6302712-141018084743-conversion-gate01

465

CHAPTER 12

Time Effects on Strength andDeformation

12.1 INTRODUCTION

Virtually every soil ‘‘lives’’ in that all of its propertiesundergo changes with time–some insignificant, butothers very important. Time-dependent chemical,geomicrobiological, and mechanical processes mayresult in compositional and structural changes thatlead to softening, stiffening, strength loss, strengthgain, or altered conductivity properties. The need topredict what the properties and behavior will bemonths to hundreds or thousands of years from nowbased on what we know today is a major challenge ingeoengineering. Some time-dependent changes andtheir effects as they relate to soil formation, composi-tion, weathering, postdepositional changes in sedi-ments, the evolution of soil structure, and the like areconsidered in earlier chapters of this book. Emphasisin this chapter is on how time under stress changes thestructural, deformation, and strength properties ofsoils, what can be learned from knowledge of thesechanges, and their quantification for predictive pur-poses.

When soil is subjected to a constant load, it deformsover time, and this is usually called creep. The inversephenomenon, usually termed stress relaxation, is adrop in stress over time after a soil is subjected to aparticular constant strain level. Creep and relaxationare two consequences of the same phenomenon, thatis, time-dependent changes in structure. The rate andmagnitude of these time-dependent deformations aredetermined by these changes.

Time-dependent deformations and stress relaxationare important in geotechnical problems wherein long-term behavior is of interest. These include long-termsettlement of structures on compressible ground, de-formations of earth structures, movements of naturaland excavated slopes, squeezing of soft ground aroundtunnels, and time- and stress-dependent changes in soilproperties. The time-dependent deformation responseof a soil may assume a variety of forms owing to thecomplex 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 usuallyfollow logical and often predictable patterns, at leastfor simple stress and deformation states such as uni-axial and triaxial compression, and they are describedin this chapter. Incorporation of the observed behav-ior into simple constitutive models for analytical de-scription of time-dependent deformations and stresschanges is also considered.

Time-dependent deformation and stress phenomenain soils are important not only because of the imme-diate direct application of the results to analyses ofpractical problems, but also because the results can beused to obtain fundamental information about soilstructure, interparticle bonding, and the mechanismscontrolling the strength and deformation behavior.Both microscale and macroscale phenomena are dis-cussed because understanding of microscale processescan provide a rational basis for prediction of macro-scale responses.

Copy

right

ed M

ater

ial

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

466 12 TIME EFFECTS ON STRENGTH AND DEFORMATION

Figure 12.1 Creep and stress relaxation: (a) Creep underconstant stress and (b) stress relaxation under constant strain.

12.2 GENERAL CHARACTERISTICS

1. As noted in the previous section soils exhibitboth creep1 and stress relaxation (Fig. 12.1).Creep is the development of time-dependentshear and/or volumetric strains that proceedat a rate controlled by the viscouslike resistanceof soil structure. Stress relaxation is a time-dependent decrease in stress at constant defor-mation. The relationship between creep strainand the logarithm of time may be linear, con-cave upward, or concave downward as shownby the examples in Fig. 12.2.

2. The magnitude of these effects increases withincreasing plasticity, activity, and water content

1 The term creep is used herein to refer to time-dependent shearstrains and /or volumetric strains that develop at a rate controlled bythe viscous resistance of the soil structure. Secondary compressionrefers to the special case of volumetric strain that follows primaryconsolidation. The rate of secondary compression is controlled bythe viscous resistance of the soil structure, whereas, the rate of pri-mary consolidation is controlled by hydrodynamic lag, that is, howfast water can escape from the soil.

of the soil. The most active clays usually exhibitthe greatest time-dependent responses (i.e.,smectite � illite � kaolinite). This is becausethe smaller the particle size, the greater is thespecific surface, and the greater the water ad-sorption. Thus, under a given consolidationstress or deviatoric stress, the more active andplastic clays (smectites) will be at higher watercontent and lower density than the inactive clays(kaolinites). Normally consolidated soils exhibitlarger magnitude of creep than overconsolidatedsoils. However, the basic form of behavior isessentially the same for all soils, that is, undis-turbed and remolded clay, wet clay, dry clay,normally and overconsolidated soil, and wet anddry sand.

3. An increase in deviatoric stress level results inan increased rate of creep as shown in Fig. 12.1.Some soils may fail under a sustained creepstress 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 fewminutes or hours. This is termed creep rupture,and an early illustration of its importance wasthe development of slope failures in the Cucar-acha clay shale, which began some years afterthe excavation of the Panama Canal (Casa-grande and Wilson, 1951).

4. The creep response shown by the upper curvein Fig. 12.1 is often divided into three stages.Following application of a stress, there is first aperiod of transient creep during which the strainrate decreases with time, followed by creep atnearly a constant rate for some period. For ma-terials susceptible to creep rupture, the creeprate then accelerates leading to failure. Thesethree stages are termed primary, secondary, andtertiary creep.

5. An example of strain rates as a function of stressfor undrained creep of remolded illite is shownin Fig. 12.3. At low deviator stress, creep ratesare very small and of little practical importance.The curve shapes for deviator stresses up toabout 1.0 kg/cm2 are compatible with the pre-dictions of rate process theory, discussed in Sec-tion 12.4. At deviator stress approaching thestrength of the material, the strain rates becomevery large and signal the onset of failure.

6. A characteristic relationship between strain rateand time exists for most soils, as shown, forexample, in Fig. 12.4 for drained triaxial com-pression creep of London clay (Bishop, 1966)

Copy

right

ed M

ater

ial


GENERAL CHARACTERISTICS 467

Figure 12.2 Sustained stress creep curves illustrating different forms of strain vs. logarithmof time behavior.

and Fig. 12.5 for undrained triaxial compressioncreep 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. 12.4and in kg/cm2 in Fig. 12.5), the logarithm of thestrain rate decreases linearly with increase in the

logarithm of time. The slope of this relationshipis essentially independent of the creep stress;increases in stress level shift the line verticallyupward. The slope of the log strain rate versuslog time line for drained creep is approximately�1. Undrained creep often results in a slope be-

Copy

right

ed M

ater

ial



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. Theonset of failure under higher stresses is signaledby a reversal in slope, as shown by the topmostcurve in Fig. 12.5.

7. Pore pressure may increase, decrease, or remainconstant during creep, depending on the volumechange tendencies of the soil structure andwhether or not drainage occurs during the de-formation process. In general, saturated softsensitive clays under undrained conditions aremost susceptible to strength loss during creepdue to reduction in effective stress caused byincrease in pore water pressure with time. Heav-ily overconsolidated clays under drained con-

ditions are also susceptible to creep rupture dueto softening associated with the increase in wa-ter content by dilation and swelling.

8. Although stress relaxation has been less studiedthan creep, it appears that equally regular pat-terns of deformation behavior are observed, forexample, Larcerda and Houston (1973).

9. Deformation under sustained stress ordinarilyproduces an increase in stiffness under the ac-tion of subsequent stress increase, as shownschematically in Fig. 12.6. This reflects thetime-dependent structural readjustment or ‘‘ag-ing’’ that follows changes in stress state. It isanalogous to the quasi-preconsolidation effect

Copy

right

ed M

ater

ial


GENERAL CHARACTERISTICS 469

Figure 12.4 Strain rate vs. time relationships during drained creep of London clay (datafrom 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 thevirgin compression line and the value of the pre-consolidation pressure, determined in the��,p

laboratory are influenced by the rate of loadingduring one-dimensional consolidation (Grahamet al., 1983a; Leroueil et al., 1985). Thus, esti-mations of the overconsolidation ratio of claydeposits in the field are dependent on the load-ing rates and paths used in laboratory tests fordetermination of the preconsolidation pressure.If it is assumed that the relationship betweenstrain and logarithm of time during compressionis linear over the time ranges of interest and thatthe secondary compression index C�e is constantregardless of load, the rate-dependent precon-solidation pressure at can be related to the�� p 1

axial strain rate as follows (Silvestri et al, 1986;Soga and Mitchell, 1996; Leroueil and Marques,1996):

C�e / (Cc�Cr) �� p 1 1� � (12.1)� � � �� p(ref) 1(ref) 1(ref)

where Cc is the virgin compression index, Cr isthe recompression index and is the pre-��p(ref)

consolidation pressure at a reference strain rateIn this equation, the rate effect increases� .1(ref)

with the value of � � C�e / (Cc � Cr). The var-iation of preconsolidation pressure with strainrate is shown in Fig. 12.8 (Soga and Mitchell,1996). The data define straight lines, and theslope of the lines gives the parameter �. In gen-eral, the value of � ranges between 0.011 and0.094. Leroueil and Marques (1996) report val-ues between 0.029 and 0.059 for inorganicclays.

11. The undrained strength of saturated clay in-creases with increase in rate of strain, as shownin Figs. 12.9 and 12.10. The magnitude of theeffect is about 10 percent for each order of mag-nitude increase in the strain rate. The strain rate

Copy

right

ed M

ater

ial



Figure 12.5 Strain rate vs. time relationships during undrained creep of Osaka alluvial clay(Murayama and Shibata, 1958).

Figure 12.6 Effect of sustained loading on (a) stress–strainand strength behavior and (b) one-dimensional compressionbehavior.

effect is considerably smaller for sands. In amanner similar to Eq. (12.1), a rate parameter� can be defined as the slope of a log–log plotof deviator stress at failure qƒ at a particularstrain rate relative to qƒ(ref), the strength at a�1

reference strain rate versus strain rate.� ,1(ref)

This gives the following equation:

�q �ƒ 1� (12.2)� �q �ƒ(ref) 1(ref)

The value of � ranges between 0.018 and 0.087,similar to the � rate parameter values used todefine the rate effect on consolidation pressurein Eq. (12.1). Higher values of � are associatedwith more metastable soil structures (Soga andMitchell, 1996). Rate dependency decreaseswith increasing sample disturbance, which isconsistent with this finding.

12.3 TIME-DEPENDENTDEFORMATION–STRUCTURE INTERACTION

In reality, completely smooth curves of the type shownin the preceding figures for strain and strain rate as afunction of time may not exist at all. Rather, as dis-

Copy

right

ed M

ater

ial


TIME-DEPENDENT DEFORMATION–STRUCTURE INTERACTION 471

Figure 12.7 Rate dependency on one-dimensional compression characteristics of Batiscanclay: (a) compression curves and (b) preconsolidation pressure (Leroueil et al., 1985).

Copy

right

ed M

ater

ial



Figure 12.8 Strain rate dependence on preconsolidation pressure determined from one-dimensional constant strain rate tests (Soga and Mitchell, 1996).

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 structurereorganization’’ may occur, reflecting a stochastic char-acter for the deformation, as shown in Fig. 12.11 forcreep of an undisturbed diatomaceous, lacustrine, ov-erconsolidated clay. Ter-Stepanian (1992) suggests thatthere are four levels of deformation: (1) the molecularlevel, which consists of displacement of flow units by

surmounting energy barriers, (2) mutual displacementof particles as a result of bond failures, but withoutrearrangement, (3) the structural level of soil defor-mation involving mutual rearrangements of particles,and (4) deformation at the aggregate level. Behavior atlevels 3 and 4 is discussed below; that at levels 1 and2 is treated in more detail in Section 12.4.

Copy

right

ed M

ater

ial



Figure 12.10 Strain rate dependence on undrained shear strength determined using constantstrain rate CU tests (Soga and Mitchell, 1996).

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 morestable configurations. Forces at interparticle contactshave both normal and tangential components, even ifthe macroscopic applied stress is isotropic. If, duringthe creep process, there is an increase in the proportionof applied deviator stress that is carried by interparticlenormal forces relative to interparticle tangential forces,then the creep rate will decrease. Hence, the rate atwhich deformation level 3 occurs need not be uniformowing to the particulate nature of soils. Instead it willreflect a series of structural readjustments as particlesmove up, over, and around each other, thus leading to

the somewhat irregular sequence of data points shownin Fig. 12.11.

Microscopically, creep is likely to occur in the weakclusters discussed in Section 11.6 because the contactsin them are at limiting frictional equilibrium. Anysmall perturbation in applied load at the contacts ortime-dependent loss in material strength can lead tosliding, breakage or yield at asperities. As particlesslip, 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 thatconsiders sliding at interparticle contacts to be visco-frictional. The rate at which sliding of two particlesrelative to each other occurs depends on the ratio ofshear to normal force at their contact. The relationshipbetween rate and force is formulated in terms of rateprocess theory (see Section 12.4), and the mechanisticrepresentations of the contact normal and shear forcesare 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

2 Kuhn (1987) used the following equation for rate of sliding at acontact:

t2kT F � ƒX � � exp � sinh� � � �nh RT 2kTn ƒ1

where n1 is the number of bonds per unit of normal force, ƒt is thetangential force and ƒn is the normal force. The others are parametersrelated to rate process theory as described in Section 12.4.

Copy

right

ed M

ater

ial



Figure 12.12 Normal and tangential interparticle force mod-els according to Kuhn (1987).

Figure 12.13 Creep curves developed by numerical analysisof an irregular packing of circular disks (from Kuhn andMitchell, 1993).

tangential forces at individual contacts can change dur-ing deformation even though the applied boundarystresses are constant. Small changes in the tangentialand normal force ratio at a contact can have a verylarge influence on the sliding rate at that contact. Thesechanges, when summed over all contacts in the shearzone, result in a decrease or increase in the overallcreep rate.

A numerical analysis of an irregular packing of cir-cular disks using the sinh-dashpot representation givescreep behavior comparable to that of many soils asshown in Fig. 12.13 (Kuhn and Mitchell, 1993). Thecreep rate slows if the average ratio of tangential tonormal force decreases, whereas it accelerates and mayultimately lead to failure if the ratio increases. In somecases, the structural changes that are responsible forthe decreasing strain rate and increased stiffness maycause the overall soil structure to become more meta-stable. Then, after the strain reaches some limitingvalue, the process of contact force transfer from de-creasing tangential to increasing normal force reverses.This marks the onset of creep rupture as the structurebegins to collapse. A similar result was obtained byRothenburg (1992) who performed discrete particlesimulations in which smooth elliptical particles werecemented 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) performedone-dimensional compression tests on sands, and Fig.

Copy

right

ed M

ater

ial



After5 days After

290 sBefore Test

Sieve Size (μm)0

0

20

40

60

80

100

100 200 300 400

Dry sandRD = 75%Pressure = 15.4 MPa

� ��

�

Per

cent

age

Pas

sing

(%

)

Figure 12.14 Changes in particle size distribution of sandbefore loading and after two different load durations (fromLeung, et al., 1996).

12.14 shows the particle size distribution curves forsamples before loading and after two different load du-rations. The amount of particle breakage increasedwith load duration. Microscopic observations revealedthat angular protrusions of the grains were ground off,producing fines. The fines fill the voids between largerparticles and crushed particles progressively rear-ranged themselves with time.

Aging—Time-Dependent Strengthening of SoilStructure

The structural changes that occur during creep that iscontinuing at a decreasing rate cause an increase insoil stiffness when the soil is subjected to further stressincrease 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 ofthe void ratio decrease during the sustained compres-sion period. Time-dependent changes of these types area consequence of ‘‘aging’’ effects, which alter thestructural state of the soil. The fabric obtained by creepmay 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 inFig. 12.15a. They also measured the shear wave ve-locities after different times during the tests usingbender elements and computed the small strain elasticshear modulus. Figure 12.15b shows the change inshear 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) andNakagawa et al. (1995). Figure 12.16 shows changesin shear modulus with time under a constant confiningpressure for kaolinite clay during consolidation (An-derson and Stokoe, 1978). Two distinct phases of shearmodulus–time response are evident. During primaryconsolidation, values of the shear modulus increaserapidly at the beginning and begin to level off as theexcess pore pressure dissipates. After the end of pri-mary consolidation, the modulus increases linearlywith the logarithm of time during secondary compres-sion.

The expected change in shear modulus due to voidratio change during secondary compression can be es-timated using the following empirical formula forshear modulus as a function of void ratio and confiningpressure (Hardin and Black, 1968):

2(2.97 � e) 0.5G � A p� (12.3)1 � e

where A is a unit dependant material constant, e is thevoid ratio, and p� is the mean effective stress. Thedashed line in Fig. 12.16 shows the calculated in-creases in the shear modulus due to void ratio decreaseusing Eq. (12.3). It is evident that the change in voidratio alone does not provide an explanation for the sec-ondary time-dependent increase in shear modulus. Thisaging effect has been recorded for a variety of mate-rials, ranging from clean sands to natural clays (Afifiand 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 influencethe stability of soil structure. Changes in sand fabricwith time after load application in one-dimensionalcompression were measured by Bowman and Soga(2003). Resin was used to fix sand particles after var-ious loading times. Pluviation of the sand produced ahorizontal 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 smallvoids became smaller, and particles group or cluster

Copy

right

ed M

ater

ial



2.6

2.4

2.2

2.0

1.84 6 8 10 20

Vertical Effective Stress σ�v (kPa)

1

Small-strain Stiffness G0 (MPa)

0.5 2

A A

B B

C C

D D

E

F

E

F

Stiffness ChangeDuring the PrimaryConsolidationBetween B and C

NormalConsolidation Line

Creep for120 days

Increase in Stiffnessduring Creep (C-D)

Quasi-preconsolidationPressure

DestructuringState

(a ) (b )

5

Voi

d R

atio

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 levelfindings, it appears that the movements of particleslead to interlocking zones of greater local density. Theinterlocked state may be regarded as the final state ofany one particle under a particular applied load, due tokinematic restraint. The result, with time, is a stiffer,more efficient, load-bearing structure, with areas of rel-atively large voids and neighboring areas of tightlypacked particles. The increase in stiffness is achievedby shear connections obtained by the clustering. Then,when load is applied, the increased stiffness andstrength of the granular structure provides greater re-sistance to the load and the observed aging effect isseen. 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 agingperiod, however, the strengthening effect of aging maybe less, as the load-bearing particle column directiondiffers from the load direction.

Time-Dependent Changes in PhysicochemicalInteraction 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 betweenparticles. To illustrate this, Nakagawa et al. (1995) ex-amined the physicochemical interactions between claysand pore fluid using a special consolidometer in whichthe sample resistivity and pore fluid conductivity couldbe 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 theexperiment, and changes in shear wave velocities andelectrical properties were monitored during the tests.

The test results showed that the pore fluid compo-sition and ion mobility changed with time. At eachload increment, as the effective stress increased withpore pressure dissipation, the shear wave velocities,and therefore the shear modulus, generally increasedwith time as shown in Fig. 12.17. It may be seen, how-ever, that in some cases, the shear wave velocities atthe beginning of primary consolidation decreasedslightly from the velocities obtained immediately be-fore application of the incremental load, probably as aresult of soil structure breakdown. During the subse-quent secondary compression stage, the shear wave ve-locity again increased. As was the case for the resultsin Fig.12.16, the increases in shear wave velocity dur-

Copy

right

ed M

ater

ial



10

20

30

40

50

0

0.5

1.0

1.5

2.0

Primary Consolidation Secondary Compression

Sam

ple

Hei

ght C

hang

e (m

m)

100 101 102 103 104

Time (min)

101 102 103 104

Time (min)

Ball Kaolinite

Possible Change in Gby Void Ratio DecreaseOnly Estimated UsingEq. (12.3)

Initial Consolidation Pressure = 70 kPaInitial Void Ratio e0 = 1.1

IG = ΔG per log time= 6.2 MPa

She

ar M

odul

us o

f les

s th

an γ

= 1

0-3 %

(M

Pa)

Figure 12.16 Modulus and height changes as a function oftime under constant confining pressure for kaolinite: (a) shearmodulus 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 (fromNakagawa et al., 1995).

ing secondary compression are greater than can be ac-counted for by increase in density.

The electrical conductivity of the sample measuredby filter electrodes increased during the early stages ofconsolidation, but then decreased continuously there-after as shown in Fig. 12.18. The electrical conductiv-ity is dominated by flow through the electrolytesolution in the pores. During the initial compression, abreakdown of structure releases ions into the pore wa-ter, increasing the electrical conductivity. With time,the conductivity decreased, suggesting that the releasedions are accumulating near particle surfaces. Some ofthese released ions are expelled from the specimen asconsolidation progressed as shown in Fig. 12.18b. Aslow equilibrium under a new state of effective stressis hypothesized to develop that involves both smallparticle rearrangements, associated with decrease invoid 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 otherprocesses.

Primary consolidation can be considered a result ofdrainage 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 microporesis restricted due to small pore size and physicochem-ical interactions close to the clay particle surfaces. Ak-agi (1994) did compression tests on specially preparedclay containing primarily Ca in the micropores and Nain the macropores. Concentrations of the two ions inthe expelled water at different times after the start ofconsolidation were consistent with this hypothesis.

Copy

right

ed M

ater

ial



Figure 12.18 Changes in electrical conductivity of the porewater 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 RATEPROCESS

Deformation and shear failure of soil involve time-dependent rearrangement of matter. As such, thesephenomena are amenable for study as rate processesthrough application of the theory of absolute reactionrates (Glasstone et al., 1941). This theory providesboth insights into the fundamental nature of soilstrength and functional forms for the influences of sev-eral factors on soil behavior.

Detailed development of the theory, which is basedon statistical mechanics, may be found in Eyring(1936), Glasstone et al. (1941), and elsewhere in thephysical chemistry literature. Adaptations to the studyof 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 andMitchell (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 flowunits, are constrained from movement relative to eachother 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 ofsufficient magnitude to surmount the barrier. The po-tential energy of a flow unit may be the same followingthe activation process, or higher or lower than it wasinitially. These conditions are shown by analogy withthe rotation of three blocks in Fig. 12.20. In each case,an energy barrier must be crossed. The assumption ofa steady-state condition is implicit in most applicationsto soils concerning the at-rest barrier height betweensuccessive equilibrium positions.

The magnitude of the activation energy depends onthe 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 areabout 12 to 17, 40 to 400, and 100 to 150 kJ/mol offlow units, respectively.

Activation Frequency

The energy to enable a flow unit to cross a barrier maybe provided by thermal energy and by various appliedpotentials. For a material at rest, the potential energy–displacement relationship is represented by curve A inFig. 12.21. From statistical mechanics it is known that

Copy

right

ed M

ater

ial


SOIL DEFORMATION AS A RATE PROCESS 479

Figure 12.20 Examples of activated processes: (a) steady-state, (b) increased stability, and(c) decreased stability.

Figure 12.21 Effect of a shear force on energy barriers.

Copy

right

ed M

ater

ial



the average thermal energy per flow unit is kT, wherek is Boltzmann’s constant (1.38 � 10�23 J K�1) and Tis the absolute temperature (K). Even in a material atrest, thermal vibrations occur at a frequency given bykT /h, where h is Planck’s constant (6.624 � 10�34 Js�1). The actual thermal energies are divided amongthe flow units according to a Boltzmann distribution.

It may be shown that the probability of a given unitbecoming activated, or the proportion of flow units thatare activated during any one oscillation is given by

Fp( F) � exp � (12.4)� �NkT

where N is Avogadro’s number (6.02 � 1023), and Nkis equal to R, the universal gas constant (8.3144 J K�1

mol�1). 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 thatsoftening, melting, or evaporation occurs. If, however,a directed potential, such as a shear stress, is applied,then the barrier heights become distorted as shown bycurve B in Fig. 12.21. If ƒ represents the force actingon a flow unit, then the barrier height is reduced by anamount (ƒ� /2) in the direction of the force and in-creased by a like amount in the opposite direction,where � represents the distance between successiveequilibrium positions.3 Minimums in the energy curveare displaced a distance � from their original positions,representing an elastic distortion of the material struc-ture.

The reduced barrier height in the direction of forceƒ 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 barrierheight decreases the activation frequency to

kT F /N � ƒ� /2� ← � exp � (12.7)� �h kT

3 Work (ƒ� / 2) done by the force ƒ as the flow unit drops from thepeak of the energy barrier to a new equilibrium position is assumedto be given up as heat.

The net frequency of activation in the direction ofthe force then becomes

kT F ƒ�(� →) � (� ←) � 2 exp � sinh� � � �h RT 2kT

(12.8)

Strain Rate Equation

At any instant, some of the activated flow units maysuccessfully cross the barrier; others may fall back intotheir original positions. For each unit that is successfulin crossing the barrier, there will be a displacement ��.The component of �� in a given direction times thenumber of successful jumps per unit time gives the rateof movement per unit time. If this rate of movementis expressed on a per unit length basis, then the strainrate 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), andthe rate is directly proportional to ƒ. This is the casefor 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 ƒ�N� � X exp � exp (12.13)� � � �h RT 2RT

Equation (12.13) applies except for very small stressintensities, where the exponential approximation of thehyperbolic sine is not justified. Equations (12.10) and

Copy

right

ed M

ater

ial


BONDING, EFFECTIVE STRESSES, AND STRENGTH 481

(12.13) or comparable forms have been used to obtaindashpot coefficients for rheological models, to obtainfunctional forms for the influences of different factorson 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 contactlaw in discrete element modeling as described in theprevious section. Puzrin and Houlsby (2003) used it asan internal function of a thermomechanical-basedmodel and derived a rate-dependent constitutive modelfor soils.

Soil Deformation as a Rate Process

Although there does not yet appear to be a rigorousproof of the correctness of the detailed statistical me-chanics formulation of rate process theory, even forsimple chemical reactions, the real behavior of manysystems 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 resultsdo not prove the correctness of the theory; they do,however, support the concept that soil deformation isa thermally activated process.

Arrhenius Equation

Equation (12.13) may be written

kT E� � X exp � (12.14)� �h RT

where

ƒ�NE � F � (12.15)

2

is termed the experimental activation energy. For allconditions constant except T, and assuming thatX(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 1900to describe the temperature dependence of chemicalreaction rates. It has been found suitable also forcharacterization of the temperature dependence ofprocesses such as creep, stress relaxation, secondarycompression, thixotropic strength gain, diffusion, andfluid flow.

12.5 BONDING, EFFECTIVE STRESSES, ANDSTRENGTH

Using rate process theory, the results of time-dependent stress–deformation measurements in soilscan be used to obtain fundamental information aboutsoil 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 distributeduniformly among S flow units per unit area, then

�ƒ � (12.17)

S

Displacement of a flow unit requires that interatomicor intermolecular forces be overcome so that it can bemoved. Let it be assumed that the number of flow unitsand the number of interparticle bonds are equal.

If D represents the deviator stress under triaxialstress conditions, the value of ƒ on the plane of max-imum shear stress is

Dƒ � (12.18)

2S

so Eq. (12.13) becomes

kT F D�� X exp � exp (12.19)� � � �h RT 4SkT

This equation describes creep as a steady-stateprocess. Soils do not creep at constant rate, however,because of continued structural changes during de-formation as described in Section 12.3, except forthe 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 ofcreep 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 thepossible 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 conditionsof unchanged soil structure. Thus, the value of E canbe determined from the slope of a plot of ver-ln(� /T)sus (1/T). Procedures for evaluation of strain rates for

Copy

right

ed M

ater

ial



Table 12.1 Activation Energies for Creep of Several Materials

MaterialActivation Energy

(kJ/ mol)a Reference

1. Remolded illite, saturated, water contents of30 to 43%

105–165 Mitchell, et al. (1969)

2. Dried illite: samples air-dried fromsaturation, 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 structuresSame as

waterAndersland 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 notalways clear in the references cited.

soils at different temperatures but at the same structureare given by Mitchell et al. (1968, 1969).

Determination of Number of Bonds For stresseslarge enough to justify approximating the hyperbolicsine function by a simple exponential in the creep rateequation and small enough to avoid tertiary creep, thelogarithm of strain rate varies directly with the deviatorstress. For this case, Eq. (12.19) can be written

� � K(t) exp(�D) (12.21)

where

kT FK(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 slopeof the relationship between log strain rate and stress.It is evaluated using strain rates at the same time afterthe start of creep tests at several stress intensities. With

� known, � /S is calculated as a measure of the numberof interparticle bonds.4

Activation Energies for Soil Creep

Activation energies for the creep of several soils andother materials are given in Table 12.1. The free energyof activation for creep of soils is in the range of about80 to 180 kJ/mol. Four features of the values for soilsin Table 12.1 are significant:

1. The activation energies are relatively large, muchhigher than for viscous flow of water.

2. Variations in water content (including completedrying), adsorbed cation type, consolidation pres-sure, void ratio, and pore fluid have no significanteffect on the required activation energy.

3. The values for sand and clay are about the same.4. Clays in suspension with insufficient solids to

form a continuous structure deform with an ac-tivation energy equal to that of water.

4 A procedure for evaluation of � from the results of a test at asuccession of stress levels on a single sample is given by Mitchellet al. (1969).

Copy

right

ed M

ater

ial



Figure 12.22 Interpretation of � in terms of silicate mineralsurface structure.

Number of Interparticle Bonds

Evaluation of S requires knowledge of �, the separationdistance between successive equilibrium positions inthe interparticle contact structure. A value of 0.28 nm(2.8 A) has been assumed because it is the same as thedistance separating atomic valleys in the surface of asilicate mineral. It is hypothesized that deformation in-volves the displacement of oxygen atoms along con-tacting particle surfaces, as well as periodic rupture ofbonds at interparticle contacts. Figure 12.22 shows thisinterpretation for � schematically. If the above as-sumption for � is incorrect, calculated values of S willstill be in the same correct relative proportion as longas � remains constant during deformation.

Normally Consolidated Clay Results of creep testsat different stress intensities for different consolidationpressures enable computation of S as a function of con-solidation pressure. Values obtained for undisturbedSan Francisco Bay mud are shown in Fig. 12.23. Theopen point is for remolded bay mud. An undisturbedspecimen was consolidated to 400 kPa, rebounded to

50 kPa, and then remolded at constant water content.The effective consolidation pressure dropped to 25 kPaas a result of the remolding. The drop in effectivestress was accompanied by a corresponding decreasein the number of interparticle bonds. Tests on remoldedillite gave comparable results. A continuous inverse re-lationship between the number of bonds and watercontent over a range of water contents from more than40 percent to air-dried and vacuum-desiccated clay isshown in Fig. 12.24. The dried material had a watercontent of 1 percent on the usual oven-dried basis. Thevery large number of bonds developed by drying isresponsible for the high dry strength of clay.

Overconsolidated Clay Samples of undisturbedSan Francisco Bay mud were prepared to overcon-solidation ratios of 1, 2, 4, and 8 following the stresspaths 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 thecreep 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 normallyconsolidated clay. Some of the bonds formed duringconsolidation are retained after removal of much of theconsolidation pressure.

Values of compressive strength and numbers ofbonds from Fig. 12.25 are replotted versus each otherin Fig. 12.26. The resulting relationship suggests thatstrength depends only on the number of bonds and isindependent of whether the clay is undisturbed, re-molded, normally consolidated, or overconsolidated.

Dry Sand Creep tests on oven-dried sand yieldedresults of the same type as obtained for clay, as shownin Fig. 12.27, suggesting that the strength-generatingand creep-controlling mechanisms may be similar forboth types of material.

Composite Strength-Bonding Relationship Valuesof 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 realityshould be expected, as discussed further later.

Significance of Activation Energy and Bond NumberValues

The following aspects of activation energies and num-bers of interparticle bonds are important in the under-standing of the deformation and strength behavior ofuncemented soils.

Copy

right

ed M

ater

ial



Figure 12.23 Number of interparticle bonds as a function of consolidation pressure fornormally consolidated San Francisco Bay mud.

Figure 12.24 Number of bonds as a function of water con-tent for illite.

1. The values of activation energy for deformationof soils are high in comparison with other ma-terials and indicate breaking of strong bonds.

2. Similar creep behavior for wet and dry clay andfor wet and dry sand indicates that deformationis not controlled by viscous flow of water.

3. Comparable values of activation energy for wetand dry soil indicate that water is not respon-sible for bonding.

4. Comparable values of activation energy for clayand sand support the concept that interparticlebond strengths are the same for both types ofmaterial. 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 watercontent.

6. The number of bonds is directly proportional toeffective consolidation pressure for normallyconsolidated clays.

7. Overconsolidation leads to more bonds than innormally consolidated clay at the same effectiveconsolidation 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 ofbonds.

10. There are about 100 times as many bonds in dryclay as in wet clay.

Copy

right

ed M

ater

ial



Figure 12.25 Consolidation pressure, strength, and bond numbers for San Francisco Baymud.

Although it may be possible to explain these resultsin more than one way, the following interpretation ac-counts well for them. The energy F activates a moleof flow units. The movement of each flow unit mayinvolve rupture of single bonds or the simultaneousrupture of several bonds. Shear of dilute montmoril-lonite–water pastes involves breaking single bonds(Ripple and Day, 1966). For viscous flow of water, theactivation energy is approximately that for a single hy-drogen bond rupture per flow unit displacement, eventhough each water molecule may form simultaneously

up to four hydrogen bonds with its neighbors. If thesingle-bond interpretation is also correct for soils, thenconsistency in Eq. (12.10) requires that shear force ƒpertain to the force per bond. On this basis, parameterS 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 ofthe total bonds in the system.

That the activation energy for deformation of soil iswell into the chemical reaction range (40 to 400 kJ/mol) does not prove that bonding is of the primary

Copy

right

ed M

ater

ial



Figure 12.26 Strength as a function of number of bonds forSan Francisco Bay mud.

Figure 12.28 Composite relationship between shear strengthand number of interparticle bonds (from Matsui and Ito,1977). Reprinted with permission from The Japanese Societyof SMFE.

Figure 12.27 Strength as a function of number of bonds for dry Antioch River sand.

valence type because simultaneous rupture of severalweaker 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,(2) it is the same for wet and dry soils, and (3) it isessentially the same for different adsorbed cations and

Copy

right

ed M

ater

ial



pore fluids (Andersland and Douglas, 1970) suggestthat bonding is through solid interparticle contacts.Physical evidence for the existence of solid-to-solidcontact between clay particles has been obtained in theform of photomicrographs of particle surfaces thatwere scratched during shear (Matsui et al., 1977, 1980)and acoustic emissions (Koerner et al., 1977).

Activation energy values of 125 to 190 kJ/mol areof the same order as those for solid-state diffusion ofoxygen in silicate minerals. This supports the conceptthat creep movements of individual particles could re-sult from slow diffusion of oxygen ions in and aroundinterparticle contacts. The important minerals in bothsand and clay are silicates, and their surface layersconsist 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 heldtogether by hydrogen. It is not too different from thatof 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 lesscontinuous solid structure containing water moleculesthat propagates through interparticle contacts can bevisualized.

An individual flow unit could be an atom, a groupof atoms or molecules, or a particle. The precedingarguments are based on the interpretation that individ-ual atoms are the flow units. This is consistent withboth the relative and actual values of S that have beendetermined for different soils. Furthermore, by using aformulation of the rate process equation that enabledcalculation of the flow unit volume from creep testdata, Andersland and Douglas (1970) obtained a valueof about 1.7 A3, which is of the same order as that ofindividual atoms. On the other hand, Keedwell (1984)defined flow units between quartz sand particles asconsisting of six O2� ions and six Si4� ions and be-tween two montmorillonite clay particles as consistingof four H2O molecules.

If particles were the flow units, not only would it bedifficult to visualize their thermal vibrations, but thenS would relate to the number of interparticle contacts.It is then difficult to conceive how simply drying a claycould 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 isthat drying, while causing some increase in the numberof interparticle contacts during shrinkage, causesmainly an increase in the number of bonds per contactbecause of increased effective stress.

At any value of effective stress, the value of S isabout the same for both sand and clay. The number ofinterparticle contacts should be vastly different; how-

ever, for equal numbers of contacts per particle, thenumber per unit volume should vary inversely with thecube of particle size. Thus, the number of clay particlesof 1-�m particle size should be some nine orders ofmagnitude greater than for a sand of 1-mm averageparticle size. Each contact between sand particleswould involve many bonds; in clay, the much greaternumber of contacts would mean fewer bonds per par-ticle.

The contact area required to develop bonds in thenumbers 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 shearsurface. Oxygen atoms on the surface of a silicate min-eral have a diameter of 0.28 nm. Allowing an area 0.30nm on a side for each oxygen gives 0.09 nm2, or 9 �10�16 cm2, per bonded oxygen for a total area of 9 �10�16 � 8 � 1010 � 7.2 � 10�5 cm2/cm2 of soil crosssection.

Hypothesis for Bonding, Effective Stress, andStrength

Normal effective stresses and shear stresses can betransmitted only at interparticle contacts in most soils.5

The predominant effects of the long-range physico-chemical forces of interaction are to control the initialsoil fabric and to alter the forces transmitted at contactpoints from what they would be due to applied stressesalone.

Interparticle contacts are effectively solid, and it islikely that both adsorbed water and cations in the con-tact zone participate in the structure. An interparticlecontact may contain many bonds that may be strong,approaching the primary valence type. The number ofbonds at any contact depends on the compressive forcetransmitted at the contact, and the Terzaghi–Bowdenand Tabor adhesion theory of friction presented in Sec-tion 11.4, can account for strength. The macroscopicstrength is directly proportional to the number ofbonds.

For normally consolidated soils the number of bondsis directly proportional to the effective stress. As a re-sult of particle rearrangements and contacts formedduring virgin compression, an overconsolidated soil ata given effective stress has a greater number of bondsand higher strength than a normally consolidated soil.This effect is more pronounced in clays than in sandsbecause the larger and bulky sand grains tend to re-

5 Pure sodium montmorillonite may be an exception since a part ofthe 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.

Copy

right

ed M

ater

ial



cover their original shapes when unloaded, thus rup-turing most of the bonds in excess of those needed toresist the lower stress. The strength of the interparticlecontacts can vary over a wide range, depending on thenumber 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 soilsare silicates, and they all have similar surface struc-tures.

In the absence of chemical cementation, interparticlebonds may form in response to interparticle contactforces 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. Thereshould be no difference between friction and cohesionin terms of the shearing process. Complete failure inshear involves simultaneous rupture or slipping of allbonds along the shear plane.

12.6 SHEARING RESISTANCE AS A RATEPROCESS

Deformation at large strain can approach a steady-statecondition where there is little further structural changewith time (such as at critical state). In this case, Eq.(12.19) can be used to describe the shearing resistanceas a function of strain rate and temperature. If the max-imum shear stress � is substituted for the deviator stressD, then

� � �1 3� � (12.24)2

and

kT F �� X exp � exp (12.25)� � � �h RT 2SkT

Taking logarithms of both sides of Eq. (12.25) gives

kT F ��ln � � ln X � � (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 followingrelationship between bonds per unit area and effectivestress is suggested.

S � a � b�� (12.28)ƒ

where a and b are constants and is the effective��ƒnormal stress on the shear plane. Thus, Eq. (12.27)becomes

2a F 2akT � 2b F 2bkT �� ln � � ln �� ƒ�N � B �N � B

(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 frictiondepend on the number of bonds times the bondstrength, as reflected by the activation energy, and thatthe values of c and � should depend on the rate ofdeformation and the temperature.

Strain Rate Effects

All other factors being equal, the shearing resistanceshould increase linearly with the logarithm of the rateof strain. This is shown to be the case in Fig. 12.9,which contains data for 26 clays. Additional data forseveral clays are shown in Fig. 12.29, where shearingresistance as a function of the speed of vane rotationin a vane shear test is plotted. Analysis of the relation-ship between shearing stress and angular rate of vanerotation " shows that � / log " decreases with anincrease in water content. This follows directly fromEq. (12.29) because

d� 2akT 2bkT 2kT� � �� (a � b��)ƒ ƒd ln(� /B) � � �

(12.33)

that is, d� /d ln is proportional to the number of(� /B)bonds, which decreases with increasing water content.

Copy

right

ed M

ater

ial


CREEP AND STRESS RELAXATION 489

Figure 12.29 Effect of rate of shear on shearing resistance of remolded clays as determinedby the laboratory vane apparatus (prepared from the data of Karlsson, 1963).

This interpretation of the data in Figs. 12.9 and 12.29assumes that the effective stress was unaffected bychanges in the strain rate, which may not necessarilybe true in all cases.

Effect of Temperature

Assumptions of reasonable values for parameters showthat the term is less than one (Mitchell, 1964).(� /B)Thus the quantity in Eq. (12.29) is negative,ln(� /B)and an increase in temperature should give a decreasein strength, all other factors being constant. That thisis the case is demonstrated by Fig. 12.30, which showsdeviator stress as a function of temperature for samplesof San Francisco Bay mud compared under conditionsof equal mean effective stress and structure. Other ex-amples of the influence of temperature on strength areshown in Figs. 11.6 and 11.133.

12.7 CREEP AND STRESS RELAXATION

Although the designation of a part of the strain versustime relationship as steady state or secondary creepmay be convenient for some analysis purposes, a truesteady state can exist only for conditions of constantstructure and stress. Such a set of conditions is likelyonly for a fully destructured soil, and a fully destruc-tured state is likely to persist only during deformationat a constant rate, that is, at failure. This state is oftencalled ‘‘steady state,’’ in which the soil is deformingcontinuously at constant volume under constant shearand confining stresses (Castro, 1975; Castro andPoulos, 1977).

Otherwise, bond making and bond breaking occurat different rates as a result of different internal time-and strain-dependent phenomena, which might includethixotropic hardening, viscous flows of water and ad-

Copy

right

ed M

ater

ial



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 deviatorstress 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 causechanges in the ratio of tangential to normal forces atinterparticle contacts that may be responsible for largechanges in creep rate. Because of these time depend-encies some of the parameters in Eq. (12.19) may betime dependent. For example, Feda (1989) accountedfor the time dependency of creep rate by takingchanges in the number of structural bonds into account.Therefore, application of Eq. (12.19) for the determi-nation of the bonding and effective stress relationshipsdiscussed in Section 12.5 required comparison of creeprates under conditions of comparable time and struc-ture.

The influence of creep stress magnitude on the creeprate at a given time after the application of the stressto identical samples of a soil was shown in Fig. 12.3.At low stresses the creep rates are small and of littlepractical importance. The curve shape is compatiblewith the hyperbolic sine function predicted by rateprocess theory, as given by Eq. (12.10). In the mid-range of stresses, a nearly linear relationship is foundbetween logarithm of strain rate and stress, also as pre-dicted by Eq. (12.10) for the case where the argumentof the hyperbolic sine is greater than 1. At stressesapproaching the strength of the material, the strain ratebecomes very large and signals the onset of failure.Other examples of the relationships between logarithmof strain rate and creep stress corresponding to differ-ent times after the application of the creep stress aregiven in Fig. 12.31 for drained tests on London clay

and Fig. 12.32 for undrained tests on undisturbed SanFrancisco Bay mud. Only values for the midrange ofstresses are shown in Figs. 12.31 and 12.32.

Effect of Composition

In general, the higher the clay content and the moreactive the clay, the more important are stress relaxationand creep, as illustrated by Figs. 4.22 and 4.23, wherecreep rates, approximated by steady-state values, arerelated to clay type, clay content, and plasticity. Time-dependent deformations are more important at highwater 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 strainrates may be small in sand or dry soil, the form of the

Copy

right

ed M

ater

ial



Figure 12.32 Variation of creep strain rate with deviatorstress for undrained creep of normally consolidated San Fran-cisco Bay mud.

BA

C

D

E2820 min

1450 min

90 min

20 min

2 min0

0.2

0.4

0.6

0.8

1

0.5 1.0 1.5 2.0 2.5 3.0 3.5

Deviator strain (%)

0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Strain increment ratio dεv/dεs during creep

(a)

(b)

Drained Triaxial Test

Confining Pressure σ�3 = 414 kPa

Deviator Stress from 344 to 377 kPa

Vol

umet

ric S

trai

n (%

)

Str

ess

ratio

q/

p�

Figure 12.33 Dilatancy relationship obtained from drainedcreep tests on kaolinite: (a) development of volumetric anddeviatoric strains with time and (b) effect of stress ratio onstrain increment ratio d� /d�s (from Walker, 1969).

behavior conforms with the patterns described and il-lustrated above. This is to be expected, as the basiccreep 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 mechanismof creep is the same for dry and wet materials (Losertet al., 2000). Takei et al. (2001) showed that the de-velopment of creep strains due to time-dependentbreakage of talc specimens increased more for satu-rated specimens than dry ones. However, a negligibleeffect of water on creep rate was reported by Ahn-Danet al. (2001) who performed creep tests on unsaturatedand saturated crushed gravel and by Leung et al.(1996) who performed one-dimensional compression

6 Volumetric 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 bedue to the presence or absence of impurities that maylubricate 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 shearingand volumetric plastic deformations in soils, an in-crease in either mean pressure or deviator stress cangenerate both types of deformations. Creep behavior isno exception. Time-dependent shear deformations areusually referred to as deviatoric creep or shear creep.Time-dependent deformations under constant stress re-ferred to as volumetric creep. Secondary compressionis a special case of volumetric creep.

Deviatoric creep is often accompanied by volumetriccreep. The ratio of volumetric to deviatoric creep fol-

Copy

right

ed M

ater

ial



Figure 12.34 Pore pressure development with time during undrained creep of illite.

lows a plastic dilatancy rule. Walker (1969) inves-tigated the time-dependent change of these two com-ponents from incremental drained triaxial creep testson normally consolidated kaolinite. The increase inshear strains with increase in volumetric strains at dif-ferent times is shown in Fig. 12.33a. At the beginningof the triaxial test, the deviator stress was instantane-ously increased from 344 to 377 kPa and kept constant.After an immediate increase in shear strains at constantvolume (AB in Fig. 12.33a), section BD correspondsto 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 strainswas 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 issimilar to the dilatancy rule described in Section 11.20.

Sand deforms with time in a similar manner. Underprogressive 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 confiningpressure, particularly after particle crushing becomesimportant at high stresses (Yamamuro and Lade, 1993).For dense sand under high deviator stress, dilativecreep is observed (Murayama et al., 1984; Mejia et al.,1988). The volumetric response of dense sand andgravel 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). Somedense sand samples contract initially but then dilatewith 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-independentplastic strains, and the same framework of soil plastic-ity can possibly be used. It can be argued whetherit is necessary to separate the deformation intotime-dependent and independent components. Rate-independent behavior can be considered as the limitingcase of rate-dependent behavior at a very slow rate ofloading.

Volumetric-deviatoric creep coupling implies thatrapid 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 constanttotal minor principal stress, the magnitude of the porepressure change depends on the volume change ten-dencies of the soil when subjected to shear distortions.These tendencies are, in turn, controlled by the voidratio, structure, and effective stress, and can be quan-tified in terms of the pore pressure parameter as dis-Acussed in Chapters 8 and 10. An example showing porepressure increase with time for consolidated undrainedcreep tests on illite at several stress intensities is shownin Fig. 12.34. Figure 12.35 shows a slow decrease in

Copy

right

ed M

ater

ial



Figure 12.35 Normalized pore pressure vs. time relationships during creep of kaolinite.

pore pressure during the sustained loading of kaolinite.Similar behavior was demonstrated in the measuredstress paths of undrained creep test on San FranciscoBay 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 strengthin terms of effective stresses does not change unlessthere are chemical, biological, or mineralogicalchanges during the creep period. This is illustrated bythe stress paths shown schematically in Fig. 12.37,where the pre- and postcreep strengths fall on the samefailure 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 relaxationstresses 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 deformationand stress relaxation in soils have been done on sam-ples consolidated isotropically and tested in triaxialcompression or by measurement of secondary com-pression in oedometer tests. However, most soils innature have been subjected to an anisotropic stress his-tory, and deformation conditions conform more toplane strain than triaxial compression in many cases.Some investigations of these factors have been made.

Although the general form of the stress–strain–timeand stress–strain rate–time relationships are similar tothose shown above for triaxial loading conditions, theactual values may differ considerably.

For example, undisturbed Haney clay, a gray siltyclay from British Columbia, with a sensitivity in therange of 6 to 10, was tested both in triaxial compres-sion and plane strain (Campanella and Vaid, 1974).Samples were normally consolidated both isotropicallyand under K0 conditions to the same vertical effectivestress. Samples consolidated isotropically were testedin triaxial compression. Coefficient K0 consolidationwas 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 thedeformations. The plane strain and K0 consolidatedtriaxial samples gave about the same creep behaviorunder the same deviatoric stress, which suggests thatpreventing strain in one horizontal direction and/or theintermediate principal stress were not factors of majorimportance 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; Imaiand Tang, 1992). Following the initial large changefollowing load application, the pore pressure may ei-ther dissipate, with accompanying volume change ifdrainage is allowed, or change slowly during creep orstress relaxation, if drainage is prevented. The devel-opment of complete effective stress and void ratioequilibrium may take a long time. One illustration of

Copy

right

ed M

ater

ial



80 160 240 320 400 480 560 64000

80

160

Normal Undrained TriaxialCompression Test: EffectiveStress Path

Total Stress Path ofTriaxial Compression Test

1

3

Possible CriticalState Line

Effective Stress StateAfter 1,000 min Creep

After 20,000 min CreepEffective Stress Pathof Undrained Creep

10 20 30 40 50 60 70 8000

10

20

30

40

50Normal Undrained TriaxialCompression Test: EffectiveStress Path

Total Stress Path ofTriaxial Compression Test

1

3Possible CriticalState Line

Effective Stress StateAfter 1,000 min Creep

After 20,000 min CreepEffective Stress Pathof Undrained Creep

240

320

400

(a)

(b)

Dev

iato

r S

tres

s q

(kP

a)

Dev

iato

r S

tres

s q

(kP

a)

Mean Pressure p� (kPa)

Mean Pressure p� (kPa)

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).

Copy

right

ed M

ater

ial



Figure 12.37 Effects of undrained creep on the strength of normally consolidated clay.

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 therelationship between void ratio and effective stress isdependent on the time for compression under anygiven stress. Another is given by Fig. 12.41, whichshows pore pressures during undrained creep of SanFrancisco 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 drainageand the start of a creep test. The consolidation periodwas greater than that required for 100 percent primaryconsolidation. The curve marked 0 percent stress levelrefers 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 pressurethat 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 secondarycompression prior to the prevention of further drain-age. This is illustrated by the data in Fig. 12.42, whichshow pore pressure as a function of time for samplesthat have undergone different amounts of secondarycompression.

In summary, creep deformation depends on the ef-fective stress path followed and any changes in stress

Copy

right

ed M

ater

ial



Figure 12.39 Influence of temperature on the initial and final stresses in stress relaxationtests on Osaka clay—undrained triaxial compression (Murayama, 1969).

Figure 12.40 Creep curves for isotropically and K0-consolidated samples of undisturbedHaney clay tested in triaxial and plane strain compression (from Campanella and Vaid, 1974).Reproduced with permission from the National Research Council of Canada.

Copy

right

ed M

ater

ial


RATE EFFECTS ON STRESS–STRAIN RELATIONSHIPS 497

Figure 12.41 Pore pressure development during undrained creep of San Francisco Bay mudafter consolidation at 100 kPa for 1800 min (from Holzer et al., 1973). Reproduced withpermission from the National Research Council of Canada.

Figure 12.42 Pore pressure development under undrained conditions following differentperiods of secondary compression (from Holzer et al., 1973). Reproduced with permissionfrom the National Research Council of Canada.

with time. Furthermore, time-dependent volumetric re-sponse is governed both by the rate of volumetric creepand 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 therate of loading and drainage conditions, the separationof consolidation and creep deformations can be diffi-cult in the early stage of time-dependent deformationas given by section BD in Fig. 12.33a. In some cases,a fully coupled analysis of soil–pore fluid interactionwith 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–STRAINRELATIONSHIPS

An increase in strain rate during soil compression ismanifested by increased stiffness, as was noted in Sec-tion 12.3. In essence, the state of the soil jumps to thestress–strain curve that corresponds to the new strainrate. Commonly, this rate-dependent stress–strain

Copy

right

ed M

ater

ial



5% /h

0.5% /h

0.05% /h

Belfast Clay 4 mσ1c = σ�v0

16%/h

1%/h0.25%/h

Winnipeg Clay 11.5 mσ1c > σ�v0

CAU Triaxial Compression TestsRelaxation Tests (R)

00

0.1

0.2

0.3

0.4

0.5

0.6

4 8 12 16 20

Axial Strain

(σ1 –

σ3)/

2σ1c

R

R

R

SP1 testSP2 test

εv1 = 2.70 & 10–6 s–1.εv2 = 1.05 & 10–7 s–1

εv1

.. εv1

.

εv1.

εv1.

εv1.

εv1.

εv3.εv2

.

εv2.

εv2.

εv2.

εv3 = 1.34 & 10–5 s–1.

30

25

20

15

10

5

00 50 100 150 200 250

Effective Stress σv� (kPa)

Str

ain

ε v (

%)

Figure 12.43 Rate-dependent stress–strain relations ofclays: (a) undrained triaxial compression tests of Belfastand Winnipeg clay (Graham et al., 1983a) and (b) one-dimensional compression tests of Batiscan clay (Leroueil etal., 1985).

curve, noted by Suklje (1957), is the same as if thesoil had been loaded from the beginning at the newstrain rate. This phenomenon is often observed inclays. Examples are given in Fig. 12.43a for undrainedtriaxial compression tests of Belfast and Winnipegclays (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 decreasingstrain rate or increasing duration of testing. Preconsol-idation pressures obtained from one-dimensional con-solidation tests and undrained shear strengths obtainedfrom triaxial tests are just two points on a soil’s yieldenvelope in stress space. For a given metastable soilstructure, the degree of rate dependency of preconsol-idation pressure is similar to that of undrained shearstrength (Soga and Mitchell, 1996). If the apparent pre-consolidation pressure depends on the strain rate atwhich the soil is deforming, then the same analogy canbe 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 familyof strength envelopes corresponding to constant strainrates7 obtained from drained and undrained creep testson stiff plastic Mascouche clay from Quebec (Leroueiland Marques, 1996).

The effective stress failure line of soil is uniquelydefined regardless of the magnitude of the strain rateapplied in undrained compression. Figure 12.45ashows the failure line of Haney clay (Vaid and Cam-panella, 1977). The line represents the stress conditionsat the maximum ratio of The data were obtained�� /��.1 3

by various undrained tests, and a unique failure linecan be observed. Figure 12.45b shows the undrainedstress paths and the critical state line of reconstitutedmixtures of sand and clay with plasticity indices rang-ing from 10 to 30 (Nakase and Kamei, 1986). A uniquecritical state line can be observed although the rates ofshearing are different. The change in undrained shearstrength with strain rate results from a difference ingeneration of excess pore pressures. A decrease instrain rate leads to larger excess pore pressures at fail-ure due to creep deformation.

7 The strain rate is defined as where is the volu-2 2� � � � � , �vs v s vmetric strain rate and is the deviator strain rate (Leroueil and Mar-�s

ques, 1996). Whether the use of this strain rate measure is appropriateor not remains to be investigated.

Copy

right

ed M

ater

ial



Figure 12.44 Influence of strain rate on the yield surface of Masouche clay (from Leroueiland Marques, 1996).

Excess Pore Pressure Generation in NormallyConsolidated Clays

Excess pore pressure development depends primarilyon the collapse of soil structure. Accordingly, strain isthe primary factor controlling pore pressure generation.This is shown in Fig. 12.46 by the undrained stress–strain–pore pressure response of normally consolidatednatural Olga clay (Lefebvre and LeBouef, 1987). Thenatural clay specimens were normally consolidated un-der consolidation pressures larger than the field over-burden pressure and then sheared at different strainrates. Although the deviator stress at any strain in-creases with increasing strain rate, the pore pressureversus strain curves are about the same at all strainrates. At a given deviator stress, the pore pressure gen-eration was larger at slower strain rates as a result ofmore creep under slow loading. This is consistent withthe observation made in connection with undrainedcreep of clays as discussed in Section 12.7. Strain-driven pore pressure generation was also suggested byLarcerda and Houston (1973) who showed that porepressure does not change significantly during triaxial

stress relaxation tests in which the axial strain is keptconstant.

Overconsolidated Clays

Rate dependency of undrained shear strength decreaseswith increasing overconsolidation, since there is nocontraction or collapse tendency observed during creepof heavily overconsolidated clays. Sheahan et al.(1996) prepared reconstituted specimens of Bostonblue clay at different overconsolidation ratios andsheared them at different strain rates in undrained con-ditions. Figure 12.47 shows that the undrained stresspath and the strength were much more strain rate de-pendent for lightly overconsolidated clay (OCR � 1and 2) than for more heavily overconsolidated clay(OCR � 4 and 8). The results also show that thestrength failure envelope is independent of strain rateas discussed earlier.

The strain rate effects on stress–strain–pore pressureresponse of overconsolidated structured Olga clays areshown in Fig. 12.48 (Lefebvre and LeBouef, 1987).The natural samples were reconsolidated to the field

Copy

right

ed M

ater

ial



Const. Stress Creep

Const. Load CreepStep CreepThixotropic Hardened

Const. Rate of Strain ShearConst. Rate of Loading ShearAged Samples

0

0

00.1

0.1

0.2

0.2

0.2

0.3

0.3

0.4

0.4

0.4

0.5 0.6

0.6

0.7 0.8

0.8

(σ1� + σ3�)/2σ�1c

(σ1�

–σ 3

�)/2

σ�1c

(a)

(b)

1.0

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.61.0

p�/σ�vc

0 0.2 0.4 0.6 0.8 1.0p�/σ�vc

q�/σ

� vc

1.0

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

q�/σ

� vc

ε(%/min)M-15

M15 Soil (Plasticity Index = 15) M10 Soil (Plasticity Index = 10)

..

7&10-1: ε1

.7&10-2: ε2

.7&10-3: ε3

ε(%/min).

7&10-1: ε17&10-2: ε2

.

.

.7&10-3: ε3

M-10K 0

-Line

K 0-L

ine

Crit

ical

-Sta

te L

ine

Crit

ical

-Sta

te L

ine

Critical-State Line

Critical-State Line

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, withstrain softening indicating development of localizedshear failure planes. Up to the peak stress, the responsefollows what has been described previously, that is thestress–strain response is rate dependent and the porepressure generation is strain dependent but indepen-dent of rate. However, after the peak, the pore pressure

generation becomes rate dependent. This is due to localdrainage within the specimens as the deformation be-comes localized. As the time to failure increased, thereis more opportunity for local drainage toward the di-lating shear band and the measured pore pressure maynot represent the overall behavior of the specimens.The difference in softening due to swelling at the fail-

Copy

right

ed M

ater

ial



0.1 %/hr0.5 %/hr2.6 %/hr12.3 %/hr

Axial Strain Rate

Axial Strain (%)1 2 3 4 5 6 7 8

Axial Strain (%)1 2 3 4 5 6 7 8

0

20

40

60

80

100

120

0

20

40

60

80

120

140

100

0.1 %/hr0.5 %/hr2.6 %/hr12.3 %/hr

Axial Strain Rate

Normally Consolidated Olga ClayUndrained Triaxial Compression TestsInitial Isotropic Confining Pressure =137 kPa

Dev

iato

r S

tres

s q

(kP

a)E

xces

s P

ore

Pre

ssur

e Δu

(kP

a)

Figure 12.46 Stress–strain and pore pressure–strain curvesfor normally consolidated Olga clay (from Lefebvre andLeBouef, 1987).

0.1 %/hr0.5 %/hr2.6 %/hr12.3 %/hr

Axial Strain Rate

Axial Strain (%)1 2 3 4 5 6 7 8

Axial Strain (%)

1 2 3 4 5 6 7 8

0

10

20

40

50

0

10

0.1 %/hr0.5 %/hr2.5 %/hr12.3 %/hr

Axial Strain Rate

Overconsolidated Olga ClayUndrained Triaxial Compression TestsInitial Isotropic Confining Pressure = 17.6 kPa

30

60

70

20

Devia

tor

Str

ess q

(kP

a)

Excess P

ore

Pre

ssure

Δu

(kP

a)

Figure 12.48 Stress–strain and pore pressure–strain curvesfor overconsolidated Olga clay (from Lefebvre and LeBouef,1987).

OCR=1

OCR=2

OCR=4

OCR=8

OCR=2OCR=4

Effective Stress Path for AxialStrain Rate = 0.50 %/hr

0.80.60.40.20.0

0.1

0.2

0.3

0.4

-0.1

Effective Stress State at Peak forAxial Strain Rate = 0.051 %/hrAxial Strain Rate = 0.50 %/hrAxial Strain Rate = 5.0 %/hrAxial Strain Rate = 49 %/hr

Initial K0 Consolidation State

(σ�a + σ�r)/2σ�vm

(σ� a

–σ�

r)/2

σ�vm

OCR=1LargeRateEffectOCR=8

NegligibleRate Effect

Figure 12.47 Rate dependency stress path and strength ofoverconsolidated Boston blue clay (from Sheahan et al.,1996).

ure plane results in apparent rate dependency at largestrains. Similar observations were made by Atkinsonand Richardson (1987) who examined local drainageeffects by measuring the angles of intersection of shearbands 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 arequite small in many cases (Tatsuoka et al., 1997; DiBenedetto et al., 2002). An example of time depend-ency observed for drained plane strain compressiontests of Hostun sand is shown in Fig. 12.49 (Matsushitaet al., 1999). The stress–strain curves for three differ-ent strain rates (1.25 � 10�1, 1.25 � 10�2, and 1.25 �10�3 %/min) are very similar, indicating very smallrate effects when the specimens are sheared at a con-stant strain rate. On the other hand, the change fromone rate to another temporarily increases or decreasesthe resistance to shear. The influence of accelerationrather than the rate is reflected by the significant creepdeformation and stress relaxation of this rate-insensitive material as shown the figure. This is differ-

Copy

right

ed M

ater

ial



0 1 2 3 4 5 6 7 83.0

3.5

4.0

4.5

5.0

5.5

6.0

Variation of Stress-strain curve by ConstantStrain Rate Tests at Axial Strain Rates = 0.125,0.0125 and 0.00125 %/min. A Very Small RateEffect Is Observed for Continuous Loading.

CRS at 0.125%/min

Creep

CRS at 0.125%/min

Creep

CRS at 0.00125%/min

Creep

CRS at0.125%/min

CRS at0.125%/min

Stress Relaxation

Creep

CRS at 0.00125%/min

Accidental Pressure Drop Followedby Relaxation Stage

Str

ess

Rat

io σ

� a/σ

� r

Shear Strain γ = εa – εr (%)

Figure 12.49 Creep and stress relaxation of Hostun sand(from Matsushita et al., 1999).

NSF-ClayIsotropicallyConsolidatedp�0 = 300 kPaEmax = 239 MPa

Esec = Δq/εa, Eeq = (Δq)SA / (εa)SA

0

200

100

300

You

ng's

Mod

ulus

,Ese

cor

Eeq

(MP

a)

10-3 10-2 10-1 100

Axial Strain, εa orSingle Amplitude of Cyclic Axial Strain, (εa)SA (%)

Figure 12.50 Clay stiffness degradation curves at threestrain rates (from Shibuya et al., 1996).

Co

effi

cien

t o

f S

trai

n R

ate,

�(γ

)

Shear Strain, γ (%)

Figure 12.51 Strain rate parameter �G and strain level forseveral clays (from Lo Presti et al., 1996).

ent from the observations made for clays as shown inFig. 12.43 in which a unique stress–strain–strain raterelationship was observed. Hence, the modeling ofstress–strain–rate behavior of sands appears to bemore complicated than that of clays, and further in-vestigation is needed, as time-dependent behavior ofsands 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 observedeven at small strain levels for clays. The stiffness in-creases less than 6 percent per 10-fold increase instrain rate (Leroueil and Marques, 1996). The rate de-pendency on stiffness degradation curves measured bymonotonic loading of a reconstituted clay is shown inFig. 12.50 (Shibuya et al., 1996). At different strainlevels, the increase in the secant shear modulus withshear strain rate is often expressed by the followingequation (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 withincrease in log strain rate log and is the, G(, )ref

secant shear modulus at strain and reference strainrate 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-

Copy

right

ed M

ater

ial


MODELING OF STRESS–STRAIN–TIME BEHAVIOR 503

250

200

150

100

50

010-2 10-1 100 101 102

Frequency (Hz)

Sandy Silty Clay

Sandy Elastic Silt

Kaolin

Vallencca Clay

Kaolin

Sandy Lean Clay

Kaolin SubgradeFat Clay

She

ar M

odul

us G

(M

Pa)

Figure 12.52 Rate dependency of cyclic small strain stiffness of a sandy elastic silt (fromMeng and Rix, 2004).

ferent plasticity clays (Lo Presti et al., 1996). Themagnitude of rate dependency increases with strainlevel, 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 issubjected can vary widely. For example, the frequencyof sea and ocean waves is in the range of 10�2 to 10�1

Hz, earthquakes are in the range of 0.1 to a few hertz,and machine foundations are in the range of 10 to 100Hz. Similarly to monotonic loading, the effect of load-ing frequency on shear modulus degradation is smallin clean, coarse-grained soils (Bolton and Wilson,1989; Stokoe et al., 1995), but the effect becomes moresignificant 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 effectson a shear modulus degradation curve for a clay ob-tained from cyclic loading is shown in Fig. 12.50 alongwith the monotonic data. Figure 12.52 shows the effectof frequency on shear modulus of several soils at verysmall shear strains (less than 10�3 percent) measuredby 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 strainrate applied to a soil increases with applied shear strainas shown by the equation below:

� 4ƒ (12.35)c

where ƒ is the frequency and c is the cyclic shearstrain amplitude. Using Eq. (12.35), Matesic and Vu-cetic (2003) report values of �G of 2 to 11 percent forclays and 0.2 to 6 percent for sands as the strain rateincreases 10-fold. The values of �G in general de-creased when the applied cyclic shear strain increasedfrom 5 � 10�4 percent to 1 � 10�2 percent. It shouldbe noted that the strain range examined is within thenon-linear elastic range (zone 1 to zone 2 in Section11.17). 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 deformationsbecome more significant. Hence, it is possible that thefundamental mechanisms of rate dependency are dif-ferent at small elastic strain levels than at larger plasticstrains.

Small strain damping shows more complex fre-quency dependency, as shown in Fig. 12.53 (Shibuyaet al., 1995; Meng and Rix, 2004). At a frequency ofmore than 10 Hz, the damping ratio increases withincreased frequency, possibly due to pore fluid viscos-ity effects. As the applied frequency decreases, thedamping ratio decreases. However, at a frequency lessthan 0.1 Hz, the damping ratio starts to increase withdecreasing frequency. This may result from creep ofthe soil (Shibuya et al., 1995).

12.9 MODELING OF STRESS–STRAIN–TIMEBEHAVIOR

Constitutive models are needed for the solution of geo-technical problems requiring the determination of de-

Copy

right

ed M

ater

ial



10210110010-110-2

Frequency (Hz)

1

2

3

4

5

6

Sandy Silty Clay

Sandy Elastic Silt

Kaolin

Vallencca Clay Fat Clay

Sandy Lean Clay

Clayey Subgrades

Pisa Clay

Augusta ClayD

ampi

ng (

%)

Figure 12.53 Effect of strain rate of damping ratio of soils (from Shibuya et al., 1995 andMeng and Rix, 2004).

formations, displacements, and strength and stabilitychanges that occur over time periods of differentlengths. Various approaches have been used, includingempirical curve fitting, extensions of rate processtheory, rheological models, and advanced theoriesof viscoelasticity and viscoplasticity. Owing to thecomplexity 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 aremany times undergoing consolidation as well as sec-ondary compression or creep, it is not surprising thatdevelopment 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 beapplied for simple analyses and comparisons, and oneof 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 isbeyond the scope of this book. Comprehensive reviewsof many models for representation of the time-dependent plastic response of soils are given in Adachiet al. (1996).

General Stress–Strain–Time Function

Strain Rate Relationships between axial strain rateand time t of the type shown in Figs. 12.4 and 12.5�

can be expressed by

� tln � �m ln (12.36)� �

�(t ,D) t1 1

or

tln � � ln �(t ,D) � m ln (12.37)� �1 t1

where is the axial strain rate at unit time and�(t ,D)1

is a function of stress intensity D, m is the absolutevalue of the slope of the straight line on the log strainrate versus log time plot, and t1 is a reference time, forexample, 1 min. Values of m generally fall in the rangeof 0.7 to 1.3 for triaxial creep tests; lower values arereported for undrained conditions than for drained con-ditions. For the development shown here, the stressintensity D is taken as the deviator stress (�1 � �3). Ashear 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 is a fictitious value of strain rate at�(t,D )0

D � 0, a function of time after start of creep, and �is the slope of the linear part of the log strain rateversus stress plot. From Eqs. (12.37) and (12.39)

tln �(t ,D) � m ln � ln �(t,D ) � �D (12.40)� �1 0t1

For D � 0,

Copy

right

ed M

ater

ial



Figure 12.54 Influence of creep stress magnitude on thecreep rate at a given time after stress application.

tln �(t,D ) � ln �(t ,D ) � m ln (12.41)� �0 1 0 t1

in which is the value of strain rate obtained by�(t ,D )1 0

projecting the straight-line portion of the relationshipbetween log strain rate and deviator stress at unit timeto a value of D � 0. Designation of this value by Aand substitution of Eq. (12.41) into Eq. (12.39) gives

tln � � ln A � �D � m ln (12.42)� �t1

which may be written

mt1�D� � Ae (12.43)� �tThis simple three-parameter equation has been

found suitable for the description of the creep rate be-havior of a wide variety of soils. The parameter A isshown in Fig. 12.54. Since it reflects an order of mag-nitude for the creep rate under a given set of condi-tions, it is in a sense a soil property. A minimum oftwo creep tests are needed to establish the values of A,�, and m for a soil. If identical specimens are testedusing different creep stress intensities, a plot of logstrain rate versus log time yields the value of m, anda plot of log strain rate versus stress for different valuesof time can be used to find � and A from the slopeand the intercept at unit time, respectively.

The parameter � has units of reciprocal stress. Ifstress is expressed as the ratio of creep stress tostrength at the beginning of creep, D /Dmax, then thedimensionless quantity �Dmax should be used. For agiven soil and test type, values of �Dmax do not varygreatly for different water contents, as the change in �with water content is compensated by a change inDmax. Thus the strain rate versus time behavior for anystress at any water content can be predicted from theresults of creep tests at any other water content, pro-

vided the variation of strength with water content isknown. Since normal strength tests are considerablysimpler and less time consuming than creep tests, theuniqueness of the quantity �Dmax can be useful becausethe results of a limited number of tests can be used topredict behavior over a range of conditions. A furthergeneralization of Eq. (12.43) then is

mt1� � A exp(�D) (12.44)� �twhere

D� � �D D � (12.45)max Dmax

Strain A general relationship between strain � andtime is obtained by integration of Eq. (12.43). Twosolutions are obtained, depending on the value of m.If � � �1 at t � t1 � 1, then

A 1�m� � � � exp (�D)(t � 1) when m 11 1 � m

(12.46)

and

� � � � A exp (�D)ln t when m � 1 (12.47)1

Creep curve shapes corresponding to these relation-ships are shown in Fig. 12.55. These curves encompassthe variety of shapes shown in Fig. 12.2. A similarequation to Eq. (12.46) was developed by Mesri et al.(1981) from Eq. (12.43). The initial time-independentstrain was neglected, and the resulting equation is

1�mAt t1� � exp(�D) (12.48)� �1 � m t1

It may be seen in Fig. 12.56 that this equation de-scribes the uniaxial creep behavior of three clays verywell. Data for both drained and undrained creep areshown.

Stress Relaxation Stress decay during stress relax-ation is approximately linear with logarithm of timeuntil it levels off at some residual stress after a longtime. There is equivalency between creep and stressrelaxation in that a general phenomenological modelthat predicts one can be used to predict the other, asshown by Akai et al. (1975), Lacerda (1976), Borja(1992), and others. For example, Eq. (12.44) takes thefollowing form when stress relaxation is started afterdeformation at constant rate of strain (Lacerda andHouston, 1973):

Copy

right

ed M

ater

ial



Figure 12.55 Creep curve shapes predicted by the general stress–strain–time function ofEqs. (12.46) and (12.47).

D D t� � 1 � s log (t � t ) (12.49)0D tD0 00

where s is the slope of the stress relaxation curve, andthe zero subscript refers to conditions at the start ofstress relaxation. Also

#s � (12.50)

D

where

2.3(1 � m)# � (12.51)

�

The validity of this equation has been establishedfor m � 1.0. Pore pressures decrease slightly duringundrained stress relaxation.

Stresses may not begin to relax immediately afterthe strain rate is reduced to zero. The time t0 betweenthe time that the strain rate is reduced to zero and thebeginning of relaxation is a variable that depends onthe soil type and the prior strain rate. This is shownschematically in Fig. 12.57. The greater the initial rateof strain to a given deformation, the more quickly re-laxation begins. This is a direct reflection of the rela-tive differences in equilibrium soil structures duringand after deformation. Values of t0 as a function ofprior strain rate are shown in Fig. 12.58 for severalsoils. These curves can be described empirically by

h0t � (12.52)0 �

where h0 is the strain rate to give a delay time of t0 �1 min before stresses begin to relax. The data presentedby Lacerda and Houston (1973) indicate that the valuesof # and h0 increase with increasing plasticity of thesoil.

Constitutive Models

Different rheological models have been proposed forthe mathematical description of the stress–strain–timebehavior of soils that are made up of combinations oflinear springs, viscous dashpots, and sliders. In theMurayama and Shibata (1958), Christensen and Wu(1964), and Abdel-Hady and Herrin (1966) models, thedashpots are nonlinear, with stress–flow rate responsegoverned by rate process theory. Rheological modelsare useful conceptually to aid in recognition of elasticand plastic components of deformation. They are help-ful for visualization by analogy of viscous flow thataccompanies time-dependent change of structure to amore stable state. Mathematical relationships can bedeveloped in a straightforward manner for the descrip-tion of creep, stress relaxation, steady-state deforma-tion, and the like in terms of the model constants. Inmost cases, these relationships are complex and neces-sitate the evaluation of several parameters that may notbe valid for different stress intensities or soil states.Only one-dimensional stresses and deformations are

Copy

right

ed M

ater

ial



Figure 12.56 Correspondence between creep strain predicted by Eq. (12.48) and measuredvalues. Diagrams are from Mesri et al. (1981), which were based on analyses by Semple(1973).

Copy

right

ed M

ater

ial



Figure 12.57 Influence of prior strain rate on stress relaxa-tion.

Figure 12.58 Influence of prior strain rate on the time tostart of stress relaxation (adapted from Lacerda and Houston,1973).

considered. None appears to exist that has the gener-ality and simplicity of the three-parameter creep Eqs.(12.43), (12.46), and (12.47).

Both plasticity and creep are controlled by the mo-tion of dislocations or breakage among soil particles,so it may be physically more correct to predict bothplastic and creep deformations with one equation. Twoparticularly promising approaches are based on an ex-tension of the Cam-clay model to take into accounttime-dependent volumetric and deviatoric deforma-tions (Kavazanjian and Mitchell, 1980; Borja and Ka-vazanjian, 1985; Kaliakin and Dafalias, 1990; Borja,1992; Al-Shamrani and Sture, 1998; Hashiguchi andOkayasu, 2000) and on an elasto-viscoplastic equationdeveloped using flow surface theory (Sekiguchi, 1977,1984; Matsui and Abe, 1985, 1986, 1988; Matsui etal., 1989; Yin and Graham, 1999) and overstress theory(Adachi and Oka, 1982; Katona, 1984: Kutter andSathialingham, 1992; Rocchi et al., 2003).

12.10 CREEP RUPTURE

As discussed in Section 12.2 and shown in Fig. 12.6,the strength of a soil and the stress–strain curve maybe changed as a result of creep. In some cases, suchas the drained creep of a compressive soil, the strengthmay be increased. Changes in strength may be as muchas 50 percent or more of the strength measured in nor-mal undrained tests prior to creep.

Causes of Strength Loss During Creep

Loss of strength during creep is particularly importantin soft clays deformed under undrained conditions andheavily overconsolidated clays in drained shear. Bothof these conditions are pertinent to certain types ofengineering problems: the former in connection withstability of soft clays immediately after construction,and the latter in connection with problems of long-termstability.

The loss of strength as a result of creep may beexplained in terms of the following principles of be-havior:

1. If a significant portion of the strength of a soil isdue to cementation, and creep deformationscause failure of cemented bonds, then strengthwill be lost.

2. In the absence of chemical or mineralogicalchanges the strength depends on effectivestresses. If creep causes changes in effectivestress, then strength changes will also occur.

3. In almost all soils, shear causes changes in porepressure during undrained deformation andchanges in water content during drained defor-mation.

4. Water content changes cause strength changes.

These processes are illustrated by the stress pathsand effective stress envelope shown schematically inFig. 12.37.

Strength loss in saturated, heavily overconsolidatedclays tested under undrained conditions has also beenreported, for example, Casagrande and Wilson (1951),Goldstein and Ter-Stepanian (1957), and Vialov andSkibitsky (1957). This may be explained as follows.Shear deformations cause dilation and the developmentof negative pore pressures, which do not develop uni-formly throughout the sample but concentrate alongplanes where the greatest shearing stresses and strainsdevelop. With time during sustained loading, water mi-grates into zones of high negative pore pressures lead-ing to softening and strength decrease relative to thestrength in ‘‘normal’’ undrained strength tests. Thisleads to shear band formation.

Copy

right

ed M

ater

ial


CREEP RUPTURE 509

Figure 12.59 Stress paths for normal undrained shear and drained creep of heavily over-consolidated clay.

This process is shown in Fig. 12.59 with referenceto an effective stress failure envelope for a heavily ov-erconsolidated clay. The effective stress path is repre-sented by AB, and AC represents the total stress pathin a conventional consolidated-undrained (CU) test.The negative pore pressure at failure is CB. If a creepstress DE is applied to the same clay, a negative porepressure EF is induced. This negative pore pressuredissipates during creep, and the clay in the shear zoneswells. At the end of the creep period, the effectivestress will be as represented by point E. Further shearstarting from these conditions leads to strength G,which is less than the original value at B. It is evidentalso that if the negative pore water pressure is largeenough, and the sustained load is applied long enough,then point E could reach the failure envelope. Thisappears to have been the conditions that developed inseveral cuts in heavily overconsolidated brown Londonclay, which failed some 40 to 70 years after excava-tions were made (Skempton, 1977).

Time to Failure

The time to failure of soils susceptible to strength lossunder sustained stresses depends on the rates at whichpore pressures develop and at which water can migrateinto or out of the critical shear zone. These rates are,

in turn, a function of deformation rates, the hydraulicconductivity, and the surrounding water pressure anddrainage conditions. The time to failure of heavily ov-erconsolidated clays in which negative pore water pres-sures develop as a result of unloading is best estimatedon the basis of drained strengths, effective stresses, andconsideration of the rate of swelling that is possiblefor the particular clay and ambient stress and ground-water conditions. An exception would be whenstrength loss results from the time-dependent ruptureof cementing bonds. In this case, sustained load creeptests in the laboratory may allow establishment of astress level versus time-to-failure relationship.

For soils subject to failure during undrained creep,the time to failure is usually a negative exponentialfunction of the stress, for stresses greater than somelimiting value below which no failure develops evenafter very long times.8 The relationship between devi-ator stress, normalized to the pretest major principaleffective stress, and time to failure for Haney clay isshown in Fig. 12.60. These and similar data define cer-

8 This critical stress below which creep rupture does not occur hasbeen termed the upper yield, the lower yield being the stress belowwhich deformations are elastic (Murayama and Shibata 1958, 1964).

Copy

right

ed M

ater

ial



Figure 12.60 Time to rupture as a function of creep stressfor Haney clay (Campanella and Vaid, 1972).

Figure 12.61 Creep rate behavior of K0-consolidated, undisturbed Haney clay under axi-symmetric loading (Campanella and Vaid, 1972).

tain principles relating to the probability of creep rup-ture and the time to failure:

1. Values of the parameter m less than 1.0 in Eqs.(12.43) through (12.46) are indicative of a highpotential strength loss during creep and eventualfailure (Singh and Mitchell, 1969).

2. The minimum strain rate prior to the onset�min

of creep rupture decreases, and the time to failureincreases, as the stress intensity decreases, asshown in Fig. 12.61 for Haney clay. The rela-tionship is unique, as may be seen in Fig. 12.62,which shows that

Ct � (12.53)ƒ �min

Values of the constant C accurate to about �0.2log cycles are given in Table 12.2.

3. The strain at failure is a constant independent ofstress level, as shown in Fig. 12.63. The failure

Copy

right

ed M

ater

ial


SAND AGING EFFECTS AND THEIR SIGNIFICANCE 511

Figure 12.62 Relationship between time to failure and min-imum creep rate (from Campanella and Vaid, 1974). Repro-duced with permission from the National Research Councilof Canada.

strain is taken as the strain corresponding to theminimum strain rate. For the case of undrainedcreep rupture, this is consistent with the conceptthat pore pressure development is uniquely re-lated to strain and independent of the rate atwhich it accumulates (Lo, 1969a, 1969b).

The relationship expressed by Eq. (12.53) results di-rectly from the fact that the strain at the point of min-imum strain rate is a constant independent of stress orstrain rate. The general stress–strain rate–time function[see Eq. (12.43)] describes the strain rate–time behav-ior until is reached. For t1 � 1 and � � 0 at t ��min

0, the corresponding strain–time equation is

A 1�m� � exp(�D)t (12.54)1 � m

By setting � � 0 at t � 0, the assumption is madethat there is no instantaneous deformation. Substitutionfor A exp(�D) in Eq. (12.54) gives

1 m 1�m� � �t t (12.55)1 � m

which at the point of minimum strain rate becomes

1 C� � constant � � t � (12.56)ƒ min ƒ1 � m 1 � m

Thus, the constant in Eq. (12.53) is defined by

C � (1 � m)� (12.57)ƒ

Values of �ƒ for Haney clay tested in three ways areshown in Fig. 12.63, and values of C and m are inTable 12.3. The agreement between predicted and mea-sured values of C is reasonable.

Predictions of the time to failure under a given stressmay be made in the following way. Strain at failurecan be determined by either a creep rupture test or bya normal shear or compression test. If a normalstrength test is used, then the rate of strain must beslow enough to allow pore pressure equalization ordrainage, depending on the conditions of interest, andthe stress history and stress system should simulatethose in the field. Parameter m can be established froma creep test, and then C can be computed from Eq.(12.57). Values of A and � are established from creeptests at two stress intensities. Then, for t1 � 1,

1�mC � � t � A exp(�D)t (12.58)min ƒ ƒ

and corresponding values of D and tƒ can be calculatedusing Eq. (12.58) rewritten as

1 Cln t � ln � �D (12.59) � � �ƒ 1 � m A

Other constitutive models are available to model thecomplex time-dependent behavior under various load-ing conditions. For example, Sekiguchi (1977) de-veloped a viscoplastic model that gives excellentrepresentations of strain rate effects on undrainedstress–strain behavior, stress relaxation, and creep rup-ture of normally consolidated clays. Other modelslisted in Section 12.9 are able to simulate time-dependent behavior in a similar manner.

12.11 SAND AGING EFFECTS AND THEIRSIGNIFICANCE

Over geological time, lithification and chemical reac-tions can change sand into sandstone or clay into mud-stone or shale. However, even over engineering time,behavior of soils can alter as stresses redistribute afterconstruction (Fookes et al., 1988). As discussed in theprevious sections, it is well established that fine-grained soils and clays have properties and behaviorthat change over time as a result of consolidation,

Copy

right

ed M

ater

ial



Table 12.2 Creep Rupture Parameters for Several Clays

SoilTest

Typea

CreepRate

Parameter,m

C � (� t )min ƒ

(�0.2 logcycles)

Undisturbed Haneyclay, N.C.b

ICU 0.7 1.2


ACU 0.4� 0.2


ACU-PS 0.5 0.3

Undisturbed Seattleclay, O.C.c

ICU 0.5 0.6

UndisturbedTonegawa loamc

U 0.8 1.6

UndisturbedRedwood Cityclay, N.C.c

ICU 0.75 2.8

Undisturbed Bangkokmudc

ICU 0.70 1.4

Undisturbed Osakaclayc

1.0 0.07

aICU, isotropic consolidated, undrained triaxial; ACU, K0 consoli-dated, undrained triaxial; ACU-PS, K0 consolidated, plane strain; andU, compression test.

bData from Campanella and Vaid (1974).cData from Singh and Mitchell (1969).

Figure 12.63 Axial strain at minimum strain rate as a function of creep stress for undis-turbed Haney clay (from Campanella and Vaid, 1974). Reproduced with permission fromthe National Research Council of Canada.

Copy

right

ed M

ater

ial



Table 12.3 Predicted and Measured Values of C forHaney Clay

TestCondition

Creep RateParameterm (from

Table12.2)

�ƒ

(fromFig.

12.63)

CPredicted

by Eq.(12.57)

CMeasured

ICUa 0.7 2.8 0.84 1.2ACUb 0.4 0.3 0.18 0.20ACU-PSc 0.5 0.5 0.25 0.30

aIsotropic consolidated, undrained triaxial.bAnisotropic, consolidated, undrained triaxial.cAnisotropic consolidated, undrained, plane strain.Data from Campanella and Vaid (1974).

0 20 40 60 80 1000.00

0.05

0.10

0.15

0.20

0.25

0.30

ΔG/G1000 = 0.03PI 0.5

ΔG : Modulus Increase in Every 10-fold Time IncreaseG1000 : Modulus at 1,000 min

Marcuson et al. (1972)Afifi et al. (1973)Trudeau et al. (1973)Anderson et al. (1973)Zen et al. (1978)Kokusho et al. (1982)Umehara et al. (1985)

Jamiolkowski (1996)

a, b

cd

e

f

a = Ticino Sand (Silica)b = Hokksund Sand (Silica)c = Messina Sand and Gravel (Silica)d = Glauconite Sand(Quartz/Glauconite)e = Quiou Sand (Carbonate)f = Kenya Sand (Carbonate)

Sand

Clay

Mod

ulus

Incr

ease

Rat

io Δ

G/

G10

00

Plasticity Index PI

Figure 12.64 Modulus increase ratio for clays (from Ko-kusho, 1987), supplemented by the data for sands (fromJamiolkowski, 1996).

shear, swelling, chemical and biological changes, andthe like. Until recently it has not been appreciated thatcohesionless soils exhibit this behavior as well. Muchrecent field evidence of the changing properties ofgranular soils over time is now available and these datasuggest that recently disturbed or deposited granularsoils gain stiffness and strength over time at constanteffective stress—a phenomenon called aging. The ev-idence includes the time-dependent increase in stiff-ness and strength of densified sands as measured bycone penetration resistance (Mitchell and Solymer,1984; Thomann and Hryciw, 1992; Ng et al., 1998)and the setup of displacement piles in granular mate-rials (Astedt et al., 1992; York et al., 1994; Chow etal., 1998; Jardine and Standing, 1999; Axelsson,2000). Hypotheses to explain this phenomenon includeboth creep processes and chemical and biological ce-mentation processes. The discussion in this section isfocused primarily on granular soils as the relevant as-pects for clays are treated in detail throughout othersections of the book.

Increase in Shear Modulus with Time

As discussed in Section 12.3, the shear modulus atsmall strain is known to increase with time under aconfining stress, and this is considered to be the con-sequence of aging. This behavior can be quantified bya coefficient of shear modulus increase with time usingthe following formula (Anderson and Stokoe, 1978):

I � G / log(t / t ) (12.60)G 2 1

N � I /G (12.61)G G 1000

where IG is the coefficient of shear modulus increasewith time, t1 is a reference time after primary consol-

idation, t2 is some time of interest thereafter, G is thechange in small strain shear modulus from t1 to t2,G1000 is the shear modulus measured after 1000 min ofconstant confining pressure, which must be after com-pletion of primary consolidation, and NG is the nor-malized shear modulus increase with time. Largeincrease in stiffness due to aging is represented bylarge values of IG or NG. In general, the measured NG

value for clays ranges between 0.05 and 0.25. The ag-ing effect also increases with an increasing plasticityindex as shown in Fig. 12.64 (Kokusho, 1987). Thedata in the figure have been supplemented by valuesof G /G for several sands compiled by Jamiolkowski(1996). Mesri et al. (1990) report that NG for sandsvaries between 0.01 and 0.03 and increases as the soilbecomes finer. Jamiolkowski and Manassero (1995)give values of 0.01 to 0.03 for silica sands, 0.039 forsand with 50 percent mica, and 0.05 to 0.12 for car-bonate sand. Experimental results show that the rate ofincrease in stiffness with time for very loose carbonatesand increases as the stress level increases (Howie etal., 2002). Isotropic stress state resulted in a slowerrate of increase in stiffness.

There is only limited field data that shows evidenceof aging effects on stiffness. Troncoso and Garces(2000) measured shear wave velocities using downhole

Copy

right

ed M

ater

ial



Figure 12.65 Normalized shear modulus as function of ag-ing of tailings (from Troncoso and Garces, 2000).

Figure 12.66 Effect of time on the cone penetration resis-tance of sand following blast densification at the Jebba Damsite.

wave propagation tests in low-plasticity silts with finescontents from 50 to 99 percent at four abandoned tail-ing dams in Chile. The shear modulus normalized bythe vertical effective stress is plotted against the ageof the deposit in Fig. 12.65. The age of the deposits isexpressed as the time since deposition. Although thesoil properties vary to some degree at the four sites,9

very significant increase in stiffness at small strains canbe observed after 10 to 40 years of aging. The degreeto which secondary compression could have contrib-uted to this increase is not known.

Time-Dependent Behavior after GroundImprovement

Stiffness and strength of sand increase with time afterdisturbance and densification by mechanical processessuch as blasting and vibrocompaction. Up to 50 per-cent or more increase in strength has been observedover 6 months (Mitchell and Solymer, 1984; Thomannand Hryciw, 1992; Charlie et al., 1992; Ng et al., 1998;Ashford, et al., 2004) as measured by cone penetrationtesting.

The Jebba Dam project on the Niger River, Nigeria,was an early well-documented field case where agingeffects in sands were both significant and widespread(Mitchell and Solymer, 1984). The project involved thetreatment of foundation soils beneath a 42-m-high dam

9 The four sites identified by Troncoso and Garces (2000) are calledBarahona, Cauquenes, La Cocinera, and Veta del Agua and the agingtimes between abandonment and testing were 28, 19, 5, and 2 years,respectively. The tailing deposits at Barahona had a liquid limit of41 percent and a plastic limit of 14 percent, whereas those at theother three sites had liquid limits of 23 to 29 percent and plasticlimits of 2 to 6 percent.

and seepage blanket. Due to large depths of the loosesand deposit requiring densification, a two-stage den-sification program was performed. The upper 25 m ofsand (and a 5- to 10-m-thick sand pad placed by hy-draulic filling of the river) was densified using vibro-compaction. Deposits between depths of 25 to 40 mwere densified by deep blasting.

During the blasting operations, it was observed thatthe sand exhibited both sensitivity—that is, strengthloss on disturbance—and aging effects. A typical ex-ample of the initial decrease in penetration resistanceafter blasting densification and subsequent increasewith time is shown in Fig. 12.66. Initially after im-provement, there was in some cases a decrease in pen-etration resistance, despite the fact that surfacesettlements ranging from 0.3 to 1.1 m were measured.With time (measured up to 124 days after improve-ment), however, the cone penetration resistance was

Copy

right

ed M

ater

ial



Figure 12.68 Effect of time on the cone penetration resis-tance of hydraulic fill sand after placement at the Jebba Damsite.

-10 0 10 20 30 400.02

0.1

1.0

Temperature (°C)

Jefferies et al. (1988)

Charlie et al. (1992)

Mitchell andSolymer (1984)

Schmertmann (1987) andFordham et al. (1991)

Em

piric

al C

onst

ant K

Figure 12.69 Rate of increase of normalized CPT tip resis-tance against temperature for different cases of reported ag-ing effects after blasting (by Charlie et al., 1992).

Figure 12.67 Effect of time on the cone penetration resis-tance of sand following vibrocompaction densification at theJebba Dam site.

found to increase by approximately 50 to 100 percentof the original values. Similar behavior was found fol-lowing blast densification of hydraulic fill sand thathad been placed for construction of Treasure Island inSan Francisco Bay more than 60 years previously(Ashford et al., 2004).

Aging effects were also observed after placement ofhydraulic fill working platforms in the river at theJebba Dam site and after densification by vibrocom-paction as shown in Figs. 12.67 and 12.68. In the caseof vibrocompaction, however, there was considerablevariability in the magnitude of aging effects throughoutthe site. Because of the greater density increase causedby vibrocompaction than by blast densification, no in-itial decrease in the penetration resistance was ob-served at the end of the compaction process.

Charlie et al. (1992) found a greater rate of agingafter densification by blasting for sands in hotter cli-mates than in cooler climates and suggested a corre-lation between the rate of aging and mean annual airtemperature for available field data as shown in Fig.12.69. In the figure, the increase in the CPT tip resis-tance (qc) with time is expressed by the followingequation:

q (N weeks)c � 1 � K log N (12.62)q (1 week)c

where N is the number of weeks since disturbance andK expresses the rate of increase in tip resistance inlogarithmic time.

Schmertmann (1991) postulated that a ‘‘complicatedsoil structure’’ is present in freshly deposited soil. Thestructure then becomes more stable by ‘‘drained dis-persive movements’’ of soil particles. He suggests thatstresses would arch from softer, weaker areas to stifferzones with time, leading to an increase in K0 with time.Mitchell and Solymar (1984) suggested that the ce-mentation of particles may be the mechanism of aging

Copy

right

ed M

ater

ial



Figure 12.70 Increase in total and shaft capacity with timefor displacement piles in sand (from Chow et al., 1998 andBowman, 2003).

of sands, similar to diagenesis in locked sands andyoung rocks (Dusseault and Morgenstern, 1979; Bar-ton, 1993) in which grain overgrowth has been ob-served. However, others have questioned whethersignificant chemical reactions can occur over the shorttime of observations. In addition, there is some evi-dence of aging in dry sands wherein chemical proc-esses would be anticipated to be very slow.

Setup of Displacement Piles

Much field data indicates that the load-carrying capac-ity of a pile driven into sand may increase dramaticallyover several months, long after pore pressures havedissipated (e.g., Chow et al., 1998; Jardine and Stand-ing, 1999). The amount of increase is highly variable,ranging from 20 to 170 percent per log cycle of timeas shown in Fig. 12.70 (Chow et al., 1998; Bowman,2002). Most of the increase in capacity occurs alongthe shaft of the pile as the radial stress at rest increases

with time (Axelsson, 2000). Evidence suggests thatpiles in silts and find sands set up more than those incoarse sands and gravels (York et al., 1994). Bothdriven and jacked piles exhibit setup, whereas boredpiles do not. Hence, the stress–strain state achievedduring the construction processes of pile driving havean influence on this time-dependent behavior and var-ious mechanisms have been suggested to explain this(Astedt et al., 1992; Chow et al., 1998; Bowman,2002). Unfortunately, at present, there is no conclusiveevidence to confirm any of the proposed hypotheses.

Despite the many field examples and laboratorystudies on aging effects, there is still uncertainty aboutthe mechanism(s) responsible for the phenomenon.Understanding the mechanism(s) that cause aging is ofdirect practical importance in the design and evaluationof ground improvement, driven pile capacity, and sta-bility problems where strength and deformation prop-erties and their potential changes with time areimportant. Mechanical, chemical, and biological fac-tors have been hypothesized for the cause of aging.Biological processes have so far been little studied;however, mechanical and chemical phenomena havebeen investigated in more detail, and some current un-derstanding is summarized below.

12.12 MECHANICAL PROCESSES OF AGING

Creep is hypothesized as the dominant mechanism ofaging of granular systems on an engineering timescaleby Mesri et al. (1990) and Schmertmann (1991). In-creased strength and stiffness does not occur solelyfrom the change in density that occurs during second-ary compression. Rather, it is due to a continued re-arrangement of particles resulting in the increasedmacrointerlocking of particles and the increased mi-crointerlocking of surface roughness. This is supportedby the existence of locked sands (Barton, 1993; Rich-ards and Barton, 1999), which exhibit a tensile strengtheven without the presence of binding cement. Somemicromechanical explanations of the process are givenin Section 12.3.

Although no increase in stiffness was detected whenglass balls were loaded isotropically (Losert et al.,2000), sand has been found to increase in strength andstiffness under isotropic stress conditions (Daramola,1980; Human, 1992). These increases develop even un-der isotropic confinement because the angular particlescan lock together in an anisotropic fabric. It has beenshown that more angular particles produce materialsmore susceptible to creep deformations (Mejia et al.,1988, Human, 1992, Leung et al., 1996). Isotropic

Copy

right

ed M

ater

ial


CHEMICAL PROCESSES OF AGING 517

Time (s)

100 101 102 103 104 105

101 102 103 104 105

0.00

-0.05

0.05

-0.10

-0.15

-0.20

-0.25

-0.30

0.00

0.05

0.10

0.15

0.20

0.25

Glass Ball

Leighton BuzzardUniform Silica Sand

Glass Ball

Dilation

MontpellierNatural Sand

Leighton BuzzardUniform Silica Sand

Montpellier Natural Sand

Stress State at Creep: p� = 600 kPa and q = 800 kPaAll Samples Were Prepared With Relative Density ofApproximately 70%.

Vol

umet

ric S

trai

n (%

)D

evia

toric

Str

ain

(%)

Figure 12.71 Dilative creep observed in triaxial creep testsof dense fine sand (by Bowman and Soga, 2003).

compression tests by Kuwano (1999) showed that ra-dial creep strains were greater than axial strains in soilswith angular particles than in soils with rounded par-ticles due to a more anisotropic initial fabric. Angularparticles can result in longer duration of creep and agreater aging effect since they have a larger range ofstable contacts and the particles can interlock. Asspherical particles rearrange more easily than elon-gated ones (Oda, 1972a), rounder particles initiallycreep at a higher rate before settling into a stable state.Hence, any aging effect on rounded particles tends todisappear quickly when the soil is subjected to newstress state.

When a constant shear stress is applied to loosesand, large creep accompanied by volumetric contrac-tion is observed (Bopp and Lade, 1997). Higher con-tact forces due to loose assemblies contribute toincreased particle crushing, contributing to contractionbehavior. Hence, decrease in volume by soil crushingleads to increase in stiffness and strength.

Field data suggest that displacement piles inmedium-dense to dense sands set up more than thosein loose sand (York et al., 1994). Dense granular ma-terials may dilate with time depending on the appliedstress level during creep as shown in Fig. 12.71 (Bow-man and Soga, 2003). Initially, the soil contracts withtime, but then at some point the creep vector rotatesand the dilation follows. Similar observations weremade by Murayama et al. (1984) and Lade and Lui(1998). This implies that sands at a high relative den-sity will set up more as more interlock between par-ticles may occur (Bowman, 2002). The laboratoryobservation of initial contraction followed by dilationconveniently explains the field data of dynamic com-paction where the greater initial losses and eventualgains in stiffness and strength of sands are found closeto the point of application where larger shear stressesare applied to give dilation (Dowding and Hryciw,1986; Thomann and Hryciw, 1992; Charlie et al.,1992).

Increased strength and stiffness due to mechanicalaging occurs predominantly in the direction of previ-ously applied stress during creep (Howie et al., 2002).No increase was observed when the sand was loadedin a direction orthogonal to that of the applied shearstress during creep (Losert et al., 2000).

12.13 CHEMICAL PROCESSES OF AGING

Chemical processes are a possible cause of aging. His-torically, the most widespread theory used to explainaging effects in sand has involved interparticle bond-

ing. Terzaghi originally referred to a ‘‘bond strength’’in connection with the presence of a quasi-preconsolidation pressure in the field (Schmertmann,1991). Generally, this mechanism has been thought ofas type of cementation, which would increase the co-hesion of a soil without affecting its friction angle.

Denisov and Reltov (1961) showed that quartz sandgrains adhered to a glass plate over time. They placedindividual sand grains on a vibrating quartz or glassplate and measured the force necessary to move thegrains as shown in Fig. 12.72. The dry grains wereallowed to sit on the plate for varying times and thenthe plate was submerged, also for varying times, beforevibrating began. It was found that the force requiredto move the sand grains continued to increase up toabout 15 days of immersion in water. The cementatingagent was thought to be silica-acid gel, which has anamorphous structure and would form a precipitate at

Copy

right

ed M

ater

ial



�

�

�

��

�

�

& &

&

&�

ff0––– 3.0

2.0

1.0

0

4

3

21

10 min 2h 20hTime

(1) Without Soaking(2) 42-hour Soaking(3) 6-day Soaking(4) 14-day Soaking

(1) Glass or Quartz Plate(2) To the Oscillation Generator

1

2

Figure 12.72 Results of vibrating plate experiment fromDenisov and Reltov (1961). Term ƒ/ƒ0 is a measure of thebonding force between sand and glass or quartz plate.

Figure 12.73 Effect of aging on the penetration resistance of River sand (from Joshi et al.,1995).

particle contacts (Mitchell and Solymer, 1984). The in-creased strength is derived from crystal overgrowthscaused by pressure solution and compaction.

Strong evidence of a chemical mechanism being re-sponsible for some aging was obtained by Joshi et al.(1995). A laboratory study was made of the effect oftime on penetration resistance of specimens preparedwith different sands (river sand and sea sand) and porefluid compositions (air, distilled water, and seawater).After loading under a vertical stress of 100 kPa, thevalues of penetration resistance were obtained afterdifferent times up to 2 years. Strength and stiffnessincreases were observed in all cases, and a typical plotof load–displacement curves at various times is shownin Fig. 12.73. The effects of aging were greater for thesubmerged sand than for the dry specimens. Scanningelectron micrographs of the aged specimens in distilledwater and seawater showed precipitates on and in be-tween sand grains. For the river sand in distilled water,the precipitates were composed of calcium (the solublefraction of the sand) and possibly silica. For the riversand in seawater, the precipitates were composed ofsodium chloride.

However, there are several reported cases in whichcementation was an unlikely mechanism of aging, atleast in the short term. For example, dry granular soilscan show an increase in stiffness and strength withtime (Human, 1992; Joshi et al., 1995; Losert et al.,2000). Cementation in dry sand is unlikely, as moistureis required to drive solution and precipitation reactionsinvolving silica or other cementation agents.

Mesri et al. (1990) used the triaxial test data fromDaramola (1980) to argue against a chemical mecha-

Copy

right

ed M

ater

ial


CHEMICAL PROCESSES OF AGING 519

Figure 12.74 Effect of aging on stress–strain relationship of Ham River sand (from Dara-mola, 1980).

nism responsible for aging effects in sands. Figure12.74 shows the effects of aging on both the stiffnessand shear strength of Ham River sand. Four consoli-dated drained triaxial tests were performed on sampleswith the same relative density and confining pressure(400 kPa) but consolidated for different periods of time(0, 10, 30, and 152 days) prior to the start of the tri-axial tests. The results showed that the stiffness in-creased and the strain to failure decreased withincreasing time of consolidation. Although increasedvalues of modulus were observed, the strain at failureis approximately 3 percent. Mesri et al. (1990) arguethat this large strain would destroy any cementation,and therefore another less brittle mechanism must beresponsible for the increase in stiffness.

In summary, experimental evidence indicates thatmechanical aging behavior is enhanced by shear stress

application in denser materials. It is also associatedwith the microinterlocking occurring during the gen-eration of creep strain. The increase in stiffness andstrength is observed in the direction of the appliedstresses, but the aging effect disappears rather quicklywhen loads are applied in other directions. Chemicalaging can also occur within days depending on suchfactors as chemical environment and temperature.

Some conditions in natural deposits are not repli-cated in small-scale laboratory testing. Most laboratorytests are done using clean granular materials, whereasin the field there will be impurities, biological activity,and heterogeneity of void ratio and fabric. Further-more, the introduction of air and other gases duringground improvement may have consequences that haveso far not been fully evaluated. Arching associatedwith dissipation of blast gases and the redistribution of

Copy

right

ed M

ater

ial



stresses through the soil skeleton may also play a role(Baxter and Mitchell, 2004). The boundary conditionsassociated with penetration testing in rigid-wall cylin-ders in the laboratory may prevent detection of time-dependent increases in penetration resistance that aremeasured under the free-field conditions in the field.

12.14 CONCLUDING COMMENTS

With exception of settlement rate predictions, most soilmechanics analyses used in geotechnical engineeringassume limit equilibrium and are based on the as-sumption of time-independent properties and defor-mations. In reality, time-dependent deformations andstress changes that result from the time-dependent orviscous rearrangement of the soil structure may be re-sponsible for a significant part of the total ground re-sponse.

Rate process theory has proven a particularly fruitfulapproach for the study of time-dependent phenomenain soils at consistencies of most interest in engineeringproblems, that is, at water contents from about theplastic limit to the liquid limit. From an analysis of theinfluences of stress and temperature on deformationrates and other evidence, it has been possible to deducethat interparticle contacts are essentially solid and thatclay strength derives from interatomic bonding in thesecontacts. The strength depends on the number of bondsper unit area, and the constant of proportionality be-tween number of bonds and strength is essentially thesame for all silicate minerals, probably because of theirsimilar surface structures.

Recognition of the fact that any macroscopic stressapplied to a soil mass induces both tangential and nor-mal forces at the interparticle contacts is essential tothe understanding of rheological behavior. The resultsof discrete particle simulations show that changes increep rate with time can be explained by changes inthe tangential and normal force ratio at interparticlecontacts that result from particle rearrangement duringdeformation. The change in microfabric in relation tostrong particle networks and weak clusters leads topossible explanation of the mechanical aging process.

Time-dependent deformations and stress relaxationfollow predictable patterns that are essentially the samefor all soil types. Simple constitutive equations can rea-sonably describe time-dependent behavior under lim-ited conditions. Much remains to be learned, however,about the influences of combined stress states, stresshistory and transient drainage conditions on creep,stress relaxation, and creep rupture before reliable

analyses and predictions can be made for large andcomplex geotechnical structures.

QUESTIONS AND PROBLEMS

1. Find an article about a problem, project, or issuethat involves some aspect of the long-term behaviorof a soil as an important component. The articlemay be from a technical journal or magazine orelsewhere. The only requirement is that it involvesconsideration of time-dependent ground behavior insome way.a. Prepare a one-page informative abstract of the

article.b. Summarize the important geotechnical issues in

the article and write down what you believe youwould need to know to understand them wellenough to solve the problem, resolve the issue,advise a client, and so forth. Do not exceed twopages.

c. Identify topics, figures, equations, and other ma-terial in Chapter 12, if any, that might be usefulin addressing the problems.

2. The figure below shows relationships between (1)number of interparticle bonds and effective consol-idation pressure and (2) compressive strength andnumber of interparticle bonds for three soils as de-termined using rate process theory. Determine ��theangle of internal friction in terms of effectivestresses (as determined from CU tests with porepressure measurements), for each soil. Assume Aƒ

� 0, 0.3, and 0.3 for the sand, illite, and Bay mud,respectively, in the range 0 � (�1 � �3)ƒ � 500kPa, where is the ratio of pore pressure at failureAƒ

to the deviator stress at failure (�1 � �3)ƒ.

00

10

20

30

40

100 200 300 400 500

Num

ber

of B

onds

- 1

010cm

- 2

σ�c = Effective Consolidation Pressure (kPa)

Sand

Bay Mud

IlliteCopy

right

ed M

ater

ial


QUESTIONS AND PROBLEMS 521

0 5 10 15 20 25

Number of Bonds - 1010cm-2

0

200

400

600

800

1000

Sand, Illite, Bay Mud

(σ1

–σ 3

) max

(kP

a)

3. Equation (12.43) is a simple three-parameter equa-tion for strain rate during constant stress creep ofsoils.a. Show the meaning of �, D, and m on a clearly

labeled sketch.b. Modify Eq. (12.43) and indicate the information

needed to permit prediction of creep rates for agiven soil at any value of water content andstress intensity from a knowledge of creep ratesat a single water content corresponding to dif-ferent stress intensities.

c. Develop a relationship between stress intensityand time to failure for a soil subject to strengthloss under the application of a sustained stress.

4. The results of triaxial compression creep tests onsamples of overconsolidated Bay mud at three stress

D = 100 kPa

D = 85 kPa

D = 68 kPa

RuptureX

10 100 1000 10,00010

20

30

40

50

Axi

al S

trai

n (%

) Water Content = 60%Dmax = 125 kPa

Time (min)

90

80

70

60

50

40

30

Wat

er C

onte

nt (

%)

Compressive Strength (kPa)20 40 80 100 200 400

intensities are shown below, as is the variation ofcompressive strength with water content. A tem-porary excavation is planned that will create a slopewith an average factor of safety of 1.5. The averagewater content of the clay in the vicinity of the cutis 50 percent. The excavation is planned to remainopen for a period of 4 months. Prepare a plot ofstrain rate versus time for an element of clay andassess the probability of a creep rupture failure oc-curring during this period.

5. Given thata. The creep rate of a soil, for times up to the onset

of failure, can be expressed by Eq. (12.43), inwhich D is the deviator stress, and

b. The time to failure by creep rupture, tƒ, can betaken as the time corresponding to minimumstrain rate, prior to acceleration of defor-� ,min

mation and failure, and tests have shown that

� t � constantmin ƒ

If a test embankment designed at a factor ofsafety of 1.05 based on shear strength deter-mined in a short-term test fails in creep ruptureafter 3 months, how long should it be beforefailure of a prototype embankment having a fac-tor of safety of 1.3? From a plot of deformationrate versus time for the test embankment, it hasbeen found that m � 0.75. The results of short-term creep tests have shown also that �Dmax �6.0. The factor of safety is defined as thestrength available divided by the strength thatmust be mobilized for stability.

Copy

right

ed M

ater

ial



6. Would you expect that creep and stress relaxationwill be significant contributors to the stress–deformation and long-term strength of soils on theMoon? Why?

7. List possible causes of sand aging wherein the stiff-ness and strength (usually as determined by pene-

tration tests) can increase significantly over time pe-riods as short as weeks or months following depo-sition and/or densification. Outline a test programthat might be done to test the validity of one ofthese causes.

Copy

right

ed M

ater

ial


6302712-141018084743-conversion-gate01

Documents

timedependent deformations

time effects

stressdependent changes

timedependent chemical

deformsover time

stress changes thestructural

stress history

postdepositional changes