Microstructural model for delayed deformation of clay: loading … · 2017-12-17 ·...

Microstructural model for delayed deformation ofclay: loading history effects

Eduardo E. Alonso and Vicente Navarro

Abstract: Microstructural observations of clayey soils indicate that clay particles are aggregated in units of low inter-nal porosity (microporosity). Larger voids (macroporosity) between those units are associated with the flow of free wa-ter. Within this context, the secondary deformation of clay is interpreted as being a consequence of local mass transferof water from the micropores to the macropores. The capability of a model based on this idea to reproduce the effectsof loading history on the secondary compression of clay is explored. An experimental test program has been per-formed, secondary compression records have been interpreted within the proposed theoretical framework, and materialparameters have been derived. The simulation of additional validation tests has been successful. The theory and modeldeveloped provide a unified framework for compression, as well as swelling secondary deformation.

Key words: secondary consolidation, creep, preloading, overconsolidation ratio, oedometer tests, swelling.

Résumé : Les observations de la microstructure des sols argileux indique que les particules d’argile sont aggloméréesdans des unités de faible porosité interne (microporosité). Les pores plus grands (macroporosité) entre ces unités sontassociés à l’écoulement de l’eau libre. Dans ce contexte, la déformation secondaire de l’argile est interprétée commeune conséquence du transfert de masse d’eau de la microscopie vers la macroscopie. En se basant sur cette idée, onexplore la capacité d’un modèle de reproduire les effets de l’histoire du chargement sur la compression secondaire del’argile. Un programme expérimental d’essais a été réalisé, des données de compression secondaire ont été interprétéesdans le cadre théorique proposé, et des paramètres du matériau ont été dérivés. La simulation d’essais additionnels devalidation a été couronnée de succès. La théorie et le modèle développés fournissent un cadre de travail unifié pour lacompression de même que pour les déformations secondaires de gonflement.

Mots clés : consolidation secondaire, fluage, préchargement, rapport de surconsolidation, essais d’oedomètre, gonfle-ment.

[Traduit par la Rédaction] Alonso and Navarro 392

Introduction

Secondary compression of clays has usually been mod-elled with rheological models (Gibson and Lo 1961;Bjerrum 1967, 1972; Nash 2001) and other phenomen-ological formulations that have been discussed in Imai(1995) and Tatsuoka et al. (2000). From a different perspec-tive, some authors have understood secondary compressionas a local mass transfer of water between the soil macro-structure and the associations of clay particles(microstructure) (de Jong 1968; Berry and Poskitt 1972;Mitchell 1993; Sills 1995). On the basis of similar ideas,Navarro and Alonso (2001) presented a framework for thelong-term compression of clays. The model was successfullyapplied to the analysis of simple records of primary and sec-

ondary compression of three different clay soils. Neverthe-less, the capability of the model to describe the secondaryresponse induced by a general preloading history was leftunexplored. The objective of this paper is to extend the ex-isting framework to address the effect of loading–unloadinghistories on subsequent secondary deformations. Model pre-dictions are compared with the results of long-term com-pression, and swelling was recorded.

Theoretical framework

Within a given representative volume of soil, water occu-pying the larger open voids (macrostructural water) andwater filling the small voids inside the clay aggregates(microstructural water) may be interchanged as a result oflocal differences in their chemical potentials (µM and µm re-spectively). This process may be interpreted also as a phasechange between two states of the same fluid. If a linear ap-proach is used (de Groot and Mazur 1984), the mass ex-change of water may be determined by

[1] cm = α(µM – µm)

where α is a phenomenological transfer coefficient; and cm isthe mass-transfer rate per unit volume. If the macrostructuralwater is assumed to be an ideal solution, its chemical poten-tial may be deduced from Edlefsen and Anderson (1943):

Can. Geotech. J. 42: 381–392 (2005) doi: 10.1139/T04-097 © 2005 NRC Canada

381

Received 14 March 2003. Accepted 4 September 2004.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 16 April 2005.

E.E. Alonso.1 Department of Geotechnical Engineering andGeosciences, Universitat Politècnica de Catalunya, GranCapitán s/n, 08034 Barcelona, Spain.V. Navarro. Civil Engineering School, Universidad deCastilla la Mancha, Camilo José Cela s/n, 13071 CiudadReal, Spain.

1Corresponding author (e-mail: [email protected]).

[2] µ µρM L M Vo

L Lo

W WM

MW( , , ) ( )

( )lnT p x T

p p RTx= + − +

where µVo( )T Vo is the chemical potential of water vapour inequilibrium with macrostructural water at temperature T andreference pressure pLo; MWW is the molecular weight of wa-ter; ρW is the bulk water density; pL is the pressure of theliquid in the macrostructure; R is the universal gas constant:and xM is the mole fraction of the macropore water (xM =nM/nL, where nM is the number of moles of macrostructuralwater and nL is the total number of moles of macrostructuralliquid (water plus dissolved salts)).

The chemical potential of the microstructural water wasshown to be (Navarro and Alonso 2001) the following:

[3] µ σ µ σ πρm m Vo

m

m

( , , ) ( )( , )

T e TT e= + −

where the function π is the swelling pressure; ρm is the den-sity of the micropore water; and em is the microstructuralvoid ratio (defined as volume of voids in the microstructure/volume of clay minerals). Note that em should be distin-guished from both the macrostructural void ratio, eM(defined as volume of voids in the macrostructure/volume ofclay minerals), and the conventional void ratio e (e = eM +em).

The function π defines the stress that must be applied toensure that under temperature T, the void ratio of the micro-structure is em when the macrostructural water is at the refer-ence pressure (usually atmospheric). Equation [3] can beviewed as a generalization of the classical expression of Lowand Anderson (1958) for when an arbitrary stress referencestate, σ, is introduced. The effect of temperature on π is notdiscussed in this paper. Equation [3] was derived for a con-stant mass of solid. Therefore, both the mass of clay miner-als and the mass of exchangeable cations are assumed to beconstant. This assumption does not hold for ion exchangeprocesses, where both the amount and the kind of exchange-able cations would change. These exchanges will affect thesoil microfabric, modifying its swelling potential and, as aconsequence, the chemical potential of its micropore water.Although the structure of eq. [3] will hold if substantialchanges in exchangeable cations are expected, the expres-sion used to compute the swelling pressure should be pro-gressively updated to maintain the value of π consistent withthe current microfabric.

Swelling pressure determined in natural and compactedclayey soils is often exponentially related to dry densityor the void ratio. In contrast, swelling strains measured insoaking-under-load tests seem to depend linearly on the log-arithm of applied confining stress (Alonso et al. 1987).These experimental observations suggest that an appropriaterelationship between π and em should be of the followingform:

[4] emf – emo = –D ln(πf /πo)

where the constant D depends on the soil type and tempera-ture and describes the microstructural deformability; and thesubscripts “f” and “o” refer to two reference states (final andinitial, respectively).

Using eqs. [2] and [3], eq. [1] may be expressed as

[5] cm = G (π – πB)

where G is the microstructural–macrostructural water trans-fer coefficient; and πB is defined as

[6] π σ ρρ

ρB

m

WL Lo

m

Wm

MW= − − −( ) lnp p RT x

where πB may be interpreted as the water pressure potentialthat the “boundary” (loading and state of the macrostructuralwater) imposes on the microstructure. Note that in the caseof pure water (xM = 1), if the reference pressure, pLo, istaken as zero for convenience, the stress πB becomes essen-tially the effective stress. It will be exactly the effectivestress if the densities of adsorbed and free water are equal.

In eq. [5], the term G describes the microstructure tomacrostructure water transfer properties. Navarro andAlonso (2001) performed an analysis of a standard consoli-dation test on an undisturbed clay sample from Ares Estuaryin Galicia, Spain (wL = 51%; wp = 38%; w = 40%, where wLis the liquid limit, wp is the plastic limit, and w is the naturalwater content) to compute the value of G/ρm. The obtainedvalues for the steps 80–150, 150–300, and 300–600 kPa ofthe secondary consolidation stage are shown in Fig. 1. It canbe seen that G/ρm decreases with the reduction of void ratio.This reduction is initially fast, but it slows down as the soildeforms. The three records plotted in Fig. 1 are similar, andthey may be approximated by exponentially decaying func-tions. Figure 1 also shows that whenever a new load is ap-plied, the secondary deformation process proceeds at ratessimilar to the rates recorded at the start of previous loadingsteps. There is, however, a slow absolute decrease of thetransfer coefficient G/ρm as the deformation accumulates.These trends suggest that the local dehydration processmay be approximated by microconsolidation phenomena inwhich the clay clusters or clay aggregates lose water towardthe macropores. If aggregates are described by some regulargeometry, such as microlayers with double drainage, theclassical theory of consolidation offers the possibility offinding an expression for G/ρm.

Within this analogy, the excess pore water pressure insidea consolidating microlayer will be a measure of the differ-ence between the effective stress inside the aggregates, π,and the stress imposed on their boundaries, πB. Then, thefollowing relationship may be found between the volumetricdeformation rate (rate of water expelled from the aggregates,qm = cm/ρm) and the average excess pore water pressure inthe consolidating layer, u:

[7]G q

uS

KH

S

m

m T

n

m T

n

ρρ

γρ

m

mm

Wm

2

e

e≈ =

−

=

∞

−

=

∞

∑

∑

2

2

0

0

where S is the specific surface of the clay; K is the layer per-meability; H is the layer thickness; and m = (2n + 1)(π/2).The following classical expression for the degree of consoli-dation provides the deformation (εm) as a function of time:

[8]εε

m e

∞

−

=

∞

= − ∑1 22

20

m T

n m

© 2005 NRC Canada

382 Can. Geotech. J. Vol. 42, 2005

where ε∞ is the final deformation of the microlayer. Equa-tions [7] and [8] have been used to represent G/ρm as a func-tion of εm/ε∞ in Fig. 2. A comparison of Fig. 2 with theexperimental plots of Fig. 1 suggests that consolidation ofthe clay aggregates may be a suitable physical explanationfor the delayed deformations. It should be added that thisanalogy incorporates in a natural way the observed recoveryof initial secondary strain rates whenever a new incrementalload is applied to the soil. If these results are taken intoaccount, the following relationship between G/ρm and thechange in microstructural void ratio is suggested:

[9] G g e c/ exp( / )ρm m= −0 ∆

where c is a material parameter; and the decrease in micro-void ratio (∆em) starts at the current void ratio at the begin-ning of every new loading increment. If eq. [4] is taken intoaccount, eq. [9] may be rewritten as

[10] G g

g H

D c

D c D c C

/ ( / )

( / ) ( / ) ( / )

/

/ /

ρ π π

π π π π π πm

B B

=

= =0 0

0 0 Β

The material parameter c is a global measure of the shape,size, permeability, and stiffness of the clay aggregates. Equa-tions 7 and 8 provide an interpretation of this relationship.Therefore, C (= D/c) synthesizes the effect of clay aggregatestructure on the evolution of the rate of mass interchange.On the other hand, the initial value of the transfer term, g0 =(G /ρm)0, seems to depend on the current value of themicrovoid ratio (Fig. 1). This dependence appears to beweak, but it has an obvious sense: one should expect a pro-gressive difficulty inmicrostructural–macrostructural watertransfer as the soil structure becomes more compressed.However, a sufficiently satisfactory fit was obtained for the

Ares clay using a constant g0 (see Fig. 3, where D = 3.38 ×10–2, c = 2.78 × 10–3, and g0 = 1.05 × 10–6 (s·kPa)–1). If g0 isassumed to be constant, the material parameter H is also aconstant; it defines the mass-transfer coefficient at the be-ginning of the secondary compression (eq. [10], for π = πB).

The conventional secondary index, Cα, can be derivedin terms of the parameters of the model described here(Navarro and Alonso 2001). It was found that Cα depends ontime and on the particular loading step considered. Furtherdiscussion concerning Cα and its relationship with the theorypresented is given in the same reference.

To solve general one-dimensional consolidation problems,eq. [5] should be introduced into the balance equations forboth macrostructural water and microstructural water(Navarro and Alonso 2001):

[11]1

10

++ + =

eD

DS M M m

W

et

qz

c∂∂ ρ

( )

[12]D

DeS M

mm

et

c= +1ρ

where the operator DS(•)/Dt is the material derivative withrespect to the movement of the solid skeleton; and qM is themass flow of macrostructural water. In the present paperonly thin samples were tested, and a traditional uncoupledapproach has been adopted. Therefore, attention is paid tosecondary compression only. A parameter estimation pro-gram that identifies secondary compression parameters waswritten in connection with a finite difference program thatsolves the differential equation (eq. [12]) for secondary de-formations. The “search space” for the identification proce-dure is the three-dimesional space D–C–H of all feasibleparameters. This is a large space, and the existence of local

© 2005 NRC Canada

Alonso and Navarro 383

Fig. 1. Experimental values of G/ρm determined in loading steps 80–150, 150–300, and 300–600 kPa of an oedometer test on the Aresclay. emo, microstructural void ratio at the beginning of step 80–150 kPa.

minima for the objective function is very likely. If thereis no reliable first estimation of parameters, minimizationmethods based on gradient techniques (Gauss–Newton orLevenberg–Marquardt) or even Powell’s method will proba-bly end in a local minimum. To avoid these difficulties, anidentification criterion based on minimum root mean squareerrors was put forward. The “shape” of the root mean squareerror was characterized along the search space by means ofan algorithm for systematic sampling. A dense population ofparameter vectors was generated, and the estimation error

associated with each of them was computed. The bestapproximation was then identified, and the optimum set ofparameters was determined.

Stress path effects

The outlined theoretical framework provides the basis fordiscussing the expected secondary response of a soil sub-jected to loading paths inspired by the usual preloadingstrategies. It will be assumed, for simplicity, that no salts are

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Fig. 2. Computed deformation rates for a finite layer.

Fig. 3. Variation of G/ρm as a function of the change in microporosity, ∆em, for loading steps 80–150, 150–300 and 300–600 kPa in anoedometer test on Ares clay.

present in the macrostructural water and that the densities offree water and microstructural water are equal. The potentialof the free water vapour, µVo, will be taken as the reference(zero) value.

Consider first the effect of a loading–unloading cyclecharacterized by a given value of the overconsolidation ratio(OCR). The void ratio – effective stress compression curveis indicated in Fig. 4a. The sample will be initially equili-brated under stress σ1′ at point A. If the stress σ1′ has beenacting for a long time, the swelling pressure, π, will also beequal to σ1′ . This is indicated in Fig. 4b. A loading step( )σ σ12′ − ′ is now applied. Once the transient hydrodynamic(primary) period is completed the new position of equilib-rium will be point B in Fig. 4a. However, this is a fastchange, and therefore the microstructural equilibrium stress,π, will not change. The sample is now maintained under aconstant effective stress, σ2′ , at point B for a certain period.Secondary (microstructural) deformations develop, B → Bα(Fig. 4a). Then the sample is unloaded, Bα → C; and oncethe primary expansion is completed, a new equilibriumpoint, C, is reached. The sample is finally maintained atpoint C, and as time increases, secondary deformations de-velop, C → Cα.

Note that the creep stage at point B generates micro-structural deformations (∆emi), which will increase themicrostructural equilibrium stress to point Bα (Fig. 4b).

However, the (fast) unloading, Bα → C, does not introducechanges in the microstructural stress π. Therefore, the creepstage at point C will start when the effective stress on thesample is σ3′ , whereas its microstructural equilibrium stressis given by πΒα

in Fig. 4b. Notice that σ π2 Βα′ > , and there-

fore the microstructural–macrostructural potential gradientimplies a dehydration of the microstructure and an addi-tional secondary compression, which will progressivelymove the equilibrium state of the microstructure toward po-sition Cα.

Consider, however, the same test in Figs. 4c and 4d, withthe sole difference of allowing a longer secondary creepstage at point B. The result of this longer creep is that thenew microstructural equilibrium stress at Bα (πΒα

) exceedsthe unloading stress σ3′ (σ πΒα3′ < ). Then, the water poten-tial gradient established between microstructures and macro-structures at point C after unloading reverses sign, andhydration of the microstructure is predicted. This phenome-non leads to delayed soil swelling. This example illustratesthe fundamental role played in the model by the micro-structural equilibrium stress: controlling long-term compres-sion or swelling phenomena as a response to changes inconfining stress.

The unloading step (σ2′ – σ3′ ) may be characterized by aformal OCR (OCR = σ σ2′ ′/ 3 ). It is clear that an increase inOCR leads first to progressive reduction of delayed com-

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Fig. 4. Interpretation of primary and secondary deformations in a compression test. (a), (c) Variation of total void ratio with effectivestress. (b), (d) Variation of microstructural void ratio with swelling pressure.

pression and eventually to long-term soil expansion. Thetransition is, however, smooth and continuous.

Other paths may now be easily analysed. Consider, for in-stance, the three alternative paths plotted in Fig. 5a. PathABCDD1 implies rapid loading–unloading. It is a primarypath. At point D1 = C1, the microstructure has not been de-hydrated, and the equilibrium microstructural stress remainsat its original value (Fig. 5b). The other alternative paths(path ABB1B2, which accumulates the higher creep strain;and the intermediate path ABCC1) imply periods of creepbefore the target point C1 is reached: they imply different πstresses on the microstructure, as indicated in Fig. 5b. There-fore, once the specimen is maintained under constant effec-tive stress σc′ , the long-term reaction of the three casesconsidered will be different, as illustrated in Figs. 5c and 5d.The largest secondary strains are predicted for pathABCDD1, because this path implies the largest unbalancebetween the boundary stress, σc′ , and the current micro-structural equilibrium stress (the swelling pressure), πA. Theminimum development of creep strain is found for the pathABB1B2, because the soil microstructure is loaded withthe maximum equilibrium stress, πB1 (and the appliedeffective — or boundary — stress remains the same).

The results of a series of laboratory tests following thestress paths qualitatively indicated in Figs. 4 and 5 will nowbe compared with model predictions.

Test results and model predictions

First series of tests: effect of OCR and preloading timeSamples from the Llobregat Delta silty clay, taken in

boreholes, were selected for several long-term oedometertests. The idea was to investigate the soil response under thetype of stress paths usually imposed by preloading, as de-scribed before. The soil tested is a low plasticity silty clay(wL = 32%; wP = 18%). Values of the virgin compression in-

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Fig. 5. Interpretation of the effect of alternative stress–time paths. (a) Variation of total void ratio with effective stress. (b) Variation ofmicrostructural void ratio with swelling pressure. (c) Change in void ratio during the creep stage. (d) Expected creep deformation his-tories.

Test No. σ2′ (kPa) σ3′ (kPa)OCR(σ σ2 3′ ′/ )

Time elapsedunder σ σ2 2

′ ′( )t

500.200 400 200 2.00 500 h500.120 240 200 1.20500.105 210 200 1.05100.200 400 200 2.00 100 h100.120 240 200 1.20100.105 210 200 1.05

10.200 400 200 2.00 10 min10.120 240 200 1.2010.105 210 200 1.05

Table 1. Definition of testing program (remoulded samples ofsilty clay from the Llobregat Delta, Barcelona).

dex, Cc, varied between 0.1 and 0.3. The ratio Cs/Cc wasfound to be 0.12 on average. The Cα values were determinedin situ by interpreting long-term extensometer data of thementioned preload test. These Cα values were found to varybetween 2 and 8 × 10–3 under normally consolidated condi-tions. For the purposes of the present investigation, naturalsoil with a water content close to the liquid limit was re-moulded to initially prepare the specimens. Samples 8 mmthick were tested in oedometer cells. For this thickness, pri-mary consolidation was essentially completed within thefirst 100 s of loading–unloading. All the deformation histo-

ries presented in the paper correspond to measurementstaken at times in excess of the first 100 s after loading or un-loading.

Nine consolidation tests were performed. They are definedin Table 1 in terms of consolidation stress, σ2′ (refer toFigs. 4a and 4c); unloading stress, σ3′ ; and time spent underσ2′ , t ′σ2

. During the first part of a given test, samples wereloaded in small steps until a vertical stress of 100 kPa wasreached. This stress was maintained for 1 h. Then the stressσ2′ and the rest of the testing protocol were applied. The fi-nal part of all tests involved the recording of sample settle-

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Fig. 6. Recorded strain histories for three specimens of remoulded silty clay in the 100 h preloading time series.

Fig. 7. Secondary settlements rates recorded for all the samples tested under stress σ3′ = 200 kPa.

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ment under σ3′ . The OCR of the specimens tested is definedas σ σ2 3′ ′/ . Tests were performed in series of three specimensmanufactured from a common batch of remoulded soil. Eachone of these series is characterized by a common consolida-tion time under σ2′ and varying OCR, as indicated in Ta-ble 1. Once a test series was completed, a new batch ofremoulded soil was prepared for the next series of threetests.

Figure 6 shows the complete deformation–time history ofthe three specimens corresponding to a consolidation time,t ′σ2

, of 100 h under σ2′ . After unloading, when the specimenis maintained under σ3′ = 200 kPa, the secondary strains re-corded (that is, deformations experienced for times in excessof the first 100 s) show the effect of OCR. For a low value(OCR = 1.05) the specimen exhibits a long-term compres-sion creep. However, at OCR = 2, a long-term expansionwhose rate decreases with time is recorded. When the sec-ondary strain–time behaviour recorded in all specimens un-der the common σ3′ = 200 kPa is plotted together with acommon time origin (Fig. 7), the joint effect of OCR andpreloading time is observed. Long-term compression, whichcorresponds to low OCR and short preloading times, gradu-ally evolves towards a long-term expansive behaviour whenOCR and preloading time increase.

The parameter estimation procedure described before wasapplied to the set of nine long-term tests performed; theidentified parameters are given in Table 2 for each of thetests performed. The presence of salts in the macrostructuralwater was disregarded. A temperature of 20 °C was main-tained almost constant during the tests.

A comparison between measured secondary deformationsunder σ3′ = 200 kPa and model computations is given inFigs. 8a–8c. The accuracy of the simulations of each indi-vidual test is very good, and this shows the flexibility of themodel to reproduce the observed secondary behaviour. How-ever, a satisfactory prediction requires that a single set ofmodel parameters be able to reproduce the whole set of testsperformed. Variability between different samples is alwaysunavoidable, but some of the adjustments between modeland experiment had a low correlation coefficient. This is thecase with tests 500.105 and 100.105, which correspond toa low OCR (OCR = 1.05). In those two tests the pre-consolidation load was maintained for a relatively long time:500 and 100 h, respectively. The values of the parameter Didentified for these two tests were significantly higher thanthe rest of the D values (see Table 2). The actual deforma-tion records for these tests performed at a very low OCR(see Fig. 8c) suggest that friction could have had a signifi-cant effect, reducing the measured effect of unloading. Ap-parently, the short preloading time for test 10.105 (10 h),performed also at OCR = 1.05, reduced this frictional effect.In tests 100.120 and 10.200, swelling develops in steps, aphenomenon that may also be associated with side friction.In tests 100.120 and 10.200, unlike in the rest of the experi-ments, swelling is recorded first and a (small) compressionis measured at the end.

Despite these observations, the set of identified parame-

ters shown in Table 2 exhibits significant similarities. A sin-gle set of parameters was identified for the tests not affectedby the preceding comments: 500.200, 500.120, 100.200,10.120, and 10.105. In this case a joint optimization proce-dure was followed, and the derived parameters are shown inTable 3. Figure 9 shows a comparison between measuredsecondary deformations and model predictions. The correla-tion coefficient between calculated and measured values isvery high: 96.5% (Fig. 10).

The set of tests represented in Fig. 9 covers the full rangeof observed responses, which evolve from a marked second-ary compression for low OCR and short preloading time(test 10.105) to a significant secondary swelling for highOCR and long preloading times. This transition is related tothe evolution of the pressure potentials π and πB (essentially,the effective stress). Model behaviour may be convenientlyfollowed if these two potentials are calculated along time forthe five selected tests. This was done through the finite dif-ference program previously mentioned. Results are repre-sented in Fig. 11. Whenever π > πB, a swelling behaviourwill develop because a transfer of free (macrostructural) wa-ter toward the microstructure takes place. This is the casewith test 500.200 (see Fig. 11) when the stress on the speci-men is reduced to 200 kPa after a period of 500 h of second-ary creep. By that time, the swelling pressure of themicrostructure has reached a value of 397 kPa > 200 kPa,and water will be adsorbed onto the clay microstructure. Intest 10.105 (see Fig. 11), the development of swelling pres-sure during the fast creep interval before unloading is verysmall (π has reached a value of 105 kPa), and further com-pression creep should be expected for the final stress (σ3′ =200 kPa) because π < πB and water will abandon the micro-structure.

Fig. 8. Comparison of measured secondary settlements (dots) and model computation (bold lines) under σ3′ = 200 kPa. (a) Specimensof the 500 h test series. (b) Specimens of the 100 h test series. (c) Specimens of the 10 min test series.

TestNo.

H × 1011

(s·kPa)–1 C D × 103Correlation(R2 index) (%)

500.200 1.114 7.750 7.111 98.1500.120 1.240 7.700 5.000 76.7500.105 1.000 6.266 10.000 40.7100.200 0.800 8.000 4.700 95.9100.120 1.152 1.450 1.000 96.2100.105 1.000 7.764 10.000 80.3

10.200 7.480 5.044 1.000 79.310.120 0.927 3.247 4.319 99.410.105 0.625 4.250 7.243 99.5

Table 2. Identified model parameters.

H (s·kPa)–1 C D

1.043×10–11 7.734 5.366×10–3

Table 3. Identified model parameters: jointestimation for tests 500.200, 500.120,100.200, 10.120, and 10.105.

Validation tests: alternative stress–deformation pathsending in the same state

In the case plotted in Fig. 5a, the idea was to check themodel by performing new creep tests, of a different nature,once the basic soil parameters had been identified throughanalysis of the first series of tests. Parameters given in Ta-ble 3 were assumed to represent the secondary behaviour ofthe Llobregat Delta silty clay. In fact, the tests performed in-volved a double check, because the tests had first to be de-fined. Once a target stress to measure creep is selected (σC′ =200 kPa), the values of preloading stresses σB′ and σD′ andthe associated preloading creep times had to be found so that

the deformation plot shown in Fig. 5a could be producedwith a common final point in C1. The test was therefore de-signed with the help of the program written to solve the pri-mary and secondary consolidation problem, following thedeveloped theory. The derived preloading stresses and creeptimes are indicated in Table 4.

Three specimens of the reference soil were then preparedunder conditions identical to those for the first series oftests. The specimens were subjected to the stress–time pathsdefined by Fig. 5a and Table 4. Once the specimens hadreached the common state at point C1, the secondary strainswere measured for an additional period of 11 days. The

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Fig. 9. Measured (light line, bold symbol) and simulated (bold line, light symbol) secondary settlements of specimens of the 500.200,500.120, 100.200, 10.120, and 10.105 tests (see Table 1) under σ3′ = 200 kPa.

Fig. 10. Correlation between measured and simulated secondary compression for tests analysed in Fig. 9. Displacements are normal-ized with respect to the maximum value measured at each individual test.

measured secondary behaviour and the model prediction us-ing the parameters of Table 3 are plotted in Fig. 12. Themodel seems to capture the soil behaviour in a consistentway. The measured intensity of the secondary strains is or-dered in the manner predicted by the theory. The correspon-dence is also good from a quantitative perspective, and thisprovides encouraging support to the proposed model.

Summary and conclusions

Long-term deformations (expansion or compression) havebeen associated with local hydration or dehydration of thesoil microstructure. A workable model in terms of themacrostructural liquid pressure and the microstructural voidratio has been proposed. In this model, three material param-

eters control the delayed deformation. Their physical naturehas been discussed through an interpretation of the local wa-ter transfer in terms of microconsolidation phenomena.

Model calculations have been compared with the resultsof three series of oedometer tests designed to investigate the

© 2005 NRC Canada


Fig. 11. Evolution of microstructural swelling pressure, π, for tests modelled in Fig. 9.

Path i σ i′ (kPa)Preloading timeunder σ i′

1 σD′ = 210 100 s2 σC′ = 200 1 h 30 min3 σB′ = 195 10 days 9 h

Table 4. Definition of the stress–time path(Fig. 5a).

Fig. 12. Simulated and experimental creep histories for three different preloading stress–time paths (see Table 4).

effect of OCR and preloading times. The model is capableof making satisfactory quantitative predictions from long-term compression to long-term expansion when OCR andpreloading times increase. This is not common in existingmodels for secondary deformations, which usually areoriented to compressive strains. Perhaps more relevant isthe fact that this model provides a consistent interpretationof the experimental observations. Examples have also beengiven of the evolution of the basic constitutive functions(such as water potentials and swelling pressure) as the soildeforms. A validation series of tests has also been presented,and these compared satisfactorily with model predictions.

The model seems to provide a comprehensive and fairlysimple interpretation of delayed deformations of clayey soilsunder general preloading–time (or strain) histories.

References

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Berry, P.L., and Poskitt, T.J. 1972. The consolidation of peat.Géotechnique, 22(1): 27–52.

Bjerrum, L. 1967. Engineering geology of Norwegian normally-consolidated marine clays as related to settlement of buildings.7th Rankine Lecture. Géotechnique, 17(2): 81–118.

Bjerrum, L. 1972. Embankments on soft ground. State of the Artreport. In Proceedings, Specialty Conference on Performance ofEarth and Earth-supported Structures. Purdue University, WestLafayette, Ind. ASCE, New York. Vol. 2, pp. 1–54.

de Groot, S.R., and Mazur, P. 1984. Non-equilibrium thermody-namics. Dover, London.

de Jong, G.J. 1968. Consolidation models consisting of an assem-bly of viscous elements or a cavity channel network. Géo-technique, 18: 195–228.

Edlefsen, N.E., and Anderson, A.B.C. 1943. Thermodynamics ofsoil moisture. Hilgardia, 15(2): 31–298.

Gibson, R.E., and Lo, K.Y. 1961. A theory of consolidation forsoils exhibiting secondary compression. Norwegian Geo-technical Institute, Oslo, Publication No. 41. pp. 1–16.

Imai, G. 1995. Analytical examinations of the foundations toformulate consolidation phenomena with inherent time-dependence. In Compression and consolidation of clayey soils.Edited by H. Yoshikuni and O. Kusakabe. A.A. Balkema, Rot-terdam. pp. 891–935.

Low, P.F., and Anderson, D.M. 1958. Osmotic pressure equationsfor determining thermodynamic properties of soil water. SoilScience, 86: 251–253.

Mitchell, J.K. 1993. Fundamentals of soil behaviour. 2nd ed. JohnWiley & Sons Ltd., New York.

Nash, D. 2001. Discussion: “Precompression design for secondarysettlement reduction”. Géotechnique, 51(9): 822–823.

Navarro, V., and Alonso, E.E. 2001. Secondary compression ofclays as a local dehydration process. Géotechnique, 51(10):859–869.

Sills, G. 1995. Time dependent processes in soil consolidation. InCompression and consolidation of clayey soils. Edited byH. Yoshikuni and O. Kusakabe. A.A. Balkema, Rotterdam.pp. 875–889.

Tatsuoka, F., Santucci de Magistris, F., Hayano, K., Momoya, Y.,and Koseki, J. 2000. Some new aspects of time effects on thestress–strain behaviour of stiff geomaterials. In The geotechnicsof hard soils – soft rocks. Proceedings of the 2nd InternationalSymposium on Hard Soils – Soft Rocks, Naples, Italy, 12–14 October 1998. Edited by A. Evangelista and L. Picarelli.A.A. Balkema, Rotterdam. Vol. 3, pp. 1285–1372.

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