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Simple modelling of stress-strain behaviour of unsaturated soils

M. Kikumoto, H. Kyokawa & T. NakaiDepartment of Civil Engineering, Nagoya Institute of Technology, Japan

ABSTRACT: A simple constitutive model taking account of the necessary features of unsaturated soils isproposed by extending an elastoplastic model for saturated soils. The proposed model is formulated using theBishops effective stress and the critical state friction angle is thus assumed to be constant. In the model, thedecrease (or increase) in the degree of saturation is linked with upward (or downward) movement of normallyconsolidation line in the plane of mean effective stress and void ratio, by which the typical volumetric behav-iours of unsaturated soils are properly described. A simple method for extending a classical water characteris-tics curve assuming a unique relationship between suction and degree of saturation to incorporate the effectsof hydraulic hysteresis and density is also proposed. In this paper, mechanisms of hydraulic collapse andcompaction of soils are discussed through a series of simulations by the proposed model.

1 INTRODUCTONIt is usually indicated that suction increases thestiffness and unsaturated soils are hence able toexist in a looser state compared with saturatedones. Compression behaviour of unsaturated soil iscompared with that of saturated one in Fig. 1, inwhich vertical component of the effective stress "defined by Bishop (1959) as eq. (1) is plottedagainst void ratio e and degree of saturation Sr.

( ) ijrnetijijwaijaijij sSuuu +=+= (1)

is a variable given as a function of Sr and is as-sumed to be equal to Sr here for simplicity. It isseen from Fig. 1 that: the unsaturated sampleshows relatively high stiffness and retains a largervoid ratio compared with the saturated onein thebeginning stage of compression; the unsaturatedsample however exhibits significant compressionfrom a stress level of around 200 kPa, Sr then in-creases due to the compression (though suction iskept constant) and the compression line asymptoti-cally approaches to the normally consolidation lineof the saturated one (NCLsat). Meanwhile Fig. 2

Fig. 1 Oedometer test under constant suction (drainedpore-air and drained pore-water) on catalpo clay (afterHonda, 2000).

shows the results of compression and soaking testson air dried silt. Although the air dried silts canstay in a looser region above the NCLsat during thecompression process, once samples are soaked un-der constant applied stress or void ratio, they ex-hibit hydraulic collapse or stress relaxation untilthey come close to the NCLsat. In this study, amodel for saturated soils is extended in order toincorporate such behaviour of unsaturated soils.

2 MODEL FOR SATURATED SOILSOutline of an elastoplastic constitutive model forsaturated soils considering the effect of density isexplained here. Fig. 3 shows a typical compressioncurve of saturated, over consolidated soilin e-lnrelation. It is assumed in the model that: soil ex-hibits elastoplastic deformation even under overconsolidated state and its state gradually ap-proaches to the NCLsat with increase in the stresslevel; soil having a larger value of a state variable, which is defined as the difference of void ratiosof current state and normally consolidated state

under same confining stress, shows stiffer

Applied pressure [t/ft2]

0.1 1.0 10.0 30.00.65

0.70

0.75

0.80

Soaked at constantapplied pressure

Soaked at constantvoid ratio

Saturated sample (slurry)Saturated sample

Compression line ofair-dried sample (8 tests)

Figure 2 One-dimensional compression and soakingtests on air dried silt (after Jennings and Burland, 1962).

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Fig. 3 Typical relationship between void ratio e and ef-fective stress of saturated, over consolidated soil.

Fig. 4 Modelling of the volumetric behaviour of satu-rated, over consolidated soil.

behaviour. Fig. 4 illustrates the variation of voidratio from the initial state I (= 0, e = e0) to thecurrent state P (= , e = e). Here, eN0 and eN arevoid ratios on the NCLsat under same stress level.The recoverable elastic change of void ratio ee isassumed to follow a usual elastic relationship as:

0

ln= ee (2)

where is the swelling index. The change of voidratio e and its plastic component are given as:

00

0 ln

+== eee (3)

( ) ( ) ( )00

ln

+== ep eee (4)

where is the compression index. From eq. (4), ayield functionfis introduced as eq. (5).

( ) ( ) 0ln 00

=++= pef

(5)

As the state variable may decrease with the de-velopment of the plastic deformation and finallyconverges to zero, eq. (6) is applied.

( )( ( pp deadeGd == (6)Here, a is a parameter controlling the effect ofdensity. A one-dimensional constitutive relation-ship of saturated, over consolidated soil is finallyobtained from the consistency condition (df= 0).

( )( )

d

Gdedede ep

++

=+=

1(7)

Suctions

s

Fig. 5 Modelling of hydraulic hysteresis.

As volumetric behaviours are mainly focused on inthis paper, formulation of the simplified, one-dimensional model has been explained. This modelcan however be extended to multi-dimensional oneeasily by adding a term of stress ratio to the yieldfunction and assuming the associated flow rule.

3 MODEL FOR WATER RETENTION CURVEIt is well-known that the stress-strain behaviour ofunsaturated soil remarkably changes due to thevariation ofSr. A proper model for soil water char-acteristic curve (SWCC) is thus necessary to for-mulate a constitutive model for unsaturated soils.In this section, a model for SWCC incorporatingthe effects of hysteresis and density is proposed.

SWCCs usually trace different paths accordingto drying and wetting histories. Two curves in Fig.

5 represent the highest and lowest boundaries ofSr, which are usually referred to as main dryingand wetting curves. In the proposed model, theyare firstly described by a classical SWCC modelproposed by van Genuchten (1980) as follows.

( ) ( ) ( )[ ]( ) 0

1, minmaxmin

==

++=

rd

r

mndrd

SsF

SsSSSsSf (8)

( ) ( ) ( )[ ]( ) 0

1, minmaxmin

==

++=

rw

r

mnwrw

SsF

SsSSSsSf (9)

Smin, Smax, n, m and are material parameters and

the subscripts dand w denote the main drying andwetting curves, respectively. As any state of waterretention lies between two main curves, a variable

Iw defined as the ratio of interior division of thecurrent state between two reference states on themain curves is introduced as an intermediary.

( ( 1= wrdrwrrw SSSSI (10)

Here, Srd

and Srw

are degrees of saturation of thereference states on the main drying and wettingcurves under current suction, respectively. Iwbound by 0 for the main wetting curve and 1 for

the main drying curve provides a simplified, scalarrepresentation of the current state of hydraulic hys-teresis. It increases monotonically from 0 to 1 as Sr

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Table 1 Parameters for water retention curve.

decreases, and vice versa. An evolution law for thestate variableIw is thus given as:

( )

>

=

0

013

3

rrw

rrww

dSwhendSI

dSwhendSIdI

(11)

where is a material constant. From eqs. (9) to(11), a function fc representing the water retention

surface through the current state (s, Sr,Iw) is givenas follow.

( ) ( ) ( ) ( ) 01,, =+= rwwdwwrc SsFIsFIISsf (12)

As any hydraulic state of unsaturated soil alwayslocates on this surface fc = 0, an incremental formof the SWCCconsidering the influence of the hy-draulic hysteresis is given by applying the consis-tency condition (dfc = 0) to eq. (12).

It is also indicated through the past experimen-tal studies (e.g. Tarantino and Tombolato, 2005)that volumetric behaviour also influences the

SWCC and denser soil tends to retain higher Sr.The increase ofSr under a constant suction (Fig. 1)is presumably a consequence of the volumetriccompression. In this study, a modified suction s

*,

which is dependent on void ratio e as well as ordi-nary suction s, is applied to describe the effect ofvolume change. A tentative form ofs

*considering

the effect of density is given as:

e

satN

ess

=* (13)

where e is a parameter controlling the effect ofdensity andNsat is the reference void ratio of satu-rated, normally consolidation soil under atmos-pheric pressure. By replacing the ordinary suctionin eq. (12) with the modified suction s

*, the effect

of the density can automatically be incorporated.Material parameters for SWCCof two soils are

summarised in Table 1. Calculated results by themodel are compared with experimental ones inFig. 6. It is known that the proposed model accu-rately predicts the hysteretic soil water characteris-tic behaviour under complicated suction histories.It is also seen from Fig. 7 showing the calculated

SWCCs of sands of different densities that the pro-posed model properly describes the influence ofdensity.

0

5

10

15

20

25

0 20 40 60 80 100

Suctions

[kPa]

Degree of saturationSr[%]

Fig. 6 Water retention curve of White silica sand (afterHuang, 2005) and corresponding results of simulation.

0

5

10

15

20

25

0 20 40 60 80 100

Suctions[kPa]

Degree of saturation Sr[%]

Fig. 7 Calculated water retention curves of White silicasands of different densities.

4 MODEL FOR UNSATURATED SOILSIn this section, the one-dimensional model for sa-turated soil is extended to one incorporating theunsaturated soil behaviour.

Firstly, we propose to formulate the modelbased on the Bishops single effective stress de-fined by eq. (1). It has been indicated by the pastexperimental studies (e.g. Sivakumar, 1993; Khali-li et al., 1998) that critical state friction angle isuniquely defined by stress ratio of the Bishops ef-fective stress regardless of the degree of saturationSr. It has also been mentioned through the experi-mental evidences (Kawai et al., 2000) that, pro-vided that the effective degree of saturation Sre(Karube et al., 1996) is applied as the effectivestress parameter , a unique relationship betweenstress ratio of Bishops stress and dilatancy exists

irrespective of Sr. It can hence be stated that aunique critical state friction angle and a unique

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0.2

0.4

0.6

0.8

1.0

100 1000 10000

NCLsat

60

70

80

90

100

100 1000 10000

Fig. 10 Simulations of constant suction consolidationtestson over consolidated soils of different Sr.in Sr due to volumetric contraction though the suc-tion remains constant. It is also known that theproposed model can consider the experimentaltendency reported by Wheeler and Sivakumar(1995) that soil having lower Sr shows stiffer be-haviour.

Second, soaking tests have been performed ontwo kinds of samples (initial void ratio e0 = 0.8,0.9). In the simulation, suction s is firstly increasedfrom zero to 294 kPa under constant net stress netof 49 kPa, net is then increased to prescribed val-ues (49, 294, 588, 1017, 1960, 2940, 9800 kPa)under constant s of 294 kPa and samples are fi-nally soaked by decreasing s to zero. It is seenfrom Fig. 11 that the proposed model is able tosimulate typical soaking collapse behaviour suchas shown in Fig. 2. The volume changes due to

soaking process are plotted against net stress levelin Fig. 12. The proposed model properly predictsthe magnitude of the hydraulic collapse consider-ing the effects of stress level and density. The pro-posed model can also describe the soaking-inducedexpansion of heavily over consolidated soil re-ported by Sun et al. (2007).

Third, compaction behaviour has been investi-gated (Figs. 13 and 14). Suction s is firstly in-creased under constant applied stress of 50 kPauntil prescribed water content w is achieved. Be-cause compaction of soil is generally regarded as

the expulsion of pore-air without significant drain-age of pore-water, total stress is increased from50 kPa to predetermined maximum value

0.2

0.4

0.6

0.8

1.0

100 1000 10000

70

80

90

100

0 50 100 150 200 250 300

soaking(decreaseins)

Fig. 11 Simulations of constant suction consolidationand subsequent soaking tests (initial void ratio e0 = 0.8).

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

100 1000 10000Decreaseofvoidratio

e

duringsoaking

Net stress net[kPa]

compression

66.5

66.7

67.368.6

73.1

78.8

98.7

72.9

73.0

73.276.1

80.5

73.7

98.8

Srbefore soaking

e0=0.8

e0=

0.9

expansion Figure 12 Decreases in void ratio of dense and loosesamples (e0 = 0.8, 0.9) due to soaking process.

(max = 100, 200, 400, 800, 1600, 3200, 6400 kPa)with keeping constant water content and backagain in this calculation. Fig. 13 shows compac-

tion behaviours of three samples of different watercontent. It is seen from this figure that: soil samplehaving larger water content is more compressiblein the beginning stage of compaction; soil is hardlycompressed regardless of its water content once itis saturated as there remains little pore-air to bedrained to compact soil. Results of 80 compactiontests, in which water content w and maximum ap-plied stress max are varied, are summarised interms of dry density (void ratio) and water contentin Fig. 14. It is pointed out from this figure that theproposed model can express typical compaction

curves having optimum water content and it canalso consider the transition of compaction curvesowing to the increase in the applied pressure.

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