effects of viscous property and wetting on 1-d …

897

i) Shanghai Jiao Tong University, China, Formerly, Research Fellow, Tokyo University of Science and Saitama University.ii) Bandung Institute of Technology, Indonesia.iii) Tokyo University of Science, Japan (tatsuoka＠rs.noda.tus.ac.jp).

The manuscript for this paper was received for review on June 25, 2010; approved on May 20, 2011.Written discussions on this paper should be submitted before May 1, 2012 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku,Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

Fig. 1. Three-component model

897

SOILS AND FOUNDATIONS Vol. 51, No. 5, 897–913, Oct. 2011Japanese Geotechnical Society

EFFECTS OF VISCOUS PROPERTY AND WETTING ON 1-DCOMPRESSION OF CLAY AND MODEL SIMULATION

JIANLIANG DENGi), HASBULLAH NAWIRii) and FUMIO TATSUOKAiii)

ABSTRACT

A series of one-dimensional (1D) compression tests on compacted kaolin powder were performed to evaluate thecombined eŠects of the viscous property and wetting on the elasto-viscoplastic deformation of soil. In the tests, bothcreep deformation and collapse deformation due to wetting were allowed to take place at various ˆxed stress states dur-ing otherwise monotonic loading at a ˆxed strain rate. Combined eŠects of the viscous property and wetting on thestress-strain behaviour observed during 1-D compression were described by incorporating the wetting eŠects into anon-linear three-component elasto-viscoplastic model (a 3C model). Based on the experimental results, the eŠects ofwetting on the inviscid stress and the irreversible strain relation of the plastic component of the 3C model and theproperty of the viscous component, having an Isotach property, are formulated as a function of the degree of satura-tion. Complicated rate- and time-dependent stress-strain behaviour observed during saturation at a ˆxed stress stateand subsequent monotonic loading at a constant strain rate were successfully simulated.

Key words: clay, collapse, creep deformation, elasto-viscoplastic property, oedometer tests, simulation (IGC:D5/E13)

INTRODUCTION

Settlement in soft soil deposits is one of the most com-mon geotechnical engineering issues. The eŠects of wet-ting comprise one of the major factors of the one-dimen-sional compression of unsaturated soil (e.g., Jenningsand Burland, 1962; Jotisankasa et al., 2007; De Gennaroet al., 2009). Since Jennings and Knight (1957) proposedan experimental method to evaluate the collapse defor-mation during the wetting of soil, a number of resear-chers have proposed diŠerent models to describe the col-lapse deformation for diŠerent types of geomaterials un-der various loading conditions (e.g., Wheeler et al., 2003;Gallipoli et al., 2003). Although the viscous property isnot taken into account in most of these models, its sig-niˆcant eŠects have been observed not only in saturatedclay (e.g., Kawabe et al., 2009), but also in air-dried orunsaturated compacted specimens of clay powder (e.g.,Li et al., 2004; Deng and Tatsuoka, 2004, 2007). There-fore, it is necessary to properly evaluate the eŠects of theviscous property on deformation during the wetting proc-ess. De Gennaro et al. (2009) considered the eŠects ofboth the viscous property and the wetting of ‰uid-ˆlledporous chalks. The framework of the model employed inthe present study (explained below in detail) is similar totheir model. However, the yield stress becomes zero whenthe strain rate becomes zero in their model, and there-

fore, the behaviour during cyclic loading cannot be ad-dressed. On the other hand, the above is not the case inthe model used in the present study, and thus, the modelcan simulate the elasto-viscoplastic behaviour under cy-clic loading conditions in which the sign for the strain ratechanges arbitrarily including zero strain rate states(Kawabe et al., 2009). Furthermore, the details are diŠer-ent in many respects due to the diŠerent trends of behav-iour of the diŠerent types of geomaterials dealt with intheir study and ours.

Di Benedetto et al. (2002, 2005), Tatsuoka et al. (2002,2008a) and Tatsuoka (2007) showed that the non-linearthree-component elasto-viscoplastic model described inFig. 1 can properly simulate the viscous eŠects on thestress-strain behaviour, including creep deformation and

898898 DENG ET AL.

the eŠects of strain rate on the monotonic loading (ML)stress-strain relation of a wide variety of geomaterials, in-cluding soft and stiŠ clay, sand, gravel, sedimentary softrock and cement-mixed soil, observed in various labora-tory stress-strain tests, including triaxial and plane straincompression tests, torsional and direct shear tests andone-dimensional (1D) compression tests. Moreover, Tat-suoka et al. (2008b) and Ezaoui et al. (2010) showed thatthe eŠects of ageing on the elasto-viscoplastic stress-strain behaviour of young cement-mixed soil can be prop-erly simulated by incorporating these eŠects into theproperties of the hypo-elastic component of the modeland the inviscid stress and irreversible strain relation ofthe plastic component.

In parallel to the above, a long-term research programhas been undertaken to evaluate the combined eŠects ofviscous properties, wetting and ageing on the drainedstress-strain behaviour of soil. The present paper reportsthe results of the following studies performed as part ofthis research program:1) The 1D deformation of compacted kaolin powder

during the wetting process, from air-dried conditionsto diŠerent degrees of saturation at diŠerent ˆxedstress states, was experimentally evaluated. The eŠectsof this process on the stress-strain behaviour duringsubsequent ML at a ˆxed strain rate were also evaluat-ed.

2) The three-component model was modiêd to describethe combined eŠects of the viscous property and wet-ting on the stress-strain behavior of kaolin. In sodoing, the eŠects of wetting were treated as negativeeŠects on the inviscid stress and irreversible strain be-haviour.

The issue of the consolidation associated with the dissipa-tion of excess pore water pressure is out of the scope ofthe present study. Despite the above, the model proposedin this paper can be applied to cases where this consolida-tion phenomenon takes place. The net stress, deˆned asthe applied total stress minus the pore air pressure for un-saturated soil, is equivalent to the eŠective stress, deˆnedas the applied total stress minus the pore water pressurefor fully saturated soil. In the present study, the relation-ship between the net stress and the strain was evaluatedfor the following reasons:1) The net stress can be evaluated independently of the

matrix suction (deˆned as the pore air pressure minusthe pore water pressure), which is not measured in thepresent study. In addition, it is rather di‹cult to ob-tain an accurate measurement of the matrix suction inusual êld cases.

2) The relationship between isotropic average skeletonstress p! (deˆned as p!＝(s!1＋s!2＋s!3)/3, where s!i＝si－uair－x(uair－uwater); x is a function of the degree ofsaturation) and the void ratio is usually not unique fordiŠerent degrees of saturation or for diŠerent valuesof matrix suction (e.g., Gallipoli et al., 2003).

The degree of saturation was selected as the basicparameter to describe the eŠects of the wet condition onthe net stress—strain relation, in place of the matrix suc-

tion for the following reasons:1) The eŠects of matrix suction are uniquely related to

those of the degree of saturation when only the wet-ting process is dealt with (e.g., Oka et al., 2010).

2) The degree of saturation can be easily evaluated inmost cases.

3) It was not possible in this research to evaluate withconˆdence the matrix suction in the specimen over avery wide range of saturation (from nearly zero tonearly 100z).

MODELLING OF VISCOUS PROPERTY IN 1DCOMPRESSION

Non-linear Three-component ModelThe general three-component model framework (Fig.

1: Di Benedetto, 1987; Di Benedetto and Tatsuoka, 1997;Di Benedetto et al., 1999, 2001, 2002; Tatsuoka et al.,2002, 2008a) was employed in the present study. Accord-ing to this framework, with or without cohesion, ir-respective of the type of viscous property and whetherpositive or negative ageing is active, the total strain rate ·ecan be broken down as

·e＝`èlastic component ·ee''＋`ìrreversible component ·eir''. (1a)

where ·eir may be called visco-plastic strain rate ·evp. Elasticstrain ee is obtained as ee＝fdee, where dee is the elasticstrain increment. Stress s, which is the vertical net stressin the present study, is broken down as

s＝`ìnviscid component sf''＋``viscous component sv'' (1b)

where sf is the inviscid stress activated in the nonlinear in-viscid component (Fig. 1), which is a non-linear functionof eir (and others). It is assumed that, when the eŠects ofageing, wetting and other factors are not taken into ac-count, for a given history of strain (eir), there exists aunique sf¿eir relation (called the reference relation) forML whatever strain (or stress) rate is currently takingplace, while diŠerent sf¿eir relations are formed forloading, unloading, reloading and so on. As the strainrate becomes zero, the respective s¿eir relations ap-proach the corresponding sf¿eir relation, i.e., the elasto-viscoplastic model becomes the elasto-plastic model. It isassumed that there is a unique s¿ee relation for thehypo-elastic component. On the other hand, even underML conditions, the sv¿eir relation of the non-linear vis-cous component is not unique; the current value of sv de-pends not only on the instantaneous value of sf (or eir),but also on instantaneous irreversible strain rate ·eir andthe history of eir for viscous property types other than theIsotach (as explained below).

Isotach Model for Clay in 1D CompressionAmong the various viscous property types that have

been found for geomaterials (Tatsuoka, 2007; Tatsuokaet al., 2008a), the Isotach type is the simplest and themost classical, while this type is known to be relevant to

899

Fig. 2. Saturation process in tests on kaolin (batch A, category c)

Fig. 3. Test methods for kaolin

8991D COMPRESSION OF CLAY

many types of saturated clay in 1D compression (e.g.,Imai, 1995; Niemunis and Krieg, 1996; Leroueil and Mar-ques, 1996; Leroueil, 1997). Deng and Tatsuoka (2004),Deng and Tatsuoka (2007) and Kawabe et al. (2007, 2009)showed that the following speciˆc type of Isotach modelcan properly simulate the stress-strain behaviour of manytypes of clay in 1D compression tests in which sustainedloading at ˆxed stress states and stepwise changes in thestrain rates were performed many times during otherwiseML at a constant strain rate.

sv(eir, ·eir)＝sf(eir)･gv( ·eir) (1c)gv( ·eir)＝a･[1－exp s1－(` ·eir`/ ·eir

r＋1)mt] (1d)

where gv( ·eir) is the viscosity function (Di Benedetto et al.,2002; Tatsuoka et al., 2002), which is a highly non-linearfunction of ·eir, becoming zero when eir

r＝0 and a when eirr

becomes inˆnitive, and m and eirr are positive constants.

In the present study, this Isotach model was modiˆed toincorporate the eŠects of wetting in an analogous way asa method that incorporates the negative eŠects of ageinginto the model (Tatsuoka et al., 2008b).

EXPERIMENTS

Two series of 1D compression tests were performed.The ˆrst series was performed using kaolin from batch A(D50＝0.0013 mm; PI＝41.6; LL＝79.6z) for the periodof 2003–2004 at the University of Tokyo and the secondseries was performed using kaolin from batch B (D50＝0.0034 mm; PI＝21.3; LL＝44.1z) for the period of2008–2010 at Tokyo University of Science. The initialdimensions of the specimens were 2 cm in height and 6 cmin diameter. The vertical strains were measured externallyand then corrected for system compliance and compres-sion of the ˆlter paper (one sheet at each end of the speci-men for batch A and one sheet of PTFE ˆlter paper at thetop and one sheet of membrane ˆlter paper at the bottomof the specimen for batch B). The ˆlter paper used in thebatch A tests is 0.26 mm-thick industrial ˆlter paper(Type no. 2, Tokyo Roshi Kaisha, Ltd.). In the tests us-ing batch B, air-permeable and water-impermeable PTFEˆlter paper (Tokyo Roshi Kaisha, Ltd.) and air-imperme-able and water-permeable membrane ˆlter paper (PallCo. Ltd.) were used to control the water content muchmore precisely and in a much more delicate way than thetests using batch A.

Tests on the Kaolin in Batch AA number of specimens, as categorized below, were

prepared by compacting air-dried powder in an oedome-ter ring in six layers with six blows of about 100 N per lay-er using a hammer with a diameter of 50 mm:

a) Six specimens (Aa1¿Aa6) were kept air-driedthroughout the 1D compression.

b) Four specimens (Ab1¿Ab4) were saturated by in-troducing water from the specimen bottom and leftfor one day at sv＝0.

c) Three specimens were saturated and left for one dayat sv＝160 kPa (Ac1) or 330 kPa (Ac2) or 950 kPa

(Ac3).The specimens were kept air-dried throughout loading(category a) and were tested using an ordinary oedometerring placed under atmospheric pressure without applyingany back pressure. Therefore, net stress sv-n in this case isequal to applied vertical stress sv. The specimens saturat-ed at sv＝0 or at sustained loading stages (category b)were tested in a triaxial cell ˆlled with tap water. Smallsettlement took place due to the gravity force during thesaturation at sv＝0. However, there was no record ofstrain history during this process.

In all the tests on the specimens saturated at sv largerthan zero (category c), shown in Fig. 2, a high degree ofsaturation was ensured as follows. Referring to Fig. 3(a),after the specimen in an oedometer ring was submerged intap water, a negative cell pressure of about －85 kPa wasapplied. Then, the cell pressure was increased to 200 kPa,

900

Table 1. Test conditions

TestNo. e0

a) tab)

(H)tw

c)

(H)Sr0

d)

(z)Sr–w

e)

(z)·e0

f)

(×10－7/s)b

value

Aa1 1.57 — — 0.5 — 12 —

Aa2g) 1.59 — — 0.5 — 7.7 0.034

Aa3 1.61 — — 0.4 — 220 —

Aa4g) 1.43 — — 0.9 — 7.6 0.033

Aa5g) 1.63 — — 0.7 — 19 0.034

Aa6g) 1.59 — — 0.6 — 20 0.033

Ab1 1.55 — 24 0.7 100 250 —

Ab2 1.56 — 24 0.7 100 12 —

Ab3g) 1.63 — 24 0.7 100 18 0.067

Ab4 1.68 — 24 0.6 100 250 —

Ac1 1.62 0.05 23 0.7 100 220 —

Ac2 1.62 0.05 23 0.7 100 220 —

Ac3 1.63 0.05 22 0.7 100 220 —

B1g) 3.30 14 24 0.3 43 13 0.054

B2g) 1.52 14 49 0.7 19 9.5 0.050

B3 3.81 14 49 0.3 6 130 —

B4 2.83 12 48 0.4 40 190 —

B5 3.43 13 48 0.3 100 160 —

B6 3.60 13 23 0.3 100 100 —

a) Initial void ratiob) Sustained loading period at air-dried statec) Sustained loading period during wettingd) Initial saturation degreee) Prescribed ˆnal saturation degree immediately after wettingf) Lowest strain rate during MLg) Test with step changes in the vertical strain rate Fig. 4. Experimental results, kaolin (batch A)

900 DENG ET AL.

which was subsequently kept throughout each test. Thesaturated specimens were drained from the top, where thepore water pressure (upore-a) was equal to the cell pressure(200 kPa), while the pore pressure at the bottom of thespecimen was measured (upore-b). The degree of saturationof the specimens at the end of the 1D compression testswas obtained using the water content measured after thetests and conˆrmed to be nearly 100z. In this case, netstress sv-n is the applied total stress minus the average ofthe back pressures measured at the top and the bottom ofthe specimen (upore-a and upore-b).

In all the tests, an automatically controlled precise gearsystem (Tatsuoka et al., 1994; Santucci de Magistris etal., 1999) was used to axially load the specimen. With thissystem, it is possible to largely change the strain ratewithout a delay and to perform sustained loading at anarbitrary constant vertical stress unless the strain rate thatis to take place is very large.

Table 1 lists the test conditions. More than half of thebatch A specimens were subjected to continuous ML at aconstant vertical strain rate, while ˆve of the specimenswere subjected to several step changes in the vertical

strain rate and sustained loading lasting for one day dur-ing otherwise ML at a constant strain rate. The details ofthis test method are described in Deng and Tatsuoka(2004).

Figures 4(a) and (b) summarize the overall relation-ships between void ratio e and net vertical stress sv-n onthe logarithmic scale and between sv-n and vertical strainev from the experiments of categories a, b and c, as previ-ously outlined. Figure 4(c) shows the time histories ofvertical strain. These experimental results are simulatedby the three-component model as shown later in thispaper. The specimens of category c exhibited large com-

901

Fig. 5. sv-n-ev relation for kaolin (batch A)Fig. 6. Evaluation of b: (a) air-dried kaolin and (b) saturated kaolin

(batch A)


pression at a high rate when saturated at diŠerent non-zero sv-n values. The compression rate became smaller assv-n became larger than about 150 kPa. Although verticalloading was programmed to keep the sv-n value constantduring this wetting process, as seen from Fig. 2, the load-ing system using a gear-type loading system, of which themaximum applicable strain rate is 4.9×10－5/s§2 ·e0,could not follow the high compression rate of the speci-men during the wetting stage and the sv-n value tem-

porarily dropped in a large way. Similar, but smaller,compression of the specimen and the corresponding tem-porary drop in the sv-n value took place when the cellpressure (i.e., the back pressure) was increased from －85kPa to 200 kPa to make the wetted specimens fully satu-rated. As seen from Figs. 4(a) and (b), upon the restart ofML after the wetting stage, the e-sv-n curves tended tojoin those of the specimens saturated at sv-n＝0 before thestart of ML at a constant strain rate (category b).

Figures 5(a) and (b) show the sv-n-ev relations from thetests on specimens Aa5 (category a) and Ab3 (category b),in which the strain rate was stepwise changed many timesduring otherwise ML at a constant strain rate in order toevaluate the viscous property of the material. The viscousproperty was quantiˆed based on stress jumps, Dsv-n, tak-ing place upon a step change in the strain rate, as illustrat-ed in Fig. 5(c) (Di Benedetto et al., 2002; Tatsuoka et al.,2002, 2008a). It was conˆrmed that Dsv-n was alwaysproportional to instantaneous stress Dsv-n for a given ra-tio of the irreversible strain rates before and after achange, ( ·ev)after/( ·ev)before, in all the tests. Based on this fact,the Dsv-n/sv-n－log10 [( ·ev)after/( ·ev)before] relations were plot-ted in Figs. 6(a) and (b). A very small scatter in the datameans that normalization Dsv-n/sv-n is relevant. It mayalso be seen that the relation is rather linear. The slope of

902

Fig. 7. Experimental results, kaolin (batch B)

902 DENG ET AL.

the respective ˆtted linear relations is deˆned as the rate-sensitivity coe‹cient, b.

Tests on the Kaolin in Batch BExcept for specimen B2, the specimens were produced

by the air-pluviation method, which made the initial voidratios of the specimens much higher than those of batchA (prepared by compaction; Fig. 4(a)). Therefore, thespecimens of batch B were under nearly normally consoli-dated (NC) states during primary loading (Fig. 7(a)). Onthe other hand, specimen B2 was produced by the samemethod as the specimens of batch A. So, the specimens ofbatch A and specimen B2 were initially under overcon-solidated (OC) states and entered into NC states after thestart of global yielding, as the other batch A specimens(Fig. 4(a)). Referring to Fig. 3(b), back pressure was notapplied to the specimens placed in an ordinary oedometerring under atmospheric pressure. The specimens were ini-tially air-dried and then made wet to diŠerent degrees ofsaturation at ˆxed vertical stress sv. In this case, net stresssv-n is equal to applied vertical stress sv.

The vertical loading on the batch B specimens (exceptfor B1 and B2) was performed by automatically controll-ing the air-pressure supplied to an air-cylinder so that MLcould be precisely performed at a speciˆed constant strainrate. The details are reported in Kongkitkul et al. (2011).Unlike the tests using the kaolin in batch A (Fig. 4(a)),

because of the characteristic feature of a pneumatic load-ing system, it was easy to keep the total vertical stressconstant even when the specimen exhibited large com-pression rates due to wetting, as shown in Fig. 7(a). Onlyin tests B1 and B2, a gear-type of loading system was usedas in the tests on batch A. In these two tests, unlike theother tests on batch B, the speed of the addition of waterwas controlled to be slow enough to keep the sv-n valueconstant during wetting, as shown in Fig. 7(d).

After the sv-n value reached the prescribed value, sv0＝33, 113 or 503 kPa, the air-dried specimens were subject-ed to sustained loading (SL) for a period of 12 to 14 hours(Table 1). As typically seen from Fig. 7(d), the creepdeformation during this SL stage was generally verysmall. Subsequently, within ˆve minutes during SL, dis-tilled water was percolated to make the specimen wettoward diŠerent prescribed degrees of saturation, Sr-w

(Table 1). The Sr-w values obtained from the water con-tents measured after the respective tests are listed in Table1. In the tests in which the specimens were fully saturated,the 1D compression cell was ˆlled with distilled water inthe course of the saturation process. The relationship be-tween the void ratio and the degree of saturation present-ed in the ˆgure inset in Fig. 7(d) was obtained from theamount of water added and the vertical strain. As an at-tempt to obtain highly homogeneous wet states, the speci-mens were left for one or two days under the sustained

903

Fig. 8. (a) sv-n-ev relation and (b) evaluation of b for partially saturat-ed kaolin (batch B) Fig. 9. Deˆnition of C e

c, Cr and Cc


load (Table 1). Further research is necessary to evaluatethe homogeneity of the water content as a function ofelapsed time.

In test B1, SL was performed on a partially saturatedspecimen at sv-n＝800 kPa for one day during otherwiseML at a constant strain rate (Fig. 8(a)). More than a halfof the specimens were subjected to continuous ML at aconstant vertical strain rate (Table 1). In two tests, thevertical strain rate was stepwise changed several timesduring otherwise ML (Fig. 8(a)) to evaluate rate-sensitivi-ty coe‹cient b (Fig. 8(b)). With the data plotted in thisˆgure, the range of Sr is from 63z to 94z (average of79z). A small scatter among these data points and thefact that the b value of this partially saturated specimen issimilar to the value of the saturated compacted specimen(Fig. 6(b)) indicate that the eŠect of Sr on the b value issmall, at least when Sr is larger than 63z. This issue isanalyzed in detail later in this paper. Unloading andreloading cycles were applied to batch B to evaluate theelastic property (e.g., Fig. 13 shown later). However, thecyclic loading behavior is beyond the scope of thisresearch.

Discussion on Vertical Stress Parameter for UnsaturatedSpecimens

For the tests on batch A, in which air and water werefreely drained, net vertical stress sv-n is deˆned as

sv-n＝(sv)total－uback (2)

where (sv)total is the applied total stress and uback is theaverage of the pore water pressure at the specimen top,upore-a, equal to the cell pressure, and the value measuredat the specimen bottom, upore-b (Fig. 3(a)). Before the startof the saturation process, uback is essentially the same asthe atmospheric air pressure (i.e., uair＝0), because thetubes connecting the specimen and the measurementdevice were ˆlled with air. Then, we have

sv-n＝(sv)total－uback＝(sv)total (3)

Equation (3) is also valid for the tests on batch B (Fig.3(b)), as the uback value was always equal to the at-mospheric pressure (i.e., uback＝uair＝0).

Net stress sv-n, deˆned above and used in the presentstudy, is essentially the same as the stress diŠerence (s-ua)(Bishop and Blight, 1963) and the net stress used byJotisankasa et al. (2007), Nuth and Laloui (2008) andother researchers. With fully saturated specimens, theabove-deˆned net stress sv-n becomes the eŠective verticalstress, as seen by replacing uback with uwater in Eq. (2).

ISOTACH MODEL IN THE THREE-COMPONENTMODEL FRAMEWORK

Stress-strain RelationsThe results of the ML tests at constant strain rates,

shown above, can be approximated by the following seg-mental linear e¿log sv-n relations, illustrated in Fig. 9(a):

For primary compression: －De＝Cc･D log sv-n (4a)For recompression: －De＝Cr･D log sv-n (4b)

where Cc is the slope of the primary compression relation,

904

Fig. 10. Monotonic loading from an initial state of either normallyconsolidated (NC) or overconsolidated (OC) specimens

904 DENG ET AL.

equal to the coe‹cient of compressibility under normallyconsolidated (NC) conditions, and Cr is the slope of therecompression relation from an overconsolidated (OC)state, equal to the coe‹cient of compressibility forrecompression. Note that the stress-strain behaviour dur-ing recompression is not purely elastic; it exhibits noticea-ble yielding. It is very di‹cult to accurately evaluate thebehaviour of the hypo-elastic component, because elasticvoid ratio increment De e can be purely observed for onlya stress increment Dsv-n that is very small. The followinglinear relation is assumed for a ˆnite change in Dsv-n dur-ing loading, unloading, reloading and so on (Fig. 9(a)):

－De e＝C ec･D log sv-n (5)

Then, the following relations are assumed for the inviscidcomponent subjected to ML at a constant strain rate:

For primary compression:－Deir＝C ir

c (sfv-n)･D log sf

v-n (6a)For recompression from an OC state:

－Deir＝C irr (s

fv-n)･D log sf

v-n (6b)

where De ir is the irreversible void ratio increment, sfv-n is

the net inviscid vertical stress and C irc and C ir

r are thecoe‹cients of compressibility, which are functions of thecurrent value for sf

v-n. Referring to Eq. (1a), we have

·e＝ ·e ir＋ ·e e (7)

where ·e＝－(1＋e0)･ ·ev, ·e ir＝－(1＋e0)･ ·eirv , ·e e＝－(1＋e0)･

·eev and e0 is the initial void ratio. By integrating Eq. (7)

with respect to time, we have

De＝De ir＋De e (8)

From Eqs. (1b) and (1c), we have

sv-n＝s1＋gv( ·eirv )t･s

fv-n (9a)

log sv-n＝log s1＋gv( ·eirv )t＋log sf

v-n (9b)

where svv-n is the net viscous vertical stress. When ·eir

v§con-stant, we have

D log sv-n＝D log sfv (10)

By introducing Eqs. (5), (6a), (8) and (10) into Eq. (4a),we have

C irc (s

fv-n)＝Cc－C e

c (11)

Equation (11) shows that C irc is constant during primary

compression at a constant irreversible strain rate, ·eirv ,

which is nearly the same as the strain rate, ·ev. In Fig.9(b), the relationships in terms of the total stress and theinviscid stress are depicted considering that sv-n is largerthan sf

v-n (i.e., sv-n＝sfv-n＋sv

v-n and Eq. (1b)).In summary, the constitutive relations during the pri-

mary loading of the three components are as follows:

Hypo-elastic component:－Dee＝C e

c･D log sv-n (12a)Inviscid component:

－Deir＝(Cc－C ec)･D log sf

v-n (12b)Viscous component:

svv-n＝gv( ·eir

v )･sfv-n (12c)

It should be noted that as Cc:C ec, －Deir§Cc･D log

(sfv(eir)).Similarly, the relations for recompression from an OC

state are as follows:

Hypo-elastic component:－Dee＝C e

r･D log sv-n (13a)Inviscid component:

－Deir＝(Cr－C er)･D log sf

v-n (13b)Viscous component:

svv-n＝gv( ·eir

v )･sfv-n (13c)

where C er＝C e

c. Figure 10 schematically shows e－log sv

relations when ML starts at diŠerent constant total strainrates from a common initial state, where no viscous eŠectis included, obtained by following the above formula-tion. These relations have the following features:

a) Due to the speciˆc non-linearity of the viscosityfunction (Eq. (1d)), a fast increase in the irreversi-ble strain is obstructed by the viscous componentwhen the total strain rate suddenly becomes a cer-tain ˆnite value from zero. Therefore, the strainrate immediately after the start of ML is nearly elas-tic, which makes the initial slope essentially thesame as C e

c whether ML starts from an OC state ora primary loading state.

b) As ML proceeds, the irreversible strain rate in-creases and the total strain rate becomes nearly thesame as the irreversible strain rate. Then, all the e－log sv-n relations for ML at diŠerent strain ratesbecome parallel to each other. All these relationsare also parallel to the relation when the irreversiblestrain rate is equal to zero (i.e., the reference rela-tion). The slope of all these relations is equal toeither Cr when ML starts from an OC state, or Cc

when ML starts from a primary loading state.c) The reference relation exhibits a large change in the

slope from Cr to Cc upon the start of large scaleyielding when entering the primary loading statefrom the OC state.

WETTING EFFECTS ON THE THREECOMPONENTS

For a given soil element having a given volume, the

905

Fig. 11. Deir－log sfv-n relation for diŠerent Sr values: (a) for OC clay

and (b) for NC clay at high pressure


degree of saturation Sr is uniquely related to the watercontent, which also represents the wetting condition. Inthis study, the eŠects of wetting on the behaviour of thethree components are represented by the relationships be-tween Sr and 1) the viscosity function gv and 2) C e

c, Cc andCr. For the ˆrst approximation, possible interactionsamong these relations were ignored and their relation-ships were deˆned independently, as shown below.

gv-Sr RelationThe parameters for the viscosity function gv( ·eir) (Eq.

(1d)) can be determined from a given measured value ofrate-sensitivity coe‹cient b (Figs. 6 and 8; Di Benedettoet al., 2002). Therefore, the eŠect of Sr on gv comes fromthe eŠect of Sr on the b value, which is represented byfunction b(Sr) (e.g., Deng and Tatsuoka, 2004). In thepresent study, the following relationship between gv and Sr was assumed:

gv( ·eir, Sr)＝a(b(Sr))･[1－exp s1－(` ·eir`/ ·eirr＋1)mt] (14)

It is assumed that parameters m and eirr are independent

of Sr. The introduced function a(b(Sr)) and experimental-ly obtained b(Sr) are shown later.

C ec-Sr RelationC e

c should be a function of Sr and the stress level atleast. However, it is very di‹cult to experimentally deˆnethis function. As stated later, C e

c is assumed to be in-dependent of Sr and the stress level for the ˆrst approxi-mation.

C irc -Sr and C ir

r -Sr RelationsAs shown in Fig. 11(a), it is assumed that the reference

relation starting from an OC state under dried conditions(Sr＝0) consists of three sections, namely, PT1P0 (a linearrecompression relation), P0P2 (a transition) and P2PT2 (alinear primary compression relation). The slope of PT1P0

is the recompression index C irr (＝tan aT1) and the slope of

P2PT2 is the compression index C irc (＝tan aT2). P0P1 and

P1P2, which are the extrapolations of PT1P0 and P2PT2, arereplaced by transition curve P0P2 to describe morerealistically the experimental results (e.g., Fig. 4(a)).Transition curve P0P2 was obtained as a B áezier's curve de-ˆned by P0, P1 and P2, explained in the APPENDIX.

Similarly, the reference relation, starting from an OCstate under fully saturated conditions (Sr＝1.0), consistsof three sections, PT1P?0 (a linear recompression relation),P?0P?2 (a transition relation) and P?2PT2 (a linear primarycompression relation). It is assumed that initial point PT1,where the total stress is nearly zero, is the same when d-ried as when fully saturated. That is, point PT1 is an inter-section of the two recompression curves when dried andwhen fully saturated. At point PT1, therefore, no settle-ment takes place by wetting from Sr＝0 to Sr＝1.0. PointPT1 is an imaginary state, which cannot actually bereached. The ultimate point, PT2, is an intersection of thetwo compression curves when dried and when fully satu-rated, where the total stress is a certain extremely largevalue. It is also assumed that point PT2 is the same when

dried and when fully saturated. Point PT2 is also an imagi-nary state. In actuality, the void ratio cannot becomesmaller than a certain small positive value and compres-sion curve Deir－log sf

v-n cannot be maintained to belinear when sf

v-n becomes extremely large, as illustrated inFig. 11(b). Figure 11(b) was prepared only to illustratehow the compression curves would be at extremely largesf

v-n values, but not referred to in the simulation of actualtests.

To formulate the transition sections of the relationswhen dried and when fully saturated, _P0P2 and _P?0P?2, it isassumed for the sake of simplicity that three lines, P0P?0,P1P?1 and P2P?2, are parallel to each other with a slope oftan (aP1). Then, the reference relation when partiallysaturated with a constant value of Sr between 0 and 1.0,

_PT1P!0P!2PT2, is located between those when dried andwhen fully saturated, _PT1P0P2PT2 and _PT1P?0P?2PT2, asshown in Fig. 11(a). The wetting eŠect is represented by aclockwise rotation of the recompression curve aboutpoint PT1 and an anti-clockwise rotation of the primarycompressions curve about point PT2. That is, the com-pression curve when partially saturated, ( _PT1P!0P!2PT2), isobtained as follows:1) The primary compression line, P!1PT2, is determined by

the anti-clockwise rotating of line P1PT2 about pointPT2 to have a slope of tan (a!T2) that is determined from

906

Fig. 12. Strain increments during and after the addition of water

906 DENG ET AL.

an experimentally deˆned Sr-C irc relation (i.e., Eq.

(24), shown later).2) Point P!1 is obtained as the point where compression

line P!1PT2 intersects line P1P?1. The recompression line,P!1PT1, is then obtained by connecting P!1 and PT1.

3) Point P!0 is obtained as the point where line P!1PT1 in-tersects line P0P?0 (determined from the ending pointsof the recompression lines when dried and when fullysaturated).

4) Point P!2 is obtained as the point where primary com-pression line P!1PT2 intersects line P2P?2 (determinedfrom the starting points of the primary compressionlines when dried and when fully saturated).

5) The transition curve between P!0 and P!2 is then ob-tained as a B áezier's curve ( see the APPENDIX).

In order to determine the inviscid compression curves fordiŠerent Sr values, by following the above-describedprocedure, the coordinates at points PT1, PT2, P0, P1, P2

and inclination angle aP1, together with a function ofaT2(Sr) (or C ir

c (Sr)), should be known. These parametersand function were determined based on the experimentalresults as follows. It may be seen from Fig. 4(a) that thesv-n value at intersection point PT1, between the recom-pression curves starting from an OC state under air-driedand fully saturated conditions, (Fig. 11(a)) is very small.The value obtained by extrapolating the test results(batch A) is equal to 0.032 kPa. On the other hand, itmay be seen from Figs. 4(a) and 8(a) that the sv-n value atintersection point PT2 between the primary compressioncurves under air-dried and fully saturated conditions(Fig. 11(a)) is very large. The value obtained by ex-trapolating the test results (batch A) is equal to 30 MPa.Function aT2(Sr), which controls the primary part of theprimary compression curve for diŠerent Sr values, wasdetermined experimentally as shown later.

SIMULATIONS OF THE EXPERIMENTS

Typical experimental results on kaolin (batches A andB) were simulated as follows:1) The ML tests were simulated by strain control using

the time histories of measured ev. The time histories ofstress during ML (including the stress drop during thesaturation process in the tests on batch A) were simu-lated until the vertical stress became the prescribed lar-gest value. The simulation was started using the ·ev

value measured at the start of loading where sv-nº12kPa.

2) The time history of ev during respective creep stageswas obtained by a stress control simulation at a givenconstant stress for the actual period (i.e., 24¿48hours). The simulation of the subsequent ML phasewas performed by strain control, as described above.

3) In the simulation of the respective tests, the sv-n valueat the start of the respective events (i.e., creep stageand ML) was set to be the same as the one measured inthe experiment. Therefore, the discrepancy betweenthe measured and simulated strain levels graduallybecame larger as the strain increased in each event. In

this way, the accuracy of the simulation was evaluatedconsistently in terms of strain.

Saturation FunctionIn the simulation of the respective tests, the degree of

saturation until the end of the wetting process was givenas a function of time and vertical strain as

Sr(t, ev)＝w0×rc

e0－(1＋e0)×evfor tºtp

Sr(t, ev)＝w0×rc

e0－(1＋e0)×ev＋

t－tp

t－tp

(wfinal－w0)×rc

e0－(1＋e0)×ev

＋1

Cfr

for tÆtp (15)

where w0 is the initial water content at the start of thetests (where ev＝0), wfinal is the water content at the ˆnalstate of wetting, evaluated based on the water contentmeasured after the test, rc is the density of the kaolin clayparticles (n.b., batches A and B have diŠerent values), e0

is the initial void ratio at the start of the tests, tp is thetime at the start of the water percolation into the speci-men and Cfr is a parameter showing the water ‰ow rateduring the saturation process, namely, 0.002 for batch Aand 0.001 for batch B. It should be noted that the Sr valuecalculated by Eq. (15) may become larger than unity, be-cause this value assumes that the squeezed water duringloading is always constrained within the soil. In actuality,however, when Sr tends to become larger than unity, thepore water is drained from the soil. Thus, the maximumvalue for Sr was set to be unity in the simulation.

Figure 12 shows the measured and the simulated timehistories of vertical strain during and after the process ofsupplying water from outside the oedometer ring at a ˆx-ed vertical stress (Fig. 3) from a typical test (batch B). Alarge settlement took place during wetting. The settle-ment continued after the end of the water supply, whichis creep deformation due to the viscous property that hasbeen enhanced by wetting. The simulation shown in thisˆgure is part of the simulation results shown later in Fig.21(c).

907

Fig. 13. Evaluation of C ec in test B2

Fig. 14. EŠects of saturation degree on b value of clay

Fig. 15. Irreversible decrease in void ratio in idealized experimentswith a nonlinear Deir－log sf

v-n relation


Elastic StrainThe following relationship between ee and sv-n is ob-

tained from Eq. (12a) or (13a):

sv-n＝10(1＋e0)･ee

v

C ec

＋log sv0 (16)

where sv0 (kPa) is the initial value for sv-n (kPa), which islarger than zero. It was assumed that C e

c＝0.001 indepen-dent of the stress level and Sr. In Fig. 13, for test B2, thisvalue is compared with the stress-strain relation immedi-ately after the start of recompression. As seen from thisˆgure, it was not possible to conˆdently evaluate thevalue of C e

c and the dependency of C ec on the stress level

or Sr based on the experiment results. Despite the above,as this value is much smaller than both C ir

c and C irr , it is

likely that errors in the simulation by this approximationare very small.

Viscosity FunctionFigure 14 shows the relationships between the b value

and Sr from 1D compression tests on compacted speci-mens of Fujinomori clay (Li et al., 2004) and those onkaolin (batches A and B) from the present study (seeFigs. 6 and 8 for typical test results). The relations for thetwo types of clay are very similar. Equation (17) was ˆt-ted to the data for the kaolin (batches A and B):

b(Sr)＝0.0805－0.034×exp (－Sr/1.8)－0.015×exp (－Sr/0.05) (17)

where Sr is in decimals. Then, by following the methodexplained in Di Benedetto et al. (2002), the value for a ofviscosity function gv (Eq. (14)) was determined for diŠer-ent values of b in the range shown in Fig. 14. By usingconstant values m＝0.032 and ·eir

r＝10－9/sec, which areboth assumed to be independent of the pressure level andSr, the following empirical equation was obtained fora(b(Sr)) in Eq. (14) by ˆtting it to a set of a and b valuesobtained as above:

a(b(Sr))＝16.3×b(Sr) (18a)

From Eqs. (14), (17) and (18a), the following equation isobtained:

gv( ·eirv , Sr)＝16.3×b(Sr)×«1－exp{1－Ø` ·eir`

·eirr＋1»

m

}$(18b)

where m＝0.032 and ·eirr＝10－9/sec.

Reference CurvesThe reference relations (i.e., the Deir－log sf

v-n rela-tions, Fig. 11), representing the stress-strain behaviour ofthe nonlinear inviscid component (Fig. 1) for diŠerent Sr

values, with diŠerent values for C irc and C ir

r , were ob-tained as follows. Figure 15 illustrates the changes in voidratio before, during and after the wetting process in rela-tion to the reference curves under dried conditions (Ref0for Sr＝0), fully saturated conditions (Ref1 for Sr＝1.0)and certain partially saturated conditions (Ref for Sr be-tween 0 and 1). Suppose that during SL, for a period ofTcreep0 under the initial dried conditions, the state movesfrom point B to point C with a void ratio decrease in theamount of Decreep.0. Subsequently, the addition of waterto the specimen starts at state C and continues until the Sr

value reaches a prescribed ˆnal value at state E (listed in

908

Fig. 16. Irreversible decrease in void ratio in idealized experimentswith a linear Deir－log sf

v-n relation

Fig. 17. RDem-Sr relation at ˆnal state of saturation process under sus-tained load of essentially NC specimens of batch B (including dataotherwise not reported in this paper)

Table 2. Parameters and functions for the inviscid component

Parameters and functions

Simulationsof batch A

C irc (Sr)*) 0.526－0.241×((Sr/(Sr＋0.143))＋0.125×Sr)

PointPT2

sfv–n＝30 MPa

e0＋De ir＝0.375

PointPT1

sfv–n＝0.032 kPa

e0＋De ir＝1.641

aP1 14.09

Simulationsof batch B

C irc (Sr)*) 0.905－0.495×((Sr/(Sr＋0.143))＋0.125×Sr)

PointPT2

sfv–n＝3.3 MPa

e0＋De ir＝0.533

PointPT1

Varies with diŠerent specimens due to diŠerentinitial void ratio

aP1 28.89

*): Sr in decimal

908 DENG ET AL.

Table 1), if Srº1.0, or at state E?, if Sr＝1.0. The corre-sponding decrease in void ratio is Dem (until state E if Sr

º1.0) and Dem, f (until state E? if Sr＝1.0). Due to theeŠects of wetting, the reference curve at the end of creepperiod Tcreep1 changes from Ref0 to Ref if Srº1 or Ref1 ifSr＝1. The ultimate change in void ratio by wetting, afterthe creep deformation has fully taken place at a constantvalue of sf

v-n, is Dew (from state D to state G) and Dew, f

(from state D to state G?) in Fig. 15. Then, the wettingeŠects on the reference relation can be represented by theparameter as follows:

RDe＝Dew

Dew, f(19)

Equation (19) means that RDe＝ 0 if the ˆnal Sr＝0, andRDe＝1 if the ˆnal Sr＝1. As seen from Fig. 16, parameterRDe is related to the coe‹cients of compression when Sr＝0, Sr＝1 and 0ºSrº1 as

RDe＝Dew

Dew, f＝

C irc0･L－C ir

c･LC ir

c0･L－C irc1･L

＝C ir

c0－C irc

C irc0－C ir

c1(20)

where L is the stress diŠerence indicated in Fig. 16, andC ir

c0 and C irc1 are the values of C ir

c when Sr＝0 and Sr＝1, re-spectively. The value when 0ºSrº1, C ir

c , can be obtainedfrom the known values for RDe, C ir

c0 and C irc1 based on Eq.

(20).As the relationship between parameter RDe and the ˆnal

Sr value cannot be obtained directly from the experi-ments, the relation is obtained by the following approxi-mated method. That is, the initial water content at state B(Fig. 15) in the respective tests is assumed to be zero. RDe

is related to the measured void ratio changes Dem andDem, f, as follows:

RDe＝Dew

Dew, f＝

Dem－(De1－De2)Dem, f－(De1－De3)

(21)

where De1, De2 and De3 are the decreases in void ratio thatwould have taken place if SL, starting from states C, Eand E?, had continued for an inˆnitive time under dried,partially saturated or fully saturated conditions. In theexperiments performed in the present study, the changes

in void ratio induced by the eŠects of wetting were muchlarger than those induced by creep deformation, whileDe1, De2 and De3 are not largely diŠerent from eachother. The above means that Dem:(De1－De2) and Dem, f

:(De1－De3). Therefore, without losing su‹cient ac-curacy, Eq. (20) can be approximated as

RDe＝Dew

Dew, f§

Dem

Dem, f＝RDem (22)

Figure 17 shows the relationship between the RDem valueand the degree of saturation Sr from all the related testson kaolin (batch B), in which Tcreep0＝12¿14 hours andTcreep1＝24¿48 hours. As the sustained vertical stress atwhich the specimen is made wet or fully saturatedbecomes higher, RDem tends to become slightly larger. Inthe simulations of the all tests, the eŠects of the stress lev-el during SL on this relation were ignored for the sake ofsimplicity and the following function was used forRDe(Sr):

RDe(Sr)＝RDem(Sr)＝(Sr/(Sr＋0.143))＋0.125×Sr (23)

909

Fig. 18. Comparison of experiment and simulation, kaolin (batch A) Fig. 19. Comparison of experiment and simulation, kaolin (batch B)


From Eqs. (20), (22) and (23), the relationship betweenC ir

c and Sr was determined. For example, for given valuesof C ir

c0＝0.905 and C irc1＝0.41, we obtain the following:

C irc＝0.905－0.495×Ø Sr

Sr＋0.143＋0.125×Sr» (24)

The function of C irc used in the simulations is listed in

Table 2. For a known C irc (＝tan (aT2)), the curve

_PT1P!0P!2PT2 (Fig. 11(a)) was determined using the func-tions and the parameters listed in Table 2.

Simulation ResultsFigures 18 through 21 compare the measured and the

simulated stress-strain relations. In the respective ˆgures,the thin curves represent the experimental results, whilethe thick curves represent the simulations. All the testswere simulated using the same model parameters andfunctions for batches A and B respectively (Table 2) andthe common ones for batches A and B (Table 3). It can beseen from these ˆgures that the Isotach model, modiˆed

910

Fig. 20. Simulation of Ac1

Table 3. Common parameters and functions for all simulations

Parameters and functions

Hypo-elasticcomponent

C ec(Sr)*) 0.001

Nonlinearviscouscomponent

b(Sr)*) 0.0805－0.034×exp(－Sr/1.8)－0.015×exp(－Sr/0.05)

a(Sr)*) 16.3×b(Sr)

m 0.032

·e irr 10－9/sec

*): Sr in decimal

910 DENG ET AL.

to take into account the eŠects of wetting, simulates themeasured behavior very well, particularly in the follow-ing aspects:1) The decrease in the void ratio during the respective

wetting stages at a constant net vertical stress is simu-lated well (e.g., Figs. 19 and 21). Figure 22 comparesthe measured and the simulated increments in verticalstrain due to wetting at a constant vertical stress. Thevertical strain increment during creep, with a virtuallyconstant Sr, is smaller than that during the saturationstage (e.g., Fig. 21(b)). Yet, this creep strain is wellsimulated by the Isotach model (e.g., Fig. 21(b)).

2) A temporary drop in stress due to a very fast reductionin volume during wetting, which could not be fol-lowed by the displacement control loading device, isvery well simulated (Figs. 18 and 20). Figure 20(d)shows the details of the process for a typical case.

3) A jump in stress when the strain rate is stepwisechanged, due to the viscous property of the testmaterial, is simulated well (e.g., Fig. 18 and 21).

4) Very stiŠ behaviour upon the restart of ML at a con-stant strain rate after the end of creep stage, which isalso due to the viscous property, is simulated accurate-ly too (e.g., Figs. 18, 19 and 21).

5) Although the moist conditions in the specimen duringand immediately after the wetting process shouldsomehow be non-uniform, it was assumed in the simu-lations that the water content is always uniform.Despite the above, the time histories for the averagestrain during and after the wetting process are simulat-ed well (Figs. 18(c), 19(c) , 20(c) and 21(c)). This maybe due to the fact that the eŠects of average Sr on theaverage stress—average strain relations assumed in thesimulations accurately represent the average behav-iour of the specimen because of a small thickness ofthe specimen (i.e., about 2 cm).

6) There is inevitable scatter in the test results due to avariance that could not be controlled among the diŠer-

911

Fig. 21. Simulation of B1

Fig. 22. Comparison of measured and simulated strain incrementsduring wetting


ent tests. Despite this scatter, the discrepancy of thesimulations using common parameters and functionsdeˆned based on the average of the observed behav-iour is rather small. The simulation level is improvedif parameters ˆtted to the respective test data are used.

The above-mentioned trends of behaviour indicate thatthe basic structure of the model proposed in this paper isrelevant.

CONCLUSIONS

The following conclusions can be obtained from thepresent study:1) The elasto-viscoplastic behaviour in the one-dimen-

sional (1D) compression of kaolin powder is sig-niˆcantly aŠected by wetting.

2) Several assumptions (e.g., a constant elastic compo-nent) were used in the simulation. Despite the above,the observed very complicated trends of stress-strainbehavior, namely, a) during monotonic loading atdiŠerent strain rates, b) during and after a step changein the strain rate, and c) during and after wetting at aˆxed net stress and during subsequent ML, can besimulated rather well by incorporating the wettingeŠects into the stress-strain properties of the elasticand plastic components of a non-linear three-compo-nent model.

3) The wetting eŠects on the average net stress—averagestrain relations and the time histories of average netstress and strain can be predicted accurately by in-troducing basic parameters and functions that are afunction of the average degree of saturation.

912912 DENG ET AL.

ACKNOWLEDGMENTS

The authors are grateful to the Japan Society for thePromotion of Science for the Fellowship (ID No.P08391) supporting this research.

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APPENDIX

B áezier's CurveAs shown in Fig. A-1, after the coordinates of P0, P1

and P2 have been determined, the coordinates of anypoint (such as point R) on the curve _P0RP2 can be deter-

913

Fig. A–1. B áezier's curve


mined by the following equations:

W0x(s)＝P0x＋s･(P1x－P0x); W0y(s)＝P0y＋s･(P1y－P0y);W1x(s)＝P1x＋s･(P2x－P1x); W1y(s)＝P1y＋s･(P2y－P1y);Rx(s)＝W0x(s)＋s･(W1x(s)－W0x(s));Ry(s)＝W0y(s)＋s･(W1y(s)－W0y(s)).

Curve _P0RP2 is formed by the locus of point R whenparameter s increases from zero to 1.

The tangents of _P0RP2 at P0 and P1 are P0P1 and P2P1,respectively.

effects of viscous property and wetting on 1-d …

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