63027 12

465
CHAPTER 12
Copyrighted Material
Time Effects on Strength and
Deformation
12.1 INTRODUCTION
Virtually every soil ‘‘lives’’ in that all of its properties
undergo changes with time–some insignificant, but
others very important. Time-dependent chemical,
geomicrobiological, and mechanical processes may
result in compositional and structural changes that
lead to softening, stiffening, strength loss, strength
gain, or altered conductivity properties. The need to
predict what the properties and behavior will be
months to hundreds or thousands of years from now
based on what we know today is a major challenge in
geoengineering. Some time-dependent changes and
their effects as they relate to soil formation, composi-tion,
weathering, postdepositional changes in sedi-ments,
the evolution of soil structure, and the like are
considered in earlier chapters of this book. Emphasis
in this chapter is on how time under stress changes the
structural, deformation, and strength properties of
soils, what can be learned from knowledge of these
changes, and their quantification for predictive pur-poses.
When soil is subjected to a constant load, it deforms
over time, and this is usually called creep. The inverse
phenomenon, usually termed stress relaxation, is a
drop in stress over time after a soil is subjected to a
particular constant strain level. Creep and relaxation
are two consequences of the same phenomenon, that
is, time-dependent changes in structure. The rate and
magnitude of these time-dependent deformations are
determined by these changes.
Time-dependent deformations and stress relaxation
are important in geotechnical problems wherein long-term
behavior is of interest. These include long-term
settlement of structures on compressible ground, de-formations
of earth structures, movements of natural
and excavated slopes, squeezing of soft ground around
tunnels, and time- and stress-dependent changes in soil
properties. The time-dependent deformation response
of a soil may assume a variety of forms owing to the
complex interplays among soil structure, stress history,
drainage conditions, and changes in temperature, pres-sure,
and biochemical environment with time. Time-dependent
deformations and stress relaxations usually
follow logical and often predictable patterns, at least
for simple stress and deformation states such as uni-axial
and triaxial compression, and they are described
in this chapter. Incorporation of the observed behav-ior
into simple constitutive models for analytical de-scription
of time-dependent deformations and stress
changes is also considered.
Time-dependent deformation and stress phenomena
in soils are important not only because of the imme-diate
direct application of the results to analyses of
practical problems, but also because the results can be
used to obtain fundamental information about soil
structure, interparticle bonding, and the mechanisms
controlling the strength and deformation behavior.
Both microscale and macroscale phenomena are dis-cussed
because understanding of microscale processes
can provide a rational basis for prediction of macro-scale
responses.
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466 12 TIME EFFECTS ON STRENGTH AND DEFORMATION
12.2 GENERAL CHARACTERISTICS
1. As noted in the previous section soils exhibit
both creep1 and stress relaxation (Fig. 12.1).
Creep is the development of time-dependent
shear and/or volumetric strains that proceed
at a rate controlled by the viscouslike resistance
of soil structure. Stress relaxation is a time-dependent
decrease in stress at constant defor-mation.
The relationship between creep strain
Material
Copyrighted and the logarithm of time may be linear, con-cave
upward, or concave downward as shown
by the examples in Fig. 12.2.
2. The magnitude of these effects increases with
increasing plasticity, activity, and water content
Figure 12.1 Creep and stress relaxation: (a) Creep under
constant stress and (b) stress relaxation under constant strain.
1The term creep is used herein to refer to time-dependent shear
strains and/ or volumetric strains that develop at a rate controlled by
the viscous resistance of the soil structure. Secondary compression
refers to the special case of volumetric strain that follows primary
consolidation. The rate of secondary compression is controlled by
the viscous resistance of the soil structure, whereas, the rate of pri-mary
consolidation is controlled by hydrodynamic lag, that is, how
fast water can escape from the soil.
of the soil. The most active clays usually exhibit
the greatest time-dependent responses (i.e.,
smectite illite kaolinite). This is because
the smaller the particle size, the greater is the
specific surface, and the greater the water ad-sorption.
Thus, under a given consolidation
stress or deviatoric stress, the more active and
plastic clays (smectites) will be at higher water
content and lower density than the inactive clays
(kaolinites). Normally consolidated soils exhibit
larger magnitude of creep than overconsolidated
soils. However, the basic form of behavior is
essentially the same for all soils, that is, undis-turbed
and remolded clay, wet clay, dry clay,
normally and overconsolidated soil, and wet and
dry sand.
3. An increase in deviatoric stress level results in
an increased rate of creep as shown in Fig. 12.1.
Some soils may fail under a sustained creep
stress significantly less (as little as 50 percent)
than the peak stress measured in a shear test,
wherein a sample is loaded to failure in a few
minutes or hours. This is termed creep rupture,
and an early illustration of its importance was
the development of slope failures in the Cucar-acha
clay shale, which began some years after
the excavation of the Panama Canal (Casa-grande
and Wilson, 1951).
4. The creep response shown by the upper curve
in Fig. 12.1 is often divided into three stages.
Following application of a stress, there is first a
period of transient creep during which the strain
rate decreases with time, followed by creep at
nearly a constant rate for some period. For ma-terials
susceptible to creep rupture, the creep
rate then accelerates leading to failure. These
three stages are termed primary, secondary, and
tertiary creep.
5. An example of strain rates as a function of stress
for undrained creep of remolded illite is shown
in Fig. 12.3. At low deviator stress, creep rates
are very small and of little practical importance.
The curve shapes for deviator stresses up to
about 1.0 kg/cm2 are compatible with the pre-dictions
of rate process theory, discussed in Sec-tion
12.4. At deviator stress approaching the
strength of the material, the strain rates become
very large and signal the onset of failure.
6. A characteristic relationship between strain rate
and time exists for most soils, as shown, for
example, in Fig. 12.4 for drained triaxial com-pression
creep of London clay (Bishop, 1966)
Copyright © 2005 John Wiley Sons Retrieved from: www.knovel.com

GENERAL CHARACTERISTICS 467
Material
Copyrighted Figure 12.2 Sustained stress creep curves illustrating different forms of strain vs. logarithm
of time behavior.
and Fig. 12.5 for undrained triaxial compression
creep of soft Osaka clay (Murayama and Shi-bata,
1958). At any stress level (shown as a per-centage
of the strength before creep in Fig. 12.4
and in kg/cm2 in Fig. 12.5), the logarithm of the
strain rate decreases linearly with increase in the
logarithm of time. The slope of this relationship
is essentially independent of the creep stress;
increases in stress level shift the line vertically
upward. The slope of the log strain rate versus
log time line for drained creep is approximately
1. Undrained creep often results in a slope be-

Material
Copyrighted Figure 12.3 Variation of creep strain rate with deviator stress for undrained creep of re-molded
illite.
tween 0.8 and 1 for this relationship. The
onset of failure under higher stresses is signaled
by a reversal in slope, as shown by the topmost
curve in Fig. 12.5.
7. Pore pressure may increase, decrease, or remain
constant during creep, depending on the volume
change tendencies of the soil structure and
whether or not drainage occurs during the de-formation
process. In general, saturated soft
sensitive clays under undrained conditions are
most susceptible to strength loss during creep
due to reduction in effective stress caused by
increase in pore water pressure with time. Heav-ily
overconsolidated clays under drained con-ditions
are also susceptible to creep rupture due
to softening associated with the increase in wa-ter
content by dilation and swelling.
8. Although stress relaxation has been less studied
than creep, it appears that equally regular pat-terns
of deformation behavior are observed, for
example, Larcerda and Houston (1973).
9. Deformation under sustained stress ordinarily
produces an increase in stiffness under the ac-tion
of subsequent stress increase, as shown
schematically in Fig. 12.6. This reflects the
time-dependent structural readjustment or ‘‘ag-ing’’
that follows changes in stress state. It is
analogous to the quasi-preconsolidation effect

GENERAL CHARACTERISTICS 469
Material
Copyrighted Figure 12.4 Strain rate vs. time relationships during drained creep of London clay (data
from Bishop, 1966).
due to secondary compression discussed in Sec-tion
8.11; however, it may develop under un-drained
as well as drained conditions.
10. As shown in Fig. 12.7, the locations of both the
virgin compression line and the value of the pre-consolidation
pressure, , determined in the p
laboratory are influenced by the rate of loading
during one-dimensional consolidation (Graham
et al., 1983a; Leroueil et al., 1985). Thus, esti-mations
of the overconsolidation ratio of clay
deposits in the field are dependent on the load-ing
rates and paths used in laboratory tests for
determination of the preconsolidation pressure.
If it is assumed that the relationship between
strain and logarithm of time during compression
is linear over the time ranges of interest and that
the secondary compression index Ce is constant
regardless of load, the rate-dependent precon-solidation
pressure p at ˙1 can be related to the
axial strain rate as follows (Silvestri et al, 1986;
Soga and Mitchell, 1996; Leroueil and Marques,
1996):
Ce / (CcCr) p ˙1 ˙1 (12.1) p(ref) ˙1(ref) ˙1(ref)
where Cc is the virgin compression index, Cr is
the recompression index and p(ref) is the pre-consolidation
pressure at a reference strain rate
˙ . In this equation, the rate effect increases 1(ref)
with the value of Ce/(Cc Cr). The var-iation
of preconsolidation pressure with strain
rate is shown in Fig. 12.8 (Soga and Mitchell,
1996). The data define straight lines, and the
slope of the lines gives the parameter . In gen-eral,
the value of ranges between 0.011 and
0.094. Leroueil and Marques (1996) report val-ues
between 0.029 and 0.059 for inorganic
clays.
11. The undrained strength of saturated clay in-creases
with increase in rate of strain, as shown
in Figs. 12.9 and 12.10. The magnitude of the
effect is about 10 percent for each order of mag-nitude
increase in the strain rate. The strain rate

Material
Figure 12.5 Strain rate vs. time relationships during undrained creep of Osaka alluvial clay
Copyrighted (Murayama and Shibata, 1958).
Figure 12.6 Effect of sustained loading on (a) stress–strain
and strength behavior and (b) one-dimensional compression
behavior.
effect is considerably smaller for sands. In a
manner similar to Eq. (12.1), a rate parameter
can be defined as the slope of a log–log plot
of deviator stress at failure qƒ at a particular
strain rate ˙1 relative to qƒ(ref), the strength at a
reference strain rate ˙ , versus strain rate. 1(ref)
This gives the following equation:
qƒ ˙1 (12.2) qƒ(ref) ˙1(ref)
The value of ranges between 0.018 and 0.087,
similar to the rate parameter values used to
define the rate effect on consolidation pressure
in Eq. (12.1). Higher values of are associated
with more metastable soil structures (Soga and
Mitchell, 1996). Rate dependency decreases
with increasing sample disturbance, which is
consistent with this finding.
12.3 TIME-DEPENDENT
DEFORMATION–STRUCTURE INTERACTION
In reality, completely smooth curves of the type shown
in the preceding figures for strain and strain rate as a
function of time may not exist at all. Rather, as dis-

TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION 471
Figure 12.7 Rate dependency on one-dimensional compression characteristics of Batiscan
clay: (a) compression curves and (b) preconsolidation pressure (Leroueil et al., 1985).

Material
Figure 12.8 Strain rate dependence on preconsolidation pressure determined from one-dimensional
constant strain rate tests (Soga and Mitchell, 1996).
Copyrighted Figure 12.9 Effect of strain rate on undrained strength (Kulhawy and Mayne 1990). Re-printed
with permission from EPRI.
cussed by Ter-Stepanian (1992), a ‘‘jump-like structure
reorganization’’ may occur, reflecting a stochastic char-acter
for the deformation, as shown in Fig. 12.11 for
creep of an undisturbed diatomaceous, lacustrine, ov-erconsolidated
clay. Ter-Stepanian (1992) suggests that
there are four levels of deformation: (1) the molecular
level, which consists of displacement of flow units by
surmounting energy barriers, (2) mutual displacement
of particles as a result of bond failures, but without
rearrangement, (3) the structural level of soil defor-mation
involving mutual rearrangements of particles,
and (4) deformation at the aggregate level. Behavior at
levels 3 and 4 is discussed below; that at levels 1 and
2 is treated in more detail in Section 12.4.

Material
Figure 12.10 Strain rate dependence on undrained shear strength determined using constant
strain rate CU tests (Soga and Mitchell, 1996).
Copyrighted Figure 12.11 Nonuniformity of creep in an undisturbed, di-atomaceous,
lacustrine, overconsolidated clay (from Ter-
Stepanian, 1992).
Time-Dependent Process of Particle Rearrangement
Creep can lead to rearrangement of particles into more
stable configurations. Forces at interparticle contacts
have both normal and tangential components, even if
the macroscopic applied stress is isotropic. If, during
the creep process, there is an increase in the proportion
of applied deviator stress that is carried by interparticle
normal forces relative to interparticle tangential forces,
then the creep rate will decrease. Hence, the rate at
which deformation level 3 occurs need not be uniform
owing to the particulate nature of soils. Instead it will
reflect a series of structural readjustments as particles
move up, over, and around each other, thus leading to
the somewhat irregular sequence of data points shown
in Fig. 12.11.
Microscopically, creep is likely to occur in the weak
clusters discussed in Section 11.6 because the contacts
in them are at limiting frictional equilibrium. Any
small perturbation in applied load at the contacts or
time-dependent loss in material strength can lead to
sliding, breakage or yield at asperities. As particles
slip, propped strong-force network columns are dis-turbed,
and these buckle via particle rolling as dis-cussed
in Section 11.6.
To examine the effects of particle rearrangement,
Kuhn (1987) developed a discrete element model that
considers sliding at interparticle contacts to be visco-frictional.
The rate at which sliding of two particles
relative to each other occurs depends on the ratio of
shear to normal force at their contact. The relationship
between rate and force is formulated in terms of rate
process theory (see Section 12.4), and the mechanistic
representations of the contact normal and shear forces
are shown in Fig. 12.12. The time-dependent compo-nent
in the tangential forces model is given as a ‘‘sinh-dashpot’’.
2 The average magnitudes of both normal and
2Kuhn (1987) used the following equation for rate of sliding at a
contact:
2kT
F ƒt ˙X
exp sinh h RT 2kTn ƒn 1
where n1 is the number of bonds per unit of normal force, ƒt is the
tangential force and ƒn is the normal force. The others are parameters
related to rate process theory as described in Section 12.4.

Figure 12.12 Normal and tangential interparticle force mod-els
according to Kuhn (1987).
Figure 12.13 Creep curves developed by numerical analysis
of an irregular packing of circular disks (from Kuhn and
Mitchell, 1993).
tangential forces at individual contacts can change dur-ing
deformation even though the applied boundary
stresses are constant. Small changes in the tangential
and normal force ratio at a contact can have a very
large influence on the sliding rate at that contact. These
changes, when summed over all contacts in the shear
zone, result in a decrease or increase in the overall
creep rate.
A numerical analysis of an irregular packing of cir-cular
disks using the sinh-dashpot representation gives
creep behavior comparable to that of many soils as
shown in Fig. 12.13 (Kuhn and Mitchell, 1993). The
creep rate slows if the average ratio of tangential to
normal force decreases, whereas it accelerates and may
ultimately lead to failure if the ratio increases. In some
cases, the structural changes that are responsible for
the decreasing strain rate and increased stiffness may
cause the overall soil structure to become more meta-stable.
Then, after the strain reaches some limiting
value, the process of contact force transfer from de-creasing
tangential to increasing normal force reverses.
This marks the onset of creep rupture as the structure
begins to collapse. A similar result was obtained by
Rothenburg (1992) who performed discrete particle
simulations in which smooth elliptical particles were
cemented with a model exhibiting viscous character-istics
in both normal and tangential directions.
Particle Breakage During Creep
Particle breakage can contribute to time-dependent de-formation
of sands (Leung et al., 1996; Takei et al.,
2001; McDowell, 2003). Leung et al. (1996) performed
one-dimensional compression tests on sands, and Fig.

Dry sand
RD = 75%
Pressure = 15.4 MPa

After
5 days After
Before
290 s

Test
Material
Copyrighted
Sieve Size (μm)
100
80
60
40
20
0
0

100 200 300 400

Percentage Passing (%)
Figure 12.14 Changes in particle size distribution of sand
before loading and after two different load durations (from
Leung, et al., 1996).
12.14 shows the particle size distribution curves for
samples before loading and after two different load du-rations.
The amount of particle breakage increased
with load duration. Microscopic observations revealed
that angular protrusions of the grains were ground off,
producing fines. The fines fill the voids between larger
particles and crushed particles progressively rear-ranged
themselves with time.
Aging—Time-Dependent Strengthening of Soil
Structure
The structural changes that occur during creep that is
continuing at a decreasing rate cause an increase in
soil stiffness when the soil is subjected to further stress
increase as shown in Fig. 12.6. Leonards and Alt-schaeffl
(1964) showed that this increase in preconso-lidation
pressure cannot be accounted for in terms of
the void ratio decrease during the sustained compres-sion
period. Time-dependent changes of these types are
a consequence of ‘‘aging’’ effects, which alter the
structural state of the soil. The fabric obtained by creep
may be different from that caused by increase in stress,
even though both samples arrive at the same void ratio.
Leroueil et al. (1996) report a similar result for an ar-tificially
sedimented clay from Quebec, as shown in
Fig. 12.15a. They also measured the shear wave ve-locities
after different times during the tests using
bender elements and computed the small strain elastic
shear modulus. Figure 12.15b shows the change in
shear modulus with void ratio.
Additional insight into the structural changes ac-companying
the aging of clays is provided by the re-sults
of studies by Anderson and Stokoe (1978) and
Nakagawa et al. (1995). Figure 12.16 shows changes
in shear modulus with time under a constant confining
pressure for kaolinite clay during consolidation (An-derson
and Stokoe, 1978). Two distinct phases of shear
modulus–time response are evident. During primary
consolidation, values of the shear modulus increase
rapidly at the beginning and begin to level off as the
excess pore pressure dissipates. After the end of pri-mary
consolidation, the modulus increases linearly
with the logarithm of time during secondary compres-sion.
The expected change in shear modulus due to void
ratio change during secondary compression can be es-timated
using the following empirical formula for
shear modulus as a function of void ratio and confining
pressure (Hardin and Black, 1968):
(2.97 e)2 G A p0.5 (12.3)
1 e
where A is a unit dependant material constant, e is the
void ratio, and p is the mean effective stress. The
dashed line in Fig. 12.16 shows the calculated in-creases
in the shear modulus due to void ratio decrease
using Eq. (12.3). It is evident that the change in void
ratio alone does not provide an explanation for the sec-ondary
time-dependent increase in shear modulus. This
aging effect has been recorded for a variety of mate-rials,
ranging from clean sands to natural clays (Afifi
and Richart, 1973; Kokusho, 1987; Mesri et al., 1990,
and many others). Further discussion of aging phenom-ena
is given in Section 12.11.
Time-Dependent Changes in Soil Fabric
Changes in soil fabric with time under stress influence
the stability of soil structure. Changes in sand fabric
with time after load application in one-dimensional
compression were measured by Bowman and Soga
(2003). Resin was used to fix sand particles after var-ious
loading times. Pluviation of the sand produced a
horizontal preferred particle orientation of soil grains,
and increased vertical loading resulted in a greater ori-entation
of particle long axes parallel to the horizontal,
which is in agreement with the findings of Oda (1972a,
b, c), Mitchell et al. (1976), and Jang and Frost (1998).
Over time, however, the loading of sand caused parti-cle
long axes to rotate toward the vertical direction
(i.e., more isotropic fabric).
Experimental evidence (Bowman and Soga, 2003)
showed that large voids became larger, whereas small
voids became smaller, and particles group or cluster

2.6
2.4
2.2
2.0
Material
1.8
Copyrighted A A
B B
C C
D D
4 6 8 10 20
Vertical Effective Stress σv (kPa)
0.5 1
2
E
Small-strain Stiffness G0 (MPa)
F
E
F
Stiffness Change
During the Primary
Consolidation
Between B and C
Normal
Consolidation Line
Creep for
120 days
Increase in Stiffness
during Creep (C-D)
Quasi-preconsolidation
Pressure
Destructuring
State
(a) (b )
5
Void Ratio e
Figure 12.15 (a) Compression curve and (b) variation of the maximum shear modulus G0
with void ratio for artificially sedimented Jonquiere clay (from Leroueil et al., 1996).
together with time. Based on these particulate level
findings, it appears that the movements of particles
lead to interlocking zones of greater local density. The
interlocked state may be regarded as the final state of
any one particle under a particular applied load, due to
kinematic restraint. The result, with time, is a stiffer,
more efficient, load-bearing structure, with areas of rel-atively
large voids and neighboring areas of tightly
packed particles. The increase in stiffness is achieved
by shear connections obtained by the clustering. Then,
when load is applied, the increased stiffness and
strength of the granular structure provides greater re-sistance
to the load and the observed aging effect is
seen. The numerical analysis in Kuhn and Mitchell
(1993) led to a similar hypothesis for how a more
‘‘braced’’ structure develops with time. For load appli-cation
in a direction different to that during the aging
period, however, the strengthening effect of aging may
be less, as the load-bearing particle column direction
differs from the load direction.
Time-Dependent Changes in Physicochemical
Interaction of Clay and Pore Fluid
A portion of the shear modulus increase during sec-ondary
compression of clays is believed to result from
a strengthening of physicochemical bonds between
particles. To illustrate this, Nakagawa et al. (1995) ex-amined
the physicochemical interactions between clays
and pore fluid using a special consolidometer in which
the sample resistivity and pore fluid conductivity could
be measured. Shear wave velocities were obtained us-ing
bender elements to determine changes in the stiff-ness
characteristics of the clay during consolidation.
Kaolinite clay mixed with saltwater was used for the
experiment, and changes in shear wave velocities and
electrical properties were monitored during the tests.
The test results showed that the pore fluid compo-sition
and ion mobility changed with time. At each
load increment, as the effective stress increased with
pore pressure dissipation, the shear wave velocities,
and therefore the shear modulus, generally increased
with time as shown in Fig. 12.17. It may be seen, how-ever,
that in some cases, the shear wave velocities at
the beginning of primary consolidation decreased
slightly from the velocities obtained immediately be-fore
application of the incremental load, probably as a
result of soil structure breakdown. During the subse-quent
secondary compression stage, the shear wave ve-locity
again increased. As was the case for the results
in Fig.12.16, the increases in shear wave velocity dur-

50
40
30
20
10
0
0.5
1.0
1.5
2.0
IG = ΔG per log time
= 6.2 MPa
Primary Consolidation Secondary Compression
Sample Height Change (mm)
101 102 103 104
Time (min)
100 101 102 103 104
Time (min)
Ball Kaolinite
Possible Change in G
by Void Ratio Decrease
Only Estimated Using
Eq. (12.3)
Initial Consolidation Pressure = 70 kPa
Initial Void Ratio e0 = 1.1
Shear Modulus of less than γ = 10-3% (MPa)
Figure 12.16 Modulus and height changes as a function of
time under constant confining pressure for kaolinite: (a) shear
modulus and (b) height change (from Anderson and Stokoe,
1978).
Figure 12.17 Changes in shear wave velocity during pri-mary
consolidation and secondary compression of kaolinite.
Consolidation pressures: (a) 11.8 kPa and (b) 190 kPa (from
Nakagawa et al., 1995).
ing secondary compression are greater than can be ac-counted
for by increase in density.
The electrical conductivity of the sample measured
by filter electrodes increased during the early stages of
consolidation, but then decreased continuously there-after
as shown in Fig. 12.18. The electrical conductiv-ity
is dominated by flow through the electrolyte
solution in the pores. During the initial compression, a
breakdown of structure releases ions into the pore wa-ter,
increasing the electrical conductivity. With time,
the conductivity decreased, suggesting that the released
ions are accumulating near particle surfaces. Some of
these released ions are expelled from the specimen as
consolidation progressed as shown in Fig. 12.18b. A
slow equilibrium under a new state of effective stress
is hypothesized to develop that involves both small
particle rearrangements, associated with decrease in
void ratio during secondary compression, and devel-opment
of increased contact strength as a result of pre-cipitation
of salts from the pore water and/or other
processes.
Primary consolidation can be considered a result of
drainage of pore water fluid from the macropores,
whereas secondary compression is related to the de-layed
deformation of micropores in the clay aggregates
(Berry and Poskitt, 1972; Matsuo and Kamon, 1977;
Sills, 1995). The mobility of water in the micropores
is restricted due to small pore size and physicochem-ical
interactions close to the clay particle surfaces. Ak-agi
(1994) did compression tests on specially prepared
clay containing primarily Ca in the micropores and Na
in the macropores. Concentrations of the two ions in
the expelled water at different times after the start of
consolidation were consistent with this hypothesis.

Activation Frequency
The energy to enable a flow unit to cross a barrier may
be provided by thermal energy and by various applied
potentials. For a material at rest, the potential energy–
displacement relationship is represented by curve A in
Fig. 12.21. From statistical mechanics it is known that
Figure 12.18 Changes in electrical conductivity of the pore
water during primary consolidation and secondary compres-sion
of kaolinite. Consolidation pressures: (a) 95 kPa and (b)
190 kPa (from Nakagawa et al., 1995).
Figure 12.19 Energy barriers and activation energy.
12.4 SOIL DEFORMATION AS A RATE
PROCESS
Deformation and shear failure of soil involve time-dependent
rearrangement of matter. As such, these
phenomena are amenable for study as rate processes
through application of the theory of absolute reaction
rates (Glasstone et al., 1941). This theory provides
both insights into the fundamental nature of soil
strength and functional forms for the influences of sev-eral
factors on soil behavior.
Detailed development of the theory, which is based
on statistical mechanics, may be found in Eyring
(1936), Glasstone et al. (1941), and elsewhere in the
physical chemistry literature. Adaptations to the study
of soil behavior include those by Abdel-Hady and Her-rin
(1966), Andersland and Douglas (1970), Christen-sen
and Wu (1964), Mitchell (1964), Mitchell et al.
(1968, 1969), Murayama and Shibata (1958, 1961,
1964), Noble and Demirel (1969), Wu et al. (1966),
Keedwell (1984), Feda (1989, 1992), and Kuhn and
Mitchell (1993).
Concept of Activation
The basis of rate process theory is that atoms, mole-cules,
and/or particles participating in a time-dependent
flow or deformation process, termed flow
units, are constrained from movement relative to each
other by energy barriers separating adjacent equilib-rium
positions, as shown schematically by Fig. 12.19.
The displacement of flow units to new positions re-quires
the acquisition of an activation energy
F of
sufficient magnitude to surmount the barrier. The po-tential
energy of a flow unit may be the same following
the activation process, or higher or lower than it was
initially. These conditions are shown by analogy with
the rotation of three blocks in Fig. 12.20. In each case,
an energy barrier must be crossed. The assumption of
a steady-state condition is implicit in most applications
to soils concerning the at-rest barrier height between
successive equilibrium positions.
The magnitude of the activation energy depends on
the material and the type of process. For example, val-ues
of
F for viscous flow of water, chemical reac-tions,
and solid-state diffusion of atoms in silicates are
about 12 to 17, 40 to 400, and 100 to 150 kJ/mol of
flow units, respectively.

SOIL DEFORMATION AS A RATE PROCESS 479
Material
Figure 12.20 Examples of activated processes: (a) steady-state, (b) increased stability, and
(c) decreased stability.
Copyrighted Figure 12.21 Effect of a shear force on energy barriers.

the average thermal energy per flow unit is kT, where
k is Boltzmann’s constant (1.38 1023 J K1) and T
is the absolute temperature (K). Even in a material at
rest, thermal vibrations occur at a frequency given by
kT/h, where h is Planck’s constant (6.624 1034 J
s1). The actual thermal energies are divided among
the flow units according to a Boltzmann distribution.
It may be shown that the probability of a given unit
becoming activated, or the proportion of flow units that
are activated during any one oscillation is given by

F
p(
F) exp (12.4) NkT
Material
Copyrighted where N is Avogadro’s number (6.02 1023), and Nk
is equal to R, the universal gas constant (8.3144 J K1
mol1). The frequency of activation then is
kT
F
exp (12.5) h NkT
In the absence of directional potentials, energy bar-riers
are crossed with equal frequency in all directions,
and no consequences of thermal activations are ob-served
unless the temperature is sufficiently high that
softening, melting, or evaporation occurs. If, however,
a directed potential, such as a shear stress, is applied,
then the barrier heights become distorted as shown by
curve B in Fig. 12.21. If ƒ represents the force acting
on a flow unit, then the barrier height is reduced by an
amount (ƒ/2) in the direction of the force and in-creased
by a like amount in the opposite direction,
where represents the distance between successive
equilibrium positions.3 Minimums in the energy curve
are displaced a distance from their original positions,
representing an elastic distortion of the material struc-ture.
The reduced barrier height in the direction of force
ƒ increases the activation frequency in that direction to
kT
F/N ƒ/2
→ exp (12.6) h kT
and in the opposite direction, the increased barrier
height decreases the activation frequency to
kT
F/N ƒ/2
← exp (12.7) h kT
3Work (ƒ / 2) done by the force ƒ as the flow unit drops from the
peak of the energy barrier to a new equilibrium position is assumed
to be given up as heat.
The net frequency of activation in the direction of
the force then becomes
kT
F (→) (←) 2 exp ƒ
sinh h RT 2kT
(12.8)
Strain Rate Equation
At any instant, some of the activated flow units may
successfully cross the barrier; others may fall back into
their original positions. For each unit that is successful
in crossing the barrier, there will be a displacement .
The component of in a given direction times the
number of successful jumps per unit time gives the rate
of movement per unit time. If this rate of movement
is expressed on a per unit length basis, then the strain
rate ˙ is obtained.
Let X F (proportion of successful barrier cross-ings
and ) such that
˙ X[( →) ( ←)] (12.9)
Then from Eq. (12.8)
kT
F ƒ
˙ 2X exp sinh (12.10) h RT 2kT
The parameter X may be both time and structure de-pendent.
If (ƒ/2kT) 1, then sinh(ƒ/2kT) (ƒ/2kT), and
the rate is directly proportional to ƒ. This is the case
for ordinary Newtonian fluid flow and diffusion where
1
˙ (12.11)

where ˙ is the shear strain rate, is dynamic viscosity,
and is shear stress.
For most solid deformation problems, however,
(ƒ/2kT) 1 (Mitchell et al., 1968), so
ƒ 1 ƒ
sinh exp (12.12) 2kT 2 2kT
and
kT
F ƒN
˙ X expexp(12.13) h RT 2RT
Equation (12.13) applies except for very small stress
intensities, where the exponential approximation of the
hyperbolic sine is not justified. Equations (12.10) and

BONDING, EFFECTIVE STRESSES, AND STRENGTH 481
(12.13) or comparable forms have been used to obtain
dashpot coefficients for rheological models, to obtain
functional forms for the influences of different factors
on strength and deformation rate, and to study defor-mation
rates in soils. For example, Kuhn and Mitchell
(1993) used this form as part of the particle contact
law in discrete element modeling as described in the
previous section. Puzrin and Houlsby (2003) used it as
an internal function of a thermomechanical-based
model and derived a rate-dependent constitutive model
for soils.
Soil Deformation as a Rate Process
Although there does not yet appear to be a rigorous
proof of the correctness of the detailed statistical me-chanics
formulation of rate process theory, even for
simple chemical reactions, the real behavior of many
systems has been substantially in accord with it. Dif-ferent
parts of Eq. (12.13) have been tested separately
(Mitchell et al., 1968). It was found that the tempera-ture
dependence of creep rate and the stress depend-ence
of the experimental activation energy [Eq.
(12.14)] were in accord with predictions. These results
do not prove the correctness of the theory; they do,
however, support the concept that soil deformation is
a thermally activated process.
Arrhenius Equation
Equation (12.13) may be written
kT E
˙ X exp (12.14) h RT
where
ƒN
E
F (12.15)
2
is termed the experimental activation energy. For all
conditions constant except T, and assuming that
X(kT/h) constant A,
E
˙ A exp (12.16) RT
Equation (12.16) is the same as the well-known em-pirical
equation proposed by Arrhenius around 1900
to describe the temperature dependence of chemical
reaction rates. It has been found suitable also for
characterization of the temperature dependence of
processes such as creep, stress relaxation, secondary
compression, thixotropic strength gain, diffusion, and
fluid flow.
12.5 BONDING, EFFECTIVE STRESSES, AND
STRENGTH
Using rate process theory, the results of time-dependent
stress–deformation measurements in soils
can be used to obtain fundamental information about
soil structure, interparticle bonding, and the mecha-nisms
controlling strength and deformation behavior.
Deformation Parameters from Creep Test Data
If the shear stress on a material is and it is distributed
uniformly among S flow units per unit area, then

ƒ (12.17)
S
Displacement of a flow unit requires that interatomic
or intermolecular forces be overcome so that it can be
moved. Let it be assumed that the number of flow units
and the number of interparticle bonds are equal.
If D represents the deviator stress under triaxial
stress conditions, the value of ƒ on the plane of max-imum
shear stress is
D
ƒ (12.18)
2S
so Eq. (12.13) becomes
kT
F D
˙ X expexp(12.19) h RT 4SkT
This equation describes creep as a steady-state
process. Soils do not creep at constant rate, however,
because of continued structural changes during de-formation
as described in Section 12.3, except for
the special case of large deformations after mobiliza-tion
of full strength. Thus, care must be taken in ap-plication
of Eq. (12.19) to ensure that comparisons of
creep rates and evaluations of the influences of differ-ent
factors are made under conditions of equal struc-ture.
The time dependency of creep rate and the
possible time dependencies of the parameters in Eq.
(12.19) are considered in Section 12.8.
Determination of Activation Energy From Eq.
(12.14)
ln(˙/T) E
(12.20)
(1/T) R
provided strain rates are considered under conditions
of unchanged soil structure. Thus, the value of E can
be determined from the slope of a plot of ln(˙ /T) ver-sus
(1/T). Procedures for evaluation of strain rates for

soils at different temperatures but at the same structure
are given by Mitchell et al. (1968, 1969).
Determination of Number of Bonds For stresses
large enough to justify approximating the hyperbolic
sine function by a simple exponential in the creep rate
equation and small enough to avoid tertiary creep, the
logarithm of strain rate varies directly with the deviator
stress. For this case, Eq. (12.19) can be written
˙ K(t) exp(D) (12.21)
Material
4A procedure for evaluation of from the results of a test at a
succession of stress levels on a single sample is given by Mitchell
Copyrighted et al. (1969). where
kT
F
K(t) X exp (12.22) h RT

(12.23)
4SkT
Parameter is a constant for a given value of ef-fective
consolidation pressure and is given by the slope
of the relationship between log strain rate and stress.
It is evaluated using strain rates at the same time after
the start of creep tests at several stress intensities. With
known, /S is calculated as a measure of the number
of interparticle bonds.4
Activation Energies for Soil Creep
Activation energies for the creep of several soils and
other materials are given in Table 12.1. The free energy
of activation for creep of soils is in the range of about
80 to 180 kJ/mol. Four features of the values for soils
in Table 12.1 are significant:
1. The activation energies are relatively large, much
2. Variations in water content (including complete
3. The values for sand and clay are about the same.
4. Clays in suspension with insufficient solids to
Table 12.1 Activation Energies for Creep of Several Materials
Material
higher than for viscous flow of water.
drying), adsorbed cation type, consolidation pres-sure,
void ratio, and pore fluid have no significant
effect on the required activation energy.
form a continuous structure deform with an ac-tivation
Activation Energy
energy equal to that of water.
(kJ/ mol)a Reference
1. Remolded illite, saturated, water contents of
30 to 43%
105–165 Mitchell, et al. (1969)
2. Dried illite: samples air-dried from
saturation, then evacuated
155 Mitchell, et al. (1969)
3. San Francisco Bay mud, undisturbed 105–135 Mitchell, et al. (1969)
4. Dry Sacramento River sand 105 Mitchell, et al. (1969)
5. Water 16–21 Glasstone, et al. (1941)
6. Plastics 30–60 Ree and Eyring (1958)
7. Montmorillonite–water paste, dilute 84–109 Ripple and Day (1966)
8. Soil asphalt 113 Abdel-Hady and Herrin (1966)
9. Lake clay, undisturbed and remolded 96–113 Christensen and Wu (1964)
10. Osaka clay, overconsolidated 120–134 Murayama and Shibata (1961)
11. Concrete 226 Polivka and Best (1960)
12. Metals 210 Finnie and Heller (1959)
13. Frozen soils 393 Andersland and Akili (1967)
14. Sault Ste. Marie clay, suspensions,
discontinuous structures
Same as
water
Andersland and Douglas (1970)
15. Sault Ste. Marie clay, Li, Na, K forms,
in H2O and CCl4, consolidated
117 Andersland and Douglas (1970)
aThe first four values are experimental activation energies, E. Whether the remainder are values of
F or E is not
always clear in the references cited.

Number of Interparticle Bonds
Evaluation of S requires knowledge of , the separation
distance between successive equilibrium positions in
the interparticle contact structure. A value of 0.28 nm
(2.8 A˚
) has been assumed because it is the same as the
distance separating atomic valleys in the surface of a
silicate mineral. It is hypothesized that deformation in-volves
the displacement of oxygen atoms along con-tacting
particle surfaces, as well as periodic rupture of
bonds at interparticle contacts. Figure 12.22 shows this
interpretation for schematically. If the above as-sumption
for is incorrect, calculated values of S will
still be in the same correct relative proportion as long
as remains constant during deformation.
Normally Consolidated Clay Results of creep tests
at different stress intensities for different consolidation
pressures enable computation of S as a function of con-solidation
pressure. Values obtained for undisturbed
San Francisco Bay mud are shown in Fig. 12.23. The
open point is for remolded bay mud. An undisturbed
specimen was consolidated to 400 kPa, rebounded to
Figure 12.22 Interpretation of in terms of silicate mineral
surface structure.
50 kPa, and then remolded at constant water content.
The effective consolidation pressure dropped to 25 kPa
as a result of the remolding. The drop in effective
stress was accompanied by a corresponding decrease
in the number of interparticle bonds. Tests on remolded
illite gave comparable results. A continuous inverse re-lationship
between the number of bonds and water
content over a range of water contents from more than
40 percent to air-dried and vacuum-desiccated clay is
shown in Fig. 12.24. The dried material had a water
content of 1 percent on the usual oven-dried basis. The
very large number of bonds developed by drying is
responsible for the high dry strength of clay.
Overconsolidated Clay Samples of undisturbed
San Francisco Bay mud were prepared to overcon-solidation
ratios of 1, 2, 4, and 8 following the stress
paths shown in the upper part of Fig. 12.25. The sam-ple
represented by the triangular data point was re-molded
after consolidation and unloading to point d,
where it had a water content of 52.3 percent. The un-drained
compressive strength as a function of consol-idation
pressure is shown in the middle section of Fig.
12.25, and the number of bonds, deduced from the
creep tests, is shown in the lower part of the figure.
The effect of overconsolidation is to increase the num-ber
of interparticle bonds over the values for normally
consolidated clay. Some of the bonds formed during
consolidation are retained after removal of much of the
consolidation pressure.
Values of compressive strength and numbers of
bonds from Fig. 12.25 are replotted versus each other
in Fig. 12.26. The resulting relationship suggests that
strength depends only on the number of bonds and is
independent of whether the clay is undisturbed, re-molded,
normally consolidated, or overconsolidated.
Dry Sand Creep tests on oven-dried sand yielded
results of the same type as obtained for clay, as shown
in Fig. 12.27, suggesting that the strength-generating
and creep-controlling mechanisms may be similar for
both types of material.
Composite Strength-Bonding Relationship Values
of S and strength for many soils are combined in Fig.
12.28. The same proportionality exists for all the ma-terials,
which may seem surprising, but which in reality
should be expected, as discussed further later.
Significance of Activation Energy and Bond Number
Values
The following aspects of activation energies and num-bers
of interparticle bonds are important in the under-standing
of the deformation and strength behavior of
uncemented soils.

Material
Figure 12.23 Number of interparticle bonds as a function of consolidation pressure for
normally consolidated San Francisco Bay mud.
Copyrighted Figure 12.24 Number of bonds as a function of water con-tent
for illite.
1. The values of activation energy for deformation
of soils are high in comparison with other ma-terials
and indicate breaking of strong bonds.
2. Similar creep behavior for wet and dry clay and
for wet and dry sand indicates that deformation
is not controlled by viscous flow of water.
3. Comparable values of activation energy for wet
and dry soil indicate that water is not respon-sible
for bonding.
4. Comparable values of activation energy for clay
and sand support the concept that interparticle
bond strengths are the same for both types of
material. This is supported also by the unique-ness
of the strength versus number of bonds re-lationship
for all soils.
5. The activation energy and presumably, there-fore,
the bonding type are independent of con-solidation
pressure, void ratio, and water
content.
6. The number of bonds is directly proportional to
effective consolidation pressure for normally
consolidated clays.
7. Overconsolidation leads to more bonds than in
normally consolidated clay at the same effective
consolidation pressure.
8. Strength depends only on the number of bonds.
9. Remolding at constant water content causes a
decrease in the effective consolidation pressure,
which means also a decrease in the number of
bonds.
10. There are about 100 times as many bonds in dry
clay as in wet clay.

Material
Copyrighted Figure 12.25 Consolidation pressure, strength, and bond numbers for San Francisco Bay
mud.
Although it may be possible to explain these results
in more than one way, the following interpretation ac-counts
well for them. The energy
F activates a mole
of flow units. The movement of each flow unit may
involve rupture of single bonds or the simultaneous
rupture of several bonds. Shear of dilute montmoril-lonite–
water pastes involves breaking single bonds
(Ripple and Day, 1966). For viscous flow of water, the
activation energy is approximately that for a single hy-drogen
bond rupture per flow unit displacement, even
though each water molecule may form simultaneously
up to four hydrogen bonds with its neighbors. If the
single-bond interpretation is also correct for soils, then
consistency in Eq. (12.10) requires that shear force ƒ
pertain to the force per bond. On this basis, parameter
S indicates the number of single bonds per unit area.
In the event activation of a flow unit requires simul-taneous
rupture of n bonds, then S represents 1/nth of
the total bonds in the system.
That the activation energy for deformation of soil is
well into the chemical reaction range (40 to 400 kJ/
mol) does not prove that bonding is of the primary

(2) it is the same for wet and dry soils, and (3) it is
essentially the same for different adsorbed cations and Copyrighted Material
Figure 12.26 Strength as a function of number of bonds for
San Francisco Bay mud.
Figure 12.28 Composite relationship between shear strength
and number of interparticle bonds (from Matsui and Ito,
1977). Reprinted with permission from The Japanese Society
of SMFE.
valence type because simultaneous rupture of several
weaker bonds could yield values of the magnitude ob-served.
On the other hand, the facts that (1) the acti-vation
energy is much greater than for flow of water,
Figure 12.27 Strength as a function of number of bonds for dry Antioch River sand.

pore fluids (Andersland and Douglas, 1970) suggest
that bonding is through solid interparticle contacts.
Physical evidence for the existence of solid-to-solid
contact between clay particles has been obtained in the
form of photomicrographs of particle surfaces that
were scratched during shear (Matsui et al., 1977, 1980)
and acoustic emissions (Koerner et al., 1977).
Activation energy values of 125 to 190 kJ/mol are
of the same order as those for solid-state diffusion of
oxygen in silicate minerals. This supports the concept
that creep movements of individual particles could re-sult
from slow diffusion of oxygen ions in and around
interparticle contacts. The important minerals in both
sand and clay are silicates, and their surface layers
consist of oxygen atoms held together by silicon at-oms.
Water in some form is adsorbed onto these sur-faces.
The water structure consists of oxygens held
together by hydrogen. It is not too different from that
of the silicate layer in minerals. Thus, a distinct bound-ary
between particle surface and water may not be dis-cernable.
Under these conditions, a more or less
continuous solid structure containing water molecules
that propagates through interparticle contacts can be
visualized.
An individual flow unit could be an atom, a group
of atoms or molecules, or a particle. The preceding
arguments are based on the interpretation that individ-ual
atoms are the flow units. This is consistent with
both the relative and actual values of S that have been
determined for different soils. Furthermore, by using a
formulation of the rate process equation that enabled
calculation of the flow unit volume from creep test
data, Andersland and Douglas (1970) obtained a value
of about 1.7 A˚
3, which is of the same order as that of
individual atoms. On the other hand, Keedwell (1984)
defined flow units between quartz sand particles as
consisting of six O2 ions and six Si4 ions and be-tween
two montmorillonite clay particles as consisting
of four H2O molecules.
If particles were the flow units, not only would it be
difficult to visualize their thermal vibrations, but then
S would relate to the number of interparticle contacts.
It is then difficult to conceive how simply drying a clay
could give a 100-fold increase in the number of inter-particle
contacts, as would have to be the case accord-ing
to Fig. 12.27. A more plausible interpretation is
that drying, while causing some increase in the number
of interparticle contacts during shrinkage, causes
mainly an increase in the number of bonds per contact
because of increased effective stress.
At any value of effective stress, the value of S is
about the same for both sand and clay. The number of
interparticle contacts should be vastly different; how-ever,
for equal numbers of contacts per particle, the
number per unit volume should vary inversely with the
cube of particle size. Thus, the number of clay particles
of 1-m particle size should be some nine orders of
magnitude greater than for a sand of 1-mm average
particle size. Each contact between sand particles
would involve many bonds; in clay, the much greater
number of contacts would mean fewer bonds per par-ticle.
The contact area required to develop bonds in the
numbers indicted in Figs. 12.23 to 12.27 is very small.
For example, for a compressive strength of 3 kg/cm2
( 300 kPa) there are 8 1010 bonds/cm2 of shear
surface. Oxygen atoms on the surface of a silicate min-eral
have a diameter of 0.28 nm. Allowing an area 0.30
nm on a side for each oxygen gives 0.09 nm2, or 9
1016 cm2, per bonded oxygen for a total area of 9
1016 8 1010 7.2 105 cm2/cm2 of soil cross
section.
Hypothesis for Bonding, Effective Stress, and
Strength
Normal effective stresses and shear stresses can be
transmitted only at interparticle contacts in most soils.5
The predominant effects of the long-range physico-chemical
forces of interaction are to control the initial
soil fabric and to alter the forces transmitted at contact
points from what they would be due to applied stresses
alone.
Interparticle contacts are effectively solid, and it is
likely that both adsorbed water and cations in the con-tact
zone participate in the structure. An interparticle
contact may contain many bonds that may be strong,
approaching the primary valence type. The number of
bonds at any contact depends on the compressive force
transmitted at the contact, and the Terzaghi–Bowden
and Tabor adhesion theory of friction presented in Sec-tion
11.4, can account for strength. The macroscopic
strength is directly proportional to the number of
bonds.
For normally consolidated soils the number of bonds
is directly proportional to the effective stress. As a re-sult
of particle rearrangements and contacts formed
during virgin compression, an overconsolidated soil at
a given effective stress has a greater number of bonds
and higher strength than a normally consolidated soil.
This effect is more pronounced in clays than in sands
because the larger and bulky sand grains tend to re-
5 Pure sodium montmorillonite may be an exception since a part of
the normal stress can be carried by physicochemical forces of inter-action.
The true effective stress may be less than the apparent effec-tive
stress by R A as discussed in Chapter 7.

cover their original shapes when unloaded, thus rup-turing
most of the bonds in excess of those needed to
resist the lower stress. The strength of the interparticle
contacts can vary over a wide range, depending on the
number of bonds per contact.
The unique relationship between strength and num-ber
of bonds for all soils, as indicted by Fig. 12.28,
reflects the fact that the minerals comprising most soils
are silicates, and they all have similar surface struc-tures.
In the absence of chemical cementation, interparticle
bonds may form in response to interparticle contact
forces generated by either applied stresses, physico-chemical
forces of interaction, or both. Any bonds ex-isting
in the absence of applied effective stress, that is,
when 0, are responsible for true cohesion. There
should be no difference between friction and cohesion
in terms of the shearing process. Complete failure in
shear involves simultaneous rupture or slipping of all
bonds along the shear plane.
12.6 SHEARING RESISTANCE AS A RATE
PROCESS
Deformation at large strain can approach a steady-state
condition where there is little further structural change
with time (such as at critical state). In this case, Eq.
(12.19) can be used to describe the shearing resistance
as a function of strain rate and temperature. If the max-imum
shear stress is substituted for the deviator stress
D, then
1 3 (12.24)
2
and
kT ˙ X exp
F exp
(12.25) h RT 2SkT
Taking logarithms of both sides of Eq. (12.25) gives
kT
F
ln ˙ lnX (12.26) h RT 2SkT
By assuming X(kT/h) is a constant equal to B
(Mitchell, 1964), Eq. (12.26) can be rearranged to give
2S 2SkT
˙
F ln(12.27) N B
From the relationships in Section 12.5, the following
relationship between bonds per unit area and effective
stress is suggested.
S a b (12.28) ƒ
where a and b are constants and ƒ is the effective
normal stress on the shear plane. Thus, Eq. (12.27)
becomes
2a
F 2akT ˙ 2b
F 2bkT
˙ ln ln N B N B
ƒ (12.29)
Equation (12.29) is of the same form as the Cou-lomb
equation for strength:
c tan (12.30) ƒ
By analogy,
2a
F 2akT ˙
c ln (12.31)
N B
2b
F 2bkT ˙
tan ln (12.32)
N B
These equations state that both cohesion and friction
depend on the number of bonds times the bond
strength, as reflected by the activation energy, and that
the values of c and should depend on the rate of
deformation and the temperature.
Strain Rate Effects
All other factors being equal, the shearing resistance
should increase linearly with the logarithm of the rate
of strain. This is shown to be the case in Fig. 12.9,
which contains data for 26 clays. Additional data for
several clays are shown in Fig. 12.29, where shearing
resistance as a function of the speed of vane rotation
in a vane shear test is plotted. Analysis of the relation-ship
between shearing stress and angular rate of vane
rotation shows that
/
log decreases with an
increase in water content. This follows directly from
Eq. (12.29) because
d 2akT 2bkT 2kT
(a b) d ln(˙ /B) ƒ ƒ
(12.33)
that is, d /d ln (˙ /B) is proportional to the number of
bonds, which decreases with increasing water content.

CREEP AND STRESS RELAXATION 489
Material
Copyrighted Figure 12.29 Effect of rate of shear on shearing resistance of remolded clays as determined
by the laboratory vane apparatus (prepared from the data of Karlsson, 1963).
This interpretation of the data in Figs. 12.9 and 12.29
assumes that the effective stress was unaffected by
changes in the strain rate, which may not necessarily
be true in all cases.
Effect of Temperature
Assumptions of reasonable values for parameters show
that the term (˙ /B) is less than one (Mitchell, 1964).
Thus the quantity ln(˙ /B) in Eq. (12.29) is negative,
and an increase in temperature should give a decrease
in strength, all other factors being constant. That this
is the case is demonstrated by Fig. 12.30, which shows
deviator stress as a function of temperature for samples
of San Francisco Bay mud compared under conditions
of equal mean effective stress and structure. Other ex-amples
of the influence of temperature on strength are
shown in Figs. 11.6 and 11.133.
12.7 CREEP AND STRESS RELAXATION
Although the designation of a part of the strain versus
time relationship as steady state or secondary creep
may be convenient for some analysis purposes, a true
steady state can exist only for conditions of constant
structure and stress. Such a set of conditions is likely
only for a fully destructured soil, and a fully destruc-tured
state is likely to persist only during deformation
at a constant rate, that is, at failure. This state is often
called ‘‘steady state,’’ in which the soil is deforming
continuously at constant volume under constant shear
and confining stresses (Castro, 1975; Castro and
Poulos, 1977).
Otherwise, bond making and bond breaking occur
at different rates as a result of different internal time-and
strain-dependent phenomena, which might include
thixotropic hardening, viscous flows of water and ad-

Figure 12.30 Influence of temperature on the shearing re-sistance
of San Francisco Bay mud. Comparison is for sam-ples
at equal mean effective stress and at the same structure.
Figure 12.31 Variation of creep strain rate with deviator
stress for drained creep of London clay (data from Bishop,
1966).
sorbed films, chemical, and biological transformations,
and the like. Furthermore, distortions of the soil struc-ture
and relative movements between particles cause
changes in the ratio of tangential to normal forces at
interparticle contacts that may be responsible for large
changes in creep rate. Because of these time depend-encies
some of the parameters in Eq. (12.19) may be
time dependent. For example, Feda (1989) accounted
for the time dependency of creep rate by taking
changes in the number of structural bonds into account.
Therefore, application of Eq. (12.19) for the determi-nation
of the bonding and effective stress relationships
discussed in Section 12.5 required comparison of creep
rates under conditions of comparable time and struc-ture.
The influence of creep stress magnitude on the creep
rate at a given time after the application of the stress
to identical samples of a soil was shown in Fig. 12.3.
At low stresses the creep rates are small and of little
practical importance. The curve shape is compatible
with the hyperbolic sine function predicted by rate
process theory, as given by Eq. (12.10). In the mid-range
of stresses, a nearly linear relationship is found
between logarithm of strain rate and stress, also as pre-dicted
by Eq. (12.10) for the case where the argument
of the hyperbolic sine is greater than 1. At stresses
approaching the strength of the material, the strain rate
becomes very large and signals the onset of failure.
Other examples of the relationships between logarithm
of strain rate and creep stress corresponding to differ-ent
times after the application of the creep stress are
given in Fig. 12.31 for drained tests on London clay
and Fig. 12.32 for undrained tests on undisturbed San
Francisco Bay mud. Only values for the midrange of
stresses are shown in Figs. 12.31 and 12.32.
Effect of Composition
In general, the higher the clay content and the more
active the clay, the more important are stress relaxation
and creep, as illustrated by Figs. 4.22 and 4.23, where
creep rates, approximated by steady-state values, are
related to clay type, clay content, and plasticity. Time-dependent
deformations are more important at high
water contents than at low. Deviatoric creep and sec-ondary
compression are greater in normally consoli-dated
than overconsolidated soils.
Although the magnitude of creep strains and strain
rates may be small in sand or dry soil, the form of the

is a special case of volumetric creep.
Deviatoric creep is often accompanied by volumetric
creep. The ratio of volumetric to deviatoric creep fol-
Figure 12.32 Variation of creep strain rate with deviator
stress for undrained creep of normally consolidated San Fran-cisco
Bay mud.
1
0.8
0.6
0.4
0.2
Drained Triaxial Test
Confining Pressure σ3 = 414 kPa
Deviator Stress from 344 to 377 kPa
A B
C
D
E
2820 min
1450 min
90 min
20 min
2 min
0
0.5 1.0 1.5 2.0 2.5 3.0 3.5
Deviator strain (%)
1.0
0.8
0.6
0.4
0.2
0
(a)
0 1 2 3 4
Strain increment ratio dεv/dεs during creep
(b)
Volumetric Strain (%)
Stress ratio q/p
Figure 12.33 Dilatancy relationship obtained from drained
creep tests on kaolinite: (a) development of volumetric and
deviatoric strains with time and (b) effect of stress ratio on
strain increment ratio d/ds (from Walker, 1969).
behavior conforms with the patterns described and il-lustrated
above. This is to be expected, as the basic
creep mechanism is the same in all inorganic soils.6
Water may ‘‘lubricate’’ the particles and possibly in-crease
the creep rate even though the basic mechanism
of creep is the same for dry and wet materials (Losert
et al., 2000). Takei et al. (2001) showed that the de-velopment
of creep strains due to time-dependent
breakage of talc specimens increased more for satu-rated
specimens than dry ones. However, a negligible
effect of water on creep rate was reported by Ahn-Dan
et al. (2001) who performed creep tests on unsaturated
and saturated crushed gravel and by Leung et al.
(1996) who performed one-dimensional compression
6Volumetric creep and secondary compression of organic soils, peat,
and municipal waste fills can develop also as a result of decompo-sition
of organic matter.
creep tests on sands. The conflicting evidence may be
due to the presence or absence of impurities that may
lubricate or cement the soil in the presence of water
(Human, 1992; Bowman, 2003).
Volume Change and Pore Pressures
Due to the known coupling effects between shearing
and volumetric plastic deformations in soils, an in-crease
in either mean pressure or deviator stress can
generate both types of deformations. Creep behavior is
no exception. Time-dependent shear deformations are
usually referred to as deviatoric creep or shear creep.
Time-dependent deformations under constant stress re-ferred
to as volumetric creep. Secondary compression

lows a plastic dilatancy rule. Walker (1969) inves-tigated
the time-dependent change of these two com-ponents
from incremental drained triaxial creep tests
on normally consolidated kaolinite. The increase in
shear strains with increase in volumetric strains at dif-ferent
times is shown in Fig. 12.33a. At the beginning
of the triaxial test, the deviator stress was instantane-ously
increased from 344 to 377 kPa and kept constant.
pressure increase with time for consolidated undrained
creep tests on illite at several stress intensities is shown
in Fig. 12.34. Figure 12.35 shows a slow decrease in
After an immediate increase in shear strains at constant
volume (AB in Fig. 12.33a), section BD corresponds
to primary consolidation that is controlled by the dis-sipation
of pore pressures. After point D, creep oc-curred,
and the ratio of volumetric to deviatoric strains
was independent of time. This ratio decreased with in-creasing
stress ratio as shown in Fig. 12.33b. This ob-servation
led to the time-dependent flow rule, which is
similar to the dilatancy rule described in Section 11.20.
Sand deforms with time in a similar manner. Under
progressive deviatoric creep, the volumetric creep re-sponse
is highly dependent on density, the stress level,
and the stress path before creep. The rate of both vol-umetric
and deviatoric creep increases with confining
pressure, particularly after particle crushing becomes
important at high stresses (Yamamuro and Lade, 1993).
For dense sand under high deviator stress, dilative
creep is observed (Murayama et al., 1984; Mejia et al.,
1988). The volumetric response of dense sand and
gravel with time is a highly complex function of stress
history, with some samples contracting or dilating
(Lade and Liu, 1998; Ahn-Dan et al., 2001). Some
dense sand samples contract initially but then dilate
with time (Bowman and Soga, 2003). Further discus-sion
of the creep behavior of sands in relation to me-chanical
aging phenomena is given in Section 12.11.
The fundamental process of creep strain develop-ment
is therefore similar to that of time-independent
plastic strains, and the same framework of soil plastic-ity
can possibly be used. It can be argued whether
it is necessary to separate the deformation into
time-dependent and independent components. Rate-independent
behavior can be considered as the limiting
case of rate-dependent behavior at a very slow rate of
loading.
Volumetric-deviatoric creep coupling implies that
rapid application of a stress or a strain invariably re-sults
in rapid change of pore water pressures in a sat-urated
soil under undrained conditions. For a constant
total minor principal stress, the magnitude of the pore
pressure change depends on the volume change ten-dencies
of the soil when subjected to shear distortions.
These tendencies are, in turn, controlled by the void
ratio, structure, and effective stress, and can be quan-tified
in terms of the pore pressure parameter A as dis-cussed
in Chapters 8 and 10. An example showing pore
Figure 12.34 Pore pressure development with time during undrained creep of illite.

Material
Figure 12.35 Normalized pore pressure vs. time relationships during creep of kaolinite.
Copyrighted pore pressure during the sustained loading of kaolinite.
Similar behavior was demonstrated in the measured
stress paths of undrained creep test on San Francisco
Bay mud (Arulanandan et al., 1971). As shown in Fig.
12.36, the effective stress states shifted toward the fail-ure
line. At higher stress levels, the specimens even-tually
underwent creep rupture. However, soil strength
in terms of effective stresses does not change unless
there are chemical, biological, or mineralogical
changes during the creep period. This is illustrated by
the stress paths shown schematically in Fig. 12.37,
where the pre- and postcreep strengths fall on the same
failure envelope.
Effects of Temperature
An increase in temperature decreases effective stress,
increases pore pressure, and weakens the soil structure.
Creep rates ordinarily increase and the relaxation
stresses corresponding to specific values of strain de-crease
at higher temperature. These effects are illus-trated
by the data shown in Figs. 12.38 and 12.39.
Effects of Test Type, Stress System, and Stress Path
Most measurements of time-dependent deformation
and stress relaxation in soils have been done on sam-ples
consolidated isotropically and tested in triaxial
compression or by measurement of secondary com-pression
in oedometer tests. However, most soils in
nature have been subjected to an anisotropic stress his-tory,
and deformation conditions conform more to
plane strain than triaxial compression in many cases.
Some investigations of these factors have been made.
Although the general form of the stress–strain–time
and stress–strain rate–time relationships are similar to
those shown above for triaxial loading conditions, the
actual values may differ considerably.
For example, undisturbed Haney clay, a gray silty
clay from British Columbia, with a sensitivity in the
range of 6 to 10, was tested both in triaxial compres-sion
and plane strain (Campanella and Vaid, 1974).
Samples were normally consolidated both isotropically
and under K0 conditions to the same vertical effective
stress. Samples consolidated isotropically were tested
in triaxial compression. Coefficient K0 consolidation
was used for both K0 triaxial and plane strain tests.
The results shown in Fig. 12.40 indicate that the pre-creep
stress history had a significant effect on the
deformations. The plane strain and K0 consolidated
triaxial samples gave about the same creep behavior
under the same deviatoric stress, which suggests that
preventing strain in one horizontal direction and/or the
intermediate principal stress were not factors of major
importance for this soil under the test conditions used.
Interaction Between Consolidation and Creep
Experimental evidence suggests that creep occurs dur-ing
primary consolidation (Leroueil et al., 1985; Imai
and Tang, 1992). Following the initial large change
following load application, the pore pressure may ei-ther
dissipate, with accompanying volume change if
drainage is allowed, or change slowly during creep or
stress relaxation, if drainage is prevented. The devel-opment
of complete effective stress and void ratio
equilibrium may take a long time. One illustration of

Material
0 10 20 30 40 50 60 70 80
Mean Pressure p (kPa)
Copyrighted 0 80 160 240 320 400 480 560 640
Figure 12.36 Measured stress paths of undrained creep tests of San Francisco Bay mud.
Initial confining pressure: (a) 49 kPa and (b) 392 kPa (from Arulanandan et al., 1971).
50
40
30
20
10
400
320
160
80
0
Normal Undrained Triaxial
Compression Test: Effective
Stress Path
Total Stress Path of
Triaxial Compression Test
1
3
Possible Critical
State Line
Effective Stress State
After 1,000 min Creep
Effective Stress Path
of Undrained Creep
0
Normal Undrained Triaxial
Compression Test: Effective
Stress Path
Total Stress Path of
Triaxial Compression Test
1
3
Possible Critical
State Line
Effective Stress State
Effective Stress Path
of Undrained Creep
240
(a)
(b)
Deviator Stress q (kPa) Deviator Stress q (kPa)
Mean Pressure p (kPa)

Figure 12.37 Effects of undrained creep on the Material
strength of normally consolidated clay.
Copyrighted Figure 12.38 Creep curves for Osaka clay tested at different temperatures—undrained tri-axial
compression (Murayama, 1969).
this is given by Fig. 10.5, where it is shown that the
relationship between void ratio and effective stress is
dependent on the time for compression under any
given stress. Another is given by Fig. 12.41, which
shows pore pressures during undrained creep of San
Francisco Bay mud. In each sample, consolidation un-der
an effective confining pressure of 100 kPa was al-lowed
for 1800 min prior to the cessation of drainage
and the start of a creep test. The consolidation period
was greater than that required for 100 percent primary
consolidation. The curve marked 0 percent stress level
refers to a specimen maintained undrained but not sub-ject
to a deviator stress. This curve indicates that each
of the other tests was influenced by a pore pressure
that contained a contribution from the prior consoli-dation
history.
The magnitude and rate of pore pressure develop-ment
if drainage is prevented following primary con-solidation
depend on the time allowed for secondary
compression prior to the prevention of further drain-age.
This is illustrated by the data in Fig. 12.42, which
show pore pressure as a function of time for samples
that have undergone different amounts of secondary
compression.
In summary, creep deformation depends on the ef-fective
stress path followed and any changes in stress

Figure 12.39 Influence of temperature on the Material
initial and final stresses in stress relaxation
tests on Osaka clay—undrained triaxial compression (Murayama, 1969).
Copyrighted Figure 12.40 Creep curves for isotropically and K0-consolidated samples of undisturbed
Haney clay tested in triaxial and plane strain compression (from Campanella and Vaid, 1974).
Reproduced with permission from the National Research Council of Canada.

RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS 497
Figure 12.41 Pore pressure development during undrained creep of San Francisco Bay mud
after consolidation at 100 kPa for 1800 min Material
(from Holzer et al., 1973). Reproduced with
permission from the National Research Council of Canada.
Copyrighted Figure 12.42 Pore pressure development under undrained conditions following different
periods of secondary compression (from Holzer et al., 1973). Reproduced with permission
from the National Research Council of Canada.
with time. Furthermore, time-dependent volumetric re-sponse
is governed both by the rate of volumetric creep
and by the rate of consolidation. The latter is a com-plex
function of drainage conditions and material prop-erties,
especially the permeability and compressibility.
Because the effective stress path is controlled by the
rate of loading and drainage conditions, the separation
of consolidation and creep deformations can be diffi-cult
in the early stage of time-dependent deformation
as given by section BD in Fig. 12.33a. In some cases,
a fully coupled analysis of soil–pore fluid interaction
with an appropriate time-dependent constitutive model
is necessary to reconcile the time-dependent deforma-tions
observed in the field and laboratory.
12.8 RATE EFFECTS ON STRESS–STRAIN
RELATIONSHIPS
An increase in strain rate during soil compression is
manifested by increased stiffness, as was noted in Sec-tion
12.3. In essence, the state of the soil jumps to the
stress–strain curve that corresponds to the new strain
rate. Commonly, this rate-dependent stress–strain

5% /h
0.5% /h
0.05% /h
Belfast Clay 4 m
σ1c = σv0
16%/h
1%/h
0.25%/h
Winnipeg Clay 11.5 m
σ1c σv0
CAU Triaxial Compression Tests
Relaxation Tests (R)
0
0.6
0.5
0.4
0.3
0.2
0.1
0
4 8 12 16 20
Axial Strain
(σ1 – σ3)/2σ1c
R
R
R
Effective Stress σv (kPa)
εv2
.
SP1 test
SP2 test
εv1
.
εv1 . = 2.70 10–6 s–1
εv2 = 1.05 10–7 s–1
εv2 .
εv1
.
.
. εv1
εv1
εv1
.
εv1
.
.
.
εv3
.
εv2
.
εv2
ε.v3 = 1.34 10–5 s–1
0
5
10
15
20
25
30
0 50 100 150 200 250
Strain εv (%)
Figure 12.43 Rate-dependent stress–strain relations of
clays: (a) undrained triaxial compression tests of Belfast
and Winnipeg clay (Graham et al., 1983a) and (b) one-dimensional
compression tests of Batiscan clay (Leroueil et
al., 1985).
curve, noted by S˘
uklje (1957), is the same as if the
soil had been loaded from the beginning at the new
strain rate. This phenomenon is often observed in
clays. Examples are given in Fig. 12.43a for undrained
triaxial compression tests of Belfast and Winnipeg
clays (Graham et al., 1983a) and Fig. 12.43b for one-dimensional
compression tests of Batiscan clay (Ler-oueil
et al., 1985).
Yield and Strength Envelopes of Clays
The undrained shear strength and apparent precon-solidation
pressure of soils decrease with decreasing
strain rate or increasing duration of testing. Preconsol-idation
pressures obtained from one-dimensional con-solidation
tests and undrained shear strengths obtained
from triaxial tests are just two points on a soil’s yield
envelope in stress space. For a given metastable soil
structure, the degree of rate dependency of preconsol-idation
pressure is similar to that of undrained shear
strength (Soga and Mitchell, 1996). If the apparent pre-consolidation
pressure depends on the strain rate at
which the soil is deforming, then the same analogy can
be expanded to the assumption that the size of the en-tire
yield envelope is also strain rate dependent (Tav-enas
and Leroueil, 1977). Figure 12.44 shows a family
of strength envelopes corresponding to constant strain
rates7 obtained from drained and undrained creep tests
on stiff plastic Mascouche clay from Quebec (Leroueil
and Marques, 1996).
The effective stress failure line of soil is uniquely
defined regardless of the magnitude of the strain rate
applied in undrained compression. Figure 12.45a
shows the failure line of Haney clay (Vaid and Cam-panella,
1977). The line represents the stress conditions
at the maximum ratio of /. The data were obtained 1 3
by various undrained tests, and a unique failure line
can be observed. Figure 12.45b shows the undrained
stress paths and the critical state line of reconstituted
mixtures of sand and clay with plasticity indices rang-ing
from 10 to 30 (Nakase and Kamei, 1986). A unique
critical state line can be observed although the rates of
shearing are different. The change in undrained shear
strength with strain rate results from a difference in
generation of excess pore pressures. A decrease in
strain rate leads to larger excess pore pressures at fail-ure
due to creep deformation.
7The strain rate is defined as ˙ ˙2 ˙2, where ˙ is the volu- vs v s v
metric strain rate and ˙s is the deviator strain rate (Leroueil and Mar-ques,
1996). Whether the use of this strain rate measure is appropriate
or not remains to be investigated.

Material
Copyrighted Figure 12.44 Influence of strain rate on the yield surface of Masouche clay (from Leroueil
and Marques, 1996).
Excess Pore Pressure Generation in Normally
Consolidated Clays
Excess pore pressure development depends primarily
on the collapse of soil structure. Accordingly, strain is
the primary factor controlling pore pressure generation.
This is shown in Fig. 12.46 by the undrained stress–
strain–pore pressure response of normally consolidated
natural Olga clay (Lefebvre and LeBouef, 1987). The
natural clay specimens were normally consolidated un-der
consolidation pressures larger than the field over-burden
pressure and then sheared at different strain
rates. Although the deviator stress at any strain in-creases
with increasing strain rate, the pore pressure
versus strain curves are about the same at all strain
rates. At a given deviator stress, the pore pressure gen-eration
was larger at slower strain rates as a result of
more creep under slow loading. This is consistent with
the observation made in connection with undrained
creep of clays as discussed in Section 12.7. Strain-driven
pore pressure generation was also suggested by
Larcerda and Houston (1973) who showed that pore
pressure does not change significantly during triaxial
stress relaxation tests in which the axial strain is kept
constant.
Overconsolidated Clays
Rate dependency of undrained shear strength decreases
with increasing overconsolidation, since there is no
contraction or collapse tendency observed during creep
of heavily overconsolidated clays. Sheahan et al.
(1996) prepared reconstituted specimens of Boston
blue clay at different overconsolidation ratios and
sheared them at different strain rates in undrained con-ditions.
Figure 12.47 shows that the undrained stress
path and the strength were much more strain rate de-pendent
for lightly overconsolidated clay (OCR 1
and 2) than for more heavily overconsolidated clay
(OCR 4 and 8). The results also show that the
strength failure envelope is independent of strain rate
as discussed earlier.
The strain rate effects on stress–strain–pore pressure
response of overconsolidated structured Olga clays are
shown in Fig. 12.48 (Lefebvre and LeBouef, 1987).
The natural samples were reconsolidated to the field

Const. Stress Creep
Const. Rate of Strain Shear
Const. Rate of Loading Shear
Aged Samples
Const. Load Creep
Step Creep
Thixotropic Hardened
0.1
0
0
0.1
0.2
Material
Line
Copyrighted State Critical-Line
K0-Critical-State Line
0
0.2
0.2
0.3
0.3
0.4
0.4
0.4
0.5 0.6
0.6
0.7 0.8
M-15
710-1: ε1
710-2: ε.2
710-3: ε3
0.8
(σ1 + σ3)/2σ1c
(σ1 – σ3)/2σ1c
(a)
q/σvc
(b)
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
1.0
p/σvc
M-10
.
ε(%/min)
. .
710-1: ε1
710-2: ε2
710-3: ε3
Critical-State Line
K0-Line
Critical-State Line
0 0.2 0.4 0.6 0.8 1.0
p/σvc
q/σvc
1.0
0.8
0.6
0.4
0.2
0
–0.2
–0.4
–0.6
.
ε(%/min)
.
.
.
M15 Soil (Plasticity Index = 15) M10 Soil (Plasticity Index = 10)
Figure 12.45 Strain rate independent failure line: (a) Haney clay (from Vaid and Campa-nella,
1977) and (b) reconstituted mixtures of sand and clay (from Nakase and Kamei, 1986).
overburden pressure. The deformation is brittle, with
strain softening indicating development of localized
shear failure planes. Up to the peak stress, the response
follows what has been described previously, that is the
stress–strain response is rate dependent and the pore
pressure generation is strain dependent but indepen-dent
of rate. However, after the peak, the pore pressure
generation becomes rate dependent. This is due to local
drainage within the specimens as the deformation be-comes
localized. As the time to failure increased, there
is more opportunity for local drainage toward the di-lating
shear band and the measured pore pressure may
not represent the overall behavior of the specimens.
The difference in softening due to swelling at the fail-

Normally Consolidated Olga Clay
Undrained Triaxial Compression Tests
Initial Isotropic Confining Pressure =137 kPa
Axial Strain Rate
Material
Copyrighted 1 2 3 4 5 6 7 8
Axial Strain (%)
Axial Strain Rate
0.1 %/hr
0.5 %/hr
2.6 %/hr
12.3 %/hr
1 2 3 4 5 6 7 8
Axial Strain (%)
120
100
80
60
40
20
0
140
120
100
80
60
40
20
0
0.1 %/hr
0.5 %/hr
2.6 %/hr
12.3 %/hr
Excess Pore Pressure Δu (kPa) Deviator Stress q (kPa)
Figure 12.46 Stress–strain and pore pressure–strain curves
for normally consolidated Olga clay (from Lefebvre and
LeBouef, 1987).
Overconsolidated Olga Clay
Undrained Triaxial Compression Tests
Initial Isotropic Confining Pressure = 17.6 kPa
Axial Strain Rate
1 2 3 4 5 6 7 8
Axial Strain (%)
Axial Strain Rate
0.1 %/hr
0.5 %/hr
2.6 %/hr
12.3 %/hr
1 2 3 4 5 6 7 8
Axial Strain (%)
70
60
50
40
20
10
0
10
0
0.1 %/hr
0.5 %/hr
2.5 %/hr
12.3 %/hr
30
20
Excess Pore Pressure Δu (kPa) Deviator Stress q (kPa)
Figure 12.48 Stress–strain and pore pressure–strain curves
for overconsolidated Olga clay (from Lefebvre and LeBouef,
1987).
OCR=1
OCR=2
OCR=4
OCR=8
OCR=2
OCR=4
Effective Stress Path for Axial
Strain Rate = 0.50 %/hr
0.2 0.4 0.6 0.8
0.4
0.3
0.2
0.1
0.0
-0.1
Effective Stress State at Peak for
Axial Strain Rate = 0.051 %/hr
Axial Strain Rate = 49 %/hr
Initial K0 Consolidation State
(σa + σr)/2σvm
(σa – σr)/2σvm
OCR=1
Large
Rate
OCR=8 Effect
Negligible
Rate Effect
Figure 12.47 Rate dependency stress path and strength of
overconsolidated Boston blue clay (from Sheahan et al.,
1996).
ure plane results in apparent rate dependency at large
strains. Similar observations were made by Atkinson
and Richardson (1987) who examined local drainage
effects by measuring the angles of intersection of shear
bands with very different times of failure.
Rate Effects on Sands
Similar rate-dependent stress–strain behavior is ob-served
in sands (Lade et al., 1997), but the effects are
quite small in many cases (Tatsuoka et al., 1997; Di
Benedetto et al., 2002). An example of time depend-ency
observed for drained plane strain compression
tests of Hostun sand is shown in Fig. 12.49 (Matsushita
et al., 1999). The stress–strain curves for three differ-ent
strain rates (1.25 101, 1.25 102, and 1.25
103 %/min) are very similar, indicating very small
rate effects when the specimens are sheared at a con-stant
strain rate. On the other hand, the change from
one rate to another temporarily increases or decreases
the resistance to shear. The influence of acceleration
rather than the rate is reflected by the significant creep
deformation and stress relaxation of this rate-insensitive
material as shown the figure. This is differ-

6.0
5.5
5.0
4.5
4.0
3.5
3.0
Variation of Stress-strain curve by Constant
Strain Rate Tests at Axial Strain Rates = 0.125,
0.0125 and 0.00125 %/min. A Very Small Rate
Effect Is Observed for Continuous Loading.
CRS at
0.125%/min
Creep
Material
Copyrighted Creep
Creep
CRS at
0.125%/min
Stress Relaxation
CRS at 0.00125%/min
CRS at 0.125%/min
Creep
CRS at 0.125%/min
CRS at
0.00125%/min
Accidental Pressure Drop Followed
by Relaxation Stage
0 1 2 3 4 5 6 7 8
Stress Ratio σa /σr
Shear Strain γ = εa – εr (%)
Figure 12.49 Creep and stress relaxation of Hostun sand
(from Matsushita et al., 1999).
NSF-Clay
Isotropically
Consolidated
Esec = Δq/εa, Eeq = (Δq)SA / (εa)SA
Emax = 239 MPa p0 = 300 kPa
300
200
100
0
Young's Modulus, Esec or Eeq (MPa)
10-3 10-2 10-1 100
Axial Strain, εa or
Single Amplitude of Cyclic Axial Strain, (εa)SA (%)
Figure 12.50 Clay stiffness degradation curves at three
strain rates (from Shibuya et al., 1996).
Coefficient of Strain Rate, (γ)
Shear Strain, γ (%)
Figure 12.51 Strain rate parameter G and strain level for
several clays (from Lo Presti et al., 1996).
ent from the observations made for clays as shown in
Fig. 12.43 in which a unique stress–strain–strain rate
relationship was observed. Hence, the modeling of
stress–strain–rate behavior of sands appears to be
more complicated than that of clays, and further in-vestigation
is needed, as time-dependent behavior of
sands can be of significance in geotechnical construc-tion
as discussed further in Section 12.10.
Stiffness at Small and Intermediate Strains
Although the magnitude is small, the strain rate de-pendency
of the stress–strain relationship is observed
even at small strain levels for clays. The stiffness in-creases
less than 6 percent per 10-fold increase in
strain rate (Leroueil and Marques, 1996). The rate de-pendency
on stiffness degradation curves measured by
monotonic loading of a reconstituted clay is shown in
Fig. 12.50 (Shibuya et al., 1996). At different strain
levels, the increase in the secant shear modulus with
shear strain rate ˙ is often expressed by the following
equation (Akai et al., 1975; Isenhower and Stokoe,
1981; Lo Presti et al., 1996; Tatsuoka et al., 1997):

G
( ) (12.34) G
log ˙ G( , ˙ ) ref
where
G is the increase in secant shear modulus with
increase in log strain rate
log ˙ , and G( , ˙ ) is the ref
secant shear modulus at strain and reference strain
rate ˙ . The magnitude is large in clays, considerably ref
less in silty and clayey sands, and small in clean sands
(Lo Presti et al., 1996; Stokoe et al., 1999). The vari-ation
of G with strain is shown in Fig. 12.51 for dif-

MODELING OF STRESS–STRAIN–TIME BEHAVIOR 503
ferent plasticity clays (Lo Presti et al., 1996). The
magnitude of rate dependency increases with strain
level, especially for strain levels larger than 0.01 per-cent,
which is within the preplastic region zone 3 de-scribed
in Section 11.17.
Rate Effects During Cyclic Loading
The frequencies of cyclic loading to which a soil is
subjected can vary widely. For example, the frequency
of sea and ocean waves is in the range of 102 to 101
Hz, earthquakes are in the range of 0.1 to a few hertz,
and machine foundations are in the range of 10 to 100
Hz. Similarly to monotonic loading, the effect of load-ing
problems requiring the determination of de-
frequency on shear modulus degradation is small
in clean, coarse-grained soils (Bolton and Wilson,
1989; Stokoe et al., 1995), but the effect becomes more
significant in fine-grained soils (Stokoe et al., 1995;
d’Onofrio et al., 1999; Matesic and Vucetic, 2003;
Meng and Rix, 2004). An example of frequency effects
on a shear modulus degradation curve for a clay ob-tained
from cyclic loading is shown in Fig. 12.50 along
with the monotonic data. Figure 12.52 shows the effect
of frequency on shear modulus of several soils at very
small shear strains (less than 103 percent) measured
by torsional shear and resonant column apparatuses
(Meng and Rix, 2004). The effect is 10 percent in-crease
per log cycle at most.
At a given frequency of cyclic loading, the strain
rate applied to a soil increases with applied shear strain
as shown by the equation below:
˙ 4ƒ (12.35) c
250
200
150
100
50
where ƒ is the frequency and c is the cyclic shear
strain amplitude. Using Eq. (12.35), Matesic and Vu-cetic
(2003) report values of G of 2 to 11 percent for
clays and 0.2 to 6 percent for sands as the strain rate
increases 10-fold. The values of G in general de-creased
when the applied cyclic shear strain increased
from 5 104 percent to 1 102 percent. It should
be noted that the strain range examined is within the
non-linear elastic range (zone 1 to zone 2 in Section
11.17). The monotonic loading data presented in Fig.
12.51 show that the rate effect becomes more pro-nounced
at larger strain, that is, as plastic deformations
become more significant. Hence, it is possible that the
fundamental mechanisms of rate dependency are dif-ferent
at small elastic strain levels than at larger plastic
strains.
Small strain damping shows more complex fre-quency
dependency, as shown in Fig. 12.53 (Shibuya
et al., 1995; Meng and Rix, 2004). At a frequency of
more than 10 Hz, the damping ratio increases with
increased frequency, possibly due to pore fluid viscos-ity
effects. As the applied frequency decreases, the
damping ratio decreases. However, at a frequency less
than 0.1 Hz, the damping ratio starts to increase with
decreasing frequency. This may result from creep of
the soil (Shibuya et al., 1995).
12.9 MODELING OF STRESS–STRAIN–TIME
BEHAVIOR
Constitutive models are needed for the solution of geo-technical
Sandy Elastic Silt
0
10-2 10-1 100 101 102
Frequency (Hz)
Sandy Silty Clay
Kaolin
Vallencca Clay
Kaolin
Sandy Lean Clay
Kaolin Subgrade
Fat Clay
Shear Modulus G (MPa)
Figure 12.52 Rate dependency of cyclic small strain stiffness of a sandy elastic silt (from
Meng and Rix, 2004).

Clayey Subgrades
Sandy Lean Clay
10-2 10-1 100 101 102
Frequency Material
(Hz)
Copyrighted 6
5
4
3
2
1
Sandy Silty Clay
Sandy Elastic Silt
Kaolin
Vallencca Clay Fat Clay
Pisa Clay
Augusta Clay
Damping (%)
Figure 12.53 Effect of strain rate of damping ratio of soils (from Shibuya et al., 1995 and
Meng and Rix, 2004).
formations, displacements, and strength and stability
changes that occur over time periods of different
lengths. Various approaches have been used, including
empirical curve fitting, extensions of rate process
theory, rheological models, and advanced theories
of viscoelasticity and viscoplasticity. Owing to the
complexity of stress states, the many factors that influ-ence
the creep and stress relaxation properties of a soil,
and the difficulty of accounting for concurrent volu-metric
and deviatoric deformations in systems that are
many times undergoing consolidation as well as sec-ondary
compression or creep, it is not surprising that
development of general models that can be readily im-plemented
in engineering practice is a challenging un-dertaking.
Nonetheless, some progress has been made in estab-lishing
functional forms and relationships that can be
applied for simple analyses and comparisons, and one
of these is developed in this section. A complete re-view
and development of all recent theories and pro-posed
relationships for creep and stress relaxation is
beyond the scope of this book. Comprehensive reviews
of many models for representation of the time-dependent
plastic response of soils are given in Adachi
et al. (1996).
General Stress–Strain–Time Function
Strain Rate Relationships between axial strain rate
˙ and time t of the type shown in Figs. 12.4 and 12.5
can be expressed by
˙t
ln m ln (12.36) ˙(t ,D) t 1 1
or
t
ln ˙ ln ˙(t ,D) m ln (12.37) 1 t1
where ˙(t ,D) is the axial strain rate at unit time and 1
is a function of stress intensity D, m is the absolute
value of the slope of the straight line on the log strain
rate versus log time plot, and t1 is a reference time, for
example, 1 min. Values of m generally fall in the range
of 0.7 to 1.3 for triaxial creep tests; lower values are
reported for undrained conditions than for drained con-ditions.
For the development shown here, the stress
intensity D is taken as the deviator stress (1 3). A
shear stress or stress level could also be used.
The same data plotted in the form of Figs. 12.3,
12.31, and 12.32 can be expressed by

˙ln
D (12.38) ˙(t,D ) 0
or
ln ˙ ln ˙(t,D ) D (12.39) 0
in which ˙(t,D ) is a fictitious value of strain rate at 0
D 0, a function of time after start of creep, and
is the slope of the linear part of the log strain rate
versus stress plot. From Eqs. (12.37) and (12.39)
t
ln ˙(t ,D) m ln 1 ln ˙(t,D ) 0 D (12.40) t 1
For D 0,

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