. When the particle separation is more than severalnanometers, the two clay surfaces interact onlyby the hydrodynamic viscous effects of the in-terparticle fluid, and the frictional force may beestimated using fluid dynamics.

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PHYSICAL INTERACTIONS AMONG PARTICLES 393

Figure 11.32 Concept of ‘‘electrical’’ surface roughness ac-cording to Santamarina et al., (2001): (a) electrical roughnessand (b) conceptual picture of friction in fine-grained particles.

The aggregation of clay plates in calcium montmo-rillonite produces particle groups that behave more likeequidimensional particles than platy particles. There ismore physical interference and more intergrain contactthan in sodium montmorillonite since the water contentrange for the strength data shown in Fig. 11.30 wasonly about 50 to 97 percent, whereas it was about 125to 450 percent for the sodium montmorillonite. At aconsolidation pressure of about 500 kPa, the slope ofthe failure envelope for calcium montmorillonite wasabout 10�, which is in the middle of the range for non-clay sheet minerals (Table 11.1).

11.6 PHYSICAL INTERACTIONS AMONGPARTICLES

Continuum mechanics assumes that applied forces aretransmitted uniformly through a homogenized granularsystem. In reality, however, the interparticle force dis-tributions are strongly inhomogeneous, as discussed inChapter 7, and the applied load is transferred througha network of interparticle force chains. The generic

disorder of particles, (i.e., local spatial fluctuations ofcoordination number, and positions of neighboring par-ticles) produce packing constraints and disorder. Thisleads to inhomogeneous but structured force distribu-tions within the granular system. Deformation is as-sociated with buckling of these force chains, andenergy is dissipated by sliding at the clusters of par-ticles between the force chains.

Discrete particle numerical simulations, such as thediscrete (distinct) element method (Cundall and Strack,1979) and the contact dynamics method (Moreau,1994), offer physical insights into particle interactionsand load transfers that are difficult to deduce fromphysical experiments. Typical inputs for the simula-tions are particle packing conditions and interparticlecontact characteristics such as the interparticle frictionangle ��. Complete details of these numerical methodsare beyond the scope of this book; additional infor-mation can be found in Oda and Iwashita (1999).However, some of the main findings are useful fordeveloping an improved understanding of how stressesare carried through discrete particle systems such assoils and how these distributions influence the defor-mation and strength properties.

Strong Force Networks and Weak Clusters

Examples of the computed normal contact force dis-tribution in a granular system are shown in Figs.11.33a for an isotropically loaded condition and11.33b for a biaxial loaded condition (Thornton andBarnes, 1986). The thickness of the lines in the figureis proportional to the magnitude of the contact force.The external loads are transmitted through a networkof interparticle contact forces represented by thickerlines. This is called the strong force network and is thekey microscopic feature of load transfer through thegranular system. The scale of statistical homogeneityin a two-dimensional particle assembly is found to bea few tens of particle diameters (Radjai et al., 1996).Forces averaged over this distance could therefore beexpected to give a stress that is representative of themacroscopic stress state. The particles not forming apart of the strong force network are floating like a fluidwith small loads at the interparticle contacts. This canbe called the weak cluster, which has a width of 3 to10 particle diameters.

Both normal and tangential forces exist at interpar-ticle contacts. Figure 11.34 shows the probability dis-tributions (PN and PT) of normal contact forces N andtangential contact forces T for a given biaxial loadingcondition. The horizontal axis is the forces normalizedby their mean force value (�N� or �T�), which de-pend on particle size distribution (Radjai et al., 1996).The individual normal contact forces can be as great

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Figure 11.33 Normal force distributions of a two-dimensional disk particle assembly: (a) isotropic stress con-dition and (b) biaxial stress condition with maximum load inthe vertical direction (after Thornton and Barnes, 1986).

as six times the mean normal contact force, butapproximately 60 percent of contacts carry normalcontact forces below the mean (i.e., weak clusterparticles). When normal contact forces are larger thantheir mean, the distribution law of forces can be ap-proximated by an exponentially decreasing function;Radjai et al. (1996) show that PN(� � N /�N�) �ke1.4(1�� ) fits the computed data well for both two-andthree-dimensional simulations. The exponent wasfound to change very slightly with the coefficient ofinterparticle friction and to be independent of particlesize distributions.

Simulations show that applied deviator load is trans-ferred exclusively by the normal contact forces in thestrong force networks, and the contribution by theweak clusters is negligible. This is illustrated in Fig.11.35, which shows that the normal contact forces con-tribute greater than the tangential contact forces to thedevelopment of the deviator stress during axisymme-

tric compression of a dense granular assembly (Thorn-ton, 2000). The strong force network carries most ofthe whole deviator load as shown in Fig. 11.36 and isthe load-bearing part of the structure. For particles inthe strong force networks, the tangential contact forcesare much smaller than the interparticle frictional resis-tance because of the large normal contact forces. Incontrast, the numerical analysis results show that thetangential contact forces in the weak clusters are closeto the interparticle frictional resistance. Hence, the fric-tional resistance is almost fully mobilized between par-ticles in the weak clusters, and the particles are perhapsbehaving like a viscous fluid.

Buckling, Sliding, and Rolling

As particles begin to move relative to each other duringshear, particles in the strong force network do not slide,but columns of particles buckle (Cundall and Strack,1979). Particles in the strong force network collapseupon buckling, and new force chains are formed.Hence, the spatial distributions of the strong force net-work are neither static nor persistent features.

At a given time of biaxial compression loading, par-ticle sliding is occurring at almost 10 percent of thecontacts (Kuhn, 1999) and approximately 96 percentof the sliding particles are in the weak clusters (Radjaiet al., 1996). Over 90 percent of the energy dissipationoccurs at just a small percentage of the contacts (Kuhn,1999). This small number of sliding particles is asso-ciated with the ability of particles to roll rather than toslide. Particle rotations reduce contact sliding and dis-sipation rate in the granular system. If all particlescould roll upon one another, a granular assemblywould deform without energy dissipation.7 However,this is not possible owing to restrictions on particlerotations. It is impossible for all particles to move byrotation, and sliding at some contacts is inevitable dueto the random position of particles (Radjai and Roux,1995).8 Some frictional energy dissipation can there-fore be considered a consequence of disorder of par-ticle positions.

As deformation progresses, the number of particlesin the strong force network decreases, with fewer par-ticles sharing the increased loads (Kuhn, 1999). Figure

7 This assumes that the particles are rigid and rolling with a single-point contact. In reality, particles deform and exhibit rolling resis-tance. Iwashita and Oda (1998) state that the incorporation of rollingresistance is necessary in discrete particle simulations to generaterealistic localized shear bands.8 For instance, consider a chain loop of an odd number of particles.Particle rotation will involve at least one sliding contact.

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Figure 11.34 Probability distributions of interparticle contact forces: (a) normal forces and(b) tangential forces. The distributions were obtained for contact dynamic simulations of500, 1024, 1200 and 4025 particles. The effect of number of particles in the simulation onprobability distribution appears to be small (after Radjai et al., 1996).

Figure 11.35 Contributions of normal and tangential contactforces to the evolution of the deviator stress during axisym-metric compression of a dense granular assembly (afterThornton, 2000).

Figure 11.36 Contributions of strong and weak contactforces to the evolution of the deviator stress during axisym-metric compression of a dense granular assembly (afterThornton, 2000).

11.37 shows the spatial distribution of residual defor-mation, in which the computed deformation of eachparticle is subtracted from the average overall defor-mation (Williams and Rege, 1997). A group of inter-locked particles that instantaneously moves as a rigidbody in a circular manner can be observed. The outer

boundary of the group shows large residual deforma-tion, whereas the center shows very small residual de-formation. The rotating group of interlocked particles,which can be considered as a weak cluster, becomesmore apparent as applied strains increase toward fail-ure. The bands of large residual deformation [termed

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Figure 11.37 Spatial distribution of residual deformation ob-served in an elliptic particle assembly at an axial strain levelof (a) 1.1%, (b) 3.3%, (c) 5.5%, (d) 7.7%, (e) 9.8%, and(ƒ) 12.0% (after Williams and Rege, 1997).

2 4 6 8 10-2-4-6-8

0.5

1.0

1.5

-0.5

-1.0

Stress Ratio q/p�

Axial Strain (%)

2 4 6 8 10-2-4-6-8

0.10.2

Fabric Anisotropy A

Axial Strain (%)

0.30.4

-0.1-0.2

-0.3-0.4-0.5

Triaxial Compression

Triaxial Extension

More in Vertical Direction

Contact Plane Normalsin Initial State:

Same in All Directions

More in Horizontal Direction

More in Vertical Direction

Contact Plane Normalsin Initial State:

Same in All Directions

More in Horizontal Direction

(a)

(b)

A

BC

A

BC

Figure 11.38 Discrete element simulations of drained tri-axial compression and extension tests of particle assembliesprepared at different initial contact fabrics: (a) stress–strainrelationships and (b) evolution of fabric anisotropy parameterA (after Yimsiri, 2001).

microbands by Kuhn (1999)] are where particle trans-lations and rotations are intense as part of the strongforce network. Kuhn (1999) reports that their thick-nesses are 1.5D50 to 2.5D50 in the early stages of shear-ing and increase to between 1.5D50 and 4D50 asdeformation proceeds. This microband slip zone mayeventually become a localized shear band.

Fabric Anisotropy

The ability of a granular assemblage of particles tocarry deviatoric loads is attributed to its capability todevelop anisotropy in contact orientations. An initialisotropic packing of particles develops an anisotropiccontact network during compression loading. This isbecause new contacts form in the direction of com-pression loading and contacts that orient along the di-rection perpendicular to loading direction are lost.

The initial state of contact anisotropy (or fabric)plays an important role in the subsequent deformationas illustrated in Fig. 11.18. Figure 11.38 shows results

of discrete particle simulations of particle assembliesprepared at different states of initial contact anisotropyunder an isotropic stress condition (Yimsiri, 2001). Theinitial void ratios are similar (e0 � 0.75 to 0.76) andboth drained triaxial compression and extension testswere simulated. Although all specimens are initiallyisotropically loaded, the directional distributions ofcontact forces are different due to different orientationsof contact plane normals (sample A: more in the ver-tical direction; sample B: similar in all directions; sam-ple C: more in the horizontal direction). As shown inFig. 11.38a, both samples A and C showed stiffer re-sponse when the compression loading was applied inthe preferred direction of contact forces, but softer re-sponse when the loading was perpendicular to the pre-ferred direction of contact forces. The response ofsample B, which had an isotropic fabric, was in be-tween the two. Dilation was most intensive when thecontact forces were oriented preferentially in the di-

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0.1

0.05

0.0

-0.05

-0.1

1 2 3 4 5 6N/<N>

Fab

ric A

niso

trop

y P

aram

eter

A

Figure 11.39 Fabric anisotropy parameter A for differentlevels of contact force when the specimen is under biaxialcompression loading conditions (after Radjai et al., 1996).

Figure 11.40 Evolution of the fabric anisotropy parametersof strong forces and weak clusters when the specimen is un-der biaxial compression loading conditions (after Thorntonand Antony, 1998).

rection of applied compression; and experimental datapresented by Konishi et al. (1982) shows a similartrend.

Figure 11.38b shows the development of fabric an-isotropy with increasing strain. The degree of fabricanisotropy is expressed by a fabric anisotropy param-eter A; the value of A increases with more verticallyoriented contact plane normals and is negative whenthere are more horizontally oriented contact plane nor-mals.9 The fabric parameter gradually changes with in-creasing strains and reaches a steady-state value as thespecimens fail. The final steady-state value is indepen-dent of the initial fabric, indicating that the inherentanisotropy is destroyed by the shearing process. Thefinal fabric anisotropy after triaxial extension is largerthan that after triaxial compression because the addi-tional confinement by a larger intermediate stress inthe extension tests created a higher degree of fabricanisotropy.

Close examination of the contact force distributionfor the strong force network and weak clusters givesinteresting microscopic features. Figure 11.39 showsthe values of A determined for the subgroups of contact

9 The density of contact plane normals E(�) with direction � is fittedwith the following expression (Radjai, 1999):

cE(�) � {1 � A cos 2(� � � )}c�

where c is the total number of contacts, �c is the direction for whichthe maximum E is reached, and the magnitude of A indicates theamplitude of anisotropy. When the directional distribution of contactforces is independent of �, the system has an isotropic fabric andA � 0.

forces categorized by their magnitudes when the spec-imen is under a biaxial compression loading condition(Radjai, 1999). The direction of contact anisotropy ofthe weak clusters (N /�N� less than 1) is orthogonalto the direction of compression loading, whereas thatof the strong force network (N /�N� more than 2) isparallel. Figure 11.40 shows an example of fabric ev-olution with strains in biaxial loading (Thornton andAntony, 1998). The fabric anisotropy is separated intothat in the strong force networks (N /�N� of morethan 1) and that in the weak clusters (N /�N� less than1). Again the directional evolution of the fabric in theweak clusters is opposite to the direction of loading.Therefore, the stability of the strong force chainsaligned in the vertical loading direction is obtained bythe lateral forces in the surrounding weak clusters.

Changes in Number of Contacts and MicroscopicVoids

At the beginning of biaxial loading of a dense granularassembly, more contacts are created from the increasein the hydrostatic stress, and the local voids becomesmaller. As the axial stress increases, however, the lo-cal voids tend to elongate in the direction of loadingas shown in Fig. 11.41. Consequently particle contactsare lost. As loading progresses, vertically elongated lo-cal voids become more apparent, leading to dilation in

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Figure 11.41 Simulated spatial distribution of local microvoids under biaxial loading (afterIwashita and Oda, 2000): (a) 11 � 1.1% (before failure), (b) 11 � 2.2% (at failure), (c)� �

11 � 4.4% (after failure), and (d ) 11 � 5.5% (after failure).� �

terms of overall sample volume (Iwashita and Oda,2000).

Void reduction is partly associated with particlebreakage. Thus, there is a need to incorporate graincrushing in discrete particle simulations to model thecontractive behavior of soils (Cheng et al., 2003). Nor-mal contact forces in the strong force network are quitehigh, and, therefore, particle asperities, and even par-ticles themselves, are likely to break, causing the forcechains to collapse.

Local voids tend to change size even after the ap-plied stress reaches the failure stress state (Kuhn,1999). This suggests that the degrees of shearing re-quired for the stresses and void ratio to reach the crit-ical state are different. Numerical simulations byThornton (2000) show that at least 50 percent axial

strain is required to reach the critical state void ratio.Practical implication of this is discussed further in Sec-tion 11.7.

Macroscopic Friction Angle Versus InterparticleFriction Angle

Discrete particle simulations show that an increase inthe interparticle friction angle �� results in an increasein shear modulus and shear strength, in higher rates ofdilation, and in greater fabric anisotropy. Figure 11.42shows the effect of assumed interparticle friction angle�� on the mobilized macroscopic friction angle of theparticle assembly (Thornton, 2000; Yimsiri, 2001). Themacroscopic friction angle is larger than the interpar-ticle friction angle if the interparticle friction angle issmaller than 20�. As the interparticle friction becomes

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Drained (Thornton, 2000)

Drained Triaxial Compression (Yimsiri, 2001)

Undrained Triaxial Compression (Yimsiri, 2001)

Drained Triaxial Extension (Yimsiri, 2001)

Undrained Triaxial Extension (Yimsiri, 2001)

Experiment (Skinner, 1969)

10 20 30 40 50 60 70 80 9000

10

20

30

40

50

Interparticle Friction Angle (degrees)

Mac

rosc

opic

Fric

tion

Ang

le (

degr

ees)

Figure 11.42 Relationships between interparticle friction angle and macroscopic frictionangle from discrete element simulations. The macroscopic friction angle was determinedfrom simulations of drained and undrained triaxial compression (TC) and extension (TE)tests. The experimental data by Skinner (1969) is also presented (after Thornton, 2000, andYimsiri, 2001).

more than 20�, the contribution of increasing interpar-ticle friction to the macroscopic friction angle becomesrelatively small; the macroscopic friction angle rangesbetween 30� and 40�, when the interparticle frictionangle increases from 30� to 90�.10

The nonproportional relationship between macro-scopic friction angle of the particle assembly and in-terparticle friction angle results because deviatoric loadis carried by the strong force networks of normalforces and not by tangential forces, whose magnitudeis governed by interparticle friction angle. Increase ininterparticle friction results in a decrease in the per-centage of sliding contacts (Thornton, 2000). The in-terparticle friction therefore acts as a kinematicconstraint of the strong force network and not as thedirect source of macroscopic resistance to shear. If theinterparticle friction were zero, strong force chainscould not develop, and the particle assembly will be-

10 Reference to Table 11.1 shows that actually measured values of ��

for geomaterials are all less than 45�. Thus, numerical simulationsdone assuming larger values of �� appear to give unrealistic results.

have like a fluid. Increased friction at the contacts in-creases the stability of the system and reduces thenumber of contacts required to achieve a stable con-dition. As long as the strong force network can beformed, however, the magnitude of the interparticlefriction becomes of secondary importance.

The above findings from discrete particle simula-tions are partially supported by the experimental datagiven by Skinner (1969), which are also shown in Fig.11.42. He performed shear box tests on spherical par-ticles with different coefficients of interparticle frictionangle. The tested materials included glass ballotini,steel ball bearings, and lead shot. Use of glass ballotiniwas particularly attractive since the coefficient of in-terparticle friction increases by a factor of between 3.5and 30 merely by flooding the dry sample. Skinner’sdata shown in Fig. 11.42 indicate that the macroscopicfriction angle is nearly independent of interparticlefriction angle.

Effects of Particle Shape and Angularity

A lower porosity and a larger coordination number areachieved for ellipsoidal particles compared to spherical

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DeviatorStress

q = σ�a – σ�r

Mean Pressure p�

M (triaxialcompression)

ln p�

SpecificVolume v

1

Γ

λcs

CriticalState Line

IsotropicCompressionLine

λ

CriticalState Line

M (triaxialextension)

CriticalState Line

σ�a

σ�r σ�r

Compression Lines ofConstant Stress Ratio q/p�

(a)

(b)

Figure 11.43 Critical state concept: (a) p�–q plane and (b)v–ln p� plane.

particles (Lin and Ng, 1997). Hence, a denser packingcan be achieved for ellipsoidal particles. Ellipsoid par-ticles rotate less than spherical particles. An assemblyof ellipsoid particles gives larger values of shearstrength and initial modulus than an assemblage ofspherical particles, primarily because of the larger co-ordination number for ellipsoidal particles. Similarfindings result for two-dimensional particle assemblies.Circular disks give the highest dilation for a givenstress ratio and the lowest coordination number com-pared to elliptical or diamond shapes (Williams andRege, 1997). An assembly of rounded particles exhib-its greater softening behavior with fabric anisotropychange with strain, whereas an assembly of elongatedparticles requires more shearing to modify its initialfabric anisotropy to the critical state condition(Nouguier-Lehon et al., 2003).

11.7 CRITICAL STATE: A USEFUL REFERENCECONDITION

After large shear-induced volume change, a soil undera given effective confining stress will arrive ultimatelyat a unique final water content or void ratio that isindependent of its initial state. At this stage, the inter-locking achieved by densification or overconsolidationis gone in the case of dense soils, the metastable struc-ture of loose soils has collapsed, and the soil is fullydestructured. A well-defined strength value is reachedat this state, and this is often referred to as the criticalstate strength. Under undrained conditions, the criticalstate is reached when the pore pressure and the effec-tive stress remain constant during continued deforma-tion. The critical state can be considered a fundamentalstate, and it can be used as a reference state to explainthe effect of overconsolidation ratios, relative density,and different stress paths on strength properties of soils(Schofield and Wroth, 1968).

Clays

The basic concept of the critical state is that, undersustained uniform shearing, there exists a unique re-lationships among void ratio ecs (or specific volumevcs � 1 � ecs), mean effective pressure , and deviatorp�cs

stress qcs as shown in Fig. 11.43. An example of thecritical state of clay was shown in Fig. 11.4a. The crit-ical state of clay can be expressed as

q � Mp� (11.25)cs cs

v � 1 � e � % � � ln p� (11.26)cs cs cs cs

where qcs is the deviator stress at failure, is thep�cs

mean effective stress at failure, and M is the critical

state stress ratio. The critical state on the void ratio (orspecific volume)–mean pressure plane is defined bytwo material parameters: �cs, the critical state com-pression index and %, the specific volume intercept atunit pressure (p� � 1). The compression lines underconstant stress ratios are often parallel to each other,as shown in Fig. 11.43b.

Parameter M in Equation (11.25) defines the criticalstate stress ratio at failure and is similar to �� for theMohr–Coulomb failure line. However, Equation(11.25) includes the effect of intermediate principalstress because p� � � � , whereas the�� 2 1 2 3

Mohr–Coulomb failure criterion of Eq. (11.4) or (11.5)does not take the intermediate effective stress into ac-count. In triaxial conditions, and�� a r r r

for compression and extension, respectively�� r a

(see Fig. 11.43).11 Hence, Eqs. (11.4) and (11.25) canbe related to each other for these two cases as follows:

11 is the axial effective stress and is the radial effective stress.�� a r

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CRITICAL STATE: A USEFUL REFERENCE CONDITION 401

DeviatorStress q

Mean pressure p�

M

ln p�

SpecificVolume v

1

Γ

λcs

Critical State Line


λ

CriticalState Line

A

A

B

B

C

C

3

1

DeviatorStress q

Mean pressure p�

M

ln p�

SpecificVolume v

1

Γ

λcs

Critical State Line


λ

CriticalState Line

D

E

F

D

F

E

31

(a)

Drained Strength

Drained Strength

UndrainedStrength

Undrained Strength

D�

D�

G

G

Drained Peak Strength

(b)

Figure 11.44 Drained and undrained stress–strain response using the critical state concept:(a) normally consolidated clay and (b) overconsolidated clay.

6 sin ��critM � for triaxial compression (11.27)3 � sin ��crit

6 sin ��critM � for triaxial extension (11.28)3 � sin ��crit

These equations indicate that the correlation be-tween M and is not unique but depends on the��crit

stress conditions. Neither is a fundamental property ofthe soil, as discussed further in Section 11.12. None-theless, both are widely used in engineering practice,and, if interpreted properly, they can provide usefuland simple phenomenological representations of com-plex behavior.

The drained and undrained critical state strengths areillustrated in Figs. 11.44a and 11.44b for normallyconsolidated clay and heavily overconsolidated clay,respectively. The initial mean pressure–void ratio stateof the normally consolidated clay is above the criticalstate line, whereas that of the heavily overconsolidatedclay is below the critical state line. When the initialstate of the soil is normally consolidated at A (Fig.11.44a), the critical state is B for undrained loading

and C for drained triaxial compression. Hence, the de-viator stress at critical state is smaller for the undrainedcase than for the drained case. On the other hand, whenthe initial state of the soil is overconsolidated fromD� (Fig. 11.44b), the critical state becomes E for un-drained loading and F for drained triaxial compression.The deviator stress at critical state is smaller for thedrained case compared to the undrained case. It is im-portant to note that the soil state needs to satisfy bothstate equations [Eqs. (11.25) and (11.26)] to be at crit-ical state. For example, point G in Fig. 11.44b satisfies

and qcs, but not ecs; therefore, it is not at the criticalp�cs

state.Converting the void ratio in Eq. (11.26) to water

content, a normalized critical state line can be writtenusing the liquidity index (see Fig. 11.45).

w � w ln(p� /p�)cs PL PLLI � � (11.29)cs w � w ln(p� /p� )LL PL PL LL

where wcs is the water content at critical state whenthe effective mean pressure is p�. and are thep� p�LL PL

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wLL

wPL

ln(p�PL)ln(p�LL)

LI = 1

LI = 0

ln(p�PL)ln(p�LL) ln(p�)

LiquidityIndex

WaterContent

Liquid Limit

Plastic Limit

(ln(p’), LI)

ln(p�)

wLI

LICS

LIeq-1wcs

CriticalState Line

IsotropicCompression Line

CriticalState Line

Meanpressure

MeanPressure

(a) (b)

(ln(p�), w)

Figure 11.45 Normalization of the critical state line: (a) water content versus mean pressureand (b) liquidity index versus mean pressure.

mean effective pressure at liquid limit (wLL) and plasticlimit (wPL), respectively; � 1.5 to 6 kPa andp� p�LL PL

� 150 to 600 kPa are expected considering the un-drained shear strengths at liquid and plastic limits arein the ranges suLL � 1 to 3 kPa and suPL � 100 to 300kPa, respectively12 (see Fig. 8.48).

Using Eq. (11.29), a relative state in relation to thecritical state for a given effective mean pressure (i.e.,above or below the critical state line) can be definedas (see Fig. 11.45)

log(p� /p� )LLLI � LI � LI � 1 � LI � (11.30)eq cs log(p� /p� )PL LL

where LIeq is the equivalent liquidity index defined bySchofield (1980). When LIeq � 1 (i.e., LI � LIcs) andq/p� � M, the clay has reached the critical state. Figure11.46 gives the stress ratio when plastic failure (orfracture) initiates at a given water content. When LIeq

� 1 (the state is above the critical state line), and thesoil in a plastic state exhibits uniform contractive be-havior. When LIeq � 1 (the state is below the criticalstate line), and the soil in a plastic state exhibits lo-calized dilatant rupture, or possibly fracture, if thestress ratio reaches the tensile limit (q/p� � 3 for tri-axial compression and �1.5 for triaxial extension; seeFig. 11.46b). Hence, the critical state line can be usedas a reference to characterize possible soil behaviorunder plastic deformation.

12 A review by Sharma and Bora (2003) gives average values ofsuLL � 1.7 kPa and suPL � 170 kPa.

Sands

The critical state strength and relative density of sandcan be expressed as

q � Mp� (11.31)cs cs

e � e 1max csD � � (11.32)R,cs e � e ln(� /p�)max min c

where ecs is the void ratio at critical state, emax and emin

are the maximum and minimum void ratios, and �c isthe crushing strength of the particles.13 The criticalstate line based on Eq. (11.32) is plotted in Fig. 11.47.The plotted critical state lines are nonlinear in the e–ln p� plane in contrast to the linear relationship foundfor clays. This nonlinearity is confirmed by experi-mental data as shown in Fig. 11.4b.

At high confining pressure, when the effective meanpressure becomes larger than the crushing strength,many particles begin to break and the lines becomemore or less linear in the e–ln p� plane, similar to the

13 Equation (11.32) is derived from Eq. 11.36 proposed by Bolton(1986) with IR � 0 (zero dilation). Bolton’s equation is discussedfurther in Section 11.8. Other mathematical expressions to fit theexperimentally determined critical state line are possible. For exam-ple, Li et al. (1999) propose the following equation for the criticalstate line (ecs versus p�):

�p�e � e � � � �cs 0 s pa

where e0 is the void ratio at p � 0, pa is atmospheric pressure, and�s and � are material constants.

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CRITICAL STATE: A USEFUL REFERENCE CONDITION 403

q/p�

0.5 LIeq1.0

3

-1.5

MTC

MTE

DilatantRupture

Fracture

Ductile Plasticand Contractive

DilatantRuptureFracture


Triaxial Extension

(a)

p�

qTriaxial Compression

Triaxial Extension

1

3

2

3

MTC

MTE

TensileFracture

TensileFracture

(b)

Ductile Plasticand Contractive

Figure 11.46 Plastic state of clay in relation to normalized liquidity index: (a) stress ratiowhen plastic state initiates for a given LIeq and (b) definition of stress ratios used in (a) (afterSchofield, 1980).

0

0.2

0.4

0.6

0.8

1

1.20.001 0.01 0.1 1

p�/σc p�/σc

Rel

ativ

e D

ensi

ty D

r

0

0.2

0.4

0.6

0.8

1

1.20 0.1 0.2 0.3 0.4 0.5

Rel

ativ

e D

ensi

ty D

r

emax

emin

emax

emax

emin

(a) (b)

DR,cs =––

ecs

emax emin=

In (σc/p�)1

Figure 11.47 Critical state line derived from Eq. (11.32): (a) e–log p� plane and (b) e–p�plane.

behavior of clays. Coop and Lee (1993) found thatthere is a unique relationship between the amount ofparticle breakage that occurred on shearing to a criticalstate and the value of the mean normal effective stress.This implies that sand at the critical state would reacha stable grading at which the particle contact stresses

would not be sufficient to cause further breakage. Coopet al. (2004) performed ring shear tests (see Section11.11) on a carbonate sand to find a shear strain re-quired to reach the true critical state (i.e., constantparticle grading). They found that particle breakagecontinues to very large strains, far beyond those

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reached in triaxial tests. Figure 11.48a shows the vol-umetric strains measured for a selection of their tests,which were carried out at vertical stress levels in therange of 650 to 860 kPa. A constant volumetric strainis reached at a shear strain of around 2000 percent. Forspecimens at lower stress levels, more shear strains(20,000 percent or more) were required. Similar find-ings were made for quartz sand (Luzzani and Coop,2002). Figure 11.48b shows the degree of particlebreakage with shear strains in the logarithmic scale.The amount of breakage is quantified by Hardin’s(1985) relative breakage parameter Br defined in Fig.10.14. At very large strains, the value of Br finallystabilizes. The strain required for stabilization dependson applied stress level. Interestingly, less shear strainwas needed for the mobilized friction angle to reachthe steady-state value (Fig. 11.48c) than for attainmentof the constant volume condition, (Fig. 11.48a). Thecritical state friction angle was unaffected by the par-ticle breakage.

In summary, the critical state concept is very usefulto characterize the strength and deformation propertiesof soils when it is used as a reference state. That is, asoil has a tendency to contract upon shearing when itsstate is above the critical state line, whereas it has atendency to dilate when it is below the critical stateline. Various normalized state parameters have beenproposed to characterize the difference between the ac-tual state and the critical state line, as illustrated in Fig.11.49. These parameters have been used to character-ize the stiffness and strength properties of soils. Someof them are introduced later in this chapter.

11.8 STRENGTH PARAMETERS FOR SANDS

Many factors and phenomena act together to deter-mined the shearing resistance of sands, for example,mineralogy, grain size, grain shape, grain size distri-bution, (relative) density, stress state, type of tests andstress path, drainage, and the like [see Eq. (11.3)]. Inthis section, the ways in which these factors have be-come understood and have been quantified over the lastseveral decades are summarized. Several correlationsare given to provide an overview and reference fortypical values and ranges of strength parameters forsands and the influences of various factors on theseparameters.14

14 A number of additional useful correlations are given by Kulhawyand Mayne (1990).

Early Studies

The important role of volume change during shear, es-pecially dilatancy, was recognized by Taylor (1948).From direct shear box testing of dense sand specimens,he calculated the work at peak shear stress state andshowed that the energy input is dissipated by the fric-tion using the following equation:

� dx � �� dy � �� dx (11.33)peak n n

where �peak is the applied shear stress at peak, is the��nconfining normal (effective) stress on the shear plane,dx is the incremental horizontal displacement at peak,dy is the incremental vertical displacement (expansionpositive) at peak stress, and � is the friction coefficient.The energy dissipated by friction (the component inthe right-hand side) is equal to the sum of the workdone by shearing (first component in the left-handside) and that needed to increase the volume (the sec-ond component in the left-hand side). The latter com-ponent is referred to as dilatancy.

Rearranging Eq. (11.33),

� dypeak � tan �� (11.34)� �m�� dx

Thus, the peak shear stress ratio or the peak mobilizedfriction angle consists of both interlocking (dy/dx)��mand sliding friction between grains (�). This equationthat relates stress to dilation is called the stress–dilatancy rule, and it is an important relationship forcharacterizing the plastic deformation of soils, as fur-ther discussed in Section 11.20.

Rowe (1962) recognized that the mobilized frictionangle must take into account particle rearrange-��mments as well as the sliding resistance at contacts anddilation. Particle crushing, which increases in impor-tance as confining pressure increases and void ratiodecreases, should also be added to these components.The general interrelationships among the strength con-tributing factors and porosity can be represented asshown in Fig. 11.50. In this figure, is the friction��fangle corrected for the work of dilation. It is influencedby particle packing arrangements and the number ofsliding contacts. The denser the packing, the more im-portant is dilation. As the void ratio increases, the mo-bilized friction angle decreases. The critical state isdefined as the condition when there is no volumechange by shearing [i.e., (dy/dx) � 0 in Eq. (11.34)].The corresponding mobilized friction angle is .�� m crit

The ‘‘true friction’’ in the figure is associated with theresistance to interparticle sliding.

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STRENGTH PARAMETERS FOR SANDS 405

0

0

20

40

0.0

0.2

0.4

0.6

0.8

1.0

?

0

10

20

30

40

50

(a)

(b)

(c)

Shear Strain (%)

50,000 100,000 150,000

Vol

umet

ric s

trai

n (%

) RS3

RS5

RS7

RS8

RS13

RS15

RS3

RS7

RS8

RS9

RS10

RS15

RS7

800 kPaunsheared

RS8

Luzzani & Coop, 805 kPa

650-930 kPa

248-386 kPa

60-97 kPa

Shear Strain

Shear Strain (%)

10

101

100

100

1000

1000

10,000

10,000

100,000

100,000

1,000,000

Rel

ativ

e B

reak

age

Mob

ilize

d F

rictio

n A

ngle

(de

gree

s)

Figure 11.48 Ring shear test results of carbonate sand: (a) volumetric strain versus shearstrain, (b) the degree of particle breakage with shear strains, and (c) mobilized friction angleversus shear strains (after Coop et al., 2004).

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Log (Mean pressure p�)

Void ratioe

Critical stateline (p�c , e c)

LooseSand

Densesand

(p�L, eL)

ecL

1. State parameter (Been and Jefferies, 1985)

Ψ = e – ec

Loose sand Ψ = eL – ecL (>0)

Dense sand Ψ = eD – ecD (<0)

2. State index (Ishihara et al., 1998)

Is = (e0 – ec )/(e0 – e)

Loose sand Is = (e0 – ecL)/(e0 – eL) (>1)

Dense sand Is = (e0 – e cD)/(e0 – e0) (<1)

3. State pressure index (Wang et al., 2002)

Ψ>0

ecD

(p�D, eD)

Ψ<0

p�cDp�cL Loose sand Ip = p�L/p�cL (>1)

Dense sand Ip = p�D/p�cD (<1)

Ip = p�/p�c

Figure 11.49 Various parameters that relate the current state to the critical state.

Porosity n (%)

26 30 34 38 42

26

46

42

38

34

30

φ

DensestPacking

CriticalVoid Ratio

φ�crit

φ�m

φ�f

True Friction

To Zero

Crushing (estimated)Rearrangement, Fabric Development

DilationInterlocking

Figure 11.50 Contributions to shear strength of granularsoils (modified from Rowe, 1962).

Critical State Friction Angle

The specific value of the critical state angle of internalfriction depends on the uniformity of particle��crit

sizes, their shape, and mineralogy and is developed atlarge shear strains irrespective of initial conditions.Typical values are 40� for well-graded, angular quartzor feldspar sands, 36� for uniform subangular quartzsand, and 32� for uniform rounded quartz sand. Particlecrushing appears to have little effect on (Coop,��crit

1990; Yasufuku et al., 1991). This is demonstrated inthe ring shear test results shown in Figs. 11.48b and11.48c; with increasing shear strains, the critical statestrength is reached well before particle crushingceases.

Peak Friction Angle

The peak friction angle can be considered as the sumof interparticle friction, rearrangement, crushing, andthe dilation contribution. For plane strain conditions,Bolton (1986) proposed the following empirical equa-tion that relates the mobilized friction angle �� at agiven stress state to the critical state friction angle

and dilation angle � :��crit

�� 0.8� (11.35)crit

where dilation angle � is the ratio of volumetric strainincrement d�v to the axial strain d�a at the stress stateof interest. This is similar to Taylor’s equation (Eq.(11.34)). However, � in Eq. (11.34) changes withshear, whereas is a constant material property.��crit

The relative density Dr is a convenient index to char-acterize the interlocking characteristics packing struc-ture. The effects of relative density, grain size, andgradation on the peak friction angle of cohesionlesssoils are illustrated by Fig. 11.51. Similar informationin terms of void ratio, unit weight, and Unified SoilClassification is given in Fig. 11.52. The peak valuesof friction angle for quartz sands range from about 30�to more than 50�, depending on gradation, relative den-sity, and confining pressure.

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Figure 11.51 Dependence of the friction angle of cohesion-less soils on relative density and gradation (after Schmert-mann, 1978).

Figure 11.52 Dependence of friction angle of cohesionlesssoils on unit weight and relative density (after NAVFAC,1982).

0.1 0.2 0.5 1 2 5 10 20 50 100Effective Mean Pressure at Peak Failure (kPa)

20

25

30

35

40

45

50 Uniformly Graded Cambria SandInitial Relative Density = 89.5%

Triaxial Extension

Contraction at Peak FailureDilation at Peak Failure


Sec

ant F

rictio

n A

ngle

at P

eak

Fai

lure

(deg

ree)

Figure 11.53 Effect of confining pressure on peak frictionangle (after Yamamuro and Lade, 1996).

Although the values of interparticle friction angleand the critical state friction angle are essen-�� crit

tially constant for a given mineral, the magnitudes ofthe dilation angle � in Eq. (11.35) vary with effectiveconfining pressure; that is, Figs. 11.51 and 11.52 applyfor a particular value of confining pressure. In general,the contribution of dilation increases with increasingdensity and decreases with increasing confining pres-sure. The effect of confining pressure on peak frictionangle is shown in Fig. 11.53 (Yamamuro and Lade,1996). Up to confining pressures of 5 to 10 MPa, thepeak friction angle decreases with increasing confiningpressure due to suppressed dilation and particle crush-ing. At pressures greater than about 10 MPa, the fric-tion angle remains approximately constant, but thevalues in triaxial extension are smaller than those intriaxial compression.

To take effective confining pressure into account,Bolton (1986) proposed the normalized dilatancy indexIR, defined as

�cI � D (Q � ln p�) � R � D ln � R (11.36)� �R r r p�

where Dr is the relative density, and p� is the meaneffective confining pressure. The empirical parameterQ is related to the crushing strength of the soil parti-cles; that is, Q � ln �c, where �c is the crushingstrength (same dimensions as p�). The Q values (usingkPa) are 10 for quartz and feldspar, 8 for limestones,7 for anthracite, and 5.5 for chalk. Bolton (1986) foundthat R � 1 fits the available data well. The critical stateis achieved when IR � 0, and this is given as Eq.(11.32). Index IR increases as the soil density increases.The parameter characterizes the state of the soil in re-lation to the critical state, similarly to the ones illus-trated in Fig.11.49.

Using IR (between 0 and 4), Bolton (1986) deducedthe following correlations for the peak friction angleand critical state friction angle (in degrees) from theplots shown in Fig. 11.54.

for triaxial compression conditions�� 3Im crit R

(11.37)

for plane strain conditions�� 5Im crit R

(11.38)

The dilatancy contribution to sand strength, repre-sented by the difference between the peak triaxial com-pression friction angle and the critical state friction

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Eq.(11.37)

Eq.(11.38)

0.0 0.2 0.4 0.6 0.8 1.00

4

8

12

16

20Plane Strain TestsTriaxial Compression Tests

Data for p� ≈ 300 kPa

Relative Density Dr

φ�m

ax–

φ�cr

it (d

egre

es)

Figure 11.54 Difference between peak friction angle andcritical state friction angle for triaxial compression and plainstrain compression data on sands (after Bolton, 1986).

Figure 11.55 Dilatancy component as a function of meaneffective stress at critical state and relative density (modifiedfrom Bolton, 1986).

angle , as determined by Bolton (1986), is shown��crit

in Fig. 11.55. The values shown are appropriate forquartz sands (Q � 10).

Other forms to characterize the peak friction anglein relation to the initial state of a sand are available.For example, Been and Jefferies (1986) relate the peakfriction angle to the state parameter � defined in Fig.11.49, as shown in Fig. 11.56.

As shown in Fig. 11.54 and by Eqs. (11.37) and(11.38), the critical state and peak friction angles varydepending on test conditions. The difference is relatedto the magnitude of the intermediate principal stress inrelation to the major and minor principal stresses. Fur-ther details are given in Section 11.12.

Undrained Strengths

In most cases, the deformation of sands occurs underdrained conditions. However, the undrained behaviorof sands is important when flow slides or earthquakesare of concern. These events are very rapid, and rapiddeformation of loose to medium dense cohesionlesssoils can generate excess pore water pressures resultingin loss of strength or liquefaction. The stress-strain re-lationship in undrained triaxial tests of Toyoura sandat different densities are shown in Fig. 11.57a, and thecorresponding effective stress paths are shown in Fig.11.57b (Yoshimine et al., 1998). A sudden flow failurecan occur in loose sand deposits by the drop in strengthwith increase in shear strain. Typical undrained re-sponses of sand specimens at different densities areillustrated in Fig. 11.58a.

Loose sand exhibits peak strength and then softens.The peak state on the p�–q plane is termed the collapsesurface (Sladen et al., 1985),15 and the slope increaseswith increase in initial density and decrease in confin-ing pressure, as illustrated in Fig. 11.58b. In triaxialcompression, the slope for many sandy soils rangesfrom 0.62 to 0.90 with an upper bound of 1.0 (Olsonand Stark, 2003). Once the soil softens, large sheardeformation is generated by moderate shear stresses.The softened soil eventually leads to the steady state,in which there is no further contraction tendency. Thepore pressures and stresses remain constant as the soilcontinues to shear in a steady state of deformation(Castro, 1975; Poulos, 1981). The steady state occurswhen the soil continuously deforms at constant vol-ume, constant stress, and constant velocity.16 It devel-ops under stress-controlled conditions because of theflowing nature of softened soil. When the soil is veryloose, the effective stress becomes zero, indicating astatic liquefaction condition, which is the transforma-tion of a granular material from a solid to a liquefiedstate (Youd et al., 2001).

15 Similar concepts are proposed by others. For example, the criticalstress ratio (Vaid and Chern, 1985), the instability line (Lade, 1992),and the yield strength ratio (Olson and Stark, 2003).16 The basic concept of the steady state is essentially the same as thecritical state defined for clay by Schofield and Wroth (1969).

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Figure 11.56 Peak friction angle versus state parameter � (after Been and Jefferies, 1986).

Even dense sands exhibit positive excess pore pres-sures at the beginning of deformation up to smallstrain. However, after a certain stress ratio is reached,the undrained stress path reverses its direction indicat-ing contractive to dilative behavior as shown in Fig.11.58c, and the stress reversal is called the state ofphase transformation (Ishihara et al., 1975). Thestress–strain response thereafter is strain hardening anddoes not exhibit any peak. The soil eventually reachesthe ultimate steady state or the critical state as long asthe pore water does not cavitate.

Medium dense specimens initially soften after thestress state passes the collapse line as illustrated in Fig.11.58c. The stress state then reaches a point of mini-mum strength, which is called the quasi-steady state(Alarcon-Guzman et al., 1988) or flow with limited liq-uefaction (Ishihara, 1993). At this stage, the soil is inthe state of phase transformation, and the mobilizedstrength then increases gradually with further shearstrain due to increase in effective stress by negativepore water pressure development. As shearing contin-ues, the soil shows a strain-hardening behavior, climb-ing along the critical state line, and the stress statefinally reaches the critical or ultimate steady state atvery large strains. Reported data indicate that the slopeof the critical state on the p�–q plane is approximatelythe same as that of the phase transformation line (Beenet al., 1991; Ishihara, 1993; Zhang and Garga, 1997;Vaid and Sivathayalan, 2000); at least, these lines aredifficult to distinguish from each other.

For loose sand, the steady state is the minimum un-drained shear strength associated with a rapid collaps-ing of soil structure. As discussed in Section 11.7, ithas been suggested that the stress state of the steadystate is a function of void ratio, so a unique criticalstate line exists on the e–log p� plane as shown in Fig.11.4b (Castro, 1975; Poulos et al., 1985, and others).The shape depends on grain angularity and fines con-tent (Zlatovic and Ishihara, 1995). At a given initialvoid ratio, the steady state strength can be determinedfrom the critical state line. For a relatively small con-fining pressure, a small change in void ratio can givedramatic difference in undrained shear strength be-cause the critical state line on e–log p� plane is veryflat at this stress level.

For medium dense sand, the quasi-steady state canbe considered as the minimum undrained shearstrength. As the soil continues to deform, the shearingresistance increases. Although the stress ratios atquasi-steady state, and critical state are similar on thep�–q plane, the quasi-steady state on e–log p� planelies below the critical state line as shown in Fig. 11.59.For a given initial void ratio, therefore, the stress stateof quasi-steady state is smaller than that of the criticalstate.

The location of the quasi-steady state line on e–logp� plane is influenced by shear mode and sample prep-aration method (i.e., soil fabric) (Konrad, 1990; Ishi-hara, 1993; Yoshimine and Ishihara, 1998). Figure11.60 shows the undrained shear behavior of Toyoura

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Figure 11.57 Undrained stress–strain response of Toyoura sand specimens prepared at dif-ferent densities by dry pluviation (after Yoshimine et al., 1998).

sand in triaxial compression, triaxial extension, andsimple shear (Yoshimine et al., 1999). The specimenswere prepared to similar void ratios, and an initial con-fining pressure of 100 kPa was applied. The minimumundrained shear strength and the quasi-steady statevary significantly depending on the mode of shearing,which in turn leads to different quasi-steady state lineson the e–p� plane as shown in Fig. 11.61. Hence, largevariation of minimum undrained shear strengths is of-ten observed depending on shear mode, which is pri-

marily due to the anisotropic soil fabric. Further detailsare given in Section 11.12.

The slope of the collapsing surface and the mini-mum undrained strength are related to both initialdensity and confining pressure. Typical values of thecollapse surface stress ratio obtained from triaxialcompression tests are plotted against state parameter �in Fig. 11.62 (Olson and Stark, 2003). Although thedata are scattered, a general trend for a given sand isthat the slope decreases with decreasing state param-

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normal stress

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ff ff c tan

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