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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 756854 10 pageshttpdxdoiorg1011552013756854

Research ArticleStrength Theory Model of Unsaturated Soils withSuction Stress Concept

Pan Chen Changfu Wei Jie Liu and Tiantian Ma

State Key Laboratory of Geomechanics and Geotechnical Engineering Institute of Rock and Soil MechanicsChinese Academy of Sciences Xiaohongshan Wuchang Wuhan Hubei 430071 China

Correspondence should be addressed to Changfu Wei cfweiwhrsmaccn

Received 24 June 2013 Revised 12 September 2013 Accepted 18 September 2013

Academic Editor Pengcheng Fu

Copyright copy 2013 Pan Chen et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A theoretical model is developed for describing the strength property of unsaturated soilsThemodel is able to predict convenientlythe strength changes of unsaturated soils undergoing repeated changes of water content Suction stress is adopted in the newmodelin order to get the sound form of effective stress for unsaturated soils The shear strength of unsaturated soils is dependent on itssoil-moisture state based on the results of shear experiments Hence the parameters of this model are related tightly to hydraulicproperties of unsaturated soils and the strength parameters of saturated soilsThe predictive curves by the newmodel are coincidentwith experimental data that underwent single drying and dryingwetting cycle pathsHence hysteretic effect in the strength analysisis necessary to be considered to predict the change of shear strength of unsaturated soils that underwent dryingwetting cycles Oncethe new model is used to predict the change of shear strength lots of time could be saved due to avoiding heavy and complicatedstrength tests of unsaturated soils Especially the model can be suitable to evaluate the shear strength change of unsaturated soilsand the stability of slopes experienced the dryingwetting cycles

1 Introduction

The change of soil strength is very important to evaluatethe stability of slopes and road embankment In this casethe strength model of unsaturated soils is always concernedin geotechnical engineering Until now many researchershave shown that the shear strength of unsaturated soilswas tightly related to soil-moisture state [1ndash3] The strengthof unsaturated soils will increase or decrease during theintermittent precipitation and fluctuant water tables [4 5]Hence a new model should be necessarily developed inwhich the change of soil strength can be reasonably predictedunder the repeated change of water content The model willbe critical to analyze these problems that soil slopes becomedestabilized during the intermittent rainfall process

The theoretical model for shear strength of unsaturatedsoils is generally developed based on the concept of effectivestress which is similar to the equation of shear strength ofsaturated soilsThere are two types of description for effectivestress of unsaturated soils that are prevalent in the literatureOne type bases on the single stress variable [6] and the other

on the double stress variable [7] The equations of shearstrength are given in Formulas (1) and (2) respectively asfollows

120591 = 1198881015840+ [120590 minus 119901

119873+ 120594 (119901

119873minus 119901119882)] tan1205931015840 (1)

120591 = 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + (119901119873 minus 119901119882) tan120601119887 (2)

where 120591 is the shear strength 1198881015840 is the cohesion at thesaturated state 120590 is the normal total stress 1205931015840 is the effectiveinternal friction angle and 119901119873 and 119901119882 are the pore airpressure and pore water pressure respectively There are twoparameters 120594 and 120601119887 in Formulas (1) and (2)which cause thedifference of strength equations in form between saturatedsoils and unsaturated soils 120594(119901119873 minus 119901119882) tan1205931015840 and (119901119873 minus119901119882) tan120601119887 are the change of shear strength arising by the

change of matric suction (water content) in unsaturated soilsOn the one hand 120594 in single stress variable framework willchange with the degree of saturation 119878

119903 But the results of

some experiments show that the relationship is not unique

2 Journal of Applied Mathematics

between 120594 and 119878119903[1 8] 120594 is strongly depended on the soil

structure and soil-moisture state On the other hand thedefinition of the two-state variable exists some confusionin the scales from the view of continuum mechanics [9]Normal stress loads on the skeleton of soils but matricsuction stresses on the air-water interface Unfortunatelythe two stress state variables are treated as stress variablesacting on the representative elementary volume for the soils(REV) In Formula (2) the relationship of 120601119887 and matricsuction 119904

119888(119904119888= 119901119873minus 119901119882) is nonlinear for the same soil

[10] And the functional relation is different for each typeof soil between 120601119887 and 119904

119888 Vanapalli et al [10] proposed an

empirical relation for calculating 120601119887 based on experimentsof unsaturated soils Khalili and Khabbaz [11] and Kayadelenet al [12] gave different strengthmodels based on the analysisof effective stress respectively Other forms of strengthmodelare also found in the literature [13ndash15]

Although the existing models of shear strength havebeen used to analyze the geotechnical problems by manyresearchers some problems may come out in practical appli-cation Firstly the parameters 120594 and 120601119887 in these formulas arenot easy to be directly determined through the experimentsof unsaturated soils Secondly the physical basis of theseformulas is not definite Hence it could be suitable for somesoils but it is difficult to be adapted for other soils underdifferent soil-moisture conditions Thirdly matric suction isone of the main variables in the expression of soil strengthThere are some difficulty questions in the measurement ofmatric suction which limit the development of experimentaltechnology in laboratory and field conditions such as cav-itation phenomenon [16] The application of these modelsis perhaps not reasonable to evaluate some problems in thefield Last but not least most of the equations were limitedto be used under only the drying process of unsaturatedsoils But the strength characteristics of unsaturated soils aredifferent which underwent the drying and wetting processesdue to the existent of hysteretic effect at varying water contentconditions [17]

Focusing on these problems of the existing shear strengthmodels a new model will be developed to describe thestrength property of unsaturated soils A conception ofsuction stress is introduced in order to modify the formulaof effective stress of unsaturated soils Based on the modifiedformula of effective stress the formula of shear strength canbe obtained using Mohr-Coulombrsquos theory Shear strengthenvelope of unsaturated soils is unique on the plane withmodified effective stress and shear strength at differentmatricsuction or water content states Then the soil-water retentioncurve is used to replace empirical parameters 120594 and 120601119887Therefore the strength of unsaturated soils can be predictedby the measurement of water content which is easy to bedetermined both in the laboratory and field In order toexplore the strength property of unsaturated soils that under-went dryingwetting cycles the hysteretic model (ISVH) [18]is introducedThe new model is able to simulate the strengthevolution of unsaturated soils under repeated hydraulic pathsLastly the predictive results from the model are comparedwith experimental data

Modified SSCCSSCC

MC-FE under saturated state

MC-FE under unsaturated state

pN minus pW

120590998400n

120591

1206019984001

1206019984002

120601998400c9984001

c9984002

c998400

120590S0

120590S1

120590S2

120590S or 120590998400S

Figure 1 Suction stress characteristic curve from Mohr-Coulombfailure envelopes

2 The Concept of Suction Stress andSuction Strength

21The Concept of Suction Stress Based on the analysis of thecomposites of microscopic forces in details Lu and Likos [19]defined suction stress which is the macroscopic expressionof different interactions in microscale (such as physical-chemical force van der Waals force electrical double-layerrepulsion and surface tension) The theoretical basis of thesuction stress has been discussed by Lu and Likos [19] Herethe method for obtaining suction stress is given from directshear test and triaxial shear test of unsaturated soils Tradi-tionally the data of shear strength under different normalstresses are plotted on the plane of normal effective stress1205901015840

119899and shear strength 120591 or net mean effective stress 1199011015840 and

deviatoric stress 119902 seen in Figure 1 1198881015840 11988810158401 and 1198881015840

2are the

intercept that Mohr-Coulomb failure envelops (MC-FE)and cut 120591-axis 1198881015840 1198881015840

1 and 1198881015840

2are called effective cohesion

under saturated state and unsaturated state respectively 12059310158401205931015840

1 and 1205931015840

2are effective internal friction angle at different

saturated state The cohesion is the bond force or attractiveeffect among soil particles The intercept on the 120591-axis givesthe frictional action among soil particles It is puzzled thatthe intercept can be called cohesion Further the intercept onthe 120591-axis cannot fully represent the bond strength amongthe soil skeletons Actually the intersection is the attractiveeffect that MC-FE is prolonged and cuts the 1205901015840

119899-axis which

can be expressed by 120590119878 called suction stress by Lu and Likos

[19] The formula of suction stress can be given from directshear test as follows

120590119878=

minus1198881015840

tan1205931015840 (3)

The suction stress can be obtained by Formula (3) at differentwater content state The relationship of suction stress andmatric suction is called suction stress characteristic curve(SSCC) It is to be noted that suction stress is not zero forfine grain soils under saturated state called 120590

1198780 The original

SSCC needs to be modified by the suction stress 1205901198780

togo through the origin which is consistent with soil-waterretention curve (SWRC) seen in Figure 1

Journal of Applied Mathematics 3

250

200

150

100

50

00 50 100 150 200 250

Matric suction sc = 50kPaMatric suction sc = 100 kPaMatric suction sc = 200 kPa

Matric suction sc = 350kPaMatric suction sc = 500kPaFitting curves

Normal stress (kPa)

Shea

r stre

ngth

(kPa

)

(a) Shear strength envelops at drier than optimum water content

300

250

200

150

100

50

00 50 100 150 200 250



Normal stress (kPa)

Shea

r stre

ngth

(kPa

)

(b) Shear strength envelops at wetter than optimum water content

Matric suction sc = 0kPaMatric suction sc = 150 kPaMatric suction sc = 350kPa

Matric suction sc = 500kPaFitting curves

300

250

200

150

100

50

00 50 100 150 200 250

Normal stress (kPa)

Shea

r stre

ngth

(kPa

)

(c) Shear strength envelops at optimum water content

Figure 2 The curves of normal stress and shear strength at different initial water content state of soil (experimental data from Vanapalliet al [10])

The suction stress can also be obtained from triaxial sheartests according to Mohr-Coulomb criterion The formula isgiven as follows

120590119878= minus

1205901015840

1minus 119901119873

2 tan (1205874 + 12059310158402) tan1205931015840

+

(1205901015840

3minus 119901119873) tan (1205874 + 12059310158402)2 tan1205931015840

+

1198881015840

tan1205931015840

(4)

where 12059010158401and 1205901015840

3are the major and minor principle stress

respectively

The new effective stress can be defined based on suctionstress as follows

1205901015840= (120590 minus 119901

119873) minus 120590119878 (5)

where 1205901015840 is the effective stress and 120590 is the total stress Inorder to verify the expression of effective stress is reasonablethe experimental data of shear strength or deformation testsof unsaturated soils could be used

Sandy-clay till was used to do shear strength test atthree types of water content state by Vanapalli et al [10]Shear failure envelops go upward drift with matric suctionincreasing which is shown in Figure 2(a) SimultaneouslyFigures 2(a) 2(b) and 2(c) show that the shear strengths are


200

150

100

50

00 50 100 150 200 250 300 350 400

Matric suction s = 50kPaMatric suction s = 100 kPaMatric suction s = 200 kPa

Matric suction s = 350kPaMatric suction s = 500kPaThe fitting curves

Effective normal stress (kPa)

Shea

r stre

ngth

(kPa

)

(a) Modified shear strength envelops at drier than optimum watercontent

The fitting curves

200

150

100

50

00 50 100 150 200 250 300 350 400


Shea

r stre

ngth

(kPa

)

Matric suction sc = 0kPaMatric suction sc = 150kPaMatric suction sc = 250kPa

Matric suction sc = 350kPaMatric suction sc = 500kPa

(b) Modified shear strength envelops at wetter than optimum watercontent

The fitting curves

250

200

150

100

50

00 100 200 300 500400


Shea

r stre

ngth

(kPa

)


Matric suction sc = 500kPa

(c) Modified shear strength envelops at optimum water content

Figure 3 The curves of effective normal stress and shear strength at different initial water content state of soil (experimental data fromVanapalli et al [10])

not the same although the matric suction and normal stressare completely identical The unique difference is the initialstate of the soils The shear strength is largest at wetter thanoptimum water content state and it is smallest at drier thanoptimumwater content state Obviously the shear strength ofunsaturated soils 120591 is strongly depended on the state of watercontent We can obtain suction stresses from Formula (3) or(4) using the tested data from Figures 2(a) 2(b) and 2(c)Then the new effective stress is achieved The strength failureenvelops are redrawn in the plane 1205901015840 minus 120591 in Figure 3

It is interesting that these data points at failure state tendto a line The phenomenon is not coincidental There aremanymeasured data from literatures that are not given due to

the limited space of this paper which can be seen detailedlyin the literature [21 22]That is to say the critical state failureenvelop is unique under the new effective stress frameworkof unsaturated soils And the effective stress can be alsoused for saturated soils which is coincident with Terzaghirsquoseffective theory The new effective stress is the reasonableone with suction stress The problem of nonunique failureenvelops of unsaturated soils in traditional framework issolved by the new effective stress frameworkThemeaning ofeffective stress with suction stress is clear in the framework ofcontinuum mechanics And it is deduced that the propertiesin deformation and strength of unsaturated soils could bedescribed in the unified way by themathematics Formula (5)


22 Suction Strength of Unsaturated Soils The shear strengthequation of unsaturated soils is obtained based on theeffective stress Formula (5)

120591119891= 1198881015840+ (120590 minus 119901

119873minus 120590119878) tan1205931015840 (6)

where 1198881015840 and 1205931015840 are the effective cohesion and the effectiveinternal friction angle at saturated state respectively

Suction stress ismacrorepresentation of interaction of soilparticles in microscale which increases the attraction of soilskeleton and shear strength Compared with saturated soilthere is a difference that the shear strength is related to watercontent The change of strength due to the fluctuate of watercontent is defined as suction strength 1198881015840

119878

119888119878= minus120590119878tan1205931015840 (7)

Based on these literatures [11ndash13] the suction strength canbe obtained from the shear strength tests The apparentcohesion 119888 is defined as

119888 = 1198881015840+ 119888119878 (8)

The shear strength equation can be modified as follows

120591119891= 119888 + (120590 minus 119901

119873) tan1205931015840 (9)

The equation is coincident with the one of saturated soils inform Therefore the shear strength of unsaturated and satu-rated soils can be both expressed by Formula (9)The formulaof shear strength is obtained in the unified framework of soilsin which new empirical parameters are not introduced

3 Shear Strength Model of Unsaturated SoilsDepending on Hydraulic State

The suction stress is related to water content of unsaturatedsoils from the above analyses The relationship is derived byLu et al [21] based on the principles of thermodynamics

120590119878= minus119878119890(1199011198731015840

minus 119901119882) (10)

where 119901119882 is the water pressure and matric suction 119904119888is

defined as follows

119904119888= 1199011198731015840

minus 119901119882 (11)

119878119890is the effective degree of saturation

119878119890=

(119878119903minus 119878

irr119903)

(1 minus 119878irr119903)

(12)

where 119878119903is the degree of saturation and 119878irr

119903is the residual

degree of saturationIntroducing Formulas (10) (11) and (12) to Formulas (7)

and (9) the suction strength and shear strength equations canbe given as follows

119888119878= 119878119890119904119888tan1205931015840 (13)

120591119891= 1198881015840+ 119878119890119904119888+ (120590 minus 119901

119873) tan1205931015840 (14)

As seen from Formula (14) the shear strength of unsat-urated soils is only related to the degree of saturation butalso related to matric suction The relationship of the shearstrength and matric suction (the degree of saturation) canbe obtained by introducing the soil-water retention curve(SWRC) The change of soil-water state is generally notmonotonic under intermittent precipitation and fluctuatingwater tables The capillary hysteresis (hydraulic hysteresis)often exits during the increment and decrement of watercontent in the seepage process of unsaturated soils Capil-lary hysteresis refers to the nonunique relationship betweenthe degree of saturation and matric suction and describesthe irreversible changes in the degree of saturation occur-ring during the preceding sequence of drying and wettingof a porous medium The importance of hysteretic effect(hydraulic hysteresis) in the unsaturated flow has been foundin the literatures [23] Furthermore hysteretic effect canalso significantly influence the shear strength and the shearbehaviour as seen in works such as those performed byKwong [24] and Khoury and Miller [25] Kwong [24] foundthat the strengths of unsaturated soils getting wetter are lowerthan those getting drier Khoury and Miller [25] found thatshear strength following a dryingwetting process was higherthan that for the drying process alone at the similar matricsuction and net normal stressThese results give an importantconclusion that the water content and matric suction are ofequal importance to obtain the shear strength of unsaturatedsoils The two soil-water state parameters are affected bythe hydraulic hysteresis The effect of hysteresis should beconsidered in the analysis of the strength problems related tounsaturated soils

In order to conclude the hysteretic effect in the shearstrength problems the hysteretic soil-water relationshipshould be developed for constructing the shear strengthmodel of unsaturated soils There are some methods whichmay be used to consider hysteretic effect during the processof the water content change history [26ndash28] The main objectof this paper is to develop a new strength model to reproducethe change of shear strength that underwent the effect ofcapillary hysteresis in unsaturated soils Recently a capillaryhysteretic model with internal state variables (ISVH-model)was developed by Wei and Dewoolkar [18] The boundarysurface plasticity theory is used to model the hystereticbehavior of the soil water retention curves In this model thearbitrary water content variable path can be traced betweenthe main boundary curves The equations of the model arepresented here for the integrality of this paper Feng andFredlund [29] offered an equation which was used to well fitthe boundary curves of the soil water retention curves Themain drying curve is

119878119903119863=

1 + 119878irr119903119863(119904119888119887119863)119886119863

1 + (119904119888119887119863)119886119863

(15a)

and the main wetting curve is

119878119903119882=

1 + 119878irr119903119882(119904119888119887119882)119886119882

1 + (119904119888119887119882)119886119882

(15b)


500

400

200

300

100

0

05 06 07 08 09 10

Measured dataThe fitting curve

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)

Figure 4 The fitting tested curve of SWRC for the completelydecomposed granite soil (measured data from Hossain and Yin[20])

where 119878119903119863

and 119878119903119882

are the degrees of saturation of the dryingand wetting boundaries respectively 119878irr

119903119863and 119878irr

119903119882are the

residual degrees of saturation at the drying and wettingconditions respectively 119887

119863 119886119863 119887119882 and 119886

119882are the four

material parametersThe evolution equation of the degree of saturation that

underwent the drying and wetting cycles is described asfollows

119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)

where 119899 is the direction of the hydraulic path and its valueis minus1 or 1 For the wetting path 119899 = minus1 and for the dryingpath 119899 = 1 119870

119901(119878119903 119904119888 119899) is given as follow

119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)

where 119889 is a fit parameter which is an additional param-eter to describe all of the scanning curves in the hys-teretic cycle matric suction in the main drying and wettingboundary curves is expressed by 119904

119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878

119903) is the difference of the matric

suction between the main boundary curves when the soilwater state is at the degree of saturation 119878

119903 119903(119878119903) =

120581119863(119878119903) minus 120581119882(119878119903) 119870119901and 119904119888(119878119903 119899) are respectively the slope

and matric suction of the main drying and wetting boundarycurves as follows(1) Drying path (119899 = 1)

119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)

(2)Wetting path (119899 = minus1)

119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)

Introducing the hysteretic model (ISVH) suction strengthand shear strength are expressed as follows

119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)

If the current soil-water state (119904119888 119878119903) and the increment of the

degree of saturation 119878119903are given the change of shear strength

of unsaturated soils can be predicted by Formulas (20a) and(20b) during any dryingwetting cycle combined with theshear strength of saturated soils (1198881015840 and 1205931015840)

4 Model Verification

The suction strength and shear strength of unsaturated soilsare predicted based onFormulas (18a) (18b) (20a) and (20b)And the predictive curves are compared with experimentalstrength data The parameters of soil-water retention curvesfrom fitting the measured soil-water data are used to predictthe change of shear strength

41 Strength Property under Single Drying Hydraulic PathThe soil-water retention curve and direct shear strengthtests of unsaturated completely decomposed granite soil areperformed in the laboratory by Hossain and Yin [20] Themeasured soil-water retention curve is given in Figure 4 Andthe shear strength parameters at saturated state are 1198881015840 =00 kPa and 1205931015840 = 299∘ Formula (15a) is adopted to fit themeasured soil-water data for the drying process alone Thefitting curve is also shown in Figure 4 And the parametersof the soil-water retention curve are listed in Table 1 For-mulas (19b) and (20b) are adopted to predict the suctionstrength and shear strength of the soils combining with theshear strength parameters at saturated state The predictivecurves are shown in Figure 5 The coincidence of suctionstrength is well under low suction conditions And a littledeviation comes out in high suction conditions (Figure 5(a))The prediction of shear strength of the soils is acceptableat low effective stresses However the deviation betweenmeasured data and the predictive curve is large at highereffective stresses The similar results are shown by Hossainand Yin [20] The distinct dilative behavior of unsaturatedcompacted completely decomposed granite soil was observedin the measurement by Hossain and Yin [20] The apparentcohesion intercept and angle of internal friction increase withmatric suction due to the effect of dilative behavior It is to benote that the effective stress is obtained based on Formula (5)(Figure 5(b))

The samemethod is also adopted to predict shear strengthofDiyarbakir residual claysThe shear strength and soil-water


Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400

Matric suction (kPa)

Measured dataThe predictive curve

(a) Suction strength

Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400

Effective stress (kPa)

Measured data sc = 0kPaMeasured data sc = 50kPaMeasured data sc = 100 kPa

Measured data sc = 200 kPaMeasured data sc = 300kPaPredictive curve

(b) Shear strength

Figure 5 Comparison of tested data with the predictive curve of suction strength and shear strength (measured data from Hossain and Yin[20])

Table 1 Parameters of SWRC of unsaturated soil and shear strength of saturated soil for predicting suction strength of unsaturated soils thatunderwent single drying hydraulic path

Soil types Parameters of SWRC Strength parameters119899 119878

irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840

Completely decomposed granite soil 036 0120 02961 91735 0 299Diyarbakir residual clays 0581 0442 04679 48874 1482 219

1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


Figure 6 The fitting tested curve of SWRC for Diyarbakir residualclays (measured data from Kayadelen et al [12])

retention tests are performed by Kayadelen et al [12] Thefitting soil-water retention curve is shown in Figure 6 Andthe parameters of the soil-water retention curve are listed inTable 1 The shear strength parameters of Diyarbakir residual

clays at saturated state are 1198881015840 = 1482 kPa and 1205931015840 = 219∘Thepredictive curves of suction strength and shear strength of theresidual clays are shown in Figure 7As seen fromFigures 7(a)and 7(b) the suction strength and shear strength both wellmatch up to experimental data The distribution of the shearstrength data at different matric suctions is approximatelylinear again in Figure 7(b) The critical state failure is uniqueunder the new effective stress state based on suction stress Itdoes not exist that the failure envelops are nonunique basedon the new shear strength model

42 Strength Property under DryingWetting Paths The shearstrength is very important for predicting the slope stabilityunder the intermittent precipitation conditions for exampleGvirtzman et al [30]The variation in strength of unsaturatedsoils will be studied in the section under the repeating changeof water content conditions However the measured data ofshear strength that underwent dryingwetting process arelittle seen in the literature The main reasons are that the testinstruments are not well established and these experimentsare time-consuming in the laboratory

Goh et al [5] performed a series of unsaturated con-solidation drained triaxial tests under drying and wettingin which compacted sand-kaolin specimens were adoptedThe soil-water retention curves are also measured by tem-pecell and pressure plate in Goh et al [5] The 1198881015840 and 1205931015840 of


100

80

60

40

20

0

Suct

ion

stres

s (kP

a)

0 100 200 300 400 500Matric suction (kPa)

Experimental dataThe predictive curve


Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength

Figure 7 Comparison of tested data with the predictive curve of suction strength and shear strength (measured data from Kayadelen et al[12])

Table 2 Parameters of SWRC of unsaturated soil and shear strength of saturated soil for predicting suction strength of unsaturated soils thatunderwent dryingwetting paths


irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840

Sand-kaolin mixture 0470 0055 0566 152030 0055 0481 87481 196200 85 269

the compacted saturated sand-kaolin mixture are 85 kPaand 269∘ respectively The soil-water retention data and thefitting curves are given in the Figure 8 The main drying andmain wetting curve are fitted by Formulas (15a) and (15b)respectively The wetting scanning data are used to correctthe parameter 119889 in Formula (17) Then the parameter 119889 isadopted to predict the drying scanning curveTheparametersfor fitting SWRC are listed in Table 2

The parameters of the main drying and wetting curvesand the parameter d are used to predict the change ofsuction strength of the unsaturated sand-kaolin mixture thatunderwent the dryingwetting cycle The predictive resultsare presented in Figure 9The suction strength obtained fromthe drying process was predicted by these parameters of themain drying and drying scanning curves respectively Thecoincidence is well compared with the measured data Atthe low suction state the predictive curve is much nearerthe measured results using the parameters from the dryingscanning curve than those from the main drying curveHowever the tendency of the predictive curves is the same atthe high suction stateThat is due to that the drying scanningcurve and the main drying curve are coincident at the highsuction condition seen in Figure 8 The similar comparisonswere done between the measured suction strength from thewetting process with the predictive curves The predictivetendency from the wetting process is different from theone from the drying state At the high suction state the

predictive curve is much nearer the measured results usingthe parameters from the wetting scanning curve than thosefrom the main wetting curve However the tendency of thepredictive curves is also the same at the low suction stateThat is due to that the wetting scanning curve and the mainwetting curve are coincident at the high suction conditionwhich can be also seen in Figure 8 The soil-water state isnot always along the main boundary curves due to hystereticeffect The soil-water state is perhaps along the scanningdrying or wetting curve at the shear strength tests Hencethe predictive results by the scanning curves are nearer to themeasured strength data than the ones by the main boundarycurves

It noted to say that the correlation of suction strengthand soil-water state is evident seen in Figure 9 Furthermorethe difference of suction strength is increasing with theincrement of matric suction for the drying or wetting pathAnd the suction strength along the drying path is larger thanthe ones along the wetting path That is to say the shearstrength is closely related to the water content and matricsuction Due to the existence of hysteretic effect the watercontent may be different at the same matric suction throughdifferent dryingwetting paths The shear strength will bedifferent though the matric suction is the same Hence itis indispensable that the research work should be carriedout on the change of shear strength of unsaturated soils thatunderwent the repeating changes of water content

Journal of Applied Mathematics 9M

atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)

Measured main drying dataMeasured main wetting dataFitting main drying data

Fitting main wetting dataCorrected the parameter dPredicted the scanning curve

Figure 8The fitting tested curve of SWRC for sand-kaolin mixture(measured data from Goh et al [5])

150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)

Measured data along drying pathMeasured data along wetting pathPredicted curve using the main drying curvePredicted curve using the main wetting curvePredicted curve using the wetting scanning curvePredicted curve using the wetting scanning curve

Figure 9 Comparison of tested data with the predictive curve ofsuction strength under drying-wetting paths (measured data fromGoh et al [5])

5 Conclusions

The theoretical strength model is developed based on theconcept of suction stress The predictive curves of the modelare compared with experimental data And its validity of thestrength model is verified There are some conclusions asfollows(1) Suction stress is the macroscopic effect of many

different microscopic forces in unsaturated soils Redun-dant parameters need not to be introduced to describe

these microscopic forces respectively The effective stressframework of unsaturated soils is improved The failureenvelop of unsaturated soils is unique at different matricsuction and net normal stresses based on the new effectivestress framework The cohesion arisen by tension among soilparticles expresses its real concept The relation of matricsuction and suction stress can be uniquely expressed bythe suction stress characteristic curve (SSCC) which is veryimportant for describing the stress state similar with SWRCfor describing the soil-water state The uncertainty of theparameters of shear strength of soils can be avoided in theeffective stress frame(2) In the new strength model the SWRC and SSCC

are combined to predict the change of shear strength ofunsaturated soils under repeated water content (matric suc-tion) change The SWRC is used to predict the change ofsuction strength and shear strength of unsaturated soils TheSWRC is widely adopted in geotechnical engineering and soilphysicsTheparameters of SWRCaremuch easier to obtain inlaboratory or field compared with the shear strength tests ofunsaturated soils Furthermore the time can be saved largelyonce the SWRC is used to predict the strength of unsaturatedsoils especially for fine gained soils(3) The predictive curves of suction strength and shear

strength of the soils both well match up to experimen-tal data for the completely decomposed granite soil andDiyarbakir residual clays that underwent single drying pathsFurthermore the coincidence is well compared with themeasured suction strength for the sand-kaolinmixture underthe repeated change ofwater contentHence the new strengththeory model with suction effective concept is validated forpredicting the change of shear strength of unsaturated soilsthat underwent the fluctuation of water content In the newstrengthmodel only the parameters of SWRC of unsaturatedsoils and shear strength of saturated soil were used to predictthe change of shear strength of unsaturated soils under thearbitrary change of water content(4) Based on the measured strength data and predictive

curves the shear strength is closely related to the watercontent and matric suction The shear strength can bedifferent at the same matric suction due to the existenceof hysteretic effect Hence hysteretic effect in the seepageprocess should be considered to predict the change of shearstrength of unsaturated soils that underwent dryingwettingcycles

Acknowledgments

The research is supported by the National Natural ScienceFoundation of China (11302243 11072255) the Natural Sci-ence Foundation of GuangXi (no 2011 GXNSFE018004) andthe planedmajor science and technology projects of Zhejiang(no 2009C13010)

References

[1] J E B Jennings and J B Burland ldquoLimitations to the use ofeffective stresses in partly saturated soilsrdquo Geotechnique vol 12no 2 pp 125ndash144 1962


[2] D G Fredlund N R Morgenstern and R A Widger ldquoShearstrength of unsaturated soilsrdquo Canadian Geotechnical Journalvol 15 no 3 pp 313ndash321 1978

[3] S K Vanapalli D E Pufahl and D G Fredlund ldquoThe effect ofstress state on the soil-water characteristic behavior of acompacted sandy-clay tillrdquo in Proceedings of the 51st CanadianGeotechnical Conference pp 81ndash86 1998

[4] L Cao and X-Q Luo ldquoExperimental study of dry-wet circu-lation of Qianjiangping Landslidersquos unsaturated soilrdquo Rock andSoil Mechanics vol 28 pp 93ndash97 2007 (Chinese)

[5] S G Goh H Rahardjo and E C Leong ldquoShear strengthequations for unsaturated soil under drying and wettingrdquoJournal of Geotechnical and Geoenvironmental Engineering vol136 no 4 pp 594ndash606 2010

[6] AWBishop ldquoTheprinciple of effective stressrdquoTekniskUkebladvol 106 no 39 pp 859ndash863 1959

[7] DG Fredlund andN RMorgenstern ldquoStress state variables forunsaturated soilsrdquo Journal of the Geotechnical Engineering Divi-sion vol 103 no 5 pp 447ndash466 1977

[8] N Khalili F Geiser and G E Blight ldquoEffective stress inunsaturated soils review with new evidencerdquo InternationalJournal of Geomechanics vol 4 no 2 pp 115ndash126 2004

[9] N Lu ldquoIsmatric suction a stress variablerdquo Journal of Geotechni-cal and Geoenvironmental Engineering vol 134 no 7 pp 899ndash905 2008

[10] S K Vanapalli D G Fredlund D E Pufahl and A W CliftonldquoModel for the prediction of shear strength with respect to soilsuctionrdquo Canadian Geotechnical Journal vol 33 no 3 pp 379ndash392 1996

[11] N Khalili and M H Khabbaz ldquoA unique relationship for 120594 forthe determination of the shear strength of unsaturated soilsrdquoGeotechnique vol 48 no 5 pp 681ndash687 1998

[12] C Kayadelen M A Tekinsoy and T TaskIran ldquoInfluence ofmatric suction on shear strength behavior of a residual clayeysoilrdquo Environmental Geology vol 53 no 4 pp 891ndash901 2007

[13] A L Oberg and G Sallfors ldquoDetermination of shear strengthparameters of unsaturated silts and sands based on the waterretention curverdquo Geotechnical Testing Journal vol 20 no 1 pp40ndash48 1997

[14] LMiao S Liu andY Lai ldquoResearch of soil-water characteristicsand shear strength features of Nanyang expansive soilrdquo Engi-neering Geology vol 65 no 4 pp 261ndash267 2002

[15] Y F Xu ldquoFractal approach to unsaturated shear strengthrdquo Jour-nal of Geotechnical and Geoenvironmental Engineering vol 130no 3 pp 264ndash273 2004

[16] N Lu and W J Likos Unsaturated Soil Mechanics John WileyNew York NY USA 2004

[17] D Tami H Rahardjo and E-C Leong ldquoEffect of hysteresis onsteady-state infiltration in unsaturated slopesrdquo Journal ofGeotechnical and Geoenvironmental Engineering vol 130 no 9pp 956ndash967 2004

[18] C F Wei and M M Dewoolkar ldquoFormulation of capillary hys-teresis with internal state variablesrdquo Water Resources Researchvol 42 Article IDW07405 2006

[19] N Lu andW J Likos ldquoSuction stress characteristic curve for un-saturated soilrdquo Journal of Geotechnical and GeoenvironmentalEngineering vol 132 no 2 pp 131ndash142 2006

[20] M A Hossain and J-H Yin ldquoShear strength and dilative chara-cteristics of an unsaturated compacted completely decomposedgranite soilrdquo Canadian Geotechnical Journal vol 47 no 10 pp1112ndash1126 2010

[21] N Lu J W Godt and D T Wu ldquoA closed-form equation foreffective stress in unsaturated soilrdquo Water Resources Researchvol 46 Article IDW05515 2010

[22] P Chen A Study on Seepage and Slope Stability of UnsaturatedSoils Subjected to DryingWetting Cycles Institute of Rock andSoil Mechanics Chinese Academy of Sciences Wuhan China2011

[23] C YangD Sheng and J P Carter ldquoEffect of hydraulic hysteresison seepage analysis for unsaturated soilsrdquo Computers andGeotechnics vol 41 pp 36ndash56 2012

[24] H K Kwong Effect of hysteresis infiltration and tensile stress onthe strength of an unsaturated soil [PhD thesis] NanyangTechnological University Singapore 1997

[25] CNKhoury andGAMiller ldquoInfluence of hydraulic hysteresison the shear strength of unsaturated soils and interfacesrdquoGeotechnical Testing Journal vol 35 no 1 2012

[26] H Q Pham D G Fredlund and S L Barbour ldquoA study of hys-teresis models for soil-water characteristic curvesrdquo CanadianGeotechnical Journal vol 42 no 6 pp 1548ndash1568 2005

[27] H-C Huang Y-C Tan C-W Liu and C-H Chen ldquoA novelhysteresis model in unsaturated soilrdquo Hydrological Processesvol 19 no 8 pp 1653ndash1665 2005

[28] AMaqsoud B BussiereMAubertin andMMbonimpa ldquoPre-dicting hysteresis of the water retention curve from basicproperties of granular soilsrdquo Geotechnical and Geological Engi-neering vol 30 pp 1147ndash1159 2012

[29] M Feng and D G Fredlund ldquoHysteretic influence associatedwith thermal conductivity sensor measurementsrdquo in ProceedingfromTheory to the Practice of Unsaturated Soil Mechanics 52ndCanadian Geotechnical Conference and Unsaturated Soil GroupRegina 1999

[30] H Gvirtzman E Shalev O Dahan and Y H Hatzor ldquoLarge-scale infiltration experiments into unsaturated stratified loesssediments monitoring and modelingrdquo Journal of Hydrologyvol 349 no 1-2 pp 214ndash229 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


between 120594 and 119878119903[1 8] 120594 is strongly depended on the soil

structure and soil-moisture state On the other hand thedefinition of the two-state variable exists some confusionin the scales from the view of continuum mechanics [9]Normal stress loads on the skeleton of soils but matricsuction stresses on the air-water interface Unfortunatelythe two stress state variables are treated as stress variablesacting on the representative elementary volume for the soils(REV) In Formula (2) the relationship of 120601119887 and matricsuction 119904

119888(119904119888= 119901119873minus 119901119882) is nonlinear for the same soil

[10] And the functional relation is different for each typeof soil between 120601119887 and 119904

119888 Vanapalli et al [10] proposed an

empirical relation for calculating 120601119887 based on experimentsof unsaturated soils Khalili and Khabbaz [11] and Kayadelenet al [12] gave different strengthmodels based on the analysisof effective stress respectively Other forms of strengthmodelare also found in the literature [13ndash15]

Although the existing models of shear strength havebeen used to analyze the geotechnical problems by manyresearchers some problems may come out in practical appli-cation Firstly the parameters 120594 and 120601119887 in these formulas arenot easy to be directly determined through the experimentsof unsaturated soils Secondly the physical basis of theseformulas is not definite Hence it could be suitable for somesoils but it is difficult to be adapted for other soils underdifferent soil-moisture conditions Thirdly matric suction isone of the main variables in the expression of soil strengthThere are some difficulty questions in the measurement ofmatric suction which limit the development of experimentaltechnology in laboratory and field conditions such as cav-itation phenomenon [16] The application of these modelsis perhaps not reasonable to evaluate some problems in thefield Last but not least most of the equations were limitedto be used under only the drying process of unsaturatedsoils But the strength characteristics of unsaturated soils aredifferent which underwent the drying and wetting processesdue to the existent of hysteretic effect at varying water contentconditions [17]

Focusing on these problems of the existing shear strengthmodels a new model will be developed to describe thestrength property of unsaturated soils A conception ofsuction stress is introduced in order to modify the formulaof effective stress of unsaturated soils Based on the modifiedformula of effective stress the formula of shear strength canbe obtained using Mohr-Coulombrsquos theory Shear strengthenvelope of unsaturated soils is unique on the plane withmodified effective stress and shear strength at differentmatricsuction or water content states Then the soil-water retentioncurve is used to replace empirical parameters 120594 and 120601119887Therefore the strength of unsaturated soils can be predictedby the measurement of water content which is easy to bedetermined both in the laboratory and field In order toexplore the strength property of unsaturated soils that under-went dryingwetting cycles the hysteretic model (ISVH) [18]is introducedThe new model is able to simulate the strengthevolution of unsaturated soils under repeated hydraulic pathsLastly the predictive results from the model are comparedwith experimental data

Modified SSCCSSCC

MC-FE under saturated state

MC-FE under unsaturated state

pN minus pW

120590998400n

120591

1206019984001

1206019984002

120601998400c9984001

c9984002

c998400

120590S0

120590S1

120590S2

120590S or 120590998400S

Figure 1 Suction stress characteristic curve from Mohr-Coulombfailure envelopes

2 The Concept of Suction Stress andSuction Strength

21The Concept of Suction Stress Based on the analysis of thecomposites of microscopic forces in details Lu and Likos [19]defined suction stress which is the macroscopic expressionof different interactions in microscale (such as physical-chemical force van der Waals force electrical double-layerrepulsion and surface tension) The theoretical basis of thesuction stress has been discussed by Lu and Likos [19] Herethe method for obtaining suction stress is given from directshear test and triaxial shear test of unsaturated soils Tradi-tionally the data of shear strength under different normalstresses are plotted on the plane of normal effective stress1205901015840

119899and shear strength 120591 or net mean effective stress 1199011015840 and

deviatoric stress 119902 seen in Figure 1 1198881015840 11988810158401 and 1198881015840

2are the

intercept that Mohr-Coulomb failure envelops (MC-FE)and cut 120591-axis 1198881015840 1198881015840

1 and 1198881015840

2are called effective cohesion

under saturated state and unsaturated state respectively 12059310158401205931015840

1 and 1205931015840

2are effective internal friction angle at different

saturated state The cohesion is the bond force or attractiveeffect among soil particles The intercept on the 120591-axis givesthe frictional action among soil particles It is puzzled thatthe intercept can be called cohesion Further the intercept onthe 120591-axis cannot fully represent the bond strength amongthe soil skeletons Actually the intersection is the attractiveeffect that MC-FE is prolonged and cuts the 1205901015840

119899-axis which

can be expressed by 120590119878 called suction stress by Lu and Likos

[19] The formula of suction stress can be given from directshear test as follows

120590119878=

minus1198881015840

tan1205931015840 (3)

The suction stress can be obtained by Formula (3) at differentwater content state The relationship of suction stress andmatric suction is called suction stress characteristic curve(SSCC) It is to be noted that suction stress is not zero forfine grain soils under saturated state called 120590

1198780 The original

SSCC needs to be modified by the suction stress 1205901198780

togo through the origin which is consistent with soil-waterretention curve (SWRC) seen in Figure 1


250

200

150

100

50

00 50 100 150 200 250



Normal stress (kPa)

Shea

r stre

ngth

(kPa

)


300

250

200

150

100

50

00 50 100 150 200 250



Normal stress (kPa)

Shea

r stre

ngth

(kPa

)




300

250

200

150

100

50

00 50 100 150 200 250

Normal stress (kPa)

Shea

r stre

ngth

(kPa

)




120590119878= minus

1205901015840

1minus 119901119873

2 tan (1205874 + 12059310158402) tan1205931015840

+

(1205901015840

3minus 119901119873) tan (1205874 + 12059310158402)2 tan1205931015840

+

1198881015840

tan1205931015840

(4)

where 12059010158401and 1205901015840


respectively


1205901015840= (120590 minus 119901

119873) minus 120590119878 (5)




200

150

100

50

00 50 100 150 200 250 300 350 400




Shea

r stre

ngth

(kPa

)


The fitting curves

200

150

100

50

00 50 100 150 200 250 300 350 400


Shea

r stre

ngth

(kPa

)




The fitting curves

250

200

150

100

50

00 100 200 300 500400


Shea

r stre

ngth

(kPa

)










120591119891= 1198881015840+ (120590 minus 119901

119873minus 120590119878) tan1205931015840 (6)



119878

119888119878= minus120590119878tan1205931015840 (7)


119888 = 1198881015840+ 119888119878 (8)


120591119891= 119888 + (120590 minus 119901

119873) tan1205931015840 (9)




120590119878= minus119878119890(1199011198731015840

minus 119901119882) (10)


defined as follows

119904119888= 1199011198731015840

minus 119901119882 (11)


119878119890=

(119878119903minus 119878

irr119903)

(1 minus 119878irr119903)

(12)





119888119878= 119878119890119904119888tan1205931015840 (13)

120591119891= 1198881015840+ 119878119890119904119888+ (120590 minus 119901

119873) tan1205931015840 (14)



119878119903119863=

1 + 119878irr119903119863(119904119888119887119863)119886119863

1 + (119904119888119887119863)119886119863

(15a)


119878119903119882=

1 + 119878irr119903119882(119904119888119887119882)119886119882

1 + (119904119888119887119882)119886119882

(15b)


500

400

200

300

100

0

05 06 07 08 09 10


Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


where 119878119903119863

and 119878119903119882


119903119863and 119878irr

119903119882are the


119863 119886119863 119887119882 and 119886

119882are the four



119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)



119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)


119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878



119903 119903(119878119903) =



119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)


119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)


119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)









Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










250

200

150

100

50

00 50 100 150 200 250



Normal stress (kPa)

Shea

r stre

ngth

(kPa

)


300

250

200

150

100

50

00 50 100 150 200 250



Normal stress (kPa)

Shea

r stre

ngth

(kPa

)




300

250

200

150

100

50

00 50 100 150 200 250

Normal stress (kPa)

Shea

r stre

ngth

(kPa

)




120590119878= minus

1205901015840

1minus 119901119873

2 tan (1205874 + 12059310158402) tan1205931015840

+

(1205901015840

3minus 119901119873) tan (1205874 + 12059310158402)2 tan1205931015840

+

1198881015840

tan1205931015840

(4)

where 12059010158401and 1205901015840


respectively


1205901015840= (120590 minus 119901

119873) minus 120590119878 (5)




200

150

100

50

00 50 100 150 200 250 300 350 400




Shea

r stre

ngth

(kPa

)


The fitting curves

200

150

100

50

00 50 100 150 200 250 300 350 400


Shea

r stre

ngth

(kPa

)




The fitting curves

250

200

150

100

50

00 100 200 300 500400


Shea

r stre

ngth

(kPa

)










120591119891= 1198881015840+ (120590 minus 119901

119873minus 120590119878) tan1205931015840 (6)



119878

119888119878= minus120590119878tan1205931015840 (7)


119888 = 1198881015840+ 119888119878 (8)


120591119891= 119888 + (120590 minus 119901

119873) tan1205931015840 (9)




120590119878= minus119878119890(1199011198731015840

minus 119901119882) (10)


defined as follows

119904119888= 1199011198731015840

minus 119901119882 (11)


119878119890=

(119878119903minus 119878

irr119903)

(1 minus 119878irr119903)

(12)





119888119878= 119878119890119904119888tan1205931015840 (13)

120591119891= 1198881015840+ 119878119890119904119888+ (120590 minus 119901

119873) tan1205931015840 (14)



119878119903119863=

1 + 119878irr119903119863(119904119888119887119863)119886119863

1 + (119904119888119887119863)119886119863

(15a)


119878119903119882=

1 + 119878irr119903119882(119904119888119887119882)119886119882

1 + (119904119888119887119882)119886119882

(15b)


500

400

200

300

100

0

05 06 07 08 09 10


Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


where 119878119903119863

and 119878119903119882


119903119863and 119878irr

119903119882are the


119863 119886119863 119887119882 and 119886

119882are the four



119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)



119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)


119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878



119903 119903(119878119903) =



119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)


119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)


119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)









Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200

150

100

50

00 50 100 150 200 250 300 350 400




Shea

r stre

ngth

(kPa

)


The fitting curves

200

150

100

50

00 50 100 150 200 250 300 350 400


Shea

r stre

ngth

(kPa

)




The fitting curves

250

200

150

100

50

00 100 200 300 500400


Shea

r stre

ngth

(kPa

)










120591119891= 1198881015840+ (120590 minus 119901

119873minus 120590119878) tan1205931015840 (6)



119878

119888119878= minus120590119878tan1205931015840 (7)


119888 = 1198881015840+ 119888119878 (8)


120591119891= 119888 + (120590 minus 119901

119873) tan1205931015840 (9)




120590119878= minus119878119890(1199011198731015840

minus 119901119882) (10)


defined as follows

119904119888= 1199011198731015840

minus 119901119882 (11)


119878119890=

(119878119903minus 119878

irr119903)

(1 minus 119878irr119903)

(12)





119888119878= 119878119890119904119888tan1205931015840 (13)

120591119891= 1198881015840+ 119878119890119904119888+ (120590 minus 119901

119873) tan1205931015840 (14)



119878119903119863=

1 + 119878irr119903119863(119904119888119887119863)119886119863

1 + (119904119888119887119863)119886119863

(15a)


119878119903119882=

1 + 119878irr119903119882(119904119888119887119882)119886119882

1 + (119904119888119887119882)119886119882

(15b)


500

400

200

300

100

0

05 06 07 08 09 10


Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


where 119878119903119863

and 119878119903119882


119903119863and 119878irr

119903119882are the


119863 119886119863 119887119882 and 119886

119882are the four



119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)



119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)


119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878



119903 119903(119878119903) =



119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)


119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)


119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)









Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591119891= 1198881015840+ (120590 minus 119901

119873minus 120590119878) tan1205931015840 (6)



119878

119888119878= minus120590119878tan1205931015840 (7)


119888 = 1198881015840+ 119888119878 (8)


120591119891= 119888 + (120590 minus 119901

119873) tan1205931015840 (9)




120590119878= minus119878119890(1199011198731015840

minus 119901119882) (10)


defined as follows

119904119888= 1199011198731015840

minus 119901119882 (11)


119878119890=

(119878119903minus 119878

irr119903)

(1 minus 119878irr119903)

(12)





119888119878= 119878119890119904119888tan1205931015840 (13)

120591119891= 1198881015840+ 119878119890119904119888+ (120590 minus 119901

119873) tan1205931015840 (14)



119878119903119863=

1 + 119878irr119903119863(119904119888119887119863)119886119863

1 + (119904119888119887119863)119886119863

(15a)


119878119903119882=

1 + 119878irr119903119882(119904119888119887119882)119886119882

1 + (119904119888119887119882)119886119882

(15b)


500

400

200

300

100

0

05 06 07 08 09 10


Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


where 119878119903119863

and 119878119903119882


119903119863and 119878irr

119903119882are the


119863 119886119863 119887119882 and 119886

119882are the four



119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)



119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)


119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878



119903 119903(119878119903) =



119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)


119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)


119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)









Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










500

400

200

300

100

0

05 06 07 08 09 10


Mat

ric su

ctio

n (k

Pa)

Saturation (deg)


where 119878119903119863

and 119878119903119882


119903119863and 119878irr

119903119882are the


119863 119886119863 119887119882 and 119886

119882are the four



119878119903= minus

119904119888

119870119901(119878119903 119904119888 119899)

(16)



119870119901(119878119903 119904119888 119899) = 119870

119901(119878119903 119899) +

1198891003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

119903 (119878119882

119903) minus1003816100381610038161003816119904119888minus 119904119888(119878119903 119899)1003816100381610038161003816

(17)


119888= 120581119863(119878119903) and 119904

119888=

120581119882(119878119903) respectively 119903(119878



119903 119903(119878119903) =



119904119888(119878119903 1) = 120581

119863(119878119903) 119870

119901(119878119903 1) =

119889120581119863(119878119903)

119889119878119903

(18a)


119904119888(119878119903 minus1) = 120581

119882(119878119903) 119870

119901(119878119903 minus1) =

119889120581119882(119878119903)

119889119878119882

119903

(18b)


119888119878= 119904119888tan1205931015840 119904

119888le 0 (19a)

119888119878= 119878119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0 (19b)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119904

119888tan1205931015840 119904

119888le 0 (20a)

120591119891= 1198881015840+ (120590 minus 119901

119873) tan1205931015840 + 119878

119890( 119878119903) 119904119888tan1205931015840 119904

119888gt 0

(20b)









Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Suct

ion

stren

gth

(kPa

)120

100

80

60

40

20

00 100 200 300 400




Suct

ion

stren

gth

(kPa

)

400

300

200

100

00 100 200 300 500400




(b) Shear strength




irr119903

119886 119887 (kPa) 1198881015840 (kPa) 120593

1015840


1600

2000

1200

800

400

006 08 10

Mat

ric su

ctio

n (k

Pa)

Saturation (deg)








100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

80

60

40

20

0

Suct

ion

stres

s (kP

a)




Suct

ion

stres

s (kP

a)

0

100

200

300

400

0 100 200 300 500400 700600




(b) Shear strength




irr119903119863

119886119863119887119863(kPa) 119878

irr119903119882

119886119882119887119882

(kPa) 119889 1198881015840 (kPa) 120593

1015840







atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










atric

suct

ion

(kPa

)

1200

1000

800

600

400

200

005 06 07 08 09 10

Saturation (deg)




150

100

50

00 50 100 150 200 250 300


Suct

ion

stren

gth

(kPa

)



5 Conclusions







Acknowledgments


References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

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