Unified Model for Small-Strain Shear Modulus of Variably Saturated Soil

Transcription

1 Unified Model for Small-Strain Shear Modulus of Variably Saturated Soil Yi Dong, A.M.ASCE 1 ; Ning Lu, F.ASCE 2 ; John S. McCartney, Ph.D., P.E., M.ASCE 3 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Abstract: The small-strain shear modulus G is a property fundamental to the deformation response of soils to dynamic loading. It depends on soil mineralogy, particle characteristics, void ratio, effective stress, and the degree of saturation. Some theories have been established to predict G of soils under variably saturated conditions. Accordingly, a new unified model was developed for G of different soils under variably saturated conditions. The model is based on two conceptual mechanisms: material hardening/softening, which is considered using the degree of saturation as a variable, and interparticle contact forces carried by the soil skeleton, which is considered using the effective stress as a variable. By utilizing suction stress theory, it is demonstrated that G can be correlated to the parameters of the soil-water retention curve. Results from the literature for 22 soils along with new results for 7 soils were used to validate the model under variably saturated conditions and a range of total stresses. DOI: 1.161/(ASCE)GT American Society of Civil Engineers. Author keywords: Small-strain shear modulus; Unsaturated soils; Bender element; Soil-water retention; Suction stress. Introduction The shear modulus of soils at strain levels less than.1% is referred to as the initial, maximum, or small-strain shear modulus and is typically denoted as G or G max. It is a fundamental parameter of soils in geotechnical problems such as earthquake ground response analysis, static and dynamic soil-structure interactions, and liquefaction potential evaluation (Clayton 211; Likitlersuang et al. 213; Yang and Gu 213). The small-strain shear modulus, G, represents the stiffness of the soil skeleton under the conditions where slippage at particle contacts is negligible and the soil fabric is fixed. In this case, the soil exhibits an elastic response where the deformation is governed by contact behavior and is reversible without hysteresis (Cho and Santamarina 21; Santamarina et al. 21). A large number of investigations have been conducted on dry or saturated soils examining the effects of void ratio, overconsolidation ratio (OCR), stress-strain history, and strain rate on G (Hardin and Richart 1963; Viggiani and Atkinson 1995; Rampello et al. 1997; Sorensen et al. 21). Several studies have also been conducted to characterize the G of unsaturated soils considering the effect of particle size, compaction energy, degree of saturation, matric suction, and hysteresis (Cho and Santamarina 21; Shibuya and Mitachi 1994; Santagata et al. 25; Sawangsuriya et al. 29; Ng et al. 29; Sorensen et al. 21; Khosravi and McCartney 212; Yang and Gu 213; Heitor et al. 213; 1 Postdoctoral Fellow, Dept. of Civil and Environmental Engineering, Colorado School of Mines, th St., Golden, CO ydong@mines.edu 2 Professor, Dept. of Civil and Environmental Engineering, Colorado School of Mines, th St., Golden, CO 841 (corresponding author). ninglu@mines.edu 3 Associate Professor, Dept. of Structural Engineering, Univ. of California, San Diego, 95 Gilman Dr., La Jolla, CA mccartney@ucsd.edu Note. This manuscript was submitted on March 6, 215; approved on January 25, 216; published online on April 26, 216. Discussion period open until September 26, 216; separate discussions must be submitted for individual papers. This paper is part of the Journal of Geotechnical and Geoenvironmental Engineering, ASCE, ISSN Oh and Vanapalli 214). However, there is not a universal model that can generalize the small-strain shear modulus for soils with changes in effective stress under both saturated and unsaturated conditions. The small-strain shear modulus can be measured by using various types of tests, both in the field and in the laboratory (Kurtulus and Stokoe 28; Rampello et al. 1997; Viggiani and Atkinson 1995). The bender element method, among all techniques, is the most versatile and simple laboratory test to determine G for all soils (Blewett et al. 2; Lee and Santamarina 25; Leong et al. 25; Clayton 211). Based on current testing procedures, however, the measurement of G for unsaturated soils is timeconsuming and cumbersome to have the soil samples reach equilibrium under the applied net normal stress and matric suction. Hence, the experimental investigations of G for unsaturated soils in the literature cannot cover the complete soil spectrum. In this case, empirical equations available for estimating G are limited in that they cannot be extended to other soil types without additional experimental testing. Lu and Kaya (213) developed a simple and rapid approach, namely the drying cake (DC) method, to dry soil specimens and characterize their mechanical and hydraulic properties over a wider range of degree of saturation than any other method. In this study, the DC test was modified to incorporate shear-wave measurements using the bender element so that the evolution of G with changes in degree of saturation following the primary drainage path can be assessed for different soil types. The experimental results from these tests along with experimental results from the literature were also used to verify a newly proposed unified model of small-strain shear modulus for all types of soils under saturated and unsaturated conditions. Suction Stress Based Unified Conceptual Model of G Micromechanical analyses suggest that the macroscale small-strain stiffness is stress dependent and can be fitted well by the general power equation of the following form (Hardin and Richart 1963): G ¼ A FðeÞðσ Þ γ 1 ð1þ ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

2 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. where A and γ 1 = fitting parameters; FðeÞ = empirical function of void ratio e; and σ = effective stress. This equation applies for dry and fully saturated soils. To extend this principle to unsaturated soils, several investigators used Bishop s (1959) effective stress concept and introduced the effect of net normal stress and matric suction (Mancuso et al. 22; Mendoza et al. 25; Ng et al. 29; Sawangsuriya et al. 29; Khosravi and McCartney 212; Heitor et al. 213; Oh and Vanapalli 214) to study the dependence of G on saturation and matric suction for limited types of soil. For instance, Sawangsuriya et al. (29) proposed a G model for unsaturated soils in the following form: G ¼ A FðeÞ½ðσ u a ÞþS κ ðu a u w ÞŠ γ 2 where S = degree of saturation; u a and u w = pore air and water pressures, respectively; and κ and γ 2 are empirical parameters. To avoid the difficulty of determination of Bishop s effective stress parameter χ, the authors adopted the experimental relationship χ ¼ S κ that was originally proposed by Vanapalli and Fredlund (2) to evaluate the effective stress in unsaturated soils (i.e., the term in brackets). However, Eq. (2) only works well for a limited range of net confining stress values. Another example is the G model for unsaturated soils proposed by Oh and Vanapalli (214), which considers the nonlinear relationship between G and matric suction ð2þ G unsat ¼ G sat ½1 þ ζðu a u w ÞS ξ Š ð3þ where G sat and G unsat represent the small-strain shear modulus under saturated and unsaturated conditions, respectively, and ζ and ξ are fitting parameters. This model applies only to nonplastic sandy soils under a certain range of net confining stress. Khosravi and McCartney (212) proposed a model for G to consider the effects of hydraulic hysteresis involving the concept of material hardening proposed by Wheeler et al. (23). However, this model requires several additional parameters beyond those in the models mentioned previously, so it was not used in this study for comparisons. Despite the fact that matric suction is intercorrelated with the net normal stress in describing the state of stress in unsaturated soils, matric suction is not a stress state variable because it does not directly generate the interparticle stress acting on particle contacts (Lu 28). External confining and matric suction are intrinsically coupled when both are applied to unsaturated soils, which means that it is necessary to upscale these two variables acting on the solid particles and air-water interfaces into stress state variables acting on the faces of the air-water-solid representative elementary volume (Lu 28). Because the models in Eqs. (2) and (3) do not consider this upscaling, they cannot uncouple the effect of matric suction (or degree of saturation) from that of the external-confining stress for unsaturated soils. Lu and Likos (26) and Lu et al. (21) proposed the concept of suction stress, which upscales and combines different types of interparticle forces dependent on matric suction and is intrinsically related to the soil-water retention curve (SWRC) of a given type of soil. Through the suction stress framework, the effective stress calculated in the following equations can truly represent the contribution of contact behavior to the small-strain shear modulus for unsaturated soils (Lu et al. 21) σ ¼ðσ u a Þ σ s ð4þ σ s ¼ S e ðu a u w Þ¼ ðs e =αþ½s n=ð1 nþ e 1Š 1=n ð5þ where σ s = suction stress; S e = effective degree of saturation; and α and n are empirical fitting parameters in the van Genuchten (198) SWRC model S e ¼ S S r ¼f1þ½αðu 1 S a u w Þ n Šg 1=n 1 ð6þ r where S r = residual degree of saturation. Parameter α is related to the inverse of the air-entry suction of a soil (kpa 1 ), and parameter n is a dimensionless pore-size distribution parameter of a soil. Although the degree of saturation plays a role in definition of effective stress (Lu et al. 21), Khalili et al. (24) found that changes in degree of saturation may have independent effects on material hardening/softening and effective stress. To investigate the role of degree of saturation on material hardening/softening, Lu and Kaya (214) performed compression tests on specimens brought to equilibrium at different initial degrees of saturation using the DC method to develop an empirical relationship for the elastic moduli of unsaturated soils at finite-strain levels ( 1%). Sandy, silty and clayey soils at different degrees of saturation were compressed under vertical stresses up to 1 kpa to determine the finite-strain Young s and shear moduli values. The empirical relationship indicates that the finite-strain shear modulus is proportional to a power of the effective degree of saturation as follows: G ðs e Þ ð1=s e Þ β where β = experimental fitting parameter, and the inverse of S e is shown in parentheses to reflect that increases in the effective degree of saturation result in a material softening mechanism. Several other studies have also observed material hardening/softening during decreases and increases in the effective degree of saturation under constant external stress (Ng et al. 29; Khosravi and McCartney 212). Although their approach may have successfully quantified the general trend between S and G, the finite-strain shear moduli values measured by Lu and Kaya (214) are much smaller in magnitude than the value of G because of significant degradation in stiffness with the greater strain level in the compression tests (e.g., <.1 1%). In this case, the contact or skeleton forces may have redistributed during the loading to high strain levels caused by the change of soil texture and fabric. This suggests that the enhancement in G stemming from transmission of contact forces through the soil skeleton without redistribution of stresses needs to be considered separately from the material hardening/ softening. The effects of material hardening/softening better represent the G for a constant soil fabric regime at small strains. Based on the above discussion of different types of existing models for G and the role of effective stress in unsaturated soils, a new unified model for G is proposed extended from the finite-strain power law in Eq. (7) and covering all soils under both saturated and unsaturated conditions. For the small-strain shear modulus, the role of effective stress acting on the soil skeleton is considered as a separate mechanism from the role of material softening/hardening because of changes in the degree of saturation. A schematic principle of the conceptual model in a representative elementary volume (REV) of the soil is shown in Fig. 1. Specifically, illustrations of the stiffness development attributable to material softening/hardening of soil matrix and the contact forces formed by hydration or capillary water at the two different soilwater regimes, hydration water and capillary water regime, are shown in Figs. 1(a and b), respectively. Suction stress incorporates the effects of soil-water interaction in both regimes and has been used to calculate the effective stress for unsaturated soils. The effective stress is assumed to primarily affect the shear modulus of unsaturated soils through the soil skeleton or network of particle contacts. A conceptual model for the proposed mechanisms contributing to the shear modulus development in a REV is shown in Fig. 1(c). In this figure, G ðs e Þ and G ðσ Þ represent the ð7þ ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

3 External loading Effective stress Soil matrix Soil water hydration around and within the particles External loading Capillary water at contacts saturated ðs e ¼ 1Þ, no external-confining stress is applied (σ ¼ ), and the suction stress is zero ðσ ¼ σ s ¼ Þ, A can be defined as the small-strain shear modulus of saturated soil without confinement G sat. The value of Gsat incorporates the effect of void ratio, particle interlocking, initial conditions, and stress history. By replacing A with G sat, Eq. (9) can be rewritten as follows: 1 β σ G ¼ G sat γ þ 1 ð1þ P atm S e When soil is saturated ðs e ¼ 1Þ, the proposed equation reduces to a form similar to Hardin s equation [Eq. (1)] Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Matrix stiffness G S) Contact stiffness G ) External loading Fig. 1. Schematic illustration of cementation attributable to soil-water at hydration regime; capillary regime; (c) the conceptual model consists of two counterparts of small-strain shear modulus attributable to soil grain skeleton and contact force components of the shear modulus arising from the role of the effective degree of saturation and from the role of the effective stress, respectively. In such a REV, the two components can be linked together in a multiplicative form. As the effective degree of saturation varies under constant external-confining stress, the soil matrix changes in stiffness. This component can be quantified using a form of Eq. (7)(Lu and Kaya 214). The externally applied net stress and the suction stress σ s control the stresses at the particle contacts. For unsaturated soils, the effective stress can be determined using Eq. (4)(Lu et al. 21). Thus, the second component can be quantified as a similar proportionality of power law of effective stress, consistent with the Hardin s equation as follows: G ðσ Þ ðσ =P atm þ 1Þ γ where effective stress σ is normalized by the value of the standard absolute atmospheric pressure P atm ð 11.3 kpaþ, and γ is an empirical fitting parameter. Although the standard absolute atmospheric pressure is used as a constant normalization pressure, laboratory tests such as those in this study commonly use the gauge pressure as the pore air pressure (e.g., for a specimen with S e ¼ 1, pore air pressure u a ¼ ). Based on the conceptualized principles illustrated in Fig. 1, the small-strain shear modulus can be determined by a soil matrix hardening/softening component related to the effects of the effective degree of saturation and a contact force component solely attributable to the effective stress. The proposed model can be written as follows: 1 β σ G ¼ G ðs e Þ G ðσ γ Þ¼A þ 1 ð9þ S e P atm where A is an experimental fitting parameter having the same units as the small-strain shear modulus (typically MPa). When soil is (c) ð8þ G ¼ G sat σ P atm þ 1 γ ð11þ where the effective stress reduces to Terzaghi s effective stress in saturated soil, σ ¼ σ u w. Therefore, the proposed equation unifies the small-strain shear modulus extended from saturated soil to unsaturated soil. Experimental Program To examine the evolution of G for different soils with respect to degree of saturation, tests on seven soils were conducted by incorporating shear-wave velocity measurement following the DC method of Lu and Kaya (213). A schematic experimental setup of the shear-wave velocity measuring system is shown in Fig. 2. First, soil cake specimens were statically compacted using a GeoTac loading frame and then were saturated by submersion in water within a desiccator and applying vacuum for sufficient time (usually 1 2 days). All soil cake specimens were prepared with a diameter of 76.2 mm and a thickness of 19 mm. Next, soil cake specimens were dried in a chamber with limited opening for lowering the evaporation rate and suction gradient between soil sample and surrounding air to ensure the soil has homogeneous distribution in degree of saturation throughout the cake during the evaporation drying process. The system was maintained at a constant room temperature and a relative humidity of 1%, which is equivalent to a boundary total suction of approximately 317 MPa applied on the DC system, according to the Kelvin s equation (e.g., Lu and Likos 24). To monitor the weight change, the soil cake with the supporting plate was placed on a digital balance for monitoring of changes in the degree of saturation. Soil cake Support Humidity controlled Bender element Digital balance Soil water content Digital filter/ Amplifier Oscilloscope/ Function generator Fig. 2. Schematic illustrations of the soil cake specimen, bender element sensors, and peripheral electronics for small-strain shear modulus measurement ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

4 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Table 1. Physical Properties and Fitting Parameters of the Studied Soils Number Soil name Unified soil classification system Porosity Void ratio van Genuchten parameters Plasticity index Testing conditions Net confining stress Fitting parameters of proposed model ϕ e θ r α (kpa 1 ) n σ u a (kpa) G sat (MPa) β γ R 2 Reference 1 Bonny silt ML Unsaturated This study 2 Hopi silt SC Unsaturated This study 3 BALT silt ML Unsaturated This study 4 Iowa silt ML Unsaturated This study 5 Denver claystone CL Unsaturated This study 6 Denver bentonite CH Unsaturated This study 7 Missouri clay Unsaturated This study 8 SC-Std-Opt SC Unsaturated Sawangsuriya et al. (29) 9 ML-Std-Opt ML Unsaturated Sawangsuriya et al. (29) 1 CL1-Std-Opt CL Unsaturated Sawangsuriya et al. (29) 11 CL2-Std-Opt CL Unsaturated Sawangsuriya et al. (29) 12 CH-Std-Opt CH Unsaturated Sawangsuriya et al. (29) 13 d-p1 ML Unsaturated Khosravi and McCartney (212) 14 d-p15 ML Unsaturated Khosravi and McCartney (212) 15 d-p2 ML Unsaturated Khosravi and McCartney (212) 16 w-p1 ML Unsaturated Khosravi and McCartney (212) 17 w-p15 ML Unsaturated Khosravi and McCartney (212) 18 w-p2 ML Unsaturated Khosravi and McCartney (212) 19 dw-p11 ML Unsaturated Ng et al. (29) 2 dw-p3 ML Unsaturated Ng et al. (29) 21 S1 MH Unsaturated Mendoza and Colmenares (26) 22 S2 MH Unsaturated Mendoza and Colmenares (26) 23 S3 MH Unsaturated Mendoza and Colmenares (26) 24 S4 MH Unsaturated Mendoza and Colmenares (26) 25 S5 MH Unsaturated Mendoza and Colmenares (26) 26 TS-d SP.41.7 Saturated 5,1,2, Yoon et al. (28) 27 TS-l SP.44.8 Saturated 5,1,2, Yoon et al. (28) 28 SS-d SP.41.7 Saturated 5,1,2, Yoon et al. (28) 29 SS-l SP Saturated 5,1,2, Yoon et al. (28) Note: CH = clay of high plasticity; CL = clay of low plasticity; MH = silt of high plasticity; ML = silt; SC = clayey sand; SP = poorly graded sand; Std = standard proctor; Opt = optimum moisture. ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

5 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. The shear-wave measuring system includes a Tektronix MDO312 oscilloscope (Tektronix) with an AFG (Arbitrary Function Generator) function generator module and a Krohn-Hite 394 digital filter/amplifier. A 5-Hz step function was sent to the transmitter, and the received signal was passed through a 1-Hz high-pass channel and a 5-kHz low-pass channel, with 2 db of gain amplification for each channel. The signals are digitized at a sampling rate of 2.5 GHz. Data points of 1k were recorded for each signal to determine the first arrival of the shear-wave. The stacking of 256 signals permits reduction of the noncoherent noise. A pair of bender elements was installed at the center of the top and bottom surfaces of the soil cake. The transmitter sent an S-wave through the soil cake, and the vibration was detected by the receiver. The shear wave velocity V s was obtained from the first arrival of the S-wave. The G values can be calculated using G ¼ ρ ðv s Þ 2,whereρ is the mass density of soil. Validation of the Proposed Model A classification of all types of soils studied in this paper according to their sources is shown in Table 1, which includes 29 soils in total: (1) 7 soils tested in this study using bender elements with suction applied using the DC method [soil number (No.) 1 7, reference a]; (2) 5 soils from Sawangsuriya et al. (29) (soil No. 8 12, reference b) with suction control using axis translation; (3) 6 soils from Khosravi and McCartney (212) (soil No , reference c) with suction control using axis translation; (4) 2 soils from Ng et al. (29) (soil No. 19 2, reference d) with suction control with axis translation; (5) 5 soils from Mendoza and Colmenares (26) (soil No , reference e) with suction control using air drying; and (6) 4 soils from Yoon et al. (28) (soil No , reference f) with suction control using axis translation. The soil classifications, index Matric suction, [kpa] Matric suction, [kpa] SC-Std-Opt data SC-Std-Opt fitted ML-Std-Opt data ML-Opt-Std fitted CL1-Std-Opt data CL1-Std-Opt fitted dw-p11 data dw-p11 fitted S4 data S4 fitted CH-Std-Opt data CH-Std-Opt fitted properties (porosity, void ratio, and plasticity index), loading condition and applied net confining stress, van Genuchten (198) SWRC model parameters, and the fitting parameters for G in the proposed unified model are all summarized in Table 1. The SWRCs of soils 1 25 were experimentally obtained or adopted from the literature. The different data sets from Khosravi and McCartney (212) and Ng et al. (29) are all for the same soil, but for wetting and drying SWRC paths under different net normal stresses. The least squares fitting technique was used for determination of the fitting parameters for Eq. (1) for each soil. The soils studied cover a wide range of soils, from sandy to silty to clayey soils, and considered different ranges of externally applied stresses. Data from references a and e are under no net normal stress and only experience changes in matric suction; data from references b, c, and d are under constant net normal stress and experience changes in matric suction; and data from reference f are under fully saturated conditions under different net normal stresses. The general behavior of proposed unified model fits well with the experimental results. The value of G sat varies from 15 to 125 MPa, depending on soil type and initial void ratio. The parameter β varies from.1 to 3, whereas the parameter γ varies from.1 to.78. All soils have the coefficient of correlation R 2 value between.88 and.99, indicating that the proposed Eq. (1) can reasonably represent the small-strain shear modulus. The SWRCs of the selected soils types (e.g., SC, ML, CL, CH, MH) examined in this study are shown in Fig. 3. All data points from the original papers shown in Table 1 are replotted in Figs. 3(a and c), along with the fitted van Genuchten (198) SWRC curves. The suction stress characteristic curves (SSCC) defined by Eq. (5) using the fitted SWRC parameters from Figs. 3(a and c) are shown in Figs. 3(b and d). As shown, the suction stress changes significantly, spanning the full saturation range Suction stress, s [-kpa] Suction stress, s [kpa] SC-Std-Opt ML-Opt-Std CL1-Std-Opt dw-p11 S4 CH-Std-Opt (c) (d) Fig. 3. Soil-water retention curves (a and c) and suction stress characteristic curves (b and d) of selected soils ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

6 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (c) Hopi silt data Hopi silt fitted Iowa silt data Iowa silt fitted Denver claystone data Denver claystone fitted S1 data S1 fitted S3 data S3 fitted S4 data S4 fitted S5 data S5 fitted with values ranging over 2 4 orders of magnitude. This implies dramatic variation in effective stress under unsaturated conditions and potentially strong influences of material hardening/softening. Comparisons between the measured small-strain shear modulus and numerically fitted values by using proposed unified model for selected soils according to different external stresses and saturation conditions are shown in Figs The relationships between G and the effective degree of saturation are shown in Figs. 4(a and c), whereas the relationships between G and the effective stress are shown in Figs. 4(b and d). All of these specimens were tested under no external-confining stress using the DC method. For soils tested under constant confining stress with varying matric suction, the relationships between G and the effective degree of saturation are shown in Figs. 5(a and c), whereas the relationships between G and the effective stress are shown in Figs. 5(b and d). For sandy soils at fully saturated conditions and various constant external confining stresses, the relationship between G and the effective stress for measured data and model fittings is shown in Fig. 6. The data and curves shown in Figs. 4 6 indicate that the proposed unified model fits well with experimental data. The G evolution with respect to the effective degree of saturation has different patterns (concave upward or downward, or S-shape), depending on the soil type and confining stresses. In general, it is clear that G increases with the increasing effective stress and with the reducing degree of saturation. Correlation with Soil-Water Retention Parameters The intrinsic correlation between soil-water retention and suction stress implies that the effective stress dependency of small-strain shear modulus is interrelated to soil-water retention. In the SWRC model of van Genuchten (198), the α parameter (the inverse of the Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (d) S1 data S1 fitted S3 data S3 fitted S4 data S4 fitted S5 data S5 fitted Hopi silt data Hopi silt fitted Iowa silt data Iowa silt fitted Denver claystone data Denver claystone fitted Fig. 4. Comparison of measured small-strain shear modulus with calculated values by proposed equation for selected soils in terms of degree of saturation and effective stress, unsaturated and no external-confining conditions air-entry suction) reflects the largest pore size and also the density of soil samples. Conversely, the effect of initial porosity or void ratio has been embedded into parameter G sat. Hence, the correlation between fitting parameters of the proposed unified model and α was not observed, and only correlations with the soil type or pore size distribution parameter n were considered in this paper. The relationship between G sat and n is shown in Fig. 7. In this case, lower values of n which correspond to fine-grained cohesive soils have higher G values under saturated conditions. The dependency of G sat on initial void ratio is shown in Fig. 7 with soils labeled by the Soil No. in Table 1. The results indicate that denser soils with lower initial e generally have higher G sat values, although the trend may not be clear because of overconsolidation effects and the fact that the initial void ratio before application of the confining stress is plotted in this figure. Specifically, for a given soil type, a constant initial e with increasing external confinement leads to increasing G sat values, whereas constant external confinement with increasing initial e results in a decrease in G sat. However, both trends are not strong enough when considering all of the soils together to develop a quantitative empirical relationship. Such correlations between G sat and n or e for an individual soil may show stronger trends, and quantitative relationships can be considered on a project-specific basis. In general, because the parameter G sat depends on initial soil relative density or void ratio and solid particle properties, it needs to be experimentally determined as well for a given set of initial conditions for a soil. The relationships between the other fitting parameters of the proposed model and the n parameter of the SWRC are shown in Fig. 8. Parameter β was observed to follow a power-law correlation with n in Fig. 8. Lower β values were observed for high n values (sandy soils), while the β value increases as the n value increases and the pore size distribution decreases (i.e., a greater clay ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

7 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (c) SC-Std-Opt data SC-Std-Opt fitted ML-Std-Opt data ML-Std-Opt fitted CL1-Std-Opt data CL1-Std-Opt fitted dw-p11 data dw-p11 fitted d-p2 data d-p2 fitted dw-p3 data dw-p3 fitted fraction). An empirical fitting equation was identified as β ¼ 9.6 n 6., which can be used for estimation of G of unsaturated soil using the van Genuchten (198) SWRCmodel for different soil types. This intercorrelation between material hardening/softening parameter β and soil pore size gradation parameter n implies that the material softening/hardening mechanism stem from the cementation effect of hydration of fine content of the soil. More clayey soil with lower n value has higher parameter β numbers, reflecting higher sensitivity of G to the effective degree of saturation for fine soils and higher material softening/hardening Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (d) SC-Std-Opt data SC-Std-Opt fitted ML-Std-Opt data ML-Std-Opt fitted CL1-Std-Opt data CL1-Std-Opt fitted dw-p11 data dw-p11 fitted d-p2 data d-p2 fitted dw-p3 data dw-p3 fitted Fig. 5. Comparison of measured small-strain shear modulus with calculated values by proposed equation for selected soils in terms of degree of saturation and effective stress, unsaturated and constant external-confining conditions Small-strain shear modulus, G [MPa] TS-d (d) data TS-d (d) fitted TS-s (d) data TS-s (d) fitted 5 TS-d (l) data TS-d (l) fitted TS-s (l) data TS-s (l) fitted Small-strain shear modulus, G [MPa] 2 15 SS-d (d) data SS-d (d) fitted SS-s (d) data 1 SS-s (d) fitted SS-d (l) data SS-d (l) fitted SS-s (l) data SS-s (l) fitted Fig. 6. Comparison of measured small-strain shear modulus with calculated values by proposed equation for fully saturated sandy soils in terms of effective stress, saturated and various external-confining stresses effect during wetting/drying. The effective stress term is not only determined by suction stress in unsaturated conditions, but also depends on the external net normal stress. For this reason, the power γ of the effective stress term does not show clear correlation with the n parameter as shown in Fig. 8 and should be experimentally determined to estimate the G of soils for saturated and unsaturated cases. The average value of γ for the soils investigated in this study is.5, which corresponds with the theoretical value expected from the Hertzian contact theory, and the range of parameters is consistent with those observed by Stokoe et al. (1999) for different soils. ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

8 G sat [MPa] G sat [MPa] Soil No.1 Soil No.3 Soil No.5 Soil No.7 Soil No.9 Soil No.12 Soil No.19-2 Soil No Soil No.2 Soil No.4 Soil No.6 Soil No.8 Soil No.1-11 Soil No Soil No Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. [-] n -6. R = n [-] n Fig. 8. Correlation between proposed model and soil-water retention curve of unsaturated soils: β versus n; γ versus n Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (c) n Hopi silt R 2 =.99 Sawangsuriya et al. model R 2 =.98 R 2 =.98 S1 Sawangsuriya et al. model R 2 =.97 R 2 =.95 R 2 = Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (d) Initial void ratio, e Fig. 7. Dependency of G sat on soil pore size structure and initial void ratio: G sat versus n; G sat versus e Denver claystone R 2 =.99 Sawangsuriya et al. model R 2 =.94 R 2 =.94 S2 R 2 =.89 Sawangsuriya et al. model R 2 =.86 R 2 = Fig. 9. Comparison of proposed model prediction using empirical correlation between β and n value, with prediction behavior of Eqs. (2) and (3) for selected soils, unsaturated and no external-confining conditions ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

9 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. Comparison with Other G Models Based on the correlation between proposed unified model and soilwater retention, given the van Genuchten (198) SWRC model parameters, the unified model can be written into the following equation form for prediction of G of saturated or unsaturated soils: G ¼ G sat Small-strain shear modulus, G [MPa] 1 Small-strain shear modulus, G [MPa] (c) S e n 6. ML-Std-Opt Sawangsuriya et al. model dw-p11 Sawangsuriya et al. model σ þ 1 P atm γ R 2 =.97 R 2 =.97 R 2 = R 2 =.99 R 2 =.97 R 2 =.98 ð12þ The G of all 29 soils were predicted using Eq. (12) and were compared with fitted curves for the models in Eqs. (2) and (3). Comparisons of the predicted values of G from the proposed unified model prediction with those from the other two fitted models are shown in Figs. 9 and 1. Generally, the newly proposed model has similar performance with the models described by Eqs. (2) and (3). For most of the soils, the proposed model has a R 2 value that is slightly higher than that for the models in Eqs. (2) and (3). However, the proposed unified model correlates with the fitting parameters to the SWRC of the soils, whereas the fitting parameters of the models in Eqs. (2) and (3) have no solid physical meaning. A statistical comparison shows that for the total of 25 unsaturated soils under investigation, the proposed unified model has better or equivalent prediction for 15 soils, has middle predictability for 3 soils, and has relatively worse prediction behavior in the other 7 soils. Most of the R 2 values for the unified model prediction are in the range of.9 to 1., with only few values of approximately.88. For Eq. (1), the model is established based on Terzaghi s effective stress theory for saturated soils. The effect of matric suction or suction stress when soil is desaturated is not taken in account. Hence, this model cannot predict well the material softening/ hardening effect for unsaturated soils. For Eqs. (2) and (3), the Small-strain shear modulus, G [MPa] Small-strain shear modulus, G [MPa] (d) matric suction is multiplied by a power function of the effective degree of saturation as an independent variable. In this case, matric suction is intrinsically interrelated to the effective degree of saturation. However, the changes of matric suction and degree of saturation have converse effects: matric suction increases as the effective degree of saturation decreasing and vice versa. This nature makes the fitting parameters [i.e., the exponent κ of S of Eq. (2)] compete with the exponent γ 2 of the effective stress term. Thus, the fitting results for the parameters κ and γ 2 for the model in Eq. (2) do not have a solid physical meaning or correlate with constitutive soil properties (e.g., the SWRC). The same case is also observed for the fitting parameters ζ and ξ for the model in Eq. (3). Note that while Eq. (12) can be used to predict G for all types of soil under both mechanical (external stress) and environmental (degree of saturation) loading conditions, it cannot predict G under a completely dry condition, (i.e., S or S e ¼ ), because of the appearance of S e in the denominator. However, should such situation arise as in the commonly encountered case of dry sand, the denominator S e in the second term in Eq. (12) can be substituted by S e þ S with S set to be a dimensionless nonzero constant. Conclusions w-p1 Sawangsuriya et al. model dw-p3 Sawangsuriya et al. model R 2 =.91 R 2 =.87 R 2 =.82 R 2 =.98 R 2 =.98 R 2 =.98 Fig. 1. Comparison of proposed model prediction using empirical correlation between β and n value, with prediction behavior of Eqs. (2) and (3) for selected soils, unsaturated and constant external-confining conditions This study addresses the roles of material hardening/softening and effective stress using suction stress theory on the small-strain shear modulus across a wide range of soil types, degrees of saturation, and externally applied stresses. A conceptual model was established that consists of two individual mechanisms related to the impacts of effective stress and material hardening/softening on the soil matrix and contact behavior. Following the conceptual model, a quantitative model was proposed by extending the ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639

10 Downloaded from ascelibrary.org by Colorado School of Mines on 7/25/16. Copyright ASCE. For personal use only; all rights reserved. power-law approach for estimating small-strain shear modulus under different effective stress levels to both fully saturated and unsaturated soils. An advantage of the proposed unified model is that its parameters have physical meaning and can be correlated with the soil-water retention curve to predict the small-strain shear modulus for unsaturated soils. Specifically, the material softening/ hardening effect quantified using the effective degree of saturation was found to correlate well with the pore-size distribution parameter of the van Genuchten SWRC model. The evolution of smallstrain shear modulus with the effective degree of saturation was observed to be more sensitive for silty and clayey soils than sandy soils. Further, the model was found to apply for cases of different loading conditions, which include both no external stress and various external net confining stress conditions. Acknowledgments This research is supported by grant from the National Science Foundation (NSF CMMI ) to Ning Lu and John S. McCartney. References Bishop, A. W. (1959). The principle of effective stress. Teknisk Ukeblad I Samarbeide Med Teknikk, 16(39), Blewett, J., Blewett, I. J., and Woodward, P. K. (2). Phase and amplitude responses associated with the measurement of shear-wave velocity in sand by bender elements. Can. Geotech. J., 37(6), Cho, G. C., and Santamarina, J. C. (21). Unsaturated particulate materials: Particle-level studies. J. Geotech. Geoenviron. Eng., 1.161/ (ASCE)19-241(21)127:1(84), Clayton, C. R. I. (211). Stiffness at small strain: Research and practice. Géotechnique, 61(1), Hardin, B. O., and Richart, F. E. (1963). Elastic wave velocities in granular soils. J. Soil Mech. and Found. 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Measurement of smallstrain shear modulus G max of dry and saturated sands by bender element, resonant column, and torsional shear tests. Can. Geotech. J., 45(1), ASCE J. Geotech. Geoenviron. Eng. J. Geotech. Geoenviron. Eng., 41639