63027 11a

369
CHAPTER 11
Copyrighted Material
Strength and Deformation
Behavior
11.1 INTRODUCTION
All aspects of soil stability—bearing capacity, slope
stability, the supporting capacity of deep foundations,
and penetration resistance, to name a few—depend on
soil strength. The stress–deformation and stress–
deformation–time behavior of soils are important in
any problem where ground movements are of interest.
Most relationships for the characterization of the
stress–deformation and strength properties of soils are
empirical and based on phenomenological descriptions
of soil behavior. The Mohr–Coulomb equation is by
far 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 the
failure plane, and is a friction angle. Equation (11.1)
applies for ff defined as a total stress, and c and are
referred to as total stress parameters. Equation (11.2)
applies for ff defined as an effective stress, and c and
are effective stress parameters. As the shear resis-tance
of soil originates mainly from actions at inter-particle
contacts, the second equation is the more
fundamental.
In reality, the shearing resistance of a soil depends
on many factors, and a complete equation might be of
the form
Shearing resistance F(e, c, , , C, H, T, , ˙, S)
(11.3)
in which e is the void ratio, C is the composition, H
is 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, and
the 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 confining
pressures, 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 residual
strength. The shear resistance values applicable in
practice depend on factors such as whether or not the
problem is one of loading or unloading, whether or not
short-term or long-term stability is of interest, and
stress orientations.
Emphasis in this chapter is on the fundamental fac-tors
controlling the strength and stress–deformation
behavior of soils. Following a review of the general
characteristics of strength and deformation, some re-
Copyright © 2005 John Wiley Sons Retrieved from: www.knovel.com

370 11 STRENGTH AND DEFORMATION BEHAVIOR
lationships among fabric, structure, and strength are
examined. 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 from
discrete particle simulations. Typical values of strength
parameters are listed. The concept of yielding is intro-duced,
for many clays at residual state. When
and the deformation behavior in both the pre-yield
(including small strain stiffness) and post-yield
regions is summarized. Time-dependent deformations
and aging effects are discussed separately in Chapter
12. The details of strength determination by means of
laboratory and in situ tests and the detailed constitutive
modeling of soil deformation and strength for use in
numerical analyses are outside the scope of this book.
11.2 GENERAL CHARACTERISTICS OF
STRENGTH AND DEFORMATION
Strength
1. In the absence of chemical cementation between
grains, the strength (stress state at failure or the
ultimate stress state) of sand and clay is ap-proximated
by a linear relationship with stress:
tan (11.4) ff ff
c
(a)
Strain
Peak
b
Critical State
Residual
Shear
Stress τ
φcritical state
φresidual
Normal effective stress σ
φpeak
Secant Peak
Strength Envelope
Tangent Peak
Strength Envelope
Critical state
Strength Envelope
Residual Strength Envelope
(b)
Peak Strength
At Large Strains
a
b
c
d
a d
Dense or
Overconsolidated
d
a
Loose or Normally
Consolidated
Shear
Stress τ
or Stress
Ratio τ/σ
b, c
Figure 11.1 Peak, critical, and residual strength and associated friction angle: (a) a typical
stress–strain curve and (b) stress states.
or
( ) ( )sin (11.5) 1ff 3ff 1ff 3ff
where the primes designate effective stresses
1ff and 3ff are the major and minor principal
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 soil
particles associated with changes in the soil fab-ric.
The magnitude of these contributions de-pends
on the effective stress and the volume
change tendencies of the soil. For such materials
the stress–strain curve from a shearing test is
typically of the form shown in Fig. 11.1a. The
maximum 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 plane
after large shear strain or shear displacement,
and the strength decreases even further to the
residual strength (point d). The corresponding
three failure envelopes can be defined as shown
in Fig. 11.1b, with peak, critical, and residual
friction angles (or states) as indicated.
3. Peak failure envelopes are usually curved in the
manner shown in Fig. 4.16 and schematically in
Fig. 11.1b. This behavior is caused by dilatancy
suppression and grain crushing at higher
stresses. Curved failure envelopes are also ob-served

GENERAL CHARACTERISTICS OF STRENGTH AND DEFORMATION 371
expressed in terms of the shear strength nor-malized
by the effective normal stress as a func-tion
of effective normal stress, curves of the
type 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 confining
pressures, test type, and sample preparation
methods. For dense sand, the secant peak fric-tion
angle (point b in Fig. 11.1b) consists in part
Figure 11.2 Variation of residual strength with stress level (after Bishop et al., 1971): (a)
Brown London clay and (b) Weald clay.

5. The peak strength of saturated clay is influenced
6. During critical state deformation a soil is com-pletely
Void Ratio e
Water Content w
of internal rolling and sliding friction between
grains and in part of interlocking of particles
(Taylor, 1948). The interlocking necessitates
either volume expansion (dilatancy) or grain
fracture and/or crushing if there is to be
deformation. For loose sand, the peak friction
angle (point b in Fig. 11.1b) normally coincides
with the critical-state friction angle (point c),
and there is no peak in the stress–strain curve.
most by overconsolidation ratio, drainage con-ditions,
structure, disturbance (which causes a change in
effective stress and a loss of cementation), and
creep or deformation rate effects. Overconsoli-dated
a given effective stress than normally consoli-dated
stress histories and the different water contents
at peak. For comparisons at the same water con-tent
A and A, the Hvorslev strength parameters ce
and e are obtained (Hvorslev, 1937, 1960).
Further details are given in Section 11.9.
the critical state friction angle values are inde-pendent
for a given set of testing conditions the shearing
eff
ce
0
effective confining pressures, original
clays usually have higher peak strength at
clays, as shown in Fig. 11.3. The differ-ences
in strength result from both the different
but different effective stress, as for points
destructured. As illustrated in Fig. 11.1b,
of stress history and original structure;
Normally Consolidated
Virgin Compression
A A
Rebound
σe σff
Peak Strength Envelope
Normal Effective Stress σ
τ
φe
φcrit
Overconsolidated
A A
σff
Hvorslev Envelope
Overconsolidated
Shear Stress τ
Figure 11.3 Effect of overconsolidation on effective stress
strength envelope.
resistance depends only on composition and ef-fective
stress. The basic concept of the critical
state is that under sustained uniform shearing at
failure, there exists a unique combination of
void ratio e, mean pressure p, and deviator
stress q.1 The critical states of reconstituted
Weald clay and Toyoura sand are shown in Fig.
11.4. The critical state line on the p–q plane is
linear,2 whereas that on an e-ln p (or e-log p)
plane tends to be linear for clays and nonlinear
for sands.
7. At failure, dense sands and heavily overconsol-idated
clays have a greater volume after drained
shear or a higher effective stress after undrained
shear than at the start of deformation. This is
due to its dilative tendency upon shearing. At
failure, loose sands and normally consolidated
to moderately overconsolidated clays (OCR up
to about 4) have a smaller volume after drained
shear or a lower effective stress after undrained
shear than they had initially. This is due to its
contractive tendency upon shearing.
8. Under further deformation, platy clay particles
begin to align along the failure plane and the
shear resistance may further decrease from the
critical state condition. The angle of shear re-sistance
at this condition is called the residual
friction angle, as illustrated in Fig. 11.1b. The
postpeak shearing displacement required to
cause a reduction in friction angle from the crit-ical
state value to the residual value varies with
the 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 of
only 1 or 2 mm is sufficient to give residual
strength.4 For soil against soil, a slip along the
1 In three-dimensional stress space , xy, yz, zx ( x, y, z ) or
the equivalent principal stresses ( 1, 2, 3), the mean effective
stress p, and the deviator stress q is defined as
p ( )/3 ( )/3 x y z 1 2 3
q (1 / 2)
( )2 ( )2 ( )2 6 2 6 2 6 2 x y y z z x xy yz zx
(1/ 2) ( )2 ( )2 ( )2 1 2 2 3 3 1
For triaxial compression condition ( 1 2 3), p ( 1
2 2) /3, q 1 2
2The critical state failure slope on p–q plane is related to friction
angle , as described in Section 11.10.
3A rock that has undergone differential movements at high temper-ature
and pressure in which the mineral grains are crushed against
one another. The rock shows a series of lamination planes.
4D. U. Deere, personal communication (1974).

Critical State Line Critical State Line
0 100 200 300 400 500 600
Mean Pressure p(kPa)
(a-1) p versus q
4
3
2
1
0.02 0.05 0.1 0.5 1 5
Critical State Line
Isotropic Normal
Compression Line
(a) (b)
500
400
300
200
100
0
0.7
0.6
0.5
0.4
0.3
100 200 300 400 500
Mean Pressure p (kPa)
0 1 2 3 4
0
Mean Pressure p(MPa)
(b-1) p versus q
Initial State
Mean Pressure p(MPa)
0.95
0.90
0.85
0.80
0.75
Overconsolidated
Overconsolidated
Critical State Line
Deviator Stress q (MPa)
Void ratio e (a-2) e versus lnp
(b-2) e versus logp
Void ratio e Deviator Stress q (kPa)
Figure 11.4 Critical states of clay and sand: (a) Critical state of Weald clay obtained by
drained 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) Critical
state of Toyoura sand obtained by undrained triaxial compression tests of loose and dense
specimens 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 may
be required, as shown by Fig. 11.5. However,
significant softening can be caused by strain
localization and development of shear bands,
especially for dense samples under low confine-ment.
9. Strength anisotropy may result from both stress
and fabric anisotropy. In the absence of chemi-cal
cementation, the differences in the strength
of two samples of the same soil at the same void
ratio but with different fabrics are accountable
in terms of different effective stresses as dis-cussed
in Chapter 8.
10. Undrained strength in triaxial compression may
differ significantly from the strength in triaxial
extension. However, the influence of type of test
(triaxial compression versus extension) on the
effective stress parameters c and is relatively

Material
Figure 11.5 Development of residual strength with increasing shear displacement (after
Copyrighted Bishop et al., 1971).
Figure 11.6 Effect of temperature on undrained strength of
kaolinite in unconfined compression (after Sherif and Bur-rous,
1969).
small. Effective stress friction angles measured
in plane strain are typically about 10 percent
greater than those determined by triaxial com-pression.
11. A change in temperature causes either a change
in void ratio or a change in effective stress (or
a combination of both) in saturated clay, as dis-cussed
in Chapter 10. Thus, a change in tem-perature
can cause a strength increase or a
strength decrease, depending on the circum-stances,
as illustrated by Fig. 11.6. For the tests
on kaolinite shown in Fig. 11.6, all samples
were prepared by isotropic triaxial consolidation
at 75F. 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 in
strength accompanied the increases in temper-ature.
Stress–Strain Behavior
1. Stress–strain behavior ranges from very brittle
for some quick clays, cemented soils, heavily
overconsolidated clays, and dense sands to duc-tile
for insensitive and remolded clays and loose
sands, as illustrated by Fig. 11.7. An increase in

Figure 11.7 Types of stress–strain behavior.
(a) Typical Strain Ranges in the Field
Retaining Walls
Foundations
Tunnels
10-4 10-3 10-2 10-1 100 101
Strain %
Dynamic Methods
Local Gauges
Conventional Soil Testing
Linear Elastic
Nonlinear Elastic
Preyield Plastic
Full Plastic
(b) Typical Strain Ranges for Laboratory Tests
Stiffness G or E
Figure 11.9 Stiffness degradation curve: stiffness plotted
against logarithm of strains. Also shown are (a) the strain
levels observed during construction of typical geotechnical
structures (after Mair, 1993) and (b) the strain levels that can
be measured by various techniques (after Atkinson, 2000).
confining pressure causes an increase in the de-formation
modulus as well as an increase in
strength, as shown by Fig. 11.8.
2. Stress–strain relationships are usually nonlin-ear;
soil stiffness (often expressed in terms of
tangent or secant modulus) generally decreases
with increasing shear strain or stress level up to
peak failure stress. Figure 11.9 shows a typical
stiffness degradation curve, in terms of shear
modulus G and Young’s modulus E, along with
typical strain levels developed in geotechnical
construction (Mair, 1993) and as associated with
different laboratory testing techniques used to
measure the stiffness (Atkinson, 2000). For ex-ample,
Fig. 11.10 shows the stiffness degrada-tion
of sands and clay subjected to increase in
shear strain. As illustrated in Fig. 11.9, the stiff-ness
degradation curve can be separated into
Figure 11.8 Effect of confining pressure on the consoli-dated-
drained stress–strain behavior of soils.
four zones: (1) linear elastic zone, (2) nonlinear
elastic zone, (3) pre-yield plastic zone, and (4)
full plastic zone.
3. In the linear elastic zone, soil particles do not
slide relative to each other under a small stress
increment, and the stiffness is at its maximum.
The soil stiffness depends on contact interac-tions,
particle packing arrangement, and elastic
stiffness of the solids. Low strain stiffness val-ues
can be determined using elastic wave veloc-ity
measurements, resonant column testing, or
local strain transducer measurements. The mag-nitudes
of the small strain shear modulus (Gmax)
and Young’s modulus (Emax) depend on applied
confining pressure and the packing conditions
of soil particles. The following empirical equa-tions
are often employed to express these de-pendencies:
G A F (e)pnG (11.6) max G G
E A F (e)nE (11.7) i(max) E E i
where FG(e) and FE(e) are functions of void
ratio, p is the mean effective confining pres-sure,
is the effective stress in the i direction, i
and the other parameters are material constants.

TC PSC
Toyoura Sand
Ticino Sand
140
120
100
80
60
40
20
or residual excess pore pressures in undrained
conditions after unloading. Available experi-
10-4 10-3 10-2 10-1 100
Shear Strain (%)
(a)
120
100
80
60
40
Confining Pressures
σc = 400 kPa
σc = 200 kPa
σc = 100 kPa
10-5 10-4 10-3 10-2 10-1 100
Shear Strain (%)
20
σc = 30 kPa
Confining
Pressure
78.4 kPa
49 kPa
(b)
Secant Shear Modulus G (MPa)
Secant Shear Modulus G (MPa)
Figure 11.10 Stiffness degradation curve at different confining pressures: (a) Toyoura and
Ticino sands (TC: triaxial compression tests, PSC: plain strain compression tests) (after
Tatsuoka et al., 1997) and (b) reconstituted Kaolin clay (after Soga et al., 1996).
Figure 11.11 shows examples of the fitting of
the above equations to experimental data.
4. The stiffness begins to decrease from the linear
elastic value as the applied strains or stresses
increase, and the deformation moves into the
nonlinear elastic zone. However, a complete cy-cle
of loading, unloading, and reloading within
this zone shows full recovery of strains. The
strain at the onset of the nonlinear elastic zone
ranges from less than 5 104 percent for non-plastic
104
103
102
101
100
Undisturbed
Remolded
Remolded with CaCO3
nG = 0.13
nG = 0.65
nG = 0.63
100 101 102 103 104
Confining pressure, p (kPa)
soils at low confining pressure conditions
to greater than 5 102 percent at high confin-ing
pressure or in soils with high plasticity (San-tamarina
et al., 2001).
5. Irrecoverable strains develop in the pre-yield
plastic zone. The initiation of plastic strains can
be determined by examining the onset of per-manent
volumetric strain in drained conditions
At each vertical effective stress,
horizontal effective stress σh (kPa)
was varied between 98 kPa and
196 kPa
100 150 200 250 300
500
450
400
350
300
250
Vertical Effective Stress, σv (kPa)
(a) (b)
nE = 0.49
Shear Modulus,Gmax MPa
Vertical Young's Modulus
Evmax/FE(e) (MPa)
Figure 11.11 Small strain stiffness versus confining pressure: (a) Shear modulus Gmax of
cemented 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).

mental data suggest that the strain level that in-itiates
plastic strains ranges between 7 103
and 7 102 percent, with the lower limit for
uncemented normally consolidated sands and
the upper limit for high plasticity clays and ce-mented
6. A distinctive kink in the stress–strain relation-ship
defines yielding, beyond which full plastic
confining pressure p, but have different void 0
ratios due to the different stress history (Fig.
11.14a). Drained tests on normally consolidated
clays and lightly overconsolidated clays show
ductile behavior with volume contraction (Fig.
11.14b). Heavily overconsolidated clays exhibit
a stiff response initially until the stress state
reaches the yield envelope giving the peak
strength and volume dilation. The state of the
strains are generated. A locus of stress states
that initiate yielding defines the yield envelope.
Typical yield envelopes for sand and natural
clay are shown in Fig. 11.12. The yield envelope
expands, shrinks, and rotates as plastic strains
develop. It is usually considered that expansion
is related to plastic volumetric strains; the sur-face
expands when the soil compresses and
shrinks 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 state
moves in the pre-yield zone, the inner envelopes
move with the stress state. This multienvelope
concept allows modeling of complex deforma-tions
observed for different stress paths (Mroz,
1967; Pre´vost, 1977; Dafalias and Herrman,
1982; Atkinson et al., 1990; Jardine, 1992).
7. Plastic irrecoverable shear deformations of
saturated soils are accompanied by volume
Initial Condition
Yield State
sands.
Failure Line Stress Path
0.2 0.4 0.6 0.8 1.0
q = σa-σr
MPa
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
MPa
Failure Line
Yield Envelope
changes when drainage is allowed or changes in
pore water pressure and effective stress when
drainage is prevented. The general nature of this
behavior is shown in Figs. 11.13a and 11.13b
for drained and undrained conditions, respec-tively.
The volume and pore water pressure
changes depend on interactions between fabric
and stress state and the ease with which shear
deformations can develop without overall
changes in volume or transfer of normal stress
from the soil structure to the pore water.
8. The stress–strain relation of clays depends
largely on overconsolidation ratio, effective
confining 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 to
different overconsolidation ratios before shear-ing.
The specimens are consolidated at the same
0.2 0.4 0.6
MPa
q = σa-σr
MPa
0.6
0.4
0.2
0.0
p = (σa + 2σr)/3 p = (σa + 2σr)/3
-0.2
Yield State
Yield Envelope
Preyield Boundary
Pre-yield State
Initial State Surrounded by
Linear Elastic Boundary
Linear Elastic
Boundary
(a) (b)
Figure 11.12 Yield envelopes: (a) Aoi sand (Yasufuku et al., 1991) and (b) Bothkennar clay
(from Smith et al., 1992).

Same Initial Confining Pressure
Dense Soil
Loose Soil
Critical State
Deviator
Stress
Metastable Fabric
Axial or Deviator Strain
Dense Soil
+ΔV/V0
0
Loose Soil
Material
-ΔV/V0
Metastable Fabric
(a)
Copyrighted Same Initial Confining Pressure
Dense Soil
Cavitation
Loose Soil
Critical State
Deviator
Stress
Metastable Fabric
Loose soil
Dense Soil
Metastable Fabric
-Δu
0
+Δu
Axial or Deviator Strain
(b)
Critical State
Cavitation
Figure 11.13 Volume and pore pressure changes during shear: (a) drained conditions and
(b) undrained conditions.
Initial State
Failure at Critical State
(D: Drained, U: Undrained)
Virgin Compression Line
1 Normally consolidated
2 Lightly Overconsolidated
3 Heavily
Overconsolidated
U1
U2
U3
D
Critical
State Line
3 Heavily
Overconsolidated
2 Lightly
Overconsolidated
1 Normally Consolidated
Axial or Deviatoric Strain
3 Heavily Overconsolidated
Deviator
Stress
+ΔV/V0
-ΔV/V0
U3
U2
Axial or Deviatoric Strain
Deviator
Stress
-Δu
+Δu
Void
Ratio
log p
D Critical State
(a) (b) (c)
U1
p0
Figure 11.14 Stress–strain relationship of normally consolidated, lightly overconsolidated,
and heavily overconsolidated clays: (a) void ratio versus mean effective stress, (b) drained
tests, and (c) undrained tests.

FABRIC, STRUCTURE, AND STRENGTH 379
Figure 11.16 Effect of temperature on the stiffness of Osaka
clay in undrained triaxial compression (Murayama, 1969).
soil then progressively moves toward the critical
state exhibiting softening behavior. Undrained
shearing of normally consolidated and lightly
overconsolidated clays generates positive excess
pore pressures, whereas shear of heavily over-consolidated
(MPa)
0.3
0.2
0.0
Failure Line in
Triaxial Compression
σr/σa = 0.54
0.1 0.2 0.3 0.4
0.1
-0.1
-0.2
-0.3
(MPa)
Mean Pressure
p = (σa+ 2σr)/3
σr/σa = 1.84
Failure Line in
Triaxial Extension
Initial At Failure
Anisotropically Consolidated σr/ σa = 0.54
Isotropically Consolidated
Deviator Stress q = σa + σrσ
Anisotropically Consolidated σr/σa = 1.84
Figure 11.15 Undrained effective stress paths of anisotrop-ically
and isotropically consolidated specimens (after Ladd
and Varallyay, 1965).
clays generates negative excess
pore pressures (Fig. 11.14c).
9. The magnitudes of pore pressure that are de-veloped
in undrained loading depend on initial
consolidation stresses, overconsolidation ratio,
density, and soil fabric. Figure 11.15 shows the
undrained effective stress paths of anisotropi-cally
and isotropically consolidated specimens
(Ladd and Varallyay, 1965). The difference in
undrained shear strength is primarily due to dif-ferent
excess pore pressure development asso-ciated
with the change in soil fabric. At large
strains, the stress paths correspond to the same
friction 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 of
stress is shown in Fig. 11.16 for Osaka clay
tested in undrained triaxial compression at dif-ferent
temperatures. Increase in temperature
causes consolidation under drained conditions
and softening under undrained conditions.
11.3 FABRIC, STRUCTURE, AND STRENGTH
Fabric Changes During Shear of Cohesionless
Materials
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 changes
associated with the sliding and rolling of grains during
triaxial compression were determined using a uniform
sand composed of rounded to subrounded grains with
sizes in the range of 0.84 to 1.19 mm and a mean axial
length 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. A
delayed setting water–resin solution was used as the
pore fluid. Samples prepared by each method were
tested to successively higher strains. The resin was
then allowed to set, and thin sections were prepared.
The differences in initial fabrics gave the markedly dif-ferent
stress–strain and volumetric strain curves shown
in Fig. 11.17, where the plunging method refers to

Material
Figure 11.17 Stress–strain and volumetric strain relationships for sand at a void ratio of
0.64 but with different initial fabrics (after Oda, 1972a). (a) Sample saturated with water
and (b) sample saturated with water–resin solution.
Copyrighted tamping. There is similarity between these curves and
those for Monterey No. 0 sand shown in Fig. 8.23. A
statistical analysis of the changes in particle orientation
with increase in axial strain showed:
1. For samples prepared by tapping, the initial fab-ric
tended toward some preferred orientation of
long axes parallel to the horizontal plane, and the
intensity of orientation increased slightly during
deformation.
2. For samples prepared by tamping, there was very
weak 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 appear
until after peak stress had been reached; however, the
distribution of normals to the interparticle contact
planes E() (a measure of fabric anisotropy) did
change with strain, as may be seen in Fig. 11.18. This
figure shows different initial distributions for samples
prepared by the two methods and a concentration of
contact plane normals within 50 of the vertical as de-formation
progresses. Thus, the fabric tended toward
greater anisotropy in each case in terms of contact
plane orientations. There was little additional change
in E() after the peak stress had been reached, which
implies that particle rearrangement was proceeding
without significant change in the overall fabric.
As the stress state approaches failure, a direct shear-induced
fabric forms that is generally composed of
regions of homogeneous fabric separated by discon-tinuities.
No discontinuities develop before peak
strength is reached, although there is some particle ro-tation
in the direction of motion. Near-perfect preferred
orientation develops during yield after peak strength is
reached, but large deformations may be required to
reach this state.
Compaction Versus Overconsolidation of Sand
Specimens at the same void ratio and stress state be-fore
shearing, but having different fabrics, can exhibit
different stress–strain behavior. For example, consider
a case in which one specimen is overconsolidated,
whereas the other is compacted. The two specimens
are prepared in such a way that the initial void ratio is
the same for a given initial isotropic confining pres-sure.
Coop (1990) performed undrained triaxial com-pression
tests of carbonate sand specimens that were
either overconsolidated or compacted, as illustrated in
Fig. 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 overconsolidated
sample was initially stiffer than the compacted speci-men.
The difference can be attributed to (i) different
soil fabrics developed by different stress paths prior to
shearing and (ii) different degrees of particle crushing
prior to shearing (i.e., some breakage has occurred dur-

FABRIC, STRUCTURE, AND STRENGTH 381
Material
Copyrighted Figure 11.18 Distribution of interparticle contact normals as a function of axial strain for
sand samples prepared in two ways (after Oda, 1972a): (a) specimens prepared by tapping
and (b) specimens prepared by tamping.
ing the preconsolidation stage for the overconsolidated
specimen). Therefore, overconsolidation and compac-tion
produced materials with different mechanical
properties. However, at large deformations, both spec-imens
exhibited similar strengths because the initial
fabrics were destroyed.
Effect of Clay Structure on Deformations
The high sensitivity of quick clays illustrates the prin-ciple
that flocculated, open microfabrics are more rigid
but more unstable than deflocculated fabrics. Similar
behavior may be observed in compacted fine-grained
soils, 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 kaolinite
compacted using kneading and static methods are
shown in Fig. 11.20. The high shear strain associated
with kneading compaction wet of optimum breaks
down flocculated structures, and this accounts for the
much lower peak strength for the sample prepared by
kneading compaction.
The recoverable deformation of compacted kaolinite
with flocculent structure ranges between 60 and 90
percent, 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 the
braced-box type of fabric that remains after static com-paction
to withstand stress without permanent defor-mation
than is possible with the broken-down fabric
associated 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 Bothkennar
clay, 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 burrowing
mollusks and worms (bioturbation), gave the stiffest
response, whereas samples with distinct laminated fea-tures
showed the softest response.

2
1.5
1
Material
Copyrighted Normal Compression Line
Overconsolidated
Sample
0.1 1
Mean Pressure p
Void
Ratio
Compacted Sample
(MPa)
Overconsolidated
Compacted
0 0.2 0.4 0.6
1.0
0.8
0.6
0.4
0.2
0
p (MPa)
q (MPa)
Compacted
Overconsolidated
0 4 8 12 16 20
1.0
0.75
0.5
0.25
0
Axial strain εa(%)
(a)
(b)
(c)
q (MPa)
Figure 11.19 Undrained response of compacted specimen and overconsolidated specimen
of 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 elongated
particles align with their long axes in the direction of
slip. By then, the basal planes of the platy clay parti-cles
are enclosed between two highly oriented bands
of particles on opposite sides of the shear plane. The
dominant mechanism of deformation in the displace-ment
shear zone is basal plane slip, and the overall
thickness of the shear zone is on the order of 50 m.
Fabrics associated with shear planes and zones have
been 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 strength
associated with these fabrics is treated in more detail
in 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 pore
pressures developed during shear. The undrained
strength loss associated with remolding undisturbed
clay can also be accounted for in terms of differences
in effective stress, provided part of the undisturbed
strength does not result from cementation. Remolding
breaks 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, which
shows the results of incremental loading triaxial com-pression
tests on two samples of undisturbed and re-molded
San Francisco Bay mud. In these tests, the
undisturbed sample was first brought to equilibrium
under an isotropic consolidation pressure of 80 kPa.
After undrained loading to failure, the triaxial cell was
disassembled, and the sample was remolded in place.
The apparatus was reassembled, and pore pressure was
measured. 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 two
samples are shown in Figs. 11.23a and 11.23b, and
stress 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

FRICTION BETWEEN SOLID SURFACES 383
Figure 11.20 Stress–strain behavior of kaolinite compacted
by two methods.
Figure 11.21 Ratio of recoverable to total strain for samples
of kaolinite with different structure.
Stress-Strain Relationships Stress Paths
Facies
0.4 0.6 0.8 1.0
0.6
0.4
0.2
0.0
Facies
0 2 4
Axial Strain (%)
0.6
0.4
0.2
0.0
Mottled
Bedded
Laminated
Mottled
Bedded
Laminated
(σa + σr)/2σao
(σa – σr)/2 σao
(σa – σr)/2 σao
Figure 11.22 Effect of macrofabric on undrained response
of Bothkennar clay in Scotland (after Hight and Leroueil,
2003).
way. Thus, in the absence of chemical or mineralogical
changes, different strengths in two samples of the same
soil at the same void ratio can be accounted for in
terms 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 of
grains in contact, resistance to volume change (dila-tancy),
grain rearrangement, and grain crushing. The
true friction coefficient is shown in Fig. 11.24 and is
represented by
T
tan (11.8) N
where N is the normal load on the shear surface, T is
the shear force, and , the intergrain sliding friction
angle, is a compositional property that is determined
by the type of soil minerals.

Material
Copyrighted Figure 11.23 (a) and (b) Effect of remolding on undrained strength and pore water pressure
in San Francisco Bay mud. (c) Stress paths for triaxial compression tests on undisturbed and
remolded samples of San Francisco Bay mud.
Basic ‘‘Laws’’ of Friction
Two laws of friction are recognized, beginning with
Leonardo da Vinci in about 1500. They were restated
by Amontons in 1699 and are frequently referred to as
Amontons’ laws. They are:
1. The frictional force is directly proportional to the
normal force, as illustrated by Eq. (11.8) and Fig.
11.24.
2. The frictional resistance between two bodies is
independent of the size of the bodies. In Fig.
11.24, the value of T is the same for a given value
of N regardless of the size of the sliding block.
Although these principles of frictional resistance
have long been known, suitable explanations came
much later. It was at one time thought that interlocking
between irregular surfaces could account for the be-havior.
On this basis, would be given by the tangent
of the average inclination of surface irregularities on
the sliding plane. This cannot be the case, however,
because such an explanation would require that de-

Material
Figure 11.23
(Continued)
Copyrighted Figure 11.24 Coefficient of friction for surfaces in contact.
crease as surfaces become smoother and be zero for
perfectly smooth surfaces. In fact, the coefficient of
friction can be constant over a range of surface rough-ness.
Hardy (1936) suggested instead that static
friction originates from cohesive forces between
contacting surfaces. He observed that the actual area
of contact is very small because of surface irregulari-ties,
and thus the cohesive forces must be large.
The foundation for the present understanding of the
mobilization of friction between surfaces in contact
was laid by Terzaghi (1920). He hypothesized that the
normal load N acting between two bodies in contact
causes yielding at asperities, which are local ‘‘hills’’
on the surface, where the actual interbody solid contact
develops. The actual contact area Ac is given by
N
A (11.9) c y
where y is the yield strength of the material. The
shearing strength of the material in the yielded zone is
assumed to have a value m. The maximum shearing
force 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 Acm m (11.11)
N Acy y
This concept of frictional resistance was subse-quently
further developed by Bowden and Tabor (1950,

shown in Fig. 11.26. The conclusion to be drawn from
this figure is that adsorbed layers are present on the
surface of soil particles in the terrestrial environment,
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 the
adhesion theory of friction: roughness and surface ad-sorption.
Surface Roughness
The surfaces of most solids are rough on a molecular
scale, with successions of asperities and depressions
ranging from 10 nm to over 100 nm in height. The
slopes of the nanoscale asperities are rather flat, with
individual angles ranging from about 120 to 175 as
shown in Fig. 11.25. The average slope of asperities
on metal surfaces is an included angle of 150; on
rough 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 area
is only a small fraction of the total surface area.
Quartz surfaces polished to mirror smoothness may
consist of peaks and valleys with an average height of
about 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 smoother
than most soil particles composed of bulky minerals.
The actual surface texture of sand particles depends on
geologic history as well as mineralogy, as shown in
Fig. 2.12.
The cleavage faces of mica flakes are among the
smoothest naturally occurring mineral surfaces. Even
in 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 the
particle.
Thus, large areas of solid contact between grains are
not probable in soils. Solid-to-solid contact is through
asperities, and the corresponding interparticle contact
stresses 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 of
solids, the surface structure may differ from that in the
interior, and material may be adsorbed from adjacent
phases. Even ‘‘clean’’ surfaces, prepared by fracture of
a 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 time
for adsorption of a monolayer tm is given by
1
t (11.12) m SZ
where is the area occupied per molecule, S is the
fraction of molecules striking the surface that stick to
it, and Z is the number of molecules per second strik-ing
a square centimeter of surface. For a value of S
equal to 1, which is reasonable for a high-energy sur-face,
the relationship between tm and gas pressure is

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 shear
strength of the contact. Plastic and/or elastic defor-mations
determine the contact area at asperities.
Plastic Junctions If asperities yield and undergo
plastic deformation, then the contact area is propor-tional
to the normal load on the asperity as shown by
Eq. (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 as
shown in Fig. 11.27. If the contaminant film strength
is 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 provide
a 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 and
other platy minerals such as mica, even though the
surface structure is similar for all the silicate minerals.
The small particle size of clays means that the load
per particle, for a given effective stress, will be small
relative to that in silts and sands composed of the bulky
minerals. The surfaces of platy silt and sand size par-ticles
are smoother than those of bulky mineral parti-
Figure 11.27 Plastic junction between asperities with ad-sorbed
surface films.
5 Conditions may be different on the Moon, where ultrahigh vacuum
exists. This vacuum produces cleaner surfaces. In the absence of
suitable adsorbate, clean surfaces can reduce their surface energy by
cohering 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, are
more regular but not as high as those for the bulky
minerals.
Thus, it can be postulated that for a given number
of contacts per particle, the load per asperity decreases
with decreasing particle size and, for particles of the
same size, is less for platy minerals than for bulky
minerals. Because should increase as the normal load
per asperity increases, and it is reasonable to assume
that the adsorbed film strength is less than the strength
of the solid material (c m), it follows that the true
friction angle () is less for small and platy particles
than for large and bulky particles. In the event that two
platy 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 giving
a lower value of .
In reality, the behavior of plastic junctions is more
complex. Under combined compression and shear
stresses, deformation follows the von Mises–Henky
criterion, which, for two dimensions, is
2 3 2 2 (11.14) y
For asperities loaded initially to y, the appli-cation
of a shear stress requires that become less
than y. The only way that this can happen is for the
contact area to increase. Continued increase in leads
to continued increase in contact area. This phenome-non
is called junction growth and is responsible for
cold welding in some materials (Bowden and Tabor,
1964). If the shear strength of the junction equals that
of the bulk solid, then gross seizure occurs. For the
case where the ratio of junction strength to bulk ma-terial
strength is less than 0.9, the amount of junction
growth 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 in
terms of plastic yield. For two smooth spheres in con-tact,
application of the Hertz theory leads to
d (NR)1 / 3 (11.15)
where d is the diameter of a plane circular area of
contact; is a function of geometry, Poisson’s ratio,
and Young’s modulus6; and R is the sphere radius. The
contact area is
6For a sphere in contact with a plane surface 12(1 2) /E.

A (NR)2 / 3 (11.16) c 4
If the shear strength of the contact is i, then
T A (11.17) i c
and
T (R)2 / 3 N1 / 3 (11.18) N i 4
Material
Copyrighted According to these relationships, the friction coef-ficient
for two elastic asperities in contact should de-crease
with increasing load. Nonetheless, the adhesion
theory would still apply to the strength of the junction,
with the frictional force proportional to the area of real
contact.
If it is assumed that the number of contacting as-perities
in a soil mass is independent of particle size
and effective stress, then the influences of particle size
and effective stress on the frictional resistance of a soil
with asperities deforming elastically may be analyzed.
For uniform spheres arranged in a regular packing, the
gross area covered by one sphere along a potential
plane of sliding is 4R2. The normal load per contacting
asperity, assuming one asperity per contact, is
N 4R2 (11.19)
Using Eq. (11.16), the area per contact becomes
A (4R3)2 / 3 (11.20) c 4
and the total contact area per unit gross area is
1 (A ) R2(4)2 / 3 (4)2 / 3 c T 4R2 4 16 (11.21)
The total shearing resistance of is equal to the
contact area times i, so
2 / 3 i 1 / 3 (4) K() (11.22) 16 i
where K (4)2 / 3 /16. On this basis, the coefficient
of friction should decrease with increasing , but it
should 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 contact
with 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 with
increasing normal load up to some limiting value of
normal stress is evident for mica and the clay minerals
in Fig. 11.28 and has been found also for several clays
and clay shales (Bishop et al., 1971), for diamond
(Bowden and Tabor, 1964), and for solid lubricants
such 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 the
normal stress independence of the frictional resistance
of quartz, feldspar, and calcite:
1. As the load per particle increases, the number of
asperities in contact increases proportionally, and
the deformation of each asperity remains essen-tially
constant. In this case, the assumption of one
asperity 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 of
contact is approximately proportional to the ap-plied
load and hence the coefficient of friction is
constant 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 film
contact. Thus, the average strength per contact
increases 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 both
elastic and plastic deformation. A normal pressure of
11 GPa (1,500,000 psi) is required to produce plastic
deformation, and brittle failure usually occurs before
plastic deformation. Plastic deformations are evidently
restricted 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 of
surface texture on a microscale and characteristics of
the adsorbed films.
With the exception of some data for quartz, there
appears to be little information concerning possible
variations 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-

FRICTIONAL BEHAVIOR OF MINERALS 389
Figure 11.28 Variation in friction angle with normal stress for different minerals (after
Kenney, 1967).
Material
Copyrighted creased from 31 for coarse silt to 22 for coarse sand.
This is an apparent contradiction to the independence
of 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 on
a microscale could have been size dependent. Further-more,
there could have been different amounts of par-ticle
rearrangement and rolling in the tests on the
different size fractions.
Sliding Friction
The frictional resistance, once sliding has been initi-ated,
may be equal to or less than the resistance that
had to be overcome to initiate movement; that is, the
coefficient of sliding friction can be less than the co-efficient
of static friction. A higher value of static
friction 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 minerals
involving single contacts (Procter and Barton, 1974).
Stick–slip is not observed during shear of assemblages
of large numbers of particles because the slip of indi-vidual
contacts is masked by the behavior of the mass
as a whole. However, it may be an important mecha-nism
of energy dissipation for cyclic loading at very
small strains when particles are not moving relative to
each 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 sliding
relative to each other, and test results for particle as-semblages
are influenced by particle rearrangements,
volume changes, surface preparation factors, and the
like. Some values are available, however, and they are
presented and discussed in this section.
Nonclay Minerals
Values of the true friction angle for several minerals
are listed in Table 11.1, along with the type of test and
conditions used for their determination. A pronounced
antilubricating effect of water is evident for polished
surfaces of the bulky minerals quartz, feldspar, and cal-cite.
This apparently results from a disruptive effect of
water on adsorbed films that may have acted as a lu-bricant
for dry surfaces. Evidence for this is shown in
Fig. 11.29, where it may be seen that the presence of
water had no effect on the frictional resistance of
quartz surfaces that had been chemically cleaned prior
to the measurement of the friction coefficient. The
samples tested by Horn and Deere (1962) in Table 11.1
had not been chemically cleaned.
An apparent antilubrication effect by water might
also arise from attack of the silica surface (quartz and
feldspar) or carbonate surface (calcite) and the for-mation
of silica and carbonate cement at interparticle
contacts. Many sand deposits exhibit ‘‘aging’’ effects
wherein their strength and stiffness increase noticeably
within 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 percent
have been measured in some cases. The relative im-portance
of chemical factors, such as precipitation at

Table 11.1 Values of Friction Angle () Between Mineral Surfaces
Mineral Type of Test Conditions (deg) Comments Reference
Quartz Block over particle
set in mortar
Dry 6 Dried over CaCl2 before
testing
Tschebotarioff and
Welch (1948)
Moist 24.5
Water saturated 24.5
Quartz Three fixed particles
over block
Material
Copyrighted Water saturated 21.7 Normal load per particle
increasing from 1 to
100 g
Hafiz (1950)
Quartz Block on block Dry 7.4 Polished surfaces Horn and Deere (1962)
Quartz Particles on
polished block
Water saturated 22–31 decreasing with in-creasing
particle size
Rowe (1962)
Quartz Block on block Variable 0–45 Depends on roughness
and cleanliness
Bromwell (1966)
Quartz Particle–particle Saturated 26 Single-point contact Procter and Barton
(1974)
Particle–plane Saturated 22.2
Particle–plane Dry 17.4
Feldspar Block on block Dry 6.8 Polished surfaces Horn and Deere (1962)
Feldspar Free particles on flat
surface
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)
Muscovite Along cleavage
faces
Dry 23.3 Oven dry Horn and Deere (1962)
Dry 16.7 Air equilibrated
Saturated 13.0
Phlogopite Along cleavage
faces
Saturated 8.5
Biotite Along cleavage
faces
Saturated 7.4
Chlorite Along cleavage
faces
Saturated 12.4
interparticle contacts, changes in surface characteris-tics,
and mechanical factors, such as time-dependent
stress redistribution and particle reorientations, in caus-ing
the observed behavior is not known. Further details
of aging effects are given in Chapter 12.
As surface roughness increases, the apparent anti-lubricating
effect of water decreases. This is shown
in Fig. 11.29 for quartz surfaces that had not been
cleaned. Chemically cleaned quartz surfaces, which
give the same value of friction when both dry and wet,

FRICTIONAL BEHAVIOR OF MINERALS 391
Material
Figure 11.29 Friction of quartz (data from Bromwell, 1966 and Dickey, 1966).
Copyrighted show a loss in frictional resistance with increasing sur-face
roughness. Evidently, increased roughness makes
it 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 that
the cleaning process was not effective on the rough
surfaces.
For soils in nature, the surfaces of bulky mineral
particles are most probably rough relative to the scale
in Fig. 11.29, and they will not be chemically clean.
Thus, values of 0.5 and 26 are reasonable
for 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, and
surface ions are not fully hydrated. Thus, the adsorbed
layer is not easily disrupted. Observations have shown
that the surfaces of the sheet minerals are scratched
when tested in air (Horn and Deere, 1962). When the
surfaces of the layer silicates are wetted, the mobility
of the surface films is increased because of their in-creased
thickness and because of greater surface ion
hydration and dissociation. Thus, the values of
listed in Table 11.1 for the sheet minerals under satu-rated
conditions (7–13) are probably appropriate for
sheet mineral particles in soils.
Clay Minerals
Few, if any, directly measured values of for the clay
minerals are available. However, because their surface
structures are similar to those of the layer silicates dis-cussed
previously, approximately the same values
would be anticipated, and the ranges of residual fric-tion
angles measured for highly plastic clays and clay
minerals support this. In very active colloidal pure
clays, such as montmorillonite, even lower friction an-gles
have been measured. Residual values as low as 4
for sodium montmorillonite are indicated by the data
in Fig. 11.28.
The effective stress failure envelopes for calcium
and sodium montmorillonite are different, as shown by
Fig. 11.30, and the friction angles are stress dependent.
For each material the effective stress failure envelope
was the same in drained and undrained triaxial com-pression
and unaffected by electrolyte concentration
over the range investigated, which was 0.001 N to 0.1
N. 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 that
described in Chapter 10. Interlayer expansion in cal-cium
montmorillonite is restricted to a c-axis spacing
of 1.9 nm, leading to formation of domains or layer
aggregates of several unit layers. The interlayer spac-

Figure 11.30 Effective stress failure diagrams for calcium and sodium montmorillonite (af-ter
Mesri and Olson, 1970).
Material
Copyrighted Figure 11.31 Shear and consolidation behavior of sodium
montmorillonite (after Mesri and Olson, 1970).
ing of sodium montmorillonite is sensitive to double-layer
repulsions, which, in turn, depend on the
electrolyte concentration. The influence of the electro-lyte
concentration on the behavior of sodium mont-morillonite
is to change the water content, but not the
strength, at any effective consolidation pressure. This
suggests that the strength generating mechanism is in-dependent
of the system chemistry.
The platelets of sodium montmorillonite act as thin
films held apart by high repulsive forces that carry the
effective stress. For this case, if it is assumed that there
is essentially no intergranular contact, then Eq. (7.29)
becomes
A u R 0 (11.23) i 0
Since u0 is the conventionally defined effective
stress , 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 strength
because the shearing strength of water and solutions is
essentially independent of hydrostatic pressure. The
small friction angle that is observed for sodium mont-morillonite
at low effective stresses can be ascribed
mainly to the few interparticle contacts that resist par-ticle
rearrangement. Resistance from this source evi-dently
approaches a constant value at the higher
effective stresses, as evidenced by the nearly horizontal
failure envelope at values of average effective stress
greater than about 50 psi (350 kPa), as shown in Fig.
11.30. The viscous resistance of the pore fluid may
contribute 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 given
by Santamarina et al. (2001) using the concept of
‘‘electrical’’ surface roughness as shown in Fig. 11.32.
Consider two clay surfaces with interparticle fluid as
shown in Fig. 11.32b. The clay surfaces have a number
of discrete charges, so a series of potential energy
wells exists along the clay surfaces. Two cases can be
considered:
1. When the particle separation is less than several
nanometers, there are multiple wells of minimum
energy between nearby surfaces and a force is
required to overcome the energy barrier between
the wells when the particles move relative to each
other. Shearing involves interaction of the mole-cules
of the interparticle fluid. Due to the multi-ple
energy wells, the interparticle fluid molecules
go through successive solidlike pinned states.
This stick–slip motion contributes to frictional
resistance and energy dissipation.
2. When the particle separation is more than several
nanometers, the two clay surfaces interact only
by the hydrodynamic viscous effects of the in-terparticle
fluid, and the frictional force may be
estimated using fluid dynamics.

PHYSICAL INTERACTIONS AMONG PARTICLES 393
Figure 11.32 Concept of ‘‘electrical’’ surface roughness ac-cording
to Santamarina et al., (2001): (a) electrical roughness
and (b) conceptual picture of friction in fine-grained particles.
The aggregation of clay plates in calcium montmo-rillonite
produces particle groups that behave more like
equidimensional particles than platy particles. There is
more physical interference and more intergrain contact
than in sodium montmorillonite since the water content
range for the strength data shown in Fig. 11.30 was
only about 50 to 97 percent, whereas it was about 125
to 450 percent for the sodium montmorillonite. At a
consolidation pressure of about 500 kPa, the slope of
the failure envelope for calcium montmorillonite was
about 10, which is in the middle of the range for non-clay
sheet minerals (Table 11.1).
11.6 PHYSICAL INTERACTIONS AMONG
PARTICLES
Continuum mechanics assumes that applied forces are
transmitted uniformly through a homogenized granular
system. In reality, however, the interparticle force dis-tributions
are strongly inhomogeneous, as discussed in
Chapter 7, and the applied load is transferred through
a network of interparticle force chains. The generic
disorder of particles, (i.e., local spatial fluctuations of
coordination number, and positions of neighboring par-ticles)
produce packing constraints and disorder. This
leads to inhomogeneous but structured force distribu-tions
within the granular system. Deformation is as-sociated
with buckling of these force chains, and
energy is dissipated by sliding at the clusters of par-ticles
between the force chains.
Discrete particle numerical simulations, such as the
discrete (distinct) element method (Cundall and Strack,
1979) and the contact dynamics method (Moreau,
1994), offer physical insights into particle interactions
and load transfers that are difficult to deduce from
physical experiments. Typical inputs for the simula-tions
are particle packing conditions and interparticle
contact characteristics such as the interparticle friction
angle . Complete details of these numerical methods
are 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 for
developing an improved understanding of how stresses
are carried through discrete particle systems such as
soils 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 and
11.33b for a biaxial loaded condition (Thornton and
Barnes, 1986). The thickness of the lines in the figure
is proportional to the magnitude of the contact force.
The external loads are transmitted through a network
of interparticle contact forces represented by thicker
lines. This is called the strong force network and is the
key microscopic feature of load transfer through the
granular system. The scale of statistical homogeneity
in a two-dimensional particle assembly is found to be
a few tens of particle diameters (Radjai et al., 1996).
Forces averaged over this distance could therefore be
expected to give a stress that is representative of the
macroscopic stress state. The particles not forming a
part of the strong force network are floating like a fluid
with small loads at the interparticle contacts. This can
be called the weak cluster, which has a width of 3 to
10 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 and
tangential contact forces T for a given biaxial loading
condition. The horizontal axis is the forces normalized
by 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

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 in
the vertical direction (after Thornton and Barnes, 1986).
as six times the mean normal contact force, but
approximately 60 percent of contacts carry normal
contact forces below the mean (i.e., weak cluster
particles). When normal contact forces are larger than
their 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-and
three-dimensional simulations. The exponent was
found to change very slightly with the coefficient of
interparticle friction and to be independent of particle
size distributions.
Simulations show that applied deviator load is trans-ferred
exclusively by the normal contact forces in the
strong force networks, and the contribution by the
weak 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 the
development of the deviator stress during axisymme-tric
compression of a dense granular assembly (Thorn-ton,
2000). The strong force network carries most of
the whole deviator load as shown in Fig. 11.36 and is
the load-bearing part of the structure. For particles in
the strong force networks, the tangential contact forces
are much smaller than the interparticle frictional resis-tance
because of the large normal contact forces. In
contrast, the numerical analysis results show that the
tangential contact forces in the weak clusters are close
to the interparticle frictional resistance. Hence, the fric-tional
resistance is almost fully mobilized between par-ticles
in the weak clusters, and the particles are perhaps
behaving like a viscous fluid.
Buckling, Sliding, and Rolling
As particles begin to move relative to each other during
shear, particles in the strong force network do not slide,
but columns of particles buckle (Cundall and Strack,
1979). Particles in the strong force network collapse
upon 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 the
contacts (Kuhn, 1999) and approximately 96 percent
of the sliding particles are in the weak clusters (Radjai
et al., 1996). Over 90 percent of the energy dissipation
occurs 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 to
slide. Particle rotations reduce contact sliding and dis-sipation
rate in the granular system. If all particles
could roll upon one another, a granular assembly
would deform without energy dissipation.7 However,
this is not possible owing to restrictions on particle
rotations. It is impossible for all particles to move by
rotation, and sliding at some contacts is inevitable due
to 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 particles
in 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 rolling
resistance is necessary in discrete particle simulations to generate
realistic localized shear bands.
8For instance, consider a chain loop of an odd number of particles.
Particle rotation will involve at least one sliding contact.

Figure 11.34 Probability distributions of interparticle contact forces: (a) normal forces and
(b) tangential forces. The distributions were Material
obtained for contact dynamic simulations of
500, 1024, 1200 and 4025 particles. The effect of number of particles in the simulation on
probability distribution appears to be small (after Radjai et al., 1996).
Copyrighted Figure 11.35 Contributions of normal and tangential contact
forces to the evolution of the deviator stress during axisym-metric
compression of a dense granular assembly (after
Thornton, 2000).
Figure 11.36 Contributions of strong and weak contact
forces to the evolution of the deviator stress during axisym-metric
compression of a dense granular assembly (after
Thornton, 2000).
11.37 shows the spatial distribution of residual defor-mation,
in which the computed deformation of each
particle is subtracted from the average overall defor-mation
(Williams and Rege, 1997). A group of inter-locked
particles that instantaneously moves as a rigid
body 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, becomes
more apparent as applied strains increase toward fail-ure.
The bands of large residual deformation [termed

Figure 11.37 Spatial distribution of residual deformation ob-served
in an elliptic particle assembly at an axial strain level
of (a) 1.1%, (b) 3.3%, (c) 5.5%, (d) 7.7%, (e) 9.8%, and
(ƒ) 12.0% (after Williams and Rege, 1997).
Stress Ratio q/p
1.5
1.0
0.5
Contact Plane Normals
in Initial State:
More in Vertical Direction
Same in All Directions
More in Horizontal Direction
-8 -6 -4 -2 2 4 6 8 10
-0.5
-1.0
Axial Strain (%)
Fabric Anisotropy A
0.4
0.2
0.1
Contact Plane Normals
in Initial State:
-8 -6 -4 -2 2 4 6 8 10
Axial Strain (%)
0.3
-0.1
-0.2
-0.3
-0.4
-0.5
Triaxial Extension
More in Vertical Direction
Same in All Directions
More in Horizontal Direction
(a)
(b)
A
B
C
A
B
C
Figure 11.38 Discrete element simulations of drained tri-axial
compression and extension tests of particle assemblies
prepared at different initial contact fabrics: (a) stress–strain
relationships and (b) evolution of fabric anisotropy parameter
A (after Yimsiri, 2001).
microbands by Kuhn (1999)] are where particle trans-lations
and rotations are intense as part of the strong
force 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 as
deformation proceeds. This microband slip zone may
eventually become a localized shear band.
Fabric Anisotropy
The ability of a granular assemblage of particles to
carry deviatoric loads is attributed to its capability to
develop anisotropy in contact orientations. An initial
isotropic packing of particles develops an anisotropic
contact network during compression loading. This is
because 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 deformation
as illustrated in Fig. 11.18. Figure 11.38 shows results
of discrete particle simulations of particle assemblies
prepared at different states of initial contact anisotropy
under an isotropic stress condition (Yimsiri, 2001). The
initial void ratios are similar (e0 0.75 to 0.76) and
both drained triaxial compression and extension tests
were simulated. Although all specimens are initially
isotropically loaded, the directional distributions of
contact forces are different due to different orientations
of 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 in
Fig. 11.38a, both samples A and C showed stiffer re-sponse
when the compression loading was applied in
the preferred direction of contact forces, but softer re-sponse
when the loading was perpendicular to the pre-ferred
direction of contact forces. The response of
sample B, which had an isotropic fabric, was in be-tween
the two. Dilation was most intensive when the
contact forces were oriented preferentially in the di-

0.1
0.05
0.0
-0.05
-0.1
Material
Copyrighted voids become more apparent, leading to dilation in
1 2 3 4 5 6
N/N
Fabric Anisotropy Parameter A
Figure 11.39 Fabric anisotropy parameter A for different
levels of contact force when the specimen is under biaxial
compression loading conditions (after Radjai et al., 1996).
Figure 11.40 Evolution of the fabric anisotropy parameters
of strong forces and weak clusters when the specimen is un-der
biaxial compression loading conditions (after Thornton
and Antony, 1998).
rection of applied compression; and experimental data
presented by Konishi et al. (1982) shows a similar
trend.
Figure 11.38b shows the development of fabric an-isotropy
with increasing strain. The degree of fabric
anisotropy is expressed by a fabric anisotropy param-eter
A; the value of A increases with more vertically
oriented contact plane normals and is negative when
there 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 the
specimens fail. The final steady-state value is indepen-dent
of the initial fabric, indicating that the inherent
anisotropy is destroyed by the shearing process. The
final fabric anisotropy after triaxial extension is larger
than that after triaxial compression because the addi-tional
confinement by a larger intermediate stress in
the extension tests created a higher degree of fabric
anisotropy.
Close examination of the contact force distribution
for the strong force network and weak clusters gives
interesting microscopic features. Figure 11.39 shows
the values of A determined for the subgroups of contact
9The density of contact plane normals E(

is fitted
with the following expression (Radjai, 1999):
c
E(

)} c
where c is the total number of contacts,

c is the direction for which
the maximum E is reached, and the magnitude of A indicates the
amplitude of anisotropy. When the directional distribution of contact
forces is independent of

, the system has an isotropic fabric and
A 0.
forces categorized by their magnitudes when the spec-imen
is under a biaxial compression loading condition
(Radjai, 1999). The direction of contact anisotropy of
the weak clusters (N/N less than 1) is orthogonal
to the direction of compression loading, whereas that
of the strong force network (N/N more than 2) is
parallel. Figure 11.40 shows an example of fabric ev-olution
with strains in biaxial loading (Thornton and
Antony, 1998). The fabric anisotropy is separated into
that in the strong force networks (N/N of more
than 1) and that in the weak clusters (N/N less than
1). Again the directional evolution of the fabric in the
weak clusters is opposite to the direction of loading.
Therefore, the stability of the strong force chains
aligned in the vertical loading direction is obtained by
the lateral forces in the surrounding weak clusters.
Changes in Number of Contacts and Microscopic
Voids
At the beginning of biaxial loading of a dense granular
assembly, more contacts are created from the increase
in the hydrostatic stress, and the local voids become
smaller. As the axial stress increases, however, the lo-cal
voids tend to elongate in the direction of loading
as shown in Fig. 11.41. Consequently particle contacts
are lost. As loading progresses, vertically elongated lo-cal

Material
Copyrighted Figure 11.41 Simulated spatial distribution of local microvoids under biaxial loading (after
Iwashita and Oda, 2000): (a) 1.1% (before failure), (b) 2.2% (at failure), (c)
11 11 4.4% (after failure), and (d) 11 5.5% (after failure).
11 terms of overall sample volume (Iwashita and Oda,
2000).
Void reduction is partly associated with particle
breakage. Thus, there is a need to incorporate grain
crushing in discrete particle simulations to model the
contractive behavior of soils (Cheng et al., 2003). Nor-mal
contact forces in the strong force network are quite
high, and, therefore, particle asperities, and even par-ticles
themselves, are likely to break, causing the force
chains 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 by
Thornton (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 Interparticle
Friction Angle
Discrete particle simulations show that an increase in
the interparticle friction angle results in an increase
in shear modulus and shear strength, in higher rates of
dilation, and in greater fabric anisotropy. Figure 11.42
shows the effect of assumed interparticle friction angle
on the mobilized macroscopic friction angle of the
particle assembly (Thornton, 2000; Yimsiri, 2001). The
macroscopic friction angle is larger than the interpar-ticle
friction angle if the interparticle friction angle is
smaller than 20. As the interparticle friction becomes

Material
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)
0 10 20 30 40 50 60 70 80 90
Copyrighted 50
40
30
20
10
0
Interparticle Friction Angle (degrees)
Macroscopic Friction Angle (degrees)
Figure 11.42 Relationships between interparticle friction angle and macroscopic friction
angle from discrete element simulations. The macroscopic friction angle was determined
from simulations of drained and undrained triaxial compression (TC) and extension (TE)
tests. The experimental data by Skinner (1969) is also presented (after Thornton, 2000, and
Yimsiri, 2001).
more than 20, the contribution of increasing interpar-ticle
friction to the macroscopic friction angle becomes
relatively small; the macroscopic friction angle ranges
between 30 and 40, when the interparticle friction
angle 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 load
is carried by the strong force networks of normal
forces and not by tangential forces, whose magnitude
is governed by interparticle friction angle. Increase in
interparticle friction results in a decrease in the per-centage
of sliding contacts (Thornton, 2000). The in-terparticle
friction therefore acts as a kinematic
constraint of the strong force network and not as the
direct source of macroscopic resistance to shear. If the
interparticle friction were zero, strong force chains
could 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 simulations
done 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 the
number of contacts required to achieve a stable con-dition.
As long as the strong force network can be
formed, however, the magnitude of the interparticle
friction becomes of secondary importance.
The above findings from discrete particle simula-tions
are partially supported by the experimental data
given 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 friction
angle. The tested materials included glass ballotini,
steel ball bearings, and lead shot. Use of glass ballotini
was particularly attractive since the coefficient of in-terparticle
friction increases by a factor of between 3.5
and 30 merely by flooding the dry sample. Skinner’s
data shown in Fig. 11.42 indicate that the macroscopic
friction angle is nearly independent of interparticle
friction angle.
Effects of Particle Shape and Angularity
A lower porosity and a larger coordination number are
achieved for ellipsoidal particles compared to spherical

Deviator
Stress
q = σa – σr
M (triaxial
compression)
Mean Pressure p
Isotropic
Compression
Line
ln p
Specific
Volume v
1
Γ
Critical
State Line
λcs
(a)
Critical
State Line
λ
M (triaxial
extension)
Critical
State Line
σa
σr σr
Compression Lines of
Constant Stress Ratio q/p
(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 packing
can be achieved for ellipsoidal particles. Ellipsoid par-ticles
rotate less than spherical particles. An assembly
of ellipsoid particles gives larger values of shear
strength and initial modulus than an assemblage of
spherical particles, primarily because of the larger co-ordination
number for ellipsoidal particles. Similar
findings result for two-dimensional particle assemblies.
Circular disks give the highest dilation for a given
stress ratio and the lowest coordination number com-pared
to elliptical or diamond shapes (Williams and
Rege, 1997). An assembly of rounded particles exhib-its
greater softening behavior with fabric anisotropy
change with strain, whereas an assembly of elongated
particles requires more shearing to modify its initial
fabric anisotropy to the critical state condition
(Nouguier-Lehon et al., 2003).
11.7 CRITICAL STATE: A USEFUL REFERENCE
CONDITION
After large shear-induced volume change, a soil under
a given effective confining stress will arrive ultimately
at a unique final water content or void ratio that is
independent of its initial state. At this stage, the inter-locking
achieved by densification or overconsolidation
is gone in the case of dense soils, the metastable struc-ture
of loose soils has collapsed, and the soil is fully
destructured. A well-defined strength value is reached
at this state, and this is often referred to as the critical
state strength. Under undrained conditions, the critical
state is reached when the pore pressure and the effec-tive
stress remain constant during continued deforma-tion.
The critical state can be considered a fundamental
state, and it can be used as a reference state to explain
the 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, under
sustained uniform shearing, there exists a unique re-lationships
among void ratio ecs (or specific volume
vcs 1 ecs), mean effective pressure pcs, and deviator
stress qcs as shown in Fig. 11.43. An example of the
critical 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, pcs is the
mean effective stress at failure, and M is the critical
state stress ratio. The critical state on the void ratio (or
specific volume)–mean pressure plane is defined by
two material parameters: cs, the critical state com-pression
index and %, the specific volume intercept at
unit pressure (p 1). The compression lines under
constant stress ratios are often parallel to each other,
as shown in Fig. 11.43b.
Parameter M in Equation (11.25) defines the critical
state stress ratio at failure and is similar to for the
Mohr–Coulomb failure line. However, Equation
(11.25) includes the effect of intermediate principal
stress 2 because p 1 2 3, whereas the
Mohr–Coulomb failure criterion of Eq. (11.4) or (11.5)
does not take the intermediate effective stress into ac-count.
In triaxial conditions, a r r and r
r a for compression and extension, respectively
(see Fig. 11.43).11 Hence, Eqs. (11.4) and (11.25) can
be related to each other for these two cases as follows:
11 is the axial effective stress and is the radial effective stress. a r

CRITICAL STATE: A USEFUL REFERENCE CONDITION 401
6 sin M crit for triaxial compression (11.27)
3 sin crit
6 sin crit M for triaxial extension (11.28)
3 sin crit
These equations indicate that the correlation be-tween
M and crit is not unique but depends on the
where wcs is the water content at critical state when
the effective mean pressure is p. pLL and pPL are the
stress conditions. Neither is a fundamental property of
the soil, as discussed further in Section 11.12. None-theless,
both are widely used in engineering practice,
and, if interpreted properly, they can provide useful
and simple phenomenological representations of com-plex
behavior.
The drained and undrained critical state strengths are
illustrated in Figs. 11.44a and 11.44b for normally
consolidated clay and heavily overconsolidated clay,
respectively. The initial mean pressure–void ratio state
of the normally consolidated clay is above the critical
state line, whereas that of the heavily overconsolidated
clay is below the critical state line. When the initial
state of the soil is normally consolidated at A (Fig.
11.44a), the critical state is B for undrained loading
Deviator
Stress q
M
Critical
State Line
Mean pressure p
ln p
Specific
Volume v
1
Γ
λcs
A
B
Critical State Line
Isotropic
Compression
Line
λ
A
B
C
C
3
1
and C for drained triaxial compression. Hence, the de-viator
stress at critical state is smaller for the undrained
case than for the drained case. On the other hand, when
the initial state of the soil is overconsolidated from
D (Fig. 11.44b), the critical state becomes E for un-drained
The deviator stress at critical state is smaller for the
drained case compared to the undrained case. It is im-portant
state 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 ecspcs ; therefore, it is not at the critical
state.
Converting the void ratio in Eq. (11.26) to water
content, a normalized critical state line can be written
using the liquidity index (see Fig. 11.45).
Deviator
Stress q
loading and F for drained triaxial compression.
to note that the soil state needs to satisfy both
w w ln(p /p) LI cs PL PL (11.29) cs w w ln(p /p ) LL PL PL LL
Critical
State Line
M
Mean pressure p
ln p
Specific
Volume v
Drained Peak Strength
1
Γ
λcs
Critical State Line
Isotropic
Compression
Line
λ
D
E
F
D
F
E
3
1
(a)
Drained Strength
Drained Strength
Undrained
Strength
Undrained Strength
D
D
G
G
(b)
Figure 11.44 Drained and undrained stress–strain response using the critical state concept:
(a) normally consolidated clay and (b) overconsolidated clay.

wLL
wPL
Material
Copyrighted (ln(p), w)
ln(pLL) ln(pPL)
LI = 1
LI = 0
ln(pLL) ln(p) ln(pPL)
Liquidity
Index
Water
Content
Liquid Limit
Plastic Limit
(ln(p’), LI)
ln(p)
w
LI
LICS
LIeq-1
wcs
Critical
State Line
Isotropic
Compression Line
Critical
State Line
Mean
pressure
Mean
Pressure
(a) (b)
Figure 11.45 Normalization of the critical state line: (a) water content versus mean pressure
and (b) liquidity index versus mean pressure.
mean effective pressure at liquid limit (wLL) and plastic
limit (wPL), respectively; pLL 1.5 to 6 kPa and pPL
150 to 600 kPa are expected considering the un-drained
shear strengths at liquid and plastic limits are
in the ranges suLL 1 to 3 kPa and suPL 100 to 300
kPa, respectively12 (see Fig. 8.48).
Using Eq. (11.29), a relative state in relation to the
critical state for a given effective mean pressure (i.e.,
above or below the critical state line) can be defined
as (see Fig. 11.45)
log(p /p ) LI LI LI 1 LI LL (11.30) eq cs log(p /p ) PL LL
where LIeq is the equivalent liquidity index defined by
Schofield (1980). When LIeq 1 (i.e., LI LIcs) and
q/p M, the clay has reached the critical state. Figure
11.46 gives the stress ratio when plastic failure (or
fracture) initiates at a given water content. When LIeq
1 (the state is above the critical state line), and the
soil in a plastic state exhibits uniform contractive be-havior.
When LIeq 1 (the state is below the critical
state line), and the soil in a plastic state exhibits lo-calized
dilatant rupture, or possibly fracture, if the
stress ratio reaches the tensile limit (q/p 3 for tri-axial
compression and 1.5 for triaxial extension; see
Fig. 11.46b). Hence, the critical state line can be used
as a reference to characterize possible soil behavior
under plastic deformation.
12A review by Sharma and Bora (2003) gives average values of
suLL 1.7 kPa and suPL 170 kPa.
Sands
The critical state strength and relative density of sand
can be expressed as
q Mp (11.31) cs cs
e e 1 D max cs (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 is
the crushing strength of the particles.13 The critical
state 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 found
for clays. This nonlinearity is confirmed by experi-mental
data as shown in Fig. 11.4b.
At high confining pressure, when the effective mean
pressure becomes larger than the crushing strength,
many particles begin to break and the lines become
more 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 discussed
further in Section 11.8. Other mathematical expressions to fit the
experimentally determined critical state line are possible. For exam-ple,
Li et al. (1999) propose the following equation for the critical
state line (ecs versus p):
e e cs 0 s pa
p
where e0 is the void ratio at p 0, pa is atmospheric pressure, and
s and are material constants.

CRITICAL STATE: A USEFUL REFERENCE CONDITION 403
q/p
0.5 1.0 LIeq
Material
Copyrighted 3
MTC
MTE
-1.5
Dilatant
Rupture
Fracture
Ductile Plastic
and Contractive
Dilatant
Fracture Rupture
Triaxial Extension
(a)
p
q
Triaxial Extension
1
3
2
3
MTC
MTE
Tensile
Fracture
Tensile
Fracture
(b)
Ductile Plastic
and Contractive
Figure 11.46 Plastic state of clay in relation to normalized liquidity index: (a) stress ratio
when plastic state initiates for a given LIeq and (b) definition of stress ratios used in (a) (after
Schofield, 1980).
0
0.2
0.4
0.6
0.8
1
DR,cs =
–
–
ecs
emax emin
1.2
0.001 0.01 0.1 1
=
1
In (σc/p)
p/σc p/σc
Relative Density Dr
0
0.2
0.4
0.6
0.8
1
1.2
0 0.1 0.2 0.3 0.4 0.5
Relative Density Dr
emax
emin
emax
emax
emin
(a) (b)
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 that
there is a unique relationship between the amount of
particle breakage that occurred on shearing to a critical
state and the value of the mean normal effective stress.
This implies that sand at the critical state would reach
a stable grading at which the particle contact stresses
would not be sufficient to cause further breakage. Coop
et al. (2004) performed ring shear tests (see Section
11.11) on a carbonate sand to find a shear strain re-quired
to reach the true critical state (i.e., constant
particle grading). They found that particle breakage
continues to very large strains, far beyond those

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