power law for elastic moduli of unsaturated soil

Power Law for Elastic Moduli of Unsaturated SoilNing Lu, F.ASCE1; and Murat Kaya2

Abstract: Elasticmoduli (Young’s modulus and shear modulus) are material properties that describe a material’s elastic stress-strain relation,and are therefore two of themost important properties in geotechnical engineering design and analysis. For soils, in addition to their well-knowndependence on stress, these moduli depend on volumetric water content and/or matric suction, particularly for silty and clayey soils. This studyproposes a simple power law to describe the dependence of these two moduli on volumetric water content for all types of soils. A series ofuniaxial compression tests are conducted on various compacted soils under varying volumetricwater content. Young’smoduli aremeasured andused to test the validity of the proposed power-law relationship. Additional validation and comparative analyses are conducted using othercompacted soils studied by previous investigators and empirical models for shear modulus from the literature. It is shown that the proposedpower law agreeswell with the othermodels, but ismuch simpler because the othermodels use bothmatric suction and volumetricwater contentas independent variables and involvemore fitting parameters. A practical three-point testing procedure is provided to determine the singlefittingparameter that defines the power law for any type of soil. The procedure involves measuring elastic moduli at three states of water content: dry,wet (nearly saturated), and themiddle points. Test results for 16 soils demonstrate that the proposed three-point testing procedure can accuratelycapture the dependence of Young’s moduli for all types of soils from sandy to silty to clayey soils (R2 . 95% for most soils). The proposedpower law provides a simple and practical way to describe changes in elastic moduli of soils under variably saturated conditions.DOI:10.1061/(ASCE)GT.1943-5606.0000990. © 2014 American Society of Civil Engineers.

Author keywords:Elasticmodulus;Young’smodulus; Shearmodulus;Unsaturated soils;Matric suction;Volumetricwater content; Specificsurface area.

Introduction

Elastic moduli (Young’s modulus and shear modulus) refer tothe stiffness of materials in response to recoverable deformation.Young’s modulus can be defined as the ratio of the applied normalstress to the responding normal strain. Shear modulus can be definedas the ratio of the applied shear stress to the responding shear strain.Young’s modulus has been widely used as the basis for calculatingsettlements for a variety of foundation types bearing on compactedor over consolidated soils after construction. Shear modulus hasbeen used as an effective parameter for examining stiffness of thebase course and subgrade in pavementmaterials [Yoder andWitczak1975;Nazarian et al. 2003;National CooperativeHighwayResearchProgram (NCHRP) 2004; Schuettpelz et al. 2010].

To conduct geotechnical design and analysis, it is often desirableto utilize models for these elastic moduli. In the classical theories,elastic modulus is mainly considered to be a function of stressconditions (normal and deviator stresses), soil type (grain size,composition, and pore structure), and environmental factors (volu-metric water content and/or matric suction). Historically, stress andenvironmental conditions have been considered to be the main

factors in some early theories (Edil 1973; Fredlund et al. 1975;Moossazadeh and Witczak 1981; Edil et al. 1981). In recent years,environmental factors such as volumetric water content and matricsuction have been explicitly incorporated in describing dependenceof elastic modulus on these factors (Ng et al. 2009; Sawangsuriyaet al. 2009; Schuettpelz et al. 2010; Khosravi andMcCartney 2011).Despite the strong dependence of elastic modulus on environmentalfactors shown in many silty and clayey soils tested in recent studies(Mancuso et al. 2002; Khoury and Zaman 2004), few theories ormodels effectively describing such dependence have been rigor-ously tested for different types of soils under variably saturatedconditions.

Some new evidence is presented for the dependence of Young’smodulus on volumetric water content for various soils ranging fromsand, to silt and clays. Previous experimental data from the literatureare presented to further illustrate the dependence of shear moduluson both matric water content and soil suction. Based on examiningthese data and some previous experimental data and theories, apower law that describes the dependence of elastic modulus or shearmodulus on volumetric water content is proposed, and its accuracyfor various soils under variably saturated conditions is assessed. Apractical testing procedure is also proposed and evaluated to de-termine a single parameter defining the power law for any type of soil.

Experimental Evidence on Elastic Modulus–WaterContent Relation

Drying Cake Test and Results for Young’s Modulus

The drying cake (DC) test was recently invented by Lu and Kaya(2013) for measuring the soil water retention curve (SWRC) andhydraulic conductivity function (HCF), aswell asYoung’smodulus,of a compacted disk-shape soil specimen (cake) while drying. The

1Professor, Dept. of Civil and Environmental Engineering, ColoradoSchool of Mines, Golden, CO 80401 (corresponding author). E-mail:[email protected]

2Graduate Student, Dept. of Civil and Environmental Engineering,Colorado School of Mines, Golden, CO 80401. E-mail: [email protected]

Note. This manuscript was submitted on June 7, 2012; approved on June25, 2013; published online on June 27, 2013. Discussion period open untilJune 1, 2014; separate discussions must be submitted for individual papers.This paper is part of the Journal of Geotechnical and GeoenvironmentalEngineering, Vol. 140, No. 1, January 1, 2014. ©ASCE, ISSN 1090-0241/2014/1-46–56/$25.00.

46 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2014

J. Geotech. Geoenviron. Eng. 2014.140:46-56.

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http://dx.doi.org/10.1061/(ASCE)GT.1943-5606.0000990


mailto:[email protected]



principle and procedure ofmeasuringYoung’smodulus is illustratedin Fig. 1. For the flexibility in achieving desired porosity and watercontent range, the standard Proctor compaction technique [ASTMD698 (ASTM 1993)] is not followed. For a soil, two identicalspecimens with diameters ranging from 50 to 200mmand a thicknessranging from 10 to 20 mm are prepared by using static compaction tocreate a specified porosity at nearly saturated conditions. The com-paction process ensures that the consequent testing of elastic modulusduring drying is within the overly consolidated soils in the elasticrange. The specimens are then placed on a smooth and flat base(much larger than the specimens) coated with a thin layer of Vaselineor vacuum grease so that the friction between the cake and its sup-porting base is minimized. The drying occurs through the top freesurface of the cake. Thus, during the drying step, the cake is underzero or nearly zero total stress conditions, and the suction stress ateach point within the cake is the stress responsible for soil shrinkingand/or swelling (Lu and Kaya 2013).

Volumetric water content of the DC is closely recorded by anelectronic balance. Of the two specimens, one is used to monitor itsvolume change using a particle image velocimetry (PIV) techniqueat different water contents, whereas the other is used to measureYoung’s modulus at different water contents using a uniaxial com-pression frame. The setup for measuring Young’s modulus usinga uniaxial compression loading frame is shown in Fig. 1. The dryingprocess typically starts with the nearly saturated cake. After drying toa desired water content, one of the two cakes is taken out of thechamber and placed in the loading frame for uniaxial compression.A loading rate of 0:013mm=min based on standard testing and de-termined by conducting several different rates is applied and ensuresthat there is no difference in modulus value between this rate and a

rate 10 times smaller. Stress-strain data are recorded for identifyingYoung’s modulus, with the applied strain typically being less than1.5%or the correspondinguniaxial stress less than100kPa [Fig. 1(b)].The Young’s modulus test terminates at whichever aforementionedcriterion occurs first. The cake is then unloaded and moved out of theloading frame and placed back on the electronic balance in thechamber for continuing drying. This drying-loading-unloading pro-cess is repeated until the cake is nearly dry, or there is little change inwater content under the laboratory ambient conditions and Young’smodulus as a function of water content is observed. For sandy soils,a restraining ring is used so the loading is under the zero-horizontalstrain or K0 conditions. A detailed description of the specimenpreparation and testing procedure, as well the measurement of SWRCand HCF by the PIV technique, can be found in Lu and Kaya (2013).

Eleven different soil types ranging from sandy to silty to clayeysoils were used for the DC and modulus tests. The geotechnicalproperties of these soils are summarized in Table 1. The testingresults for Young’s modulus as a function of volumetric watercontent for these soils are shown in Figs. 2(a and b). Two patternscan be seen from these results. First, Young’s modulus generallymonotonically decreases with increasing volumetric water content.Second, soil types with low or no plasticity (e.g., sands and somesilts) exhibit less change [Figs. 2(a and b)] than those soils with highplasticity (e.g., plastic silts and clays) [Fig. 2(b)].

For comparisons with some recent models for shear modulus,the SWRCs of these soils are necessary because these models(described later) explicitly employed SWRC to quantify shear mo-dulus. The unsaturated soil parameters were obtained by fitting thevan Genuchten’s (1980) model to the SWRCs measured by Lu andKaya (2013) and are summarized in Table 2. The van Genuchten’s

Fig. 1. Illustration of measurement of Young’s modulus from the uniaxial DC test: (a) uniaxial compression setup; (b) data for stress-strain for Goldensilt; (c) Young’s modulus as a function of volumetric water content for Golden silt

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / JANUARY 2014 / 47


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model established a closed-form relation betweenmatric suction andthe equivalent degree of saturation Se

Se ¼ u2 urus 2 ur

¼�

11þ ðajua2 uwjÞn

�121=n

(1)

whereua 2 uw5matric suction;us5porosity or saturatedvolumetricwater content; ur 5 residual volumetric water content; a5 inverse ofthe air entry suction when soil starts to desaturate; and n is related tothe pore size distribution. In recent years, the SWRC has been pro-posed and used as the fundamental property for some empiricalmodels of elastic modulus (Alramahi et al. 2008; Sawangsuriya et al.2009; Schuettpelz et al. 2010).TheSWRCcanalsobeused toquantifyother fundamental properties, such as the plasticity index and thespecific surface area (Lu and Likos 2004), which is shown to beimportant to the elastic modulus.

Small Strain Test Results for Shear Modulus

Small-strain shear moduli of pavement soils obtained using a suction-control bender element testing system (Sawangsuriya et al. 2009)were also used to understand the dependency of the elastic moduluson volumetric water content. The bender element test employs anelastic wave propagation technique to measure the shear modulus ofspecimens under either suction-control or water content-controlconditions (Kawaguchi et al. 2001; Lee and Santamarina 2005).The soils used for comparison in this study are fine-grained subgradesoils classified as clayey sand (SC), low-plasticity silt (ML), low-plasticity clay (CL), and high-plasticity clay (CH). These soil speci-mens were prepared using the standard Proctor compaction (ASTMD698) at the optimum water content for the given soils. The geo-technical properties of these soils are listed in Table 1. The SWRCswere measured for the model prediction of shear modulus. In thisstudy, the unsaturated soil properties of these soils, fitted from the dataof SWRCs using vanGenuchten’s equation [Eq. (1)] for later analysis,are listed in Table 2. The shear modulus as a function of volumetricwater content is shown in Fig. 2(c). It is evident that the shear modulusfor all of these soils is a highly nonlinear function of volumetric watercontent. The small-strain elastic moduli measured for these soils willbe used for model comparisons and analysis in the “Model Com-parisons” section.

Previous Models for Shear Modulus

Variables affecting the elastic modulus in unsaturated soil havebeen an important and long-standing research subject. Recently,Sawangsuriya et al. (2009) proposed two empirical models for de-scribing the dependence of shear modulus on total stress, soil matricsuction, and water content. The first model (Model 1 hereafter) wasestablished from an earlier conception by Oloo and Fredlund (1998)using two independent state variables (matric suction and net normalstress), and is based on a shear strength model for unsaturated soil byVanapalli et al. (1996). The shearmodulusG has been presented in thefollowing form (Sawangsuriya et al. 2009):

G ¼ Aðs2 uaÞn1 þ C

�uus

�kðua2 uwÞ (2)

where A, n1, k, and C 5 curve-fitting parameters.The second proposedmodel bySawangsuriya et al. (2009) (Model

2 hereafter) was also established from an earlier conception by Olooand Fredlund (1998), but it is from the viewpoint of effective stressin unsaturated soils by Bishop (1959) (Khalili and Khabbaz 1998;Lu and Griffiths 2004; Lu and Likos 2006). The effective stressparameter by Bishop (1959) was replaced by a shear strength modelfor unsaturated soil by Vanapalli et al. (1996), leading to the fol-lowing empirical form for the shear modulus:

G ¼ A

�ðs2 uwÞ þ

�uus

�kðua 2 uwÞ

�n2(3)

where A, n2, and k 5 curve-fitting parameters. Using fine-grainedsubgrade soils compacted under different energies and volumetricwater content, Sawangsuriya et al. (2009) showed that both models[Eqs. (2) and (3)] satisfactorily describe the dependency of the shearmodulus on matric suction and water content within the testingrange of matric suction and volumetric water content. As shown inEqs. (2) and (3), to predict the shear modulus of a specific soil,either model requires four fitting parameters. Both models employknowledge of two independent variables of matric suction andvolumetric water content or SWRC [e.g., Eq. (1)] if only oneindependent variable (either matric suction or volumetric watercontent) is employed.

Table 1. Fundamental Geotechnical Properties of Various Soils Used for Elastic Modulus Analysis

Soils Classification Liquid limit Plastic limit Plasticity index PorosityWater contentat compaction

Specific surface area

Ss ðm2=gÞEsperance sand SP — — — 0.390 0.25 11.25Ottawa sand SP — — — 0.380 0.25 2.04Elliot Forest silt OL 29.0 26.0 3.0 0.390 0.22 14.34Bonny silt ML 25.0 21.0 4.0 0.470 0.29 22.38BALT silt ML 27.4 21.7 5.8 0.470 0.29 36.31Iowa silt ML 33.7 22.4 11.3 0.450 0.27 30.26Golden silt ML 31.0 17.0 14.0 0.440 0.22 83.68Hopi silt SC 36.0 23.0 13.0 0.480 0.32 53.61Denver claystone CL 44.0 23.0 21.0 0.550 0.39 48.33Georgia kaolinite CL 44.0 26.0 18.0 0.580 0.45 57.81Denver bentonite CH 118.0 45.0 73.0 0.530 0.32 122.16SC-Std-Opta SC 28.0 14.0 14.0 0.269 0.14 29.57ML-Std-Opta CL 28.0 17.0 11.0 0.338 0.15 17.21CL-2-Std-Opta CL 26.0 17.0 9.0 0.325 0.17 45.27CL-1-Std-Opta CL 42.0 18.0 24.0 0.375 0.22 49.44CH-Std-Opta CH 85.0 33.0 52.0 0.535 0.27 85.42

Note: Std-Opt 5 standard proctor compaction at optimum water content.aSawangsuriya et al. (2009).



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Proposed Power Law for Elastic Moduli

Thedependencyof elasticmodulus on either total or effective stress hasbeen a major focus for saturated soils because stress can modify poreand soil skeleton structures, and thus, the stiffness of soils. However,under unsaturated conditions, effective stress is affected not only by thetotal stress, but is also affected by interparticle stresses called suctionstress (Lu and Likos 2004, 2006). Suction stress is the effective stressbecause of variation in volumetric water content or matric suction (Luand Likos 2006; Lu 2008; Lu et al. 2010). Suction stress consists ofinterparticle physicochemical forces, pore water attraction caused bymatric suction, and surface tension. Therefore, in silty and clayey soils,suction stress is highly governed by volumetric water content ormatricsuction. Because the relationship between volumetric water contentand matric suction can be established by SWRC, volumetric watercontentmay be sufficient as the sole independent variable in describingvariation in elastic modulus for unsaturated soil.

FrombothYoung’s modulus and the shear modulus results shownin Figs. 2(a and b), and the experimental results from Sawangsuriyaet al. (2009) shown inFig. 2(c), aswell as other recent studies (Nget al.2009; Schuettpelz et al. 2010; Khosravi and McCartney 2011), it isclear that the elastic modulus of soils follows a power law decay asvolumetric water content increases. It is also observed that the elasticmodulus is maximum when the specimen is dry and minimum whenthe specimen is wet. As such, the following power law is proposed forYoung’s modulus:

E ¼ Ed þ ðEw 2EdÞ�

u2 uduw 2 ud

�m(4a)

whereE5Young’smodulus; subscript d5 dry state;w5wet state;and m 5 empirical fitting parameter. Similarly, the shear moduluscan be expressed as

G ¼ Gd þ ðGw 2GdÞ�

u2 uduw 2 ud

�m(4b)

where G 5 shear modulus. A dry state can be set at practically thedriest environment of interest, and a wet state can be set at nearsaturated or saturated conditions. Compared with Eqs. (2) and (3),there are only three parameters in Eqs. (4a) or (4b) to fully define theelastic modulus of any soil. Furthermore, two of the three param-eters, namely, Ed and Ew in Eq. (4a) andGd andGw in Eq. (4b), havedirect physical meaning because they represent the elastic moduli atthe wet and dry states. Thus, only one fitting parameter m in thepower law [Eqs. (4a) and (4b)] remains to be identified for any soilonce the elastic moduli at the dry and wet states are known. Ina normalized form, Eqs. (4a) and (4b) become

E2Ed

Ew 2Ed¼ G2Gd

Gw 2Gd¼

�u2 uduw 2 ud

�m(5)

Shear modulus (G) is often linked to Young’s modulus (E) by

G ¼ ð1þ mÞE=2 (6)

where m 5 Poisson’s ratio of soils.Fig. 3 shows the least-squared fit of the Young’s modulus of the

11 soil types as predicted by Eqs. (4a) and (4b). Fig. 3(a) shows twosandy soils and two silty soils. The sandy soils have relatively smallvaluesofm (0.01 forOttawa sandand0.2 forEsperance sand), reflectingthe insensitivity of Young’s modulus of the sandy soil with the volu-metric water content. In contrast, the two silty soils have moderatevaluesofm [0.84 forBALT(BayAreaLandslideTasks) silt and0.80 forBonny silt], reflecting the relatively sensitivity of the silty soil to the

Fig. 2. Test results of elastic modulus for various compacted soils asa function of water content from nearly saturated state to dry state for (a)Young’s modulus for Esperance sand, Ottawa sand, BALT silt, Hopisilt, and Bonny silt; (b) Young’s modulus for Golden silt, Elliot Forestsilt, Iowa silt, Denver claystone, Georgia kaolinite, and Denver ben-tonite; (c) shear modulus for SC soil, ML soil, CL1 soil, CL2 soil, andCH soil



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volumetric water content. For example, the Young’s modulus forBALTsilt is 0.9MPaat avolumetricwater content of0.42, but increasesto 3.0MPa at a volumetric water content of 0.04. As shown, the elasticmodulus of all these four soils can be well represented by Eqs. (4a) and(4b).

For the four silty soils shown in Fig. 3(b), the variation ofYoung’s modulus with water content shows a similar pattern withthe two silty soils shown in Fig. 3(a). However, the magnitude isslightly higher for two of the four silty soils (Golden silt and Hopisilt), reflecting high values of m (1.08 for Golden silt and 1.46 forHopi silt). Iowa silt has a similar amount of change in the Young’smodulus compared with the two silty soils shown in Fig. 3(a); them value (0.78) for the Iowa silt is similar to that of BALT and Bonnysilts. Overall, the Young’s modulus of all four soils are well rep-resented by Eqs. (4a) and (4b) because the coefficients of de-termination (R2) are all greater than or equal to 0.97.

For the three clayey soils shown in Fig. 3(c), Young’s modulusvaries highly nonlinearly with the water content, particularly at highvalues of water content. Denver claystone, a predominately non-expansive illite, has a moderate m value (0.56), and its variationin Young’s modulus is also moderate, varying from 1.05 MPa ata volumetric water content of 0.5–4.5 MPa to a volumetric watercontent of 0.06. The least-squared fit between Eqs. (4a) and (4b) andthe data are excellent; the coefficients of determination (R2) aregreater than 0.93 for all three soils.

Model Comparisons

Comparisons with the Young’s Modulus Tests

Because the previous models [Eqs. (2) and (3)] conceptualize theshear modulus as a function of both matric suction and volumetricwater content, and the proposed power law conceptualizes the shearmodulus as a sole function of volumetric water content, it is nec-essary to introduce the SWRC to rewrite Eqs. (2) and (3) in terms ofvolumetric water content. By assuming the residual water content uris zero, the fitting parameter n1 is 0.5, the fitting parameter k is unity(Sawangsuriya et al. 2009), and the total stress is negligible uponunloading, and by substituting Eq. (1) into Eqs. (2), (3), and (6), thefollowing two equations were obtained:

E ¼ 2G1þ m

¼ 2A1þ m

ðs2 uaÞ þ 2C1þ m

1auus

��uus

�n=12n

2 1

�1=n

(7)

E ¼ 2G1þ m

¼ 2A1þ m

�ðs2 uwÞ þ 1

auus

��uus

�n=12n

2 1

�1=n�n2

(8)

For comparison purposes, Poisson’s ratio m is assumed to be aconstant of 0.25 for all fine-grained subgrade soils when convertingshear modulus into Young’s modulus. Based on the tests of Young’smodulus and shear modulus, an average total stress (seua) of35 kPa was used. As shown in the previous equations, there are fourparameters [A,C, a, and n for Model 1 in Eq. (7), and A, n2, a, and nfor Model 2 in Eq. (8)] involved in defining shear modulus G interms of volumetric water content.

The parameters defined in Eqs. (4a), (4b), (7), and (8) wereobtained byfitting these equationswith the tests data on elasticmodulishown in Fig. 2. The least-squared fit parameters defining Model 1,Model 2, and the proposedpower law [Eqs. (4a) and (4b)] forYoung’smodulus tests of the 11 soils are shown in Table 2. The comparisonsamong these 2models and Eq. (4a) are shown in Figs. 4 and 5. For thetwo sandy soils, all three models compare well with the experimentaldata with little visible differences among the models (not shown).

From the comparisons among the six silty soils shown in Figs.4(a–c), all three models capture the trend of diminishing Young’smodulus with an increase in the volumetric water content, but theproposedmodel consistently fits better than the previous twomodelswith much higher coefficient of determination (R2) values (0.97–0.99 versus 0.63–0.99 for the two previous models). The compar-isons for the three models for BALT silt and Bonny silt [(Fig. 4(a)]are close. However, for the other four silty soils [Golden silt andElliot Forest silt in Fig. 4(b), and Hopi silt and Iowa silt in Fig. 4(c)],the proposed power law outperforms both Model 1 and Model 2.

From the comparison among the three clayey soils shown inFig. 5, the three models follow the test data closely for both Denverclaystone (predominately illite) andDenver bentonite (predominatelymontomorillinite) with high coefficients of determination (R2) greater

Table 2. Unsaturated Hydromechanical Properties of Various Soils Used for Elastic Modulus Analysis

Soils a ðkPa21Þ n ur us

Proposed modelModel 1 Model 2

m A (Eq. 2) B (Eq. 3) A (Eq. 2) N2

Esperance sand 0.220 2.52 0.018 0.39 0.20 78.00 0.17 439.00 0.01Ottawa sand 0.230 4.00 0.004 0.38 0.01 70.00 1.00 414.00 0.00Elliot Forest silt 0.010 1.64 0.010 0.39 6.12 0.10 30.13 11.22 0.82Bonny silt 0.060 1.50 0.020 0.47 0.80 18.35 1.76 23.05 0.42BALT silt 0.080 1.40 0.030 0.47 0.84 19.11 6.87 5.07 0.87Iowa silt 0.040 1.90 0.040 0.45 0.78 22.32 8.17 6.34 0.85Golden silt 0.030 1.21 0.020 0.44 1.08 48.98 0.78 67.14 0.30Hopi silt 0.030 1.90 0.080 0.48 1.46 1.00 18.27 0.01 2.44Denver claystone 0.020 1.35 0.010 0.55 0.56 26.38 2.52 20.99 0.51Georgia kaolinite 0.006 2.05 0.050 0.58 3.75 1:003 1027 11.65 1:423 10205 3.35Denver bentonite 0.010 1.46 0.100 0.53 0.96 5.24 3.08 0.98 1.02SC-Std-Opta 0.008 1.40 0.040 0.27 1.05 9,792.0 681.00 14,534.00 0.37ML-Std-Opta 0.031 1.34 0.010 0.34 0.31 4,232.0 528.00 1,590.00 0.72CL-2-Std-Opta 0.003 1.38 0.030 0.33 1.37 10,418.0 475.00 12,503.00 0.40CL-1-Std-Opta 0.003 1.38 0.030 0.38 1.63 5,931.0 283.00 5,499.00 0.47CH-Std-Opta 0.002 1.40 0.040 0.54 0.79 2,984.0 176.00 1,412.00 0.64

Note: Std-Opt 5 standard proctor compaction at optimum water content.aSawangsuriya et al. (2009).



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Fig. 3. Fitting of Young’s modulus data with the proposed model forvarious compacted soils: (a) Esperance sand, Ottawa sand, BALT silt,and Bonny silt; (b) Golden silt, Hopi silt, Elliot Forest silt, and Iowa silt;(c) Denver claystone, Georgia kaolinite, and Denver bentonite

Fig. 4. Comparative analysis of Young’s modulus prediction by dif-ferent models for (a) BALT silt and Bonny silt; (b) Golden silt and ElliotForest silt; (c) Hopi silt and Iowa silt



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than 0.91 [Figs. 5(a and c)]. The proposed model has the highestR2 values, ranging from 0.97 to 0.99. For Georgia kaolinite (pre-dominately kaolinite), the proposed model accords the experimentaldata for the entire water content range of the experimental data witha high value of the coefficient of determination (R2 5 0:98), whereasboth Model 1 and Model 2 compare poorly at low or high values ofvolumetric water content [Fig. 5(b)].

Comparisons with the Shear Modulus Tests

Comparisons of the fitted shear modulus among the three models tothe experimental data from the compacted fine-grained subgradesoils, as obtained by Sawangsuriya et al. (2009), are shown in Fig. 6.The original shear modulus data were first converted to Young’smodulus using Eqs. (7) and (8), and using an assumed constantPoisson’s ratio value (0.25) for these soils. Overall, all three modelsagree well with the experimental data, except for the CL2 soil andthe CH soil, where Model 2 yields relative low coefficients of de-termination (R2 5 0:88 for the CL2 soil and 0.54 for the CH soil).The water content range for these five soils is relatively narrow incomparison with the 11 soils used in the Young’s modulus tests,which may be attributed to the excellent overall comparisons. Thewater content ranges for these tests were all constrained within thefeasible compaction range by the optimum water contents.

In summary of comparisons, it is shown that the proposed powerlaw[Eqs. (4a) and (4b)] predicts theYoung’smodulus equally or betterthan the previous models. The proposed power law accurately rep-resents the dependence of elastic moduli on volumetric water content.

Controls on Parameter m

To make the proposed power law useful, a natural question is: whatare the controls for the fitting parameter m? By examining Table 2,the value of m is inversely proportional to sizes of soil grains. Forexample, sandy soils have the smallestm values (less than or equal to0.2), silty soils havem values between 0.3 and 1.08, and clayey soilshave m values greater than 0.79. However, statistical analyses forthe correlations between parameterm and the plasticity index do notyield a practically useful correlation, as shown in Fig. 7(a). This in-dicates that the parameter m cannot be determined from that of thegrain sizes and the plasticity index alone.

Also from Table 2, it clear that there is no correlation betweenparameter m and unsaturated property n. However, a correlation doesexist between parameterm and unsaturated property a (R2 5 0:42, andfine-grained soils consistently have higherm values and smalla values),yet this correlation is not strong enough for practical use, as shown inFig. 7(b). The slight correlation may be caused by the fact that fine-grained soils often exhibit small a values or high air entry pressures.

Because the SWRCs of all 16 soils are available, it is possible tocompute the specific surface area from each soil. Lu and Likos(2004) presented a theory to calculate capillary pore size distributionand specific surface area from SWRC data. The results of the cal-culated specific surface areas of these 16 soils are listed in Table 1,and the correlation of the specific surface area with parameter m isshown in Fig. 7(c). Other than the organic soil (Elliot Forest silt,shown as circled bullets in Fig. 7), parameterm follows a trend withthe specific surface area, with a coefficient of determination (R2) of0.76. As described in Lu and Likos (2004), the specific surface areacan be obtained by integrating the pore size distribution of soil. Thus,the specific surface area of soil plays an important role in controllingthe elastic modulus of the soil as a function of volumetric watercontent. Alternatively, stemming from practical viewpoints, a directlaboratory measurement procedure to define the power law [Eqs.(4a) and (4b)] for any given soil is proposed in the following.

Fig. 5. Comparative analysis of Young’s modulus prediction by dif-ferentmodels for (a) Denver claystone; (b) Georgia kaolinite; (c) Denverbentonite



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Proposed Practical Procedure for MeasuringElastic Moduli

Proposed Three-Point Laboratory Testing Procedure

Based on the preceding comparisons with different models and 16different soils, the proposed power law is recommended for repre-senting elastic moduli (either Young’s or shear modulus) for variablysaturated soils because it shows superior performance compared with

Fig. 6. Comparative analysis of Young’s modulus prediction by dif-ferent models for pavement soils compacted to optimum water content:(a) SC soil; (b) ML soil and CL1 soil; (c) CL2 soil and CH soil; Std-Opt 5 standard proctor compaction at optimum water content [datafrom Sawangsuriya et al. (2009)]

Fig. 7. Illustrations of controls of parameter m: (a) as a function ofplasticity index; (b) as a function of parameter a; (c) as a function of thespecific surface area



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some recentmodels thatmodel variety of soilswith fewer parametersand independent variables. As shown in Eqs. (4a), (4b), or (5), onlyone fitting parameter m is involved. A simple and practical testingprocedure was devised for a complete description of elastic modulusunder variably saturated conditions.

The procedure involves elastic modulus testing under three dif-ferent water contents: dry, wet, and a middle point between the twoend states. Both the wet and dry states should be chosen based on therepresentative environmental conditions. Under extreme cases, thewet state should be full saturation and the dry state should be underthe residual water content state. According to Eq. (5), elastic modulimeasured under these two end members provide values for Ew andEd . A third test, conducted under a water content um around theaverage of these two end states, provides a measure for Em anda unique determination of the sole fitting parameter m shown in theproposed power law [Eq. (5)]

m ¼log

�Em2Ed

Ew 2Ed

�

log

�um2 uduw 2 ud

� (9)

Thus, suction control or volumetric water content control testingprocedures as suggested in the previous models are unnecessary.Instead, simple constant water content specimens can be preparedby static compaction to a desired combination of porosity and watercontent. The effectiveness and accuracy of the suggested testing pro-cedure by using the proposed power law is assessed in the following.

Assessment of the Proposed LaboratoryTesting Procedure

Following the suggested three-point testing procedure, Figs. 8(a and b)show comparisons of Young’s modulus between Eq. (5) and the 11soils tested for Young’s modulus, whereas Fig. 8(c) shows com-parisons of Young’s modulus between Eq. (5) and the five fine-grained subgrade soils from the shear modulus measurements. Forthe two sandy soils, Eq. (5) closely follows the experimental data,although the variation in Young’s modulus is small in theses soils[Fig. 8(a)]. For the six silty soils [Figs. 8(a and b], the coefficients ofdetermination (R2) are all greater than or equal to 0.93, indicatingexcellent correlations between Eq. (5) and the experimental data. Forthe three clayey soils [Fig. 8(b)], the measured Young’s moduli canbe well represented by Eq. (5) because the coefficients of de-termination (R2) are all greater than or equal to 0.95.

Comparisons of Young’s modulus calculated by Eq. (5) andthe Young’s modulus data converted from the small-strain shearmodulus data are excellent because all of the coefficients of de-termination are greater than or equal to 0.97, as shown in Fig. 8(c). Theassessment concludes that the proposed three-point testing procedureprovides excellent predictions for elastic modulus under variably sat-urated conditions for different types of soils. Thus, it is concludedthat the proposed power law with the three-point testing procedureprovidesasimple, accurate, andpracticalway toquantifyelasticmodulusvariation under variably saturated conditions for all types of soils.

Summary and Conclusions

The high dependency of elastic moduli (Young’s modulus and shearmodulus) with the volumetric water content for silty and clayey soilshas long been recognized, yet quantitative models describing suchdependency have not been established until recently. Recent modelsrely on matric suction and volumetric water content control tests and

Fig. 8. Assessments of the proposed three-point testing procedure indetermining the single fitting parameter m for Eq. (5): (a) Esperancesand, Ottawa sand, BALT silt, and Bonny silt, Elliot Forest silt, andGolden silt; (b) Hopi silt, Iowa silt, Denver claystone, Georgia kaolinite,and Denver bentonite; (c) SC soil, ML soil, CL1, CL2 soil, and CH soil



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require knowledge of the independent variables of matric suction andvolumetric water content or the SWRC if only one independentvariable of either matric suction or soil water content is employed.Although these models can reasonably represent the variation of theelastic modulus under different volumetric water contents for fine-grained subgrade soils, these models require a number of empiricalparameters and series of sophisticated modulus testing for parameteridentification.

A systematic testing program on Young’s modulus was con-ducted for a wide range of compacted soils under drying conditions,for sandy, silty, and clayey soils. Based on examination of these dataand some previous experimental data and models, an empiricalpower law with one fitting parameter is proposed for either Young’smodulus or shear modulus as a sole function of soil water content.

Using the elastic moduli data from 16 different soils, it wasshown that the proposed power law could accurately represent thedependency of the elastic modulus on volumetric water content. Thecomparative analysis with other recent models demonstrated thatthe proposed power law performed as well or better, in some cases,in representing the elastic modulus as a function of volumetric watercontent. Furthermore, the proposed power law only involved thedetermination of one fitting parameter, whereas the previous modelsinvolved at least four parameters.

The sole fitting parameter in the proposed power law can befundamentally linked to some geotechnical properties. Althoughthis parameter is not affected by the plasticity index, or the un-saturated parameter a, it is correlated to the specific surface area ofsoils. Additional research is needed to confirm such correlation or torelate parameter m to some other fundamental properties of soils.

A simple and practical three-point laboratory testing procedurewas also proposed in identifying the fitting parameter in the pro-posed power law. The assessment of such a procedure using the 16soils indicates that the proposed power law and the three-pointtesting procedure accurately describes elastic moduli as a functionof volumetric water content for all types of soils.

This investigation emphasizes the dependency of elastic modulion volumetric water content. The dependency of elastic moduli ontotal stress or net normal stress is not investigated because the ap-plied total stress in both Young’s modulus and shear modulus testsare small (,100 kPa), but this dependency has been addressed in theclassical theories under saturated conditions. However, total stressmay play important, yet different, roles in controlling the elasticmodulus under unsaturated conditions than that under saturatedconditions. It is also possible that the initial state of volumetric watercontent and the drying or wetting path can affect values of elasticmoduli, that is, elastic moduli can behave hysterically because of thehysteretic behavior of SWRC. The loading on specimens witha small aspect ratio employed in this study may also cause someerrors or uncertainties in measuring the Young’s modulus. Futurework is needed in constructing simple and accurate theories incombining effects of both total stress and initial volumetric watercontent for predicting elastic moduli of foundation soils undervariably saturated conditions.

Acknowledgments

Financial support for this research from the National Science Foun-dation (NSF award CMMI 1233063) is greatly appreciated.

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