Recommendations for extension and re-calibration of an existing sand constitutive model taking into...

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Recommendations for extension and re-calibration of an existingsand constitutive model taking into account varying non-plasticfines content

Ali Lashkari n

Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran

a r t i c l e i n f o

Article history:Received 23 August 2013Received in revised form10 January 2014Accepted 19 February 2014Available online 27 March 2014

Keywords:Constitutive modelSilty sandCritical stateAnisotropyEquivalent intergranular void ratio

a b s t r a c t

Experimental findings have revealed that up to a certain transitional threshold, adding non-plastic silt tocoarse soils like sands leads to the increase in susceptibility of liquefaction. In silty sands, silt grains fillvoids between the coarse constituent, but do not actively participate in stress transmitting micro-structure. To consider the partial participation of fines in load bearing structure, an equivalentintergranular void ratio is suggested. The proposed empirical relationship, takes into account thecombined influence of fines content, soil gradation, and the average shape of coarse and fineconstituents. An equivalent intergranular state parameter is employed in the model formulation todefine soil state uniquely. Moreover, recent experimental studies indicate that the consequences of initialanisotropy on the mechanical behavior of silty sands are mitigated with the increase in fines content. Toconsider this phenomenon, vector magnitude, a measurable index of anisotropy, is related to finescontent. Then, proper constitutive equations are introduced to modify plastic hardening modulus andcritical state line in loading paths involving rotation of principal stress axes. The simulative capability ofthe model is evaluated by direct comparison of its predictions with experimental data of triaxial andhollow cylindrical cells reported by four research teams.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Uniformly graded loose saturated sands exhibit pore waterpressure build up, sizable loss of shear strength, and large sheardeformations when subjected to undrained shear. Liquefaction is atechnical term usually attributed to the above-mentioned phe-nomena. For many years, the liquefaction induced events werethought to be only related to clean sands. As a result, the majorityof past experimental studies were dedicated to clean sands[3,17,27,50,63,66,75]. However, according to a comprehensivereview by Yamamuro and Lade [71], the majority of the docu-mented historic cases of liquefaction induced catastrophic inci-dents have occurred in natural and man made deposits of siltysands. Unexpectedly, subsequent inspections revealed that densesands containing large amounts of non-plastic fines (e.g., non-plastic silts) are very prone to flow liquefaction because of theincomplete participation of fines in load transmitting microstruc-ture (e.g., [1,15,31,48,59,70]). For these reasons, the question ofnon-plastic fines influence on the liquefaction susceptibility of

granular soils has received increasing attention in the last twodecades (e.g., [1,15,21,31–33,35,36,39,46,47,52,56,59,69,70,73,77]among others). Recent studies have indicated that adding non-plastic fines to coarse granular structures leads to a rise inliquefaction susceptibility up to a transitional threshold finescontent generally in vicinity of 25–40% by weight of solid phase.Further increase in fines yields the rise in resistance againstliquefaction (e.g., [37,46,59,77]).

Initial fabric also has a strong weight on the mechanicalbehavior of clean sands (e.g., [62,63,74,75]). On the subject ofsand–silt mixtures, recent studies have reported that initial fabriceffects decrease with the rise in fines content (e.g., [1,7,44,45,55]).

Muir Wood [28–30] was among the first who addressed theprofound engineering influence of erosion-induced movement/relocation of fines within the structure of coarse granular soils.WAC Bennett Dam in British Columbia is an earthfill dam with acore formed of broadly graded non-plastic silty sand that wascompleted in 1968. In 1996, two sinkholes were found in thedam. Analysis of data provides evidence to support the hypothesisof the slow movement of fines from the core into the downstream.The migration of fines leads to time-dependent changes in the localpermeability and shear strength of different dam zones. As a result,taking into account the influence of the movement of fines on the

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Soil Dynamics and Earthquake Engineering

http://dx.doi.org/10.1016/j.soildyn.2014.02.0120267-7261 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ98 9173153205.E-mail addresses: [email protected], [email protected]

Soil Dynamics and Earthquake Engineering 61-62 (2014) 212–238


mechanical response of dam requires an elaborated coupled analy-sis for seepage and fines transport combined with a constitutivemodel for simulation of the mechanical behavior of soil taking intoaccount varying fines content (Muir Wood and Maeda [30]). Theexisting state-of-the-art sand constitutive models are utterly for-mulated for modeling of clean sands. In such models, soil behavioris greatly dependent on sand state measured with respect to aunique critical/steady state line [11,12,23-25,35]. This hypothesisworks reasonably well for clean sands; however, its application forsands containing considerable fractions of fines is troublesome (e.g.,[31,59,70]). Yamamuro and Lade [71] was the first who suggested aconstitutive model for silty sands. However, the need for re-calibration of yield function and model parameters for differentfines contents, densities, and stress levels limit the model applic-ability in practice. Recently, Chang and Yin [6] introduced micro-mechanical constitutive models for simulation of the mechanicalbehavior of sand–silt mixtures.

This study focuses on establishing a unified critical statecompatible framework for constitutive modeling of both cleanand silty sands. The proposed method is generic and can beapplied to other existing frameworks. The conventional elasto-plastic models of granular soils consist of the following ingredi-ents: constitutive equations describing the elastic behavior, plastic

flow rule and stress–dilation constitutive equations, and constitu-tive equations for the plastic hardening response. Furthermore, therecent constitutive models are usually state-dependent and cantake into account the influence of soil anisotropy on stiffness,dilation, and strength [11,12,23-25,35]. As a result, critical stateline, state parameter, and fabric tensor are usually introduced intheir formulations. Throughout the paper, a comprehensive litera-ture review regarding the influence of adding non-plastic fines oneach of the above ingredients is presented first. For each element,relevant new/modified constitutive equations with the possibilityof smooth transition from the clean state (zero fines content) tothe threshold fines content are presented. Then, the new/modifiedelements are implemented within the general formulation of thebounding surface plasticity platform of Dafalias et al. [12] that is apure phenomenological constitutive model. According to Hegel(1724–1804), phenomenology is an approach to philosophy thatbegins with an exploration of phenomena (i.e., what presents itselfto us in conscious experience) as a means to finally grasp theabsolute, logical, ontological and metaphysical spirit that is behindthe phenomena. Similar to the approach of Dafalias et al. [12] inestablishment of the original platform, a phenomenological viewis considered here in modification/suggestion of constitutiveequations taking into account the influence of non-plastic fines

Nomenclature

1(Bold faced symbols are used for tensor quantities.)

second order identity tensora fitting parameter for Eq. (4)A anisotropic state parameter (Eqs. (13) and (15))Ac, Ae anisotropic state parameter values under the com-

pression and extension modes of triaxialb intermediate principal stress ratio

{¼ðs2�s3Þ=ðs1�s3Þ}c ¼Me/Mc {see g(θ,c)}Cu uniformity coefficientD Dilatancy (Eqs. (21) and (33))D10 effective size of host sandd50 mean size of finese deviator part of strain tensore global void ratioen intergranular void ratio (Eq. (3))ecs critical state void ratioemax, emin maximum and minimum void ratiosesk skeleton void ratio (Eq. (2))f1, f2 fitting constants for Eqs. (12) and (20)f(r, α) yield surface (Eq. (25))F second order fabric tensor (Eq. (14))FC fines contentFCth threshold fines content (Eq. (1))F(e) void ratio function (Eq. (17))G elastic shear modulus (see Section 4.4)G0 fitting constant for Eq. (17)g(θ,c) interpolation function (¼2c/[(1þc)�(1�c)cos3θ]H0, HA see Eqs. (20) and (34)H0c, H0e values of H0 under the compression and extension

modes of triaxialH(e) void ratio function (Eq. (34))K elastic bulk modulus (Eq. (27))Kp plastic hardening modulus (Eq. (34))L normal to yield surface (Eq. (29))m a model parameter (Eq. (25))

Mc, Me slopes of critical state lines in q–p plane under thecompression and extension modes of triaxial

n fitting constant for Eq. (17)n deviator part of L (Eqs. (29) and (30))nb, nd fitting constants for Eq. (32)N ¼n:r (Eq. (30))p mean principal effective stresspin initial values of mean principal effective stresspref reference pressure (¼101 kPa)q deviator stressr roundness ratio (¼Rc/Rf)r shear stress ratio tensor (¼s/p)Rc, Rf average roundness values of coarse and fines

constituentsR average roundness of soils deviator part of effective stress tensorα back stress ratio tensorαb, αc, αd back stress ratios corresponding to bounding sur-

face, critical state surface, and dilatancy surface (Eq.(32))

β fines influence factor (Eqs. (3) and (4))β0 fitting parameter for Eq. (4)Δ vector magnitude (Eq. (14))Δc, Δf vector magnitudes of clean coarse and fines

constituentsε, εe, εp total, elastic, and plastic strain tensorsε1, ε3 major and minor strainsθ Lode angleλ fitting constants for Eq. (11)_Λ loading indexν Poisson's ratio (Eq. (27))ξ fitting constants for Eq. (11)r effective stress tensor (¼sþp1)s1, s2, s3 major, intermediate, and minor principal effective

stress componentsω1, ω2 fitting constants for Eq. (19)ϕcs critical state friction angleχ particle size ratio (¼D10/d50)ψ state parameter (Eq. (9))ψn intergranular state parameter (Eq. (10))

A. Lashkari / Soil Dynamics and Earthquake Engineering 61-62 (2014) 212–238 213


on soil behavior. Eventually, predictive capacity of the modifiedmodel is evaluated by direct comparison of its simulations withexperimental data of four independent research teams on differentsilty sands. It is shown that the model is capable of predictingvarious aspects of the mechanical behavior of silty sands using aunique set of model parameters.

2. Threshold fines content

The mechanical behavior of granular soils is highly influencedby grains contact characteristics. According to Thevanayagam andMartin [58] and Thevanayagam et al. [59], three major limitingcategories may be applied for binary mixture of granular soilsbased on their contact configuration: (a) coarse grains are primar-ily in contact [see cases i to iii in Fig. 1a], (b) fine grains areprimarily in contact with each other [see cases iv-1 and iv-2 inFig. 1b], and (c) layered configurations [see Fig. 1c]. In the firstcategory, coarse grains participate actively in load bearing struc-ture and the overall mechanical response is highly influenced bythe coarse constituent. Coarse grains are isolated in the secondcategory and as a result, soils belonging to this category behavelike the fine phase. Category (b) is out of the interest of this study,because from a behavioral viewpoint, category (a) representscoarse soils containing fines (e.g., sandy silts), and fine soilscontaining a coarse fraction (e.g., sandy silts) are indicated bycategory (b). Furthermore, due to the lack of homogeneity,category (c) is beyond the current discourse of constitutivemodeling.

In Fig. 1, the concept of threshold fines content, FCth, has beenintroduced to distinct the regime of “fines in coarse” from “coarsein fines” soil mixtures (e.g., [46,58,59,73]). Recently, Rahman andLo [46] proposed an empirical relationship for threshold finescontent

FCth � 0:401

1þexpð0:50�0:13χÞþ1χ

� �ð1Þ

where, χ is particle size ratio defined by χ¼D10/d50, in which D10 isthe effective size of the host sand and d50 is the mean size of fines.Using Eq. (1), the predicted trend for FCth is compared withmeasured data in Fig. 2.

3. Participation of fines in load bearing microstructure

By ignoring the involvement of fines in load bearing micro-structure, the first order intergranular void ratio, widely known asthe skeleton void ratio, is obtained (e.g., [19,26,32,56,64])

esk ¼eþFC1�FC

ð2Þ

where, e is global void ratio defined in terms of voids betweensolid phase and FC is fines content. In this definition, fines act asfiller and hence, esk indicates the void ratio of the coarse matrix.For the same host sands containing different values of finescontent, Thevanayagam [56] was the first who noted that thecritical state lines drawn in terms of the skeleton void ratio versusmean principal effective stress are located closer when comparedto those drawn in terms of global void ratio.

Further studies revealed that the assumption of the unemploy-ment of fines is unrealistic and fine grains usually settle betweencoarse grains [see cases ii and iii of Fig. 1a] and may play an active

Fig. 1. Intergranular soil mix classification [58,59].

0.2

0.3

0.4

0.5

0 10 20 30 40 50

Particle Size Ratio, χ = D10 / d50

Thre

shol

d Fi

nes

Con

tent

, FC

thHuang et al. (2004) Polito (1999)

Yang et al. (2006) Polito & Martin (2001)

Thevanayagam & Martin (2002) Thevanayagam et al. (2002)

Thevanayagam (2003) Rahman et al. (2011)

Prakash & Chandrasekaran (2005) Prediction

Fig. 2. Evaluation of Eq. (2) for a number of sand–silt mixtures (experimental datafrom [15,38,39,41,48,58-60,73]).

A. Lashkari / Soil Dynamics and Earthquake Engineering 61-62 (2014) 212–238214


role in load carrying microstructure (e.g., [33]). To take intoaccount the participation of fines, Thevanayagam and Martin[58] suggested the concept of equivalent intergranular void ratiodefined by

en ¼ eþð1�βÞFC1�ð1�βÞFC ð3Þ

where β, the fines influence factor, is the fraction of fines thattakes part actively in load bearing microstructure. In the absenceof electro-chemical interaction between particles in granular soils,it is reasonable to take 0rβr1 (e.g., [33,59]). Eq. (3) reduces toesk (Eq. (2)) when β¼0. On the other hand, en¼ewhen β¼1. Whilein the first limit fines do not strictly participate in load bearingskeleton, their active role in soil force carrying microstructure isindistinguishable from the coarse particles in the second limit. Forpartial involvement of fines in load bearing microstructure (i.e.,0oβo1), one has eoenoesk.

In the literature, a number of researchers suggested to use afixed value for β (e.g., E0.25) in the whole domain of finecontents (e.g., [33,58,59]). However, Yang et al. [73] used β¼0.25for samples with FCr0.2, and β¼0.40 for samples with FCE0.4.On the other hand, Rahman and Lo [46] reported the increase in βwith FC. A review of the existing data indicates that higher valuesof β irrespective of fines content are usually reported in mixtureswith well-rounded and rounded host sands together with angularor sub-angular fines. For each grain, roundness can be defined asthe average radius of curvature of the surface features relative tothe radius of the maximum sphere encircling the particle [8].

Detailed measurement of roundness was not mostly reported inthe literature for sand–silt mixtures; however, qualitative char-acters of both host and fines constituents have been described inalmost all of the available experimental studies. Powers [40]introduced a qualitative grading classification to describe particlesby means of roundness. The Powers' [40] grading procedure ispresented in Fig. 3 in conjunction with Table 1. Furthermore, Niet al. [33] reported experimental evidence on the dependence of βon the particle size ratio (χ¼D10/d50). Considering the above-mentioned phenomena, the following empirical relationship for βis suggested here

β¼ β0ðr; FCÞ FCχa ð4Þwhere, the term β0(r, FC) considers the combined influence of thecoarse and fine constituents roundness. Using the available data inthe literature, β0(r, FC) is calculated here by

β0ðr; FCÞ ¼ ð1:93þ0:04⟨r�1⟩2Þ � ð1þ3:2⟨r�1⟩2expð�22 FCÞÞ ð5Þwhere, r¼Rc/Rf is roundness ratio in which Rc and Rf are, respec-tively, average roundness of the coarse and fine fractions. o4 areMacualey brackets. For a typical scalar X, oX4¼X, when X40,and zero otherwise. Finally, “a” is a material parameter. aE�0.2 isa reasonable estimation for various silty sands. The calibra-tion procedure and experimental data used for calibration ofEqs. (4) and (5) are presented in Appendix A and Table 2.

In a majority of the existing experimental studies, fines wereproduced by crushing/grinding of coarse soils (e.g., [7,31,33,53,59]). In practice, most crushed granular soils have similar

Fig. 3. Visual shape classification [40].

Table 1Description of grains shape and the associated roundness intervals [40].

Description Very angular Angular Sub-angular Sub-rounded Rounded Well rounded

Roundness, R 0.12rRo0.17 0.17rRo0.25 0.25rRo0.35 0.35rRo0.49 0.49rRo0.70 0.70rRo1.0

Table 2Data used for calibration of Eqs. (4) and (5).

Mixture/Reference r (¼Rc/Rf) χ FC β {Back-calculated} β {Predicted by using Eqs. (4) and (5)}

Sydney sand-Majura silt (Rahman et al. [48]) 1 37 0.0 0 01 37 0.10 0.097 0.0941 37 0.15 0.15 0.141 37 0.20 0.195 0.191 37 0.30 0.29 0.28

Mai Liao silty sand (Huang et al. [15]) 1 2 0.0 0 01 2 0.15 0.30 0.261 2 0.30 0.50 0.515



Table 2 (continued )

Mixture/Reference r (¼Rc/Rf) χ FC β {Back-calculated} β {Predicted by using Eqs. (4) and (5)}

Toyoura sand-Toyoura silt (Zlatović and Ishihara [77]) 1 11.8 0 0 01 11.8 0.05 0.065 0.061 11.8 0.10 0.13 0.121 11.8 0.15 0.17 0.181 11.8 0.25 0.15 0.291 11.8 0.30 0.20 0.35

F55 Foundry sand-Sil-Co-Sil 40 (silt) (Thevanayagam et al. [59]) 2.8 16.9 0 0 02.8 16.9 0.07 0.30 0.262.8 16.9 0.15 0.25 0.242.8 16.9 0.25 0.25 0.30

M31 sand and Assyros silt (Papadopoulou and Tika [36]) 3.4 11.4 0 0 03.4 11.4 0.05 0.45 0.473.4 11.4 0.10 0.44 0.403.4 11.4 0.15 0.40 0.333.4 11.4 0.25 0.38 0.363.4 11.4 0.35 0.46 0.47

Ottawa sand- Sil-Co-Sil 106 (silt) (Murthy et al. [31]) 3.6 9.7 0 0 03.6 9.7 0.05 0.60 0.573.6 9.7 0.10 0.40 0.473.6 9.7 0.15 0.375 0.38

Hokksund sand-Chengbei silt (Yang et al. [73]) 2.6 6.9 0 0 02.6 6.9 0.05 0.25 0.262.6 6.9 0.10 0.25 0.262.6 6.9 0.15 0.27 0.272.6 6.9 0.20 0.33 0.30

Old alluvium sand (Ni et al. [33]) 2.4 4.4 0 0 02.4 4.4 0.09 0.26 0.25

Egyptian desert sand-Assirou silt (Stamatopoulos [53]) 3.6 8.8 0 0 03.6 8.8 0.15 0.46 0.383.6 8.8 0.25 0.50 0.36

0

0.2

0.4

0.6

0.8

Fines Content, FC [-]

Fine

s Pa

rtic

ipat

ion

Rat

io, β

[-]

Fine

s Pa

rtic

ipat

ion

Rat

io,β

[-]

Fine

s Pa

rtic

ipat

ion

Rat

io,β

[-]

Fine

s Pa

rtic

ipat

ion

Rat

io,β

[-]

Fine

s Pa

rtic

ipat

ion

Rat

io,β

[-]

Fine

s Pa

rtic

ipat

ion

Rat

io,β

[-]

This Study (Rc=0.9; Rf=0.25)Rahman et al. (2011)Murthy et al. (2007)

0

0.2

0.4

0.6

0.8


This Study (Rc=0.85; Rf=0.25)Rahman et al. (2011)Papadopoulou & Tika (2008)

0

0.2

0.4

0.6


This Study (Rc=0.7; Rf=0.25)Rahman et al. (2011)Thevanayagam et al. (2002)

0

0.2

0.4

0.6


This Study (Rc=0.65; Rf=0.25)Rahman et al. (2011)Yang et al. (2006)

0

0.2

0.4


This Study (Rc=Rf)Rahman et al. (2011)Rahman et al. (2008)

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4

0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4Fines Content, FC [-]

This Study (Rc=Rf)Rahman et al. (2011)Huang et al. (2004)

Fig. 4. Comparison of the predicted β with the corresponding back calculated values: (a) Ottawa sand-Sil-Co-Sil 106 mixture; (b) M31 quartz sand-Assyros silt; (c) F55,Foundry sand-Sil-Co-Sil 40; (d) Hokksund sand-Chengbei silt; (e) Sydney sand-Majura river silt; and (f) Mai Liao sand-Mai Liao silt.


0.4

0.5

0.6

0.7

0.8

10

Mean Prinipal Effective Stress, p [kPa]

Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0Fc = 0.05FC = 0.10FC= 0.15

0.4

0.5

0.6

0.7

0.8


Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-] FC = 0Fc = 0.05FC = 0.10FC= 0.15

0.4

0.5

0.6

0.7

0.8

0.9

1


Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0Fc = 0.05FC = 0.10FC= 0.15FC= 0.20

0.4

0.5

0.6

0.7

0.8

0.9

1


Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-] FC = 0Fc = 0.05FC = 0.10FC= 0.15FC= 0.20

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10


Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0Fc = 0.05FC = 0.10FC= 0.15FC= 0.25FC= 0.35

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10


Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-]

FC = 0Fc = 0.05FC = 0.10FC= 0.15FC= 0.25FC= 0.35

0.4

0.5

0.6

0.7

0.8

0.9

1

10


Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0Fc = 0.10FC = 0.15FC= 0.20FC= 0.30

0.4

0.5

0.6

0.7

0.8

0.9

1

10


Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-]

FC = 0Fc = 0.10FC = 0.15FC= 0.20FC= 0.30

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0

Fc = 0.07

FC = 0.10

FC= 0.15

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Mean Prinipal Effective Stress, p [kPa]

Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-] FC = 0

Fc = 0.07

FC = 0.10

FC= 0.15

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Crit

ical

Glo

bal V

oid

Rat

io, e

[-]

FC = 0

Fc = 0.15

FC = 0.30

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1


Crit

ical

Equ

ival

ent I

nter

gran

ular

Voi

d R

atio

, e* [

-]

FC = 0

Fc = 0.15

FC = 0.30

100 1000 10000 10 100 1000 10000 1 10 100 1000 10000 1 10 100 1000 10000

100 1000 10000 100 1000 10000 100 1000 10000 100 1000 10000

10 100 1000 10000 10 100 1000 10000 100 1000 10000 100 1000 10000

Fig. 5. Critical state lines in terms of global and equivalent intergranular void ratios: (a) and (b) Ottawa sand-Sil-Co-Sil 106 mixture [31]; (c) and (d) M31 quartz sand-Assyros silt [36]; (e) and (f) F55, Foundry sand-Sil-Co-Sil 40 [59];(g) and (h) Hokksund sand-Chengbei silt [73]; (i) and (j) Sydney sand-Majura river silt [46]; and (k) and (l) Mai Liao silty sand [15].

A.Lashkari

/Soil

Dynam

icsand

EarthquakeEngineering

61-62(2014)

212–238

217


shapes with roundness in the range 0.2–0.3 [8]. Hence, an averagevalue 0.25 is assigned for such fines. In addition, the existence ofmicroscopic photos obtained from Scanning Electron Microscope(e.g., [1,7,15]) in conjunction with the Powers' [40] approach is aninfluential tool to estimate grains roundness. In the absence ofsuch data, one can invoke to description of grains (e.g., angular,sub-angular, sub-round, or round) and estimate roundness usingTable 1.

In Fig. 4, predictions obtained from Eqs. (4) and (5) arecompared with the back-calculated values of β from six indepen-dent studies in the literature. For the aim of comparison, predic-tions by the recent empirical relationship of Rahman and Lo [46][see Eq. (6)] are also shown in Fig. 4

β¼ 1�exp �0:30ðFC=FCthÞ1�χ �0:25

� �� 1χ� FC

FCth

� �1=χ

ð6Þ

In general, roundness of grains in the coarse constituentdecreases steadily from part (a) to (f) of Fig. 4. However, finesare angular in all cases. A pronounced peak is observed in back-calculated β of those soils with well-rounded to sub-roundedcoarse constituents. Moreover, one can find that the mentionedpeak decreases and eventually disappears with the increase ofangularity. To explain the latter point, it must be noted that forhost sands with very angular to sub-angular grains [parts e and f ofFig. 4], rE1 or less. As a result, Eq. (4) is reduced to β¼1.93 FCχ�0.20. In this case, β increases linearly with FC which is consistentwith the findings of Rahman and Lo [46]. For well-rounded to sub-rounded coarse phase [parts a–d of Fig. 4], the term (1þ3.2⟨r�1⟩2

exp(�22 FC)) in β0 leads to the generation of a transient peak inthe calculated β. In all six cases, a reasonable agreement betweenthe predicted [using Eqs. (4) and (5)] and data is observed.Surprisingly, the method of this study and that of Rahman andLo [46] tend to the same direction for larger fines contents orangular constituents.

4. Effect of fines on essential elements of silty sands behavior

4.1. Effect of fines on critical state in the compressionmode of triaxial

The critical state soil mechanics has been an influential frame-work for understanding soils behavior and development of state-of-the-art soil constitutive models. Critical state is defined as thestate at which soil shears continuously at constant effective stressand density. From a historical view, a similar situation underundrained condition is often called steady state in the literature.Been et al. [3] and Verdugo and Ishihara [66] demonstrated thatcritical state and steady state are identical in clean sands. Regard-ing sands with non-plastic fines, a similar evidence has beenreported by Murthy et al. [31], Papadopoulos and Tika [36], andBobei et al. [4] in the recent years.

A theoretical study using Discrete Element Method has con-firmed the downward movement of critical state line in e–p planewith the increase of the coefficient of uniformity [72]. It is worthyto note that adding non-plastic fines to clean uniform sands havethe same influence on the increase of the coefficient of uniformity.In accordance with experimental findings, it has been widelyaccepted that adding non-plastic fines to clean sands results in aclear gradual downward transfer of critical state line in e–p plane(e.g., [9,15,31,32,36,47,48,53,58,59,65,77]). However, when drawnin terms of equivalent intergranular void ratio, en, in lieu of globalvoid ratio, e, similar sands having different non-plastic fines define anapproximately unique critical state line (e.g., [33,47,48,58,59,73]).Using the proposed en (Eqs. (3)–(5)), critical state lines of various

sands containing non-plastic silts are depicted in e–p and en–p planesin Fig. 5. It is observed that critical states illustrated in terms ofintergranular equivalent void ratio, en, specify a unique narrow regionin e–p plane.

Debates exist in the literature regarding the influence of fineson the slope of critical state line in q–p plane (or equally, constantvolume friction angle). A number of researchers reported that theslope of critical state line measured in the compression mode oftriaxial, Mc, increases with fines content (e.g., [7,31,33,36]). On theother hand, other researchers published data supporting that Mc isindependent of fines content (e.g., [5,15,48,77]). Once again,considering the particles shape can resolve the controversy. Usingthe compiled data of 85 different sands, Cho et al. [8] reported thefollowing empirical relationship between the critical state frictionangle and soil average roundness (R)

ϕcs � 42�17 R ð7Þ

Eq. (7) implies that the decrease in soil average roundness leadsto the rise in the critical state friction angle and vice versa. Thus,adding angular fines to sub-rounded or rounded host sands results

1

1.25

1.5

1.75

2


Crit

ical

Sta

te S

tres

s R

atio

, Mc

[-] Silica sand (coarse)- Glass beads (fines)

Prediction

Mc(FC=0)=1.9, ΔM= -0.75, FCth=0.45

1

1.2

1.4

1.6


Crit

ical

Sta

te S

tres

s R

atio

, Mc

[-] Murthy et al. (2007)Carraro et al. (2009)Prediction

Mc(FC=0)=1.21, ΔM=+0.95, FCth=0.31

1.2

1.3

1.4

1.5

0 0.2 0.4 0.6 0.8 1

0 0.05 0.1 0.15 0.2 0.25

0 0.1 0.2 0.3 0.4Fines Content, FC [-]

Crit

ical

Sta

te S

tres

s R

atio

, Mc

[-] Stamatopoulos (2010)

Prediction

Mc(FC=0)=1.30, ΔM=+0.115, FCth=0.315

Fig. 6. Comparison of the predicted triaxial compression critical state stress ratio withexperiments: (a) angular Silica sand-Glass bead mixture [61]; (b) Ottawa sand-Sil-Co-Sil 106 mixture [7,31]; (c) Egyptian quartz sand-Assirou silt mixture [53].



in a gradual increase in ϕcs and Mc as reported in the literature (e.g., [31,33,36,53]). On the other hand, adding crushed fines toangular or sub-angular host sands does not lead to a concretechange in soil overall angularity and hence, ϕcs and Mc remainunchanged with adding such fines. The latter case is consistentwith finings of Zlatović and Ishihara [77], Bouckovalas et al. [5],Huang et al. [15], and Rahman et al. [48]. According to Eq. (7), onemust expect a decrease in ϕcs by adding rounded fines to angularcoarse host granular media. Recently, Ueda et al. [61] studied theinfluence of adding well-rounded fine Glass beads to Silica sand.Observing sizable decrease in ϕcs with the increase in Glass beadsfraction confirms our conclusion.

To model the possible evolution of Mc with the change in finescontent, the following relationship is suggested here

McðFCÞ ¼McðFC ¼ 0Þþ34ΔM

FCFCth

� �2

�14ΔM

FCFCth

� �3

ð8Þ

In Eqs. (8), ΔM¼2[Mc(FC¼FCth)�Mc(FC¼0)], where Mc(FC¼0)and Mc(FC¼FCth) are the values of Mc at zero and threshold finescontents, respectively.

In granular soils containing a small fraction of fines, smallparticles only fill voids between grains of the coarse phase withoutactive participation in load bearing structure. On the other hand,fine grains become dominant at large fines contents for whichcoarse grains float in fines. Based on the experimental evidence(e.g., [61]), FC¼2FCth is selected here as a fines content beyondwhich any further increase in fines content does not affect M.Thus, it is assumed that the continuous mathematical functiondescribing the evolution of M with FC should have two extremaaround FC¼0 and FC¼2FCth. Furthermore, the rate of change inM is in its peak at FC¼FCth where a phase transformation fromfines in coarse to coarse in fines takes place. Hence, M should havean inflection point at FC¼FCth. The simplest polynomial functionthat satisfies the above physical requirements is Eq. (8). Thesimulative capability of Eq. (8) for three different soil mixturesis shown in Fig. 6.

4.2. Equivalent intergranular state parameter

The mechanical behavior of clean sands is affected by thecombined influence of density and mean principal effective stresswhich is technically called state [2,3,16,66]). State in granular soilsis quantified uniquely by the so-called state parameter (see [24]).Been and Jefferies [2] introduced an effective state parameter, ψ,for clean sands in the following form

ψ ¼ e�ecs ð9Þ

where, e is the global void ratio and ecs is the void ratio on criticalstate line corresponding to the current value of mean principaleffective stress.

In the previous section, it was discussed that the global voidratio in granular soils containing non-plastic fines is not alegitimate index to describe how much load bearing microstruc-ture is compact. However, according to the current knowledge, theequivalent intergranular void ratio, en, is the most proper candi-date for this role. Based on this conclusion, the existing stateparameters essentially proposed for clean sands can be general-ized to cover silty sands if global void ratio is replaced byequivalent intergranular void ratio in their formulations.

Supported by a reasonable correspondence with experimentaldata, recently, Rahman et al. [48], and Rahman and Lo [49] re-defined the state parameter of Been and Jefferies [2] in terms of

equivalent intergranular void ratio

ψ n ¼ en�ecs ð10Þ

For clean sands, the above definition is reduced to that of Beenand Jefferies [2]. In this study, ψn undertakes the role of stateparameter for both clean and silty sands.

4.3. Effect of fines on initial fabric anisotropy

In naturally deposited clean sands, the mechanical behavioris highly dependent on direction of principal effective stresseswith respect to deposition plane due to the influence of initialfabric anisotropy. For example, under the extension mode oftriaxial, clean sands tend to contract and their ultimate (say

-200

-100

0

100

200

300

Mean Principal Effective Stress, p [kPa]

σv -

σh [k

Pa]

σv -

σh [k

Pa]

σv -

σh [k

Pa]

σv -

σh [k

Pa]

Compression

Extension

-200

-100

0

100

200

300


Compression

Extension

-200

-100

0

100

200

300


Compression

Extension

-200

-100

0

100

200

300

0 50 100 150 200 250 300

0 50 100 150 200 250 300

0 50 100 150 200 250 300

0 50 100 150 200 250 300


Compression

Extension

Fig. 7. Comparison of stress paths of the samples of the same density, but withdifferent fines contents under the compression and extension modes of triaxial [sv:vertical stress, sh: horizontal stress]: (a) FC¼0 (clean sand); (b) FC¼0.15; (c)FC¼0.30; (d) FC¼0.50 (data from Rafiee Dehkharghani [44]).



critical state) strength is much lower than that under thetriaxial compression. Vaid and Chern [63] reported that thecritical state line obtained from triaxial extension tests illu-strated in e–p plane is much lower compared with that oftriaxial compression. Later studies by Riemer and Seed [50],Yoshimine et al. [75], Mooney et al. [27], and Yang et al. [74]confirmed the finding of Vaid and Chern [63]. Based on theabove considerations and similar to Dafalias et al. [12], critical

state line in en–p is defined by

ecs ¼ e0�λp

pref

� �ξ

ð11Þ

where, λ and ξ are parameters that depend on soil type. pref is areference pressure that is usually taken as the atmosphericpressure (E101 kPa). e0 is the intercept of critical state linewith the en-axis in en–p plane. To calculate e0 and similar toYang et al. [73] the following equation is suggested here

e0 ¼ eA½1� f 1A� ð12Þ

where, eA and f1 are material parameters and A is the aniso-tropic state parameter. According to Dafalias et al. [12], A isdefined by

A¼ gðθ; cÞ F : n ð13Þ

In Eq. (13), g(θ,c) is a proper interpolation function defining theactual shape of the failure points in π-plane as a function of theLode angle, θ, and c is a soil parameter. In the current model,n (trn¼0, n:n¼1) is deviatoric part of normal to the yield surface[Eq. (30) of Appendix B]. F is a second order fabric tensor. For

0

0.9

1.8

2.7

0 0.3 0.6 0.9Axial Strain, ε1 [%]

Volu

met

ric S

trai

n, ε

v [%

]

FC=0.05FC=0.15Isotropic Trend

1

3

Fig. 8. Volumetric-axial strain relationship under isotropic compression for Ottawasand samples containing non-plastic silt (data from Carraro et al. [7]).

0

20

40

60

Global Void Ratio, e [-]

Shea

r Mod

ulus

, G [k

Pa]

FC=0FC=0.15FC=0.30Eq. (17) with G0=180

σ3 = 100 kPa

0

50

100

150

Global Void Ratio, e [-]

Shea

r Mod

ulus

, G [k

Pa]

FC=0FC=0.05FC=0.10FC=0.20Eq. (17) with G0=630

σ3 = 100 kPa

0

20

40

60

Equivalent Intergranular Void Ratio, e* [-]

Shea

r Mod

ulus

, G [k

Pa]

FC=0FC=0.15FC=0.30Eq. (17) with G0=180

σ3 = 100 kPa

0

50

100

150

Equivalent Intergranular Void Ratio, e* [-]

Shea

r Mod

ulus

, G [k

Pa] FC=0

FC=0.05FC=0.10FC=0.20Eq. (17) with G0=630

σ3 = 100 kPa

0

20

40

60

Skeleton Void Ratio, esk [-]Skeleton Void Ratio, esk [-]

Shea

r Mod

ulus

, G [k

Pa]

FC=0FC=0.15FC=0.30Eq. (17) with G0=180

σ3 = 100 kPa

0

50

100

150

0.5 0.6 0.7 0.8 0.9 1 1.1 0.3 0.4 0.5 0.6 0.7 0.8

0.6 0.7 0.8 0.9 1 1.1 1.2 0.3 0.4 0.5 0.6 0.7 0.8

0.6 0.8 1 1.2 1.4 1.6 1.8 0.3 0.4 0.5 0.6 0.7 0.8

Shea

r Mod

ulus

, G [k

Pa]

FC=0FC=0.05FC=0.10FC=0.20Eq. (17) with G0=630

σ3 = 100 kPa

Fig. 9. Illustration of elastic shear modulus for Mai Liao [parts a–c] and Ottawa [parts d–f] silty sands in terms of: (a) global void ratio, (b) equivalent intergranular void ratio,(c) skeleton void ratio, (d) global void ratio, (e) equivalent intergranular void ratio, and (f) skeleton void ratio (data from Huang et al. [15] and Carraro et al. [7]).



transversely isotropic soils, F is in the following form [12,34,74]

F¼ 13þΔ

1�Δ 0 00 1þΔ 00 0 1þΔ

0B@

1CA ð14Þ

where, Δ, the vector magnitude, is a measurable index of aniso-tropy [10,34,74]. For isotropic granular media, Δ¼0 and F¼1/3I.

For loadings with fixed direction of principal stresses, Eq. (13)becomes (see [12])

A¼ �ffiffiffi6

pgðθ; cÞ cos

2α�ð1=3Þð1þbÞðb2�bþ1Þ1=2

� Δ3þΔ

� �ð15Þ

where, α is inclination of the major principal stress directionwith respect to the depositional (i.e., vertical) direction, andb¼(s2�s3)/(s1�s3) is the coefficient of intermediate principalstress. Values of A corresponding to triaxial compression (b¼0 andα¼01), and triaxial extension (b¼1 and α¼901) are, respectively,Ac ¼ �

ffiffiffiffiffiffiffiffi8=3

pðΔ=ðΔþ3ÞÞ, and Ae ¼ �c Ac .

Dafalias et al. [12] used e0 ¼ eA expð�AÞ as a tool to back-calculate vector magnitude (Δ). Since the pioneering work of Yanget al. [73], Δ can be measured directly by using the image analysistechniques. A careful inspection reveals that the back-calculatedvalues of Δ in Dafalias et al. [12] for Toyoura sand are not identicalto the measured values of Yang et al. [73]. This issue is fixed hereby adding a parameter (i.e., f1) to the constitutive equations linkinge0 to Δ (via the anisotropic state parameter A) for furtherrefinement.

Recent experimental findings indicate that adding non-plasticfines to clean sands mitigates the initial fabric effects (e.g.,[1,7,22,44,45,55]). Preferred orientation of grains and associatedcontacts are the main source of the anisotropic behavior ofgranular soils. Adding non-plastic fines brings coarse grainstoward a floating state in which the preferred orientation of coarsegrains becomes relatively unimportant on the general soilresponse. To make this issue clear, stress paths of eight undrainedtriaxial tests on samples containing different fines contents arepresented in Fig. 7. For the mixture under consideration,FCth¼0.30–0.35. Shown in part (a), tests results represent thetypical behavior of clean sands (FC¼0) subjected to triaxialcompression and extension. The noteworthy difference betweenthe behaviors in the compression and extension sides is attributedto the initial fabric anisotropy (e.g., [49,63,74,75]). For sampleswith FC¼0.15, the tendency towards contraction increasedin both sides. Unexpectedly, the samples with FC¼0.30 andalso 0.50 exhibit similar stress paths. The similarity of stresspaths under the compression and extension modes of triaxialproves the mitigation of initial anisotropy for thresholdfines content and beyond (i.e., FCZ0.3). A similar trend hasbeen reported in Kuwano et al. [22], Bahadori et al. [1], andTastan [50]. Ishihara and Okada [17] suggested an interesting

Table

3Ph

ysical

prope

rtiesof

siltysandsco

nsidered

inthis

study.

Referen

ceSo

ilSa

nd

Fines

χ¼D10/d

50

Grain

shap

ee m

axe m

inD10

(mm)

D50(m

m)

Grain

shap

ee m

axe m

ind 5

0(m

m)

Murthy

etal.[31

]Carraro

etal.[7]

Ottaw

asilic

a(san

d)-Sil-Co-Sil10

6(silt)

Rounded

0.78

0.48

0.23

20.39

Angu

lar(crush

edOttaw

asand)

––

0.02

49.7

Papad

opou

louan

dTika

[36]

M31

quartz

(san

d)-Assyros

(silt)

Rounded

0.84

10.58

20.23

60.30

0Angu

lar(crush

edAssyros

quartz)

1.66

30.65

80.02

011.8

Thev

anay

agam

etal.[59

]F5

5,Fo

undry

quartz

(san

d)-Sil-Col-Sil40

(silt)

Sub-rounded

0.80

0.60

80.16

00.25

0Angu

lar(crush

edsilic

asand)

2.1

0.62

70.01

16.9

Yanget

al.[73

]Hok

ksund(san

d)-Chen

gbei

(silt)

Sub-an

gular

0.94

90.57

20.22

10.440

Angu

larto

sub-an

gular

1.41

30.73

10.03

26.9

Rah

man

etal.[48

]Sy

dney

(san

d)-Majura

rive

r(silt)

Angu

lar

0.85

50.56

50.22

50.27

Angu

lar

––

0.006

37.0

Huan

get

al.[15

]Mai

Liao

siltysand(natural)

Angu

lar

1.12

50.64

60.08

0.09

8Angu

lar

––

0.04

41.8

Bah

adoriet

al.[1]

Firooz

kuhsandysilt(natural)

Angu

lar

0.87

40.54

80.18

0.27

Angu

lar

––

0.03

6.0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3Fines Content, FC [-]

Expo

nent

n, [

-]

Salgado et al. (2000)Carraro et al. (2009)Prediction by Eq. (19)

ω 1=0.5, ω 2=0.05; Cu=1.9

Fig. 10. Effect of fines on exponent n (Eq. (17)) under fixed values of mean principalstress and density.



approach to prove the existence of fabric anisotropy in granularsoils. According to their theory, in isotropic compression, theratio εv/ε1 [ε1 and εv are, respectively, the axial and volumetricstrains] equals 3.0 for isotropic soils. The more deviation fromthe ratio 3.0 is an indication of the more deviation fromisotropy. Carraro et al. [7] reported the results of isotropiccompression tests on samples of Ottawa sand containingdifferent non-plastic silt contents. In Fig. 8, a significantdeviation of the εv�ε1 data for FC¼0.05 samples from theisotropic condition (i.e., 3:1 line) approves the existence ofinitial anisotropy. On the other hand, coincidence of the εv�ε1data for FC¼0.15 case with the line of isotropy indicates thatthe state of fabric is nearly isotropic. Adopting the suggestionof Ishihara and Okada [17], data shown in Fig. 8 imply thatanisotropy of sand–silt mixture is reduced with the increase infines content. Based on the above experimental observations,variation of vector magnitude is linked to fines content throughthe following relationship

Δ�Δc 1� FCFCth

� �þΔf

⟨FC�FCth⟩

1�FCthð16Þ

where Δc and Δf are, respectively, the initial vector magnitudeof clean sand and the initial vector magnitude of clean non-plastic silt. According to Zhang et al. [76] larger grains havegreater participation in anisotropic behavior of granular mediaand hence, it is expected that Δc4Δf. For FC¼0, Eq. (15) isreduced to the anisotropic state parameter of Dafalias et al. [12]for clean sands. For threshold fines content (FC¼FCth), one hasΔE0 and soil mixture behaves nearly isotropic. For finescontents beyond the threshold value, fine grains take graduallythe lead role in load bearing structure and hence, anisotropicbehavior associated with the fines constituent appears (e.g.,[1,44,45]). As a rational consequence, one obtains Δ¼Δf forFC¼1.

4.4. Effect of fines on elastic shear modulus

Using resonant column or bender element tests, the so-calledsmall strain shear modulus of granular soils is usually measuredin laboratory. According to extensive experimental studies, thefollowing empirical relationship has been widely adopted in theliterature for the elastic shear modulus of clean sands (e.g.,

[13,14,20,67])

G¼ G0prefFðeÞp

pref

� �n

ð17Þ

where, G0 and n are material parameters. F(e) imposes the influenceof void ratio on elastic shear modulus: FðeÞ ¼ ð2:97�eÞ2=ð1þeÞ andFðeÞ ¼ ð2:17�eÞ2=ð1þeÞ are, respectively, suggested by Hardin andBlack [14] for soils with angular and rounded grains. Mean principaleffective stress has a great influence on the elastic shear modulusthrough the term (p/pref)n. Hardin and Black [14] recommendednE0.5. Recently, Wichtmann and Triantafyllidis [67] suggestedn� 0:40 C0:18

u for clean sands.Under fixed global void ratio, recent experimental studies have

revealed that adding non-plastic fines leads to a continuousdecrease in G0 up to a minimum value at the threshold finescontent. Further increase of fines has no influence on G0 for fines

Table 4Unified model parameters used in simulation of experiments reported by Rahman et al. [48], Huang et al. [15], and Murthy et al. [31].

Category Parameter Rahman et al. [48] Huang et al. [15] Murthy et al. [31]

Elasticity G0 90 180 630ν 0.15 0.15 0.15n 0.5 0.5 Cu¼1.9; ω1¼0.5; ω2¼0.05 [see Eq. (19) and Fig. 10]

Yield surface m 0.01 0.01 0.01Critical state line Mc(FC¼0) 1.305 1.30 1.21

ΔM 0 0 0.95 [see Fig. 6(b)]e0 0.92 1.00 0.745λ 0.0375 0.0746 0.0366ξ 0.60 0.768 0.31

Plastic Hardening modulus H(e) 1–1.02 en 1.0 1.0H0 2.0 0.42 0.16nb 0.8 1.0 1.4

Dilatancy A0 0.50 0.20 0.40nd 2.0 3.5 1.0

Gradation χ 37.0 1.8 9.7Grains shape Rc 0.25 0.25 0.90

Rf 0.25 0.25 0.25

nIn simulation of tests reported by Rahman et al. [48] and Huang et al. [15], F(en)¼(2.97�en)2/(1þen) is used instead of F(e). In simulation of tests reported by Murthy et al. [31],F(esk)¼(2.17�esk)2/(1þesk) is assumed.

Table 5Unified model parameters used in simulation of experiments reported by Bahadoriet al. [1].

Category Parameter Bahadori et al. [1]

Elasticitya G0 95ν 0.10n 0.5

Yield surface m 0.01Critical state line Mc(FC¼0) 1.30

c (¼Me/Mc) 0.75ΔM 0eA 0.885f1 0.52λ 0.022ξ 0.75

Plastic Hardening modulus H(e) 1–1.116 en

HA 1.024f2 2.5nb 2.4

Dilatancy A0 0.40nd 1.0

Gradation χ 6.0Grains shape Rc 0.25

Rf 0.25Initial anisotropy Δc 0.20

a F(e)¼(2.97�en)2/(1þen) is used.



contents beyond the threshold value (e.g., [7,15,18,57,68]). Theva-nayagam and Liang [57] and Thevanayagam andMartin [58] showedthat if e in Eq. (17) is replaced by en, a unique G0 can be obtainedirrespective of fines content. Using a micromechanical approach tostudy the governing stress transmission mechanisms in granularsoils, Radjai and Wolf [42] and Radjai et al. [43] have shown thatonly about 28% of mean principal effective stress is transmittedthrough the so-called weak contacts (i.e., active fines) in polydis-perse granular media. Moreover, they have shown that even smallerportion of weak contacts participate in force chains transmittingshear stress.

In Fig. 9(a–c), elastic shear modulus of Mai Liao sand–siltmixtures are plotted in terms of global void ratio, equivalentintergranular void ratio, and skeleton void ratio. For various finescontents, a unique narrow region is obtained if data of elasticshear modulus are depicted against equivalent intergranularvoid ratio. Illustrations in a similar manner for Ottawa sand-SilCo Sil 106 mixtures are presented in Fig. 9(d and e). Unlike thelast case, the least scatter is obtained when data are plottedversus skeleton void ratio. In this regard, further inspections arenecessary to indicate the most appropriate criterion. However,from a statistical view, the best agreements are usually achievedwhen data are shown in terms of equivalent intergranularvoid ratio.

Cho et al. [8] pointed out that the exponent n in Eq. (17)decreases with the rise in soil average roundness by

n� 0:42�0:24 log R ð18Þ

Therefore, change in n is not expected when coarse and fineconstituents were subjected to the same formation history and thesame shape (e.g., roundness) as a result. On the other hand, it isanticipated that adding angular fines to rounded hosts leads to agradual decrease in overall roundness of the packing and eventually

the rise in n. Recent studies by Salgado et al. [51] and Carraro et al.[7] confirm this conclusion. For the latter case, a modified form ofthe Wichtmann and Triantafyllidis [67] correlation in the followingform is suggested here

n� 0:40C0:18u 1þ ω1FC

ω2þFC

� �ð19Þ

where, ω1 and ω2 are material parameters. Performance of Eq. (19)for Ottawa sand (well-rounded)-Sil Co Sil 106 (angular) mixture isevaluated in Fig. 10.

4.5. Dependence of plastic hardening modulus on A

According to Dafalias et al. [12], the relocation of critical stateline for different A-values is not sufficiently effective for propermodeling of the anisotropic effects. To overcome this deficiency,the term H0 in plastic hardening modulus [Eq. (34)] should berelated to A. Similar to Lashkari and Latifi [23], the followingconstitutive equation is suggested for H0 in this study

H0 ¼HA 1þ f 2Δ cos πAc�AAc�Ae

� �� ð20Þ

where, HA and f2 (both gain positive values) are material para-meters and A, Ac, and Ae are defined in Eq. (15). For the compres-sion and extension modes of triaxial, Eq. (20) attains the formsH0c ¼HA½1þ f 2 Δ� and H0e ¼HA½1� f 2 Δ�, respectively. At thresholdfines content (i.e., ΔE0), one has H0cZH0¼HAZH0e.

In the platform of Dafalias et al. [12], H0 ¼HA ½ 1þkh�kh

ðAe �AÞ=ðAe �AcÞ� is assumed. According to the existing experimentalevidence (see Section 4.3), the presence of non-plastic fines in acoarse granular structure mitigates the anisotropic effects and themixture becomes nearly isotropic (ΔE0) around the threshold finescontent. For ΔE0, one has AEAeEAcE0. In this case, the inter-polation function of Dafalias et al. [12] becomes mathematically

0

0.5

1

1.5

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

e=0.740; pin=850 kPae=0.770; pin=600 kPae=0.821; pin=600 kPa

Experiments

0

0.5

1

1.5

Normalized Mean Principal Effective Stress, p / pin [-]


e=0.740; pin=850 kPa

e=0.770; pin=600 kPa

e=0.821; pin=600 kPa

Predictions

0

0.5

1

1.5

Axial Strain, ε1 [%] Axial Strain, ε1 [%]


Experiments

0

0.5

1

1.5

0 0.5 1.5 2 0 0.5 1 1.5 2

0 5 10 15 20 25 30 0 5 10 15 20 25 30


Predictions

1

Fig. 11. The model predictions versus experiments for FC¼0.05 (data from Rahman et al. [48]).



unstable and still predicts an anisotropic direction-dependentresponse. Nevertheless, the H0 of this study [i.e., Eq. (20)] behavesisotropically at the threshold fines content since the termf 2Δ cos ½πðAc�AÞ=ðAc�AeÞ� of Eq. (20) becomes zero for Δ¼0 andH¼HA is obtained irrespective to shearing direction.

5. Generalization of an existing platform for silty sands

Manzari and Dafalias [29] introduced a critical statebounding surface plasticity model for the state dependentbehavior of clean sands. Later, this model was improved by

0

1

2

3

4

5

FC=0.0 ; e=0.743; pin=350 kPaFC=0.15; e=0.535; pin=600 kPaFC=0.20; e=0.573; pin=600 kPa

Experiments

0

1

2

3

4

5

Normalized Mean Principal Effective Stress, p/pin [-]Normalized Mean Principal Effective Stress, p/pin [-]

Nor

mal

ized

Dev

iato

r Str

ess,

q /

p in

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q /

p in

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q /

p in

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q /

p in

[-]

FC=0.0 ; e=0.743; pin=350 kPa

FC=0.15; e=0.535; pin=600 kPa

FC=0.20; e=0.573; pin=600 kPa

Predictions

0

1

2

3

4

5



Experiments

0

1

2

3

4

5

0 2 3 4 0 2 3 4

0 5 10 15 20 25 30 0 5 10 15 20 25 30


Predictions

1 1

Fig. 13. The model predictions versus experiments for three tests with FC¼0.0, 0.15, and 0.20 (data from Rahman et al. [48]).

0

1

2

3

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]



Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]

Nor

mal

ized

Dev

iato

r Str

ess,

q

/ pin

[-]


Experiments

0

1

2

3

e=0.535; pin=600 kPa

e=0.569; pin=1300 kPa

e=0.645; pin=1100 kPa

Predictions

0

1

2

3


Experiments

0

1

2

3

0 0.5 1 1.5 2 0 0.5 1 1.5 2

0 5 10 15 20 25 30 0 5 10 15 20 25 30



Predictions

Fig. 12. The model predictions versus experiments for FC¼0.15 (data from Rahman et al. [48]).



Dafalias et al. [12] to consider the effects of inherent anisotropyon behavior. The general formulation of this platform is outlinedin Appendix B.

To obtain a unified model for clean and silty sands, theequivalent intergranular state parameter, ψn [Eq. (10)], is imple-mented in the basic framework of the model to describe the stateof soil mixture uniquely. In calculation of ψn, equivalent inter-granular void ratio is calculated using Eq. (3) in conjunction withthe newly proposed empirical Eqs. (4) and (5). It is worthy tonote that any possible dependence of plastic hardening moduluson density (e.g., the term H(e) in Eq. (34)) must be describedby means of en instead of e. In calculation of back stress ratios

(Eq. (32)), the possible variation of critical state stress ratio(especially when angular fines are added to rounded host sandsand vice versa) must be taken into account using Eq. (8). Once Mc

is determined, the slope of critical state for shearing modes otherthan triaxial compression can be determined using a properinterpolation function whose role is capturing the actual shape offailure states in multiaxial stress space [see Eq. (32) of AppendixB]. Effect of non-plastic fines on vector magnitude is consideredusing Eq. (16). Then, the influence of initial anisotropy on thelocation of critical state line and plastic hardening modulus istaken into consideration using Eqs. (12) and (20). Finally, theinfluence of fines on elastic shear modulus must be accounted

0

200

400

600

800


Shea

r Str

ess,

q [k

Pa]

e=0.85e=0.82e=0.78e=0.77

0

200

400

600

800


Shea

r Str

ess,

q [k

Pa]

e=0.85e=0.82e=0.78e=0.77

0

200

400

600

800

0 100 200 300 400 500

0 2 4 6 8 10

0 100 200 300 400 500

0 2 4 6 8 10


Shea

r Str

ess,

q [k

Pa]

e=0.85e=0.82e=0.78e=0.77

0

200

400

600

800

Shea

r Str

ess,

q [k

Pa]

e=0.85e=0.82e=0.78e=0.77

Fig. 14. The model predictions versus experiments for FC¼0.0 (data from Huang et al. [15]). (a and c) FC¼0% experiments, (b and d) FC¼0% Predictions.

0

100

200

300

400


Shea

r Str

ess,

q [k

Pa]

e=0.73e=0.70e=0.65

0

100

200

300

400

0 50 100 150 200 250


Shea

r Str

ess,

q [k

Pa]

e=0.73e=0.70e=0.65

0

100

200

300

400

0 100 200 300

0 2 4 6 8 10 0 2 4 6 8 10


Shea

r Str

ess,

q [k

Pa]

e=0.73e=0.70e=0.65

0

100

200

300

400

Shea

r Str

ess,

q [k

Pa]

e=0.73e=0.70e=0.65

Fig. 15. The model predictions versus experiments for FC¼0.15 (data from Huang et al. [15]). (a) FC¼0.15 experiments, (b, c and d) FC¼0.15 predictions.



based on the recommendations given in the sub-section entitledEffect of fines on elastic shear modulus. Detailed procedures forcalibration of the model new parameters are described inAppendix C.

The model performance is evaluated against experimentaldata of four independent research teams. The first three sets ofdata are obtained from conventional triaxial cell. More elabo-rate stress paths are considered in the fourth series where themodel simulations are compared against experimental data ofhollow cylindrical apparatus. To better manifest the model act,a number of parametric studies are presented.

Physical properties of soils are given in Table 3. In addition, themodel parameters used in simulations are presented in Tables 4 and 5.

5.1. Evaluation of the model versus experiments ofRahman et al. [48]

Rahman et al. [48] published an extensive study on Sydney sand-Majura river silt mixed with low plasticity Kaolin. In this study, moisttamped samples were subjected to shear under the undrainedcompression mode of triaxial. In Fig. 11, the model predictions forFC¼0.05 are depicted together with the corresponding data. Using the

0

200

400

600


Dev

iato

r Str

ess,

q [k

Pa]

e=0.67e=0.67e=0.63e=0.59

0

200

400

600


Dev

iato

r Str

ess,

q [k

Pa]

e=0.67e=0.67e=0.63e=0.59

0

200

400

600

0 100 200 300 400 500 600

0 2 4 6 8 10

0 100 200 300 400 500 600

0 2 4 6 8 10


Dev

iato

r Str

ess,

q [k

Pa]

e=0.67e=0.67e=0.63e=0.59

0

200

400

600

Dev

iato

r Str

ess,

q [k

Pa] e=0.67

e=0.67e=0.63e=0.59

Fig. 16. The model predictions versus experiments for FC¼0.30 (data from Huang et al. [15]). (a and c) FC¼0.30 experiments, (b and d) FC¼0.30 predictions.

0

500

1000

1500

2000


Shea

r Str

ess,

q [k

Pa]

e=0.631e=0.652

0

500

1000

1500

2000


Shea

r Str

ess,

q [k

Pa]

e=0.631e=0.652

0

500

1000

1500

2000

0 500 1000 1500

0 5 10 15 20 25 30

0 500 1000 1500

0 5 10 15 20 25 30


Shea

r Str

ess,

q [k

Pa]

e=0.631e=0.652

0

500

1000

1500

2000

Shea

r Str

ess,

q [k

Pa]

e=0.631e=0.652

Fig. 17. The model predictions versus experiments for FC¼0.0 (data from Murthy et al. [31]). (a and c) FC¼0% experiments, (b and d) FC¼0% predictions.



same model parameters, similar comparisons for FC¼0.15 are pre-sented in Fig. 12. Finally, the model predictions for three samples withFC¼0, 0.15, and 0.20 are compared with the corresponding experi-ments in Fig. 13. In all cases, a reasonable correspondence can beobserved between the model predictions and the data. Refer-ring to Fig. 11, the sample with e¼0.770 (FC¼0.05) exhibits flowwith limited deformation. A similar behavior, is observed for thesample with e¼0.569 (FC¼0.15) in Fig. 12 which is much denserthan its counterpart in Fig. 11. Even a stranger response can befound in Fig. 13. While the medium dense sample with e¼0.743(FC¼0) demonstrated a hardening response, a dense samplewith e¼0.535 (FC¼0.15) reached an ultimate strength

significantly less than the medium sample. Furthermore,despite its high density, another dense sample (e¼0.573) withFC¼0.20 failed due to flow liquefaction which is the worst typeof liquefaction. The initial values of equivalent intergranularstate parameters (ψn

inE�0.10, �0.05, and þ0.07, respectively,for samples with FC¼0.0, 0.15, and 0.20) can help to resolve theapparent ambiguity in Fig. 13. Considering the positive value ofψn

in for the sample with FC¼0.20, this sample is very prone toflow liquefaction in spite of its high density. Nevertheless, flowliquefaction is not expected for other samples (ψn

ino0 forboth). Besides, a more dilatant behavior is predictable for theloosest sample (i.e., FC¼0.0) because of its high negative value

0

500

1000

1500

2000

0 500 1000 1500


Shea

r Str

ess,

q [k

Pa]

e=0.630e=0.639e=0.648

0

500

1000

1500

2000

0 500 1000 1500


Shea

r Str

ess,

q [k

Pa]

e=0.630e=0.639e=0.648

0

500

1000

1500

2000

0 5 10 15 20 25 30


Shea

r Str

ess,

q [k

Pa] e=0.630

e=0.639e=0.648

0

500

1000

1500

2000

0 5 10 15 20 25 30Sh

ear S

tres

s, q

[kPa

] e=0.630e=0.639e=0.648

Fig. 18. The model predictions versus experiments for FC¼0.05 (data from Murthy et al. [31]).

0

300

600

900

0 200 400 600 800 0 200 400 600 800


Shea

r Str

ess,

q [k

Pa]

e=0.588; FC=0.10e=0.523; FC=0.15

0

300

600

900


Shea

r Str

ess,

q [k

Pa]

e=0.588; FC=0.10e=0.517; FC=0.15

0

300

600

900

0 5 10 15 20 25 30 0 5 10 15 20 25 30


Shea

r Str

ess,

q [k

Pa]

e=0.588; FC=0.10e=0.523; FC=0.15

0

300

600

900

Shea

r Str

ess,

q [k

Pa]

e=0.588; FC=0.10e=0.517; FC=0.15

Fig. 19. The model predictions versus experiments for two samples with FC¼0.10 and 0.15 (data from Murthy et al. [31]). (a and c) experiments, (b and d) predictions.



of ψnin. Fig. 13 indicates that our conclusions agree with the

measured behaviors.

5.2. Evaluation of the model versus experiments of Huang et al. [15]

The Chi Chi earthquake (Mw¼7.6) of 1999 caused a wide-spread liquefaction in Central Western Taiwan. Huang et al. [15]

conducted a series of triaxial compression tests on Mai Liao siltysand which is a typical soil of the region. All samples wereprepared using moist tamping method which yields to the mostpossible wide range of void ratios. Through Figs. 14–16, themodel predictions are compared with experimental data forFC¼0.0–0.30. A reasonable agreement between the modelpredictions and data is observed in figures.

0

100

200

300


Max

. Dev

iato

r Str

ess,

q [k

Pa]

Alpha=15; e=0.738Alpha=30; e=0.728Alpha=60; e=0.720Alpha=75, e=0.730

0

100

200

300


Max

. Dev

iato

r Str

ess,

q [k

Pa]


0

100

200

300

0 100 200 300

0 5 10 15 20

0 100 200 300

0 5 10 15 20

Max. Deviator Strain, ε1 - ε3 [%]

Max

. Dev

iato

r Str

ess,

q [k

Pa] Alpha=15; e=0.738

Alpha=30; e=0.728Alpha=60; e=0.720Alpha=75; e=0.730

0

100

200

300


Max

. Dev

iato

r Stre

ss, q

[kP

a]

Alpha=15; e=0.738Alpha=30; e=0.728Alpha=60; e=0.720Alpha=75; e=0.730

Fig. 20. The model simulations versus experiments on the influence of principal stress directions for FC¼0.0 and pin¼100 kPa. (data from Bahadori et al. [1]). (a) FC¼0experiments and (b) FC¼0 simulations.

0

100

200

300

400

0 100 200 300 400


Max

. Dev

iato

r Stre

ss, q

[kPa

]


0

100

200

300

400

0 100 200 300 400


0

100

200

300

400


Max

. Dev

iato

r Stre

ss, q

[kPa

]

Max

. Dev

iato

r Stre

ss, q

[kPa

]M

ax. D

evia

tor S

tress

, q [k

Pa]Alpha=15; e=0.719


0

100

200

300

400

0



Alpha=15; e=0.719Alpha=30; e=0.707Alpha=60; e=0.721Alpha=75; e=0.717

5 10 15 200 5 10 15 20

Fig. 21. The model simulations versus experiments on the influence of principal stress directions for FC¼0.0 and pin¼200 kPa. (data from Bahadori et al. [1]). (a) FC¼0experiments and (b) FC¼0 simulations.



5.3. Evaluation of the model versus experiments of Murthy et al. [31]

Murthy et al. [31] studied the undrained behavior ofOttawa sand-Sil Co Sil 106 mixtures prepared by dry slurrymethod in triaxial apparatus. For clean sand, comparison ofthe model predictions with experiments is presented inFig. 17. Similar comparisons for silty sand samples withFC¼0.05, 0.10, and 0.15 are presented in Figs. 18 and 19. A

reasonable agreement between the model predictions anddata is observed in figures.

5.4. Evaluation of the model versus experiments of Bahadori et al. [1]

Previous experimental studies using hollow cylindrical apparatushave revealed a dramatic influence of rotation of principal stressaxes on the undrained behavior of clean sands [62,74,75]). Recently,

0

50

100

150

0 50 100 150


Max

. Dev

iato

r Stre

ss, q

[kP

a]


0

50

100

150

0 50 100 150


Max

. Dev

iato

r Stre

ss, q

[kPa

]


0

50

100

150

0 5 10 15 20


Max

. Dev

iato

r Stre

ss, q

[kPa

] Alpha=15; e=0.621Alpha=30; e=0.623Alpha=60; e=0.626Alpha=75; e=0.614

0

50

100

150

0 5 10 15 20


Max

. Dev

iato

r Stre

ss, q

[kPa


Fig. 22. The model simulations versus experiments on the influence of principal stress directions for FC¼0.15 and pin¼100 kPa. (data from Bahadori et al. [1]). (a andc) FC¼0.15 experiments and (b and d) simulations.

0

50

100

150


Max

. Dev

iato

r Str

ess,

q [

kPa]


0

50

100

150

0 50 100 150 0 50 100 150


Max

. Dev

iato

r Str

ess,

q [

kPa]


0

50

100

150

0 5 10 15 20


Max

. Dev

iato

r Str

ess,

q [k

Pa]


0

50

100

150

0 5 10 15 20


Max

. Dev

iato

r Str

ess,

q [k

Pa]


Fig. 23. The model simulations versus experiments on the influence of principal stress directions for FC¼0.15 and pin¼100 kPa. (data from Bahadori et al. [1]). (a andc) FC¼0.15 experiments and (b and d) simulations.



Bahadori et al. [1] conducted a series of tests on Firoozkuh sandsamples containing different non-plastic Firoozkuh silt contents. Allsamples were prepared using dry deposition method. In testingprogram, inclination of the major principal stress direction withrespect to depositional direction (i.e., vertical), α, was selected 151,301, 601, and 751. Moreover, the coefficient of intermediate principalstress, b¼(s2–s3)/(s1–s3), was 0.5 in all tests.

For eight medium-loose samples of clean sands, the modelpredictions are depicted together with the corresponding data inFigs. 20 and 21. It can be observed that the model is capable ofreplicating the typical response of clean sands subjected torotation of principal stress axes. For silty sand samples of mediumdensity with FC¼0.15, similar comparisons are presented inFigs. 22 and 23. Finally, for very dense silty sands with FC¼0.30,

0

100

200

300


Max

. Dev

iato

r Stre

ss, q

[kP

a]


0

100

200

300


Max

. Dev

iato

r Stre

ss, q

[kP

a]


0

100

200

300


Max

. Dev

iato

r Stre

ss, q

[kPa


0

100

200

300

0 100 200 300 0 100 200 300

0 5 10 15 0 5 10 15


Max

. Dev

iato

r Stre

ss, q

[kPa


Fig. 24. The model simulations versus experiments on the influence of principal stress directions for FC¼0.30 and pin¼200 kPa. (data from Bahadori et al. [1]). (a and c)FC¼0.30 experiments and (b and d) simulations.

0

50

100

150

0 50 100 150


Max

. Dev

iato

r Stre

ss, q

[kP

a]

Alpha=15; e=0.521Alpha=30; e=0.527

0

50

100

150

0 50 100 150


Max

. Dev

iato

r Stre

ss, q

[kP

a]

Alpha=15; e=0.521Alpha=30; e=0.527

0

50

100

150

0 5 10 15


Max

. Dev

iato

r Stre

ss, q

[kPa

]

Alpha=15; e=0.521Alpha=30; e=0.527

0

50

100

150

0 5 10 15


Max

. Dev

iato

r Stre

ss, q

[kPa

]

Alpha=15; e=0.521Alpha=30; e=0.527

Fig. 25. The model simulations versus experiments on the influence of principal stress directions for FC¼0.30 and pin¼100 kPa. (data from Bahadori et al. [1]). (a and c)FC¼0.30 experiments and (b and d) simulations.



comparisons are shown in Figs. 24 and 25. Silty sands are generallydenser than clean ones. However, a remarkable tendency towardscontraction and eventual drop in the ultimate strength is observedin their behavior with the rise in FCwhich is due to the incompleteparticipation of fines in stress bearing microstructure. In addition,it is observed that the increase in FC gradually eliminates theinfluence of initial anisotropy in the mechanical behavior of thesamples. In clean samples (Figs. 20 and 21), a noteworthy diversitybetween the mechanical behaviors with the change in direction of

major principal stress axes is observed. The presence of finesresults in a relatively unique response for FC¼0.3 samples. It isworthy to note that the small discrepancy observed in Figs. 22 and23 is related to the difference in initial densities (void ratios).

5.5. Parametric studies

Numerical studies are presented in this sub-section to show theimportance/necessity of each modification introduced in the pre-vious sections. To this aim, novel features of the modified model of

0

1

2

3

Normalized Mean Principal Effective Stress, p/pin [-]

Nor

mal

ized

Dev

iato

r Str

ess,

q /

p in [-

]N

orm

aliz

ed D

evia

tor S

tres

s, q

/ p in

[-]

Proposed model: e=0.535; pin=600 kPaProposed model: e=0.645; pin=1100 kPaDisabled model: e=0.535; pin=600 kPaDisabled model: e=0.645; pin=1100 kPa

0

1

2

3

0 0.5 1 1.5 2

0 5 10 15 20 25

Axial Strain,ε1 [%]

Proposed model: e=0.535; pin=600 kPaProposed model: e=0.645; pin=1100 kPaDisabled model: e=0.535; pin=600 kPaDisabled model: e=0.645; pin=1100 kPa

Fig. 26. Numerical analyses on the influence of using either en or e in modelformulation. (a and b) FC¼0.15.

0

300

600

900


Shea

r Str

ess,

q [k

Pa]

Proposed model: e=0.588; FC=0.10


Disabled model: e=0.588; FC=0.10Disabled model: e=0.517; FC=0.15

0

300

600

900

0 200 400 600 800

0 5 10 15 20 25 30

Axial Strain, ε1 - ε3 [%]

Shea

r Str

ess,

q [k

Pa]

Proposed model: e=0.588; FC=0.10Proposed model: e=0.517; FC=0.15


Fig. 27. Numerical analyses on the influence of fines content on the elastic shearmodulus.

0

300

600

900


Shea

r Str

ess,

q [k

Pa]




0

300

600

900

0 200 400 600 800

0 5 10 15 20 25 30

Shea

r Str

ess,

q [k

Pa]

Proposed model: e=0.588; FC=0.10Proposed model: e=0.517; FC=0.15


Axial Strain, ε1 - ε3 [%]

Fig. 28. Numerical analyses on the influence of fines content on the slope of criticalstate line in q–p plane.

0

50

100

150


Max

. Dev

iato

r Str

ess,

q [

kPa]

Proposed model: Alpha=15; e=0.521

Proposed model: Alpha=30; e=0.527

Disabled model: Alpha=15; e=0.521Disabled model: Alpha=30; e=0.527

0

50

100

150

0 50 100 150

0 5 10 15


Max

. Dev

iato

r Str

ess,

q [k

Pa]

Proposed model: Alpha=15; e=0.521Proposed model: Alpha=30; e=0.527Disabled model: Alpha=15; e=0.521Disabled model: Alpha=30; e=0.527

Fig. 29. Numerical analyses on the influence of fines content on the anisotropicbehavior of silty sands. (a) FC¼0.30.



this study are ignored in sequel and for each case, the outcome isstudied. Throughout this sub-section the term “disabled model” isattributed to the so-called downgraded versions of the modifiedmodel of this study that is disabled to use one or more elementsespecially proposed in this paper for modeling of the silty sandsbehavior. The modified model of this study is reduced to aconstitutive model very similar to the original platform of Dafaliaset al. [12] when all novel constitutive equations of this study areomitted or FC¼0 is assumed. The parameters used by eachdisabled model are identical to those of the modified model ofthis study; however, it should be noted that a number ofparameters may be omitted as a direct consequence of theelimination of one or more constitutive equations.

Supported by experimental evidence, the equivalent intergra-nular void ratio was adopted in Section 3 to describe the density ofsoil load carrying structure. Using a disabled model that takesglobal void ratio instead of equivalent intergranular void ratio,the mechanical behavior of two samples borrowed from Fig. 12is studied in Fig. 26. In disabled model, participation of thefines constituent in load bearing structure is assumed identicalto that of the coarse constituent. Due to this unrealisticassumption, the predicted behaviors by the disabled modelbecome highly dilative and the ultimate strengths are remarkablyoverestimated.

In Section 4.4, it was discussed that adding non-plastic finesleads to the reduction of the elastic shear modulus. To show the

-200

-100

0

100

200

0 50 100 150 200


σv

- σh

[kPa

]

Triaxial Compression: FC=0.0

Triaxial Extension: FC=0.0

-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-200

-100

0

100

200

0 50 100 150 200


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-200

-100

0

100

200

0 50 100 150 200


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-100

-50

0

50

100

0 50 100 150 200


σv

- σh

[kPa

]



-100

-50

0

50

100

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



Fig. 30. Comparative parametric studies on the influence of fines content on the undrained behavior of clean and silty sands with the same global void ratio (e¼0.55)subjected to the compression (α¼01, b¼0) and extension (α¼901, b¼1) modes of triaxial. (a,b) e = en = 0.55; (c,d) e = 0.55, en = 0.696; (e,f) e = 0.55, en= 0.811; (g and h)e¼0.55, en = 0.851.



importance of this phenomenon, the proposed model is disabledto consider the influence of fines on G. For two tests considered inFig. 19, predictions of the proposed model are compared withthose of the counterpart disabled model in Fig. 27. It can beobserved that ignoring the influence of fines on G results in anuntrue increase in stiffness.

In Section 4.1, it was discussed that adding non-plastic fines maylead to the change in the slope of critical state line in q–p plane.Adding angular silt to sands with round grains leads to the gradualincrease in M. The proposed model is disabled to consider theinfluence of fines on M. For two tests considered in Fig. 19,

predictions of the proposed model are compared with those of thecounterpart disabled model in Fig. 28. It can be found that ignoringthe effect of FC on M results in underestimating shear strength inboth simulations of the disabled model. It must be noted that addingsilts with round particles to angular sands leads to the gradualdecrease in M. In the latter case, ignoring the dependence ofM on FCleads to overestimating shear strength.

Finally, based on experimental evidence, the influence of addingnon-plastic silt on anisotropic behavior of silty sands is consideredthrough Eqs. (15), (16) and (20). The propose model is disabled toconsider the effect of the rise in fines content on anisotropic behavior

-200

-100

0

100

200

0 100 200 300


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-200

-100

0

100

200

0 100 200 300


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-200

-100

0

100

200

0 100 200 300


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



-200

-100

0

100

200

0 100 200 300


σv

- σh

[kPa

]



-200

-100

0

100

200

-15 -10 -5 0 5 10 15

εv - εh [%]

σv

- σh

[kPa

]



Fig. 31. Comparative parametric studies on the influence of fines content on the undrained behavior of clean and silty sands with the same equivalent intergranular voidratio (en¼0.85) subjected to the compression (α¼01, b¼0) and extension (α¼90, b¼1) modes of triaxial. (a,b) e = en = 0.85; (c,d) e = 0.690, en = 0.85; (e,f) e = 0.585, en =0.85; (g and h) e = 0.549, en = 0.85.



[Eq. (16)]. For two tests studied in Fig. 25, predictions by the disabledmodel are depicted against those of the proposed model in Fig. 29.Comparing with experimental data shown in Fig. 25, a highly dilativeresponse is predicted by the disabled model which does not inagreement with data. Furthermore, experimental studies have

revealed that at high fines contents, soil response is the same withit sheared along different directions of principal stress axes. Forexample, stress paths and stress–strain behaviors of two samples arethe same in parts “a” and “b” of Fig. 25. Considering predictionsshown in Fig. 29, one can observe that ignoring Eq. (16) leads to a

0

100

200

300

0 50 100 150 200


Max

. Dev

iato

r Stre

ss, q

[kP

a] α=0º

α=90º

0

100

200

300

0 5 10 15


Max

. Dev

iato

r Stre

ss, q

[kPa

]

α=90º

α= 0ºα=30º α=60º

0

100

200

300

0 50 100 150 200


Max

. Dev

iato

r Stre

ss, q

[kP

a]

α= 90ºα= 0º

0

100

200

300

0 5 10 15Max. Deviator Strain, ε1 - ε3 [%]

Max

. Dev

iato

r Str

ess,

q [k

Pa]

α= 90ºα= 60º

α= 30ºα= 0º

0

100

200

300

0 50 100 150 200


Max

. Dev

iato

r Stre

ss, q

[kP

a]

α= 90ºα= 0º

0

100

200

300

0 5 10 15Max. Deviator Strain, ε1 - ε3 [%]

Max

. Dev

iato

r Str

ess,

q [k

Pa]

α= 90º

α= 60ºα= 30º

α= 0º

0

100

200

300

0 50 100 150 200


Max

. Dev

iato

r Stre

ss, q

[kPa

]

α= 0º - 90º

0

100

200

300

0 5 10 15


Max

. Dev

iato

r Str

ess,

q [k

Pa]

α= 0º - 90º

Fig. 32. Comparative parametric studies on the influence of fines content on the anisotropic behavior of samples with the same equivalent intergranular void ratio(en¼0.75): ((a) and (b)) FC¼0.0, e¼0.75; ((c) and (d)) FC¼0.10, e¼0.60; ((e) and (f)) FC¼0.20, e¼0.498; ((g) and (h)) FC¼0.30, e¼0.445.



significant diversity in the predicted behaviors that is not in agree-ment with the experimental data.

To better understand the model performance, two series ofparametric studies are conducted here. Comparative reviews onthe influence of fines content on the undrained response of siltysands subjected to the compression and extension modes oftriaxial are presented in Figs. 30 and 31. The global void ratio iskept unchanged in all simulations and only fines content isgradually risen up to 0.25 in Fig. 30. Similar studies are presentedin Fig. 31 for a constant value of equivalent intergranular voidratio. For both modes in Figs. 30 and 31, the tendency towardcontraction increases with the rise in fines content. Comparing theresponses, it is found that the so-called relative softer response inthe extension side disappears gently with the increase in finescontent. As a result, for high fine contents close enough to thethreshold value (FCthE0.31), the difference between the predictedbehaviors in the compression and extension modes becomesnegligible [see parts g and h of Figs. 30 and 31]. This observationis in agreement with the finings of Rafiee Dehkharghani [44]shown in Fig. 7. The subscripts “v” and “h” in Figs. 30 and 31 standfor the “vertical” and “horizontal” stress and strain components.The issue is further studied in Fig. 32 for samples of the sameequivalent intergranular void ratio (en¼0.75) containing differentfines contents (FC¼0, 0.10, 0.20, and 0.30). A noteworthy diversityis observed in stress paths and stress–strain responses ofclean sand samples subjected to shear in different directions ofmajor principal stress with respect to deposition direction. The

mentioned diversity is eliminated smoothly with the rise of FC.When fines content gains a value close to the threshold value, aunique response is observed for shearing in different directions.The latter behavior is in agreement with the finings of Bahadori elal. [1].

6. Summary and concluding remarks

In silty sands, silt grains fill voids between the larger sandgrains and produce a denser packing. Nevertheless, fine grains donot participate actively in stress transmission mechanism is siltysands. As a result, to avoid the misinterpretation of the fallaciousrise of density, researchers suggest to use equivalent intergranularvoid ratio defined based on the voids between the active grains.In this study, an equivalent intergranular void ratio was suggested.The proposed empirical relationship, takes into account thecombined influence of fines content, soil gradation, and theaverage shapes of coarse and fine constituents. Three importantscales (i.e., roundness, sphericity, and irregularity) have beensuggested in the literature to characterize particles shape ingranular media. Considering that almost all of the existing studieson the mechanical behavior of silty sands have reported descrip-tion or micro photos of grains, roundness is the simplest scale todetermine. For this reason, roundness is selected in this study todescribe the average shape of particles. In the presence of detaileddata on the particles shape, alternative correlations between β andother scales may be considered in future works. Moreover,inherent anisotropic effects are mitigated by increase in finescontent. In fact, inherent anisotropy is related to the initialpreference of grains orientation remained from the early deposi-tion era. Adding non-plastic fines brings coarse grains toward afloating state in which the preferred orientation of coarse grainsbecomes unimportant on the general soil response. To considerthe above-mentioned phenomena, a modified state parameterbased on the equivalent intergranular void ratio was used in themodel. Besides, the vector magnitude was related to fines content.Then, new proper constitutive equations were introduced tomodify plastic hardening modulus, and critical state line in loadingpaths involving rotation of principal stress axes.

Predictive capacity of the model was evaluated using theresults of 47 triaxial and hollow cylindrical tests reported by fourindependent research teams. Soils used in evaluations wereselected from different geological origins, particle shapes,

0

10

20

30

40

0.5 1.5 2.5 3.5Roundness Ratio, r = Rc/Rf [-]

Freq

uenc

y [%

]

0

10

20

30

40

50

60

4 12 20 28 36

Particle Size Ratio, χ = D10 / d50 [-]

Freq

uenc

y [%

]

0

10

20

30

40

50

0.05 0.15 0.25 0.35Fines Content, FC [-]

Freq

uenc

y [%

]

Fig. 33. Statistical frequencies of: (a) roundness ratio; (b) particle size ratio; and(c) fines content; of data presented in Table 1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

β :Back-calculated [-]

β : P

redi

cted

[-]

β (predicted) = β (back-calculated)

Fig. 34. Comparison between the predicted and back-calculated values of β.



and preparation methods. In each case, a reasonable agree-ment with experiments was observed using a single set ofparameters.

Appendix A

The equivalent intergranular void ratio, en, describes thecompactness of the load carrying structure of silty sands.As mentioned in Section 4.1, adding non-plastic fines to sandsleads to the downward movement of critical state line in e–p planedue to the insufficient participation of fine grains in load bearingstructure of the mixture when FCoFCth. However, a unique criticalstate line is obtained when critical states of mixtures withdifferent fines contents are depicted in terms of the equivalentintergranular void ratio in en–p plane. The concept of uniquenessof critical state in terms of the equivalent intergranular void ratiohas been used by many researchers to back-calculate the partici-pation of fines, β, in the load carrying structure of soils (forinstance see Thevanayagam et al. [59]; Ni et al. [33]; Rahmanand Lo [46]; and Rahman et al. [48]). The same methodology hasbeen adopted in this study. Herein, in order to back-calculate β insand–silt mixtures, the critical state line of clean sand is consid-ered as benchmark. By considering the concept of the uniquenessof critical state in en–p plane for 0oFCrFCth, all critical state linesin en–p plane should be identical to the critical state line of theclean sand in e–p plane. It should be noted that the representationof critical states in en–p and e–p planes are identical for clean sandssince e¼en for FC¼0 (i.e., clean) condition. For a particularsand–silt mixture (say fines content), a proper value of β that yieldsa critical state line identical to that of the clean sand can be foundsimply by using Eq. (3) together with a trial and error procedure. Inthis study, the results of 296 triaxial tests of nine differentsand–silt mixtures that reached the actual critical state were usedfor back-calculation of the participation factor of fines {29 tests fromRahman et al. [48]; 32 tests from Zlatović and Ishihara [77]; 25 testsfrom Huang et al. [15]; 35 tests from Thevanayagam et al. [59]; 40tests from Papadopoulou and Tika [36]; 45 tests fromMurthy et al. [31]; 38 tests from Yang et al. [73]; 6 tests from Niet al. [33]; and 46 tests from Stamatopoulos [53]}. Processing thecritical state data of 296 test results in a total of 38 data for theevolution of βwith FC, χ, and r corresponding to nine different sand–silt mixtures (see Table 2 for details). The frequency distributions of r,χ, and FC (variables used for calibration of Eqs. (4) and (5)) arepresented in Fig. 33. The Pearson Correlation Coefficient

PCC¼∑ni ¼ 1ðXi�XÞ ðYi�YÞ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑n

i ¼ 1ðXi�XÞ2q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

∑ni ¼ 1ðYi�YÞ2

qÞ is a

measure of the linear correlation between two variables X (e.g., back-calculated values of β) and Y (e.g., predicted values of β using Eqs.(4) and (5)), giving a value between þ1 and �1. PCC¼þ1, 0, and �1indicate, respectively, total positive correlation, no correlation, andtotal negative correlation. In a curve fitting procedure, Root Mean

Square of prediction Errors (RMSE ½%� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=n ∑n

i ¼ 1ðYi�Xi=XiÞ2q

�100%) quantifies the difference between an estimator (i.e., Y) and thetrue values of the quantity being estimated (X). Using data presentedin Table 2, PCC¼þ0.961 and RMSE¼24.4% are obtained for Eq. (4)together with Eq. (5). The comparison between the predicted andback-calculated values of β is illustrated in Fig. 34.

Appendix B

Manzari and Dafalias [29], Dafalias and Manzari [11], Dafaliaset al. [12], and Lashkari [24], and Papadimitriou and Boukovalas[35] introduced a family of state dependent bounding surfaceconstitutive models based on the following general expression for

dilatancy

D¼Dðα;ψ ; F;CÞ ð21Þwhere the above definition should satisfy the following condition

Dðα¼αb ¼αc ¼αd;ψ ¼ 0; F;CÞ ¼ 0 ð22Þ

In Eqs. (21) and (22), D indicates dilatancy. α is the back stressratio tensor associated with the yield surface [Eq. (25)]. αb, αc, andαd are back stress ratios corresponding to the bounding, criticalstate, and dilatancy surfaces [Eq. (32)], respectively. F is a secondorder fabric tensor and C represents intrinsic material constants.

According to the continuum mechanics principles, effectivestress (r) and total strain (ε) tensors can be written in form

r¼ sþp1 ; ε¼ eþð1=3Þεv1 ð23Þwhere, s and e are, respectively, shear stress and shear straintensors. p is mean principal effective stress and εv is the volu-metric strain. 1 is the second order identity tensor. In elastic–plastic models, the total strain rate is written in the followingadditive form

_ε¼ _εeþ _εp ð24Þwhere, superscripts “e” and “p” indicate the elastic and plasticportions of total strain rate.

In model, a yield surface characterize domain of pure elasticityin multiaxial stress space

f ðr; αÞ ¼ ½ðs�pαÞ : ðs�pαÞ�1=2�ffiffiffiffiffiffiffiffi2=3

pmp¼ 0 ð25Þ

where, α is back stress ratio and mE0.01 is a model parameterindicating the yield surface size. The following constitutive equa-tions describe soil response when stress state is located inside theyield surface (i.e., f(r, α)o0) and soil behaves fully elastic

_p¼ K _εev ; _s¼ 2G_ee ð26Þwhere, G is the elastic shear modulus and K is the elastic bulkmodulus defined by

K ¼ 23G

1þν1�2ν

� �ð27Þ

In above definition, K is related to G through the Poisson's ratio, ν.When stress state reaches the yield surface and attempts to

move beyond, plastic strains may be generated which are calcu-lated by

_εp ¼ ⟨ _Λ⟩½nþð1=3ÞD1� ð28Þ

In Eq. (28), n is deviatoric part of normal to the yield surface (L)

L¼ ∂f ðr;αÞ∂r

¼ n�13N 1 ð29Þ

in which

n¼ r�αffiffiffiffiffiffiffiffi2=3

pm

and N¼ n : r ð30Þ

In Eq. (30), r¼s/p is shear stress ratio tensor. _Λ in Eq. (28) is theloading index

_Λ¼ 1Kp

L : _r¼ 2Gn : _e�KN _εvKpþ2G�KND

ð31Þ

where, D and Kp are, respectively, dilatancy and plastic hardeningmodulus.

The model employs three other constitutive surfaces: A bound-ing surface defining domain of permissible stress states, a criticalstate surface describing the ultimate states at critical state, and adilatncy surface differentiating contractive states from dilative



ones. Back stress ratios corresponding to the above surfaces are

αb ¼ffiffiffiffiffiffiffiffi2=3

pðgðθ; cÞMcexpð�nbψ Þ�mÞ n

αc ¼ffiffiffiffiffiffiffiffi2=3

pðgðθ; cÞMc�mÞ n

αd ¼ffiffiffiffiffiffiffiffi2=3

pðgðθ; cÞMcexpðndψ Þ�mÞ n

ð32Þ

where, αb, αc, and αd are defined in Eq. (22). Mc, c, nb, and nd aremodel parameters. In the modified model of this study ψn

{Eq. (10)} and Mc(FC) {Eq. (8)} should be used in Eq. (32). Now,dilatancy and plastic hardening modulus are defined by

D¼ Aðαd�αÞ : n ð33Þand

Kp ¼H0HðeÞGðαb�αÞ : nðαd�αinÞ : n

ð34Þ

where, H0 was defined through Eq. (20) and H(e) is a possiblefunction of void ratio. αin is the value of α when the most recentshear loading started.

Appendix C

The modified model may require a total of 22 parameters forsimulation of the mechanical behavior of sands containing non-plastic silt. Herein, calibration methods for those parametersrequired for simulation of the silty sands behavior (i.e., ΔM, χ, Rc,Rf,ω1,ω2), the parameters belong to the new constitutive equations(i.e., eA, HA, f1, f2), and the parameters of the original platform ofManzari and Dafalias [29] and Dafalias et al. [12] are described.

G0, n and ν can be determined from data of resonant column orbender element tests. In the absence of such data, G0 and n can becalculated by having data of the tangents to the very beginning partof q–εq data of triaxial tests. Eq. (17) can be re-written in the formlnðG=pref Þ ¼ ln ½G0 FðeÞ�þn lnðp=pref Þ. As a result, the intercept andthe slope of the best straight line fitted to the experimental data of ln(G/pref) vs. ln(p/pref) indicate ln[G0 F(e)] and n, respectively. For silicasands, ν is expected to be in the range 0.1–0.2 and in the absence ofthe high-resolution experiments, a representative value of 0.15 canbe used. Yield function of granular soils is quite small. According tothe suggestion by Dafalias and Manzari [11], Dafalias et al. [12],m¼0.01 is a reasonable selection. Plastic hardening modulus [Eq.(34)] is zero at peak shear strength of conventional drained triaxialtests. Considering Eqs. (32) and (34), nb can be determined bynb ¼ 1=ψ nðηpeakÞlnðMc=ηpeakÞ where ηpeak and ψn(ηpeak) are, respec-tively, the instantaneous values of stress ratio (¼q/p) and intergra-nular state parameter at drained peak shear strength under thecompression mode of triaxial. In a similar fashion, dilation [Eq. (33)]becomes temporarily zero at phase transformation (where the initialcontraction turns into dilation) in undrained triaxial tests. Considera-tion of Eqs. (32) and (33) yields, nd ¼ �1=ψ nðηPTÞlnðMc=ηPTÞ inwhich ηPT and ψn(ηPT) are, respectively, the instantaneous values ofstress ratio and intergranular state parameter at phase transforma-tion under the compression mode of undrained triaxial. Under thecompression mode of triaxial, dilatancy can be calculated byD¼ ð_εpv=_εpqÞ � ððΔεv�Δp=KÞ=ðΔεq�Δq=ð3GÞÞÞ. It should be pointedout that Δεv¼0 under the undrained condition. In the model, thestress–dilation response is expressed through Eq. (33). By using theestimated value of dilation in conjunction with Eq. (33), “A” can beestimated. However, “A” can be estimated simply by trial and errorprocedures. ΔM¼2[Mc(FC¼FCth)�Mc(FC¼0)] in which Mc(FC¼0)and Mc(FC¼FCth) are, respectively, the values of Mc at zero andthreshold fines contents. Examples of calibration of Eq. (8) arepresented in Fig. 6. Having the effective size of host sand (D10) andthe mean size of non-plastic silt (d50), the particle size ratio iscalculated by χ¼D10/d50. For each particle, roundness is defined as

the ratio of the surface features average radius of curvature to theradius of the largest inscribed circle (e.g., [54]). For a number ofparticles taken from the coarse and fine ingredients, the averagevalue can be used, respectively, as Rc and Rf. Furthermore, havingmicro-photos of grains, one can apply Fig. 3 in conjunction withTable 1 to estimate Rc and Rf. e0 is the intercept of critical state line[Eqs. (11,12)] with the en-axis. One can re-write Eq. (11) in the formlnðe0�ecsÞ ¼ ln λþξ lnðp=pref Þ. Thus, the intercept and the slope ofthe best straight line fitted to the experimental data of ln(e0�ecs) vs.ln(p/pref) indicate lnλ and ξ, respectively. Using Eq. (12) and con-sidering Ae¼�c Ac, one has f 1 ¼ ðe0e�e0cÞ=½Acðe0eþe0cÞ� in which e0cand e0e are, respectively, the values of e0 for the compression andextension modes of triaxial. Once f1 is determined, Eq. (12) yieldseA ¼ e0c=ð1� f 1AcÞ ¼ e0e=ð1� f 1AeÞ in triaxial tests. HA and f2[Eq. (20)] scale plastic hardening modulus for different shearingmodes. f2 is calculated by f 2 ¼ ðH0c�H0eÞ=½ΔðH0eþH0cÞ� in which H0c

and H0e are, respectively, the values of H0 [Eq. (20)] measured in thecompression and extension modes of triaxial. Knowing f2, HA can bedetermined by HA ¼H0c=ð1þ f 2ΔÞ ¼H0e=ð1� f 2ΔÞ. Further examplesfor the calibration and evaluation of the plastic hardening modulusparameters [i.e., Eq. (20)] can be found in Lashkari and Latifi [23]. Theinfluence of fines content on the evolution of exponent n [Eq. (17)] isimposed by means of ω1 and ω2. By fitting Eq. (19) to experimentaldata of n vs. FC, ω1 and ω2 can be determined (see Fig. 10 forinstance).

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Recommendations for extension and re-calibration of an existing sand constitutive model taking into...

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